MAGNETISM AND ELECTRICITY

WORKS BY A. W. POYSER, M.A.

MAGNETISM AND ELECTRICITY.
Stage I. With 235 Illustrations. Crown 8vo,

2S. 6d.

ADVANCED MAGNETISM AND ELEC-
TRICITY. With 317 Illustrations. Crown
8vo, 45. 6d.

LONGMANS, GREEN AND CO.

LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS

MAGNETISM AND
ELECTRICITY

A MANUAL FOR STUDENTS IN
ADVANCED CLASSES

BY

E. E. BROOKS, B.Sc. (LoND.), A.M.I.E.E.

HEAD OF THE DEPARTMENT OF PHYSICS AND ELECTRICAL ENGINEERING
IN THE MUNICIPAL TECHNICAL SCHOOL, LEICESTER

AND

A. W. POYSER, M.A. (Turn. COLL. DUEL.)

HEAD MASTER OF THE GRAMMAR SCHOOL, WISBECH

WITH 420 ILLUSTRATIONS

SECOND EDITION

LONGMANS, GREEN AND CO.

39 PATERNOSTER ROW, LONDON

FOURTH AVENUE & 30TH STREET, NEW YORK

BOMBAY, CALCUTTA, AND MADRAS

1914

All rights reserved

PREFACE

OWING to the enormous progress made in scientific research during
the last twenty years, and to the consequent developments in electri-
cal theory, this volume has been prepared to replace Poyser's Advanced
Magnetism and Electricity, originally published in 1892. Practically,
the whole of the subject-matter has been rewritten on modern lines ;
indeed, little of the old book remains except its experimental form.
This has been retained because it is important that a beginner should
learn to recognise that all theory is based upon a groundwork of
experimental fact.

The present work is intended to afford such a range of general
reading in the subject as is desirable for the majority of students,
before they begin to specialise either in pure science or in the various
branches of electrical engineering. They are assumed to have an
elementary knowledge of algebra, geometry, trigonometry, and
mechanics.

The scope of the volume is that required for the pass examinations
of the Universities, and for the examinations of the Board of Educa-
tion. The chapters marked with an asterisk need not be read for
the Lower Examination of the latter body.

The sequence adopted is based upon a long experience in teach-
ing. We freely admit that several alternatives are possible, each
possessing its own advantages ; and that room exists for wide differ-
ences of opinion as to the most logical form of treatment.

Again, we are conscious that many important subjects have been
inadequately treated, but we hope that the various references and
allusions will be regarded as indicating directions for further study.

Although the electron has received a considerable amount of atten-
tion, we have thought it undesirable (in the present state of uncertainty
as to the true nature of a positive charge) to base an elementary
treatise entirely upon its properties.

On the other hand, experience has shown that it is exceedingly

V

vi PREFACE

desirable for a student to acquire the habit, at an early stage, of
thinking out problems in terms of the properties of lines of force
(electric and magnetic), and of applying the various mechanical
analogies, which may be employed — within due limits — to invest
mathematical expressions with a tangible physical meaning.

It will be seen that we have introduced K and ju, into certain
formulae, where they are usually omitted as being equal to unity.
If they are suppressed, the equations are incorrect as regards
dimensions, and the student may encounter difficulties later which
will not arise if they are used throughout.

Of the 420 illustrations, nearly three hundred have been engraved
from our own drawings; the remainder have been taken by permission
from well-known books, of which acknowledgment is made in the
text.

We have to acknowledge most valuable assistance from Mr. Francia
Hewson, of the Postal Telegraphs, for revising and correcting the
chapter on Telegraphy and Telephony.

E. E. B.
A. W. P.

May 1912.

CONTENTS

CHAP. PAGE

I. ELECTRIFICATION — CONDUCTORS AND INSULATORS —

ELECTROSCOPES ...... 1

II. INDUCTION —LINES OF ELECTRIC FORCE. . . 9

III. DISTRIBUTION — COULOMB'S LAW — ELECTRIC FIELD

AND ELECTRIC FORCE 18

IV. POTENTIAL — CAPACITY — SURFACE DENSITY . . 31
V. CONDENSERS AND CAPACITY — ENERGY OF CHARGED

CONDENSERS — SPECIFIC INDUCTIVE CAPACITY . 50

VI. ELECTRICAL MACHINES 73

VII. ELECTROMETERS 86

VIII. ATMOSPHERIC ELECTRICITY 102

IX. MAGNETIC ATTRACTION AND REPULSION — LODE-
STONES — ARTIFICIAL MAGNETS — POLES OF A

MAGNET . j . . / . . . Ill
X. MAGNETIC INDUCTION . . . * . .120
_XI. MAGNETIC FIELDS AND LINES OF FORCE . .125
XII. MAGNETIC MEASUREMENTS, . . . . .135

XIII. TERRESTRIAL MAGNETISM . . . .174

XIV. VOLTAIC CELLS — LOCAL ACTION — POLARISATION . 195
XV. OHM'S LAW APPLIED TO CIRCUITS — GROUPING

OF CELLS 216

XVI. ENERGY — POWER — EFFICIENCY OF A CELL —

HEATING EFFECT — JOULE'S LAW . . . 230
XVII. MAGNETIC PROPERTIES OF A CURRENT — ELECTRO-
MAGNETS— ELECTRIC BELL .... 240
XVIII. RESISTANCE AND ITS MEASUREMENT . . . 255
XIX. GALVANOMETERS ,..,„,. 282

vii

viii CONTENTS

CHAP.

XX. MEASUREMENT OF ELECTROMOTIVE FORCE
XXI. ELECTROLYSIS AND ITS PRACTICAL APPLICATIONS —

ACCUMULATORS

XXII. ELECTRO-MAGNETISM — INDUCTION COILS — TRANS-
FORMERS . . . . . . ,

XXIII. ELEMENTARY THEORY OF INDUCTION . .

XXIV. BALLISTIC GALVANOMETERS .....
\j XXV. THEORY OF MAGNETISATION — VALUES OF H AND B —

PERMEABILITY — INTENSITY AND SUSCEPTIBILITY
—HYSTERESIS — DIAMAGNETISM

*XXVI. ALTERNATING CURRENTS

*XXVIL MEASUREMENT OF SELF AND MUTUAL INDUCTANCE
XXVIII. THERMO-ELECTRICITY ......

*XXIX. PASSAGE OF A DISCHARGE THROUGH GASES .
XXX. TELEGRAPHY AND TELEPHONY

XXXI. DYNAMOS AND MOTORS

XXXII. LAMPS .

XXXIII. MEASURING INSTRUMENTS

XXXIV. UNITS AND DIMENSIONS ....
*XXXV. ELECTRIC OSCILLATIONS — RADIATION — WIRELESS

TELEGRAPHY AND TELEPHONY ....

APPENDIX (ON WIRELESS TELEGRAPHY) .
ANSWERS . . . . . . .

INDEX

ELECTROSTATICS

CHx\PTER I

ELECTRIFICATION

Electrical Attraction. — Exp. 1. Warm and dry1 a glass rod and a
piece of silk. Riib the glass with the silk and then present it to any light
bodies, c.g, pieces of paper, pith, or bran. Observe that they are attracted
towards the rod.

Thus by friction an additional and curious property has been
imparted to the rod, due to what is called electricity ; and the rod
itself is said to be electrified, excited, or
charged.

Thales, a Greek philosopher (600 B.C.),
was the first to record that this power was
possessed by rubbed amber, the Greek for
which — t")X.€KTpov (electron) — is the origin
of our word electricity.

Adopting the precaution mentioned in
the foot-note, this property may be ex-
hibited with a large number of bodies,
such as sealing-wax, resin, shellac, or sul-
phur, rubbed with flannel ; hot brown paper
brushed with an ordinary clothes brush ;
ebonite rubbed with flannel or even with
the dry hand.

Electrical Repulsion.— Exp. 2. Sus-
pend a small pith ball by a fine silk thread to a
suitable support (Fig. 1). (Elder pith is best for
this purpose. After cutting the pith into shape YIG. 1.

with a sharp knife, it is advisable to press it
slightly between the fingers in order to remove any projecting points.)

Place this apparatus, which is called the pith-ball pendulum, before a fire to
dry the silk. Bring an electrified rod near the ball. Observe that attraction

1 To exhibit this property all rods, rubbers, and apparatus muse be warm and
dry, and it is therefore advisable to place them in front of an ordinary coal fire
or a gas reflecting stove for some time before use.

A

ELECTROSTATICS

first takes place, but after contact with the rod the ball is violently repelled, nor
\vill it again approach the rod (unless indeed it has been in contact with the
metallic or wooden support).

Two lessons may be learnt from this experiment —

1. That a body becomes charged by contact with an electrified
body.

2. That when two bodies are charged by contact, they repel one
another.

TWO States Of Electrification.— Exp. 3. Bend a piece of wire,
as shown in Fig. 2, and suspend it by a silk thread from a suitable support.

Electrify a glass rod with silk, and place it
across the stirrup.

1. Electrify another glass rod with silk,
and hold it near the suspended rod. Re-
pulsion takes place.

2. Repeat this experiment with a rod
of sealing-wax or of ebonite rubbed with
flannel. Notice attraction.

This experiment teaches us that
there are two different states or kinds
of electrification —

(1) That developed on glass
rubbed with silk ;

(2) That on sealing-wax rubbed
with flannel.

Electrification developed on glass
rubbed with silk was once called
vitreous (vitrum, Lat., glass); that developed on sealing-wax or resin
rubbed with flannel, resinous. These terms have, however, been dis-
carded, as observers soon found that vitreous electricity could be
developed on resinous substances, and vice versa, by merely altering
the material of the rubber.

Vitreous electricity is now called positive (or + ) electricity.

Resinous electricity is now called negative (or - ) electricity.

We also learn from Experiment 3 that

(a) bodies whose electrifications are of opposite kinds mutually
attract one another j

(b) bodies whose electrifications are of the same kind repel one
another.

Conductors and Insulators.— Exp. 4. Rub a brass rod, held in
the hand, with warm silk. Bring it near a pith-ball pendulum, or better still,
near an electroscope (see p. 5), and observe that the ball is unaffected.

Exp. 5. Mount the brass rod on a glass handle, or hold it with a sheet of
india-rubber. Dry the handle and rub the brass. Present the rod to a nega-
tively charged pith-ball pendulum and observe repulsion. It is therefore
negatively charged.

Exp. 6. Repeat the last experiment, but, before bringing it to the pendulum,
touch it with the linger. There is, now, no sign of electrification

FlG. 2.

ELECTRIFICATION 3

For many years it was thought that only a certain number of
bodies — which were called electrics — were capable of being electrified.
All other bodies were called non-electrics, because they did not
exhibit any signs of electrification even after violent friction. This
distinction was erroneous, as all bodies are capable of being electrically
excited, though in different degrees, if proper precautions are adopted.
The reason of this difference in the action of various bodies escaped
the early observers from the fact that, with non-electrics, the electricity
is discharged to the earth through the hand and body of the experi-
menter, but that with electrics it is not so discharged.

Bodies, such as brass, which allow electrification to spread readily
over them and to carry it away to other bodies, are called conductors,
while those bodies, such as glass, sealing-wax, ebonite, &c., which do
not allow the electricity to escape as soon as it is developed, are called
non-conductors, insulators, or dielectrics.

Exp. 7. Insert a wooden rod into a cork, and then pass the cork into a glass
tube ; rub the tube close to the cork, and then present the end of the rod to any
light body. Notice attraction.

Exp. 8. Fasten a key or a metal
ball to a cotton thread, and then tie
the free end round the wooden rod.
Electrify the tube as before, and
observe that light bodies are attracted
to the key (Fig. 3).

Exp. 9. Repeat the last experi-
ment with silk instead of cotton.
The light bodies are not attracted.

Wood, cotton, and metal are
therefore conductors; silk is a
non-conductor.

Different bodies have differ-
ent conducting powers. It must
be remembered that all bodies
offer some resistance to the pas-
sage of electricity, although with
good conductors the resistance
is almost inappreciable, while
with the best non-conductors it is so great that practically no elec-
tricity passes from one point to another.

The best conductors are the metals (of which silver stands first),
then follow charcoal, acids, water, the human body, cotton. The best
non-conductors are dry air, fused quartz, glass, paraffin, ebonite,
shellac, sulphur, gutta-percha, resin, silk, wool, porcelain, oils. These
substances are given in their approximate order of conduction and
insulation respectively. Dry wood, marble, paper, straw occupy an
intermediate place, and are therefore sometimes called partial con-
ductors.

4 ELECTROSTATICS

Rods made from fused quartz insulate remarkably well, and are
not hygroscopic. This material can be drawn out into very fine fibres,
suitable for delicate suspensions in measuring instruments, which give
much better insulation than silk. (See its use in an electrometer,
p. 87.)

Conduction is affected by temperature, e.g. glass loses its power of
insulation when it is made very hot.

Electroscopes. — An instrument which will detect the presence,
and determine the kind of electrification of a body is called an electro-
scope. The first electroscope was made and used by Gilbert in the
year 1600, and merely consisted of a straw balanced on a fine point.

We have already used a pith-ball pendulum as an electroscope.

For rough experiments a pith-ball electroscope is sometimes
employed. This consists of two small balls of elder pith, suspended
by cotton threads from one end o£ a brass wire, the other end being
terminated by a knob. A hole, somewhat larger in diameter than the
wire, is bored through a cork, which fits the mouth of a glass vessel.
The wire is passed through the hole in the cork, and then fastened in
its place with hot shellac.

The Gold-leaf Electroscope. — Another useful instrument
is that known as the gold-leaf electroscope (Fig. 4). A metal rod is
supported by a paraffin- wax plug fitting into the neck of a wide-
mouthed glass vessel without a bottom and standing on a wooden
base. The upper end of the rod is fitted either with a metal ball or
a plate, the latter being more generally useful. To the lower end is
soldered l a flat strip of metal, about half an inch wide, which carries
two gold leaves (each about half an inch wide and two inches long).
The leaves are thus parallel and hang very close together. On opposite
sides of the interior of the vessel are generally placed two strips of
tinfoil, which touch the wooden base and which are themselves touched
by the leaves when too great a divergence is produced. These strips,
besides preventing the leaves from becoming broken by contact with
the sides of the vessel, have another important function, which will
be explained in later experiments.

1 Soldering is easily performed as follows : — Make a soldering iron by point-
ing a stout copper wire, and then inserting the other end into a handle. Now
place some zinc in a bottle, pour over it a small quantity of dilute hydrochloric
acid, and allow the action to go on for some time ; the solution is then known
as "killed spirits." Having heated the "iron," dip it for an instant in the
killed spirits, and then hold the bright point in a small piece of solder. This
gives the point a shining silver-white surface, and the operation is technically
called "tinning." The easiest method of soldering the brass disc to the copper
wire is to make both pieces bright and clean, then place a little killed spirits
on them, and a small piece of solder on the disc. Hold the iron in a Bunsen's
flame, an inch or two from the "tinned" end. When the end has become
sufficiently hot, touch the solder on the disc with it, also holding the wire on
the required place. The heat melts the solder and, by careful manipulation,
the solder flows over the two surfaces. The w re is held motionless until the
solder is set. Afterwards wash with plenty of water to prevent rusting.

ELECTRIFICATION 5

A later pattern is shown in Fig. 5. The containing vessel is a
light rectangular metal box, having two of its opposite sides fitted
with glass. The rod is supported by means of a plug of paraffin
wax, which fits in the neck, and its lower end terminates in a fixed
brass plate. One leaf — which for ordinary purposes may consist of
dutch metal or of aluminium foil instead of gold-leaf — is attached to
the upper part of the fixed plate.

(Since the discovery of radio-activity the electroscope has become
extremely important, and, in some of its later forms, it is an exceed-
ingly sensitive and accurate instrument of measurement.)

FIG. 4.

FIG. 5.

Uses Of a Gold-leaf Electroscope.— The electroscope is
used (1) to indicate the presence of a charge, and, roughly, the amount of
charge on a body ; (2) to determine the kind of charge. (See also p. 33.)

(a) Experiments to indicate the existence of a charge and to esti-
mate its amount.

Exp. 10. Rub a rod of sealing-wax, ebonite, or shellac with flannel, and bring
it gradually o^er the cap of the electroscope. Notice that the leaves diverge ; if
the body be slightly charged the divergence is small, if it be highly charged, the
divergence is greater.

Exp. 11. Gently strike the disc with fur, and observe the divergence of the
leaves.

Exp. 12. Grind roll sulphur to powder in a mortar, and notice the divergence
of the leaves when a small quantity is dropped on the disc.

(b} Experiments to determine the kind of electrification of a body.

The following rule, which will be easily understood after the
chapter on induction has been read, must be learnt. If, when the
leaves are divergent, the approach of an electrified body causes them
to diverge moret the leaves and the approaching body are similarly
electrified ; if they diverge less, the leaves and the approaching body

6 ELECTROSTATICS

are either oppositely electrified, or else the approaching body is an
uncharged conductor.

Take care to observe the first movement of the leaves as the body
approaches the electroscope.

Exp. 13. Repeat Exp. 11. Bring up a negatively charged rod, and observe
that the divergence of the leaves is greater. Tin- (list-
is therefore negatively electrified.

Exp. 14. In a similar manner prove that the sul-
phur in Exp. 12 is negatively electrified.

Exp. 15. Place a flannel cap (Fig. 6), to which a
silk thread is attached, over one end of an ebonite rod,
both having been previously warmed. Rub the. cap
round the rod, and then, by means of the silk thread,
place it on the disc, (a) Bring up a positively charged
rod, and observe that there is a further divergence of
the leaves. The flannel is therefore positive. (6) Bring
up a negatively charged rod, and observe that the
leaves collapse, (c) Place the hand over the disc, and
again notice that the leaves diverge less than at first.

Repulsion is the only sure Test of Electrification. —

From the last experiment we learn that there is increased repulsion
between the leaves, when they and the approaching body are similarly
electrified; and that both an oppositely charged body and an un-
charged conductor cause a partial, or total, collapse of the leaves.
In order, therefore, to ascertain the kind of charge we are testing, it
is essential to rely solely on repulsion.

Simultaneous and Equal Development of Both Kinds

Of Electricity.— Ezp. 16. (1) Repeat Exp. 15, by means of which the
flannel cap is proved to be charged positively. (2) Charge a gold-leaf electro-
scope negatively. Observe that when the rod is brought near, there is an
increased divergence of the leaves. The rod is therefore negatively charged.
(3) Discharge the rod (by passing it through a flame), and also the cap. Show
that they are neutral by bringing each in turn near the electroscope. Replace
the cap, and rub again. Without removing the cap, present them to an un-
charged electroscope. The leaves remain at rest.

This experiment conclusively proves that (1) positive and negative
electrifications are produced simultaneously, and (2) the positive
electrification is exactly equal in amount to the negative.

As this fact is so important, it may be advisable to demonstrate it
with greater accuracy.

Exp. 17. Insulate a metal can, connect it by a fine wire to the cap of an
electroscope some little distance away, and place a pad of flannel at the bottom
of the can. Test an ebonite rod to make sure that it is uncharged, and then nib
the flannel gently with the end of it. Observe that no effect is produced on the
leaves while the rod remains on the pad, but that they diverge when it is re-
moved. Again introduce the rod, and if no leakage has occurred along it, notice
that the divergence disappears.

Bodies not absolutely Positive or Negative.— Exp. is.

Charge a gold-leaf electroscope negatively. Excite a hot glass rod with fur and

ELECTRIFICATION 7

bring it near the electroscope. Notice that there is increased divergence, prov-
ing that the glass is negatively electrified.

Thus we learn that the kind of electrification developed by
friction depends not? only on the body rubbed, but also upon the
rubber.

The following list has been prepared so that if any two bodies
be chosen, the one standing first becomes positive, the other
negative : —

Fur

Flannel

ivory

Glass

Cotton

Paper

Silk

Paraffin-wax

Ebonite

The hand

Metals

Sulphur

Celluloid

Rubber tubing

It must be remarked, however, that the results of experiments are
somewhat uncertain with those substances which stand close together
on the list, as a slight difference in chemical composition, or in their
physical properties, may alter their behaviour. The order of the last
seven substances on the list is given from experiments made at the
time of writing. For instance, an ebonite rod (first cleaned with
emery cloth), which becomes negatively charged when rubbed on the
coat sleeve, became positively charged when flicked with a piece of
rubber tubing, and also when drawn through the dry hand. This
piece of rubber tubing became negative to every other substance on
the list, whereas another specimen (a photographer's rubber squeegee),
was found to be positive to metals, sulphur, and a pocket comb, its
position on the list lying between "the hand" and "metals."

Exp. 19. Test as many of these bodies as possible by means of a gold-leaf
electroscope, electrified from a known source.

Discharging an Electrified Body.— Exp. 20. Rub an ebonite rod

with flannel. Show tlmt it is electrified by bringing it near an uncharged gold-
leaf electroscope. Pass the rod through a Bunsen's or spirit-lamp flame. Test
as before. There is no divergence of the leaves, showing that the rod is com-
pletely discharged.

Exp. 21. Charge a gold-leaf electroscope. Hold a lighted taper above, but
not in contact with, the disc. Observe that the leaves collapse ; the electro-
scope is therefore discharged.

See also the experiments on the discharge by points, pp. 22 and 82.

Electrification by Pressure and Cleavage.— Electrification

may be produced by many methods other than friction, e.g. by chemical action,
by heat, by mere contact of dissimilar metals (all of which will be treated of in
due course), and by pressure and cleavage. v.

(a) If a piece of insulated cork be pressed on india-rubber, gutta-percha,
amber, or metals, it becomes positively electrified.

(6) If sulphur be placed on a piece of india-rubber, then broken by a hammer
and allowed to fall on the disc of a gold-leaf electroscope, the leaves diverge with
negative electricity.

8

ELECTROSTATICS

(c) Lumps of sugar broken in the dark emit a feeble light, due to the re-
combination of the two electrifications.

PyrO-EleCtricity. — When certain minerals are cooled or heated, elec-
tricity is produced. Such electricity is called pyro-electricity. It is best studied

with a suitable crystal of tourma-
line (Fig. 7), suspended by a fine
wire over a metal plate, heated by
a spirit lamp. In a short time,
the ends will be oppositely charged.
If now the plate be removed, and
the crystal discharged by passing
a flame over it, it will be found
that as it cools, the end, which
when heated was positive, becomes
negative, and that which was nega-
tive becomes positive. The points at which the free electricity is at a maximum
are called the poles ; that which was positive when the temperature was rising
is called the analogous, and that which was negative, the antilogous pole. In
Fig 7, A is the analogue, and B the antilogue.

J?IG. 7.

EXERCISE I

1. If a hot sheet of paper be brushed with an ordinary clothes brush, it will
cling to the wall of a room. The drier the air is, the longer it will cling. Why
is this ?

2. Describe an experiment to demonstrate the existence of two kinds of
electrification.

3. How would you show that these two kinds are always produced in equal
amounts ?

4. Is it possible to electrify a metal rod by friction ? If so, show how it can
be done.

5. What will be the electrical state of a silk glove after being drawn off the
hand ?

CHAPTER II

INDUCTION— LINES OF ELECTRIC FORCE

Exp. 22. Place two insulated brass spheres in contact. (Two spherical bedstead

knobs, two or three inches in diameter, supported on ebonite penholders are

excellent, and may be easily made at a cost of a few pence.) Bring a positively

charged rod near. Negative electricity is found on the sphere next the rod, and

positive on the remote one (Fig. 8).

Prove the truth of this statement

by removing the sphere remote from

the rod— taking care to touch the

insulating support only — and testing

its charge. Remove the rod, and

then test the charge on the other

sphere.

When we bring an electrified
body near, but not in contact
with, an insulated conductor —
the two brass spheres men-
tioned above form one con-
ductor when they are placed
in contact — we find that an
action takes place across the
dielectric (in this case, air), with the result that the near side of the
conductor becomes charged with the opposite electrification and the
remote side with the same kind of electrification, which is present on
the electrified body.

Such electrical action is called induction. The electrified body
producing the action is called the inducing body, and the charges
produced by the action are called induced charges.

Before doing any further experiments on induction, we must
describe an exceedingly useful instrument, called a proof-plane, which
we may use when dealing with large charges of electricity.

A Proof-Plane (Fig. 9) merely consists of a small conductor,
mounted on an insulating handle. An excellent one is made from
a disc of metal, having the edges well rounded off, fastened to a rod
of sealing-wax or of ebonite.

Sometimes the conductor is a small brass ball suspended by a
silk thread. It is then commonly called a carrier ball.

If such an instrument be brought in contact with a charged
body, it will receive part of its charge, which can therefore be

FIG. 8.

10

ELECTROSTATICS

removed, and the kind of electrification tested, without incurring the
danger of fracturing the gold leaves of an electroscope.

Exp. 23. Take an insulated, unelectrified, cylindrical conductor with
rounded ends. Bring a negatively charged rod near it, and touch the other
end with a proof-plane, (a) Prove that the charge on
the proof-plane is negative (use a negatively charged
electroscope, and observe greater divergence of the
leaves), (b) Touch the end near the rod with a proof-
plane, and show that the divergence of the leaves of
a positively charged electroscope is increased, when
the proof-plane is brought near. (c) Test a point
midway between the two ends. There is no divergence
of the leaves of an unelectrified electroscope.

Exp. 24. Repeat Exp. 22, and show that the
induced charges are equal in amount by bringing the
two spheres into contact, and then testing that there
is no residual charge. (Especially good insulation is
required, or the experiment may fail on account of
unequal leakage.)

From experiments similar to these, we infer
that a charged body acts inductively in all
FIG. 9. directions, any insulated conductor near it

acquiring two induced charges, that of opposite

sign being nearer the charged body, and that of the same sign farther
away.

Induction on a Conductor connected with the Earth.—
We are now in a position to understand the effect of placing a con-
ductor, when under inductive influence, in connection with the earth.

Exp. 25. Bring a positively electrified rod near an insulated conductor.
Touch the conductor with the finger or, indeed, with any uninsulated conductor.
Remove the hand or conductor, and then the inducing rod, and test the remain-
ing charge. We find that it is negative ; the positive has therefore disappeared.

The reason is this : — the human body and the floor of the room
are conductors, so that the positive electrification induced in this
experiment has escaped through them to the earth and is practically
lost. The conductor, however, is not discharged, but retains a negative
charge, which becomes sensible when the inducing-rod is removed.

Theory of Electrostatics. — Some working theory must now
be given to co-ordinate the experimental facts mentioned in this and
the previous chapter. It may be well to point out, however, that
theories must not be regarded as final explanations, but merely as
aids to thought, so that we can form mental pictures of facts. As
examples of this statement, we may mention that —

(1) Until quite recently it was impossible to form any definite
idea regarding the meaning of the term electric charge, and in order to
express the facts in an orderly manner, it was usual to speak vaguely of
hypothetical " electric fluids." We now know that a negative electric
charge is an assemblage of small particles (known as "electrons"),

INDUCTION— LINES OF ELECTRIC FORCE 11

which have an existence as definite and real as that of the atoms of
matter, although they are very much smaller in size (see p. 488), and
which are able to move with great freedom in the bodies we call good
conductors, and with much less freedom, or with practically no freedom
at all, in the best insulators. The nature of a positive charge is as
yet an open question, and need not be discussed here.1

(2) The facts of induction used to be explained by supposing that
electric charges attracted or repelled each other across the intervening
dielectric, and that all neutral conductors contained an inexhaustible
supply of the two kinds of electricity in equal quantities. On this
supposition, the explanation of inductive action was easy, for, if an
electrified body be placed near an insulated uncharged conductor, it may
be regarded as attracting one kind of charge and repelling the other ;
and, if the conductor be " earthed," the repelled charge (i.e. the free
charge) passes away, and leaves the attracted charge behind, which was
then said to be " bound " by the inducing charge. This hypothesis,
although it leads to perfectly correct results in most cases, does not, lioiv-
ever, bring into sufficient prominence the part played by the surrounding
dielectric. A more correct and clearer idea of the actions involved can
be obtained by introducing the conception of lines of electric force.

Lines of Electric Force. — From what has been said, it is
obvious that, if a small body containing (say) a unit positive charge
be brought into the neighbourhood of another charge, the small body
will be acted upon by a mechanical force, which has at any point a
definite magnitude and direction. The direction in which the positive
charge moves, or tends to move, is (for convenience) arbitrarily defined
as being the direction of the line of force at that point.

Such lines of force will, therefore, extend from any charged body to
an oppositely charged body, and the space around the charge — or a
number of charges — containing such lines, is called an electric field?
We can represent such a condition by regarding the charged body as
the centre of a diverging number of lines of force (always the same
number for the same charge), which terminate on the surface of a con-
ductor, or of surrounding conductors, in a charge of the opposite hind.

We also suppose (1) that these lines are in a state of tension,
always tending to contract in length, and (2) that they repel one
another laterally. All attractions or repulsions may then be regarded as
due to the endways pull or the sideways repulsion of these lines of force.

1 All substances contain these electric charges, and we may regard all electric
generators — from the old Motional machine to the modern dynamo — as being
different devices for obtaining the same result, viz. the separation of electric
charges. When a charge is at rest upon the surfaces of a body, it is commonly
known as frictional or static electricity ; when the charges are allowed to rejoin
each other by flowing through conductors, their properties, whilst in this state
of motion, are included in the general term voltaic or current electricity.

2 The strength of the field will be defined later as being measured by the
number of lines of force per square centimetre.

12 ELECTROSTATICS

A charge is present on a conductor whenever the lines of force
pass from it or into it ; in fact, from this point of view, charges are
merely the ends of lines of electric force.

Let us consider an insulated positively charged body. Lines of
force pass from it to adjacent conductors, e.g. the table, the ceiling,
and the walls of the room, and end there in a negative charge, which,
although widely distributed, is necessarily equal in total amount to the
original positive charge. Now place an insulated uncharged conductor
near, so that it is under induction. As it is in an electric field, some
lines of force must of necessity pass through it. But inside a conductor
such a state of strain cannot exist, and there the field completely dis-
appears, leaving the conductor charged with equal amounts of opposite
sign, simply because as many lines are leaving it as are entering it.

The real meaning of the terms free and bound now becomes
evident, for let A (Fig. 10) be a positively charged body acting induc-
tively on the body BC.
The induced charge at
JB is connected by lines
of force to A, and its
tendency is simply to
get as near as possible
to A. Hence, connect-
ing the body with the
earth will have no effect

10. °n the charge at B, but

that at C will be removed
because it is linked to the earth by lines of force.

Beginners are apt to think that a charge has some mysterious tend-
ency to flow to earth ; they must, however, remember that this occurs
only when the opposite charge to which it is linked is there already.
We also learn from Fig. 10 that, although the induced charges at
B and C are -necessarily equal in amount, they are each less than the
inducing charge on A, for only a portion of the total lines from A
pass through the conductor BC. We further see that the charges at
B and C will increase in amount as the distance is decreased, and
vice versa, but, under ordinary conditions, they can never become equal
to that at A, because even when the conductors are close together,
some of the lines from A will take an independent path to earth.

Faraday's Ice-Pail Experiment. — If all the lines from A
(Fig. 10) could be compelled to pass through the second conductor,
each of the induced charges would be numerically equal to that on A.
This fact was first demonstrated by Faraday in his historical "ice-
pail " 1 experiment.

Exp, 26. Insulate a metal vessel A, Fig. 11. Connect its outside, by
means of a wire, to the disc of a distant electroscope E. Charge a metal ball C,

1 So called because an ice-pail was originally used.

FIG. 11.

INDUCTION— LINES OF ELECTRIC FORCE 13

suspended by a dry silk thread, and lower it into the vessel. Owing to induc-
tion, the leaves immediately di-
verge. Observe that, when the
ball has reached a short distance
from the top of the vessel, the
leaves are at their greatest
divergence, and that they do
not alter when the ball is
allowed to descend lower. We
see from the diagram that, as
soon as the ball reaches a suffi-
cient depth, practically all its
lines of force pass through the
metal vessel on their way to the
opposite charge on the walls of
the room. Hence, their ends
inside the vessel must consti-
tute a negative charge, and on
the outside there must be an
equal positive charge. For instance, if the ball has a charge of +10 units, there
must be - 10 units inside the vessel and + 10 units on* the outside (partly on the
vessel and partly on the electroscope).

Bring the ball in contact with the interior of the vessel, and observe that the
leaves reir an divergent to the same extent as before. Remove the ball, and show
that it is completely discharged. At the moment of contact, the charge inside and
that on the ball just neutralise each other, leaving the leaves outside unaffected.

Whence, we conclude that the induced charges are each exactly equal to the
inducing positive charge.

We may arrive at this result in a slightly different manner.

Exp. 27. Again charge the ball positively and lower it into the vessel. Touch
the electroscope with the finger. Observe that the leaves collapse, owing to the
removal of the positive induced charge. Now remove the finger, and touch the
vessel with the ball. Withdraw the ball, and .notice that the leaves show no
evidence of electrification, proving that the induced negative and the inducing
positive charges were equal in amount.

Faraday confirmed this result by using four such vessels (Fig. 12),

insulated from each other by
blocks of shellac. Precisely
similar results occurred, each
pail becoming inductively
charged with opposite electri-
cities, all equal to the charge
on the inducing body.

Thus, when the electrified
ball was lowered into pail (4),
inductive action took place
through the series, pail (1)
producing the same result as

that obtained in the previous experiments with one pail.

Evidently, the lines of force diverging from C pass through the

whole arrangement, each vessel becoming charged, negatively on its

inner side and positively on its outer side.

FIG. 12.

14

ELECTROSTATICS

Inductive Process of Charging" a Gold-leaf Electro-
scope.— From what has been said, it will be easily understood that

we can electrify a body by
charge
of the

VVUx

Earth

rEarth

Earth

induction with a
opposite to that
inducing body. In fact,
this is the common method
of charging an electro-
scope.

In Fig. 13, a, we have
a negatively charged rod
brought near the disc. The
lines of force pass from the

. — ^_^ disc to the rod, and from

^ V the leaves to the earth-con-

nected strips at the base.

In ft, the disc is touched
by the hand. The leaves
\4\4.yV/ collapse because this really

connects the leaves to earth,
and that portion of the field
disappears.

In c, the earth connec-
tion, i.e. the hand, is re-
moved.

In d, the rod is with-
drawn, and part of the
FIG. 13. + charge on the disc

spreads on the leaves, both
of which send out lines of force to the earth-connected strips.

The Electrophorus. — By means of this instrument, a series
of charges may be conveniently obtained
from a single original charge. It was in-
vented by Voltain 1775, and is essentially
the same in principle as the powerful influ-
ence machines to be subsequently described.
It consists of (1) a non-conducting
plate B (Fig. 14), which can be easily
excited by friction. This is most con-
veniently made of ebonite (although
originally such plates were made by melt-
ing together resin and shellac, and pouring
the mixture into a shallow tin).

(2) A metal disc A, provided with an
insrifefeingJitHidl&L This disc must be smaller in diameter than the
plate B. j

Earth

FIG. 14.

INDUCTION— LINES OF ELECTRIC FORCE 15

There is sometimes (3) a brass or tinfoil base called the sole, but
this is not essential.

Method of Using- the Electrophorus.— Exp. 28. (i) Warm

the plate B until it is quite dry. Rub or
strike it with warm flannel or fur. This,
of course, develops a negative charge on its
surface (Fig. 15, a).

(2) By means of the insulating handle,
place the metal disc on the plate. On
account of slight irregularities on the sur-
faces the two plates touch only at a few
points. Fig. 15, b, shows the action, with
the air space exaggerated for clearness. The
charge on the plate does not pass on to the
disc, which is merely under induction in the
electric field produced by the plate.

(3) Touch the disc with the finger. The
induced negative charge on the upper surface
then disappears (Fig. 15, c).

(4) Remove the finger, and raise the disc
by the handle. It will be charged positively
(Fig. 15, d), and probably a spark can be
obtained from it.

These operations may be repeated
many times without again exciting the
plate, the energy represented by the
induced charges being obtained at the
expense of the operator, who does
work in separating the disc and the
plate against the pull of the lines of
electric force.

Exp. 29; Place a pith-ball on the disc.
It will remain there quietly during opera-
tions 2 and 3, but, on raising the disc, it
will jump off, tlms indicating the change of
distribution which occurs.

Exp. 30. If a spark can be obtained from
the disc, bring it over a gas jet from Avhich
gas is escaping. The ]atter will be ignited
by the spark.

Further Properties of Lines
of Force. — Several consequences of
the properties of lines of force must pIG ^g

be kept in view. In the first place,

the ends can move freely on a conductor ; and when the lines pass
through a conductor, they may arrange themselves differently on
opposite sides. For example, let us imagine that a charged body is
placed inside a hollow uncharged sphere, but not at the centre.
Inside the sphere, the lines of force will have the distribution shown
in Fig. 16 • they are generally curved, although at their ends they

16

ELECTROSTATICS

are perpendicular to the surface. On the outside (assuming no other
conductors are near) they are radial and uniformly distributed,

although the total number
is unaltered. Again, if a
number of charged bodies
were placed inside, the
actual distribution of the
field inside would be
rather complicated, but
the charge on the outside
would be the algebraic
sum of the internal
charges, and the field out-
side would be radial and
symmetrical as before.

The student should
notice that the external
charge may be altered or
removed without affecting
the distribution inside.
FIG. 16. In the second place,

although lines of electric

force are in general curved in space, they always end at right angles
to the surface of a conductor. This is a necessary consequence of the
tension along them and the freedom of movement at the ends.

As a mechanical analogue, think of a piece of wood, floating on the
surface of water, with a string attached to it, which is kept under
slight tension. Then, the wood will always move until the pull is
normal to the surface of the water. (The tension is assumed to be
insufficient to raise the wood.)

It is shown later (see p. 41), how the actual path of lines of
force may be obtained in simple cases, but it must be remembered
that their direction at any point (more correctly, the tangent to the
curve at that point) is simply the direction of the mechanical force
acting on any small charged body placed at that point. From which
may be deduced the important corollary, two lines of force cannot
intersect, for if they did, there would be two directions of the resultant
force at that point, and this is impossible.

\

EXERCISE II

1. If a glass rod, strongly electrified by rubbing with silk, is held at some
distance from a pith ball hung by a silk thread and having a slight positive
charge, the ball is repelled by the rod. But if the rod is brought sufficiently
near to the ball die latter is attracted. Explain this.

2. A metal pot A is placed on an insulating stand ; a smaller metal pot B is
put inside A but insulated from it. A positively electrif

A positively electrified metal ball is hung by

INDUCTION— LINES OF ELECTRIC FORCE 17

a silk thread inside B without touching it. The pot B is now connected for a
moment with the earth, the ball is then removed ; next, the pot A is connected
for a moment with the earth ; lastly, B is taken out from A without connecting
either A or B with the earth. What is finally the kind and degree of electrifi-
cation of the two pots as compared with the ball ?

3. A glass rod which has been rubbed with silk is held just below the
spout of a metal funnel from which shot drop one by one, without hitting
the glass rod, into a cup of the same metal as the funnel. State and explain the
result which may be observed (1) if the funnel and the cup are each connected
with a separate electroscope, (2) if they are both connected with the same
electroscope.

4. You have two metal pots on separate insulating stands ; also a metal ball
carried by an insulating stem ; also a wire connected with the earth. Suppose
the ball, or one of the pots, to be slightly electrified. Describe and explain a
process by the repetition of which you can electrify two pots more and more
strongly, one positively, and the other negatively.

5. A cake of shellac is rubbed with catskin. Show how to obtain from the
shellac either a + or a - charge on an insulated conductor. How can you
ascertain whether a given charge is + or -- without changing its amount ?

(Loud. Univ. Matric., 1900.)

6. Water escapes from a small earth-connected metal jet directed vertically
downwards breaking into separate drops immediately upon leaving it. Near the
jet, with its centre in a horizontal line with it, is a positively electrified sphere.
The drops fall into an insulated can, and this is found to become more and more
strongly electrified. Explain this.

If the insulation of the sphere and can were perfect, the drops would after a
time cease to fall into the can. Explain this, and show where they would
fall. (Lond. Univ. Matric., 1905.)

7. Describe Faraday's method of showing that, when a charged body is
introduced into a nearly closed conductor, the charges produced by electrostatic
induction are equal in magnitude to the inducing charge.

(Camb. Local, Senior, 1907.)

8. What is meant by a line of electric force ? Prove that a line of electric
force must cut the surface of a conductor normally. (B. of E., 1904.)

CHAPTER III

DISTRIBUTION— COULOMB'S LAW

IT follows as a necessary consequence of the properties of lines of
force, that a charge is in general confined to the outer surface of a
conductor. The charge is linked to an equal charge of opposite sign
by lines of force under tension, and as this opposite charge is usually
exterior to the conductor — the table, the walls of the room, «fcc. — the

original charge will naturally be pulled
to the outer surface of the conductor
carrying it. This fact can be easily
demonstrated in various ways.

Exp. 31. Take a hollow insulated metal
sphere, having a circular aperture at the top
of about one to one and a half inches in
diameter (Fig. 17). This can be cheaply
made from a bedstead knob insulated on an
ebonite penholder. Having charged the
sphere with positive electricity, ap]5ly a
proof-plane, c, to the interior, and bring it
in contact with an uncharged electroscope ;
no action takes place ; there is, therefore, no
electrification inside the conductor. Now
touch the outside with the proof-plane, and,
on presenting it to the electroscope, observe
that the leaves diverge.

Exp. 32. (Commonly called Cavendish's
or Biot's experiment). Charge an insulated
metal ball by means of an electrical machine
(to be described later). Place two hemi-
spherical envelopes, furnished with glass
handles, on the outside (Fig. 18). (Again, the spherical conductor may be
made by mounting a bedstead knob on an ebonite penholder. For the hemi-
spheres, cut a slightly larger knob in two, and mount each half, by means of
sealing-wax, on ebonite penholders. They answer quite as well as the more
expensive apparatus.) After contact with the sphere remove the hemispheres,
and present each of them to an uncharged gold-leaf electroscope. Notice the
divergence of the leaves. Now bring the sphere near the electroscope ; there
is no divergence of the leaves, thus proving that, although the sphere was
originally charged, the electricity has passed to the outer surface, i.e. to the
two hemispheres.

Exp. 33. (Faraday's butterfly-net experiment). Make a conical muslin bag
and fasten it to a brass ring supported on a glass stem (Fig. 19). To the apex
of the bag attach two silk threads, by means of which the bag can be turned,
inside out. After charging the bag (a) test the inside by means of a proof- piano

FIG. 17.

DISTRIBUTION— COULOMB'S LAW

19

and an electroscope. There is no action. (6) Touch the outside with the
proof-plane, and present it to the electroscope ; the leaves immediately diverge.
Discharge both proof-plane and electroscope, and then (c) turn the bag inside
out, and again show that there is no electrification on the inside.

FiG. 18.

It may be mentioned as a further illustration of this fact, that
Faraday had a room built, each side of which measured twelve feet.
He describes it as follows : " A slight cubical wooden frame was
constructed, and copper wire passed along
and across it in various directions, so as
to make the sides a large network, and
then all was covered in with paper, placed
in close connection with the wires, and
supplied in every direction with bands of
tinfoil, that the whole might be brought
into good metallic communication, and
rendered a free conductor in every part.
This chamber was insulated in the lecture-
room of the lloyal Institution. „ . . I went
into the cube and lived in it, and used
lighted candles, electrometers, and all other
tests of electrical states. I could not find
the least influence upon them, or indication
of anything particular given by them, though
all the time the outside of the cube was
powerfully charged, and large sparks and brushes were darting off
from every part of its outer surface." l

1 Experimental Researches, 1173, 1174.

FIG. 19.

20 ELECTROSTATICS

Exp. 34, as an illustration of Faraday's room. Take a cubical tin box — say
a large biscuit tin — with a small hole (about two inches square) cut in each of
two opposite sides. Insulate the box — blocks of paraffin-wax are excellent for
this purpose — and place an electroscope inside, so that any movement of the
leaves can be seen through the holes. Now connect the box to an electrical
machine, and charge it until long sparks can be drawn from it. While the lid
is on, notice that there is not the slightest divergence of the leaves. When the
lid is off, there is usually, as might be expected, a slight divergence which
would, however, be diminished by using a deeper box.

From the facts proved in this chapter we can easily understand
why delicate instruments are often covered with gauze or muslin
during experimental work.

Several points connected with this subject are worthy of mention.

When we bring a charged conductor into contact with an insulated
uncharged conductor, the charge is, as a general .rule, shared between
them. Indeed there is only one case in which a charge can be com-
pletely transferred from one conductor to another, and it appears
from the preceding experiments that this occurs when the uncharged
conductor is hollow and the charged conductor is lowered into it to a
sufficient depth before contact.

Again, it follows that a charge may exist on the inside of a
hollow conductor, if the conditions are suitable. As previously
stated, the charge is usually on the outside because it is linked to
an opposite charge on surrounding bodies. If, however, an earth-
connected conductor be supported inside it without contact, it is
equivalent to partly putting the earth inside, and now a great part
of the lines of force will pass from the inside of the vessel to earth,
and a charge may be taken (by means of a proof-plane) from either
the inside walls, or from the earthed conductor. These charges will,
of course, be opposite in sign.

Electric Surface Density. — When the fluid theory was in
vogue, electricity was considered to accumulate to certain depths on
the surface of conductors, and electricians used the term electric
density to indicate this accumulation. It must, however, be care-
fully remembered that electric density at a point is the charge per
unit of area in the neighbourhood of the point, and we must rid our
minds of the idea of depth. If a charge of Q units be uniformly
distributed over an area A, the density (p) is given by

By means of the following experiments, although they are not
sufficiently accurate as exact quantitative measurements, we may
show that the distribution of electricity varies on differently shaped
conductors, and that the density becomes greater as the conductors
become pointed.

DISTRIBUTION— COULOMB'S LAW 21

Exp. 35. Charge an insulated metal sphere with positive electricity, (a)
Touch any point on the surface with a, proof- plane, and bring it in contact with
an uncharged gold-leaf electroscope. Notice the amount of divergence, (b)
Discharge both the proof-plane and the electroscope, (c) Touch a different
point on the sphere with the proof-plane, and touch the electroscope as before.
Notice that there is an equal divergence of the leaves.

Exp. 36. Electrify an insulated pear-shaped conductor (Fig. 20). (a)
Touch the rounded end of the conductor with a proof-plane, and bring it in
contact with the disc of a gold-leaf electroscope. Observe the amount of diver-

FIG. 20.

gence of the leaves, (b) After discharging both the proof-plane and the electro-
scope, touch the pointed end a, and notice that we obtain a greater divergence
than before.

Exp. 37. Charge a hollow can, placed on a dry glass tumbler or on a cake of
paraffin-wax. Touch (a) the middle of one side, and (b) the edge with a proof-
plane, and observe that a greater divergence of the leaves of an electroscope is
obtained in the latter case.

From these experiments we, therefore, learn that the electricity
is of equal density on the sphere, but of unequal density on the
pear-shaped conductor and on the can.

Electrical Density on differently shaped Conductors.—

We have learnt that density varies on conductors of different shapes.
We may represent this variation to the eye by dotted lines at various
distances from the conductors (Fig. 21).

Coulomb performed many quantitative experiments on distribu-
tion. He found that on a cylinder/having rounded ends, 30 inches
long and 2 inches in diameter, if the density at the middle was
represented by 1, that at the ends was 2-3, while at 1 inch and
2 inches from the ends, it was 1*8 and 1*25 respectively.

Riess gave the density at the middle of the edge of a cube

22

ELECTROSTATICS

2J times as great, and that at the corners 4 times as great, as that
at the middle of a face.

It is advisable to point out that the distribution indicated in

FIG. 21.

Fig. 21 is correct only when the conductors are remote from the
influence of other electrified bodies.

Action of Points. — From what has been said regarding the
density of electricity on various conductors, it will be seen that the
charge tends to accumulate at the edges, corners, and points. This
means that the lines of force naturally tend to concentrate in these
places, and in fact when their number per unit area exceeds a
certain limiting value, the charge will be detached, the lines ter-
minating then on dust or air particles, which move as if repelled
by the body. It will be shown later that this tension is propor-
tional to the square of the surface density (p. 46), and thus it
increases very rapidly near a point. Hence, a point on a strongly
charged conductor acts as a leak, and the effect may be great enough
to practically discharge it.

Exp. 38. Electrify an ebonite rod, or other non-conductor, by friction.
Pass the point of a sharp needle over the surface two or three times without
contact. Show, by means of an uncharged electroscope, that the non-conductor
is almost completely discharged.

Coulomb's Law. — The earliest application of measurement
to electrical forces is due to Coulomb, who, about the year 1785,
measured the attractions and repulsions of electric charges by
means of his torsion balance. He established the law that, when the
charges are on bodies so small that they may be regarded as points in
comparison with the distance between them, the attraction or repulsion
varies directly as the product of the charges and inversely as the
square of the distance between them. His form of instrument is now
practically obsolete. At the best, it gives very rough results, and a
much more correct method of verifying the inverse-square law will be
given later. In view, however, of its historical importance, we

DISTRIBUTION— COULOMB'S LAW

23

append a brief description of tfte instrument and the method of
using it.

* *» '

It consists of a light rod of shellac, p (Fig. 22), having at one end a
small flisc' n, and suspended horizontally within a cylindrical glass vessel,
A, by a very fine silver wire. The upper end of the wire is fastened to a brass
button, t, in the centre o^a graduated torsion head, e. which is capable of
moving round the tube, d ; a being a
fixed index which shows the number of
degrees the head is turned. A glass rod,
i, passes through the aperture, r, and
is terminated in a gilt pith-ball, m (the
carrier ball). A scale, oc, graduated in
degrees, is fixed round the glass cylinder
on a level with the pith-ball.

Method of using the instrument. The
torsion head is turned until the disc, w,
and the pith-ball, m, are in contact. The
% glass rod, i, is removed, the ball, m,
charged, and then quickly replaced.
When m and n touch, n receives part
of the charge of w, and is therefore re-
pelled. This causes the wire to become
twisted. The force of repulsion becomes
smaller as the distance between m and n
increases, while the force of torsion becomes
greater. Hence at a certain distance
these two forces balance each other. Now,
when this is the case, as the force of tor-
sion is proportional to the angle of torsion,
the number of degrees on the scale c is
read.

In one of Coulomb's experiments the
angle between m and n was 36°, i.e. the
torsion on the wire was 36°. The torsion
head was then turned until m and n

were 18° apart (i.e. the distance was halved). This was accomplished by turning
the disc through 126° in the opposite direction, which, together with the twist
of 18° at the lower end of the wire, made a total twist of 144°. To bring m
and n 9° apart (i.e. to make the distance one quarter of the original distance) it
was necessary to turn the disc through 567° ; in this case, therefore, the total
torsion was 576°, whence

the distance being i : f : J

the force of repulsion was 36 : 144 : 576
i.e. „ ,, 1 : 4 : 16

Now these numbers are obtained by squaring the distances and then inverting ;
proving that the force of repulsion varies inversely as the square of the distance.

By a somewhat modified experiment, the force of attraction between two
oppositely electrified bodies was proved to follow the same law.

Coulomb also proved by means of this instrument that, the distances
remaining constant, the force of attraction or repulsion between two small electrified
bodies is proportional to the product of the quantities with which they are
charged.

A charge was given to m. When n was brought in contact with m, repulsion
ensued, and the angle (say 60°) between them was observed. Afterwards half
the charge on m was removed by touching it with an insulated uncharged

FlG. 22.

24 ELECTROSTATICS

ball of equal size. It was then found that when m was again introduced
(without contact with n) that the torsion head had to be twisted 30° to bring
m and n to their original angular distance of 60°. The torsion on the wire
is therefore 60° -30° = 30°, i.e. half the original torsion. Thus the repulsive
force is halved on halving the charge on m.

The student should carefully notice the conditions under which
the inverse-square law holds good.

It is essentially a point-source law, but beginners are apt to apply
it indiscriminately to all cases of attraction. It would be absurd to
use it to calculate the force between two charged spheres, say 1 inch
in diameter and 1 inch apart, although it would give a fairly exact
result if the spheres were 1 foot apart, and a still more accurate
one at greater distances, the shape of the conductors being then
immaterial.

Coulomb's law may be written in the form

*«f a)

where p = the force of attraction or repulsion,
q = quantity of one charge,
ql = quantity of the other charge,
and d = distance between them.

The first step towards obtaining a system of electrical units may
now be taken by using this law to define unit quantity of electricity
or unit charge.

We may define unit charge — as was done years ago by a com-
mittee of the British Association — in such a way that the above
expression becomes an equality when the force is measured in dynes
(see p. 573), and the distance in centimetres. We then have

Force of attraction or repulsion (in dynes) = ^1 (2)

Evidently this means that unit charge is that charge which at unit dis-
tance (1 centimetre) from an equal charge — both concentrated at points —
attracts or repels it with a force of one dyne. It is also evident that,
in numerical calculations, we need not concern ourselves about positive
or negative signs, because, if the signs are unlike, the force is one of
attraction ; and, if they are alike, it is one of repulsion.

In one respect our definition is incomplete, for experiment shows
that the force depends, not merely upon the magnitude of the charges
and the distance between them, but also upon the nature of the
surrounding medium. For example, if the air be replaced by, say,
carbon disulphide, it will be found that the force is less than half
what it was before, although the charges and the distance apart are
unaltered. Hence, in defining the unit of charge, it is necessary to
state that the force is measured in air.

DISTRIBUTION- COULOMB'S LAW 25

In order that equation (2) may be of general application, we
must write

p=m (3)

where K is a number — called the dielectric constant of the medium —
whose value depends upon the nature of the surrounding dielectric,
and which, for air, is taken as unity. (The reason of our putting K
in the denominator will not yet be understood by the student, but it
may be remarked in passing, that, if it were placed in the numerator,
the values would be almost always fractional.)

Equation (3) may now be used as the basis of an extensive
mathematical theory leading to quite correct results ; the only objec-
tion is that the method completely disguises the real nature of the
actions involved, and hence it is advisable to consider the matter from
an alternative point of view, viz. that of the electric field.

Let us consider a simple case. Imagine two insulated spheres
placed near each other, charged positive!}7 and negatively respectively.
Some lines of force will pass from one to the other, and some will
extend from the spheres to the surrounding objects or to the walls of
the room. Now, if a charged pith-ball be placed between the spheres,
we know that it will be attracted by one sphere and repelled by the
other, i.e. it will move in a certain direction. Instead, however, of
regarding this action as being caused by the attraction and repulsion
of the charges, we may regard it as being due to the fact that the
pith-ball is placed in an electric field, without concerning ourselves
about the charges, which, as we have mentioned, are merely the
terminations of the lines of force on the conductors forming the
boundaries of the portion of the dielectric between them. This can
be expressed by saying that, when an electric charge is placed in an
electric field, a mechanical force is exerted on the conductor, which
carries the charge, tending to move it along a line of force.

If, however, the nature of the surrounding dielectric is altered,
without any other change being made, experiment shows that the
mechanical force is altered in amount, although the strength of the
field is the same as before. Hence, we are obliged to distinguish
carefully between two quite different quantities — (1) the strength of
an electric field at a given point in it, and (2) the electric force at the
same point.

Definition. — The strength of an electric field at a given point
in it is measured by the number of lines of force per square centimetre
at that point, the area being supposed to be at right angles to the lines
of force.1

The student will easily understand that this definition applies
equally well when the field is not uniform in strength, for, in that
case, it is understood that by the strength of a field at a given point is
1 See remarks on magnetic fields, p. 133.

20 ELECTROSTATICS

meant the number of lines of force which ivould pass through a square
centimetre, if the field had everywhere the same strength that it
actually possesses at that point. It is advisable to notice that an
electric field has a definite direction as well as a definite strength, i.e.
it is a vector l quantity.

Definition. — The electric force at a given point in an electric
field is measured by the mechanical force in dynes exerted upon an
electric charge of unit strength pi aced at that point?

The three quantities — electric field, electric force, and charge — are
linked together by the very important relation : — •

Strength of electric field = K x Force on unit charge (4)
where it is understood that K is to be taken as unity for air (or more
strictly for a vacuum).

We are now in a position to understand the real meaning of
Coulomb's equation. For, imagine a charge of q units placed at a
point in space distant from other bodies. According to our ideas, a
definite numbsr of lines of force must radiate uniformly from it. If
we suppose that the unit charge possesses n lines of force, then nq
lines start from the point in question. (It will be convenient to regard
both the value of n and the magnitude of the unit charge as unde-
termined at this stage, except in so far that they must be in accordance
with the definition given above).

Now consider any other point P at distance d from the charge q.
As the lines of force are spread over the surface of a sphere of radius
d, we have

Field strength at P = — ^9, kird- being the area of a sphere,

but Field strength = K x Force on unit charge.

-r, ., , Field nq

. . I orce on unit charge = —

K

.
Now, suppose that any charge ql of opposite sign is placed at P,

then Force on <?]_ = ,,-. -^ x qv

IV. 4:TTCl

but the force due to the two charges q and ql is one of attraction,
.'. Attraction in dynes = --[^i -- (5)

1 Any quantity, e.g. a mass or a volume, which is completely defined by a
number only, is called a scalar quantity. A force or a velocity cannot be com-
pletely denned in that way ; we require in addition to know the direction in
which it acts. Such a quantity is called a vector. In this work it will be
assumed that the student is familiar with the theorem known as the "parallelo-
gram of forces," which shows how vectors may be added together or resolved
into components.

2 In all ordinary cases, the charge is carried by some conductor, and it is the
mechanical force on the conductor that we really measure.

DISTRIBUTION— COULOMB'S LAW 27

This is evidently the true form of the law discovered experimentally
by Coulomb. It is true whatever wre may decide to take as the unit
charge, and from it we are in a position to define unit charge in
several different ways. But, as we have already remarked, this has
been done in such a way that

Attraction (or repulsion) in dynes = 4

Comparing equations 5 and 6, and remembering that in both of
them K = 1 for air, we see that they become identical if we make
n =* 47T. Hence, the accepted definition of unit charge also implies
that such a charge must be regarded as possessing 4?r lines of force.1
The student should also notice that, in the accepted system of

units, the field strength at distance d from a charge q is ^2 (for at
that distance a total number of lines 4?r^ is spread over an area
47nZ2), and also that the force on unit charge at that distance is TT~M'

It would be a great convenience if some generally accepted nota-
tion could be adopted for the fundamental ideas mentioned in the last
few paragraphs. The student will find that similar conceptions are
required in dealing with magnetism, but in that case it fortunately
happens that certain symbols have received an international signifi-
cance. No such agreement, however, has been arrived at in statics,
and hence much confusion is felt by the student, who finds the idea
force on unit charge expressed in many different ways by different
writers (e.g. intensity of electric force). There is some objection to
every plan, but we shall in future adopt the following terminology : —

F = Field strength or number of lines of force per square

centimetre.

U = Force on unit charge (dynes).
K = Dielectric constant (numerically identical with specific in-

ductive capacity).
p = Mechanical force, i.e. attraction or repulsion (dynes).

Hence, we have

F = KU, (7)

The total number of lines of force passing through a given area

1 This definition of unit charge leads to troublesome constants in certain
equations of frequent use in electrical engineering calculations. If it were
possible to begin de novo, most probably the definition would be altered.

28 ELECTROSTATICS

will be called (in accordance with the accepted usage in magnetism)
the Flux. In a uniform field

Flux = F x Area. (9)

TllbeS Of Force. — The phenomena of electric (and magnetic) fields
may also be represented in terms of the properties of tubes of force. Consider a
small area on a charged conductor, carrying a small portion q of the total charge.
Evidently ±irq lines of force emanate from this area, and, after passing through
some dielectric medium, end finally in an equal and opposite charge on another
body. If we think of these lines as a single bundle, their bounding surface-
traces out a "tube of force," and the whole electric field can be divided up in
this way into a number of such tubes, which will completely fill the dielectric
space traversed by the field. If we choose the original area so that it carries
one unit charge, the result is a "unit tube," and the attractions and repulsions
of electric charges may be regarded as due to the tendency of such tubes to
contract in length (due to the tension along them), and to expand laterally. We
do not consider it necessary to develop this point of view, because the conception
of lines of force is universally used by electrical engineers in actual calculations,
and experience has shown that it is sufficient for all practical requirements,
besides being more convenient in practice than methods based upon the con-
ception of tubes of force.

The Inverse-square Law for Point Charges.— AS this law

forms the basis of the mathematical theory of electrostatics, a proof of its
correctness is of considerable importance. For this purpose, the method of
Coulomb cannot be regarded as satisfactory — at the best it affords merely a
rough verification.

An indirect, but much more satisfactory, proof may, however, be obtained
as follows : — We know as a fact that there is no electric force inside a hollow
conductor, and in 1773 Cavendish 1 proved mathematically that this is a neces-
sary consequence of the in verse -square law for point charges, and is inconsistent
with any other rate of variation with the distance. Laplace, Bertrand, Clerk-
Maxwell, and others have also given proofs of this theorem, for which the
student must be referred to more advanced works. Now, the fact itself can be
verified experimentally with great exactness, and such experiments constitute
the real proof of the law.

It should be noticed that similar facts are met with in connection with
gravitation. If the earth were a hollow sphere, there would be no gravita-
tional force inside the cavity, and a body placed therein would have no weight
— a result which depends upon the inverse-square law of gravitational attrac-
tion.

This resemblance appears to have been first noticed by Priestley, who also
investigated the properties of hollow conductors, and who, in 1767, suggested
(but without proof) that they might be due to an inverse-square law such as that
known to hold good in the case of gravitation.

Another way of regarding the question is instructive. Consider a uniformly
charged hollow sphere, and think of the charge as being made up of an infinite
number of very small point charges symmetrically distributed upon it. Each of
these point charges possesses a definite number of lines of force, and, in the
absence of all disturbing conditions, these lines would radiate from it symmetri-
cally in straight lines. Imagine such a system of lines to be drawn from every
one of the point charges, then, if the resultant of all these superposed lines be
taken (as for instance in Fig. 28), it will be found that there is no field within

1 Cavendish did not publish his results, and Coulomb's work (about 1785)
was performed independently.

DISTRIBUTION— COULOMB'S LAW 29

the sphere, and that the field outride is radial and of the required strength.
Now, in assuming that the lines or force radiate in straight lines from each
point charge, we have really assumed the inverse-square law. Any other law
would involve a different distribution, and the resultant field would not vanish
inside the sphere.

EXERCISE III

1. An electrical machine is placed in an insulated chamber which is lined
inside with tinfoil. The rubber of the machine is connected with the tinfoil.
What will be the effect upon an electroscope placed outside and connected with
the chamber when the machine is in action ? Explain your answer.

2. A wire is fastened by one end to the inside of a deep insulated metal jar,
and by the other end to an electroscope. When the jar is electrified, the leaves
of the electroscope diverge ; but no charge is given to a proof-plane put in
contact with the side of the jar. " Explain these results.

3. A hollow metal vessel is insulated, connected by a wire with a gold-leaf
electroscope, and charged with electricity. The leaves diverge. An uncharged
metal ball is lowered into the vessel without touching it, (1) by a silk thread,
and (2) by a wire. What is the effect on the gold leaves in each case ?

4. Two insulated metal spheres, charged respectively with +5 and -5 units,
are placed one metre apart. What is the*flirection of the resultant electric force
exerted on a small + charge at a point one metre distant from the centres of
each of the spheres ?

5. The force of attraction between two small balls was 8 dynes, when they
were placed 6 centimetres apart. What is the charge on each, if the + charge
was twice the - charge ?

6. Define unit (electrostatic) quantity of electricity, and find the force
exerted between two equal small spheres with centres 8 centimetres apart, one
charged with +16 and the other with -20 units of electricity. If the two
spheres are momentarily joined by a thin copper wire without being moved,
find the force between them. (Lond. Univ. Matric., 1903.)

7. Electric charges of 10 and 5 units are given to two bodies which are at
a distance of 50 centimetres apart. At what point on the straight line joining
the charges is the electric force zero ? (0. of. P., Senior, 1906.)

8. Equal electrical charges of 10 units each are placed on small conductors
at two opposite corners of a square of 10 centimetres side. Calculate the
electric force at either of the remaining corners. (C. of P., Senior, 1908.)

9. Two small spherical pith-balls, each 1 decigram in weight, are sus-
pended from a point by threads 50 centimetres long, and are equally charged
so as to repel each other to a distance of 20 centimetres. Find the charge on
each in electrostatic units (0 = 980). (B. of K, 1901.)

10. Define unit charge of electricity. Two charged conducting spheres repel
each other with a force equal to the weight of a milligram when placed at
a certain distance from each other. If the charge on one of the spheres is
doubled and the distance between the spheres is also doubled, what is the
amount of the repulsion ? (B. of E., 1903.)

11. Two equally charged spheres repel each other when their centres are
half a metre apart with a force equal to the weight of 6 milligrammes. What
is the charge on each, in electrostatic units ? (B. of E., 1892.)

12. A small pith-ball weighing 1 decigram, suspended by a silk fibre and
charged with positive electricity, is repelled whan a charged glass rod is brought
near it. If the direction of the electric field of the glass rod near the ball
is horizontal and its magnitude equal to 20 C.G.S. electrostatic units, when
the deflection of the fibre is 45°, what is the charge on the ball ?

(B. ofE., 1906.)

13. Two copper spheres, each 1 millimetre in diameter, are suspended from

30 ELECTROSTATICS

the same point by silk fibres 1 metre long, and when equally charged are at
a distance of 1 centimetre from centre to centre. Determine the charge on each
sphere, the density of copper being 8 '9 and the acceleration of gravity 980.

(B. of E., 1907.)

14. A short ebonite rod, with a small electrified knob at one end, is mounted
so as to turn freely about its centre in a horizontal plane. In a horizontal line
with this centre, and at distances from it of a quarter and half a metre respec-
tively, are placed insulated balls that are also charged. The rod makes ten
vibrations in a given time, but makes thirty vibrations in the same time if the
balls are interchanged. Compare the charges on the two balls.

(B.of E., 1893.)

CHAPTER IV

POTENTIAL

Exp. 39. Attach one end of a long fine wire to the disc of an electroscope, and
the other to an ebonite rod, so that the wire can be moved about and yet remain
insulated. Charge an insulated cone-shaped conductor, and by means of the rod
bring the end of the wire in contact with it. Notice that it is quite immaterial
where contact is made ; whether it be at the pointed or blunt end, a certain
deflection is obtained at the moment of contact, and this remains unaltered in
amount as the wire is moved over the surface of the conductor.

Now in Experiment 36, when a proof-plane was used in the
ordinary way, a larger charge was obtained from the pointed end than
from any other part of the conductor, and we therefore infer from
these experiments that, although the charge is not uniformly dis-
tributed over the surface of the conductor, there is some condition
which is constant at all points upon it.

This fact can be still more clearly brought out by using a hollow
insulated metal can. We know that normally the charge is entirely
on the outside, and yet, when the end of the wire connected to the
electroscope is brought into contact with the can, exactly the same
deflection is obtained whether the contact is made inside or outside.

These results are expressed by saying that the potential of a
charged conductor is the same at all points, and is quite independent
of the distribution of the charge upon it.

As the idea of potential often presents some difficulty to beginners,
it is useful to give a simple analogy. Imagine a tank filled with
water, so shaped that the water is deep in some parts and shallow in
others. This is the analogue of a non-symmetrical, charged body.
Imagine further that the electroscope is represented by a tall, gradu-
ated glass jar, and the proof -plane by a little bucket fastened to the
end of a rod. Evidently if we scoop up the water in different places
in turn and empty the contents in each case into the graduated jar, we
shall be likely to find more water in some parts than in others, because
in some places there may not be sufficient depth to fill the bucket.

This is the result obtained in Experiment 36.

If, however, we join the tank to the graduated jar by a pipe or
tube, the water will at once rise to a definite level in the jar, and
the reading will be the same wherever the connection is made, pro-
vided that it is below the surface of the water.

This is the result obtained in Experiment 39.

31

32 ELECTROSTATICS

From this analogy it will be seen that the relation between the
ideas expressed by the terms potential and charge in electricity is
similar to the relation between the terms height or level and quantity
in the case of liquids. Again, in the science of heat, temperature is
not the same thing as amount of heat, but it is the relative condition
of a body which determines the direction in which heat flows when
the body is connected to another body. So, potential is not charge or
quantity of electricity, but is a relative condition which determines the
direction a charge tends to flow when the charged body is put in con-
ducting communication with another body.

Although it is convenient to speak of the temperature or the height
of a body, the expressions really mean the difference of temperature or
difference of level between that body and some arbitrary zero — the
freezing-point of water in the first case, and the height above the
earth's surface or the sea-level in the second case. Similarly, in
electrical theory, we are really always concerned with the difference of
potential between two bodies (usually written Potential Difference, or
P.D. for brevity), and when the term potential of a body is used alone
it is understood that it really signifies the P.D. between that body and
some other body whose potential is taken as zero. Although the
state of a body infinitely distant from all electrified bodies would be
the ideal realisation of zero potential, in practice it is sufficient to
regard the potential of the earth as zero.

With conducting bodies, we may summarise the previous state-
ments as follows : —

(1) If a flow takes place between two bodies when they are
connected by a conductor, a P.D. must have existed between them
before they were connected.

(2) The effect of such flow is to equalise their potentials, hence

(3) Two bodies in conducting communication are necessarily at
the same potential.

(4) If no flow takes place when the two bodies are connected, they
must have been at the same potential before they were connected.

As corollaries to the above, we may note —

(5) A body connected to earth is at zero potential, whether
charged or not.

(6) All points on the same conductor must necessarily be at the
same potential, for if not, a flow would instantly take place inside its
substance tending to equalise the potentials. If two points on it are
kept permanently at different potentials by some external means, the
result is a continuous flow or current, a case studied later as " Voltaic "
electricity.

So far, we have said nothing about the direction of flow, and,
strictly speaking, a flow must be regarded as involving a passage of
equal amounts of positive and negative charges moving in opposite

POTENTIAL 33

directions. If we find that when two bodies, A and B, are connected,
a positive charge passes from A to B, then we may either say that A
is at a positive potential with respect to B, or that B is at a negative
potential with respect to A. Such statements are definite and
satisfactory, but it has become, customary, by analogy with the use
of the terms high or low in the case of gravitational level, to speak of
A as being at a higher potential than B, when we really mean that it
is at a positive potential with respect to B. Hence, an isolated posi-
tively charged conductor is at a higher potential than the earth, and a
similar negatively charged conductor is at a lower potential than the
earth. These terms are sometimes convenient, and are perfectly
intelligible, but the student must remember that they have no real
physical meaning.

An Electroscope Measures Differences of Potential.—

In explaining the result of Experiment 39, an electroscope was
likened to a tall, graduated glass jar for measuring quantities of a
liquid, and it follows from that experiment that when the cap of an
electroscope is connected to a body by a wire, the deflection really
measures the P.D. between that body and the tinfoil strips at the base
of the instrument, i.e. the earth.1

If this statement is correct, no deflection should be obtained when
the cap and strips are connected together by a conductor, i.e. when
they are at the same potential.

Exp. 40. Place an electroscope on an insulating stand, connect the cap to the
tinfoil at the base by a thin wire. Hold a strongly charged rod over the disc,
and notice that there is no deflection. (If the rod be brought near the side of
the glass vessel, it may be possible to produce a deflection, although cap and base
are at the same potential. Strictly speaking, it is understood that only the cap is
to be exposed to external electrification, and to satisfy this condition the glass
vessel may be enclosed in a wire cage, or more conveniently, vertical tinfoil strips
may be pasted on the outside, making contact with the base, space merely being
left for the strips to be visible. Such devices are, however, unnecessary for
ordinary purposes. )

The fact that the deflections measure differences of potential may
be well shown as follows. If any source giving a P.D. of about
200 volts is available (e.g. electric lighting mains), connect one
terminal to the cap and the other to the base of the instrument. The
result is a perfectly definite and steady deflection, whether the supply
is direct or alternating. (Care must be taken not to " short circuit "
the mains ; a piece of very fine wire or " fuse " should always be in
the circuit as an additional security.)

1 This is the great advantage obtained by having such strips, or having
metal sides to the vessel as in the more exact type of instrument. An electro-
scope will work well when the leaves are inside a glass flask or bottle, but it is
then uncertain what the deflection really measures. Made as above, its deflec-
tions measure the P.D. between the cap and the earth, and the readings have a
definite quantitative meaning.

C

34 ELECTROSTATICS

Exp. 41. Insulate an electroscope and charge the base — conveniently bv
induction. Observe that the leaves diverge. Touch the cap and notice that
the divergence is increased.

In the first instance — assuming that the base is positively charged
— both the base and the leaves and, cap are at a positive potential,
but not at the same potential, that of the base being the higher, hence
a deflection, which is a measure of the P.D. between them. When
the cap is connected to earth, its potential falls to zero, so that the P.D.
between it and the leaves is increased — hence the increased divergence.
(The student will learn later that the P.D. of the base was also lowered
by touching the cap; i.e. both potentials are altered, but it would need-
lessly complicate the question to enter into details now. As stated,
the nett result is that the P.D. is increased by earthing the cap.)

Difference of Potential Defined Quantitatively. — There
is another way of regarding these matters, which is very helpful and
instructive. We may express it briefly by remarking that when we say
a difference of potential exists between two bodies, it is really equivalent
to saying that they are connected by lines of electric force. If they are not
connected by lines of electric force, there is no difference of potential
between them.

If an electric charge be placed between the bodies it will be in an
electrical field, and there will be a mechanical force acting on it and
tending to move it in one direction or the other, according to the sign of
the charge. Again, if we carry the charge from one body to the other
against a force, a definite amount of work must be done — which, as in
the case of gravitation, is quite independent of the path traversed, as
long as we begin and end at the same points. Thus, charged bodies
are capable of either doing work or requiring work to be done on them,
and it is this idea of wrork that is involved in the definition of potential.

Definition. — The difference of potential between any two points is
measured numerically by the work in ergs done in moving a charge of
unit strength from one point to the other against the force. If the
charge is moved in the reverse direction, an equal amount of work is
done, but in this case the work is done by the charge, instead of on it.

It must be remembered that although it is convenient to measure
a difference of potential in terms of work, yet potential and work are
not the same thing. The difference of level between the floor and the
ceiling of a room is numerically equal to the work done when unit
mass is carried from one to the other, but obviously this difference of
level is not the same thing as work.

Although we are always concerned with differences of potential, it
is frequently convenient to speak of the potential at a point without
further qualification, just as we are always concerned with differences
of level, although we often speak of the height of a point. In both
cases, the term evidently means the difference between the point in
question and some accepted point of zero potential or zero level.

POTENTIAL 35

The absolute potential at a point is usually defined as being the
P.D. between that point and "infinity." This is convenient as a
definition, but there is no way of ascertaining its actual value in any
given case.

As we have already mentioned, we usually take the earth as a
standard, and hence by potential we really mean the P.D. between a
body and the earth. The fact that the earth has a very variable
potential does not introduce any practical difficulty.

Electrical Potential at a Point near a Charged Body.—
If we have a charge of Q units at a point in space, the potential it

produces at any other point at a distance r centimetres is equal to ^ •

The following elementary proof of this formula may be given : —
Let us consider .that M (Fig. 23) is a point charged with Q
units of positive electricity.

FIG. 23.

It is required to find the potential at A at distance r. By defini-
tion, this will be numerically equal to the work done in bringing a
unit positive charge from " infinity '' to A against the repulsive force
due to Q. (It will not affect our argument if we write " earth " in
place of "infinity." Such an alteration would merely imply that the
value obtained is relative to that of the earth.)

For convenience we shall determine first the P.D. between the
point A (at distance r), and the point B at distance r, i.e. the work
done in moving unit charge from B to A.

Divide the distance r' — r into a very large number of equal
parts, A a, alt be . . . . riB.

Then, Force on unit charge at A = — — 2 (p. 27).

Similarly, Force on unit charge at a — _

(a being the distance between M and the point a, Fig. 23).
Similarly, Force on unit charge at fr = ^.,2

„ »= ^

. Q

36 ELECTROSTATICS

Now, as the spaces between A and a, a and b, &c., may be made as
— small as we please, there will be no material difference if we call
r x a the mean between r2 and a2,1

i.e. the mean force between r and a = £*-

Kra

.*. work done in moving the unit charge from a to A

Kra K\r a,

Similarly, the work done in moving it from I to a

K\a I)'

Q /I 1

and from c and b, the work done = ^J-( - — -

K\b c

finally, from B to n = — { ~ —
K\n r

whence, adding these equations, the work done in passing from
B to A

KVr r)

(i.)

where VA, YB are the potentials at A and B respectively.

If B is at an infinite distance from M, then r is infinite, so that

!,=.,

whence, from equation (i.), VA = ^ (ii.)

1 A numerical example will perhaps make this statement clear. Suppose
each of the small spaces between A and B='01 centimetre, and that r = 10
centimetres, then a = 10 '01 centimetres,

then r2 = 100
and a2 = 100 '2001

so that the mean between r2 and a2 = 20Q'2Q01 = 100'10005

but r x « = 10 x lO'Ol = lOO'l

so that the difference between the mean of r2 and a2 and r x a is only '00005.
If the distance between A and B had been still smaller (say '001 centimetre) the
difference between the two results would have been only '0000005.

POTENTIAL

37

Students acquainted with the calculus will obtain this result by
writing

Potential at distance r =

[V-sfV

J r J r

_Q

Kr

Of course, it will be noticed that if the medium is air, K= 1, and

. *. Potential at distance r in air = — *

r

Application to an Isolated Sphere. — The lines of force

from a charged sphere, which is placed at a distance from other
conductors, are uniform and radial. Hence, they diverge as if from
a point charge at the centre of the sphere, and the potential at the

surface will be ^ ' where r is the radius of the sphere.
r

Potential at a Point due to
a number of Charges.— If there
are a number of charges Q15 Q2,
Q3, &c., at distances rv r2, r3, &c.,
respectively from the point A (Fig
24), then

vA=Si+9i+9»+&c.-2S

where 2 is the sign of the summa-
tion of the series.

FIG. 24.

Example.— As an example of the kind of problem depending upon this
section, we append the following : —

Within a spherical vessel of brass 1 centimetre thick, the external diameter of
which is 14 centimetres, a brass ball 8 centimetres in diameter is hung by a silk
thread so that the centres of the two spheres coincide. If the ball is charged
with 36 units of positive electricity, and if the potential of the vessel is 7, what
is the potential of the ball ? (B. of E., 1896. )

As the potential of the outer vessel is + 7,

7 we have (Fig. 25) 7 = ^ = ^, i.e. ?3 = 49.

Also, if there is a charge of +36 units
on the enclosed ball, there must be a charge
of - 36 on the inner side of the outer sphere,
for there the ends of the lines from the former
must terminate,

.'. If V= potential of the ball

+36 -36
~ ~~~

FIG. 25.

+49
6~" 7
=9-6+7
= 10 static units of potential.

38

ELECTROSTATICS

Equipotential Surfaces.— Let a charge Q be placed at a point
in space, distant from other bodies or charges. It may, for conveni-
ence, be supposed to be carried by a conductor of negligible dimensions.
We have shown that the potential at any point at distance d from

this charge is -5?. Now all points situated at this distance d must

have the same potential, i.e. the potential has the same value all over
the surface of an imaginary sphere of radius d. Such a surface is
known as an equipotential surface. Evidently any sphere having
the point charge at its centre is an equipotential surface ; and this
will also be true when the charge is carried by a conducting sphere of
any radius, provided no other bodies or charges are near.

The characteristic property of such surfaces is that a charge may
be moved about on them without doing work against electric force,
from which it follows that they are everywhere at right angles to the

lines of force. It will be seen that
they are analogous to horizontal
surfaces in relation to gravitation,
for a body can be moved any-
where on the level without doing
work against gravity.

The surface of a conductor, of
any shape whatever, is obviously
an equipotential surface, as was
proved in Experiment 39.

We may draw a system of
equipotential surfaces, due to a
charged sphere, so that there is
a unit difference of potential be-
tween each surface, i.e. so that
one erg of work is done by carry-
ing a unit of positive electricity from one surface to the next, as
follows : —

Let M (Fig. 26) be a small sphere charged with twelve units of
positive electricity.

FIG. 26.

To obtain the surface where V = 1 with this charge,

we, therefore, have r = 1 2
Where V = 2 we have r = 6

„ V = 4 „ „ r = 3,&c.

Thus the distances between the surfaces, between which there is
unit difference of potential, become greater and greater as we recede
from the charged body.

POTENTIAL

39

The charge may be thought of as the summit of a mountain, and
the equal potential lines drawn round it as indicating equal decreases
in height as its slopes are descended. If the charge is negative, it
may be thought of as a hole in the ground with sloping sides. The
lines shown in Fig. 26 would then resemble the "contour lines"
frequently drawn on maps to connect points at the same level.

When the charged body is not spherical, or when several charges
are present, the equipotential surfaces or lines are very complicated.

FIG. 27.

Fig. 27 is a reduced copy of a diagram drawn to show how their form
may be obtained in a comparatively simple case, viz. around two
charges (supposed concentrated at points) of + 3 and - 2 units respec-
tively, placed 5 centimetres apart.

The first step1 is to draw, for each charge separately, a series of equi-
potential lines, differing in potential by a convenient amount (in the
figure, T\j- of a unit wa« used). These will form a series of concentric

1 The student is advised to follow the instructions on an imperial sheet of
cartridge paper.

40 ELECTROSTATICS

circles around each charge. For instance, V = + 1 at a distance of
3 centimetres from the charge of + 3 units at A, and -hence a circle
of radius 3 centimetres is described around A as centre. Similarly
V = - 1 at a distance of 2 centimetres from the charge of - 2 units
at B, and hence a circle is described around B with a radius of 2
centimetres. Outside these, other circles are described, corresponding
to potentials + '9, + '8, + -7,&c., and - '9, - -8, - '7, &c. Within
the first two circles, others corresponding to potentials 1-1, 1'2, &c.,
might be drawn, but these lie so close together that it is difficult to
show them properly without confusing the diagram. Hence they
are omitted, but it must be remembered that this is done only for
convenience.

At the point on the axis where the two first circles touch, the
potential is + 1 due to one charge, and - 1 due to the other, the
resultant being zero. The next two circles cut in two points, and
as they represent potentials of + *9 and - -9 respectively, the re-
sultant is again zero at their points of intersection. The method of
drawing the equipotential line V = 0 is now obvious. It is only
necessary to draw a line through the intersections of the succes-
sive four-sided figures as shown in the diagram. When the point
P is reached, the student may be doubtful what to do next. Now
Pj is evidently a point on the line, because there the potentials
are + 2 and - 2 respectively. Hence, a freehand line might be
drawn from P to Pp but in the present case, in order to obtain
greater accuracy, two other points, indicated by the intersections
of the dotted arcs, have been determined. These correspond to
potentials + -25, - -25, and + -225, - -225 respectively, the radii

being calculated by writing V= -g, or + -25 = -rp from which d = 12

o
centimetres ; and similarly - "25 = — from which d^ = 8 centimetres.

Ctj

Again, in tracing the line V= +'1, no points of intersection are
obtained between R and Rp and therefore three intermediate points,
indicated by dotted arcs, have been inserted.

When drawing the line V'= + '2, it is uncertain what to do next
when the point T is reached. In this case, it is sufficient to find where
the line cuts the axis. Let this be at distance d from the charge
+ 3 units, then we have

+ •2-4-

d 5 + d

On solving this quadratic and taking the positive value, it is found
that rf = 8'66 centimetres.

In the same way, points on the axis corresponding to V= -'2,
and - '3 can be determined.

Sometimes it is desirable to find the point of intersection between

POTENTIAL

41

A and B. For instance, let the line V = + '5 cut the axis at distance
d from A, and between A and B, then

+ .5=.?— -?-
d 5-d

or d= 2-4 centimetres.

Construction for Lines of Force.— The chief difficulty in
drawing a diagram of lines of force is due to the fact that we have to
represent on a plane surface what is really a distribution of lines in
space. For our present purpose, it will be best to simplify the problem

FIG. 28.

by assuming that all the lines are confined to one plane, viz. the plane
of the paper, although this necessarily involves a loss of accuracy.

With this assumption, Fig. 28 has been drawn, showing the dis-
tribution of the lines of force in the case of the two charges for which
the equipotential lines have just been obtained. As these charges are

42 ELECTROSTATICS

+ 3 and - 2, there are 4?r x 3 lines diverging from A, and 4?r X 2 lines
from B, but it is sufficient to draw any convenient number of lines in
each case, provided wre make their ratio 3 : 2. For instance, 24 radial
lines have been drawn from A, and 16 (drawn as broken lines) from B.
All the lines diverging from A should be marked with arrows pointing
from A (because a + charge would move from A), and all lines
diverging from B, with arrows pointing towards B. Four-sided (or
triangular) figures are produced by the intersection of these radial
lines, and it will be evident on inspection that, at each intersection,
the resultant, or actual line of force, will pass through that point
roughly in the direction of the diagonal. If the lines were very much
more numerous, each quadrilateral would be approximately a parallelo-
gram, and the resultant would be its diagonal; but, as it is, we are
only certain that the resultants pass through the opposite corners of
each space ; thus, although their general direction between A and B
can be quickly traced, no pretence is made to great accuracy in the
details. From the method of procedure adopted, it will be evident
that the figure applies generally to any two charges, whatever their
actual value, provided they are in the ratio 3:2, and + and - respec-
tively. This means that we obtain the general appearance of the
field, but not the correct number of lines per square centimetre. If
the actual number of lines of force be drawn, the figure may be made
quantitatively correct. For instance, as we have used 24 and 16 lines,

24
Fig. 28 may be regarded as strictly applying to charges of + — and

1 f>
- 2~ , apart from the error caused by drawing them on a plane surface.

There will be a point on the axis, at which the field is zero.
Remembering that the field at distance d from a charge Q is ^2, we

see that, if d be the distance of this point from B, its position will be
determined by the equation

^ 2

- -x = 0, which gives d=22'3 centimetres.

This point is too far to the right to be shown on the diagram.1

Electrical Capacity. — When we charge an insulated conductor

with a certain quantity of electricity, we raise the potential of the

conductor, but the extent to which it is raised depends upon the

capacity of the conductor.

<~ Definition. — The capacity of any conductor is measured by the

L quantity of electricity with which it must be charged in order to raise its

potential from zero to unity. Thus, a small insulated sphere would

1 Similar methods apply in the case of lines of force due to magnetic poles.

POTENTIAL 43

require a small quantity of electricity to raise its potential from zero to
unity, i.e. it has small capacity. On the other hand, a large insulated
sphere must be charged with a large quantity of electricity in order
to raise its potential from zero to unity, i.e. it has a large capacity.
We therefore learn, that the potential of a conductor depends both
upon its amount of charge and upon its capacity; in fact, if C be the
capacity of a conductor, Q the quantity of electricity with which it is
charged, and V the potential, then it follows from the definition that

By aid of this formula, we are able to obtain the potential of a
number of conductors, placed at considerable distances apart, whose
capacities are Clt C2, C3, &c., and whose initial potentials are V1? V2,
V3, &c., respectively, when they are joined by fine wires (the capacities
of which we may neglect). They all acquire the same potential, V,
so that we have

+ C2 + C3 + <fec.) = V^! + V2C2 + V3C3 + &c.

Unit Capacity. — The capacity of a sphere is measured by its
radius in centimetres ; for suppose we have an insulated sphere,
removed from the influence of other charged bodies, of radius r, and
charged with a quantity of electricity, -Q, then

the potential (V) at the surface = —

Kr

whence, substituting this value in the formula C = — we have

«-
Kr

If r is 1, C is also 1 ; hence the unit of electrostatic capacity is the
capacity of a sphere of one centimetre radius in air (or in a vacuum),
when placed at a sufficiently great distance from other conductors or
charged bodies.

Subdivision of Charges on Spheres. — We are now in a
position to understand the subdivision of charges on spheres when
they are put in conducting communication with each other. The
quantity of electricity taken by each will depend upon its capacity.

1. Spheres of equal capacity, i.e. of equal radius.

(a) If we charge an insulated sphere with a certain quantity of

44 ELECTROSTATICS.

electricity, and then bring another insulated sphere of equal size in
contact with it, each one will contain half the original charge ; if, on
separating them, another be brought in contact with either, it will
receive half its charge, i.e. a quarter of the charge originally imparted
to the first sphere.

(b) If two equal insulated spheres be charged, one with ten units
of positive electricity, and the other with twenty units of positive
electricity, and then placed in contact, each will have half the sum of
the two charges, i.e. fifteen units.

(c) Similarly, if one of them has originally twenty units of positive
electricity, and the other ten units of negative electricity, after contact

+ 20-10
each has - — ~ -- = 5 units of positive electricity.

2. Spheres of unequal capacity, i.e. of unequal radius.

If a large insulated metal sphere of radius r has a charge of Q
units, and a smaller insulated sphere of radius i\ be brought in contact,
and afterwards separated ; or if the two spheres be merely connected
by a fine long wire (whose capacity we may neglect) and then charged
with Q units, the quantities (q and q1 respectively) contained by each
may be easily ascertained, for

(1)

and the two conductors will be at one potential ;

whence ? = ?! (2)

r

Substituting the value of ql from equation (1) we have

, Or

whence q = — —
r-fr,

Or

and similarly ql = -^-1

This relation is true, and the proof is the same, for any two con-
ductors of capacity 0 and Cj respectively, irrespective of shape,

and then 2 = ^ and 9l = ^V

Surface Density on Spheres.— (l) One sphere. We have

POTENTIAL 45

already learnt that the density of electricity on any insulated con-
ductor, when uniform, varies directly as the quantity and inversely as

the surface, i.e. p = ~.
A

Now, the surface of a sphere, whose radius is r, is to*2, whence,
on a sphere,

(2) Two spheres joined by a long fine wire.
Let q and ql be the charges on the spheres of radii r and
respectively, and let Q be the total charge,

Then Q = q + qv and by equation (2) on p. 44,

i.e. the quantities are directly as the radii.

The density will, however, be inversely proportional to their radii,
for from the equation given above

Pi

but ? =1

Relation between Density and Potential on Spheres» —

The relation between density and potential on a sphere may be easily
obtained, for

6-m

,:.l|* $ alld •

whence, by substitution, we have V =

Field and Force very close to a Charged Conductor.—

As the density is the number of unit charges per square centimetre,
and as unit charge has 4?r lines of force proceeding from it, it follows

46 ELECTROSTATICS

that 4:7rp lines of electric force must emerge from one square centi-
metre of any charged conductor, whatever its shape may be, provided
p be understood to mean the surface density at the place in question.
Again, as the field strength at any point is, by definition, the
number of lines per square centimetre, it follows that very close to the
surface — before the lines appreciably diverge —

Field = F = 47T/3, and consequently
Force on unit charge = U = — ^° dynes.

Jx

The latter result holds good as long as we are considering the
force on an independent charge, close to the surface, but not actually
in contact with it.

Force on the Conductor itself. — Evidently there must be an
outwards pull due to the tension of the lines of force, and it is
required to determine its amount. Here, it must first be pointed out
that when a charge Q is placed in a field of strength F, the field due
to Q must in reality be superposed upon F, producing a resultant
field. According to our definitions, however, the force acting on

TBt

Q is Q x — , from which we see that in such calculations the field

due to Q itself must be ignored, i.e. we must not regard F as being
the resultant field. Hence, if we use this method to find the pull per
unit area on a surface charged with p units per square centimetre — in
which case the actual field is obviously a resultant — it is evident that
we must think of the charge p as being under the influence of a field
partly due to itself and partly due to the remainder of the charge on
the body; and it is this latter component which must be used to

Tjl

determine the electric force —

&.

Now, according to the point of view suggested on p. 28, if we
consider the charge p alone, we must regard it as made up of point
sources, each possessing a radial and symmetrical field. If so, half of
its total lines must radiate outwardly, and the other half inwardly,
the forces due to the two halves being obviously in opposite directions.
Hence (assuming the charge to be positive), the external field due to
p is + 27T/3, and the internal field is — 2ny>. But we know that the
resultant field inside the conductor is zero, and, therefore, the com-
ponent due to the other charges on it must be just equal in amount
but opposite in sign to — 2irp. Moreover, this component is super-
posed upon both halves alike, and thus it follows that the resultant
external field is 47T/3, as already deduced by simple reasoning. It will
be seen that the force on the charge p is the same as it would be on
an independent charge of that value placed in a field of strength
27T/3, i.e. the radial pull is 2w/o2 dynes.

POTENTIAL 47

This important result may be stated more generally by saying that
when a charge Q forms the bounding surface of an electric field, in

which the electric force is U, then the force acting on Q is Q x — .

2

EXERCISE IV

1. What is the difference of potential between the points A and B (Fig. 23),
if M be charged with 72 units of positive electricity ; the distance of A being
8 centimetres, and that of B 12 centimetres ?

2. Charges of +10, +15, -5, -4 units of electricity are placed at the
corners A, B, C, D respectively, of a square, whose side is 10 centimetres long.
Find the potential at the middle point of CD.

3. Twenty units of + electricity are placed at the middle points of the sides
of an equilateral triangle, the sides of which are 9 centimetres long. Find the
potential at the centre of the inscribed circle.

4. Charges of 10, 20, 30 units of + electricity are placed at the corners A, B,
C respectively, of a square, whose sides are 10 centimetres long. Find the
potential at the corner D, and at the centre 0. Find the amount of work
necessary to be done in order to bring a + unit from D to 0.

5. Find the quantity of electricity which must be given to an insulated sphere
of 6 centimetres diameter, so that its potential may be raised from zero to 15.

6. Three insulated metal spheres placed at considerable distances apart are
charged with electricity till their potentials are 2, 5, 7 respectively. If their
radii are 2, 3, 4 respectively, find the potential of the whole system when they
are connected by a fine wire.

7. If the radii of the spheres were 4, 5, 6 centimetres respectively, and their
initial potentials were 6, 7, 8 respectively, find the potential of the whole system
when joined by a wire.

8. Two insulated metal balls, one being 1 centimetre radius, and the other
1'5 centimetre radius, were each charged to a potential 70. Find the force of
repulsion between the two balls when placed half a metre apart.

9. Two insulated brass balls are joined by a long fine wire ; one has a diameter
of 3 inches, and the other a diameter of 1 inch. A charge of 48 units of + elec-
tricity is given to them. How will the charge be distributed ?

10. A large insulated metal sphere is charged with 20 units of + electricity ;
another sphere of one-ninth the radius of the first is brought in contact. How is
the charge distributed when they are separated ?

11. Two insulated metal balls are connected by a fine wire ;. one has a radius
of 5 centimetres, and the other a radius of 8 centimetres. They are charged, and
on testing the larger one, it is found to have a charge of 16 units. What was
the total charge ?

12. To what potential must we charge an insulated sphere of 7 centimetres
radius, so that its surface-density may be represented by 2 ?

13. Two insulated brass balls are connected by a long fine wire. They are
charged to a potential 40. If the diameter of one is fourteen times that of
the other, compare their densities.

14. How much energy is expended in carrying a charge of 50 units of elec-
tricity from a place where the potential is 20 to another where it is 30 ? What
is meant by saying that the potential of a conductor is 20 ? (B. of E., 1894.)

15. Two similar deep metal jars are placed on the caps of two similar electro-
scopes at some distance apart and the caps are connected by a fine wire ; a
positively electrified ball is lowered into one of the jars without contact. Explain
the effect as to potential and divergence on both sets of leaves, and also that
which occurs on breaking the wire connection by means of a silk thread and then
removing the ball without allowing it to touch the jar. (B. of E., 1898.)

48 ELECTROSTATICS

16. A conducting sphere, of diameter 6, is electrified with 105 units ; it is
then enclosed concentrically within an insulated and unelectrified hollow con-
ducting sphere formed of two hemispheres, of thickness i and internal diameter 7.
The outer sphere is then put to earth. Determine the potential of the inner
sphere before and after the outer sphere is earth-connected. (B. of E., 1899.)

17. A hollow metal vessel is insulated and charged to potential V, and the
following operations are successively performed : (1) an insulated metal ball is
lowered into the jar without touching it, (2) the ball is momentarily earth con-
nected, (3) the jar is momentarily earth-connected, and (4) the ball is removed to
a distance. State the changes of potential of the jar and the ball at each stage.

(B. ofE., 1901.)

18. Two insulated spheres having radii of 3 and 1 centimetres respectively
are placed a long way apart ; a charge of 15 units is given to the larger sphere :
what charge must be given to the smaller in order that the larger sphere may
neither gain nor lose charge when the two spheres are connected by an insulated
wire? (B. of E., 1903.)

19. Give a careful freehand drawing of the lines of force due to a charge of
4 units of positive electricity at A, and one of 1 unit of negative at B, if the
distance between A and B is 2 '5 centimetres. (B. of E., 1904.)

20. The charge and potential of an isolated sphere are each numerically equal
to 10. Draw as correctly as you can the equipotential surfaces for potentials 2,
4, 6, 8. (B. of E., 1902.)

21. A brass ball, 7 centimetres in radius, is suspended concentrically inside
a spherical brass vessel of internal radius 9 centimetres and external radius
10 centimetres. If the charge on the ball is 56 units and the potential of the
outer vessel 5, what is the potential of the ball ? (B. of E., 1904.)

22. The difference of potential between two points A and B in a uniform
electric field is 100, A and B being 4 centimetres apart and lying upon the same
line of force. A body charged with ten units of positive electricity is placed
upon the line AB. What force does it experience ?

(Lond. Univ. Matric., 1906.)

23. Two insulated metal spheres of equal size are equally charged and placed
so that the distance between their centres is about three times the diameter of
either. Draw diagrams to represent the lines of force round the spheres when
their charges are of the same and of opposite sign, respectively. Represent upon
other diagrams the effect of placing an earth-connected metal sphere, of the same
radius, midway between the two spheres. (Lond. Univ. Matric., 1908.)

24. Explain what is meant by electrical potential, and describe some experi-
ment which proves that the potential of a body can be altered without altering
its charge.

25. Water is allowed to escape, drop by drop, from a tube connected with an
insulated vessel. The knob of a charged Leyden jar is placed near the end of
the tube from which the water drops, but so that the water does not touch it.
What will be the effect on an electroscope connected with the water- vessel ? On
what does the ultimate electrical condition of the water- vessel and electroscope
depend ?

26. One pole of a battery of many cells is earth-connected, and a long
insulated wire projects from the other end. Two insulated metal balls, of 1 inch
and 5 inch diameter respectively, are put one after the other in contact with the
end of the insulated projecting wire. What are the comparative quantities and
densities of the electricities on the two balls ?

27. A large, strongly electrified metal ball is brought towards a similar
unelectrified ball supported by a dry glass stem, as near to it as possible without
a spark passing between the balls. The balls remaining at this distance, the
unelectrified one is touched with the finger, and immediately there is a spark
between it and the other ball. Explain this.

28. A gold-leaf electroscope is put inside a tin can, which is hung up by silk

POTENTIAL 49

cords so as to be insulated. On holding a strongly electrified glass rod below the
can, no divergence of the gold leaves takes place ; but on touching the cap of
the electroscope with the finger (without touching the can) the leaves diverge.
Explain these results.

29. An insulated conductor, rounded at one end and pointed at the other, is
charged. Two small equal spheres supported by insulating stands are made to
touch the two ends and then removed. Will electricity pass from one sphere to
the other if they are connected by a wire, (1) when they are in contact with the
conductor, (2) when they are removed to a distance from it ?

30. Two equal soap-bubbles, equally and similarly electrified, coalesce into a
single larger bubble. If the potential of each bubble while at a distance from the
other and from all other conductors was P, what is the potential of the bubble
formed by their union ? (N.B. — Volume (v) of a sphere is proportional to cube
of radius (r), or v — |-7rr3.)

31. Three metal balls, the diameters of which are respectively 5, 7, 12 inches,
are all connected together by a fine wire, but are otherwise insulated. If the
smallest ball has a charge of 10, what are the charges on the other balls ?

32. Two insulated and widely separated metallic spheres receive charges of
positive electricity, which raise their potentials to 4 and 5 respectively. The
densities of the charges being in the ratio 4 : 9, compare the radii of the balls.

33. An insulated brass sphere of 4 centimetres radius is brought into a region
where the potential is 5. It is then brought into earth-connection and removed.
What is its free charge ?

34. An insulated brass sphere was brought into a region where the potential
was 10, touched with the finger and then removed. It was found to have 40 units
of negative electricity on it. What was its radius ?

35. How much work has to be spent in charging a sphere from potential 0 to
12, the diameter being 4 centimetres ?

36. Find the strength of the electric field at a point just outside an isolated
sphere of 10 centimetres radius charged with 10 electrostatic units.

(B. of E., 1903.)

37. Explain carefully how the distribution on a conductor may be tested. If
a wire attached to an electroscope is made to touch different parts of an
irregularly-shaped conductor charged with electricity, how would you expect
the indications of the electroscope to vary ?

38. In what respects is electrical potential analogous to temperature ?

The end of a wire connected to a gold-leaf electroscope is put through a hole
in the side of a hollow charged conductor. Describe and explain what happens
when

(1) it is held there without being allowed to touch the conductor,

(2) it is withdrawn,

(3) it is again inserted and allowed to touch the inside of the conductor,

(4) it is withdrawn,

(5) it is made to touch the outside of the conductor.

(6) Would the result of (2) be different if the electroscope were badly

insulated ?

CHAPTER V

CONDENSERS AND CAPACITY
FROM the definition of capacity (p. 42) we obtained the expression

for any charged conductor. As this equation is extremely important,
it may be advisable to point out its analogies with familiar facts. In
the first place, the idea of capacity in electrical theory is, in some
ways, analogous to its ordinary meaning, and from this point of view
the above equation simply amounts to the statement that the quantity
of liquid contained in a vessel depends upon its "capacity" and the
height to which it is filled. Ordinary vessels have a definite height,
i.e. there is a definite height to which they may be charged, whereas
conductors may be charged electrically to a potential limited only by
leakage and by tendency to form brush discharge; hence, it is con-
venient to regard a conductor as
similar to a vessel of indefinite depth,
but of perfectly definite cross section.
If now we agree to define the capacity
of any vessel as being numerically
equal to a quantity of liquid required
to fill it to unit depth, the above
equation follows at once.

There is, however, one great funda-
mental difference between these two
cases, for the capacity of a vessel is
definite and unalterable, whereas the
capacity of an electrical conductor is
largely dependent upon its position
with reference to neighbouring bodies.

^IG 29 Exp. 42. Obtain two similar metal plates

on insulating stands. These may con-
veniently be circular and have a diameter

of about 10 or 12 inches. Connect one plate, M, to an electroscope (Fig. 29),
and charge the system by induction (or by any other convenient method) until
a large deflection is obtained. Bring the other insulated plate, N (not shown in
the figure), within half an inch or so, and notice that there is practically no
alteration in the deflection. (Really there is a slight decrease, but so small that
it is scarcely noticeable.) Connect N" to earth by touching it with the finger,

50

CONDENSERS AND CAPACITY

51

and notice that the deflection is greatly reduced. Remove N to a distance, and
notice that the deflection returns to its original value.

During these experiments, the charge on the first plate remains
unaltered. Let this be Q. The deflection measures its potential V,
and as Q = VC, we infer : —

(1) That the presence of the second plate does not appreciably
alter the capacity of the first as long as the former is insulated.

(2) When the second plate is earth-connected, its presence
increases the capacity of the first, for the potential falls although
its charge is constant.

Exp. 43. Use the same arrangement, but connect N to earth by a wire.
Charge M until the deflection is the same as at first. Then we know that V is
the same as at first, and, as C has been increased, Q must be greater. Demon-
strate this by removing N. At once the divergence increases until the leaves
touch the metallic sides and discharge themselves.

Hence, we find that the capacity of a conductor is considerably
increased by the mere presence of an adjacent earth-connected con-
ductor. Further experiments will show, however, that the essential

Earth

FIG. 30.

FIG. 31.

condition is not earth-connection, but the presence of an opposite
charge on the second plate, for the effect will be unaltered if the
latter is kept insulated and charged oppositely in any convenient way.
These actions become easily intelligible, if we consider the
diagrams of the electric field shown in Figs. 30 and 31. In Fig. 30
both plates are insulated, M being charged positively and N un-

52 ELECTROSTATICS

charged. Evidently 1ST is under induction, but, from its shape and
position, it scarcely alters the distribution of the field due to M. In
Fig. 31 the second plate, N, is earthed. The distribution of the
field is now completely altered. The lines from M proceed to the
earth, and as N" becomes, in effect, a portion of the earth, the lines
are therefore concentrated largely between the plates, although we
know from experiment that some lines still diverge from M. The
closer the plates are together, the greater will be the concentration
of the field between them, and the greater becomes the charge which
may be concentrated on M for a given P.D. between the plates.

Such an arrangement, which in practice takes many different
forms, is known as a condenser.1

Limit to the Charge on Condensers. — When a dielectric
is traversed by lines of force, there is a mechanical strain in it, and
when this exceeds a certain value (depending on the nature of the
dielectric), the insulation breaks down, a spark or brush discharge
passing across it. For this reason, air condensers are used only for
standards of capacity ; in ordinary cases a more rigid dielectric, e.g.
glass, mica, or paraffined paper, is used, which enables the plates to be
brought much nearer together without undue risk of puncture, and
which has a further influence — to be discussed later — depending on
the value of K.

The Leyden Jar. — The earliest form of condenser was the
well-known Leyden jar, and is made as follows : — Obtain a wide-
mouthed glass bottle about 6 or 7 inches high. First,
paste tinfoil on the inside of the bottle, leaving 1 J or
2 inches of glass uncovered at the top (Fig. 32).
This may be done by cutting a strip of tinfoil of the
necessary width and length, and after pasting or gum-
ming the bottle, rolling it up and dropping it inside
the bottle ; now unroll the foil, taking great pains to
secure a smooth surface (a piece of wood, enlarged at
one end and covered with linen, is useful for this
purpose) ; and then place a circular piece over the
bottom. Next, cover the exterior with tinfoil, leaving
32 a similar margin at the top. The efficiency of the
jar is increased, and the risk of puncture is decreased,
by making thorough contact (avoiding air bubbles, (fee.) between the
coatings and the glass. Coat the glass margin with shellac varnish.
Fit a wooden stopper into the mouth, having first passed a stout
brass wire through the centre. Now solder a brass knob to the end

1 In applying the equation Q = VC to condensers, it must be understood that
V is the difference of potential between the conductors. When one of them is
earthed, V is the potential of the other, but in order to simplify ^he notation
employed in this work, V will (unless otherwise stated) always be used in the
more general sense, i.e. it will denote the P.D. between the conductors.

CONDENSERS AND CAPACITY

53

FIG. 33.

of the wire outside the jar, and to the inner end fasten a chain, which
must be of sufficient length to lie on the bottom of the jar.

A Leyden jar is said to be charged with the kind of electricity
accumulated on its inner coating.

In the following experiments, it is assumed that an electrical
machine of some kind is available, although for convenience, such
machines are dealt with in the next chapter. -

The Discharging Tongs or Discharger consists of two
curved brass rods terminated in knobs, and joined by a hinge
attached to one or two insulating handles (Fig.
33). It is used to discharge a condenser with
safety

Exp. 44. Charge a Leyden jar by holding the outer
coating in the hand and presenting the knob to the
prime conductor of a frictional machine, or to one of the
terminals of an influence machine. Place one knob of
the discharging tongs on the outer surface, and bring
the other knob near the knob of the Leyden jar.
Notice that a sharp crack is heard and a spark seen,
due to the neutralisation of the two opposite charges.
This method of discharging is called instantaneous dis-
charge.

Exp. 45, to illustrate the opposite electrical conditions of the tivo coatings of
a Leyden jar. Charge the jar positively.

(1) Holding the outer coating, draw a figure with the knob on a cake
of dry vulcanite.

(2) Place the jar on an insulator, and, taking hold of the knob, draw
another figure with the outer coating.

(3) A mixture of red lead and flowers of sulphur is then shaken through

a muslin bag, from a height above the
cake. By the friction between the red
lead and the sulphur, the red lead becomes
positively and the sulphur negatively elec-
trified. The red lead, therefore, seeks the
lines traced by the exterior coating of the jar,
and the sulphur the lines traced by the knob.
Fig. 34 represents the result of an ex-
periment, in which the circle was drawn
with the knob, and the cross with the outer
coating of the jar. The selection of the
red lead for the negative cross and of the
sulphur for the positive circle is due to
both attraction and repulsion, i.e. the
sulphur was attracted by the positive
circle and repelled by the negative cross.
As there is less repulsion in the space
between the arms of the cross than at the
ends, the particles of sulphur arranged
themselves as shown. Such figures are known as Lichtenberg's figures.

Franklin's Plate or Fulminating Pane is another form of

condenser consisting of sheets of tinfoil fastened on a plate of glass. The tinfoil
sheets are smaller than the glass ; in fact, if the glass plate be approximately 12
inches by 10 inches, a two-inch margin may conveniently be left between the edge

FIG. 34.

54 ELECTROSTATICS

of the tinfoil and that of the glass. The coatings are, however, much more con-
veniently made with metallic paint instead of tinfoil.

Seat Of Charge. — Exp. 46. Take a glass vessel, B (Fig. 35), having
two movable metallic coatings, C and D. Place the parts together so as to
form a Leyden jar. After charging it, place the jar on a non-conducting stand.
Remove the inner coating by means of an ebonite rod. Show that it is charged by
bringing it over the cap of an electroscope, then touch it and notice that the spark
obtained (if any) is barely perceptible. Hence, it is only very feebly charged.
Now lift out the glass vessel by holding its edge between the fingers. Place it
on a non-conductor, or better, retain it in the left hand whilst testing the outer
coating for charge. This also will be found to be very feebly charged. Replace
the glass, then the inner coating (by means of the ebonite rod), and finally dis-
charge the jar by means of the discharging tongs. Notice that the spark is
almost as strong as if the parts had not been separated.

Experiment 46 was devised by Franklin, and clearly demonstrates
the important fact that the essential portion of a condenser is the
dielectric, the coatings themselves being mere appurtenances, which

FIG. 35.

enable us to produce conveniently an electric field in the dielectric.
Hence, the energy of charge resides in the dielectric.

This statement holds good generally. For instance, if we charge
an isolated conductor, as in many previous experiments, the energy of
the charge is in the dielectric (the air) around it. In the case of a
charged Leyden jar, practically the whole energy resides in the glass,
although a very slight portion, negligible in comparison, is contained
in the air around the knob.

All this amounts to saying that electrostatic energy is always
energy in an electric field, and is therefore located in the medium in
which the field exists, i.e. in the dielectric.

Residual Charges. — If a condenser be charged slowly to a
given potential, and then discharged, i.e. brought to zero potential,
we find that after a short time it will acquire a potential of the same
sign as at first, but smaller in amount, so that it can be again dis-
charged, and so on. The rise in potential after each discharge is
owing to absorption of the electricity by the dielectric, and subsequent
conduction to the surface after the primary discharge has occurred.

CONDENSERS AND CAPACITY

55

This phenomenon has been carefully investigated. Clerk-Maxwell
accounted for the presence of such charges on the hypothesis that the
dielectric consisted of heterogeneous particles of unequal conducting
powers. Their presence is, however, usually explained on the sup-
position of the electric strain ; the molecules of the glass, being acted
upon by the stress of the opposite charges, are strained, and are there-
fore unable to recover their original form and volume immediately.
The 'discharges after the first are due to what are called residual
charges.

Exp. 47, to show the presence of residual charges. Charge a Leyden jar slowly,
and then discharge it. After allowing it to stand for a short time, place one
knob of the discharger on the outer coating and bring the other to the knob of
the jar. Observe that a second discharge occurs. If the jar be quite dry, a
third, fourth, and even fifth discharge may be obtained.

Leyden Battery. — When a powerful discharge is required, a
number of jars is generally used, having all their inner coatings in
metallic connection, and also all their outer coatings. Fig. 36 shows

FIG. 36.

this arrangement. The jars stand on a conductor, e.g. a piece of
tinfoil, thus placing all their outer coatings in conducting communi-
cation, while their inner coatings are joined by means of a wire
passing through holes in the knobs.

The jars are, however, generally placed in a box lined with tinfoil,
which is connected with a hook or handle on the outside of the box.
The inner coatings are connected as shown in Fig. 36. Such an
arrangement is called a Leyden battery.

Universal Discharger. — This discharger (Fig. 37) consists of two
movable brass arms, provided with universal joints and supported on glass legs.
The object, through which the discharge is to be passed, is placed on a small
table, between the two knobs terminating the arms.

Exp. 48. Place a small quantity of gunpowder on the table of the dis-
charger ; fasten one end of a piece of ivet string to that arm unconnected with

56

ELECTROSTATICS

the battery, and the other to one knob of the discharging tongs. Having
charged the battery, discharge it, and observe that the gunpowder is ignited.
The wet string, which is merely a bad conductor, is necessary to increase the
time required for the discharge — without it, the powder is scattered mechanically
before its temperature is raised to the ignition point.

Exp. 49, to show the disruptive effect of the discharge of the Lcyden lattery,
Take a block of shellac through which a wire has been passed, one end cut off
flush with the surface and the other terminated in a loop. Place a thin sheet
of glass on the upper surface of the shellac. Insulate another wire, pointed at
one end and terminated with a knob at the other, and then placs it vertically
opposite the wire in the shellac cake. Attach a chain or wire to the loop, and

FIG. 37.

then connect it with the outer coating of a Leyden battery. Charge tbe battery,
and by means of the discharging tongs connect the knob of the upright wire
with one of the knobs of the battery. A discharge occurs, which pierces the
glass.

Exp. 50. Instead of the glass, place a sheet of cardboard on the shellac
cake. (1) Let the two wires be opposite each other, and, after the discharge,
observe that the perforation is frayed on both sides of the sheet, as though it
were pierced from the middle outwards. (2) Let one wire be a little to the right
or left of the other. Notice that the hole is nearer the negatively charged wire.
This is known as Lulliii's experiment.

Forms of Condenser used in Practice. — The Leyden jar
type of condenser has recently become of great practical importance in

CONDENSERS AND CAPACITY 57

connection with wireless telegraphy, &c. This is because — although
its capacity is small for a given size — it is able to stand very great
differences of potential.

As we have mentioned on p. 52, the risk of the discharge piercing
the dielectric is decreased by making perfect contact between the
metal coatings and the glass, and hence, in the most recent com-
mercial form — known as the Moscicki condenser — the coatings are
of chemically deposited silver.

When large capacities are required, and the difference of potential
to be used is comparatively small (say, not more than about 100
volts), recourse must be had to mica or paraffined paper for the
dielectric. The former is an excellent material, but, when of suitable
quality, it is very expensive, and hence it is chiefly used for standards
of capacity. Paraffined paper condensers are now employed in large
quantities in ordinary telegraphy and telephony, and during recent
years great improvements have been made in their manufacture.
Perhaps the best method is that devised by Mansbridge. Paper
tinned on one side only — what is known as "silver paper" — is
employed, interleaved with plain paper as the dielectric. A machine
carries two reels of tinned and two of plain paper, the suitably inter-
leaved sheets being wound on mandrels into rolls of sufficient size to
have the desired capacity. Connections are made to the two coatings
by thin copper strips. When wound, the rolls are dried in ovens
until the moisture is completely expelled, and are then placed in
melted paraffin -wax in a good vacuum until completely saturated,
(the reduced pressure is of great service in eliminating air bubbles).
After removal, they are pressed into thin flat plates while still warm,
and as soon as possible sealed up in air-tight metal cases.

To give the student some idea of the numerical magnitude
involved, we may mention that the size most generally used has a
capacity of 2 microfarads. It weighs about 7 ounces, and has a
volume of about 7| cubic inches. Now, 1 microfarad = 900, 000
static units of capacity (see p. 591), and hence, with this small bulk,
a capacity is obtained equal to that of a sphere of radius 1,800,000
centimetres, i.e. about 22 miles in diameter. For the purpose of
comparison, it may be remarked that the capacity of an extra large
Leyden jar is not likely to be more than '002 microfarad, i.e. about
1800 static units, which is the capacity of a sphere about 40 yards
in diameter.

The Condensing" Electroscope, invented by Volta, is an
ordinary gold-leaf electroscope, provided with another disc (of the
same diameter as that of the electroscope), to which is fixed a glass
handle. The faces of the two discs are coated with shellac varnish,
which forms the dielectric between the plates. The condensing
electroscope is only useful when electricity from a weak but con-
tinuous source is to be tested.

58

ELECTROSTATICS

Exp. 51. Take a compound bar made of zinc and copper soldered together,
Hold the zinc end in the hand, and touch the disc of the electroscope with the
copper. Connect the upper disc with the earth by touching it with the other
hand. Remove the hand and the rod, and then lift the upper plate. This
diminishes the capacity of the lower plate to such an extent that its potential
rises, causing a divergence of the leaves. The charge will be found to be
negative.

This experiment was devised by Yolta to prove that electricity
was developed by the mere contact of two dissimilar metals; and
although the truth of his discovery was denied for a long time, the
fact was established beyond doubt by the following simple experiment
of Lord Kelvin. It is now known as the "Volta effect," and is
further discussed on p. 478.

He suspended a thin strip of metal so that it would turn freely

about a point A (Fig. 38), and then
charged it with a known kind of
electricity. Under it are placed
two semi-circular discs or rings of
dissimilar metals. No movement of
the charged strip takes place while
the two dissimilar metals are placed
apart. When, however, they are
placed in contact it immediately
turns, being attracted by the oppo-
sitely electrified metal and repelled
by the similarly electrified one.

From this apparatus, the quad-
FlG 3g rant electrometer (see Chap. VII.) was

afterwards developed by successive

improvements, and the experiment can be much more easily performed
with that instrument.

The following is a list of substances, which if any two are placed
in contact, the first becomes positively, and the second negatively
electrified : —

+ Sodium
Magnesium
Zinc
Tin
Lead
Iron

Copper
Silver
Gold
Platinum
— Graphite (Carbon)

Another instance of the use of the condensing electroscope will be
found on p. 197.

Energy of a Charged Body. — Consider an insulated con-
ductor of capacity C, charged with Q units which raises its potential
to V. Evidently it possesses energy, for it can attract or repel
bodies, give a spark, and so on. As in the case of a condenser, the

CONDENSERS AND CAPACITY 59

seat of this energy is the electric field ; in fact, an isolated charged
body is only a particular case of a condenser with widely separated
coatings — the second coating being formed wherever its lines of force
terminate on a conductor. Further, this energy is necessarily equal
to the work done in charging the conductor, and we may regard this
operation as equivalent to bringing a charge Q from an infinite
distance up to it. If its potential had remained at value V during
this operation, the work done would.be QV ergs, but actually, its
potential was zero at the beginning, and reached V only at the end of
the operation. By considering the charge Q to be divided into small
portions, which are brought up successively to the body, we can give
the following simple analogy to illustrate the real nature of the prob-
lem. Suppose that a wall has to be built of height V feet, for which
purpose a total weight of Q pounds of bricks is required. If this
whole weight had to be lifted through the height V, the work done
against gravity would be QV foot-lbs., but, as a matter of fact, only
the upper bricks have to be lifted through fhis" height, and the work
really done in building, which is equal to the potential energy of the
finished wall (with respect to gravity), is manifestly obtained by
multiplying the mass by the height of the centre of gravity, i.e.

Qxjv.

The electrical case is exactly similar. If the charge be increased
from zero to Q, the potential also increases from zero to V, so that
the mean potential is JV, and the energy is |QV ergs. Now,
Q = VC/whence, by substitution, we get

the three forms being numerically identical.

An exact proof is easily obtained by using the calculus, alternative
methods being either cumbersome or unsatisfactory.

/fv
We have Energy - QdV, where Q - VC

J°

C. Energy = cfVV^V^lY2C

J-°

JQ Q

V<iQ, where V =<£-

.'. Energy

Two Conductors Sharing a Charge.— If a conductor, with
charge Q and capacity C, shares its charge with another insulated
uncharged conductor of capacity Cv there will be a loss of energy,

60

ELECTROSTATICS

although the total charge is constant.

As we have seen, the energy

Q2

of the simple conductor is *-f- , but for the two together it is J

C + Cj •

Here again a simple analogy will be found useful. Consider a tank
filled with Q Ibs. of water to a depth V feet. Its gravitational energy
is JQY Ibs. If it be connected by a pipe to another tank, although
no water is lost, the level is lower than before. In this particular

case where the cross sections are equal, each will contain a charge —

and the depth will be — .
2

Hence, each contains J of the original

energy, and the other half has disappeared. It is also evident that
loss of energy must occur if the second tank be charged either to a
higher or a lower level (potential) than the first, the missing portion
being the energy of the flow which takes place between them. There
is only one case in which no such loss occurs, viz. when no flow takes
place, i.e. when the levels are equal before connection.

Capacities of Simple Forms of Condensers.— There is
one general method of calculating capacities. A charge Q is supposed
to be given to the arrangement, the difference of potential between
the coatings thereby produced is found by calculation, and then by
means of the equation Q = YC, C is deduced.
I. Spherical Condenser (Fig. 39).—
This consists of an insulated sphere A
of radius r, surrounded by an uninsulated
spherical shell B, of radius rv and having
a medium between them, whose dielectric
constant is K. If A be charged with Q
units of positive electricity, then the lines of
force from A will terminate in a charge of
— Q units on the inner wall of B.

By the example given on p. 37, the poten-
tial of the inner sphere A is

FIG. 39.

__

t Kr

_zQ=QA n

Krx KVr rj

As the potential of the outer sphere B is zero, the P.D. between the
coatings is also V, where : — ^ / ., , x

Vfc"(i -)

KVr rj

Now C = ,
V

Q/i_i\

K\r rj

K •

To illustrate this by numerical examples, let r = 9 centimetres, and
10 centimetres, then the capacity of A alone is 9 static units, but

CONDENSERS AND CAPACITY

61

by surrounding it with an earthed conductor of 10 centimetres radius,

and having air as the dielectric, this is increased to

— ^

i \j '

= 90' static

M N

units, i.e. the capacity of the condenser is the same as that of a sphere
of 90 centimetres radius. Again, if the space is filled with turpentine,
for which K - 2-2, the capacity is 2-2 x 90 = 198 static units.

II. Capacity of a Thin Circular Disc of Radius r. — It

2r
may be remarked that this is — . It can be obtained by finding the

7T

capacity of an ellipsoidal conductor, and taking the limiting case when
the length of its major axis becomes zero, but the proof is too diffi-
cult for insertion here.

III. Capacity of a Parallel Plate Condenser.— Let two

similar plates M and N", Fig. 40, of any shape, and each of area A, be
separated by a distance d. Let N be earthed,
and M charged with + Q units, and let us
suppose that the whole charge is uniformly con-
centrated on the inner sides of the plate, i.e. the
field is absolutely symmetrical, and is confined
entirely to the space between the plates. These

conditions are not exactly satisfied in any ordi-

nary arrangement, especially near 'the edges of I — =\ I" * Earth/
the plates, but they are approximately so when
d is very small compared with the area of either
plate.

Under these circumstances, the charge on
N will be — Q units, and 4?rQ lines will pass
from plate to plate,

... Field = F = 4^ FIG. 40.

A

but F = KU

. •. U = force on unit charge = — = ^ **

K AK

Again, the P.D. between the plates = V = work done in carrying unit
charge from one plate to the other against the force,

/. V = force x distance = U.d = -— * x d

AK

„ Q Q

whence C = £ = 7-—

V 47rU

x K static units

From this formula, we see that the capacity is directly proportional
to the area of either of the plates, and inversely proportional to the
distance between them.

62 ELECTROSTATICS

This result holds good for condensers of any shape, if only the
distance between the coatings is small compared with their area, anc
may, therefore, be applied to a Leyden jar, &c. For instance, let ui
apply it to the spherical condenser just considered, rl — r being sinal
compared with r or rr As the average area of either coating may b<

written 47rrr1? we have C = _A_ x K = _^Tr\ x K = rri .K, ;

47m 47r(?<1 — r) rl — r

result identical with that obtained on p. 60.

Effect of Changing the Dielectric.— It should be noticec

that the capacity of a parallel plate condenser can be written in tin

^
form C = -. If, then K = 2, the capacity is exactly the sami

A V"

. f

as it is in air, but with the plates at half the distance apart. Tha

is, we may regard the effect of a medium, for which K is greate
than unity, as equivalent to bringing the plates nearer together
This is true even if the dielectric does not occupy the whole o
the space between the plates ; for instance, if a slab of a dielectri*
of thickness t and dielectric-constant K be introduced betweei
the plates of a condenser, its effective thickness in terms of ai

is — , and the effect on the capacity is the same as if the distanci
K

between the plates had been reduced by an amount t— — For, if c

be the distance between the plates, the air distance in second case i

t ^ / t \

d — t, to which add — , therefore total is d—(t — - )

K \ K/

Exp. 52. Connect one of the insulated plates, M, used in Experiment 42, t<
an electroscope, and connect the other, N, to the earth. Charge M until a con
venient deflection is obtained Carefully introduce between the plates, bu
without contact, a large slab of paraffin- wax at least one inch thick.1 Notic
that the deflection decreases when the wax is inserted, but that it regains it
former value when it is removed.

We know that the deflection is a measure of the potential betweei
the plates, and hence we see that the effect of the wax is to reduce V
Q, of course, remains constant, so that by applying the formuh
Q = VC, we learn that the capacity has been increased by the insertioi
of the wax.

Influence of Dielectric on the Energy of a Chargec
Condenser. — As the energy of the system is JQV, it follows tha

1 Great care must be taken to ensure that the wax is uncharged. Paraffin
wax is so easily excited by friction, that merely picking it up from the table wil
probably charge it. It should be tested by means of an electroscope, and i
charged, it should be passed through a flame, and then tested again.

CONDENSERS AND CAPACITY

63

some energy is lost on inserting the wax, which is regained on its
removal. The question naturally arises, what becomes of it1? The
answer is simple. During insertion, there is a mechanical force
acting upon the wax, tending to pull it into the strongest part of
the field, and thus work is done at the expense of the energy of the
system. When it is withdrawn, an equal amount of work is done by
the experimenter in removing it against the force, and thus the energy
is restored to the system. (In fact, a substance, for which K is large,
behaves in an electric field somewhat in the same way as iron behaves
in a magnetic field.)

During Experiment 52, Q remained constant, but the conditions
might have been modified so that V was kept constant. For example,
we might have left the plate M permanently connected with some
source of constant potential V, then, on inserting the wax, a further
flow into it would have taken place in order to keep V constant. In
this case, it is evident that the effect of the wax is to increase the
energy of the system.

These results may also be deduced by considering the equivalent

Q2

expressions for energy J-^-, JV2C. The first shows that, when Q is

constant, the energy is inversely as the capacity ; and the second that,
when V is constant, the energy is directly as the7 capacity.

Capacity of Condensers in Parallel and in Series.—
Case I., in Parallel, i.e. such as the common arrangement of

Ley den jars, in which all the inner and all the outer coatings are
respectively connected together.

Let any number of condensers of capacities cv c2, c3, &c., be con-
nected in parallel, as shown in Fig. 41, and let Q be the total charge
given to them, which produces a P.D. of V units between A and B,

64 ELECTROSTATICS

the common terminals of the system. Then the charges in the con-
densers will be different, but Y will be the same for all of them.
If the charges be qlt g2, qy &c., we have

If C be the total capacity, Q = YC,
and, of course, ql = Vcj, q2 = Yc2, q3 = Vc3, &c.
... YC = Vcj. + Yc2 + Yc3 + &c.
or C = Cj + c2 + c3 + &c.

Case II., in Series, such as the old cascade arrangement of
Leyden jars, in which the inner coating of one is connected to the

FIG. 42.

outer coating of the next, all the jars except one being insulated.
(Fig. 42.)

As before, let a charge Q be given to the system, which produces
at P.D. between A and B, the terminals of the system. In this case,

n'

Q A

Q

B

Q

C2

Q

C,

v2

C

3

v,

V

FIG. 43.

the P.D.'s between the coatings of the individual condensers will not
be equal.

CONDENSERS AND CAPACITY 65

The charge Q is necessarily the same in all. This will be best
grasped by supposing that the charge entering the jar on the right of
Fig. 42 induces an equal negative charge on the outer coating of that
jar, and an equal positive charge which passes to the inner coating of
the second, and so on.

We have necessarily

where V = % V, = §, V2 = §, V3 = 9, <fec.

6'j Cii Cn

C G! c2 cz

1= 1 I l_

C cj c2 cs"

1

whence C

That is, the total capacity is less than the least of the component
capacities, and, in the particular case where we have n condensers of
equal capacity, i.e. when e1 = c2 = c3 = &c., we have, from the general
equation

i L+M+&C.

C Cl c2 CB

1111 n

= =—+-+-+ &c. --

(J Cj_ Cj Cj_ Cj_

^ capacity of one condenser.

whence C - — — , .—

number of condensers in series.

These results are not of much practical importance in electrostatics,
but they are often very useful in the application of condensers to
electrical measurements with voltaic currents, for, by suitable com-
binations, it is possible to obtain a range of values, when, as is usually
the case, only two or three condensers are available.

Why the capacity is reduced when "in series" will be understood by
noticing that such an arrange-
ment is equivalent to increasing
the thickness of the dielectric.

Consider two similar paral-
lel plate condensers in series
(Fig. 44). Evidently, the con-
necting wire MN may be shortened to any extent, and in the
limit the two plates at M and N coincide. The single plate thus

E

A I M N I B

FIG. 44.

66 ELECTROSTATICS

formed takes no active part in the result, and might be removed
without altering the effect. "We then have a single condenser with a

dielectric of double thickness, and from the equation C = — — x K

4ara

(p. 61), this would have half the capacity of either of the original
condensers.

Energy Of Charged Condensers. — The expressions already
given for energy apply to any combination of cdndensers, if C is
understood to mean the capacity of the combination. In dealing
with actual cases, the form most convenient for the purpose should be
selected.

Suppose a constant charging source is available, which can pro-
duce a P.D. of V volts. Then

(a) A single condenser of capacity 0 will, when charged, possess
energy JV'2C ergs ;

(b) If n similar condensers are connected in parallel, and charged
from the same source, the capacity is nC, and the total energy is
JV2 x nC. Evidently, each constituent condenser is charged to exactly
the same extent as the single condenser in case (a), and the com-
bination has, therefore, n times more energy, i.e. it is equivalent to
one condenser of n times greater capacity ;

(c) If the n condensers are arranged in series, the capacity is

— , and the total energy is JV2— , or -th the amount a single con-
n n n

denser would possess if used alone. The combination is now
equivalent to one condenser of n times smaller capacity. As
the total energy is equally shared by the constituent condensers,

each must possess — 9 °f the amount a single condenser would have.
This is also evident if we notice that the P.D. between the coatings

of each condenser is — , and therefore its individual store of energy
n

M / f W

(d) When condensers of unequal capacity are joined up in any
way, the total energy and its distribution can readily be found by
using the expressions for capacity given on pp. 64 and 65.

Specific Inductive Capacity. — Cavendish first discovered
that the capacity of a system depended upon the nature of the
surrounding medium, although he did not publish his researches.
Later, Faraday rediscovered the effect independently, and defined the
specific inductive capacity of any medium A as being the ratio : —

capacity of condenser when A is the dielectric.

capacity of the same condenser when air is the dielectric.
Our previous results show that this quantity is numerically identical

CONDENSERS AND CAPACITY

67

with K, the dielectric constant, although arrived at from a different
point of view

Faraday's Experiments.— To determine the specific inductive
capacity of a dielectric, Faraday used the apparatus represented completely in
Fig. 45, and in section in Fig. 46. The outer coating consisted of a hollow
brass sphere, made up of two halves, P and Q, which in every experiment was
put in earth-contact. In the interior was a hollow brass sphere C, connected
with a brass wire to the knob B, a thick layer of shellac, A, being used to insu-

FiG. 45.

FIG. 46.

late the rod from the outer sphere PQ. The space mn contained the substance
whose inductive capacity was to be determined. If the dielectric to be
examined was solid, P and Q could be pulled apart to admit it. If, however,
the substance was a gas, the space was first exhausted by screwing the foot to an
air-pump, and the gas afterwards introduced. A stop-cock enabled the passage
between mn and the base to be opened or closed at will.

Faraday employed in his experiments two exactly similar condensers, the
space mn in one being filled with the dielectric — say shellac — to be examined,,
while that in the other was filled with air. The air condenser was then
charged, and the potential (V) of its inner coating measured by means of a
torsion balance.1

In a particular experiment Faraday obtained a torsion of 290°. This con-
denser was then made to share its charge with the other condenser, which was

This method of measuring potentials is now obsolete.

68 ELECTROSTATICS

filled with shellac. The potential (VJ was again measured, a torsion being
obtained of 113'5°.

Now, if the capacity of the shellac condenser had been equal to that of the
air condenser, the torsion should have been 145°. As, however, the potential is
less, the capacity must be greater. In fact, if C be the capacity of the air
condenser, and Cl that of the shellac condenser, then

and, as both condensers are used together,

whence, from these equations, VC = V1(C +
In the experiment mentioned above

The mean of a number of experiments gave C1 = 1'5 C.

For convenience, however, Faraday filled the lower hemisphere only with
shellac, so that if K be the specific inductive capacity of shellac, and that of
air be unity, we have

C 1 + 1
but 1 = 1'5

whence K = 2.

Since Faraday's time, much work has been done to determine the
specific inductive capacity, or, as it is better termed, the dielectric
constant of different substances, and many devices have been used to
eliminate error. One -of the chief difficulties which experimentalists
have to contend with, arises from the fact that the capacity of the
condenser (containing solid or liquid dielectric) is affected by electric
absorption. If, for example, we charge a condenser, having ebonite
as the dielectric, to a certain potential, we find that in a short time
the potential has diminished. This is partly due to leakage, and
partly to an absorption of the electricity by the dielectric. To restore
the condenser to its original potential, a further charge must be
added. Again, the potential falls from further absorption. Thus,
the capactity of a condenser depends upon the time that the charge
has been accumulating, and the results vary according as the charge
is added slowly or instantaneously ; methods of measurement, in which
rapidly alternating currents are used, giving the lowest values. It
is for this reason that air condensers are employed for the most

CONDENSERS AND CAPACITY

69

accurate standards of capacity, as gaseous dielectrics are entirely
free from such a defect. The condensers and methods employed
in these researches are too complicated to admit of any satisfactory
explanation in this work, but the principle embodied is explained
on p. 91.

Again, the temperature at which the measurements are made
should always be stated, otherwise the results may be seriously
affected. This is especially the case with bodies which are liquid
at ordinary temperatures. Thus, for water or ice within the range
of ordinary temperatures, K has an enormous value of 78 to 80,
but Fleming and Dewar have shown that at — 185° C., and a fre-
quency of 100 cycles per second, it falls to between 2 and 3. (The
abnornmally large dielectric constant of water is of great importance
in connection with electrolysis.)

It is, therefore, not remarkable that the values obtained by
different investigators are somewhat discordant, as shown in the
following table, which has been prepared on the assumption that
K = 1 for air. (It would be more strictly logical to put K = 1 for
a vacuum, but as the values for gases are so nearly unity, the
alteration would be quite inappreciable.) /

Air . .

A vacuum .

Hydrogen .

Carbon dioxide .

Liquid oxygen

Different kinds of glass

Ebonite

Mica .

Shellac

Sulphur

Quartz

Paraffin-wax

Turpentine .

Carbon disulphide

Water at 15° C. .

Ice at - 23° C. .

Ice at - 185° C. .

Alcohol at 15° C.

Alcohol at - 185° C. .

Formic acid at 15° C. .

Formic acid at - 185° C.

•48

1-0

0-94

0-9997

1-0008

1-478

6 to 10

2-6 to 3-

6-6 to 8

2-74 to 3-73

2-24 to 3-84

4-49 to 4-55

1-99 to 2-29

2-1 to 2-3

2-67

80

78

Between 2 and 3

25

3-1

62

2-4

ELECTROSTATICS

EXERCISE V

1. Two Leyden jars are exactly alike, except that in one the tinfoil coatings
are separated by glass and in the other by ebonite. A charge of electricity is
given to the glass jar, and the potential of its inner coating is measured. The
charge is then shared between the two jars, and the potential falls to 0'6 of its
former value. If the specific inductive capacity of ebonite be 2, what is that of
glass? (B. of E., 1893.)

2. The inner coating of one spherical Leyden jar, whose surfaces have radii
12 and 14 respectively, is charged with 25 units of positive electricity, and the
inner coating of another, with surfaces of radii 8 and 12, is charged with 5 posi-
tive units, the outer coatings of both being earth -connected. Their inner coatings
are then momentarily joined by a fine wire ; in which direction will electricity
pass, the dielectric in both jars being air, and the distance between the jars con-
siderable ? Give full reasons for your answer. (B. of E., 1894.)

3. A Leyden jar consists of two concentric spherical surfaces of 5 and 6 centi-
metres diameter respectively, the intervening space being filled with air. The
outer sphere is uninsulated, the inner is charged with 20 units of electricity.
How much work is done when the inner sphere is put to the earth ?

(B. of E., 1895.)

4. An insulated sphere of 2 centimetres radius is connected by a long thin wire
with another insulated sphere, the radius of which is 6 centimetres, and which is
surrounded by a third sphere of 8 centimetres radius concentric with it. The
wire which connects the first and second spheres passes through a small hole
in the third so as not to touch it. All the spheres are conductors. Calculate
the capacity of the two connected spheres. (B. of E., 1896.)

5. A Leyden jar A, of capacity 3, is insulated and the outer coating is con-
nected by a wire with the inner coating of another Leyden jar B, of capacity 2,
the outer coating of which is uninsulated. If the inner coating of A be charged
so that the potential is V, what is the potential of the inner coating of B ?

(B. of E., 1899.)

6. What is meant by the capacity of a condenser ? Calculate the capacity of
a parallel plate air condenser of which each plate has an area of 400 square
centimetres, the distance between the plates being half a millimetre. Be careful
to state the unit in which you express your answer. (B. of E., 1906.)

7. Two Leyden jars are charged with quantities of electricity in the ratio of
2:3. If in the jar which receives the larger charge the tinfoil surface is twice as
great and the glass is twice as thick as in the other, compare the quantities of
heat produced by discharging them. (B. of E., 1903.)

8. Two Leyden jars, each having a capacity of 1000 centimetres, are charged
in series to a difference of potential of 10 electrostatic units. Calculate the energy
of discharge and state in what units it is expressed. (B. of E., 1907.)

9. The terminals of a condenser with mica as the dielectric are connected to
a quadrant electrometer and the condenser is charged so that the scale deflection
is 90. "When a second condenser of the same dimensions as the first, but having
paraffin-wax as the dielectric, is connected in parallel with the first, the deflection
falls to 30 divisions. Compare the dielectric constants of mica and paraffin.

(B. of E., 1907.)

10. An insulated metal sphere A is positively charged. Another insulated
sphere B of equal radius but uncharged is momentarily brought into contact with
it and then removed. What will be the ratio of (] ) the charge, (2) the potential,
(3) the electric energy of A after contact to the value of each of those quantities
before contact? (Lond. Univ. Matric., 1907.)

11. Explain what is meant by specific inductive capacity. Two plate con-
densers, A and B, are found to have the same capacity. The area of the plates

CONDENSERS AND CAPACITY 71

in A is four times as great as the area of the plates in B, and they are twice as
far apart. Compare the specific inductive capacities of the dielectrics in A and B0

(Oxford Local, Senior, 1901.)

12. An air condenser is formed of two circular metal plates, each of 5 centi-
metres radius, placed at a distance of O'f> centimetre from one another. The
collecting plate was charged to potential 4. What was its charge ?

(C. of P., Senior, 1905.)

13. A sphere of radius 40 millimetres is surrounded by a concentric sphere
of radius 42 millimetres, the space between the two being filled with air. What
is the relation between the capacity of this system and that of another similar
system in which the radii of the spheres are 50 and 52 millimetres respectively,
and the space between them is tilled with paraffin of specifie inductive capacity 2*5 ?

(B. of E., 1892.)

14. What is meant by the electrostatic unit of capacity ? The capacity of a
spherical condenser is 0*0033 microfarad, the diameters of the inner and outer
surfaces of the dielectric being 20 and 20*5 centimetres respectively. What is
the specific inductive capacity of the dielectric ?• (1 microfarad = 900, 000 electro-
static units of capacity.) (B. of E., 1908.)

15. Two insulated metallic plates are placed facing each other, and each of
them is connected with a separate gold-leaf electroscope. If one plate is charged,
the leaves of both electroscopes diverge. If now an un electrified slab of sulphur
is introduced between the plates without touching either, state and explain the
effect on each electroscope.

16. A positively electrified metal ball is hung by a silk thread above a gold-
leaf electroscope. Would the divergence of the leaves be altered (and if so — how
and why) by putting (1) an unelectrified cake of resin, or (2) a metal plate held
iu the hand between the ball and the electroscope, so as not to touch either ?

17. An electrified brass plate held over the cap of an electroscope causes the
leaves to diverge. On touching the electroscope the leaves fall together. If,
after removing the finger, an unelectrified dry glass plate is put between the
electrified metal plate and the electroscope, without touching either, the leaves
diverge again. Why is this ? How must the electrified plate be moved to make
the leaves collapse again ?

18. Two insulated brass plates, a good way apart and connected by a fine
wire, are electrified and then discharged. They are next electrified to the same
degree as before, but before being again discharged, they are moved so as to be
in contact with each other face to face. On now discharging the plates, more
heat is produced than was produced in the previous discharge. Account for the
difference.

19. Three equal similar Leydenjars are connected (1) in series, that is, so
that the outside coating of the first is in contact with the knob of the second,
the outside of the second in contact with the knob of the third, and the outside
of the third earth-connected ; (2) abreast, or with their similar coatings all con-
nected together — and in each case the set of jars is charged as fully as can be by
the same machine. What proportion does the heat produced by completely dis-
charging all the jars in the first case, bear to the heat produced by discharging
them in the second case ?

20. A large insulated metal ball at a great distance from all other conductors
is electrified, and then, by touching it with an earth-connected wire, is discharged.
It is again electrified to the same degree as before, and then discharged by bring-
ing it in contact with a \vall of the room. Why does the second discharge
produce less heat than the first ?

21. The inside of a Leyden jar is connected with the prime conductor of an
electrical machine, and the outside with the rubber of the machine, and also with
a brass ball fixed at TV inch from the prime conductor. If the jar is at first
without charge, ten turns of the machine are required to cause a spark to pass
between the conductor and the ball. If now (the inside of the jar remaining, as

72 ELECTROSTATICS

before, connected with the prime conductor) the outside is insulated and connected
with the inside of an exactly similar jar, the outside of which is connected with
the rubber and the brass ball ; show how many turns of the machine would be
required to make a spark pass.

22. Two equal insulated spheres, 8 centimetres in diameter, are placed some
distance apart, and one of them is charged with 100 units of electricity. The
two are then connected by a wire. Calculate the energy of the discharge which
then takes place between them.

23. A sphere of 6 centimetres radius is suspended within a hollow sphere of
8 centimetres radius. If it be electrified to potential 6, and the outer coating be
in earth contact, find the quantity of electricity with which it is charged.

24. Find the capacity of a spherical condenser, the radii of the two coatings
being 6 and 8 centimetres respectively, the dielectric being paraffin whose specific
inductive capacity is 2 '9.

25. If the radii be 7 and 10 centimetres, and the dielectric be shellac (specific
inductive capacity = 2), find the charge when the potential is 5.

26. A spherical condenser is formed by spheres of radii 10 and 8 centimetres
respectively. The space between the coatings is filled with paraffin of specific
inductive capacity 2. How much work must be done to charge the condenser so
that there is a difference of potential of 10 electrostatic units between the
coatings? (B. of E., 1904.)

27. Two Leyden jars, A and B, receive charges of + 50 and - 30 units
respectively, their outer coatings being earth-connected and their knobs at
potentials 20 and -10. Determine the energy of discharge of each jar after
their knobs have been in contact with each other. (B. of E., 1900.)

28. An insulating conducting sphere of radius 15 centimetres is charged to a
potential of 2000 electrostatic units, and is then caused to touch another insulated
conducting sphere of the same size, but uncharged. Calculate the electrical
energy of the two spheres before and after they have been broiight in contact.
How do you account for the difference in the energy in the two cases ?

(B. of E., 1910.)

29. Explain what the meaning is of the term "lines of electric force," and
what inference may be drawn from their distribution. How many lines of force
approximately will there be per square centimetre cross-section in the space
between two parallel plates, 10 centimetres diameter, of an air condenser charged
with 250 electrostatic units of electricity. (Lond. Univ. Inter. B.Sc., 1904.)

30. Find the electrostatic capacity of a condenser formed by two concentric
conducting spherical surfaces of radii 17 and 20 centimetres respectively, and find
the energy required to charge it, if the surfaces differ in potential by 100 electro-
static units. In what way will the energy vary, if the space between the surfaces
is filled by such a material as sulphur ?

(Lond. Univ. Inter. B.Sc., Honours, 1904.)

31. Two large parallel plates 6 centimetres apart are charged — one positively,
the other negatively — with 4 units of electricity per square centimetre. What
is the intensity of field between them ? (Lond. Univ. Inter. B.Sc., 1904.)

CHAPTER VI

ELECTRICAL MACHINES

AN electrical machine is a particular form of generator, a term which
includes all devices, from a battery to a dynamo, whose function
is the continuous separation of electrical charges. In all generators,
the nett result is to accumulate positive electrification at one point
and negative electrification at another point, energy being necessarily
expended in the process ; and all electrical phenomena are due to the
tendency of these charges to rejoin one another, an action which may
be allowed to take place in various ways. The most characteristic
property of the type of generator now under consideration is its
power of producing extremely high differences of potential.

Such machines may be divided into two classes, depending upon
(1) friction, (2) induction.

Machines depending on Friction.— The old-fashioned
machine is a natural development of the method of obtaining
charges by friction. It is now obsolete, but in view of its historical
importance, it may be briefly referred to.

The Cylinder Machine is probably the simplest form of
machine. It consists of (1) a glass cylinder, A (Fig. 47), supported
on two wooden stems, B B, and capable of rotation, by means of the
handle D, on a horizontal axis; (2) the cylinder is pressed upon by
a rubber, E, made of leather stuffed with horsehair, to which is
attached a flap of silk, F. The rubber should be covered with
powdered amalgam,1 and is supported on a wooden or glass stem, C
(the latter is preferable, as a negative charge can then be collected
from the rubber). (3) The prime conductor, G, which consists of
a conducting cylinder, always insulated on a glass leg. The end
nearest the glass cylinder carries a rod terminated by a cross piece,
provided with a number of sharp brass points, which is called the
comb.

Action of the Machine. — When the cylinder is turned, the
friction of the rubber generates positive electricity on the glass and

1 Electric amalgam is made by placing one part by weight of tin and two of
zinc in a crucible ; just fusing, and then adding six or eight parts of mercury.
Stir while cooling, and then reduce the mass to powder. It may be mixed with
lard and applied to the rubber ; or the rubber may be first smeared with lard,
and the amalgam sprinkled over it.

73

74

ELECTROSTATICS

negative on the rubber. The positive charge on the glass is carried
round until it comes opposite the metal comb. Induction is now
set up, and negative electricity is discharged from the points, electri-
fying the air between them and the cylinder, which passes across
to the glass and neutralises its positive charge, leaving the free
positive on the conductor. Students are apt to imagine that the
positive charges pass from the glass to the points. The row of
points is merely a device for obtaining a charge from a non-conductor,
and the above statement may be verified by touching the points with
a small proof-plane while the machine is working freely, and trans-
ferring the charge obtained to an electroscope. It will be found

FIG. 47.

to be negative, although only positive charges can be obtained from
other parts of the prime conductor (see Experiment 58).

If a negative charge is required, the rubber must be insulated and
provided with a metal knob at the back, the prime conductor being
placed in earth-communication. If both the rubber and the prime
conductor are perfectly insulated, a point will soon be reached when
sparks can no longer be obtained. This is because the normal
working of the machine not only involves the separation of the
charges by friction, but also the existence of a path or circuit
through which they may flow to rejoin each other. In the usual
case, a sufficiently conducting path is provided from the rubber
through the stand and the body of the operator to the insulated
prime conductor, or vice versa, but when both are insulated the
circuit is broken and the flow stops. But if the rubber and prime
conductor were each provided with metal discharging rods arching
over the machine until within sparking distance, both might be

ELECTRICAL MACHINES

75

insulated on glass or ebonite pillars, and yet the action would go
on freely.

Plate Machines have a circular disc of glass or ebonite instead
of a cylinder. This change modifies the arrangement of the parts,
but the action is the same. Electrically and mechanically considered,
they may be regarded as a better type of machine than the cylinder
machine just described.

Machines depending on Induction.— In this class of
machines we require an initial charge of electricity which acts in-
ductively on conductors placed near it.

A very large number of such machines have been devised, but
here it will be sufficient to consider one or two typical forms.

The VOSS Machine. — This machine consists of two parallel

FIG. 48.

plates, one fixed, and the other capable of rotation on a spindle
passing through its centre. In Fig. 48 the plate A is fixed, while
B is capable of rotation in front of it. On the bacJc of the fixed
plate there are attached two pieces of tinfoil covered with paper
(P, N, Fig. 49, which represents the back of the machine). These
are called armatures. Metal rods, MM', bent three times at right
angles, pass from the armatures over the edges of both plates, each
carrying a metal brush, which passes lightly over six or eight metal
studs, S, fixed on the front surface of the rotating plate. Facing the
front of the rotating plate are two horizontal brass combs, D, con-
nected by metal rods with two knobs, E (between which the discharge
occurs), as well as with the inner coating of two Leyden jars, LL.
The outer coatings of these jars are connected together. There are

76

ELECTROSTATICS

FIG. 49.

two other combs connected by a conductor, C, in each of which

the middle tooth is replaced by
a metal brush which passes over
the studs.

In this machine it is un-
necessary to give an initial
charge to the armatures, as
practically there always exists
some slight difference of poten-
tial sufficient to start the action,
and the construction of the
machine is such that the initial
charge increases very rapidly.

The explanation of the action
of the machine is somewhat
difficult, but perhaps it can be
most easily understood by the
following method : —
Suppose we have two insulated conductors (Fig. 50), one, P,
charged positively, the other, N, charged negatively. If we bring
^_^^ another insulated conductor near P, it will be acted on
Cpj inductively, and if we make an earth-contact at the
moment it is opposite P, it becomes charged negatively.
If this negative charge be removed or used in any way,
and then the conductor be brought near N, earth-contact
being made as before, it becomes positively charged, and
so on. Let the charged conductors — which we will call
armatures — be fixed
(V) on the back of a sta-

FuTsn tionary Slass Plate'
' and let another glass

plate, bearing a conductor — say
a metal stud — C (Fig. 51) rotate
in front of it. It is clear that
if we can (1) maintain (or better A
still, increase) the charges on the
armatures P and N, and also
(2) arrange an earth-contact at
the moment C is in front of P
and N, then the stud leaves P
negatively charged, and N posi-
tively charged, so that, if it
rotates in the direction of the

arrow, we shall get a constant supply of negative electricity at A,
and positive at B, and, if we arrange two collecting combs at these
points, we have at once a practical machine.

FIG. 61.

ELECTRICAL MACHINES

77

The latter of the two essential points, mentioned above — the
earth-contact — can easily be arranged, as thown in Fig. 51, by using
a neutralising rod, D, connected with metal brushes capable of
touching the stud on the revolving plate, when it is just in front
of P a ad N. It is evident that, when in action, it is immaterial
whether this rod be insulated or not, for in the machine described
on p. 75, the studs being at opposite ends of a diameter, whatever
quantity of positive electricity is carried off by the brush opposite P,
an equal quantity of negative electricity will be carried off, in the
same time, by the brush opposite N, one charge neutralising the other.

Let us now consider the other essential essential point — how to
maintain the charge on the armatures P and N. This is best done
by connecting a conductor to the armature P, which terminates in
a brush facing the rotating plate just before it reaches A, and another
similarly with N and B. In Fig. 52 the course of the conductor
is shown by dotted lines
where it passes behind
the plate, and by thicker
lines, M, where it bends
round the edges. Thus,
when the charged stud,
C, reaches the brush on
M, its charge is shared /\ c
with the armature, only
a small free charge being
retained. By this means
the charge on the arma-
tures is increased, how-
ever small it may be at
first. Besides this passage

of electricity to the arma- FIG. 52.

ture from the stud itself,

each portion of the glass, as it comes fresh from the induction of
one armature, is put in practically metallic conduction with the
opposite armature, and as the leakage from the armatures, when
once they are fully charged, is small, so little electricity is removed
from the plate by the brushes, that nearly the whole charge passes
on to its discharging terminal, where it is collected in the usual
way.

In the best mechanical form of the machine, the long conductor,
M, from the armature, is avoided by making the armature itself
extend further, and then fixing to this the short rod bent three
times at right angles, shown in Fig. 48.

The Leyden jars, LL, are used to increase the capacity of the
terminals. The machine will work, either with or without them ;
(a) without jars, the discharge is a steady flow of a diffuse brush

78

ELECTROSTATICS

typo; (b) icit/i, jam, a greater charge is required to raise the potential
difference between the terminals to the " sparking point," and so the
discharge passes at longer intervals, but in a more concentrated form,
giving a loud, bright spark. The outer coatings of the jars should be
in metallic connection. Removing this connection produces much the
same result as removing the jars themselves.

The Wimshurst Machine consists of two varnished glass
or ebonite plates (preferably the former) arranged to rotate simul-
taneously in opposite directions, and usually carrying from twelve
to sixteen narrow strips of tinfoil, known as " sectors." Both plates
are exactly alike. Two neutralising rods lie obliquely across the
plates, nearly at right angles to each other, one at the front and the
other at the back. These rods terminate in brushes, which touch the

plates as they revolve, and
provide four earth-connections.
Fig. 53 shows a simple design
due to Messrs. Newton & Co.
of Fleet Street. The discharg-
ing terminals are supported by
glass rods, which also carry
the collecting combs. To these
combs, on the other side of
the machine, are connected the
inner coatings of two small
Leyden jars. The outer coat-
ings are uninsulated, as the
jars stand on the wooden frame
of the machine, although it is
usual to provide a metallic con-
nection between them, which
may be removed at will.

In this machine, the station-
ary inductive armatures of the preceding type disappear, for, as both
plates rotate, it is possible to make the sectors themselves act as
inductors during a part of their course. In each revolution, each
sector twice receives a charge, and twice induces one ; the former
when it touches a brush, and the latter when it passes the position
of the brush on the opposite plate. This will be understood from
Fig. 54, which shows the distribution of charge when the machine is
at work. The signs marked outside the circle refer to the charges
on the plate at the back. Consider any sector, S, which has just
passed one collector. It is under induction from the negative charges
on the back plate which are descending towards the collector ; when
it touches the earth-connection E15 the repelled negative charge is
removed and it proceeds with a positive charge. It is now able to
act inductively on the back plate, and when passing E2 — the position

FIG. 53.

ELECTRICAL MACHINES

79

of the earthed brush on that plate — sectors in contact with E9 receive
a negative charge. The sector S then reaches the other collector, and
is wholly or partially discharged ; after which a similar sequence of
operations recurs.

The presence of the metal sectors is not essential ; the machine
will work perfectly well without them, and will probably give longer
sparks, but it will not be self-exciting, and it will be desirable to
extend the brushes radially in order to cover a somewhat larger
area. The greater the number of sectors, the more readily the
machine excites, and when very numerous, it is practically always
self-exciting. This is due to the fact that the greater the number,

FIG. 54.

the greater the probability of differences of potential existing between
them. At the same time, the length of spark obtainable diminishes,
for the insulation across the plates is decreased, and it is obvious that
a very large number of sectors would practically amount to a short
circuit. On the whole, it is convenient to use a moderate number of
sectors (say 12 to 16) on each plate.

It is interesting to notice that either the Voss or the Wimshurst
machine will run as a motor, when connected to another machine
used as a generator, the sequence of actions being reversed.

Experiments with Electrical Machines.— It is assumed

80

ELECTROSTATICS

that a Wimshurst machine is available, so constructed that the
Leyden jars may be disconnected at will.

Exp. 53, to illustrate the function of the accessory Leyden jars.

(a) Work the machine without the jars. Notice that there is a faint con-
tinuous discharge between the terminals, scarcely visible in a lighted room unless
the gap is very short, but appearing in the dark as a diffused bluish- violet glow.

(b) Place the jars in position, with their miter coatings unconnected by the
metal wire or tinfoil strips usually provided. On working the machine, observe
that the result is much the same as in (a), although probably some slight im-
provement in the way of concentration into distinct sparks may be noticed.

(c) Connect the miter coatings metallically, and observe, that when the machine
is worked, the action is completely different — brilliant, sharply denned sparks
being produced at regular intervals, which produce a much more striking and
impressive effect.

The apparently feeble effect obtained in (a) is evidently due to
the small capacity of the terminals. If it were possible to connect
them to insulated conductors of great size, which would, of course,

FIG. 55.

require large charges to give the potential necessary to break down
the dielectric air, the discharges would occur less frequently, but the
individual sparks would be more pronounced. Exactly the same
effect, without the disadvantage of undue bulk, is produced by the
addition of the Leyden jars.

It is perhaps less easy to understand why the metallic connection
provided in (c) should make such a great difference to the result as
compared with (6), for in both cases the outer coatings were unin-
sulated. Consider Fig. 55, which shows diagrammatically the state
of the condensers during the charging process preceding the spark.
On each outer coating two induced charges will as usual be produced,
and the repelled charge (which is not shown in the diagram) will
tend to pass away to earth. There is no difficulty about this, even

ELECTRICAL MACHINES 81

if the jars are standing merely on wood^ for the wood conducts quite
well enough to permit the repelled charges to pass away readily,
if sufficient time is allowed (really a very short time by ordinary
standards), and during the charging process the time allowance is
ample for the purpose. Meanwhile the P.D. between the terminals
is steadily rising to the critical value at which the dielectric will give
way, and just before that value is reached, the state of affairs would
be as shown in Fig. 55.

It is now evident that the discharge of the inside coatings by the
spark must simultaneously and suddenly release a practically equal
quantity on the outer coatings. Moreover, unless ample facilities
are provided for the escape of these outer charges, the main spark
will be retarded and largely suppressed. Now, wood does not con-
duct sufficiently well to permit the instantaneous discharge required,
whereas a metallic communication answers admirably. We see,
therefore, that the total flow takes place in a kind of circuit, each
spark between the main terminals involving an equal and oppositely
directed discharge between the outer coatings. These facts may be
illustrated as follows : —

Exp. 54. Remove the metal connection between the outer coatings of the
jars, and connect the outer coatings to a pair of insulated auxiliary discharging
terminals. When these terminals are widely separated, the main spark is
feeble and more or less of the type obtained in (a) Experiment 53. When they
are brought within some small distance — depending on the power of the machine
— the main sparks are bright, sharp, and crackling, and each is accompanied by
a second spark between the auxiliary terminals, due to the discharge of the
outer coatings.

This second spark, which may be of considerable length, is note-
worthy, because it is obtained from conductors which are already
connected by the wooden stand. These charges would certainly leak
through the wood, if more than the briefest time were allowed, but
under the circumstances the charges on the outer coating — both of
which are at zero potential until that instant — are released so sud-
denly that the P.D. between them rises above the sparking value
before the badly conducting wood can act effectively.

Exp. 55. Connect the outer coatings by a long iron chain, the rustier the
better. Notice thnt each spark between the main terminals is now accompanied
by flashes of light at each link — due to bad contact — but that the chain may be
handled freely without shock. Now break the chain in the middle and hold
the broken ends, one in each hand. Notice that, at each spark, a shock is felt,
whose magnitude depends upon, and is capable of regulation by, the distance
between the main terminals.

Action Of Points. — Exp. 56. Work the machine in the dark without
jars to obtain a brush discharge, and slowly separate the terminals. Observe
that at first the glow fills the whole space, with more brilliant flickering roots
emanating from the positive side, and that at greater distances it breaks up into
a well-defined positive brush (Fig. 56) separated by a dark space from a feeble
negative glow.

F

82

ELECTROSTATICS

Exp. 57. Hold a blunt piece of wire near each terminal in turn. Notice that
when it is near the negative terminal, a well-defined positive brush will form upon
it, but when it is near the positive terminal, the result will be a minute point

of light, which is equally typical of a nega-
tively charged point.

In this way it is easy to deter-
mine the kind of charge on the
terminals, without using a proof-
plane and an electroscope.

Exp. 58. Look at the points upon the
collecting combs whilst the machine is at
work. Notice that the positive brushes
proceed from the points on the negative
side of the machine, and that negative
FIG. 56. points of light are present on the positive

side of the machine. (See Experiment 57. )

Exp. 59. Attach a blunt-ended wire, about -J- inch thick, to each terminal
in turn. In both cases notice that it reduces the sparking distance ; and in the
dark, especially when the hand or other conductor is held near, that a positive
brush forms upon it when it is on the positive side, and a negative point when
it is upon the negative side. Notice also that the hand, if it be held sufficiently
near, feels as though a wind was blowing upon it.

Exp. 60. Demonstrate the existence of the stream of escaping charged
particles of air or dust by using the piece of apparatus — known as the electric
Avhirl — Fig. 57. It need scarcely be mentioned
that the whirl rotates in a direction opposite to
that in which the points are turned.

Another very striking illustration is
as follows : —

Exp. 61. Place an electroscope at a distance
of 15 to 20 feet from a machine. (A light metal
can may with advantage be placed upon the cap,
but, as this is merely to increase the effective area
of the cap, it is not absolutely necessary. ) Work
the machine in the usual way to get good sparks,
and notice that the electroscope is unaffected.
Now separate the terminals beyond sparking
distance, attach a bent wire to either terminal,
with the free end pointing towards the distant
electroscope. Work the machine, and, in a
short time, notice that the leaves will begin
to diverge, and that the divergence remains
unaltered when the machine stops. When the
point is on the positive terminal, the charge
found on the electroscope is positive and vice versa.

Exp. 62. Hold a lighted candle near the wire. Notice that, whether the
wire be positive or negative, the flame is blown about as by a wind ; but notice,
in the latter case, that one part of the flame — when fairly close to the wire — is
distinctly attracted by the wire.

This is due to the fact that a flame contains charged particles or
ions (see p. 318), there being a very large excess of positively charged

FIG. 57.

ELECTRICAL MACHINES

83

String:.

o

ions. The presence of these charges in the flamo explains why
excited non-conductors, like -glass or ebonite, are so readily and
completely discharged by holding them in the hot gases above it;
.the flame does not act merely like a point, as is often stated, but
neutralises the charges by supplying others of an opposite kind.

Experiments1 to show the Action of Lightning Con-
ductors.— It is convenient to have two circular metal plates — about
12 or 15 inches in diameter — with rounded edges. One of the plates
is laid upon the table and the other
suspended above it, at any convenient
height, by silk ribbons. Three small
wooden feet carrying narrow brass
tubes are also required (small cork
borers pushed completely through a
cork answer well); when these stand on
the lower plate, the brass tubes make
contact with it, and the tubes form
holders in which brass wires, carrying
brass balls of different sizes and slightly
bent, may be adjusted to various
heights without slipping (Fig. 58).

Exp. 63. Connect the upper plate to
the negative terminal of a machine, and the
lower plate to the positive terminal. Place

the three stands in position. Work the FlG. 58.

machine and adjust the height of the balls

by slipping the wire up or down the tube. The lower plate represents the
earth ; the three stands, buildings ; and the upper plate, a thunder cloud.
Notice that all may be struck, but that the spark shows a decided preference
for the smallest ball, striking that alone even wlieii it is much lower than the
others. With a little care, the balls may be adjusted until the spark strikes
any one of them indifferently, the largest ball being then nearest the upper
plate, and the smallest furthest away. 2

The important point to notice in this experiment is not the
preference shown by the spark for the smallest ball, but the fact
thereby indicated that the spark has the power of selecting its
path.

Exp. 64. Now remove one of the balls, leaving the end of the wire free.
On working the machine, observe thatj sparking is stopped, the point affording
complete protection to the other conductors even when it is considerably below their
let-el.

1 These experiments were devised by Sir Oliver Lodge.

2 Incidentally, this illustrates why the terminals of a Wimshurst machine
are usually fitted with balls of unequal size, experiment showing that a longer
spark can be obtained when the small ball is positive — the opposite arrangement
giving shorter sparks and much brush discharge. For this reason, the student
was advised to make the upper plate negative.

84 ELECTROSTATICS

This illustrates the proper function of a lightning conductor. Its
main purpose is not so much to be struck instead of the building
itself, but rather to prevent any spark occurring at all — the silent and
usually invisible brush discharges from its point quietly neutralising
the atmospheric charges.

The student will notice that if all spark discharges were of the
kind just investigated, the complete protection of buildings would
be a very simple matter. Any projecting point, which need not
necessarily be the highest part of the building, would safeguard
quite a large area, and the question of a good "earth" to the
conductor would not be material. Facts, however, prove that it is
extremely difficult to protect buildings completely, and that the
provision of a good earth-connection is of the first importance unless
the lightning-rod is to be a source of danger instead of a safe-
guard. At this stage, it is impossible to explain the matter ade-
quately, and, therefore, it is preferable to show experimentally that
all spark discharges are not of the same kind.

Exp. 65. Disconnect the plates from the main terminals. Join them up to
the outside coatings of the Leyden jars, removing the usual metallic connection
between the latter. Adjust the main terminals until sparks are obtained
between them. Probably it will also be necessary to diminish the gap between
the balls and the upper' plate. Observe that, at every spark between the main
terminals, another spark occurs between a ball and the upper plate, but that no
preference is now shown for any ball, the spark always striking the highest, what-
ever its size. Moreover, the point has completely lost its power of suppressing
sparking ; if it happens to be the highest it becomes struck by the spark, but
there is no brush discharge, and, if it is the merest shade lower than any other
conductor, it becomes absolutely useless. [This shows the practical necessity of
making a lightning conductor the highest point of the building, and also of
providing adequate cross-section to carry off a powerful discharge without fusion
or without the risk of its leaving the rod.]

These experimental results will be understood better after reading
the section on oscillatory discharges in Chapter XXXV. In the mean-
time, it may be remarked that the great difference revealed in the pro-
perties of the two kinds of sparks are natural consequences of the time
factor involved. In the first case, the P.D. between plate and ball
increased gradually during the time preceding the discharge, i.e. the
electric field formed gradually and had ample time to choose its path.
In the second case, we are dealing with a sudden impetuous rush from
the outer coatings ; it is like the giving way of a reservoir wall, or
the discharge of a gun — no preliminary time is given for a choice of
path, and the spark strikes whichever object is nearest, irrespective of
its size or pointedness.

Exp. 66, to show the instantaneous nature of the spark. Work the machine
until sparks are given. Notice that the rotating plates of the machine appear
to be stationary.

Exp. 67, to show the existence of two kinds of electrification. Put a paper
tassel, attached to a wire, first on one terminal and then on the other, In each

ELECTRICAL MACHINES 85

case, the paper strips diverge widely by repulsion, and are attracted by the hand
and by neighbouring objects. Hence botli are electrified and apparently are alike.
Now put tassels on both terminals ; the result is a conspicuous and decided
attraction between the two, the paper strips clinging together and forming a
bridge across.

Exp. 68, to illustrate the working of a Leyden jar. Attach pith-balls, sus-
pended by light threads about two or three inches long, one to the outside coating
of the jar and one to the brass rod between the ball at the top and the lid.
(a) Insulate the jar, place it so that its knob is nearly touching one terminal, and
work the machine. Notice that only occasional feeble sparks pass to the knob
— the better the insulation, the fewer there will be — and that both pith-balls
instantly diverge to the fullest extent. This shows that when the outer coating
is insulated, the jar does not act as a condenser, and that a very small charge
raises the potential of the inner coating to an equality with that of the machine.
Sparking then ceases, except to make up for leakage. Notice that, upon remov-
ing the jar and discharging it, the effect is very small, (b) Repeat these operations
with the outer coating earthed, and notice that sparks pass freely to the knob for a
few moments, after which the passage of sparks practically ceases. The pith-
ball attached to the rod is meanwhile slowly deflected to its full extent, reaching
its final position when the sparking nearly ceases. Observe, also, that the pith-
ball on the outer coating is not deflected at all. Upon removing the jar and
discharging it, a brilliant spark is obtained. This shows that a much larger
charge is needed to raise the potential of the inner coating to an equality with
that of the machine.

Exp. 69, to compare approximately the capacities oftivo Leyden jars of different
sizes.1 Place one of the jars with its knob in contact with the prime conductor
of an ordinary frictional machine (an inductive machine is not suitable). Pro-
vide two insulated brass balls, placed about % inch apart, and connect one to the
prime conductor and the other to the outer coating of the jar, which may stand
on the table as usual. On working the machine, the jar will charge up until the
P.D. between the brass balls reaches a certain value, then it will be discharged
by a spark, and the process will go on continuously. Turn the machine steadily
and count the number of revolutions between one spark and the next. Let there
be n revolutions. Repeat the experiment with the second jar, taking care not
to disturb the balls, and suppose n^ revolutions are now required. Then, assum-
ing that the quantity of charge supplied to the jars is the same per revolution
when rotation is steady, we have

Q = VC for the first jar, and
Q! = VCl for the second jar,
V being the same for both, as it is determined by the distance between the balls.

. J^C.
' Qi C3
Q_
Qi

1 This is an instructive experiment, as showing what can be done with simple
apparatus. It is quite capable of giving fair results, although it must not be
regarded as a serious method of measurement.

CHAPTER VII

ELECTROMETERS

ELECTROMETERS are instruments in which the attractions or repulsions
of electric charges are made use of in order to measure differences of
potential, and may, therefore, be regarded as extensions of the prin-
ciple embodied in the gold-leaf electroscope.1

The two most important and widely known types are those
originally devised by Lord Kelvin, known as (1) the quadrant, and
(2) the attracted disc electrometer.

The Quadrant Electrometer. — This consists of four metal

quadrants, usually hollow, and forming, when placed together, a
shallow circular box divided into quarters. In the earliest forms,
these were insulated on separate glass pillars, and within them a light
aluminium needle2 (Fig. 59 A) was suspended by two silk fibres — this
is known as bifilar suspension, its purpose being to secure a definite
zero position. The needle-system carries a small mirror, M (Fig. 59c),
the deflections being read by means of a lamp and scale in the
ordinary way.3 The alternate quadrants are connected permanently
by wires, as shown in Fig. 59B, and carry the terminals TT. Let us
suppose that a small P.D. is produced between TT, say by connecting
them to the terminals of a voltaic cell. Evidently the quadrants
become charged as shown in the figure, but the forces upon the needle
being equal and opposite, no deflection is produced. If, however,
the needle itself is charged, a deflecting couple will be produced,
which in magnitude must depend both upon the P.D. between the
quadrants and upon the potential of the needle, and if the latter be
high, very small differences of potential may be detected. If, how-
ever, any relative value is to be attached to the readings, it is obvious
that the potential of the needle must not alter during the observa-
tions, and this introduces a difficulty, because the small charge required
to raise the needle to a given potential would leak away very quickly

1 Various new and elaborate forms of electroscope have recently been
developed. It is now a refined and delicate instrument, capable of being used
for very exact measurements. In some cases a single strip of gold leaf is used,
only 1 millimetre wide and about 25 millimetres long, a microscope being used
to read its movements.

2 The word needle is conventionally used to denote the moving part of almost
all instruments, and does not imply any particular construction. In this case,
the needle is a piece of metal shaped like the figure eight.

3 This is explained on p. 147.

ELECTROMETERS

87

in spite of the silk suspension. In theolder forms, the difficulty was
overcome by attaching a platinum wire, P ( Fig. 59c), underneath the
needle, N, its lower end dipping into strong sulphuric acid, which
formed the inner coating of a glass Ley den jar, the outer being made
of tinfoil, and earth-connected. This enormously increased the
capacity of the needle-system, and after charging the jar with a few
sparks from an electrophorus (or in any other convenient way) its
potential would, under favourable conditions, remain fairly constant
for some time. The more elaborate forms of the instrument also
contained devices for indicating any change ;in. the potential of the
needle, and for keeping it at a steady value. In any case, the instru-
ment was difficult to use.

It is, however, unnecessary to discuss it in detail, becnuse a greatly
improved form has recently been devised by Dr. F. Dolezalek, which

HjSIU

Fio. 59.

is shown in Fig. 60 (inserted by permission of the Cambridge
Scientific Instrument Company).

The needle and quadrants are of small dimensions, thus reducing
the capacity. This is advantageous, because in measurements of
capacity, that of the instrument itself is always involved (see p. 90).
The quadrants are mounted upon ambroid pillars (made from amber),
which afford more perfect insulation than glass. The needle is made
of paper thinly covered with silver, and is suspended by a quartz
fibre, which provides excellent insulation combined with delicacy of
suspension, and the extreme lightness of the moving system gives
great sensitiveness. The needle and mirror are rigidly connected,
but the quartz fibre is provided with a hook at fiach end, and is
detachable, the hooks engaging in eyes in such a way that no slip
or back-lash can occur. . Several such fibres are supplied with an
instrument in order to obtain a wide range of sensitiveness. No

88

ELECTROSTATICS

Leyden jar is required, because the sensitiveness obtained by using
a quartz fibre is such that much lower needle potentials can be used.
It is usual to charge it to about 50 or 100 volts1 — a pressure easily
obtained from a battery or a lighting circuit. The insulation is so
good that it is generally sufficient merely to connect the needle with
the charging source for an instant, and for this purpose there is
provided the switch K, which is an insulated brass rod carrying a

Flo;. 60.

light strip of phosphor bronze at the lower end. By turning the
insulated terminal at the top, this strip can be brought into contact
with the needle and then withdrawn. In some cases, it may be
desired to keep the needle permanently connected to the charging
source, and then the quartz fibre is made to conduct by moistening

1 In all forms of instrument, the quadrants should be earth-connected while
the needle is being charged.

ELECTROMETERS 89

it with a weak solution of some hygroscopic salt, such as calcium
chloride ; the upper end, which is carried by an ebonite torsion head,
can then be connected to the charging source.

The fibre itself is about -j^Vir incn in diameter and 2J inches
long, and a difference of potential of y1^ volt between the terminals
gives a large deflection. In the figure, two adjacent quadrants are
shown swung open to allow of easy access to the needle.

Theory of the Quadrant Electrometer. — it is only possible

Co find an approximate relation between the P.D. to be measured and the
deflection produced. Hence, the quadrant electrometer is not well adapted to
measure P.D.'s absolutely. When properly designed, however, its readings are
proportional to the P. D. producing them over a wide range, and it, therefore, may
be regarded as an extremely sensitive instrument suitable for "null" methods, 1
and for the exact comparison of differences of potential. When actual values
are required it is best to calibrate it by noting the deflection produced by some
known P.D., such as that given by a standard cell.

The usual theory is as follows, and, although it is not free from objection,
it serves to indicate the nature of the relations between the various quantities
concerned.

Let V — potential of needle, i.e. P.D. between needle and earth, and let
Vj, V2 — potentials of pairs of quadrants respectively with respect to that of
earth. Assume that these values remain constant, and that V> V1>V2.

The needle may be regarded as one plate of a condenser, a pair of connected
quadrants forming the other plate, hence, there are two such condensers, whose
capacities, by symmetry, will be equal when the needle is at rest in the zero
position.

When the needle is deflected through an angle 0, the capacity of one of
these condensers will be increased and that of the other diminished, and it will
be assumed (as evidently approximately correct) that the amount of change is
the same for both and is proportional to 0.

These alterations in capacity will also alter the energy (for energy =|Y2C).

Hence, if m be some constant, the energy of one condenser is decreased by
^-(Y - y^md, and the energy of the other is increased by -J(Y - V2)2m0 (for, as
V>VX> V2, the needle moves towards the quadrant at potential V2).

. '. Nett gain of energy = $m0{(V - V2)2 - (V - Yj)2}, and this is expended in
doing work against the couple due to the torsion of the suspension.

but work — couple x angle

i.e. the expression ^m{(Y - V2)2 - (V - Y^2} is the moment of the couple acting
upon the needle when the deflection is 0. Now, by the theory of torsion for
small deflections, the angle is proportional to the couple producing it, and
omitting 4-w as no longer required, we may write

a)»-<t-

1 - V2) - (Yx + V2)]( Yx - Y2)}

In the ordinary use of the instrument, Y is very large compared with Yj or
V2, hence the term — — - is negligible, and, therefore, when V is constant, 0 is

1 See p. 305.

90 ELECTROSTATICS

proportional to the P.D. between the quadrants, which is the P.D. to be
measured. Evidently the sensitiveness depends on the magnitude of V.

There is another method of using the instrument. Instead of charging the
needle from an independent source, it is put in conducting communication with
one pair of quadrants. This makes V = Vl5 and the expression for 6 becomes

The instrument is now much less sensitive, but in this state it will give
steady readings with alternating differences of potential, for a little consideration
will show that the direction of the deflection is unchanged, when the sign of
all the charges is changed. Moreover, as the couple is at any instant pro-
portional to the square of the P.D. to be measured, the readings will be
proportional to its "virtual" or "square root of the mean square" value
(see p. 429).

Experiments with the Electrometer. — In practice, the
quadrant electrometer is useful as an instrument of research for many
purposes, which cannot be adequately dealt with at this stage. The
following experiments are suggested simply to illustrate general
principles.

Exp. 70, to tneasure the capacity of any conductor in static units, ~by com-
parison itrith the capacity of a sphere {Faraday's method, referred to on p. 67).
Connect the conductor to one terminal of the instrument by a long fine wire,
the other terminal being "earthed." Charge it until a convenient deflection is
obtained — suppose this is d divisions. Now, connect to the system an insulated
uncharged sphere, also by means of a long fine wire. (Long wires are required
because the various conductors must be kept at considerable distances from one
&>iother to avoid mutual influence.) The deflection will fall to some smaller
value — suppose dl divisions.

If C be the capacity to be measured, Cj the capacity of the sphere, q the
charge given to the first conductor, and V the potential to which it is raised by
that charge, then

V = ^ccd. (1)

In the second case, q was not altered in amount, but was simply shared
between the conductors, their common potential being (say) Vj

C rfx

C d

or -- = — ±— (where Ci = radius of sphere).
Cx d — aj

It is important to notice that the value obtained for C includes
the capacity of the electrometer itself. Hence the advantage obtained
by keeping that capacity small, as mentioned in the description of
the Dolezalek instrument. In exact work, its capacity would be
measured separately and allowed for. It is obvious from the above

ELECTROMETERS

91

argument that this can be done easily by connecting the instrument
to a charged sphere, noting the deflection, and then connecting it to
another insulated uncharged sphere, and again noting the deflection.
The use of a sphere, however, as a standard requires caution, since
its capacity is only equal to its radius in centimetres when it is at a
distance from other bodies, and in an ordinary room this value may
be seriously altered, because the walls and surrounding objects act
towards it as the outer coating of a condenser.

Measurement of Specific Inductive Capacity (or of
Dielectric Constant). — Here again it is intended only to indicate
the principle involved in the methods actually used — the complicated
arrangements required in practice to obviate many sources of error
making a full description unadvisable at present.

m

Earth

1

Earth

FIG. 61.

Exp. 71. Arrange a parallel plate condenser as in Experiment 42, using an
electrometer in preference to the electroscope as being more suitable for exact
readings. Charge the plate A (Fig. 61) until a convenient deflection is obtained
— the other terminal of electrometer E being earthed as usual. Let d be the
distance between the plates. Now insert, between the plates, but without con-
tact, a slab of the dielectric in question of thickness t — taking precautions to
avoid inadvertently charging the dielectric (see foot-note, p. 62). We know
from Experiment 52, that this will increase the capacity and decrease the P.D.
between the plates, thereby causing the deflection to be decreased. Now increase
the distance between the plates, by moving one of them, say B, parallel to itself.
Then, because the capacity is diminished, the deflection is increased. Let d± be
the distance between the plates at which the deflection is the same as before.
Then the capacity must also be the same as at first.

In case (i.) C =

4-rrd

In case (ii. ) the actual air space is
to an air space of (p. 62).

- 1, and the dielectric itself is equivalent

92

Therefore,
hut we have made Cx = Q

ELECTROSTATICS

A

whence K = -

Now, as d1 - d is the distance the plate B was moved, it can he measured with
great exactness.

Dielectric Constant Of Liquids. — The method most generally
used depends upon making two measurements of the capacity of any
conveniently shaped condenser, first with air as the dielectric, and
then with the liquid as the dielectric. Although such measurements
can be carried out in various ways, the best methods involve the use
of a galvanometer as described in Chapters XX. and XXIV.

Attraction between Parallel Plates. — Before describing

Lord Kelvin's attracted disc electrometer, it will
be necessary to investigate mathematically the law
of attraction between parallel plates.

Let M and N (Fig. 62) be two oppositely charged
plates of area A at distance, d, apart. We can
easily determine the attraction between them, pro-
vided we simplify the problem by assuming — as in
the previous theorem on p. 61 — that the whole field
is uniform, and confined to the space between the
plates. Ordinarily the field will fringe out at the
edges of the plates as shown, and some lines will
diverge from the outer side of at least one of the
plates — effects which may introduce serious errors
- — hence, we must be perfectly clear as to the con-
ditions under which our argument is valid. It will
be seen later how these conditions are satisfied in
the actual instrument.

Let one plate, M, carry a charge -fQ wholly
on one side, then by the assumptions made, the
other, N, carries an equal charge — Q.

The field between the plates (see p. 61) is
given by

Field = F = ^Q

A

-p 4. O

.*. Force on unit charge = U = — = • — — (1)

K AK

Now, if another charge q be placed anywhere in the space between the
plates, i.e. in the electric field, it follows that the force on it would

FIG. 62.

ELECTROMETERS 93

be f/U dynes (it being understood that q does not, by its mere
presence, alter the value of F).

This force will be the same at all points in the field — because the
latter is by assumption uniform — however close q may be to the plates,
but the force is modified if the charge is on one of the plates, for
then it is situated on the boundary of the field. This case has been
considered on p. 46, where it is shown that under such circum-
stances the force is q x -JU dynes. A similar argument may be applied
in the present instance, for if we regard each of the charges, + Q and
— Q, as possessing fields which extend horizontally in both directions,
it will be found — by drawing a simple diagram — that in the space
outside the plates the two fields are equal and opposite, thus neutral-
ising each other ; whereas between the plates they are in the same
direction and strengthen each other, each charge contributing one-
half of the actual field. Hence the attractive force on either plate

will be Q x JU dynes, where U = -^^
/. the pull, or attraction = Q x

Alv

, /nx

dynes (2)

This important equation may be thrown into several different forms.
Applying it, for instance, to the electrometer, we do not know Q, and
we must, therefore, express Q in terms of V. This is done at oncp
from the definition of V as force on unit charge x distance,

~ V.A.K

whence Q =

Substituting this value in equation (2), we have

attraction between the plates = —„ x ( ' — j .

V2AK

Let p = force of atttraction between plates (in dynes)

vr

then VJ = -— — x p
AK

.

This formula gives the P.D. between the plates, and is used in the

94

ELECTROSTATICS

attracted disc electrometer, an instrument in which the details are
somewhat complicated. The principle, however, will be readily
understood from the following explanation.

Lord Kelvin's Attracted Disc or Absolute Electro-
meter.— An attracted disc electrometer is one in which the attraction
between two parallel discs at different potentials, and at a certain
distance apart, is balanced by the weight of a given mass.

Sir William Snow-Harris con-
structed the first electrometer of
this kind. It resembled a balance,
having a flat disc suspended at one
end of the beam, and an ordinary
scale- pan at the other. This disc
was suspended above a similar insu-
lated disc, which was connected with
the charged body to be tested. When
attraction took place between the
discs, weights were added to the
scale-pan until equilibrium was re-
stored. This electrometer is defec-
tive in many particulars ; the chief
one arises from the irregular distri-
bution of the electricity on the plate
• — the surface density being much
greater at the edges than on the flat
surface.

In Lord Kelvin's form the
principle only was retained, the de-
tails of construction being entirely different, and the result was an
instrument capable of directly measuring difference of potential in
static units.

Its general appearance is shown in Fig. 63, taken by permission
from Electricity and Magnetism (Carey Foster and Porter), the
details of which (as far as is necessary for our purpose) will be under-
stood from Fig. 64. The attracted disc AB — made of aluminium
for lightness — is suspended

from the inner side of a -c C

circular metal box by means
of three delicate springs
(only one, S, is shown),
similar in shape to coach
springs, which cross each
other at the middle points,
where they are attached to
box and plate respectively. This disc nearly fills the circular aperture
in the " guard ring," GR, and is normally held by the springs slightly

FIG. 63.

ELECTROMETERS 95

above that level. The guard ring is supported by the glass case, CO,
and in Fig. 63 the box is seen opened out, showing the spring system
and the disc AB of Fig. 64. Below is the attracting plate, PP, which
is insulated, and can be raised or lowered by a micrometer screw.

The guard plate, disc and box form one conducting system, and if
we connect this system to one, and the plate PP to the other, of the
two points between which a P.D. is to be measured, an electric field
will be formed between them. We then see

(1) that, because the attracted disc forms part of a hollow con-
ductor, the field is entirely confined to one side of it ; and

(2) that, because of the presence of the guard ring, all fringing of
the field is confined to the latter, and the pull on the disc AB, which is
the quantity to be measured, is due to a practically uniform field.

Hence the conditions previously mentioned are fully satisfied.

A delicate gauge is provided to show when the disc AB is exactly
in the plane of the guard ring, and the force required for this purpose
is found, to begin with, by using known weights. (In the original
instrument this was 0'6 gram.) When the apparatus is connected up
as mentioned above, the height of the plate is adjusted until the gauge
shows that the disc is in the plane of the guard ring ; then, we have

where d is the distance between the plates ; w, the mass in grams
found in the preliminary experiment with known weights; # = 981;
A = area of attracted surfaces ; and K = 1 (for air).

Hence the quantity under the root sign becomes a constant for the
given apparatus, the only uncertainty being the precise value of A.
This has been shown to be approximately the mean of the areas of the
aperture and of the plate AB.

K is, of course, always unity in practice, but its presence in the
equation tells us that if the discs were immersed in some dielectric
liquid (in which case K would be greater than unity), the equilibrium
distance, d, would be increased for a given P.D. in the ratio \/K : 1.

It is, however, by no means easy to measure the distance d
accurately, on account of the difficulty of making the plates parallel
and perfectly plane, and hence the following method is actually used
in taking measurements.

By some auxiliary means a constant P.D. is maintained be-
tween the disc and the earth, the lower plate P being also connected
to earth. Then the latter plate is adjusted until the disc is in the
plane of the guard ring, and the reading of the micrometer taken.
If the corresponding distance be d (which is unknown), and if the
auxiliary P.D. be V- then

!V"Ir

96 ELECTROSTATICS

The plate P is then insulated from earth, and connected to the body
whose potential is to be measured (or to one of the points between
which a P.D. is to be measured, the other being connected to earth),
and again moved up or down until the disc is in adjustment with the
guard ring — a second reading of the micrometer being taken. If dl
be the new distance between the plates, and if Vj be the potential,
which has to be measured, then the P.D. between the plates is V — Vj
(assuming V to be the greater), and we have

^^-•Wior

Hence, by subtraction

Now (d — d-)) is a distance, which can be measured with great accuracy,
as it is the difference in the micrometer readings. Obviously it must
be expressed in centimetres.

Lord Kelvin used the term Jieterostatic to express the method of
using an electrometer when an auxiliary P.D. is made use of (as in the
above case and in the case of the quadrant electrometer), and he
applied the term idiostatic to the method in which the P.D. to be
measured is alone made use of (as in the second method we have given
in using the quadrant electrometer, and in the direct method of using
the above instrument).

Lord Kelvin also devised a smaller portable form, known as the
portable electrometer, -which is essentially the same in principle, but
which, on account of its smaller size, involves certain difficulties in
satisfying the theoretical conditions. It is usual to calibrate it by
using known potentials.

Further Consideration of the Attraction between

Parallel Plates. — Apart from its application to an electrometer, the
expression for the attraction between two parallel plates will repay further
examination.

(A) In equation (2), p. 93, we have attraction =p = -- dynes. As this

AK.

does not involve the distance between the plates, we learn that when Q is constant,
the attraction is independent of the distance. This may seem strange at first
sight, but the fact becomes obvious if we remember that our conditions require
the field to be uniform and parallel at all distances, and that the "attraction"
is really the pull due to the tension of the lines of force which terminate on a
given plate. If the number and distribution of these lines are unaltered, so is the
pull, and the distance has nothing whatever to do with it. This assumes that
the medium is also unaltered ; if it is changed, we see that the pull is inversely
proportional to K, although the field is the same as before ; i.e. we must suppose
that the tension of the lines of force is lessened as K increases.

(B) Again, equation (2) is often expressed in terms of surface density.

Now, surface density =/>=-,

ELECTROMETERS 97

whence by substitution in (2) on p. 93

Hence in air, the pull of a uniform field is 2?rp2 dynes per square centimetre, as
already found on p. 46. This is true generally, e.g. for the pull of the field on
a sphere charged to surface density p.

(G) It is instructive to obtain an expression for the pull of the field in terms
of the field strength itself. For this we have

^="AK~

Now Field = F = ^^
A

whence, eliminating Q, we obtain

* (dynes) (6)

from which wre deduce the very important fact that the pull of the field varies
directly as the square of its strength. It is in this respect chiefly that the
analogy between a line of force and a stretched string breaks down. In the latter
case, the pull on a body would depend upon the number of strings alone, and not
at all upon their concentration, i.e. upon F. In the former case, the pull depends
largely upon the concentration. For instance, if a certain number of elastic
strings pulled on one square centimetre, doubling their number on that area
would simply double the pull ; but if we do the same thing with lines of force,
the pull is four times as great. (The student will find later that this peculiarity
is also characteristic of magnetic lines of force, although they are essentially
different in nature. )

(D) We may also write F = KU in equation (6), and thus obtain an expression
for p in terms of U. This gives

Hence, from equations (6) and (7),

if F is kept constant (which means Q is constant), p varies inversely as K,
and if U is kept constant (which means V is constant), p varies directly as K.

(E) Another instructive expression is obtained for p in terms of Q and V.
We have, for this purpose,

* AK
andV =
whence, eliminating AK, we obtain

(8)

Now, ^QV is the energy of the system, and we, therefore, learn from this
equation that it resides entirely in the field (as previously indicated on p. 54).
For suppose that the plates have a constant charge of + Q and - Q respectively,
then — with the limitations already stated — we know that p is the same at all
distances. Now let the plates be placed infinitely near together, then d and V
are both nearly zero, and so is the energy. If the plates are now separated
against the constant pull, p, work is done of amount pxd, and this is the
energy expended in establishing the field, and also the potential energy of the
field when formed.

G

98 ELECTROSTATICS

Energy of Field per Unit Volume. — From equation (6) the

T?2

value of p is — — dynes per square centimetre, and if this force is overcome

OTTiV

through a distance of 1 centimetre, the result is the formation of a cubic centi-

F2

metre of field by an expenditure of energy of amount — = x 1 ergs.

STTIV

Hence, we arrive at the important result : — •
Energy of an electric field per unit volume

Example. — If the force of attraction, F, between two large parallel
plates, charged with equal and opposite charges ±Q is independent of the dis-
tance between them, find the energy of the charge, and the difference of potential
between them in terms of F and Q, when they are at a distance of 1 centimetre
apart. (B. of E., 1903.)

The full solution of this question would merely be a repetition of some of the
preceding results.

It follows from equation (8) that

.*. energy of charge = F x 1 ergs
also, P.D. between plates = V= ^.

Stress in a Dielectric.— Let us suppose that the equal and
opposite charges upon the two plates under consideration (or upon
any other pair of conductors) are steadily increased. Then the
mechanical stress on the dielectric, which is always produced by
the presence of an electric field, is also increased, and eventually it
will give way* and be pierced by a disruptive discharge or spark.1
The limiting stress varies very much with the nature of the di-
electric, and is much greater for materials like glass or ebonite than
for a gas. In the latter case, it also varies with the nature of the
gas, and with the pressure it supports. Consider, as an illustration,
the case of air at ordinary pressure. The relation between spark-
ing distance and potential difference has been measured for brass
balls of different sizes by various experimenters. Although the
spark length increases with the P.D. between the terminals, the exact
value depends very largely upon their size and shape, and is so
much affected by slight changes in the conditions, that only a
rough general agreement can be obtained. It is, however, a well-
marked fact that the P.D. required to break down a very thin
layer of air, is relatively much greater than that required for a
thicker layer. For instance, under certain experimental conditions,
it has been found that an air film 2 millimetres thick required

1 The student may find it difficult to realise how such a stress can exist in a
mobile body like a gas. It may be remarked that the stress occurs between
different parts of the atoms or molecules, and that it is compatible with freedom
of mass motion as a whole. ,

ELECTROMETERS

99

a P.D. at the rate of "190 static units per centimetre to pierce it,
whilst a layer 26 millimetres thick required only 90 static Units per
centimetre. The full discussion of this peculiarity is not advisable
at present, but it has been suggested that it may be due to the fact
that we arbitrarily measure the spark length as the shortest distance
between the terminals, whereas in short air gaps, the spark con-
spicuously avoids the shortest paths ; an effect perhaps best seen
under reduced pressure.

For our present purpose, it may be taken that the average break-
ing stress for air under ordinary pressures is a P.D. of about 130
static units per 'centimetre of length.

For a uniform field, such as that between parallel plates, it follows
from the definitions that the force on unit charge is then 130 dynes,

for

.e.

TT V _130
d~~ 1

This gives some idea of the greatest possible value of U under the
given conditions.

Now, we have already shown that the pull of an electric field

per unit area is — - — dynes, therefore, in this case, the pull is > '—

O7T O7T

or 672 dynes approximately, which is the breaking-down stress of
air at ordinary pressures.

Again, we have shown that the stress or pull is also given by

• dynes per unit area, and if we write — ^—-—672, then p= 10
J\. 1

units approximately.

Hence, the greatest possible charge which, under the circumstances,
can be given to a conductor is about 10 units of quantity per square
centimetre. When the pressure of a gas is reduced, the breaking stress
becomes greatly reduced, and
reaches a minimum value at
about '5 millimetre pressure.
Beyond this, it increases again,
and finally the gas becomes
non-conducting. The study
of the discharge under these
conditions is discussed in
Chapter XXIX.

Refraction of Lines
of Electric Force.— When

lines of force pass obliquely across

the bounding surface between two

dielectrics, for which K has

different values, say from air into glass or paraffin-wax, their direction changes

abruptly at the surface, i.e. they experience a kind of refraction.

FIG. 65.

100

ELECTROSTATICS

To understand this we must examine the conditions which hold good at the
boundary of the two media. Let PQ, Fig. 65, indicate the boundary between
(say) air and glass. We will suppose that the field inside the glass is uniform,
but not normal to the bounding surface. Let AB be 1 centimetre in length and
regarded as indicating 1 square centimetre marked out on the surface. Also,
let AC and BD indicate square centimetres marked out at right angles to the
surface in glass and in air respectively. Then, as explained on p. 174, the
number of lines of force passing through AB will be the measure of the
normal component of the field, and the number passing through AC and BD will
be the measure of the tangential component in glass and in air respectively. It
will be seen that, if the lines of force pass through the surface without change of
direction, the tangential components will be the same in the two media, but
not otherwise. On the other hand, the normal component is necessarily the
same in the two media, whether there is bending or not.

Consider the electric force (U). Remembering that F=KU, we see that,
if there is -no change of direction at the surface, i.e. if F is the same on both
sides, then U must have different values on the two sides. Now, the normal
component of U may have different values, but we can show that the
tangential components of U must be the same close to the surface on each side,
for otherwise it would be possible to get work done without expending energy,
i.e. a kind of perpetual motion could be obtained.

Let AB, Fig. 66, be any two points on the surface, and let Ui, U2 be the
tangential components of the force in air and glass respectively. Then if

U1>U2, a unit charge placed at A, just
outside the surface, might be driven from
A to B by the force, thus doing work.
When it arrived at B, a very slight (and
in the limjt, infinitesimal) force would
move it across the surface, and we could
bring it back to Aj against the weaker
force U2, thus doing work upon it. Arriving at Aa, an infinitesimal force would
move it again through the surface, and it would reach its original position with a
nett gain of energy during the cycle, for less work would have to be done in
bringing it back against the weaker force U2 . Such a gain of energy without
equivalent expenditure of work is contrary to all experience, and hence we infer
that the tangential components of the force must be the same very close to the
surface on each side of the boundary. Hence, the lines of force must be bent
or refracted at the surface by an amount sufficient
to make the tangential components of the force M

equal to each other.

Example. — Let a line of electric force,
Fig. 67, pass from a dielectric, whose constant is
Kj, into another whose constant is K2. Draw NN"!
perpendicular to the surface at the point of incidence
P, and suppose that the tangent to the curved line
of force at P makes with NP an angle dl in the first
medium, and an angle 02 in the second medium.
We can state the two conditions which must be
satisfied as follows : —

(1) The normal components of the field must
be the same in each medium.

(2) The tangential components of the force
must be the same in each medium.

(Of course, it follows that the tangential com-
ponents of the field and the normal components
of the force are necessarily unequal.)

Let FjUj, F2U2 refer to the two media respectively. Then the first conditi

- - A|- ~ '

FIG. 66.

/ K,

5

f u,

F2

/ **-

"2

i

,

Fie

\. 67.

ELECTROMETERS 101

means that F: cos ^ = F2 COP 0% ' (i)

and the second condition requires

Ul sin 0l = U2 sin 62 (2)

but F! = KlUl , and F2 = K2TJ2

.'. KjUi cos 61 = 1\.2\J2 cos 02 (3)

.-. If} -I tan^ = ltan^

(3) K! Jv2

or Ei-tan^
K2 tan 02

From which it appears that when lines of electric force enter a medium of
greater dielectric constant, they are bent away from the normal. If the second
medium is a conductor, K2=oc, in which case tan &i — 0, i.e. the lines of force
are incident normally on a conductor, as already inferred from simple reasoning.

EXERCISE VI

1. A is a gold-leaf electroscope, B a quadrant electrometer. For measuring
small potential differences B is found to be more sensitive than A : does it
necessarily follow that it will be more sensitive for the measurement of small
charges ? Give reasons for your answer. (B. of E., 1905.)

2. An insulated sphere having a diameter of 20 centimetres is charged. It
is then connected with an electrometer by a fine wire, the deflection being
50 divisions. An insulated and uncharged sphere of 16 centimetres diameter is
then joined to the first by a long wire, and the electrometer deflection falls to 32.
Calculate the capacity of the electrometer. (B. of E., 1906.)

3. Describe and explain the construction and action of the essential parts of
Lord Kelvin's quadrant electrometer. What is the instrument designed to
measure ?

4. In some forms of electrometer there is a movable disc, surrounded by
a wide flat ring in the same plane as the disc. Explain the use of this ring.

5. Describe and explain the use of any form of " attracted-disc electrometer."
Show what it measures, and explain the action of the "guard-ring."

6. An insulated plate 10 centimetres in diameter is charged with electricity
and supported horizontally at a distance of 1 millimetre below a similar plate
suspended from a balance and connected to earth. If the attraction is balanced
by the weight of 1 decigram, find the charge on the plate (y = 980 C.G.S.).

(B. of E., 1902.)

7. Define a unit of electricity and the strength of an electric field. Two
very large metal plates are parallel to one another, and at a distance of 2 centi-
metres apart ; one plate is charged with positive and the other with negative
electricity ; and the surface density of the charge on each plate is 20. Calculate
(a) the strength of the field between the plates, (b) the potential difference
between them. (Science and Agriculture, Inter., 19C5.)

8. A condenser is made of two parallel metal plates. These are maintained
at constant potentials. What effect will be produced on the attraction between
the plates when a slab of heavy glass, specific inductive capacity 10, and thick-
ness one-half of their distance apart, is inserted between them ? The faces
of the slab are parallel to the plates. (Lond. Univ. Inter. B.Sc., Hon., 1895.)

CHAPTER VIII

ATMOSPHERIC ELECTRICITY

THE early observers soon perceived that thunder and lightning
were similar in their nature to the crackling and light of the
electric discharge. Franklin proved an exact similarity between
the two discharges in (1) giving light, (2) noise, (3) conduction by
metals and moisture, (4) fusing metals, (5) rending imperfect con-
ductors, (6) killing animals, and (7) odour. He succeeded in drawing
electricity from the clouds by means of a kite having a pointed wire
attached to it. The kite was held by ordinary packthread, having
a key at the end to which a silk cord was fastened. Having tied the
silk to a tree, he held his hand to the key ; but at first he was unable
to obtain any result. A storm of rain, however, came on, which,
wetting the string, made it a good conductor, and he then obtained
sparks in sufficient quantity to charge a Leyden jar.

Since Franklin's famous experiment, numerous observations have
proved that the atmosphere is constantly in a state of electrification,
even during fine weather.

In such observations, it is usual to measure the difference of
potential between a point in the air and the earth ; or, preferably,
to obtain a series of such values for points at different heights and
vertically above each other.

If, for instance, the base of an electroscope is earth-connected as
usual, and the cap is connected by a conductor to a point (say) two
yards above the earth's surface, then, if the end of the conductor can
be made to acquire the potential of the air at that point, a deflection
will be produced when any P.D. exists between the point and the
earth; and as the leaves and conductor form a connected system
necessarily at the same potential, the magnitude of the deflection will
be a measure of the P.D. in question. The practical difficulty arises
from the fact that air is a non-conductor, for the end of the conductor
can only change in potential by giving up or by receiving a charge,
and we have seen that such an interchange does not readily occur
between a conductor and a non-conductor. The earliest observers,
some 150 years ago, used an electroscope to which a long upright
pointed rod was attached (Fig. 68), but this application of the
properties of pointed conductors is not satisfactory, for the equalisa-
tion of potentials at the tip takes place slowly, and there is never
any certainty that it is complete.

102

~

ATMOSPHERIC ELECTRICITY

103

Yolta attached a burning match or torch to the top of the rod, and
many later observers have used similar methods. This was a great
improvement, but it is somewhat inconvenient, and, moreover, there
is a certain small P.D. produced due to the combustion itself.

Lord Kelvin's water-dropping arrangement is very satisfactory, and
is now almost universally used. This is shown in Fig. 69 (which is
taken from Carey Foster and Porter's Electricity and Magnetism). A
vessel containing water is well insulated by sup-
porting it on a glass rod rising from the centre
of a vessel containing strong sulphuric acid (to
keep the rod dry). To this vessel is attached
a long tube extending through an aperture in
in the wall of the room into the open air, and
from the end of this tube the water slowly drops
away. The water-vessel and rod may be re-
garded as being under induction from the air
surrounding the jet; if the air is positively
charged, the jet will become negative and the
vessel inside positive, although of course both

FIG. 68.

FIG. 69.

are at the same potential, but the fact that the jet is negatively
charged shows that this potential is less than that of the adjacent air.
As the water drips away, each drop carries with it a small negative
charge, and in this way, in a very short time, the potentials are
equalised, so that an electroscope1 whose cap is connected to the
water- vessel will correctly indicate the P.D. between the air around
the jet and the earth.

1 For the purpose of exact measurement an electrometer is used.

104 ELECTROSTATICS

Very simple apparatus may be used to demonstrate the existence
of large differences of potential between the earth and points
above it.

Exp. 72. Select a tinned iron can of moderate size. Close to the bottom
make a small hole through which may be inserted the end of a narrow brass
tube about 6 feet in length. Let this project inside the can, and solder it to the
bottom. Also solder round the hole. Fill up with water, and partially close
the end of the tube with a hammer until the water slowly trickles from it.
Take this to the highest room in a building, and place it on a cake of paraffin-
wax so that the tube projects as far as possible through a window (or if
practicable, take it outside and work in the open air at the top of the building).
Connect the can to the cap of an electroscope, and then fill up with water. As a
rule, with an instrument of average sensitiveness a large deflection will be
obtained very quickly. Test the sign. In all probability it will be positive.

Such observations show that in fine weather the atmosphere is, as a
general rule, positive to the earth, i.e. the earth is negatively charged, the
lines of electric force extending vertically upwards and terminating
somewhere in positive charges. Where they terminate is a question
still under investigation. The potential increases rapidly with vertical
height for a considerable distance ; at still higher elevation with less
rapidity. The rate of increase of potential with height is an important
quantity known as the potential gradient. It is a very variable
quantity, the average value being something like 200 volts per metre
of vertical height above open ground. It may, however, greatly
exceed this value, and there is good reason for believing that the
average is different in various parts of the earth.

During wet weather, the potential gradient is often reversed in
sign, the earth being positive to the air, but, taking the whole year
into account, such reversals are for relatively short periods.

Surface Density Of the Earth.— Assuming that the electrification
of the air is entirely due to a uniform negative charge possessed by the earth,
we can estimate its surface density as follows : —

If p is the surface density, the field close to the earth's surface is given by

F = 47r/) (p. 46).

Also, if V is the P. D. between the earth and a point at a height of h centimetres
above it (the curvature of the earth being neglected), we have
V = UA, alsoF = KU,
and as K = 1 for air

If we take V as 300 volts per metre (for convenience),
then V=l static unit

and /o — - Tno^r s*ia^c units of charge per square centimetre.

This calculation assumes a uniform field ; if it is not uniform, we have
- - = lirp, where — — is the potential gradient.

ATMOSPHERIC ELECTRICITY 105

It is, however, very improbable that the electric field in the atmosphere is
entirely due to a charge situated upon the earth. The evidence suggests very
strongly that the air itself is often locally charged, and that as masses of air
sweep past the place of observation they carry their charges with them.

Sources of Atmospheric Electrification — The actual sources

of atmospheric electrification are still unknown, and the question is rendered
more difficult by the fact that the earth is always rapidly losing its charge.

Comparatively recent research has established the fact that the air is not a
perfect insulator, for charged bodies, even when thoroughly insulated, always
slowly lose their charge. For a long time this was attributed to the inevitable
leakage over the supports due to imperfect insulation, but it is now definitely
known to be due to a small but true conductivity of the air itself. Recently,
C. T. R. Wilson, using an instrument devised by himself, measured the rate of
loss of charge in fine weather, and found that the average value was 8 per cent,
per minute in the open air. This means that the total charge on the earth's
surface is lost (and renewed) every 12 or 13 minutes.

As it has been known for some time that rain-drops are frequently (but not
always) electrically charged— the sign of the charge being supposed to be usually
negative — Wilson suggested that districts experiencing fine weather are losing
negative electrification, and at the same time other districts are gaining negative
from falling rain, equilibrium being restored by earth currents below, and above
by currents at a considerable height in the atmosphere. This suggestion has so
far been neither confirmed nor disproved, but some additional evidence has been
obtained at Simla in India by Dr. G. C. Simpson. He invented and used an
apparatus which gave a practically continuous record of the amount of the
charge during rain, and its sign. He obtained the unexpected result that
positive electrification was in excess. During 37 per cent, of the total time of
rain, no charge \vas registered ; and when the rain was charged, the observations
showed that about three-quarters of the total amount recorded was positive.

Dr. Chree has, however, pointed out that there may be some local influence at
work, and that observations in other parts of the world are required before
any general conclusions can be drawn. It appears that at Simla rain chiefly
falls during thunderstorms, and that the excess of positive electrification was
especially pronounced during such storms.

How the rain becomes charged is another question, which still awaits a
definite answer. In this connection , a research made by Lenard, about twenty years
ago, may be mentioned. He found that, when water fell upon a solid and
broke into spray (e.g. upon the rocks and stones at the base of the waterfall),
the drops became positive, while the air around acquired a negative charge.
He also showed that when a drop of water (or of mercury) splashed on a metal
plate, the same effect was produced. A slight alteration in the purity of the
water, however, modified the result obtained ; exceptionally pure water showed
the effect very strongly, while it was almost imperceptible in the case of water
slightly less pure. On the other hand, a weak salt solution gave an opposite
effect, the air becoming positive instead of negative.

Sir J. J. Thomson found that minute traces of certain substances in the
water produced surprisingly large results, but that strong solutions were
practically inert, giving little or no electrical effect by splashing.

Lenard came to the conclusion that no electrical separation occurred when
water broke up into drops without splashing against some obstacle, a conclusion
which appears to need modification in the light of some recent experiments
made by Dr. Simpson, who broke up falling water into drops by means of a
vertical air jet, and found no perceptible action in the case of Simla tap water,
although there was a strong effect when distilled water was used, the water
becoming positive.

Now, rain-drops may be broken up in the atmosphere by air currents, and
lience it is possible that their charge may be accounted for in some such way.

106 ELECTROSTATICS

It is not easy to explain the frequent presence of a negative charge, although
Dr. Simpson suggests that it may be directly derived from air which has been
previously negatively charged by breaking drops.

Elster and Geitel have suggested that the electrification of the air may be
due to the action of ultra-violet rays from the sun upon the upper strata of the
atmosphere (see p. 613).

Such an action may be of two kinds : (1) the rays may dissipate negative
charges from ice crystals or dust in the atmosphere ; in fact, dry ice is known to
act in this way ; (2) the rays may simply ionise the air. Probably both effects
are produced, but it is difficult to estimate their real importance.

The slight conductivity of the air already mentioned indicates that it is
ionised to a very small extent, and the presence of these ions may greatly
influence the formation of water particles from vapour.1

The facts of surface tension show that there is a great difficulty in starting
condensation, and very small drops, even if formed, would tend to evaporate
rapidly. As a matter of fact, condensation appears always to commence around
some nucleus, such as a dust particle, which provides a finite radius to begin
with. Now, the effect of surface tension appears to be purely mechanical ; like
a tightly stretched skin, it increases the pressure inside the drop, and it has
been found by C. T. R. Wilson that this pressure may be compensated by the
opposite outward pull due to an electric charge, or rather to the lines of force
which extend outwards from it. As a result, ions serve as nuclei on which
moisture may condense, the negative ions being far more effective than positive
ones. Hence, Sir J. J. Thomson has suggested that such condensation may
produce a separation of electric charges in ionised air, the negative ions
being selected as nuclei in preference to positive, and brought down as negatively
charged rain.

In whatever way the water particles acquire their charge, a cloud may be
jegarded as a movable conductor, and, if it is charged, it may approach by
electrostatic attraction other clouds or the earth. If the charge is initially on
the individual particles, and then becomes gradually transferred to the cloud
as a whole, i.e. to its external surface, the potential may rise enormously, for
the external surface is very much less than the total surface of the whole of the
particles, and the final result may be a spark between two clouds, or between a
cloud and the earth.

Similarly, it is easy to see that the potential must rise if small charged
drops coalesce to form larger ones.

For suppose that n globules, each of radius r, in falling unite to form one
globule, then the radius (rx) of this globule can be obtained by the following
method : —

The mass of a sphere = volume x density

4?rr3 T
= --xrf

Now, the mass of the sphere of radius rl is n times that of a sphere of radius r,
whence L =

1 We may remark that the ionisation is probably due to the generally
diffused presence of exceedingly small amounts of radio-active matter. This
matter as it breaks up, liberates a and /3 rays, which are the direct cause of the
ionisatioii. See Chapter XXIX.

ATMOSPHERIC ELECTRICITY 107

If, for example, n = 8, the radius of- the larger globule will be twice that of each
of the eight, but the charge is'eight times as great. Now

and, therefore, its -potential is four times greater than that of the constituent
drops.

Thus the potential of a cloud may increase enormously by the coalescence of
the minute particles, but when dealing with such calculations it must be
remembered that they ignore the possible influence of neighbouring particles on
the result.

Lightning. — These discharges are flashes of lightning, of which
three kinds have been distinguished: (I) forked lightning, which is
no doubt due to the character of the air through which the discharge
occurs, certain portions offering greater resistance than others ; in
fact, its path is the one of least resistance. (2) Sheet lightning,
which is probably due to the illumination of the cloud where the
flash occurs, or is merely the reflection on the clouds of a distant
discharge. It is called heat or summer lightning, when the whole
horizon is illuminated by flashes so far distant that the thunder is not
heard. (3) Another form — known as globular or ball lightning — has
not yet been explained. It is said to consist of balls of fire, which
travel slowly, or even remain stationary, and then explode with
sudden violence. Fortunately, its occurrence is very infrequent.

Thunder is the report which follows the discharge. It is
probably due . to the f acfr that the lightning heats the air in its path,
producing sudden expansion and compression, which is followed by
an extremely rapid rush of air into the rarefied space. A thunder
clap is produced when the path of the discharge is short and straight.
The rattle is produced when the path is long and zigzag. Rumbling,
or rolling, is produced by the echoes among the clouds.

As the velocity of light is about 186,000 miles per second, we
may consider it as practically instantaneous ; and as that of sound,
at the ordinary atmospheric temperature, is 1120 feet per second, the
distance of a discharge can be ascertained by multiplying 1120 by
the number of seconds which elapse between the lightning and the
first sound of the thunder.

Lightning" Conductors consist of three essential parts : (1) a
rod, usually of copper or of galvanized iron, elevated above the
highest portion of a building and terminated at the top in a fine
point; (2) a conductor, connecting this rod with the ground, often
made of a stout copper ribbon; (3) the earth connection, which is
of very great importance. The lower end of the conductor should
terminate in several branches passing into a well or at any rate into
moist earth. Of course, the continuity of the conductor is of the
utmost importance.

The protective action of a lightning conductor is very simple ; for
suppose that a charged cloud passes over a building provided with

108 ELECTROSTATICS

a conductor; induction is set up, but the induced electrification
accumulates on, and is then discharged from, the raised point, and
so tends to neutralise the charge of the cloud. By this means a
disruptive discharge is frequently prevented. Such protective action
has been illustrated in Experiments 63-65, pp. 83, 84. But lightning
discharges are often oscillatory and of very high frequency.1 The
impedance of the conductor may then become enormous, and the
discharge strikes off the conductor with disastrous effect upon neigh
bouring bodies. This " choking " effect, due to impedance, is usefully
applied in order to protect electric cars supplied from overhead wires.
A small coil consisting of a few turns of thick copper wire (often
termed a "kicking" coil) is placed in the circuit between the trolley
wire and the motors, and an alternative path provided through a
spark gap to earth. The coil offers no appreciable resistance to the
direct current which drives the motors, whereas the spark gap offers
an infinite resistance. On the other hand, to oscillatory lightning
discharges of high frequency, the impedance of the coil acts very
much like an infinite resistance, while, as the potential is high, the
discharge readily sparks across the gap.

It is by no means a simple problem to protect a building from
risk of damage by lightning.

The late Professor Clerk-Maxwell suggested that a building
should be covered with a network of metallic wires, so that the
building forms, as it were, the interior of the conductor.

Sir Oliver Lodge condemns the use of a rod passing above the
highest point of a building, and suggests that the conductor should
consist of a number of lengths of common telegraph-wire connected
with large masses of metal, e.g. leaden roofs. The connection must
be thorough, and made at many points. Barbed wire should be
run round the eaves and ridges of the roof so that many points
are exposed. Balconies and other accessible places should not be
connected.

The Aurora, is a luminous phenomenon occurring chiefly in high
latitudes and depending upon the electrical condition of the atmosphere. If it
occurs in northern latitudes it is known as aurora borealis, or northern liyhts,
while in southern latitudes it is called aurora australis.

In the arctic regions the aurora occurs almost nightly, and it occasionally
extends over very large areas. The light assumes various forms and colours ;
e.g. it sometimes appears merely as pale and flickering streaks, occasionally
tinged with various colours, passing from the horizon towards the north
magnetic pole.

Usually the light is silvery or greenish, less frequently a red glow is seen.
The spectrum of this light invariably shows a yellowish green line, which is
highly characteristic of a true aurora. Sometimes fainter blue lines are seen,
and occasionally red ones. For a long time the origin of the chief auroral line
was a profound mystery, for it did not correspond to any known element, but

1 These terms are explained in Chapters XXVI. and XXXV.

ATMOSPHERIC ELECTRICITY 109

it is now fairly certain that it is due to krypton, one of the inert gases in the
atmosphere, discovered by Sir Wm. Ramsay. It is difficult to say why this
gas should appear so prominently. At the earth's surface it is not present in
relatively great quantity, ordinary air containing about one part in a million
by volume ; less than the amount of either hydrogen, helium, or neon normally
present, and very much less than the amount of argon (about 9000 parts in a
million). We may, however, point out that the percentage composition of the
atmosphere at extremely high altitudes is certain to differ widely from that
obtaining at the sea-level.

Possibly one of the red lines occasionally seen is due to neon, but this is by
no means certain.

Arrhenius in 1900 worked out a very important theory of the aurora, in
which he attributed the phenomenon to the presence of cathode rays, i.e.
electrons, shot off from the sun, and which under the influence of the magnetic
field of the earth (by an effect identical in nature with that described on
p. 486), tend to rotate round the lines of force and to drift towards the poles.
There is very strong evidence that some such action originating in the sun
is involved. Aurora are always most plentiful at the time of sun-spot maxima,
and are invariably accompanied by "magnetic storms" (p. 185).

As far as the visible phenomena are concerned, they are, no doubt, analogous
in many ways to vacuum discharges, occurring in the rarefied, and therefore
semi-conducting upper strata of the atmosphere. _ At the same time, observations
are on record, which seem to indicate that occasionally such visible effects occur
quite close to the earth's surface.

Note. — The student must not expect to find any direct connection
between the phenomena dealt with in Electrostatics, and those
to be discussed in the next section. Magnetism is, for all practical
purposes, a new subject. It is, therefore,. -desirable to point out that
the main object of the first two sections of the book is to develop
the ideas embodied in the terms "lines of electric force," and "lines
of magnetic force," respectively, while such lines are in a state of rest.
The nature of the relation which exists between them will become
apparent, when in the third and most important section — Voltaic,
or Current Electricity — we apply such ideas to phenomena de-
pending upon the coexistence of both kinds of field, while in a state
of relative motion.

MAGNETISM

CHAPTER IX

MAGNETIC ATTKACTION AND REPULSION
Lodestones or Natural Magnets. — Exp. 73. Plunge a piece of

lodestone into iron filings ; on withdrawal notice that tufts of filings cling to

certain parts. In Fig. 70 the piece has been ,

shaped with a hammer so that this attractive

power is most apparent near the ends.

Exp. 74. Suspend the piece of lodestone so

that it can turn freely. Either of the following

methods of suspension may be adopted. Fasten FlG. 70.

a thread of raw silk to a suitable support, and (a)

tie the other end to a piece of wire bent as shown in Fig. 71, or (.6) pass it

through the two free ends of a doubled strip of paper. We may call (a) a wire
stirrup, and (b) a paper stirrup. Observe that the lode-"
stone sets itself in a definite direction, pointing nearly
north and south. If disturbed from this position it
oscillates for a time, and then- comes to rest in exactly
the same position as before, the same end always point-
ing towards the north.

This hard, dark-coloured, stone-like body is
widely distributed in nature, being met with in
great abundance in Norway and Sweden, and in
some parts of America. From the circumstance
that it was originally found in Magnesia, in Asia
Minor, it was probably called magnes by the Greeks, whence we have
derived our words magnet, magnetism, &c. Although this substance
does not always possess the properties of attracting iron and of setting
itself in a north and south direction, it constitutes one of our most
important and valuable iron ores, known as magnetite, or magnetic
oxide of iron. Its chemical formula is Fe3O4. It is called a natural
magnet because it is found in a natural state, and lodestone (A.S.
loedan, to lead) because it has the remarkable property referred
to in Experiment 74, which caused it to be subsequently used in
navigation.

Artificial Magnets.— Exp. 75. Draw a piece of lodestone fifteen or
twenty times over an ordinary steel knitting-needle, or a piece of watch-spring,

FIG. 71.

112

MAGNETISM

FIG. 72.

taking care to move it always in the same direction, not to and fro.1 (a) Plunge
it into iron filings, and observe that, on withdrawal, tufts similar to those

obtained in Experiment 73,
cling to the ends ; (b) suspend
it in a paper stirrup, and ob-
serve that it sets itself in a
north and south direction.

We learn from these
experiments that a new
property, which manifests itself in several ways, has been imparted to
the steel needle by rubbing it with a lodestone. Such a piece of steel
is an artificial magnet. The process by which this property is acquired
is called magnetisation, and the steel is said to be magnetised.

Attraction of Iron by Magnets.— Exp. ?e. Obtain a number of

small, soft iron nails, as nearly as possible of the same size and weight. (1) Near
the end, a, Fig. 73, of a strong magnet,

support the greatest possible number a ~k c d, e

of the nails.

(2) Hang the nails at points 6 and
d, nearer the middle of the magnet.
Observe that, as* we approach the
middle, a smaller number can be sup-
ported, while (3) at the middle, e, even
a small particle of iron cannot be
supported.

Test the other half of the magnet
in a similar manner, and observe that
equal weights are supported at equal
distances from each end.

The curve drawn through the
free ends of each series is a rough
measure of the attractive force at different points along the magnet,

We therefore learn from this experiment that —

(a) The maximum attractive force of a magnet is situated in
somewhat indefinite areas near the ends. In certain cases, however,
especially when the length of a magnet is great compared with its
thickness, it is permissible to regard these areas as points. Such
parts are called the poles of the magnet. Later, the student will
realise that a magnet pole is a region from which "lines of magnetic
force " pass into the surrounding medium.

(b) As we approach the middle of -a magnet, the attractive power
becomes smaller until (c) all round the magnet, midway between the
poles, it ceases altogether. This is called the neutral line.

The line joining the poles is called the magnetic axis.

Poles of a Magnet. — We have seen that, in our latitude, one
pole of a magnet always points northwards, and the other southwards.
From this property the poles are distinguished one from the other by

Fia

1 This method is of no practical value,
magnets will be described later.

The ordinary methods of making

FIG. 74.

MAGNETIC ATTRACTION AND REPULSION 113

calling that which is directed towards the north, the north pole, while
the opposite one is called the south pole.

Magnetisation by Single Touch.— Exp. 77. Place a strip of

steel on a table. Bring one pole of a magnet in contact with one end of the
strip (Fig. 74). Move the magnet,
parallel to its first position, to the

other end, then lift it and replace IIIIIIH ~*\ \

it in its first position. After rub-
bing one side ten or twelve times,

turn the strip over, and treat the IIIHIIIIIIIIIB •

other side similarly.

The polarity of the end of
the bar, where the magnet
leaves it, is always of opposite
name to the magnetising pole. Thus, if a N pole be used, the end
where the magnet leaves the bar is S. and that where it is first
placed, N.

Magnetisation by Separate or Divided Touch.— Exp. 78.

Place the bar to be magnetised horizontally, and then place the opposite poles
of two bar magnets at the middle of the bar as shown in Fig. 75. Draw them
simultaneously from the middle to the ends. Lift them and place them again

at the middle. Repeat
this operation ten or
twelve times. Turn
the bar over and rub
the other side in a
similar manner.

~^n fg-' This process is

' rendered easier and

FlG- 75' more effectual if the

bar to be magnetised is supported on the opposite poles of two bar
magnets, so that the poles of the lower magnets are similar to those of
the magnetising magnets immediately above them (Fig. 75).

These methods were important before the discovery of electro-
magnetism. They are now only occasionally used, as steel bars may
be much more readily and strongly magnetised by electrical methods.

To make a Magnetic Needle. — («) Cut a piece of clock-spring,

with a pair of shears used for cutting metal, into either of the shapes shown
in Fig. 76. Magnetise it by the method of single touch. Balance it accurately
by placing it across a knife-edge, and then scratch a line to mark the position
of the knife.

(b) Make a glass cap in the following manner —

Take a piece of glass tubing (T\-inch bore is the most useful), and holding
it in a Bunsen's or spirit-lamp flame, turn it continually until it is quite soft,
and then pull the ends apart so that it has the appearance shown in Fig. 77, a.
Break the thin thread, and hold one piece in the flame until the end is
rounded off (Fig. 77, 6). It is often necessary to remove the bead which forms
on the end with another piece of hot glass-tubing. When the glass is cold,

H

114

MAGNETISM

make a mark (represented by the dotted line in the diagram) with a sharp
triangular file. The rounded portion, which forms the cap, can then be easily
separated from the rest of the tube by gentle pressure.

(c) Soften the central part of the strip of magnetised steel, by holding it in
a flame until it is red-hot, and then yraduully removing it from the flame so
that it cools slowly. Drill a hole through the needle at the middle of the line
about which it balanced, taking care that it is slightly smaller than the diameter
of the cap. After boring, the strip must be hardened by again making it red-
hot, and then suddenly plunging it in cold water. Now place the needle- upon
a small sheet of red-hot iron, when it will first turn yellow, and then gradually
blue. When this occurs, slide it off the iron into cold water. This is called
tempering.

(d) Fasten the cap, with a trace of glue, in the hole, so that it is perpendicular
to the needle, and then put it aside to dry.

(c) Make a support by gluing a cork to the centre of a board (6 in. x 3 in.

FIG. 76.

FIG. 77.

FIG. 78.

x^ in.), and then passing the eye of a fine sewing-needle into the cork. Take
care that the needle is quite vertical (Fig. 78).

(/) Bore a hole, sufficiently large to admit the glass cap, in the base of the
support. Place the needle on the board, with the cap in the hole, and re-
magnetise it. This is best done by separate touch.

Exp. 79. Place the needle on its support, and notice that it sets itself in a
north and south direction.1

Action of Magnetic Poles on each Other.— Exp. so.

Suspend a magnet in a wire stirrup. Mark the end which points northwards
with a piece of gummed paper. (1) Bring the marked end of the magnet near
the N pole of the needle. Observe that repulsion takes place. Therefore, two
N poles repel one another. (2) Bring the marked end near the S pole of the
needle. Attraction ensues. Therefore, a N and a S pole attract. (3) Repeat
these experiments with the unmarked end.

When performing this experiment, the magnet pole must be gradually
brought near the needle from a distance, for when a fairly strong magnet is

1 In these experiments care must be taken that the needle is removed from
the influence of other magnets and of pieces of iron.

MAGNETIC ATTRACTION AND REPULSION 115

brought quickly up to one end of tho needle, attraction may take place although
the poles are alike, and it will then be found that the polarity of the needle has
been reversed.

These results enable us to state the first law of magnetism — like
poles repel one another, unlike poles attract.

Exp 81, to find if a given piece of iron or steel is magnetised. Bring the
specimen to be tested towards one end of a compass needle, and observe whether
attraction or repulsion is produced. If repulsion, then the specimen is certainly
magnetised ; if attraction, try the effect of the other end on the same end of the
needle. If the effect is still attraction, the specimen is not magnetised, for either
end of the compass needle will be attracted by unmagnetised iron.

In performing this experiment, care must be taken to observe the effects at
the greatest possible distance ; for if a weakly magnetised piece of iron is
gradually brought near the compass needle, it will often be found that repulsion
occurs when the distance between the two like poles is considerable, and that
this changes to attraction when the distance is small. Hence, a rapid approach
may cause the initial repulsion to be overlooked. In this case, the effect is the
converse of that described above — the pole of the compass needle reversing the
polarity of the iron.

Position of Poles. — Exp 82. Place a magnet on a sheet of paper,
arid trace its outline with a pencil. Place a short compass needle near one end
of the magnet, and mark with dots the positions of the ends of the needle.
Repeat this for several positions of the compass-needle around the end of the
magnet. Remove the magnet and draw lines through each pair of points, pro-
ducing them until they intersect. Observe that the intersections do not occur

n a single point, but that they cover a small area, whose approximate centre
may be regarded as the position of the pole. This point may be a centimetre
or more from the end of the magnet.

The positions of the poles of three magnets were found by this method with

;he following results : —

Length.

Breadth.

Distance of Pole
from End.

31 '3 centimetres.

3 centimetres.

1*7 centimetre.

15-85
107

1-8 „

2 ,,

1-15

0-5

These results are important, because in numerical calculations the axis or
"length" of a magnet is the distance between its poles and not its total length.
This definition holds good with magnets of any 'shape, e.g. in the horseshoe
form the distance between the poles may be quite small, although the total
length is considerable.

The Earth a Magnet. — The direction which a horizontally
suspended magnet takes, is due to the fact that the earth itself is a
huge magnet, having its magnetic poles comparatively near the
geographical poles. From the law just enunciated, " like poles repel,
unlike attract," we can easily perceive that, as we have defined the

116

MAGNETISM

pole, which points north, the N pole, the earth's pole, situated

near the geographical north pole, is a S
pole.

An approximate representation of the
magnetic condition of the earth can be
made by placing a bar magnet within a
wooden globe, so that the centre of the
magnet coincides with the centre of the
globe, its S pole being about 15° to the
west of the point which represents the
geographical north pole.1 This will be
understood by reference to Fig. 79, in
which EQ represents the geographical
equator, and EQ' the magnetic equator.

The Earth's Action on a Magnet is merely Directive.

— Exp. 83. Pass each end of a magnetised needle through a small piece of cork,
and float on water. Observe that the N pole turns towards the north magnetic
pole of the earth, but that there is no movement towards the side of the vessel.

The size of the earth is enormously great as compared with that
of an artificial magnet, so that the attractive force exerted by the
earth's north magnetic pole upon the magnet's N pole, is equal and
opposite to the repulsive force exerted on the S pole ; and the at-
tractive force exerted by the south magnetic pole of the earth upon
the S pole of the magnet is equal and opposite to the repulsive force
exerted upon the N pole. The total effect of these forces upon the
poles of the magnet is therefore equivalent to a couple, one force
acting towards the north magnetic pole of the earth, and the other
towards the south magnetic pole. This will be more easily under-
stood when the properties of lines of force have been explained (see
Chapter XL).

Magnetic Meridian. — The magnetic meridian of any place
is sometimes defined as the plane drawn through the zenith (the
point in the heavens immediately overhead), and the magnetic north
and south points of the horizon ; it may be more accurately and
simply defined as the vertical plane in which lies the axis of a freely
suspended magnetic needle at rest.2

Exp. 84. Suspend a magnetised knitting-needle by means of a fibre of raw
silk, and allow it to rest just above a table. Mark the position of the two ends
of the needle. Remove it, and draw a line joining these two points. This line
is the intersection of two planes, viz. the magnetic meridian and the surface of
the table.

No Isolated Poles. — Exp. 85. Harden, but do not temper, a piece
of watch-spring. Magnetise the hardened steel by single touch, and then show

1 Approximately true in 1914.

2 The geographical meridian of any place is the plane passing through the
zenith, and through the geographical north and south poles.

MAGNETIC ATTRACTION AND REPULSION 117

that one end contains a N pole, and the other a S pole. Mark the former with
gummed paper. Break the newly made magnet into two pieces, and prove that
each piece is a perfect magnet. In fact, we shall find that (1) we obtain complete
and perfect magnets if the magnetised strip of steel be broken into any number
of pieces (Fig. 80), and (2) it is impossible to obtain a magnet with one pole only.

This fact, taken in conjunction with others which cannot be
discussed at present, points to the conclusion that each ultimate
particle of iron is naturally a perfect magnet of exceedingly small
dimensions. Under ordinary circumstances, we may suppose that
these atomic magnets arrange themselves under the influence of their
own mutual forces in such a way that their resultant action is zero
as far as external bodies are concerned.1 We may further suppose

that the process of magnetisation is equivalent to compelling some
of them to take up an ordered direction, with like poles pointing in
the same direction. We need not think that all the particles take
up such a configuration ; if only a small percentage do so, the iron
will behave as a feeble magnet. As a matter of fact, when a bar of
steel is treated as in Experiment 77, only the surface layers are usually
magnetised, as may be demonstrated by dissolving them off in acid.
Neither are we obliged to think of the particles as arranged exactly
end to end. If that were the case, only the ends of a bar would
exhibit magnetic properties, the sides throughout being neutral. It
must be remembered that in addition to the endwise attractions of
"unlike" poles along a chain of particles, there must exist a lateral
repulsion between adjacent chains
due to "like" poles, which, towards
the ends, will cause the chains to
widen out, some ending on the
sides, as shown diagrammatically
in Fig. 81. It is easy to see that
such lateral repulsion assists the
formation of shorter and more
nearly closed chains of particles
(never of longer and more open
chains), the final result of gradual rearrangements being the equivalent
to a tendency of the ends A and B, Fig. 81, to approach each other,

1 In elementary text-books, the atomic magnets in unmagnetised iron are
often said to be mixed up in general confusion. Such a statement completely
overlooks the existence of mutual forces between them, and, as Ewing has
shown, the particles really arrange themselves in little groups, more or less.
ring-shaped, which are stable under ordinary conditions,

AI

A

N

1 B

>*«

»wj**""

"'i,^''

" --^

x /• "

,"'

*NV >x>^

FIG. 81.

118

MAGNETISM

and similarly A3 and Bj, and so on. In pure iron this takes place
with great readiness, the particles settling down into the stable
configurations already mentioned.

In steel magnets, this action is resisted by something akin to
friction, due to the presence of carbon, tfec., and hence
we see that a magnet with free poles always has a
natural tendency to demagnetise itself. This action
is often called "the demagnetising effect of the poles."
When there are no free poles, e.g. when the particles
can form closed chains entirely in iron, it practically
disappears. It is for this reason that bar magnets,
when not in use, are arranged as shown in Fig. 82,
opposite poles being joined by short bars of soft iron,
A and B, known as keepers.

Our argument necessarily implies that the ultimate
particles of a solid piece of iron possess some freedom
of rotation — a fact strongly supported by other branches

N

N

FIG. 82.

\j Exp. 86. Partially till a small test-tube with steel filings. (Steel is required,
because we have found that once magnetised it retains its magnetism. It will
be shown later that this is not the case with soft iron). Holding the tube
horizontally, magnetise it by single touch. Observe that the filings set them-
selves end to end, having their longest directions parallel to the length of the
tube (Fig, 83). (a) Without disturbing the arrangement, bring the tube near

FIG. 83.

a horizontally suspended magnetic needle, and notice that one end of the tube
attracts one pole of the needle and repels the other. \Ve, therefore, conclude
that the filings form a magnet, (b) Now disturb the arrangement by shaking
the tube, and test again. Observe that no repulsion occurs, i.e. the filings as a
whole have lost their magnetism, although each individual filing may be as
strongly magnetised as before.

Why each particle of iron acts as a magnet, whereas the particles
of other elementary bodies, as a rule, do not (except with great
feebleness in a few cases), is a question of some difficulty.

It will be shown later that magnetism can be produced by an
electric current, and Ampere suggested that- each atom of iron may
have such a current circulating round it. This hypothesis, which
affords an adequate explanation, and which, in a modified form, is
supported by later discoveries, will be referred to subsequently.

MAGNETIC ATTRACTION AND REPULSION 119

Magnetic. Battery.— If a number of magnets, either bar or horse
shoe, be iised, having their similar poles adjacent, they form what is known as
a magnetic buttery.

Fig. 84 represents such a battery, in which there are twelve magnets

FIG. 84.

arranged in three sets, each set consisting of four magnets. Their similar poles
are bound together by pieces of soft iron, A and B.

In this way, a magnet is produced of much greater power than could be
obtained by using the same mass of steel in the solid form. For, as already
stated on p. 117, it is difficult to magnetise a single thick bar equally and
strongly throughout the whole of its thickness, whereas thin strips can be
readily magnetised and then built up into a compound bar. The principle of
this method of construction is often used in instruments where magnets are
required, e.g. in Ewing's hysteresis tester (p. 415), and in Lord Kelvin's compass
(p. 191), &c.

Consequent Poles. — Owing to irregular and imperfect mag-
netisation, a magnet sometimes contains more than two poles. In
such a case, the bar really consists of several magnets placed end to

FIG. 85.

end, having their similar poles together at intermediate points, as
shown in Fig. 85. The extra poles are called intermediate poles,
consecutive poles, or consequent poles.

The presence of consequent poles in a magnet is generally due
to accidental causes ; they may, however, be produced at will by
several methods, of which the following is one :—

Exp. 87. Place like poles of two bar magnets at the middle of a strip of
steel, draw them simultaneously to the ends, lift, and place them again at the
middle. Repeat this operation a few times, and then show, by placing the
strip in iron filings, that there are three poles — one near each end, and the
other of double strength at the middle.

Magnetic Substances. — Mutual magnetic attraction does not
take place between magnets and all bodies. Those substances which
have the property of attracting and of being attracted by a magnet,
are called magnetic bodies. Besides iron and steel, the following
bodies are recognised as magnetic — cobalt, nickel, chromium, man-
ganese, and cerium. Of the latter, cobalt and nickel are the best,
but even they are distinctly inferior to iron or steel in this respect.
Many other bodies, e.g. paper, porcelain, oxygen, and certain salts of
iron, are feebly attracted by very powerful magnets. This subject is
further discussed in Chapter XXV.

CHAPTER X

INDUCTION

Induced Magnetism. — Exp. SS. Place a piece of soft iron either in
contact with or near one pole of a magnet. Bring iron filings to the lower end

of the iron, and observe that they cling in
a tuft (Fig. 86). Kemove the magnet, the
filings immediately fall.

The magnetism thus communi-
cated to the iron is called induced
magnetism; the magnet communi-
cating it is called the inducing
8g magnet; the action is known as

magnetic induction.

Nature Of Induced Polarity. — Exp. 89. Cover a horizontally
suspended magnetic needle with a beaker.

'(a) Place a magnet, N (Fig. 87), in such a position that its N pole does not
appreciably repel the N pole of the needle.

i

FIG. 87.

(b) Bring a soft iron rod, R, between the magnet and the needle. Observe
that the N pole of the needle is immediately repelled.

We infer, therefore, from this experiment, that the end of the rod
near the needle acquires N polarity under the influence of the magnet,
the other end becoming, of course, S — i.e. a magnetic pole induce*
opposite polarity in the end of an iron rod near to it, and similar
polarity in the end remote from it.

Exp. 90. Repeat the last experiment with two or three smaller iron rods
between the needle and the magnet, and observe the repulsion.

We, therefore, learn that the inductive influence takes place
through a series of iron rods,

120

INDUCTION 121

Exp. 91. Obtain a number of small, wrought (soft) iron nails. Support
one on the end of a strong magnet. Place
another on the free end of this, and so on.
This forms what is commonly called magnetic
chain (Fig. 88).

As will be understood from the
figure, the N pole of the magnet in-
duces S magnetism in the point of
the first nail, and N in the head ; this
again induces S in the point of the
next, and so on through the whole
series. FIG. 88.

Exp. 92. Show, by using a magnetic needle, that tb.e polarity of the free end
of the series is similar to that of the inducing pole.

Coercive Force and Retentivity.— There is a marked differ-
ence between steel and iron, with regard to (1) the difficulty of
magnetisation, and (2) the retention of magnetisation. This differ-
ence may be easily shown by the following experiments : —

Exp. 93. Form a magnetic chain with pieces of well wrought (i.e. very soft)
iron. Remove the magnet from the uppermost piece, and observe that the
others fall away. Test one of the pieces by bringing it near both poles of
a magnetic needle. There is no repulsion, and, therefore, the pieces of iron
were temporarily magnetised.1

Exp. 94. Now form a chain with pieces of steel (e.g. steel pens). "When the
magnet is removed from the uppermost piece, the others do not drop off, i.e.
steel is said to retain its magnetism permanently.

Exp. 95. Break a steel knitting-needle (four or five inches long) at the middle.
Raise both pieces to a white heat. Plunge one in cold water to harden it, and
allow the other to cool slowly in order to keep it soft. Dip both pieces in iron
filings, and then bring a magnet in contact with the other ends. Notice that on
withdrawal, the mass of filings attached to the hard iron is smaller than that
attached to the soft iron. Remove the magnet, and observe that most of the
filings adhere to the hard piece, but that they drop off the soft piece.

The student will notice that, under similar conditions, soft iron
becomes more strongly magnetised than steel, but that it loses its!
magnetism more readily when disturbed or subjected to rough usage/
It is usual to denote these properties by the terms Coercive Force and
Retentivity, although they will not convey any real meaning to the
student until he has read Chapter XXV. ; in the meantime, it will be
sufficient to define coercive force as a measure of the power of the
material to resist changes of magnetisation ; e.g. steel has a much j
greater coercive force than soft iron. Retentivity is a measure of the
power of retaining magnetism, when the material is quite undis-
turbed and free from vibration or shock, and it may be greater for
soft iron than for steel. It must also be borne in mind that any

1 Generally there will be a slight action, a.s the iron would retain some
magnetism, however feeble,

122 MAGNETISM

piece of iron, once magnetised, cannot be completely brought back to
its original condition unless it is heated above redness. Otherwise
it always retains a small amount of what is termed " residual
magnetism."

Induction precedes Attraction. — We are now in a position
to explain more clearly why mutual attraction takes place between

a magnet and a magnetic substance. In
Fig. 89 a small piece of soft iron is
suspended by a thread. When the N
pole of a magnet approaches the iron,
induction is set up, the near side of the
iron acquiring S polarity and the remote
side, N. Attraction, therefore, takes

place between the two opposite polarities

- . , fr.. r .

and repulsion between similar polarities.

The distance between the two opposite poles is, however, less than
that between the two similar poles, and hence the attractive force
overcomes the repulsive force.

Influence Of Medium. — Exp. 96. Magnetise a knitting-needle (two
or three inches long), and suspend it horizontally by a fibre of raw silk. If
moved from its position of rest, it will make a certain number of oscillations in
a fixed time — say one minute. If, however, the N pole of a magnet be brought
towards the S pole of the needle after the latter has been moved from rest, it will
make a greater number of oscillations than before. Count the number of oscilla-
tions made in one minute, when a large sheet of glass, cardboard, a wooden
board, or even the body is interposed between the needle and the magnet. The
distance between the two magnets being constant in each case, observe that the
number of oscillations are equal.

Exp. 97. Now interpose a large sheet of soft iron. Observe that the number
of oscillations made by the needle in the same time is less than that in the last
experiment. If the iron were quite soft and very thick, the number would
approximate to that obtained when the needle oscillated under the earth's
influence alone.

It appears, therefore, that magnetic force acts across all media,
except iron and the other magnetic substances, and that they (or
rather the ether which surrounds the molecules of the medium),
directly transmit the force from one point to another.

The influence of the medium can be better shown by means of
a reflecting magnetometer (see Experiment 110, p. 149).

Magnetisation by an Electric Current.— This method,
although exceedingly important, will be merely mentioned here,
as a knowledge of voltaic electricity is necessary before it can be
rightly understood.

Exp. 98. "Wind a spiral coil of copper wire round a bar of soft iron. The
turns of wire must neither touch each other nor the bar. A good plan is to
paste a layer of brown paper round the bar, and then, when it is dry, to wind
cotton-covered wire upon it.

INDUCTION

123

Connect both ends of the spiral to the terminals or a voltaic battery. Bring
a piece of iron or steel to the bar, and observe that it is attracted and supported.

Magnetisation by Electro-magnets.— In the last experi-
ment we made and used an electro-magnet, which merely consists
of a core of soft iron, often of horse-shoe
shape, round which a coil of insulated
copper wire is wound. As we have
learnt, the core is magnetised during
the passage of an electric current round
the coil. This produces very powerful
magnets, and on this account they are
frequently used to magnetise bars of
steel.

Exp. 99. Hold an electro -magnet upright,
or, as is the common practice with magnet-
makers, fix it in a board (Fig. 90). During
the passage of the current, (1) move a steel
bar from end to end across one pole of the FIG. 90

electro-magnet ; (2) move it in the opposite

direction across the other pole. The reason of these movements will be under-
stood from previous explanations.

I/ LOSS of Magnetisation. — Magnetisation may be destroyed or
weakened under the following circumstances : —

(1) By arranging magnets, when not in use, with their similar
poles adjacent. The tendency of each pole is to induce opposite
polarity in the other, which, of course, weakens or destroys the
original polarity (see p. 118).

(2) By the earth's induction. The tendency of the earth is to
induce N magnetism in the lower end of the vertical or nearly vertical
bar, so that if a magnet is placed with its S pole downwards, its
polarity is weakened.

(3) By rough usage, either wilful or accidental. Such treatment,
no doubt, disturbs the molecular arrangement described on p. 117.

(4) By making a magnet red-hot (see p. 423). If it be only
slightly heated, it is weakened, but most of its original strength is
regained on cooling.

Exp. 100, to illustrate (4). Suspend a long thin iron wire — about 15 inches of
20 gauge — by a bifilar suspension, i.e. by two cotton threads, so that it niay
swing end on (Fig 91). Support a magnet NS, on a suitable stand, B, and
adjust the distance so that the wire, when it is swung forward, just adheres to
the magnet. Place a small sheet of mica (which may be kept in place by a
weight, W), on the end of the magnet, so that actual contact between the
magnet and the wire is prevented. Now, place a Bunsen flame between the
magnet and the wire, and notice that the latter oscillates. This is due to
the fact that the end becomes non-magnetic in the flame, and the wire therefore
falls away, although it cools sufficiently to become magnetic before the return

124

MAGNETISM

swing, and is again attracted. It does not merely lose its magnetism ; the
iron becomes actually a non-magnetic substance above a certain temperature.
It becomes a magnetic substance again when it cools, but is completely
demagnetised.

Effects Of Magnetisation. — (1) When an iron or steel bar is
magnetised it becomes very slightly longer. This increment is very small, for

even when a "bar is magnetised to its maximum, it merely amounts to

/ 20, 000

of its original length.

This observation is due to Joule, but later experiments of Bidwell have shown

FIG. 91.

that the effect is somewhat complicated. Iron at first elongates, and then, with
increasing magnetisation, contracts again until it is shorter than its original
length. Cobalt lirst shortens and then lengthens, whereas nickel always shortens
o.nd never lengthens.

(2) Heat is produced when a bar is rapidly magnetised and demagnetised,
apparently pointing to the conclusion that friction is set up between the molecules
of the bar during magnetisation. Another explanation is given on p. 416.

(3) During magnetisation, a twisted iron bar tends to untwist itself.

CHAPTER XI

MAGNETIC FIELDS

Magnetic Field and Lines of Force. — The space sur-
rounding a magnet through which its influence extends is called the
magnetic field of that magnet. At every point in the field the mag-
netic force has a definite strength depending upon the distance from
the poles ; and it has a well-defined direction at every point, as
indicated by what is called the line of force passing through the point.1

Exp. 101. Place a sheet of cardboard on a magnet. Sprinkle iron filings
from a muslin bag over the cardboard. Gently tap the cardboard as the filings
fall, and observe their arrangement along certain curves (Fig. 92.)

FIG. 92. — Arrangement about a Single Bar Magnet.

Each particle of iron assumes a definite direction, due to the action
of both poles of the magnet. Tapping the cardboard merely facilitates
the arrangement of the particles. The curves thus obtained' may be
taken to represent what are known as magnetic lines of force.

Exp. 102. Obtain the curves with the magnets arranged in various positions,
e.g. as shown in Figs. 93 to 98.

These lines of force can be obtained in an interesting manner by

1 The actual direction of the magnetic force at any point is a tangent to the
line of force at that point.

125

126

MAGNETISM

means of a compass-needle pivoted in a small circular case (about f of
an inch in diameter).

FlG. 93. — Arrangement about two Parallel Bar Magnets
with their Dissimilar Poles adjacent.

FlG. 94. — Arrangement about two Parallel Bar Magnets
with their Simif-ar Pofes adjacent.

FlG. 95.— Arrangement about a Horseshoe Magnet.

MAGNETIC FIELDS

127

Exp. 103- Place a bar magnet on a large shoot of drawing-paper fastened on
a board. Rule lines round the magnet to show its position. Place a small
compass-needle near one corner— say, the S pole of the magnet— and make pencil

FIG. 96. — Arrangement about the two Dissimilar Poles of
two Bar Magnets.

FIG. 97. — Arrangement about the two Similar Poles of
two Bar Magnets. -

FIG. 98. — Arrangement about One Pole of a Bar Magnet.

dots exactly opposite the poles of the needle. Move it so that its N pole is on
the S pole dot, and make another dot opposite its S pole. Contimie in this way
until a series of dots is obtained, and then join them by a freehand curve. This
curve represents a line of force. In a similar manner other curves can be obtained
until the whole held is mapped out.

128 MAGNETISM

Neutral Points.— If the student has a sufficiently large sheet
of paper, he will find that, -when the needle is removed from the
neighbourhood of the magnet, the lines traced out by the compass
needle are parallel. Such are the lines of force due to the earth.
He will also notice that, in certain parts of the field, the lines of force
due to the earth and those due to the magnet form irregular lozenge-
shaped figures (Figs. 99-101), in which the compass-needle sets itself

FIG. 99.

indifferently in any direction. These points are called neutral points
or points of zero force, because the force due to the earth and that
due to the magnet exactly balance each other. For an explanation,
see pp. 132, 133.

Lines of Force through Magnetic Substances.— When

the lines of force pass through a magnetic substance (p. 119), they
crowd into the substance, e.g. if a soft iron ring be placed near one

MAGNETIC FIELDS

129

pole of a bar magnet, the lines of force arrange themselves as
shown in Fig. 102. These lines were obtained in a manner similar
to that described in Experiment 103. It will be observed that there
are no lines of force inside the ring, i.e. a thick piece of iron acts as a
magnetic screen. If, therefore, a compass-needle be placed inside
a space enclosed by sufficiently thick iron, it is practically unin-
fluenced by the magnetic field in which it is placed.

N

FIG. 100.

Properties of Lines Of Force. — There are two fundamentally
distinct methods of dealing with magnetic problems. The first, and
oldest, was naturally suggested by the phenomena of attraction and
repulsion exhibited by magnetic poles ; it concentrates the student's
attention more particularly on the poles themselves, which for purposes
of mathematical treatment, although often in defiance of actual facts,
have to be regarded as points ; it seeks to determine the laws accord-
ing to which such poles act on each other and to deduce their con-

I

130

MAGNETISM

sequences. It does these things quite correctly, but it leaves
untouched the vital question : How can bodies act upon each other
without any tangible connection ? The method is sound in principle,
and is, for some purposes, the more useful, but it is only conveniently
applicable to a limited class of problems.

The second method was initiated by Faraday, who clearly saw that
the most important machinery of magnetism, from a practical point
of view, lay in the space outside the magnet itself, and that the poles
were more or less incidental circumstances. By introducing the idea
of " lines of magnetic force " he showed how the properties of that
space could be represented in a useful form. Clerk-Maxwell and
others developed the method, which has become the basis of nearly
all calculations in electrical engineering.

The patterns framed around magnets by iron filings have in them-

FIG. 101.

selves no special significance, and are easily explainable ; the im-
portance of the idea lies in perceiving that the facts of magnetism
can be accurately represented by assuming that a magnet carries with
it a number of "lines of force," which possess certain definite pro-
perties— the iron-filing figures merely enabling us to visualise their
distribution (over one plane) in simple cases.

Their most important characteristics may be summarised as
follows : —

1. The poles of a magnet are those regions where the lines of
force emerge from the iron. (From this point of view, a pole is the
analogue of a static charge, but whereas the lines of electric force
terminate in a charge, we must suppose that the lines of magnetic
force extend through the iron.)

MAGNETIC FIELDS

131

2. Lines of force never cut one another ; for if they did, the small
magnet, used in tracing the lines, would tend to set itself in two
directions at once at the point of intersection, which is impossible.
Two lines from independent sources merge together to form a re-
sultant line, which has one definite direction.

3. Lines of magnetic force always form closed curves.

4. Lines of magnetic force tend to contract in length, and to
repel each other laterally.

5. A piece of iron, when placed in a magnetic field, becomes
magnetised, and tends (a) to set its longer axis parallel to the lines
of force, (b) to move into the strongest part of the field.

6. As shown in Fig. 102, iron (or other magnetic substance) causes

FIG. 102.

the lines of force to deviate from their normal path through air (or
other non-magnetic substance). We may express this in a simple
way by saying that the lines of force tend to pass through iron,
wherever possible.

As we shall frequently discuss the properties of lines of force
without reference to the poles from whence they emerge, it is necessary
to adopt some convention to indicate their direction. The accepted
custom is to mark them with arrows pointing in the direction in
which a single free N pole would tend to move. It must be under-
stood that these arrows do not suggest any motion of the line, and
in fact do not have any physical meaning whatever. They are
simply convenient aids to thought.

As an illustration, consider Fig. 99, p. 128. It represents certain
facts, but no explanation has been given why the lines of force should

132

MAGNETISM

have that particular configuration. Suppose, however, that we re-
present the field of the earth by a series of uniformly spaced
straight lines. Then assuming the magnetic meridian to run verti-
cally across the page, and remembering that a free N pole would tend
to move northwards, we see that the direction of the earth's field must
be marked with arrows pointing north, as in Fig. 103. On this set
of lines is superposed another due to a bar magnet placed in the
meridian with its N pole towards the north. If we draw a few of

N

> 1 j

t

1

4

.,

^

\ S

^ •+*

X

A

N

'>

X

X

/

1

\
\

\
\

1

1

1

\

»

V

1
1

i

i i

/

f

^*

-

S

•

^

*

/ \

y

_>

-~

i

--

—

*c

**

1

i

i

I

1

S

FIG. 103.

its lines and mark them according to rule (remembering that a free N
pole would tend to move from a N pole towards a S pole), we get the
result shown. From this we see that both sets of lines of force are in
the same direction at all points on the axis of the magnet produced,
so that along this line the field is stronger than before, whereas at all
points on a line drawn through the centre of the magnet and at right
angles to its axis, the two fields are in opposite directions, and there-
fore weaken each other. But whilst the strength of the earth's field
is uniform, that of the magnet is variable, being stronger near the
magnet than that of the earth, and rapidly becoming weaker as the
distance from the magnet increases. Hence there will be some point

MAGNETIC FIELDS 133

at which the two fields are equal and opposite, and the resultant zero.
These points are indicated by the small enclosed areas in Fig. 99.

Evidently, if the magnet were to be turned round, the direction
of its lines would be reversed, which would make the two fields agree
in direction at right angles to the axis, and interfere along the axis.
This case is shown in Fig. 100, the neutral points being now on the
axis of the magnet.

The student should draw a similar diagram to represent the state
of affairs shown in Fig. 101, and he will then find the cause of the
peculiar distribution obtained with that arrangement.

Strength of a Magnetic Field. — It is convenient to represent
the strength of a field by the number of lines of force passing through
a given area, i.e. the closer the lines are together, the stronger is the
field. This method is defined quantitatively on p. 136.

It must, however, be borne in mind that this method of expressing
numerically the strength of a magnetic field docs not imply that the
magnetism only exists along certain lines and that in the space
between the lines there is no magnetism. The essential unseen forces
are continuous throughout the field, and the lines are only diagram-
matical devices for indicating their direction and magnitude. These
remarks also apply to electric fields.

For instance, the strength of the 'earth's magnetic field at
London is -47 lines per square centimetre, which simply means that
to represent it in terms of our units, we must draw, in the plane
of the moridian, a set of parallel equidistant lines inclined at an angle
of (about) 67° to the horizontal, and so spaced that there is one line
to about 2 square centimetres of surface. Everywhere in this field,
a unit magnetic pole would be subjected to a force of '47 dyne
tending to urge it in a direction parallel' to the lines. If its con-
straints allow it to move only horizontally, it behaves as if acted
upon by a horizontal force of '18 dyne, and if only vertically, as if
acted upon by a vertical force of '44 dyne.

These numbers are known as the "horizontal" and "vertical"
components of the magnetic force (or field) ; see p. 175.

Another conception, that of ."tubes of force," may be used in
order to fill up the total space (see p. 28), and this alternative
method is usually discussed in text-books. We think, however, that it
is less useful to beginners, and, therefore, we have not introduced it.

EXERCISE VII

1. Explain what is meant by a line of magnetic force.

Describe some experimental method of tracing such a line, and explain the
principles upon which the method depends. (Camb. Local, Senior, 1900.)

2. A bar magnet is placed in the meridian with its north pole pointing
that there are in general two points, on the line perpendicular

north. Show

134 MAGNETISM

to the meridian passing through the centre of the magnet, at which the iield
due to the magnet is equal and opposite to the horizontal field of the earth.

(Loud. Univ. Matric., 1907.)

3. A bar magnet is placed on a table. Describe IIOAV you would proceed to
determine the directions of the lines of magnetic force in the plane of the table
due to the magnet and to the earth's magnetism. (B. of E., 1901.)

4. A short bar magnet is placed at the centre of a circle three feet in
diameter, its axis being in the magnetic meridian. Trace the changes in the
direction in which a compass-needle points as it is carried round the circum-
ference of the circle. (B. of E., 1902.)

5. A magnet is placed horizontally in the magnetic meridian due south of
a compass-needle. How will the action on the latter be affected if (1) a plate
of soft iron is interposed between the two, (2) a rod of soft iron is placed along
the line which joins the centres ? Give reasons, (B. of E., 1891.)

CHAPTER XII

MAGNETIC MEASUREMENTS

ANY measurements of the forces between magnetic poles are neces-
sarily complicated by the fact that we cannot isolate two poles for
the purpose. Two magnets, each having two poles, must be used,
and the actual effect depends upon all four poles. The simplest way
to overcome this difficulty is to use long magnets, so that the two
disturbing poles are removed some distance away, thus making
their influence sufficiently small to be neglected in a rough experi-
ment.

Coulomb adopted this plan in the earliest quantitative measure-
ments of magnetic forces. He used the torsion balance method
(already explained on p. 23 in connection with his work on electric
forces. A long thin magnet was suspended horizontally from the
torsion head by means of a fine wire, and another similar magnet
replaced /, Fig. 22).

Without discussing the details of such experiments, it will be
sufficient to say that Coulomb found that the attraction or repulsion
between two small magnet poles varied directly as the product of
their strengths, when the distance between them was constant ; and
inversely as the square of the distance between them, when the pole
strengths were constant. Hence, if in and mx are the pole strengths,
and d the distance between them, Coulomb's law may be written in
the form

Attraction or repulsion oc — _!

It will be at once noticed that this expression resembles that
already given on p. 24 for electric charges, and that in this case,
limitations similar to those mentioned on that page are involved,
e.g. the law applies only when the poles can be treated as points
in comparison with the distance between them.

From this result, it follows that unit magnetic pole may be defined
in the same manner as unit charge (although it must not be supposed
that this is due to any similarity in their nature). Hence, a pole
is of unit strength, when it attracts or repels an equal pole, placed
at a distance from it of 1 centimetre in air, with a force of 1 dyne
(both poles being mere points).

135

136 MAGNETISM

In accordance with this definition we may write

Attraction (or repulsion) = -- ; — i dynes

When discussing the corresponding expression for electric charges,
it was stated that the attraction or repulsion depended also upon the
nature of the surrounding medium, whereas all ordinary substances
(except iron and a few other markedly magnetic bodies) behave
practically alike with regard to lines of magnetic force, and the
attractions or repulsions are, therefore, independent of the nature
of the surrounding medium. But when later, it becomes necessary
to study more fully the magnetic properties of iron, it will be found
that we have to distinguish carefully between the ideas conveyed by
the terms magnetising or magnetic force and magnetic field, just as
we had to distinguish between the ideas electric force and electric
field. Definitions similar to those given in statical electricity will
apply to magnetism, e.g. magnetising or magnetic force will be
numerically the mechanical force in dynes on a unit pole placed in
a magnetic field ; and the magnetic field itself will be measured by
the number of lines of force per square centimetre. The relation
between these two quantities is written —

Strength of Magnetic Field = /* x Magnetising Force,

where //., called the " permeability " of the medium, is practically
unity for all substances, except markedly magnetic ones. By common
agreement, it has become usual to put B for strength of fidd, and H
for magnetising force, so that we have

Evidently, in air, B and H are numerically identical. The importance
of the preceding statements will not be realised until the properties
of iron are discussed (see Chapter XXV.).

Further, the reasoning already given for electric charges on p. 26
may be applied to magnetic poles by merely writing " m " in place
of " q" We then find that the complete form of the law discovered
by Coulomb is

/ , . v n x mm,

Attraction (or repulsion) = —

^irpd*

and, therefore, our definition of magnetic pole implies that unit pole
has 4?r lines of force emerging from it, and that //, is unity for air.
Similarly it may be deduced that the field strength at distance d

from a point pole of strength m units is -- lines per square centimetre,
and that the force on unit pole is -m., dynes, Hence, it follows that

MAGNETIC MEASUREMENTS 137

the force «in dynes on unit pole is, in all practical cases, numerically
identical with field strength. For this reason, confusion often arises
as to the terms themselves. For instance, B and H are both vector
quantities, and in non-magnetisable substances they always coincide
in direction (although that is not always the case inside magnetic
substances). But this numerical identity does not mean that B and H
are identical in nature, and where necessary we shall point out to which
of the two ideas reference is made.

The letter H is also used to denote -the horizontal component of
the earth's magnetic force at a given place, i.e. the horizontal component
of the force in dynes on a unit pole at that place (see Chapter XIII.),
and in the following pages we shall frequently use it in this sense, as
there is no real danger of confusion arising in consequence of such
usage. Hence, //,H is, strictly speaking, the horizontal component of
the earth's field.

Some writers use the term "Gauss" to indicate strength of field,
e.tj. a field of 50 lines per square centimetre is termed a field of
strength 50 gausses. This term, although often used, has not
received unanimous international sanction, and we do not think that
it possesses any advantage over the word " line," which is universally
adopted in practical calculations.

The Inverse-square Law. — Coulomb's method requires con-
siderable experimental skill in order to obtain even roughly consistent
results, and it is, therefore, unsuitable for ordinary purposes. By
the nature of the case, direct experimental proof is difficult, because
it is impossible to satisfy strictly the required conditions.

It is shown later that the best proof, and one capable of great
accuracy, is an indirect one.

A Magnetic Balance. — Any experimental method, which is
capable of yielding approximately correct results, is, however, instruc-
tive, and Mr. W. Hibbert has devised a form of magnetic balance,1
which is very useful for this purpose (as well as for many others).

We have used a much less perfect form of balance, which involves
the same principle and which is easily constructed as follows : Two
long magnets are required, each ^ inch in diameter and about 1 yard
in length. One of them is fastened with copper wire to the edge of
a metre scale (the wire passing through small holes drilled in it).
The scale is then pivoted on a wooden stand by means of a pin passing
through £ hole at the middle — the arrangement being a modification
of the one commonly employed for demonstrating the properties of
levers. Two stops, A and B, Fig. 104, are fixed in some convenient
position in order to limit the play of the lever, and to facilitate the
adjustment ; a small weight, W (from 2 to 5 grams), is suspended by
a fine thread on one arm ; and one end of the second long magnet is
brought underneath the end of the first, as at S.

1 See Magnetism, by "W. Hibbert, published by Messrs. Longmans.

138

MAGNETISM

Exp. 104. Place the second magnet either endwise on, or at riglft angles to
the first, as may be the more convenient, but take care to ensure that the points,
which may be regarded as the effective poles, are in the same vertical line. This
can be done by adjusting the position of the second magnet until the best effect
is obtained.

The points in question will probably be an inch or more from the ends.

When this is done, note the position of the sliding weight giving balance for
as many distances between the two poles as is practicable. (It is not desirable
to attempt adjusting these distances to any definite value — it is far better to take
them as they come and to measure the distances as exactly as possible after
getting balance. )

Now, if p is the force between the magnet poles ; ?x the distance

A

o

, T

.U / >l

1 ij

T L \

|iii

1 1 1 1 1 1 1 1 1 1 1 I | 1 1 1

N^l 1 1 1 ! 1 1 ! 1 1 1 f^| 1 1 1 1 I 1

B

Jf J ^

d

J

6s

/

_ — v

FIG 104.

of the pole from the fulcrum ; W, mass of weight in grams ; and ?, its
distance from the fulcrum ; then, taking moments about the fulcrum,

>x = Wx981 xl

or =

Wx9SlxZ

,
dynes.

Hence, the force between the magnet poles is directly measured for
some distance d between them, the other two poles being so far away
as to exert no serious disturbing action.
. Again, we notice that W and ^ are constant, and, therefore,

peel. Hence", if p oc — , we have I oc — -, or Id2 = a constant. A
a- - d*

rough test might be applied by multiplying together these values, but
for the purpose of determining the law connecting the force with the
distance, it is more convenient to proceed as follows :—

Assume I oc _ , where n is unknown,

then ldn = constant.
Taking logs, we have

log / + n log d = another constant.

MAGNETIC MEASUREMENTS

139

If the values of log I be taken as ordinates, and those of log d as
abscissae, and plotted on squared paper, the result will be a straight
line as shown in Fig. 105, and we know
by elementary co-ordinate geometry that,
in the above equation, n is the tangent of

the angle of slope — i.e. ~^ = n-

Hence, if the experimental results give
an approximately straight line, we may
infer that the force varies inversely as some
power of the distance, and by measuring '
Ao and oB, we can determine the prob-
able value of that power.

The student will readily obtain such a
line, but it is very doubtful if he will find

•W f)

oB~

It is much more probable that

log d

FIG. 105.

he will find that the ratio lies between
1 and 2. This arises from the fact that the poles are not points, and,
therefore, the inverse-square law does not hold good over the short
distances used in the experiment.

Magnets. — When the ends of a long thin magnet are
fitted with spherical steel balls, the poles are much more definitely localised, and
may be taken without much error as being at the centres of the balls. If such
ball-ended magnets are used, a much closer approximation to the inverse-square
law will be obtained. This form of magnet was devised many years ago by
Robison, but its peculiar properties have only recently been brought into notice
by Dr. G. F. C. Searle of Cambridge.

Other experiments bearing on the inverse-square law are given on
pp. 150 and 158.

Determination of the Field Strength at a given Point
near a Bar Magnet, taking the Effect of both Poles into

Account. — Consider any point o
(Fig. 106), at distances d and dl from
the two poles of a magnet respec-
j -^ I tively. Then, if m be the strength

of each pole, the field at o due to

the N pole will be — , and will act
ct

along No, and the field due to the
S pole will be — , and will act along

N

Js

FIG. 106.

oS (these directions being taken in

accordance with our convention, i.e. the direction in which a free
N pole would tend to move). The actual field at o will be the

140 MAGNETISM

resultant of these two components, but it is not necessary here to
show how an expression for it can be obtained. When, however, the
point o is on the axis of the magnet, or when it is on the line at
right angles to the magnet and passing through its centre, it is
easy to determine the resultant field, and it will be sufficient to
consider these two cases.

CASE I. — When the point is on the axis of the magnet produced.

Let o (Fig. ' 107) be the
Point» NS the magnet, of

< ,.fl__ 3~» which the length is 21 centi-

F t metres, and the strength of

each pole in units ; d, the
distance between the point o and the middle of the magnet ;

then oN = d— I

and oS — df-f- ?

Now the field at o due to N —

" =

We must at this point introduce a new and important definition —

The moment of a magnet is the product of the strength of
one of its poles : and the distance between them — i.e. the moment (M)
of the magnet = m x 21 (21 being the length of the magnet).

.J. the resultant field at o = ,2M'^

Now, if half the length (/) of the magnet be very ainall compared
with d, I'2 may be neglected without making any appreciable differ-
ence, so that we, then, have

the resultant field at o = = —
d* d3

Hence, the field strength at any point on the axis, whose distance is
great compared with the length of the magnet, varies directly as its

MAGNETIC MEASUREMENTS

141

K Zl--

moment (which is a perfectly definite quantity) and inversely as the
rule of its distance from the centre of the magnet.

If the point is fairly near the magnet, the unsimplified form of
the expression must be used.

CASE II. — When the point is on the straight line at right angles
to the magnet and passing through its
centre.

In this case, the poles are equi-
distant from the point o (Fig. 108).
Let d} be this distance, so that
tl^ ** °N = 0S. Each pole produces at

the point o, a field of magnitude -- ,

rfj

acting respectively along No produced
and 0S.

As the two fields are equal, their
resultant must act along the line bi-
secting the angle between them, i.e.
along oil.

Draw Stf perpendicular to oK,
then by the properties of the triangle
of vectors, the component along oil
due to the S pole is found by writing

FIG. 108,

Component along
Field along 0S_

_ oC _

;. Component due to S pole along 0E, = ~— cos 9

"i"

To this must be added the component due to the N pole acting
along No produced, which evidently has the same magnitude and
direction.

9 Tn
.'. Resultant field at o = cos 0

Again, the figure shows that cos 0 = —

dl

-cv , , 2m I 2ml

.-. Field at 0 = -— x -r = -7-7T
di di di

but the moment (M) of the magnet is 2ml
..-. Field at 0 = ^

Hence, the field at o varies inversely as the cube of the slant dis-
tance from either pole.

142

MAGNETISM

If d be the direct distance from o to the centre of the magnet,
then, when d is great compared with /, we may write dl = d for a
first approximation. For points too close to the magnet to satisfy
this condition, we must write dl =

M

Field at o =

Experimental methods of verifying these results are given later
(see pp. 149 and 159).

Deflection of a Compass-Needle by a Magnetic Field.—

A freely suspended, or pivoted, magnet, e.g. a compass-needle, when
left to itself, comes to rest in the magnetic meridian under the
influence of the horizontal component of the earth's magnetic field
(see p. 175). Let H denote the strength of this field.1 If another
entirely distinct magnetic field inclined to that of the earth be
produced by any means, e.g. by a magnet or by an electric current,
then the two fields merge into a resultant field inclined to the direc-
tion of both.

Consider the simple case in which the second field is at right
angles to the direction of H.

Let F be the strength of this second field, assumed to be uniform
throughout the space occupied by the compass-needle. Then as F
and H are both vectors, their resultant is given in magnitude

and direction by oR in Fig.
109, and the needle will be
deflected through an angle 6
to point along oR. It is at
once evident from the figure
that

F = Htan d

W

FIG. 109.

and that the angle of deflection
is independent of the strength
of the needle. This gives a
method of measuring F when
H is known.

We have regarded F and
H as field strengths, but because p = 1 for air, this expression holds
good, and is the same numerically, if we consider F and H to be the
forces acting on unit pole.

Application to Previous Theorems. — I. Let a magnet of

moment M be placed at right angles to the magnetic meridian.
Consider any point o on its axis produced, and at distance d from
its centre (d being great in comparison with the length of the
magnet). Then the field at o is at right angles to the horizontal

1 Strictly, the field is /tH.

MAGNETIC MEASUREMENTS 143

component of the earth's field. At this point, let a short compass-
needle be placed (if the needle is a long one, it will extend into
regions off the axis* to which our results do not apply). The needle
will be deflected from the meridian through some angle 0, such that
F = H tan 0, where F is the field due to the magnet.

2M
Now it has been shown that F = -—

This is known as the end-on position, or the A position of
Gauss.

II. Let the point o be taken on the line at right angles to the
magnet and passing through its centre at a relatively great distance
d from its centre.

The field due to the magnet at this point is also at right angles
to the earth's field, and we have, by a repetition of the above
argument, ™ H^0

c/3

This is known as the broadside -on position, or the B position
of Gauss.

As before, in these expressions', the numerical values are the'
same whether we regard H as field or forced

In order to make use of these results, some form of small compass-
needle, arranged to facilitate exact measurements of 6, is required.

Such an instrument is known as a magnetometer. A simple form
may be made as follows : —

To make a Magnetometer. — (i) The box, A (Fig. no), is made by

gluing together four strips of
wood — 4J inches long, and
1| inches high. The sides are
then glued to the bottom,
which consists of a square piece
of looking-glass. Small cubes
of wood should be glued in the
top corners of the box, so that
a square of window-glass may
rest upon them to form the
cover of the box.

(2) Glue a small flat cork, FIG. 110.

B, at the centre of the looking-

glass, and insert a fine needle, point upwards, into it. The needle must be
fixed with great accuracy at the centre of the box.

1 It is instructive to notice that if we write -^ = fjJR tan 0, then each of the

terms, ^, uH, means " field "; and if we write — == = H tan 0, each means " force.
3 *

144 MAGNETISM

(3) Cut a piece of watch-spring to form a small noeclle (about 1\5 centimetres
long) of the shape shown in Fig. 111. Drill a small hole through the centre of
the needle. Make and fix a small glass cap in the hole, so that it is perpen-
dicular to the needle, as explained on p. 114.
Glue a fine pointer at right angles to the needle
— this may be made of any light rigid body ;
those used by the writer are very tine glass
fibres, made by heating glass rod and then
drawing it out.

(4) Make a graduated scale as follows : Con-
struct a circle of 2-inch radius on paper, and
divide the circumference into 1 -degree spaces.
Remove £he central part of the paper, so that a
ring, a $ inch wide, is left. Glue this carefully
to the bottom of the box. (A printed scale is
readily obtainable and should be used if possible.)

(5) Take a piece of wood, 4 feet long, 2|
.inches wide, and about £ inch thick, and cut

a square groove at the middle to hold the box, A
p, ,,, (Fig 110). The outstanding portions, F, may be

called the arms.

,. (6) Gum a strip of paper on each arm, and then graduate them in centimetres,
making the zero under the centre of the box, and graduating outwards.

The great disadvantage of this type of instrument is due to the
friction at the pivot, which makes it relatively insensitive. It is not
difficult, however, to modify the design in such a way that the needle
is suspended by a silk fibre. In fact, a galvanometer in which such
a needle is used, e.f/. a tangent galvanometer, or the simple form
described on p. 282, makes an excellent magnetometer for many pur-
poses of demonstration.

Exp. 105, to magnetise two pieces of steel to the same pole strength. Cut two
pieces of clock-spring, each piece being, say, 8-5- centimetres long. Thoroughly
harden them, and, placing them side to side, magnetise them together. If this
is done carefully the pole strength of the two magnets will be equal. To test
them, arrange the magnetometer so that, when the pointers are at zero, the arms
are in the magnetic meridian. Place one of the magnetised pieces across one
arm, so that its centre is on the middle line. Read the angles at both ends of
the pointer — suppose they are 10|° and 11°. Reverse the poles of the magnet,
and repeat these observations — suppose the angles are 11° and 11|°. Take the
mean of the four readings — in this case 11° — which gives the true deflection.

Repeat these operations with the other magnet. If the mean of the four
readings is the same as before, the magnets are of equal pole strength.

If they are found to vary, magnetise the weaker one until the deflections are
equal.

Exp. 106, to prove that the force exerted by a bar magnet docs not depend
merely upon its pole strength, but also upon its length, i. e. the force is proportional
to the magnetic moment of the magnet.

(1) In the last experiment, we found that the mean of the four deflections
for each magnet was 11°.

(2) Now place the two magnets end to end, with their opposite poles together
— of course, the distance being the same as in (1). Again take the mean of the
four readings. This, in an actual experiment with the two pieces of magnetised
steel, was 21°.

Both the deflections in (1) and (2) are produced by magnets of the same

MAGNETIC MEASUREMENTS

145

polo strength ; the greater deflection in (2) must therefore he produced hy the
greater length of the magnet.

Exp. 107, to find the. moment of a magnet by the A position of Gauss.
Arrange the magnetometer so that the arms are at right angles to the meridian,
and the pointer at zero.

(a) Place a short magnet on the arm which lies towards the east, and observe
the exact distance (which must be great compared with half the length of the
magnet) between the middle of the magnet and the point of suspension of the
needle.

(1) Let the N pole lie towards the needle, and read the deflections at both
ends.

(2) Reverse the magnet so that the S pole lies towards the needle, and
again read the deflections.

(b) Now place the magnet, at the same distance as before, on the arm which
lies towards the west. Repeat (1) and (2).

(c) Take the mean of the eight readings. This gives the true deflection.

(if) Repeat the eight observations at a different distance, and then apply the
formula given on p. 143.

H.d'tanS
2

In a particular experiment the following results were given with"
a magnet 15 centimetres long : —

Distance between
Centres.

Position of
Magnet.

Deflections.

Mean
Deflection.

Natural Tan-
gent of Mean
Deflection.

Value of M.

-38 centimetres -'

El

K2
AV 1
W 2

24 i 25
23" 22;
21|- 22

2U- 25

I oq<

J

•4348

2147

35 centimeti'es *

E 1

&

W2

30 30
31 301

26£ 27
30" 30

- 29-375

•5628

2171

Experiment 107 must be regarded as giving only an approximate
value. In the first place, d is only about five times greater than /,
and the use of the simplified formula introduces an error of about
10 per cent. In the second place, it assumes that the value of H is
known. It will be shown later how this may be measured, but if, in
the meantime, the accepted value for the open air be used, a serious
error may be introduced, for sometimes the value is widely different
inside a building from that outside, and indeed it often varies at
different points in the same room. This is due to the presence of
iron used in the construction of the building.

It is, however, easy to compare the moments of two magnets with-
out knowing the value of H. Either the A or the B position may
be used, although the former is mentioned in the next experiment.

K

14G

MAGNETISM

Exp. 108. Arrange the magnetometer for the A position of Gauss.

(a) Take the mean of the eight readings mentioned in Experiment 107, with
one magnet, which we will call A. Suppose that it is 20° 30'.

(b) Take the mean of the eight readings with the other magnet, which we will
call B, with its centre in exactly the same position. Suppose that it is 8° 15'.

Moment of A_tan 20° 30' _ -374
Moment of B ~ tan 8°15'~ :145

2'6
= — nearly.

Then

Another method of comparison may also be indicated : —

Exp. 109. Place the two magnets, whose moments are to be compared, on
opposite sides of the magnetometer — in either A or B position — and adjust their
distances until the deflection is zero.

Evidently the fields due to the two magnets are equal and opposite, and if
d and dl be the distances corresponding to the moments M and MI respectively,
we have OM 9M

for A position 2-£ = ~*

Uf' Cv-i

or, for B position M = Ml
d3 d^
which, for both positions, becomes

(It will be understood that, as the distances have to be^ubed, a small experi-
mental error will cause a relatively large error in the final result.) On p. 16f>
another important method of comparing the moments of two magnets is explained
by means of a worked example.

Reflecting" Magnetometer. — An enormous gain in sensitive-
ness and accuracy is obtained by applying the reflecting principle to

magnetometers, and such
instruments are always
used in actual practice.
Probably the most use-
ful form for teaching
purposes is one similar
to that shown in Fig.
112 (for which we are
indebted to Messrs. J. J.
Griffin & Co.). A small
concave mirror of about
1 metre radius of cur-
vature and f-inch dia-
meter is suspended by a
silk fibre in a strip of
wood ab, slotted out to
receive it, the height
being adjusted by a brass

FIG. 112.
upper end of the fibre is attached.

screw d, to which the
The cavity in which the mirror

MAGNETIC MEASUREMENTS

147

Fro. 113.

swings is closed by a sheet of glass, and is quite small — only slightly
wider than the mirror itself, and not deep enough to allow the latter
to turn completely round. In this way (a) the fibre is kept from
twisting, and (/>) the oscillations of the mirror are partly checked or
" damped" by air friction, so that it comes to rest more quickly.

It is usual, as shown in Fig. 113, to attach to the back of the
mirror one or more very small and light magnets made from watch-
spring. This construction works very well, but such
short strips tend to lose their magnetism readily, for
reasons explained on p. 117. Longer strips could be
magnetised more strongly, and would retain their magne-
tism better, but it is especially desirable to have the poles
close together, so that the arrangement may be regarded
as a point. Both conditions are satisfied by using the
horse-shoe type, a small and nearly circular magnet being made
from a fine knitting-needle, and magnetised by passing a current
through a temporary winding of copper wire. This is attached to the
back of the mirror, as shown in Fig. 114.1

The whole arrangement is fixed to a solid wooden
block, W (Fig. 112), provided with levelling screws, and
carrying at the back a small clamp, B, suitable for holding
an ordinary metre scale, MN. In many experiments,
short magnets made from knitting-needles are used, which
are held in position by a wooden slider, C.

Evidently, the instrument is best adapted for working with the B
position, because in the A posi-
tion the deflecting magnets would
sometimes have to be placed be-
tween the mirror and the scale,
which would be inconvenient.

A temporary wooden support
of any desired length may, how-
ever, be arranged at the back of
the instrument in order to demon-
strate the properties of the A
position ; indeed, such supports
are often useful in either position
when heavy magnets are employed.

A simple form of lamp and
scale to use with this instrument
is shown in Fig. 115, for which
(and for Fig. 116) we are indebted
to Mr. R W. Paul.

The light from an oil lamp, having its wick endwise on, passes

through a convex lens of about 4 inches focal length (in front of

1 This type of magnet appears to have been first suggested by Mr. W. Hibbert,

Fro. 11-1.

FIG. 115.

148

MAGNETISM

which is a vertical wire), then falls upon the magnetometer mirror,
and is reflected back to the scale. In use, the stand is first placed
in such a position that the distance between scale and mirror is
equal to the radius of curvature of the latter (usually about a metre).
Then, after raising or lowering the lamp to a suitable height, its
horizontal distance from the lens is adjusted until a sharp inverted
image of the flame falls upon the mirror. This gives the maximum
illumination. A real image of the surface of the lens, and of the
wire crossing it, is formed by the mirror on the scale, the result
being a circular disc of light crossed by a fine vertical dark line,
from which the readings are taken. If this image does not fall on
the scale at the first trial, search must be made for it with a piece
A of white paper, and

then the height of the
stand or lamp must be
altered to correspond.
A more recent
type of lamp and
scale is shown in Fig.
116. In this pattern
the source of light is
a small electric lamp
contained in the tube
I), which is fitted
with a convex lens
at the end. The
scale itself is trans-
parent, and is read
from behind by trans-
mitted light, the spot
of light being visible
without requiring a
darkened room.

When the mirror moves through an angle 0, the spot of light
moves through an angle 20 ; hence, if s (Fig. 117) is the deflection as
read on the scale, and I the distance of the scale from the mirror, we

FIG 116.

have =

20.

In ordinary cases, s is so small
compared to 7, that we may write

tan 6 = j • _, without sensible error,
I

and for comparative purposes we
may therefore regard the scale
reading as proportional to the
tangent of the angle of deflection.

As a further consequence of the

MAGNETIC MEASUREMENTS 149

sinallness of 0, in many cases it will be sufficiently accurate to regard
6 as identical with its tangent.

Experiments with Reflecting Magnetometer __ Exp. no,

to show the influence of medium (see p. 122). Arrange a reflecting magnetometer
so that the mirror can swing in the magnetic field of the earth. Raise the mirror
by slightly turning the screw, and level until it moves freely. Adjust the scale
and lamp until the spot of light and the dark central line are well defined.
Arrange a suitable support, on which a bar magnet can be placed in a horizontal
position about level with the mirror, and preferably due north or south of it.
Place the magnet at right angles (i.e. Gauss's B position), and move it gradually
nearer until a convenient deflection is obtained. Between the magnet and the
magnetometer hold various articles, e.g. a drawing board, a plate of glass, a
brick, or the like. Observe that no alteration in the deflection can be seen.
Now, between them, place a sheet of iron, and observe that the deflection is
decreased, but that it returns to its old value when the iron is removed. This
suggests that the iron puts some impediment in the path of the lines of force,
but the effect really depends upon the fact that the lines pass more readily
through iron than through air (see Fig. 102). The iron facilitates the return of
the lines to the opposite pole and cuts off part of them from the space beyond.

Exp. Ill, to compare the moments of two magnets. For this purpose two
small knitting-needle magnets may be used. Let the moments be M and Mx
respectively. Place one of them in the holder C, taking care that the centre is
in a line with the mirror, and adjust it until a convenient deflection is obtained,
Read this deflection. Turn the needle round^and read the new deflection on the
other side of the scale. Place the needle at exactly the same distance on the
other side of the instrument, and obtain two more readings. Let s be the mean
value of the four readings.

Repeat the observations with the second magnet, and let s1 be the mean of
the four readings.

Now, for the B position,

But, as already stated, in this case

d

or M
and M1 = d3H

• M _tan0
Mi ~ tail 0

tan

The moments of large magnets may be similarly compared with great accuracy,
using suitable supports in place of the metre scale.

Exp. 112, to verify the inverse-cube law. For this purpose it is desirable
to work with a greater range of distance than the metre scale permits, and it
is better to arrange a temporary support for the magnet to be used. It will be
sufficient to obtain readings on one side only.

Begin by placing the magnet (using B or A position) as far away as possible,
consistent with obtaining a readable deflection. Turn the magnet round and
read the deflection again. Take the mean. Repeat observations at various
distances, tabulating the values of the deflections and of the distances.

150 MAGNETISM

It will be found that as long as the distance is great compared
with the length of the magnet, halving the distance gives eight times
the deflection.

The law can, however, be more clearly brought out by employing
the method already described in connection with the inverse-square
law.

For, if F oc-^, where F is the field strength at distance dt

then Yd* = a constant.
But F ocs .'. s.d'B = a constant,
.'. log s -f 3 log d — another constant.

Hence, plot a graph with the values of log s for ordinates, and of
log d for abscissas. This will be a straight line if F varies as some
power of the distance, and the tangent of the angle of slope will be
that power.

It will be found that the ratio -^ (Fig. 105) is 3, and there will

be no difficulty in obtaining very consistent results, thus showing
• that the inverse^cube law expresses the facts with great accuracy.

The Inverse-square Law for Point Poles.— we have

already stated that direct experimental proof is not capable of any great
accuracy. The results obtained on pp. 140 and 141, however, lead to a much
more exact, but indirect, method of verification.

Instead of assuming its truth, as we did before, let us write Y = -^-l for the

law of action, where n is unknown, and again find expressions for the iield
strength at any point 0 on the axis, and also on the line at right angles to the
magnet.

In the first case, we have

Expanding binomially, this becomes

Field = m{(d~n + nd-<»+i)j +...)-- (rf-» - nd-in+"l +...)}
Avhich, neglecting terms involving I2 and higher powers, becomes

Iii the second case, we have

Field = 22OT 2 H x cos 8= ™_..m x
2ml

which, when I is small compared with d, becomes

Field =~j (2)

MAGNETIC MEASUREMENTS 151

The results (1) and (2) show that if the law of aetion for a single pole varies
inversely as the nth power of the distance, then the field at a relatively great
distance from the magnet is n times stronger at a point on the axis than at a
point at the same distance, but on the line passing through the centre and
at right angles to the magnet. Hence, to determine the vahie of n, we have
only to compare the held strengths at two such points, equidistant from the
centre of the magnet, and this can be done with great accuracy in several ways.
For instance, a reflecting magnetometer may be arranged so that a bar magnet can
be placed in either A or B position with reference to it, and the deflection noted
for equal distances. As a result, n will be found equal to 2, within a range of
experimental error which will be much less than for the method previously given.
This is the best method of verifying the inverse-square law.

Measurement of the Temperature Coefficient of a
Magnet. — When a magnet is slightly heated, its moment varies
with the temperature — decreasing uniformly as the temperature is
raised, and increasing as the temperature is lowered. We shall
now give a method of finding the relation which exists between the
temperature and the moment, for a moderate range of temperature.

Exp. 113. It is convenient to use a steel bar magnet about 6 inches long —
any shape, however, may be used, provided that some method is devised by
which it can be heated to various temperatures up to (say) 100° C. without dis-
turbing its position. Place it in a simple triangular trough made of zinc, or of
any other non-magnetic substance, and supported on brass legs. Arrange the
trough so that the magnet is in either the A or the B position — usually the
latter is the more convenient — with respect to the reflecting magnetometer.
Fill the trough with water and adjust the distance (which should be fairly great)
until a convenient deflection is obtained. Then heat with a Bunsen flame.
Notice that, as the temperature is raised, the deflection slightly decreases. Take
a number of readings both of temperature and of deflection between 0° C. and
100° C. Remove the burner, and take another series of readings as the magnet
cools. Notice that the deflection increases again ; although it is quite possible
that its original value will not be reached.

Experience has shown that in order to secure consistent results
in these measurements, the magnet must be carried through several
cycles of alternate heating and cooling, a process which appears to
bring it gradually into a definite physical state.

It will also be found that the amount of variation with tempera-
ture changes considerably with different magnets ; for those of poor
quality it is sometimes quite large, but for those of good quality
it is small.

Plot graphs for both rise and fall of temperature, taking deflec-
tions as ordinates and temperatures as abscissae. The graphs will
show that the change is linear, and hence the results can be expressed
by an equation of the form

M^M^l-aO,

where ~M.t is the moment at any temperature £°C.; M0, the moment
at 0° C. ; and a, the temperature coefficient of the magnet, i.e. the
change of moment per degree centigrade. From the graph, obtain
the deflections corresponding to 0° C. and 100° C. respectively. We
know that these are proportional to M0 and M100. Insert them in

152

MAGNETISM

the above equation and calculate the values of a (which from the
form of the equation, depends only on the ratio of M, to M,,).

Moment of Couple acting upon a deflected Compass-
needle. — Let AB, Fig. 118, indicate the direction of the magnetic
meridian, and let the needle be deflected by
some means through an angle 9.
The for.ce acting on each pole

= strength of pole x horizontal component

of the earth's magnetic force
= in x H dynes

Hence the forces acting upon the needle,
tending to bring it back into the meridian,
constitute a couple, whose moment is

iriR x (AN + BS) - mH x 2AN

Now AN = ON sin 0
.'. moment of couple = mH x 20N . sin 6

but 20 N" is the length of the magnetic
needle,

/. moment of couple = mH x I sin 9
= MH sin B

This result is of great importance.

Evidently, the moment of the couple increases from zero when
0 = 0° to a maximum value
of MH when 0 = 90°.

If the deflection is due
to the influence of a field,
F, at right angles to the
meridian, and uniform in
strength over the space
traversed by the needle, then
the deflecting forces on each

TT
pole will be m x - (where //,

= 1), and these will consti-

tute another couple (Fig.

119) whose moment is

m~F x AB = wF x 2AO

= wzF x 2ON cos 9

mH

FIG. 118.

Evidently, the moment

of this couple will have a maximum value of MF when 6 = 0°, and
will decrease to zero when 9 = 90°.

MAGNETIC MEASUREMENTS

153

If the magnet is in equilibrium under the influence of the two
opposing couples, their moments must be equal, so that

MFcos 0 = MHsin 0
or F = H tan 6

This agrees with the result obtained by simple reasoning on p. 142.
Oscillations of a Magnet in a Magnetic Field.— Before

giving any experimental work on this subject, it will be advisable
to construct a simple and useful form of vibration apparatus by
means of which the oscillation of a magnet may be studied.

Obtain a circular glass dish, A (Fig. 120), about 6 inches in
diameter and 3 inches deep. Cut a glass cover, slightly larger
than the dish, having a hole (a quarter of an inch in diameter)
drilled through the centre. Small pieces of wood should be glued

FIG. 120.

to the cover to keep it in position. Take about 5 inches of glass
tubing (half an inch in diameter) and fasten it over the hole in the
cover. This can readily be done by using the cover of an ordinary
deflagrating spoon, which consists of a circular brass plate, C, about
3 inches in diameter, having a hole at the centre, and provided with
a circular brass collar. Fit the glass tube into the collar by means
of a bored cork, and fasten the brass plate to the glass cover. Make
a wooden cap, D, for the top of the tube^ and fix a brass hook into
it. Make a stirrup of copper-foil or zinc-foil, E, and suspend it by a
fibre — or a few fibres, if a heavy magnet is to be oscillated — of
unspun silk to the hook in the cap. In most experiments only
light knitting-needle magnets are used, and then the best stirrup is a
double loop of silk.

When a magnet is balanced in the stirrup, it can be drawn out of
the meridian by bringing another magnet carefully up to it. It will

154 MAGNETISM

then oscillate about its position of rest, until finally it again becomes
stationary.

Exp. 114, to prove that, althowjh the extent of the oscillation gradually
diminishes, the time of pcrforminy each oscillation is the smnc.

Suspend a magnet in a stirrup of a magnetometer and draw it aside through
a small angle — say 4° or 5°. Determine the time of a definite number of oscilla-
tions— say 10 or 20. Draw it aside through a larger angle — say 8° to 10° — and
again determine the time of the same number. Observe that the times are the
same in both cases.

The actual time of vibration depends upon several conditions : —

1. Upon the mass and shape of the magnets.

Exp. 115. Magnetise a knitting-needle and suspend it in the vibration
apparatus. Draw it aside, and, as before, determine the time of a number
of complete oscillations.

Load each end of the needle with a piece of sheet lead, thus increasing the
mass and changing the shape, and again determine the time of the same number
of oscillations. Observe that the time in the latter case is greater than in the
first case, i.e. the greater the mass, the longer the time of vibration.

2. Upon the moment of the controlling couple, which in its turn
depends upon

(a) the moment of the magnet,

(b) the magnetic force due to the field, and hence upon the

strength of the field

Exp. 116. Determine the time of a definite number of complete oscillations
of a magnet. Now remagnetise the magnet as strongly as possible. This
increases its moment without altering its mass or shape. Observe that the
time of vibration of the remaguetised magnet is less than before.

Exp. 117. Determine the time of a definite number of oscillations (a) when
a small heavy magnet oscillates under the earth's influence alone ; (b) when the
strength of the field of force is altered by bringing the S pole of a long magnet
near the N pole of the oscillating magnet. Notice that in the second case the
oscillations are much more rapid, i.e. the time of oscillation is diminished.

A magnet in oscillation is a particular instance of a " compound
pendulum," and we can apply to it the general equation for the time
of vibration of any compound pendulum executing what are known aa
" simple harmonic vibrations.'

If t be the time of one complete vibration, then

/.

v;

moment of couple
angle of deflection

where I is a quantity depending upon the shape and mass of the
vibrating body, known as its moment of inertia, and the denominator
is the ratio between the couple producing any steady deflection and the
deflection produced. Experiment shows that this ratio is constant for
different deflections, as long as they are not very great in amplitude.

MAGNETIC MEASUREMENTS 155

In the particular case of a pivoted or suspended magnet vibrating
under the influence of a magnetic field, it has just been shown that
the moment of the couple required to deflect the magnet through an
angle 0 is MH sin 0,

MH sin 6

Now, when 0 is small, the ratio - — — — is practically unity, and

.-.we have* =

It must be very carefully remembered that this expression only
applies to oscillations of small amplitude. It may also be pointed
out that H is not the field strength, but that it is the magnetic force
i.e. the force in dynes on unit pole, although in all practical cases,
they will have the same numerical value.

The term moment of inertia will be understood only by students
who have some acquaintance with the theory of rotatory motion. It
may be defined as follows : —

If the mass of every particle of a body be multiplied by the square
of the distance from the axis of rotation, the sum of these products is
Ike moment of inertia of the body about that axis,

although this definition is not likely to convey much information.

For the present purpose, it will be sufficient to know that the
moment of inertia of bodies of simple geometrical shape can be
calculated from their dimensions.

For instance, for a rectangular bar magnet of mass m, length a,
width b, and thickness c, vibrating about an axis through its centre
perpendicular to the surface containing a and b, we have

(notice that it is independent of <•)

For a cylindrical bar magnet of mass m, length a, and radius ?•,
vibrating about a central axis perpendicular to the axis of the
cylinder,

Very frequently, bar magnets are used in the form of thin rods
(e.g. knitting-needles), and then these two expressions may be simpli-
fied. For when b is small compared with a, or r is small compared
with a, both forms reduce to

ma*

15(5

MAGNETISM

FIG 121

Bottle Form of Oscillation Apparatus.— It is often
necessary to experiment upon fields due to magnets,
&c., which are very far from uniform, and therefore,
in order to compare the strengths at different points in
a varying field, it is necessary that the oscillating magnet
should be very short and that it should be placed
with its centre exactly at the point in question.
Again, there may not be sufficient room for the
-large vessel shown in Fig. 120, and hence it is con-
venient to use a -needle about an inch long, suspended
inside a bottle just wide enough to allow it to oscillate.
Fig. 121 is a diagrammatic sketch of such an instru-
ment, in which the magnet NS has soldered to it a
brass rod, B, in order to increase its moment of inertia,
and thus to make it vibrate slowly for convenience
in counting the oscillations. A is a silk fibre attached
to a brass rod, R, which passes through the cork of
the bottle.

Comparison of Field Strengths by the

Method of Oscillation. — If the time of oscillation of the same
magnet be taken in fields of different strengths, M and I are con-
stant, and the equation on p. 155 reduces to

U or Hoc ^

where H is the force on unit pole.

This affords an easy and accurate method of comparing field
strengths, and thereby of indirectly measuring various other quantities.

Exp. 118, to compare the value of the horizontal component of the earth's fie Id
in the open air with its value at a point inside a building. Use a knitting-needle
magnet, 4 or 5 inches long, inside the apparatus shown in Fig. 120. Take its
time of vibration in the open air, at a distance from surrounding buildings. To
do this, disturb it slightly from its position of rest by bringing a piece of iron
near the side of the glass cover, wait until the oscillations are of small amplitude
(4D or 5°), then deteimine the time taken in executing, say, 20 complete vibra-
tions. Repeat this several times, and, from the results, find the mean time of
one vibration. Let this be t seconds.

Repeat the experiment at a given place inside the building, and let the mean
time be ^ seconds. The difference between ^ arid t will depend on circum-
stances, but in the majority of cases a small but distinct difference will be found.

Let H and H^ be the values of the horizontal component in the open air and
inside the building respectively, then

1
H ? t*

As the value of H is well known for different parts of the country, we can
easily deduce the actual value inside the building.

MAGNETIC MEASUREMENTS 157

This experiment also illustrates the method adopted in comparing
the values of H at different parts of the earth. Evidently, it is only
necessary to take the time of oscillation of the same magnet in
different places, although precautions must be taken to ensure that
its moment is not altered during transit from one place to the
other.

It may be pointed out that if n and ??, are the number of oscil-
lations in one second, corresponding to times of vibration t and tl
seconds respectively, then

n = — and TO, = —
t ^

and we may, therefore, write ==- = -- g, and this form is sometimes

1 ™l .
convenient. In practice, however, it is always the value of t that

is found (by determining the time taken in executing a given number
of vibrations), and not the value of n (for it is difficult to determine
with accuracy the number of vibrations in a given time), and, hence,
we shall usually express our results in terms of t.

•

Exp. 119, to compare the moments of two mar/nets. A reference to the.
equation on p. lf)f> will show that, if the two magnets have the same size and
weight, i.e. if their moments of inertia are the same, it will he sufficient to take
their respective times of vibration at the same place, for when I and H are
constant, we have

'"Vsr or M4

As a general rule, this condition will not be satisfied, and then it is necessary
to proceed as follows : —

Two slots are cut in a .small wooden block to receive the magnets, another
similar block is laid upon them, and the two blocks are 'screwed together with
brass screws. In this wray, the magnets are held rigidly one above the other
with their axes parallel to each other, but without contact. A hook is screwed
into the block, and the whole is suspended by a suitable fibre inside a glass
case (e.g. an empty balance case) to avoid draiights.

The time of vibration is then taken. (1) with both N poles pointing the same
way, (2) when one magnet is reversed.

Let M and Mx be the moments of the magnets, and let t and ti be the times
of vibration in (1) and (2) respectively.

The moment of inertia of the whole system is evidently the same in each
case, and hence we have (assuming M>Mj)

t = 2i

a)H

Therefore

158 MAGNETISM

from which

Example. — Two magnets are placed in an aluminium frame
with their axes horizontal and parallel, one being vertically over
the other. The frame is suspended and oscillates in the earth's
horizontal field, making 20 and 5 vibrations per minute respectively
when similar poles of the magnets are together or opposed. If the
moment of the stronger magnet is 300, what is that of the other ?

(B. of E., 1907).

Let M and Mj be the moments of the stronger and weaker
magnets respectively, and I the moment of inertia of the combination.
Let t and tv be the times of vibration in the two cases respectively ;
then, by the result given above,

M

Now zO vibrations in 1 minute = 1 vibration in 3 seconds, and
5 vibrations in 1 minute = 1 vibration in 1 2 seconds.

300_122 + 32_153
~Mi~122-32~135

• ^. MI = 300,^35 = 26J.

Further Experiments on the Inverse-square and

InverSe-CUbe LaWS. — Exp. 120, to determine the law of action of a
single pole. A third method of testing the in verse -square law may now be
given.

Place on a table a very long, thin bar magnet in the magnetic meridian,
with its N pole directed towards the north. Then at all points on its axis pro-
duced, its field is in the same direction as the earth's field, and the resultant
field is the sum of the two components.

Determine as nearly as possible the point which may be taken as the position
of a pole of the magnet (see p. 115).

Using the bottle form of vibration apparatus, place it about 3 inches from
the pole on the axis of the magnet produced, and then determine its time of
vibration. Move it about 3 inches further away, and again take the time of
vibration.

Repeat the observations for a number of points on the produced axis, in each
case measuring the distance from the pole.

Finally remove the long magnet, and take the time of vibration under the
influence of the earth's field alone.

Let F = Field due to magnet at one of the points where the observations arc
made, E — Field due to earth, and R = Resultant field.

Then, in this case, F + E = R, or F = R-E.

MAGNETIC MEASUREMENTS 159

But if t be the time of vibration at the point, and t, the time under the
influence of the earth alone, then

i
R = Cx where C is some constant,

and E = Cx^

or F

Hence, calculate the value of - in each case, subtract from the result the value

-p which is the same in all positions, and thus prepare a table of numbers pro-
portional to F corresponding to various distances (d) from the pole. Then, as
shown on p. 138, if the field varies as d>l, we have

Fdn = a constant,
and log F + w log d = an other constant.

Hence, as already explained, plot log F against log d, and determine n from the
tangent of the angle of slope. Again, it is unlikely that n will be found equal
to 2, but it will probably be nearer this value than it was in Experiment 104,
simply because it has been possible to work with greater distances, for which the
inverse-square law is approximately satisfied.

Exp. 121, to verify the inverse-cube law. Proceed exactly as in the last
experiment, but use a slwrt bar magnet instead of a long one, and measure the
distances from its centre. In this case, it will be found that n equals 3 (within
reasonable limits of experimental error) except for points too close to the magnet.

Absolute Measurement of M and H. — We have given
methods of comparing (1) the moments of two magnets ; and (2) the
horizontal components of the earth's field at different places, and we
must now give a method of obtaining M and H in absolute measure.

For this purpose, it is preferable to use a reflecting magnetometer.
A small magnet, about 4 inches long, is made from a knitting-needle ;
it should be of uniform section, and the ends should be ground flat,
and, in order to secure uniformity of magnetisation, it should be
magnetised by placing it inside a solenoid carrying a current, and not
by rubbing on another magnet. Either A or B position may be used,
but, with the instrument described on p. 146, the latter position is
the more convenient.

Exp. 122. (a) Put the small magnet in position at distance d from the
mirror (about 40 centimetres will probably be suitable), and take the scale read-
ing ; then turn the magnet round, thus reversing the direction of deflection, and
again take the reading. Now place the magnet at exactly the same distance on
the other side of the magnetometer, and by repeating the operations, obtain two
more readings. Take the mean of the four readings. Let this be n divisions.
Measure as accurately as possible the distance from the mirror to the scale, and
also express the mean deflection n in terms of the same unit of length. From

160

MAGNETISM

this determine (as on p. 148) tan 20, remembering that the reflected ray turns
through twice the angle of rotation of the mirror. But, for these small angles,
it will be sufficiently exact to take tan 0 as %• tan 26, so that the value, of tan d
can be written down at once in terms of the measured distances.

— = d* tan 0 (for B position)
H

We have now
where M is the moment of the knitting-needle magnet. It is better to work

this out instead of leaving it in factors.

Let this value be a, so that --- = «.
H

(b) Now suspend the magnet by a fine silk fibre in the apparatus shown in
Fig. 120. Make a double loop in the fibre, instead of using any form of stirrup,
so as not to add appreciably to the mass. Using swings of small amplitude,
determine the time of executing about 20 complete swings. Repeat carefully
several times, and obtain the time, t, of one complete oscillation.

Then, \ve have t =

~-=^
JV1H

whence MH = ( = &, say)

= 2ir+

\

For a round bar, whose radius is small compared with its length, the moment
of inertia, I, is, — — (see p. 155).

Two equations are thus obtained —

-=a, and
11

Then, by multiplication, we have

and by division,

= or
a

The method may, therefore, be used to determine either M or IT, although, for
the sake of simplicity, we have omitted numerous precautions and corrections,
which are necessary for an accurate measurement.

Experiment 122 (a) can, of course, be carried out, although with much less
accuracy, by means of the magnetometer described on p. 143, and we give
below the results of a particular experiment by this method. A rectangular bar
magnet — 10*7 centimetres long, 2 centimetres wide, and 110'8 grams in weight —
was used in the B position. When the distance was 25 centimetres, the
following eight readings were taken : —

Position.

Deflection.

Mean.

JS'atural tangent.

E 1

21V, 22°

}

E2
W 1

21°, 22°
211°, 20£°

-21° 26|'

•3926

W2

21£°, 2H°

J

MAGNETIC MEASUREMENTS 161

Whence M^tfxten*

_253x-3926
2

--=3067 nearly.

Using the same magnet in the oscillation apparatus (Fig. 120), 30 oscillations
were made in 11 minutes 20 seconds.

.". time of one oscillation was 22 '6 seconds.
Calculating the moment of inertia, we have

1 = 110-8 .-- =10M

Substituting in equation

MH-—I, we have

MH_4x(3-14)axlQ94

(22-6)2

-84

Whence MH x - = M2 = 84 x 3067
H

.'. M = 508
and M

.'. H='16 nearly.

It may be pointed out that the term magnetic moment has a much
more definite meaning than the term magnetic pole. \Vhatever may
be the shape of the magnet, its moment can be measured as an exact
numerical quantity, whereas a magnetic pole is little more than a con-
venient mathematical conception, and can be measured only approxi-
mately by indirect and doubtful processes.

Construction for Magnetic Curves. — The construction given on
p. 41 for electric lines of force can be applied to the field produced by a bar
magnet, on the assumption that its poles may be represented by points. As the
two poles are necessarily equal in strength, the two sets of radiating lines must
be equal in number ; and if n lines are drawn from each point, they will repre-

sent poles of strength — . It will be found, on carrying out the construction

4r

as already explained, that the distribution outside the magnet resembles the
figure given on p. 125. (The distribution inside the magnet must be inferred
from other considerations.)

Direction of the Field at a given point near a Magnet.

—This can easily be determined by a simple construction, although the method
is not suitable when the whole field has to be plotted. Let 0 (Fig. 122) be the
given point, and let d and rfx be its distances from the N and S poles respectively

L

162

MAGNETISM

treating them as points). Writing Ffl Yg for the fields at 0 fine to the N and
S poles respectively, we have

., R Fn = acting along NO produced

0 6 ^S=7T* actinS along os

Hence, mark off along XO produced, a length,
OA, to represent dt2 , and along OS a length,
OB, to the same scale to represent d2 ; complete
the parallelogram and draw the diagonal OR.
This will give the direction of the field at 0,
i.e. OR is the tangent to the curved line of
force at that point. .

FlG. 122.

FIG. 123.

As the direction of the line indicates the direction of the force acting upon a
magnetic pole, it follows that a single free pole placed anywhere in the field
would tend to move one way or the other along the curved line until it reached
the magnet. But such poles do not exist, and in considering the behaviour of
a piece of iron or of a small magnet (say an iron filing) in a magnetic field, we
must take into account the force on each pole.

Let A (Fig. 123) represent the iron filing in any non-uniform field, of which
only one line is drawn for clearness. Then each pole will tend to move along
the line of force passing through it, but in
opposite directions, and it is easy to see
that these two forces will have a resultant
tending to move the filing laterally towards
the magnet. As the filing is nearer to the
N pole of the magnet, the force in that
direction is the greater, and hence the
resultant is inclined to the axis of the
filing, which tends to move across the
lines of force until it reaches the magnet.
We may express these results by saying

that whereas a single pole tends to move along the lines of force, an actual piece
of iron tends to cut across them in the direction in which the field is increasing
in strength most rapidly.

Examples. — 1. Two exactly equal magnets aro attached
together at their mid points so that their axes are at right angles,
and the combination is pivoted so that the axes of the magnets
are horizontal and they can turn freely about a vertical axis.
How will the system set itself under the influence of the horizontal
component of the earth's field? If the moment of each magnet is
M, and the moment of inertia about the axis round which it can turn
is K, what will be the period of vibration of the system ?

(B. of E., 1906.)

Taking the latter part of the question first (for convenience), its
point depends on recognising that if each magnet has moment of
inertia K and magnetic moment M, the values of these quantities in
the combination are obtained by simple addition with regard to the
moments of inertia, and by vector addition with regard to the magnetic

MAGNETIC MEASUREMENTS

103

moments ; i.e. the moment of inertia of the combination is 2K, and the
moment of the equivalent magnet is */M2 + M2 = M. \/2

Hence, t = 2?r

2K

M. x/2

Considering the first part of the question, it is evident that, as the
moments are equal, the magnets will set themselves symmetrically
with respect to the meridian.

It will be instructive to discuss the more general question — how
will two magnets of moments M and M1? and rigidly fastened
together at their centres at an angle 0, set themselves when free to
move in the earth's field in a horizontal plane ?

The dotted line in Fig. 124 indicates the magnetic meridian.
The moments of the opposing couples are MH sin a and MXH sin ft,
and hence the condition of equilibrium is
given by

MH sin a = MjH sin /3

or ^ = ^ (a)
M]_ sin a

I. The simplest case is when 0 = 90°.
Then we have

or sin a = cos p
M sin 8 0

Crf^

from which both angles are determined.

II. When 0 is not 90°, equation (a) indicates a graphic solution.
Draw two lines (Fig 125), enclosing the angle 6, and mark off along
them, to any convenient scale, lengths OA and OB proportional to

M and Mj_ respectively. (Notice
that OA and OB do not represent
the lengths of the magnets, in fact
OB may refer to the shorter
magnet, although OB>OA.)
Complete, the parallelogram and
draw the diagonal OR, then

From the figure

AO_sin ARO_sin/?
AR

AO

AR

]V^ _sin /?

Mn sin a

FIG. 125.

sin AOR sin a

, , ±\u m

bllt — — = -r^r-

164 MAGNETISM

and OR, represents the moment of the single resultant magnet
equivalent to M and Mx

Astatic Needle. — The above reasoning shows that the magnetic
moment of a combination of magnets is the vector sum J of the moments of
the component magnets. Hence, if in Fig. 124 the two magnets are of equal
moment and their axes are oppositely directed and parallel to each other, the
resultant magnetic moment will be zero. As a whole, the system will remain
in any position, and will be quite free from any directive action due to the
earth's field (or any other uniform field).

Such an arrangement, which may be realised in various ways, is said to be
astatic. Fig. 126 shows a well-known form of the type once largely used in
the construction of the astatic galvanometer (see
p. 291). It consists of two magnets fixed one above
the other to a vertical spindle, which is suspended
by a silk fibre or mounted on bearings. It will be
understood that the magnets may be any distance
apart, provided that they are rigidly connected
together.

It is almost impossible to make a perfectly
astatic pair. This will be understood from the
following considerations : —

It is very difficult (1) to magnetise two needles
a.' to exactly the same strength ; (2) to fix them
parallel ; (3) to fix them so that their axes lie
FIG. 12G. i11 the same vertical plane, i.e. so as not to cross

one another.

A. If the needles are of unequal length, they will, owing to the action of the
earth's magnetism, tend to move into the magnetic meridian.

B. If the magnets are of equal size and strength, but their axes are not
quite in the same vertical plane, they will set themselves at right angles to the
magnetic meridian, i.e. east and west. The reason of this will be easily under-
stood, for a glance at Fig. 125 will show that two magnets of nearly equal
moment, having their axes oppositely directed, and not quite parallel to each
other, will be equivalent to a single magnet of very small moment inclined
practically at right angles to the combination.

2. The centres of two short magnets AB, CD, are at a distance
r apart. AB lies along the line joining their centres, and CD at

FIG. 127.
right angles to it. Show that the couple due to AB tending to make

CD twist round is — — -J where M and ]VL are the moments of the
^

magnets. (B. of E , 1898.)

1 The "vector sum" is the resultant obtained by the method generally
known in another connection as the "parallelogram of forces," and is applied
in the previous example.

MAGNETIC MEASUREMENTS 165

Let the moment of AB be M ; the moment of CD, M15 and its
pole strength vir As the magnets are short compared with r, we may
regard CD as being in a uniform field due to AB and acting along the
axis, and if F is the strength of this field, the force in dynes on each
pole is m^F and the moment of the couple tending to rotate CD is

Also F = (seep. 140)

.-. Moment of couple = 2MMl
rs

(The student can show, in the same way, that the couple due to CD
tending to rotate AB is

3. A bar magnet is suspended horizontally by a wire. When the
top of the wire is twisted through 120°, the magnet is deflected
through 30°. How much must the top be twisted in order to deflect
the magnet through 90° 1

When the magnet is deflected through some angle 0 by twisting
the wire, it is under the influence of two opposing couples — (1) a
deflecting couple due to the torsion, whose moment is within certain
limits proportional to the angle of twist, and (2) a restoring couple
due to the earth's magnetic field, whose moment is MH sin 9.
When in equilibrium, the moments of these couples must be equal.

Angle of twist oc MH sin 0
.'. 1 20° -30° oo MH sin 30° i.

Let t° = angle of twist when the deflection is 90°

then*0 oo MH sin, 90° ii.

sin 3Q°

But this is the actual twist on the wire, when the deflection is 90°,
therefore the top of the wire must be turned through 180° 4- 90° =
270°.

The above argument indicates a method of comparing the moments
of two magnets, which is of great theoretical importance. Its prin-
ciple will be sufficiently shown by the following example :--

4. Two magnets, A and B, are in turn suspended horizontally by a
vertical wire so as to hang in the magnetic meridian. To deflect the
magnet A through 45°, the upper end of the wire has to be turned
once round. To deflect B through the same angle, it has to be turned

166 MAGNETISM

round one and a half times. Compare the moments of the two
magnets.

We have, generally,

Angle of twist oo MH sin 6

.-. 360° -45° oc MjH sin 45°

and 540° -45° oc M,H sin 45°

5. Find approximately the force of attraction or repulsion between
two short bar magnets of moments 10 and 20 C.G.S. units re-
spectively, with their centres at a distance of 20 centimetres apart and
their axes pointing in the same direction along the same line.

(B. of E., 1909.)

Let us first consider the general case shown in Fig. 128,

FIG. 128.

The magnet M (of pole strength m and length 2Z) is placed in a
field due to the magnet MI} then

(a) the near pole of M is in a field whose strength is /-y^-4v-3> anc^

2\[

the repulsive force upon it is ~ j x m dynes

(a — I)

*>"M

(b) the further pole is a field of strength . _. l and the attrac-

\d-\-l)
dynes,

.,. Nett repulsive

tive force is -^ — J^3 x m dynes,

but, as the magnets are short, we may neglect I2 and higher powers,
so that the expression becomes

2M,w x 6d2Z 6MMt

Keijulsion = * — ••, = -

Q *

MAGNETIC MEASUREMENTS 167

(If the student considers the case when like poles are not opposed to
each other, he will find that the force is one of attraction and of the
same value.) Now, substituting the numbers given in the question,

T . 6 x 10 x 20
Repulsion = 2Q4

- -0075 dyne.

6. A compass-needle is placed on a table, and a bar magnet
is laid on the floor below it, the centre of the bar magnet being
directly underneath the centre of the needle. When the N pole
of the bar magnet is northward, the compass-needle, after being
disturbed, makes 100 oscillations in 16 minutes. When the N pole
is southwards, the needle makes 100 oscillations in 12 minutes.
When the bar magnet is removed, the needle oscillates under the
earth's influence alone ; how long will it take to make 100 oscillations'?

The compass-needle is vibrating in a field due partly to the bar
magnet and partly to the earth, both acting in the same direction.
It is implied that the earth's field is the greater and is weakened, but
not reversed, by that due to the magnet when acting in opposition.

Let E = strength of field due to earth,
„ F= „ „ „ „ „ magnet,
„ #1 = time in minutes of making 100 vibrations when both

fields coincide in direction,
„ f9 = time in minutes of making 100 vibrations when the two

fields are in opposition,
„ £-time in minutes of making 100 vibrations under earth's

influence alone.

Then E + F « «

iii.

,.. E + F 256
From i. and 11. r *

• E = 2 iv.

F 7

E 144
From i. and 111. — — =, = — =-

Jii + r t^

E= J^
* ' F ~" t'^—

v.

168

MAGNETISM

Equating iv. and v.

144

25

~T

From which *2 = 184-3

.-. £= 13-56 minutes.

7. Prove that the work done in twisting a magnet of moment M
through 90° from the meridian in a field of strength H is given by
the product MH. (B. of E., 1904.)

When the magnet is deflected through angle 0 (Fig. 129), the
pole N moves through an arc AX, which is equivalent to the distance
AB measured along the direction of the
earth's field. Hence, work is done in mov-
ing the pole against the magnetic force,
and as work = force x distance, we have,
Work done on N pole = wH x AB.
Now, an exactly equal amount of work
is done on the other pole, and

.-. Total work = 2niR x AB

= 2wH(AO-BO)

-cos 0)
= MH(l-cos 0)

So that when 0 = 90°, work done is Mil
dynes.

Intensity Of Magnetisation. — The intensity of magnetisation of
a magnet is the magnetic moment per unit of volume. For instance, in the case
of a uniform bar of pole strength m, area of cross section A, and length I, we have

M ml m

or, the intensity would be the number of unit poles per square centimetre of end
surface, if the magnetisation were strictly confined to the ends.

This method of representing the strength of a magnet is occasionally con-
venient in theoretical discussions, and is employed later in Chapter XXV.,
although at present it is merely necessary to indicate its meaning.

Magnetic Potential. — The idea of potential in connection with an
electrostatic field has been discussed in Chapter IV. Somewhat similar ideas
can be applied to the phenomena of magnetism, and the definition already given
will still hold good, provided that we write "magnetic pole" in place of
"electric charge." It, therefore, follows that the magnetic P.D. between any
two points is measured by the work done (in ergs) when a unit pole is carried
from one point to the other against a magnetic force ; also that when we speak
of the "magnetic potential at a point," we mean the P.D. between that point
and a point at infinite distance, as measured by carrying a unit pole from
infinity to the point in question against the force.

MAGNETIC MEASUREMENTS 1(5!)

On the other hand, there is no "flow" of magnetism analogous to the
:<flow" of current, produced (say) by joining points at different magnetic
potentials, and, in consequence, the idea of potential is not so generally useful
in magnetism as it is in electrostatics.

Potential at any Point near a Magnet Pole. — The argu-

ment given on p. 36 will apply to this case, provided that we write m, for the

fji.dz

"magnetic force" at distance d from an isolated magnetic pole (see p. 136).
Hence, we find that the potential at distance d from a single pole of strength m

is m , and we may agree to regard it as positive, when the pole is a N pole, and
negative when a S pole.

Potential at any Point near a Bar Magnet— in this case

we have simply to superpose the effects of the two poles, considered as acting
independently.

Let NS (Fig. 130) be the magnet, of pole p

strength m; and let P be the point in ,//

question. Then if 0 be the middle point of X;/Y

the magnet, the position of P is denned in ,''//

terms of length OP and the angle, 0, which ,' ,' i

OP makes with the axis of the magnet. ,<' / >'

& '

Potential at P due to N pole =+

Potential at P due to S pole= ~

Now, potential is not a vector quantity,
and hence we have X'B

Actual potential at P= ^fp-~ FKJ. 130.

Case Of a Short Magnet.— When OP is great compared with
the length of the magnet, this result can readily be simplified. Draw JX A and
SB (fig. 130) perpendicular to PO (or its direction produced). Then as the
angles NPO and SPO are very small, we have approximately NP = O1 OA,
and SP = OP + OB, where OA = OB.

... Potential at P^{op^

Now OA is very small compared with OP, and, therefore, the denominator will
not differ appreciably from OP2.

Again, in the triangle OAN we have -— =cos 6

m 20N cos 0

.*. the expression becomes — „ — -^-^ —
/* OP4

But 20N = length of magnet

. % Potential at P = 3!?J, where d = OP.

170 MAGNETISM

EXERCISE VIII

1. Two long magnets are placed vertically with their north poles (A and B)
on the same level as the north pole (C) of a compass-needle, one being magnetic
east and the other magnetic west of C. If the compass-needle is not deflected
when the distance AC is twice BC, and if all the magnets are so long that the
effects of the south poles may be neglected, show what are the relative pole
strengths of A and B.

2. A straight piece of watch-spring, 6 inches long, is magnetised and laid
on a flat cork floating on water. The spring is now bent until its ends are two
inches from each other, and they are fixed at that distance by a piece of thread ;
the spring is then replaced upon the cork. Compare the couples with which
the spring tends to make the cork take a definite direction in each case.

3. A uniformly magnetised bar of brittle steel is broken into two pieces, one
twice as long as the other, and the pieces are fastened together at right angles
to each other. How would the combination thus formed set itself, under the
action of the earth's magnetic force, if made to float on water ?

4. A magnetic needle is suspended horizontally in the magnetic meridian.
It is then drawn out of the meridian (a) through 30°, and afterwards (/3) through
45°. Compare the couples which act upon the needle to bring it again into the
meridian.

5. As in question 3, if a — 30° and /3 = 60°. , Compare the couples.

6. If a = 30°, 18 = 90°. Compare the couples.

7. A magnet pole of strength 6 is placed in a magnetic field of strength '42 :
at will be the force acting on the pole ?

8. A magnet pole of strength 7 experiences a force of 2 '9 dynes. "What is
1 the horizontal component of the magnetic field ?

9. A very long vertical magnet of pole strength 150 is placed at a perpen-
dicular distance of 10 centimetres from the centre of a horizontal magnetic
needle of length 5 centimetres and pole strength 30. Find tbe moment of
the couple acting upon the needle.

10. A very long vertical magnet is placed at a perpendicular distance of
12 centimetres from the centre of a horizontal magnetic needle of length
10 centimetres and strength 13. The moment of the couple acting upon the
needle is 60. Find the strength of the pole of the long magnet.

11. Two bar magnets, the moments of which are in the ratio of 8 to 27, are
placed with their centres 3 feet apart, their magnetic axes being in the same
straight line, which is perpendicular to the magnetic meridian. If their north
poles are turned towards each other, find the position which a small compass-
needle must occupy on the line joining the magnets in order that it may point
in the same direction as if the magnets were not there.

12. A bar magnet suspended by a fine wire points north and south (magnetic)
when the wire is not twisted. When the upper end of the wire is turned
through 100° the magnet is deflected 30° from the magnetic meridian. Show
how much the upper end of the wire must be turned to deflect the magnet 90°
from the meridian.

13. The lower end of a fine wire, which hangs vertically, is fastened to the
middle of a straight steel magnet, so that the magnet is suspended horizontally
by the wire. When the wire is without twist, the magnet comes to rest in the
magnetic meridian, but when the upper end of the wire is turned once round,
the magnet is deflected from the meridian through 30° ; how much must the
top of the wire be turned to make the magnet set at right angles to the meridian ?

14. Two magnets, A and B, are in turn suspended horizontally by a vertical
wire so as to hang in the magnetic meridian. To deflect the magnet A through
45° the upper end of the wire haM to be turned once round. To deflect B through

MAGNETIC MEASUREMENTS 171

the same angle it has to be turned round one and a half times. Compare the
moments of the two magnets.

15. Two straight pieces, one 3 inches and the other 5 inches long, are
cut from the same narrow strip of steel. After being equally magnetised they
are hung horizontally one at a time by the same fine glass thread so as to rest
in the magnetic meridian when the glass thread is not twisted. On turning
the upper end of the thread half round (through 180°) the shorter magnet is
deflected 10° from the meridian. Show how much the upper end of the thread
must be turned to deflect the larger magnet 10°.

16. A long bar magnet lies in the magnetic meridian, with its N pole
towards the south. A horizontally suspended compass-needle is placed in the
line obtained by producing the axis of the magnet. What effect will the
sliding of the magnet towards the needle have on the time of vibration ?

17. A glass tube containing four similar pieces of hard steel, which just fill
it when placed end to end, is suspended so that it can oscillate about its central
point in a horizontal plane. What will be the nature of the difference (if any)
in the times of oscillation when (1) the two outer pieces only, (2) the two inner
pieces only, are magnetised, unlike poles being in both cases nearest together ?
Xeglect the effects of induction, and give reasons for your answer.

18. A magnetic needle, balanced horizontally at its centre upon a fine pivot,
makes 11 vibrations in 2 mins. 1 sec. at a place A, and 12 vibrations in 2 mins.
at a place B. Compare the strength of the earth's horizontal force at the two
places, explaining clearly how you arrive at your result.

19. A magnetic needle was suspended in a paper stirrup by means of a fibre
of unspun silk, and made 12 oscillations in 2 mins. It was removed, and
remagrietised. When suspended as before, and moved from its position of rest,
it made 45 oscillations in 3 mins. Compare its magnetic moments in the two
cases.

20. A bar magnet, which can move only in a horizontal plane, is caused to
vibrate at three different stations, A, B, and C. At A it makes 20 vibrations
in 1 min. 30 sees. ; at B, 25 vibrations in 1 min. 40 sees. ; at C, 20 vibrations in
2 mins. Find three numbers proportional to the forces which act upon the
magnet at the three places.

21. A small magnetic needle, suspended horizontally by a fibre of raw silk,
makes 10 oscillations in 1 min. when under the influence of the earth's action.
When the S pole of a long magnet A is placed 3 inches from the N pole of
the needle, it makes 32 oscillations in a minute. Afterwards the S pole of
another magnet B is similarly placed, and then the needle makes 25 oscillations
in a minute. Compare the pole strengths of A and B.

22. A small magnetic needle, suspended horizontally by a fibre of unspun
silk, makes 97 oscillations in 8 mins. 5 sees, under the earth's influence. When
the S pole of a long magnet A is placed a few inches from the N pole of the
needle, it makes 160 oscillations in 5 mins. 20 sees. ; when, however, the S pole
of a magnet B is similarly placed, it makes 170 oscillations in 7 mins. 5 sees.
Compare the pole strengths of A and B.

23. A magnetic needle made 50 oscillations in 2 mins. 5 sees, under the
earth's influence. When the S pole of a long bar magnet was brought 4 centi-
metres from the N pole of the needle, 120 oscillations were made in 3 mins.
20 sees. When the distance between the poles of the needle and magnet was
12 centimetres, it made 65 oscillations in 2 mins. 10 sees. Compare the field
due to the bar magnet in the two positions.

24. A small magnetic needle, suspended horizontally by a silk fibre, makes
100 vibrations in 5 mins. 36 sees, under the influence of the earth's magnetism
only, and 100 vibrations in 4 mins. 54 sees, when a horizontal bar magnet is
placed with its centre vertically below the needle, and with its axis in the
magnetic meridian. Compare the field due to the bar magnet with that due
to the earth.

172 MAGNETISM

25. A uniformly magnetised steel wire, 6 inches long, is laid upon a table ;
a very short bit of soft iron wire is supported, so as to be free to turn about
its centre, at a distance of 6 inches from one end and 3 inches from the
other end of the magnetised wire. Show how to draw- a ligure which would give
the direction taken up by the bit of soft iron. [The effect of the earth's
magnetism is to be neglected.]

26. A bar magnet is suspended horizontally in the magnetic meridian by a
wire without torsion. To deflect the bar 10° from the meridian the top of the
wire has to be turned through 180°. The bar is removed, remagnetised, and
restored, and the top of the wire has now to be turned through 250° to deflect
the bar as much as before. Compare the magnetic moments of the bar before
and after remagnetisation. (B. ofE., 1894.)

27. A short bar magnet is placed on a table with its axis perpendicular to
the magnetic meridian, and passing through the centre of a compass-needle.
In London, the compass-needle is deflected through a certain angle when the
centre of the magnet is 25 inches from the centre of the needle. If the experi-
ment be repeated in Bombay, the magnet must be moved 5 inches nearer to
the -needle to produce the same deflection. Use these data to compare the
horizontal forces in London and Bombay. (B. of E., 1895.)

28. Prove that the magnetic force exerted by a short magnet at a point A
on the line passing through its centre and perpendicular to its axis, is the same
as the force exerted at a point on the axis, the distance of which from the

centre of the magnet is x/2 times the distance of A from the centre.

(B. of E., 1896.)

29. A magnet placed due east (magnetic) of a compass-needle deflects the
needle through 60° from the meridian. If at another station where the
horizontal force of the earth's magnetism is three times as great as at the first,
the same magnet be similarly placed with respect to the compass-needle, what
will be the deflection of the latter ? (B. of E., 1897.)

30. Two short bar magnets, the moments of wThich are 108 and 192 respec-
tively, are placed along two lines drawn on the table at right angles to each
other. Find the intensity of the magnetic field due to the two magnets at the
point of intersection of the lines, the centres of the magnets being respectively
30 and 40 centimetres from this point. (B. of E., 1900.)

31. A short bar magnet is placed, at Gibraltar, perpendicular to the magnetic
meridian, ami "end-on" towards a compass-needle from which it is distant
100 centimetres. When the experiment is repeated at Portsmouth, the magnet
has to be placed at a distance of 110 centimetres from the compass to produce
the same deflection of the needle. Compare the horizontal forces of the earth's
magnetism at Gibraltar and Portsmouth. (B. of E., 1900.)

32. A uniformly magnetised bar magnet 10 centimetres long, having a
moment of 200 C.G.S. units, is placed in a horizontal position with its axis in
the magnetic meridian. A small compass-needle placed at a distance of 10
centimetres east of the centre of the bar is observed to be in neutral equilibrium.
Find the horizontal intensity of the earth's field. (B. of E., 1901.)

33. A small compass-needle makes 10 oscillations per minute under the
influence of the earth's magnetism. When an iron rod 80 centimetres long is
placed vertically with its lower end on the same level with and 60 centimetres
from the needle, and due (magnetic) south of it, the number of oscillations of
the needle is 12 per minute. Calculate the strength of the pole of the iron
rod, (1) neglecting. (2) taking account of, the influence of the upper end.

(B. of E., 1901.)

34. A magnet 10 centimetres long is placed in the magnetic meridian, the
"north " end of the magnet being to the south. The force due to this magnet
just counterbalances the earth's horizontal force (0'18 C.G.S. units) at a place
35 centimetres from the centre of the magnet (along its axis produced). Find
the strength of each pole of the magnet. (B. of E., 1905.)

MAGNETIC MEASUREMENTS

to

35. Explain how to determine the intensity of a magnetic field by observing
the time of oscillation of a needle. A magnetic needle under the influence of
the earth alone makes 10 oscillations per minute. When placed at a point A
in the magnetic field it makes 25 oscillations per minute. Find the intensity
of the field at A in terms of the earth's field. (B. of E., 1905.)

36. Two bar magnets, respectively 10 and 12 centimetres long, are placed
at right angles to the magnetic meridian ; the first 10 centimetres north and
the other 12 centimetres south of a small compass-needle. Compare the
moments of the magnets if tna needle is not deflected. (B. of E., 1908.)

37. What is meant by a neutral point in a magnetic field ? There is found
to be a neutral point on the prolongation of the axis of a bar magnet at a
distance of 10 centimetres from the nearest pole. If the length of the bar be
10 centimetres, and H = 0'18 C.G.S. units, find the pole strength of the magnet,

(Camb. Local, Senior, 1902.)

38. Two magnets have the same pole strength, but one is twice as long as
the other. The shorter magnet is placed in the "end-on" position with its
centre at a distance of 20 centimetres from the axis of suspension of a magneto-
meter needle. Where may the other magnet be placed in order that there may
be no deflection of the needle ? (Camb. Local, Senior, 1908.)

39. A rectangular bar magnet, 10 centimetres long, is magnetised until the
strength of its poles is 150. Find the intensity of magnetisation if it has a
sectional area of 1 square centimetre.

40. Two short magnets, with their axes horizontal and perpendicular to the
magnetic meridian, are placed with their centres 30 centimetres east and 40
centimetres west respectively of a compass-needle. Compare the moments of
the magnets if the needle remains undetected, and show how to derive the
formula employed in the calculation. (B. of E., 1911.)

CHAPTER XIII

TERRESTRIAL MAGNETISM

Exp. 123. Place a glass beaker (to exclude draughts) over a pivoted compass-
needle two or three inches long. Hold a soft iron bar two or three feet long
(e.g. a poker), so that it points roughly east and west, and tap it with a piece of
wood. Then test it for polarity. It may prove to be veiy weakly magnetised,
or not sensibly magnetised at all. Hold it in the plane of the meridian, sloping
down at an angle of 60° or 70°, and having its lower end at the same level as the
N pole of the compass-needle and at the side of it. Tap the bar with the wood,
and notice that the needle is suddenly deflected, the N pole being repelled,
which shows that the lower end of the bar has become a N pole. Carefully
invert the bar, and show that the other end is a S pole. Now tap it again, and
observe that the attraction suddenly becomes a repulsion, showing that the
magnetism of the bar has been reversed.

Repeat the experiment with the bar (1) vertical, (2) horizontal and in the
meridian. Notice that the general effect is the same in each case, although it is
perceptibly weaker in the second position. Repeat with the bar horizontal and
at right angles to the meridian, and notice that no effect is obtained.

If a reflecting magnetometer be used, larger deflections will be
obtained, and it will then be possible to show that the greatest effect
is produced when the bar is inclined at an angle of about 70° to the
horizontal. Hence, the earth's lines of force, slope downwards in this
part of the world, and the angle they make with a horizontal line in
the magnetic meridian is known as the angle of inclination or dip.

It follows that a freely suspended magnet tends to place itself in
this position, so that a compass-needle pivoted at its centre of gravity
must be slightly weighted to keep it truly horizontal (or its point of
suspension must be slightly removed from its centre
of gravity towards the N pole).

Let the angle ABC (Fig. 131) be equal to the
angle of dip. Let AB represent the total strength
of the earth's field, (T), drawn to any convenient
scale, then if AC is the perpendicular on BC, AC
represents the vertical component of the earth's
field (V), and BC the horizontal component (H),
on the same scale.

Remembering that the strength of the field is
measured by the number of lines of force per square
FIG. 131. centimetre, the meaning of these terms may be

illustrated as follows : Suppose that a piece of card,
containing a hole 1 square centimetre in area, is held at right angles

TERRESTRIAL MAGNETISM 175

to the actual direction of the earth's field, i.e. at right angles to AB.
Then the number of lines passing through the aperture is the value
of T. (The earth's field is so weak that this number is less than
unity, see p. 133.) If the card be held vertically and at right angles
to the meridian, i.e. at right angles to EC, the lines will slope through
the aperture, and the smaller number passing through will be the
value of H. Similarly, if the card be laid horizontally on the table,
the number passing through will be the value of V.

From what has been already said in several places, it follows that
the same numbers also represent the respective forces in dynes acting
on a unit pole, and this is, in fact, the usual meaning to be attached
to these terms in dealing with the earth's field.

It follows at once from Fig. 131, that if any two of the four
quantities, T, Y, H, and 0, are known, the other two may be
calculated, for we have

H = Tcos 0

^ = tan (9, &c.
H

In practice, it is usually convenient to measure H and 0. Now,
H is the effective component acting on a magnet free to move only in
a horizontal plane, arid has, therefore, entered into various formulae
in the preceding chapters.

It will be seen from the figure that as the angle of dip increases,
the horizontal component decreases. For this reason, the movements
of a compass-needle become sluggish near the magnetic poles, and its
readings are more affected by friction, for the moment of the restoring
couple, MH sin 0, is small. On the other hand, in the regions near
the magnetic equator, it is the vertical component which becomes
smaller, and in certain places vanishes.

Declination. — A compass-needle does not usually point to the
true north and south. The angle contained between the magnetic and
the geographical meridians at any given place is called the angle of
declination at that place. In this country it is at present about 16°
west of true north. It follows that the magnetic poles of the earth
do not coincide with the geographical poles. In 1831, Sir James
Ross found that the magnetic north pole was situated in Boothia
Felix, 96° 46' W. longitude, and 70° 5' N. latitude. The south
magnetic pole was reached in the year 1909. It is situated at 154°
E. longitude and 72° 25' S. latitude.

Roughly speaking, we may represent the magnetic state of the
earth by supposing that a comparatively short bar magnet is placed
at its centre, having its axis slightly inclined to the axis of rotation
(see Fig. 79). Of course, there is no suggestion that such is the case :
the fact that iron becomes non-magnetic at a red heat, taken in con-

176

MAGNETISM

junction with the high temperature of the interior of the earth,
suggests that we must look in the surface layers to a moderate depth
for the seat of terrestrial magnetism, and the great local irregularity
in distribution adds support to this view.

Magnetic Elements. — The magnetic state at any given place
is completely determined if we know the values of

(1) The declination )

(2) The inclination - at that place.

(3) The total intensity (T) )

These are known as the terrestrial magnetic elements of the place.

As regards the third one, it is usual to measure H, and deduce T
from H and the angle of dip.

To Find the Direction of the Magnetic Meridian.—
The actual direction of the magnetic meridian at a given place (e.g.
inside a laboratory) may be roughly determined by means of a freely

FIG. 133.

suspended or pivoted magnet shaded from draughts. It is, however,
important to remember that the axis of magnetism does not of
necessity exactly coincide with the axis of length, and it is therefore
usual to make a second determination with the magnet turned over,
i.e. the upper side is now downwards. The two directions will not,
as a rule, coincide with the true magnetic meridian, e.g. if, in Figs.
132 and 133, ab is the axis of figure, and mn the magnetic axis, the
first position (Fig. 132) gives the reading Na too small, the true
declination being, Nra. After reversing the needle (Fig. 133), the
reading Na is too great, the true declination being Nw. The true
declination is, therefore, the mean of the two readings.

These diagrams are inserted here in order to show clearly the
nature of the error, but it must be remembered that in such an ex-
periment we usually wish to mark out the direction of the meridian in
space, e.g. across a table, or along the floor of the laboratory, and for
this purpose readings on a scale of degrees are useless. For a rough

TERRESTRIAL MAGNETISM

177

determination, four knitting-needles may be held vertically — two at
each end of the magnetic needle, and as far apart as convenient — and
their positions adjusted until they are exactly in a line with the needle.
This direction is then marked on the table, the needle inverted and
the process repeated. Finally, the angle between the two directions thus
obtained is bisected by a third line, which is the meridian required.

Measurement Of Declination. — To measure the declination, it
is evidently necessary to determine the directions of both magnetic and geo-
graphical meridians, and for the latter
purpose an astronomical observation of
some kind is required.

In the Ke\v pattern declination in-
strument, shown in Fig. 134 (taken from
Carey Foster and Porter's Electricity and
Magnetism), the magnet is in the form
of a hollow steel tube, suspended inside
a glass case by means of a silk fibre
attached to S. This magnet can bo
viewed end-on by means of the telescope
T. In the end of the tubular magnet
farthest away from the telescope, a
small glass- scale is fixed, and at the
other end a convex lens, having a focal
length equal to the length of the tube.
This enables the scale to be seen dis-
tinctly through the telescope. The
stand carrying the telescope and case
is turned round until the middle divi-
sion of the scale is seen on the cross
wires inside the telescope, and the direc-
tion of the latter is noted on the gradu-
ated circle C. Then the tubular magnet
is turned over and the observations
repeated, the mean of the two readings

giving the true direction of the magnetic meridian. It will be seen that the
instrument merely enables the observations, previously mentioned, to be made
with much greater accuracy.

For the determination of the direction of the geographical meridian, the
adjustable mirror m is provided, by means of which it is possible to view any
convenient star (or the sun) through the telescope, the telescope and star (or
sun) being then in the same vertical plane. The exact time at which the
observation is made is noted. The position of the star or of the sun at that
instant with reference to the geographical meridian is then found by referring to
the Nautical Almanac, and thence the actual reading of the geographical meridian
on the instrument becomes known.

Inclination or Dip. — TJie angle which a line drawn in the
actual direction of tlie earth's field makes with the horizontal line, is
called the angle of inclination or dip.

Exp. 124. Fasten a fibre of untwisted silk to the middle of a steel knitting-
needle. Cause the needle to hang horizontally, filing one end if necessary.
Magnetise it by rubbing it witli a magnet, and observe that it sets itself in the
magnetic meridian with its N pole dipping downwards.

M

Fm. 134.

178

MAGNETISM

Evidently, the needle takes up this position owing to its tendency
to point along the actual direction of the earth's field, but in order
that its direction may accurately coincide with that of the field, it is
necessary to ensure that no other couple is acting upon it.
Instruments for measuring dip are known as dip circles.

Dip Circle. One form of the instrument for ascertaining the

inclination is shown in Fig.
135, which is, however, not
sufficiently delicate for very
accurate measurements. It
consists of —

(a) A graduated hori-
zontal brass circle in, sup-
ported on three legs, which
are provided with levelling
screws ;

(It) above this is a plate
A, moving about a vertical
axis, which supports

(?) a vertical graduated
circle M, upon which the dip
is measured. This moves
with the plate A, and is
therefore capable of hori-
zontal rotation.

(d) A magnetic needle
a.b, supported on a horizontal
axis (which passes through
its centre of gravity) at the
centre of the graduated circle M, so that it is capable of moving in a
vertical plane.

(e) A spirit-level n, fixed to the plate A.

Method of Determining the Dip. — (1) Level the instru-
ment. This is done by turning the screws until the air-bubble lies
in the middle of the spirit-level.

(2) Turn the plate A on the circle m until the needle is vertical.
The plane of the needle is now at right angles to the magnetic
meridian, because the horizontal component now acts at right angles
to the plane in which the needle moves, so that its only effect is to
increase the pressure on one support of the needle, and to diminish
it on the other. The vertical component, therefore, acts alone, and
causes the needle to rest vertically.

(3) Turn A through 90° on the circle m. This, of course, brings
the needle into the meridian.

Too much reliance must not, however, be placed on this adjuat-

FIG. 135.

TERRESTRIAL MAGNETISM

ment, as the determination of the true vertical position requires
correction for various errors, e.g. 1, 2, and 3 below.

(4) (a) Read the angles between both poles and the horizontal
line passing through its point of suspension.

(b) Turn the needle so that the faces are reversed, and again
read the two angles.

(c) Turn the plate A through 180°, so that the readings are now
on the other side of the scale. Take the two readings.

(d) Turn the needle so that the faces are reversed, and again take
two readings, making eight readings in all.

(e) Remagnetise the needle so that its polarity is reversed, and
repeat operations (a) (b) (c) (d), thus obtaining eight more readings.

The true dip is found by taking the mean of the sixteen readings.

Sources of Error. — The reason of taking the eight readings
given in (a) (b) (c) (d) is due to the fact that there are several sources
of error to which the instru-
ment is liable.

(1) The error of centering ',
which is due to the axis, about
which the needle moves, not
passing through the centre of
the vertical circle. This error

is easily obviated by reading "
the angle at both ends of the
needle. The reason of this
will be easily understood by
referring to Fig. 136, in which
a simple example is shown.

Suppose that the true dip
is 60°, as shown by the dotted
line. If, however, the axis

pass through the circle at O, then the needle will rest in the position
NS. If the angle BOS be 59°, then the angle AON will be 61°, the
mean of which is 60°.

(2) The axis of figure may not coincide with the magnetic axis,
i.e. the geometrical axis may not contain the poles. This error is
corrected by reversing the needle so that the face which lies towards
the observer is turned towards the instrument. This will be under-
stood from Figs. 132 and 133, and the example thereon.

(3) The divided circle M may not be accurately set, i.e. the line
joining the 90° marks may not be truly vertical. The error thereby
caused is eliminated by turning the instrument through 180°, and
thus obtaining readings on the opposite sides of the scale.

(4) The axis, about which the needle moves, may not pass through
its centre of gravity. This causes the angle,, da on the graduated
scale (Fig. 135) to be either too large or too small; if the centre of

136.

180

MAGNETISM

gravity is below the axis of suspension, the force of gravity acting on
the needle will make the angle too large, as in Fig. 137; if it be
above the axis of suspension the angle will be too small, as shown in
Fig. 138.

To correct this defect the polarity must be reversed by remag-
netisation, each end of the needle receiving opposite magnetism to
that which it had at first. The dip is again observed, and the mean
of the readings taken.

(5) Friction between the supports and the axle of the needle must

FIG. 137

FIG. 138.

Agate knife-edges are generally

be diminished as much as possible,
employed for bearings.

It will be noticed that everything depends on the exact adjust-
ment of the needle in the magnetic meridian. If this condition is
not satisfied, an error will be introduced which cannot be eliminated.
For this reason, it is sometimes preferable to avoid the difficulty by
determining the apparent dip in two positions lying on each side of
the meridian and separated by 90°. It is then only necessary to
turn the instrument through exactly a right
angle, and this can be done with great accuracy.
But it involves taking two complete sets of
readings.

In Fig. 139 let MM be the direction of
the meridian, and let 01} 02 be the measured
values of dip when the needle moves in the
planes AB and AjBt respectively, making
angles a and {$ with the meridian, where a + /?
= 90°. Let 6 be the true angle of dip.

The couple due to the vertical component
V is not affected by the displacement, but that
due to the horizontal component is now partly
acting as a twist on the bearings, and its
effective portion resolved along the line AB is H cos a.

FIG. 139.

TERRESTRIAL MAGNETISM

181

FIG. 140.

\\cosp

Hence, we obtain the three triangles shown in Fig. 140, from
which we have —

H

tan 0

= tan $,

H cos «,

V

T^ 7> = tan ft,

H cos j3

From (1) and (2) cos a = —

tan 0j

"From (1) and (3) cos P = ±~ — n ~ = sin a

tan 0 \2

and sin2 a =

tan

x

tan 0 \2

but sin2 a + cos2 a = 1

tan

tan

(1)
(2)
(3)

tan2<9 tan*01 + tan*0a

. i.e. cot 2# = cot Wl + cot 2#2

Construction of a Simple Dip Circle (Fig. 143).— A dip circle

may be constructed on the following plan, and although the results obtained
with it are merely approximate, the experiments will give us a practical insight
into the methods usually adopted in accurately determining the dip at a
place.

(1) Cut two circular discs of cardboard, 10 inches in diameter. On each
draw two diameters, AB, CD (Fig. 141), at right angles to each other. With

182

MAGNETISM

the centre 0, describe two arcs, EF, GH ; the outer one being 1 inch, and
the inner one If inches from the circumference.

Describe similar arcs in the verti-
cally opposite quadrant ; and then
with a very sharp knife cut out the
portions shaded in the figure. Draw
two diameters, MN, KL, to bisect
the angles AOD, AOC. Now
graduate the circumference AC, BD
of the two quadrants as accurately
as possible into 1 -degree spaces,
marking the point 90° where the
dotted line KL meets the circum-
ference. The points A, C are thus
marked 45°, and M, N, 0° (see also
Fig. 143).1

(2) Cut four pieces of looking-
glass (6 inches long and 2 inches
broad), and glue them to the back
of the cardboard, in the position
shown by the dotted parallelograms.
By this means the shaded space EGHF
and the corresponding one in the
vertically opposite quadrant are completely covered by them ; the mirror side,
of course, is towards the graduated faces. Now glue the two discs, back to
back, so that the diameters AB, CD on one face are similarly situated to those
on the other.

(3) Bore a hole through two thin small corks, and insert in each hole a
small piece of narrow glass tubing, having previously heated each end to remove
the rough edges. Glue the corks to the cardboard circles so that the glass tubes
are exactly at their centres.

(4) Cut a board (12 inches long, 6 inches wide, and f inch thick) to form
the base of the instrument. Also cut two strips of wood (square in section)
8 inches long and f inch wide. Make two square holes in the base —

the outer edges of which are \ inch from each end of the board —
to admit one end of the strips. Having cut a groove down one side
of each strip (\\ inches long and about \ inch deep) to admit and
support the cardboard disc as shown in Fig. 143, glue them in
their places.

(5) Now fix the cardboard on the pillars so that the zero points
are in a horizontal line.

(6) Take two pieces of glass rod, soften one end of each piece
in a blow-pipe flame, and while hot make an indentation as shown
in Fig 142. Fix these in. the base, one on each side of the card-
board, so that if one end of a sewing-needle be placed in the glass
tubing at the centre and the other rests in the indentation, the
needle may be exactly horizontal and at right angles to the card-
board circles.

(7) Cut a piece of clock-spring to form a lozenge -shaped needle,

9| inches long. Drill a hole as nearly as possible at its centre of FlG. 142.
gravity. Using a. small piece of solder, fasten a fine sewing-needle
through the hole, one half projecting on each side — this forms an axis at right
angles to the strip. It will probably be found necessary to file one end of the
lozenge-shaped needle, until, before being 'magnetised^ it remains at rest on its
axis in any position whatever. When this is the case, its centre of gravity
coincides with its axis of rotation.

1 Cardboard scales can be bought at the cost of a few pence.

TERRESTRIAL MAGNETISM

183

(8) Bore a hole in the wooden base in order to insert the axis (the sewing-
needle), while the strip of steel is being magnetised.

(9) Magnetise the lozenge-shaped needle by the method of separate touch.
Exp. 125, to show the method of obtaining the dip ^oith this instrument.

Place the instrument in the magnetic meridian, and repeat the sixteen observa-

FlG. 143.

tions given on p. 179. The following table, which represents the results
of a particular experiment, is given as a guide. A is one face of the needle,
B the other.

Position of
Magnetic Needle.

Position of Face
of Instrument.

Deflections,
Upper, Lower.

Mean.

True I>ip.

A

East

66-5°, 67-5°

67°

B

68°, 69°

68-5°

A

West

66°, 67°

66-5°

B

? }

67 "5°, 68-5°

68°

After remag-
netisation

67° nearly.

A

East

67°, 68°

67-5°

B

66-5°, 66°

66-25°

A

West

67'5°, 68°

67-75°

B

"

66°, 65-5°

65-75°

Exp. 126, to find the angle of dip by means of a reflecting magnetometer.
After adjusting the scale so that the spot of light is at zero, place a rod of soft
iron, about 3 feet long and £ inch in diameter, horizontally, as shown in Fig. 144,
so that it lies in the magnetic meridian with one end immediately behind the
magnetometer mirror — the best distance from the magnetometer will be found
by trial. Notice that there is a deflection, but before taking a reading, tap
the rod several times with a piece of wood. This will help the rod to

184 MAGNETISM

acquire the full amount of magnetisation possible under the influence of the

horizontal component of the earth's field. Head
1 the deflection, say d divisions, and notice that
its direction indicates that A is a X pole. Raise
the other end, B, until the rod is exactly vertical,
! as determined by a simple plumb-line. Do this
and Law/' without altering the position of A, and, hence, to
obtain good results the end A should be held by
a pin or by some simple device which keeps it in
position when the end B is raised. The rod is
now under the influence of the vertical com-
ponent of the earth's field. Tap it again with
the wood, and read the deflection, say rf^ division,
tometer which will be greater than before. Repeat the
S ==» N operations a number of times until fairly con-
i sistent readings are obtained, and take the average

*--*r -* of the best results.

B A

Then we have V^^
i Hoed

FIG. 144. and ^ = tandip = ^

To Determine the Total Intensity. — As already stated, to
determine the third magnetic element, T, it is usual in magnetic
surveys to measure H, and then use the relation H = T cos 0, where

6 is the angle of dip. The measurement is carried out by means
• -» i-

of a portable instrument, which enables the ratio — to be obtained

M

by a slightly modified form of the method explained on pp. 159-161.
The time of vibration of the deflecting magnet is also taken in order
to determine the value of the product MH.

Magnetic Variations— (a) Variations in Declination.—

The declination varies at different places on the earth's surface. At present
it is west in Europe, but east in many places in Asia and in North and South
America. At some places there is no declination, i.e. the geographical and
magnetic meridians coincide.

The declination is, moreover, never constant even at the same place, in fact,
the needle of an exceedingly delicate instrument is never at rest.

Such variations are of two kinds : —

(1) Regular, including secular, annual, and diurnal ;

(2) Irregular or accidental.

Secular Variation. — There is a gradual change in the direction of
the compass-needle at any particular place ; the needle at one time pointing to
the Avest, and at another time to the east, of the true north.

The table at top of p. 185 shows the secular variation at Greenwich.

From it we see that before the year 1657 the declination was east ; in that
year the geographical and magnetic meridians coincided ; afterwards the
declination became west ; and that the greatest westerly declination was
reached in 1816. At the present time the declination is slowly decreasing,
but only at the rate of about 7' per year.

TERRESTRIAL MAGNETISM

185

Year.

Declination.

Year.

Declination.

1580

11° 17' E.

1890

17° 29' W.

1634

4° 0' E.

1893

17° 11' W.

1657

0° 0'

1895

16° 57' W.

1705

9° 0' W.

1898

16° 39' W.

1760

19° 30' W.

1899

16° 34' W.

1816

24° 30' W.

1902

16° 22' W.

1868

20° 33' W.

1906

16° 3' W.

1882

18° 22' W.

1909

15° 47' W.

1889

17° 35' W.

1913

15° 15' W.

Annual Variation. — The needle is also subject to small annual varia-
tions. In London, this variation is greatest at the vernal equinox, and smallest
at the summer solstice, after which it gradually increases during the next nine
months.

Diurnal Variation. — With very sensitive instruments the needle
is observed to have a daily motion. In England the N pole of the needle
moves westward every day from 7 A.M. to about 1 P.M. It then begins to move
eastward, and continues to do so until about 10 P.M. It approximately retains
this position until sunrise.

Irregular Variations. — It sometimes happens that the needle is
suddenly disturbed. These irregular and accidental disturbances are known
as magnetic storms, and frequently accompany such natural phenomena as
volcanic eruptions, earthquakes, and the aurora borealis.

ISOgOnic and Agonic Lines. — Charts have been prepared on
which places having equal declination are joined by a line. Such lines are
called lines of equal declination, or isorjonic lines. Similarly, the line joining
places where there is no declination is called the agonic line (see Map, p. 186).

(6) Variations in Dip. — The value of the dip, like that of declina-
tion, varies in different places on the earth's surface. At the magnetic poles
the needle is vertical, i.e. the dip is 90°. In London (lat. 51° N.) in 191 4,
the dip was 67° nearly. At the magnetic equator there is no dip. In the
southern hemisphere the inclination of the needle is similar to that in the
northern hemisphere, but the S pole is downwards.

Dip is also subject to secular change, i.e. its value alters during a very long
period of time.

The following table shows its alterations near London : —

Year.

Inclination.

Year.

Inclination.

1576

71° 50'

1890

67° 23' 0"

1676 i 73° 30'

1895

67° 15' 0"

1723

74° 42'

1898

67° 12' 0"

1800

70° 35'

1899 67° 10' 0"

1828

69° 47'

1903 67° 0' 51"

1854

68° 31'

1906

66° 55' 0"

1874

67° 43'

1910

' 66° 52' 36"

1913

66° 50' 27"

180

MAGNETISM

TERRESTRIAL MAGNETISM

187

ISOClinic and Aclinic Lines.— The imaginary lines which join
places at which the dip is of the same value, are called isodinic lines. The line
which joins places at which there is no dip, is called the aclinic line, or magnetic
equator (see Map, p. 188).

Magnetic Elements at Various Places.— The following

tahle, selected from the Report for the year 1909 of the Observatory Depart-
ment of the National Physical Laboratory, shows the mean values of the
magnetic elements at various places for the year specified : —

Place.

Year.

Declination.

Inclination.

Horizontal
Force in
C.G.S. Units.

Vertical
Force in
C.G.S. Units.

Greenwich . .

1908

15 53-5 W.

66 56'3 N.

•18528

•43518

Kew .

1909

16 10-8 W.

66 597 N.

•18506

•43588

Hamburg

1903

11 10-2 W.

67 23-5 N.

•18126

•43527

Valencia (Ireland).

1909

20 50-3 W.

68 15'1 N.

" -17877

•44812

Uccle (Brussels) .

1908

13 367 W.

66 1-6N.

•19061

•42867

Munich

1906

9 59-5 W.

63 10-0 N/

•20657

•40835

Baldwin (Kansas) .

1906

8 30-1 E.

68 45-1 N.

•21807

•56081

Athens .

1908

4 52-9 W.

52 117 N.

•26197

•33613

Hong Kong .

1908

03-9 E.

31 2'5 N.

•37047

•22292

Alibag (Bombay) .

1908

1 2-2 E.

23 21 '8 N.

•36857

•15922

Melbourne .

1901

8 26-7 E.

67 25'0 S.

•23305

•56024

Manila .

1904

0 51-4 E.

16 0-2 N.

•38215

•10960

Mauritius

1908

9 14-3 W.

53 44-9 S.

•23415

•31932

Christclmrch(N.Z.)

1903

16 18-4 E.

67 42-3 S.

•22657

•55259

The lines connecting places where the force is of the same value
are called isodynamic lines. (See Map, p. 189.)

Recording Instruments. — In magnetic observatories, instruments
are used which give a continuous photographic record of changes in H, V, and
declination, from which the changes in dip and total intensity may be deduced.

The declination instrument is a delicately suspended horizontal bar magnet
(usually made of several small magnets fixed together, see p. 119), to which a
mirror is attached, and a spot of light is reflected from this mirror upon a slowly
moving strip of photographic paper.

The instrument for measuring the horizontal force consists of a similar
horizontal bar magnet having a bifilar suspension attached to a torsion head.
This head is rotated until the magnet is at right angles to the meridian, in which
position the restoring couple MH due to the earth's field is balanced by the
couple due to the twist on the suspension. The latter may be regarded as con-
stant, and hence any change in H will cause a small displacement of the bar.
Such displacements are recorded as mentioned above.

It is not easy to convert a dip needle into an automatic recording instrument,
and hence the vertical force is measured instead. In this instrument, the bar
magnet is supported horizontally on knife-edges—not unlike the arms of a balance.
Evidently it tends to dip, and this tendency is balanced by placing weights in
a' scale-pan. Any alteration in V will upset the equilibrium, and will cause a
slight displacement of the magnet in a vertical plane.

In each instrument, a simple device is used in order to obtain a constant zero
line from which the displacements can be measured. The mirror is fixed below
the moving part, and is semicircular in shape. A similar semicircular mirror

188

MAGNETISM

TERRESTRIAL MAGNETISM

189

190

MAGNETISM

(the other half of an ordinary circular mirror) is rigidly fixed exactly below it.
Light from a suitable source falls upon both halves, each of which forms an
independent spot of light upon the sensitive paper. When there is no dis-
turbance, the trace will be two straight lines, but generally one will be a wavy
line, and the amplitude of its excursion is determined by reference to the straight
comparison line.

The Mariner's Compass. — Terrestrial magnetism has a most
important influence on navigation. As a magnetic needle always
points to the magnetic north and south poles, and as declination
charts have been prepared, the mariner is enabled to guide his vessel
from port to port by its indications.

A simple form of compass for this purpose consists of (a) a flat
circular card, on the under surface of which one or more light mag-
netic needles are fastened, which are capable of moving horizontally
on a pivot of steel or iridium, fitting into an agate cap.

The card is divided into thirty-two divisions, which are called
rhumbs or points of the compass. These divisions are obtained as
follows : —

The circle (Fig. 145) is divided (1) into four quadrants by means of two
diameters at right angles. The extremities of these diameters are marked
N, S, E, and W (north, south, east, and west).

(2) The four right angles thus formed are bisected by lines, the extremities
of which form the NE, NW, SE, and S\V points.
They are named by placing together the two
letters at the extremities of the bisected quadrant ;
e.g. NE is the point midway between N and E.

(3) These eight angles are bisected ; the ex-
tremities of the lines are named by placing together
the letters at the ends of the arms of the bisected
angles, remembering, however, to put the name
of the cardinal point first, and then the name of
the other point : e.g. the point midway between
the N and NE points is marked NNE (north-
north-east) ; that between S and SW is named
SSW, and so on.

(4) The sixteen angles are again bisected ; any
one of the points thus formed is named by placing
(a) the name of one of the nearest points to it
(precedence being given to the point first obtained in the above process— i.e. N
takes precedence of NNE ; NE takes precedence of NNE or ENE) ; and (6) the
name of the other nearest cardinal point, the two names being separated by the
letter b (by) ; e.g.

The point between N and NNE is marked N 5 E.

„ NE „ ENE „ NE5E.

E ,, ESE ,, EZ> S.

„ SW „ WSW „ SW&W.

,, NW „ WNW „ NW&W.

(I) The needle and card are enclosed in a copper bowl provided
with a strong glass cover.

(c) The bowl is supported on gimbals, i.e. two concentric brass

FIG. 145.

TERRESTRIAL MAGNETISM

191

rings, one of which, fastened to the bowl, moves about two pivots.
These two pivots are fastened into the outer ring, which rests, by
means of two similar pivots placed at right angles to the former, on
a support.

By means of the gimbals, the bowl always remains horizontal, but
in order to obtain steadiness and freedom from the oscillations set up
by the motion of the ship, the time of vibration of the magnet system
must be great compared with the average period of the ship's motion.

As ^ = 27TA/— — - and as H is constant, we see that t may be made
large either by increasing I or by diminishing M. The simplest way

FIG. 146.

of increasing I is to increase the mass of the system, and in the older
forms of compass, this plan was usually adopted — a comparatively
heavy magnet (or system of magnets) being employed. On the other
hand, it is important that the friction on the point should be small,
a condition which is satisfied when the moving system is as light as
possible.

Lord Kelvin showed how to reconcile these apparently conflicting-
conditions. His pattern of compass-needle is shown in Fig. 146. It
consists of an aluminium ring to which the central jewelled bearing
for the pivot is attached by silk threads. Eight short pieces of steel,
like pieces of knitting-needles, are fastened to two silk threads, which
are attached to the radiating silk threads. The whole arrangement

192 MAGNETISM

for a 10-inch circle weighs less than half an ounce. The magnetic
moment is small, and so also is the moment of inertia, but as the
construction places the greater part of the mass in the rim, where it
is most effective in producing moment of inertia, I is large relatively
with respect to M, which is exactly the condition required to make
t great.

Effect Of Iron Ships On the CompaSS.— The magnetism of
iron ships may be roughly ascribed to two general causes: (1) The ship acts
like a weak permanent magnet, due to the magnetism acquired during construc-
tion, its direction and strength depending on the ship's position during that
time. This does not alter in direction with the movements of the vessel. (2)
The ship may be regarded as a mass of iron acted upon inductively by the
earth's field. The direction of this induced magnetism will change with each
movement of the ship, and will also depend on its position on the earth's surface.

These errors are detected and compensated by the operation of " swinging "
the ship, i.e. the vessel is turned completely round so that its bow is directed
in succession to every point of the compass, and in each position the compass
reading is compared with the known direction of the meridian.

It will be seen that the effect of the permanent magnetism will resemble that
due to rotating a bar magnet under a compass-needle. There will be no devia-
tion wrhen the bar magnet points N and S, and the greatest deviation when
it points E and W. Hence, the error will reach a maximum twice in each
revolution (i.e. once in each semicircle), and £uch deviation is, therefore, known
as the semicircular deviation. As the permanent magnetism is not usually
lengthways, it is conveniently regarded as consisting of three rectangular com-
ponents— one lengthways, one sideways, and one vertical (the latter only affecting
the compass-needle when the vessel rolls) — and compensation is obtained by
placing three sets of permanent magnets inside the binnacle below the compass,
pointing forwards, sideways, and vertically respectively, each set being separately
adjusted by trial to neutralise the deviation due to the corresponding component.

Again, the induced magnetism . will, as the ship swings round, produce an
effect similar to that obtained by rotating a bar of soft iron beneath a compass-
needle. There will be no deviation when the bar points N and S, and, also,
none -when it points E and W, because in the latter position it is at right
angles to the earth's field, and is, therefore, not magnetised. But in the four
intermediate positions there will be deviations, reaching a maximum when the
vessel points NE, SW, NW, and SE, and hence this is known as quadrantal
deviation.

A large mass of soft iron symmetrically placed with respect to the needle
would not affect it in any way ; for instance, if two soft iron bare, fixed together
at right angles, were rotated beneath it, one bar would tend to neutralise the
deviation produced by the other. The quadrantal error is, therefore, eliminated
by fixing spheres of soft iron on each side and very close to the compass, their
exact distance being adjusted by trial. In this way, the effect of the lengthways
distribution of iron in the vessel is compensated. Lord Kelvin first made this
method of correction practicable by introducing his small magnetic needles, for
the older type would have required inconveniently large spheres of iron.

Thus far, we have merely considered 'the induced magnetism due to the
horizontal component, but the vertical component produces its effect, the result
being another semicircular error quite distinct in nature from the one already
mentioned, as well as a disturbance when the vessel rolls. Without going
into details, it will be sufficient to say that the former effect is neutralised
by placing a vertical soft iron bar in some definite position — usually in front
of the binnacle — and the second, partly in this way and partly by the influence
of the permanent magnets used for the other corrections.

TERRESTRIAL MAGNETISM 193

It is found that a large amount of the magnetism, induced in the vessel
during its construction, is lost during a long voyage, due, no doubt, to the
buifeting of the ship by the waves. In course of time, however, the conditions
of the vessel with respect to its magnetism become constant, so that it needs
no further correction. The magnetism, which is then retained in the ship,
is called permanent magnetism ; that which it loses, is called sub -permanent.

The AnSChutZ GyrO-CompaSS. — It is worth mentioning that a
novel form of mariner's compass, which does not depend upon magnetism, has
recently been introduced. It is a most interesting and important application
of the principle of the gyroscope.

A heavy wheel, driven at very high speed (20,000 revolutions per minute),
by an electric motor, is mounted in the binnacle, and under the influence of its
own rotation and that of the earth, tends to set its axis of rotation parallel to
the earth's axis.

Such compasses are now being introduced into the British and other navies.
They are unaffected by the presence of iron, and possess the great advantage of
indicating the true or geographical meridian. An obvious disadvantage is the
dependence on rotation, for if the motor fails the compass becomes useless.

EXERCISE IX

1. How may the magnetic meridian be determined by means of a dip circle ?

2. Explain why, in determining the magnetic dip at any place, it is neces-
sary to reverse the magnetism of the needle so as to make each end of it dip
in turn.

3. The N poles of two equal and equally magnetised magnets are attached
to the ends of a light bar of wood, so that the magnets are parallel to each
other and at right angles to the bar, and the S poles are on opposite sides of
it. If the whole be suspended by a thread, so that the bar and the magnets
lie in a horizontal plane, what position will the bar take up with respect to the
magnetic meridian ? Give reasons for your answer.

4. An iron ball is held due north of a compass-needle. Describe the motion
of the needle as the ball is carried round it in a circle in the directions north,
east, south, west, north.

5. What effect (if any) is produced (1) on the weight, (2) on the position of
the centre of gravity, of a piece of steel by magnetising it ? Give reasons for
your answer.

6. A small magnet hanging by a silk fibre makes ten oscillations in a
minute when acted on by the earth's magnetic force alone. When equal masses
of iron are placed at equal distances from it, one to the north and the other
to the south, the magnet makes more than ten oscillations in a minute ; but
when the same pieces of soft iron are placed at equal distances east and west
of the magnet, it makes less than ten oscillations in a minute. Why is this ?

7. Describe and explain some method by which the intensity of the hori-
zontal component of the earth's magnetism at two different places can be com-
pared.

8. A straight bit of straw is hung vertically by a fine silk fibre, and two
magnetised sewing-needles are thrust through it horizontally. Show what are
the essential conditions in order (a) that the needles may point in the same
direction as each would alone ; (b) that they may point equally well in any
direction.

9. Two soft iron rods are placed vertically, one east and the other west of
the centre of a compass-needle ; the lower end of the rod on the east and the
upper end of the rod on the west being level with the compass. Describe
and explain the effect on the compass.

N

194 MAGNETISM

. 10. A bar of very soft iron is set vertically. How will its upper and lower
ends respectively affect a compass-needle ? Would the result be the same at all
parts of the world as it is in this country ? If not, state generally how it would
differ at different places.

11. Find the total magnetic force of the earth at a place where the horizontal
component is '18 dyne and the dip 45°.

12. Describe, by the help of diagrams, the general character of the distribu-
tion of the lines of magnetic force over the earth's surface. (B. of E., 1911.)

13. Supposing an iron ship to behave like a permanent magnet, with a
north pole at the bow and a south pole at the stern, explain howT a compass on
board would behave as the ship swings through 360°. Would the effect be the
same in England and at the equator ? (B of E., 1899.)

14. What is meant by (1) declination, (2) agonic lines, (3) magnetic equator ?
Draw a map showing the general position of the agonic lines, and indicate the
direction of the declination in the regions into which these lines divide the
earth's surface. (B. of E., 1903.)

15. What is an isogonal line? Describe the general form of the isogonals
over the surface of the earth. How are the observations made which are used in
determining the isogonals ? (B. of E., 1906.)

16. A thin uniform magnet, 1 metre long, is suspended from the north end
so that it can turn freely about a horizontal axis which lies magnetic east and
west. The magnet is found to be deflected from the perpendicular through an
angle 0 (sin 6 = -1 ; cos 6= '995). If the weight of the magnet is 10 grams, the
horizontal component of the earth's field is €2 C.G.S. units, and the vertical
component '4 C.G.S., find the moment of the magnet. (B. of E., 1906.)

YOLTAIC ELECTRICITY

CHAPTER XIV

VOLTAIC CELLS

IF by some means the ends of a wire are kept at different potentials,
we produce what is known as an electric current through the wire.
Such a difference of potential is often produced by chemical action ;
but before discussing any method by which it can be done, it will be
advantageous to perform a few preliminary experiments.

Exp. 127. (a) Pour a little strong sulphuric acid into a dry test-tube, and'
drop in a small piece of zinc. Notice that the zinc is unattacked — although
there may be a few small bubbles of gas evolved at first.

(6) Repeat the experiment, using dilute acid, and notice the rapid evolution of
gas. If the gas be collected and examined, it will be found to be hydrogen.

(c) After the action has continued for a short time, pour a little of the liquid
into a porcelain evaporating dish and then evaporate it to dryness over a Bunsen's
or a spirit-lamp flame. A white solid — sulphate of zinc — remains, which has
been formed by the zinc dissolving in the acid.

If the weight of hydrogen be computed, and the loss of weight of the zinc be
determined, the two weights will be constant in proportion, viz. one part of
hydrogen will be evolved, when 31'5 parts of zinc are dissolved. These numbers
are to one another as the chemical equivalents of the elements (p. 326).

This is an example of chemical action, and is represented by the
equation —

Exp. 128. (a) Immerse a piece of pure zinc in dilute sulphuric acid. Observe
that it is unattacked. It must, however, be mentioned that pure zinc is very
difficult to obtain, so that some action will probably occur, although it will be
less than when commercial zinc is used.

(6) Touch the zinc with a copper wire, under the surface of the liquid. Notice
that a greatly increased evolution of gas takes place, and also that the bubbles
emanate from the immersed portion of the copper wire.

(c) Eepeat the experiment with commercial zinc, using very weak acid so that
the action is not great at first. Notice that, when contact is made with the copper
wire, the rate of evolution of the gas is increased, and that it is given off at both
the copper and the zinc.

The Simple Cell. — Exp. 129. (a) Immerse plates of zinc and copper,
side by side, but withmit contact, in dilute sulphuric acid. Notice that evolution
of gas takes place from the zinc at once ; the copper plate being so far inert.

(b) Let the plates touch beneath the surface. At once there is an increased
evolution of gas, which now comes off from both plates.

195

196 VOLTAIC ELECTRICITY

(c) Instead of making contact in the liquid, attach a wire to each plate
(Fig. 147), and notice that exactly a similar effect
is produced when the two wires are brought into
contact.

(d) Put the ends of the two wires on the tongue,
but without letting them touch. A peculiar taste is
perceived.

(e) Hold one wire immediately above a compass-
needle, in the direction of its length, and then touch
the end of that wire with the other wire. Notice
that the needle is deflected.

Experiment 129, (d) and (e), shows that
the connecting wire possesses new and peculiar
FIG. 147. properties.

This arrangement constitutes what is known

as a simple voltaic cell, and the wire is said to carry an electric current.
From a practical point of view, the cell has one obvious defect —
the zinc dissolves rapidly and gives off gas during the whole time it
is in the liquid, whether a current is being taken out of the cell or
not. We see that this loss might be stopped by removing the zinc
when not in use, or more conveniently by using a plate of pure zinc,
which would only be attacked when the wires are joined, i.e. when
the circuit is closed. The use of pure zinc is out of the question, but
there is a simple method — termed amalgamation — by which ordinary
zinc may be made to act like pure zinc.

Exp. 130. Amalgamate the strip of zinc used in the last experiment. This
consists of two operations, of which, with this particular strip, the first has been
performed : (1) Cleaning the zinc by dipping it in dilute sulphuric acid, and (2)
coating its surface with mercury, which is easily done by pouring a little mercury
on the zinc, and immediately spreading it over the surface with a rag or brush.
The mercury unites with the zinc, forming a silvery-white amalgam of mercury
and zinc. The unused mercury ought to be collected and put in a bottle for
future use.

Put the strip back in its place, and notice that no gas is evolved at cither
plate until the plates (a) touch in the liquid, or (b) are connected by a wire
outside the liquid. Then, gas is evolved in quantity, but it comes entirely from
the copper plate. In fact, the experiment strongly suggests that the copper plate
is being attacked and that the zinc is unattacked. If, however, the plates be
weighed before and after the experiment, it will be found that it is the zinc
which has lost weight and not the copper.

This evolution of gas at the inert plate is characteristic of voltaic
action, and when it occurs it is certain that a current is flowing in
some circuit.1

1 On this point beginners are apt to get confused. They have learnt in
chemistry to associate the evolution of gas with chemical action, and thus regard
the hydrogen as .really being evolved at the zinc plate, but in some mysterious
way coming apparently from the copper, although such an idea contradicts the
evidence of their own eyes. As a matter of fact, the hydrogen evolved during
any experiment of moderate length has never been near the zinc, and later
(p. 204) it will be seen that the evolution of gas at the zinc surface is not real,
but only apparent.

VOLTAIC CELLS

197

Now, if an electric current flows through the wire connecting the
plates, a P.D. must have existed between them before they were con-
nected ; and, as already stated, this is only another way of saying
that they are connected by lines of electric, force, which again implies
that one plate is charged positively and the other negatively. If the
current persists indefinitely after connection, it can only be due to
the fact that these charges are replaced as fast as they are removed
by conduction in the wire. It should also be noticed that a flow of
current implies a motion of lines of electric force in the dielectric
outside the conductor.

If either plate is connected to the cap of an electroscope, not
the slightest effect will be observed, because the instrument is
not sufficiently sensitive to respond to such feeble charges. If,
however, a quadrant electrometer is available, a large deflection
will be obtained on connecting up the cell to its terminals, and
by observing the direction of the deflection it can be shown that the
copper plate is charged positively, and the zinc plate, negatively.

A similar result can be obtained in a simpler way by using a
condensing electroscope (p. 57), but, with an instrument of ordinary
sensitiveness, there may be a difficulty in obtaining a sufficiently marked
effect if only one cell is
used. We must, therefore,
anticipate matters a little,
and use about half a dozen
Leclanche or dry cells
(p. 208) connected up in
series (p. 216).

Exp. 131. (a) Bring the
free ends of the wires in con-
tact with the cap and the
disc, as shown in Fig. 148,
and notice that there is no
deflection, either at the mo-
ment of contact or when the
wires are removed. (In the
figure only one cell is drawn for convenience, and the cap and disc are shown
slightly separated in order to mark the charges. Really they are in contact
except for the layer of varnish.) Lift the upper disc, and observe that the leaves
diverge. Show, in the usual way, that the charge is positive.

(b) Repeat the experiment with the wires reversed, i.e. the one from the
zinc touching the electroscope. Show that the charge • in the leaves is
negative.

We know from previous experiments that the presence of the
oppositely charged disc greatly increases the capacity of the cap of
the electroscope, and hence a much greater charge flows into it than
would otherwise be the case. On removing the disc, the capacity
diminishes, and hence, the quantity remaining the same, the potential
rises sufficiently to cause a divergence of the leaves.

CdEES

Zn.

FIG. 148.

198

VOLTAIC ELECTRICITY

Properties of a Conductor carrying- a Current— The

new properties acquired by a conductor, whilst carrying an electric
current, may be summarised as follows : —

(1) A magnetic field is produced in the space around the conductor.

(2) Heat is produced in it during the whole of the time the
current is flowing.

(3) Chemical action takes place in certain conducting liquids
when the current is passed through them.

On the first property mentioned above depends the action of
various forms of galvanometer, which may be regarded as an
instrument primarily intended to detect the existence of an electric
current. Detailed information upon this subject will be given
later; at present it is sufficient to assume that such an instrument
is available, its method of action being immaterial. A very simple

form may be made by the student as follows :

Simple Galvanometer. — (a) Make a wooden framework about
6 inches long, 1 J inch wide, and 1J inch deep. As it is preferable

not to have a top to the frame-
work, support the wooden sides
by rectangular wooden blocks,
as shown in Fig. 149. Cut a
groove about an inch wide
FIG. 149. underneath the bottom.

(b) Wind silk-covered copper

wire ten or twelve times round the frame, so that it lies evenly in
the groove.

(c) Fasten the frame to a wooden base, A (Fig. 150), having first
cut a groove in it similar to tko one in the bottom of the frame.

(d) Attach the ends of the wires to two binding-screws, B, C,
fixed to the base.

(e) Having graduated a circular piece of cardboard, D, in
degrees, glue it to the

bottom of the frame,
taking care that the
zero of the scale is
under the middle wire.

(/) Fix a sewing- A
needle vertically in a BB
small cork so that the FiG. -150.

point projects about a

quarter of an inch, and then glue the cork so that the needle forms
a pivot at the centre of the card.

(g) Place a magnetic needle, E (two inches or so long), on the pivot.

Exp. 132. Connect the wires from a cell to the terminals of the simple gal-
vanometer, and observe that, when the current passes above the needle from
north to south, the N pole of the needle is deflected towards the east, and when
it passes from south to north, the N pole is deflected towards the west.

VOLTAIC CELLS 199

Thus, it is possible to define current direction, although previous
work has shown that there is really — so far as our knowledge goes
at present — a flow of a positive charge one way and of an equal
amount of negative the other way. It is simply for convenience
that we agree to make the term " direction of current " mean
the direction in which the positive charge floics. Outside the cell,
it is from copper to zinc.

Electromotive Series. — It must not be supposed that the
production of a P.D. is peculiar to zinc and copper immersed in dilute
sulphuric acid, or that it is exceptionally strong with these materials.
Practically, any two metals will give the effect, but it is greater
in proportion to the difference in the action of the acid upon them.
Further, liquids of various kinds may be used, e.g. other acids,
a solution of common salt or of other salts, the only change being
in the magnitude of the P.D. set up. The maximum value of the
P.D. produced (which is the value when the plates are on " open
circuit," i.e. when they are not connected by a wire) is called the
electromotive force of the cell. Experiment shows that its value
is independent of the shape or size of the cell and is constant for the
same materials, i.e. it depends only upon the nature of the chemical
action involved (see p. 311).

Exp. 133. Attach the wires from a copper-and-zinc cell to the binding-
screws of the simple galvanometer, and note the direction of deflection.

Exp. 134. Replace the copper by plates of platinum, iron, silver (half a
crown), and carbon respectively. Having attached the wires to the binding-
screws, observe that, in each case, the deflection of the needle is in the same
direction as when copper was used. "We, therefore, conclude that the current
passes through the wire from each of these substances to the zinc.

Exp. 135. Repeat the experiment with iron and carbon plates. Notice that
the deflection shows that the current flows through the wire from the carbon
to the iron.

From experiments similar to these, the following substances,
when partially immersed in dilute sulphuric acid, have been arranged,
so that, any two being used, the current flows through the connecting-
wire from the latter to the former : —

1. Zinc 6. Nickel 10. Silver

2. Cadmium 7. Bismuth 11. Gold

3. Tin 8. Antimony 12. Platinum

4. Lead 9. Copper 13. Carbon

5. Iron

The position of any substance on the list varies considerably with
(a) its condition; (&) the strength of the liquid; and (c) the nature
of the liquid. The greater the distance on the list between any two
bodies, the greater the difference of potential.

The substance from which the current flows through the wire 13
called electro-negative to the other, which is electro-positive.

200

VOLTAIC ELECTRICITY

FIG. 151.

The terms electro-positive and electro-negative substances must
not be confounded with positive and negative poles. The following
consideration will make this clear : The starting-place of the current
is the portion of the electro-positive plate (e.g. the zinc) immersed in
the liquid. This portion is therefore positive, while, as we have
proved in Experiment 131, the external portion is negative. Now,
the parts outside the liquid are the poles of the cell ; thus we learn
that the negative pole is the external portion of the positive plate ;
and vice versa.

A current may, however, be produced by one metal and two
liquids. The beaker, A (Fig. 151), contains strong nitric acid, in
which is placed a porous pot, B, containing a
solution of caustic potash. If the ends of two
platinum wires are immersed respectively in the
acid and the potash, and their free ends con-
nected with a simple galvanometer, the needle
will be deflected in a direction which proves
that the current flows through the wire from the
acid to the alkali.

Local Action. — The simple cell obtained
by placing zinc and copper plates in dilute
sulphuric acid has two principal defects, to
which all cells are more or less subject. The
first is known as local action. This term signifies the useless con-
sumption of zinc, when the cell is on " open circuit." We have
already shown that this defect can be remedied by amalgamating the
zinc. At the same time, this is only a partial remedy, and in any
case frequent re-amalgamation is necessary. To explain this action,
we must refer to preceding experiments, in which it was shown that
pure zinc is not attacked by dilute sulphuric acid, unless it is touched
by a copper1 plate or wire either under or outside the liquid. From
such experiments, it may be inferred that the solution of ordinary
zinc in dilute sulphuric acid is voltaic in nature.
Such zinc contains a certain proportion of other
substances as impurities, e.g. particles of iron, carbon,
arsenic, which behave to it in much the same way as
copper.

In Fig. 152, such a particle is shown embedded in the
zinc surface. The combination forms a minute voltaic
cell, short circuited on itself, and acting like the zinc
and copper plates in Experiment 129, when they are
made to touch beneath the surface of the liquid. The current
direction, in that case, is shown in Fig. 153, i.e. the current passes
from zinc to copper inside the liquid, and from copper to zinc where

1 As we have seen, other substances, e.g. platinum, silver, gold, or carbon,
would serve -for this purpose.

FIG. 152.

VOLTAIC CELLS

201

FIG. 153.

these make contact, whether that be inside or outside (compare

Fig. 147). Hence, the direction of the current set up by the zinc

and the impurity will be as shown in Fig. 152,

and hydrogen gas will be liberated, not at the

zinc surface, but at the surface of the impurity.

As these impurities are exceedingly small and

numerous, the observed effect is the evolution

of gas at the zinc surface as a whole. The

effect of amalgamating the zinc is not easy to

explain completely; roughly, we may regard

the mercury as dissolving some zinc but not the impurities, and thus

covering the whole surface with a thin, semi-fluid layer of zinc

amalgam, which coats over the impurities and presents a uniform

surface to the acid. The efficiency of the amalgam diminishes as the

mercury soaks into the zinc, so that the operation must be repeated,

in fact, new zincs require very frequent amalgamation. After a

time, owing to their gradual absorption of a considerable amount

of mercury, they work with much less attention.

Polarisation. — The second defect has long been known as

polarisation, a term to which no special

Cv meaning should be attached, except as

_^— ~~/^J signifying the phenomenon about to be

^^ described. This is best realised by a simple
experiment.

Exp. 136. Connect a simple cell to a galvano-
meter, and notice the direction of deflection. Con-
nect a second copper plate, exactly like the first,
to the galvanometer as shown in Fig. 154. After
the cell has worked until there is a perceptible
accumulation of hydrogen, take out the zinc and
replace it by the copper. Notice that a slight

leflection, which rapidly decreases in amount, is produced in the opposite
direction.

From this experiment, we learn that a copper plate covered with
hydrogen can form a voltaic cell in conjunction with a second copper
plate — the hydrogen acting in the same way as the zinc.

Now, this tendency to generate a current in the reverse direction
must be in action during the normal working of the cell, and, although
it cannot actually send such a current, it must tend to weaken the
main copper-to-zinc current.

Later, we shall learn that it is preferable to regard the hydrogen,
not as setting up a reverse current but as giving a reverse electromolive
force. To prevent misunderstanding, we must mention that the
presence of this reverse electromotive force would, for many purposes,
be immaterial, if only it remained constant in amount. This,
however, is not the case, for we see that it will be zero at the
instant the circuit is closed, and that it will gradually increase to a

FIG. 154.

202

VOLTAIC ELECTRICITY

maximum as the gas accumulates; every disturbance, which causes
the gas to escape, necessarily altering its value. The slight variations
in current strength thereby produced give rise to difficulties, especially
in electrical measurements.

Besides the action illustrated in the last experiment, the hydrogen
has also a mechanical effect, inasmuch as its presence decreases the
available surface of the copper plate.

Hence, to prevent polarisation we must get rid of the hydrogen.
This was first successfully accomplished by Daniell, whose cell is still
largely used.

Daniell's Cell is constructed in a variety of forms. Its essential
parts are a zinc rod or plate immersed in dilute sulphuric acid,
separated by a porous cell from a solution of
copper sulphate, in which a copper rod or plate
is immersed.

It is often constructed as shown in Fig. 155,
so that the outer vessel, A, is entirely of copper,
and contains a strong solution of copper sulphate.
In order to keep the strength of the solution
constant, crystals of the substance are placed on
a perforated shelf, D, with which the copper
vessel is generally provided. Inside this, is a
cylindrical porous cell, B, containing dilute sul-
phuric acid and a rod of amalgamated zinc, C.

Theory of Voltaic Cells.— It may be
pointed out in passing, that a voltaic cell is really
a device which enables us to control the evolution of the energy due
to chemical action, and to apply it, readily and conveniently, to
various useful purposes. The solution of zinc in dilute sulphuric acid
is an action whose nature is essentially the same as that of the burn-
ing of coal. In the latter case, all the energy is liberated as heat,
and this heat must be utilised at the place where it is produced.
Zinc might be burnt in an open fire, and every pound of it would
then give out a definite amount of heat, subject to the same restric-
tion, although it would obviously be a more expensive fuel than coal.
When zinc dissolves in sulphuric acid to form zinc sulphate, the
reaction is not quite the same, although here again a definite amount
of energy is given out per pound weight. As a general rule, this
energy is liberated as heat, which merely raises the temperature of
the liquid and its surroundings, but when the zinc forms part of a
voltaic cell, other possibilities arise. It is still possible to liberate all
the energy as heat, but that heat is not now necessarily produced at
the spot where the zinc is consumed. It may be all produced in the
liquid if we wish, and in any case some heat must be liberated there,
but this may be as small in amount as we please, and the remainder
can be produced at any part of the circuit, no matter how large that

FIG. 155.

VOLTAIC CELLS 203

circuit may be. Further, the energy is not necessarily liberated as
heat ; it may be employed to do useful mechanical work at any point
in the circuit. It is this flexibility of application which makes an
electric current useful for so many purposes. In itself it does not
create energy : that must always be obtained from some source of
power — coal, a waterfall, zinc, &c. — but it serves to distribute such
energy, to transmit it without serious loss over great distances, and
to apply it at any given point to some required purpose in an efficient
and an easily controllable manner.

Evidently zinc is not the only substance that could serve as fuel
in a cell ; it is, on the whole, simply the most convenient one. Un-
fortunately, there is little prospect of utilising the oxidation of carbon
itself in a voltaic cell, although much work has been done with that
end in view. Again, many inventors have claimed that they have
made cells capable of working vigorously without any serious con-
sumption of zinc, but it is obvious from the above statements that a
certain definite relation, which is independent of the particular kind
of cell used, must exist between the weight of zinc consumed and the
amount of electrical energy produced. Further information will be
given later upon this subject.

The question naturally arises : Whence come the charges, positive
and negative, in the case of a voltaic cell 1 Of course, it is possible
to study the properties of cells and currents without attempting to
give an answer, but it is almost impossible to think correctly about
the matter, unless we have some working theory as a guide. The
following brief sketch is, therefore, given,, although admittedly im-
perfect and open to criticism in many respects.

When a simple salt, e.g. common salt, or an acid (which for our
purpose may be regarded as a salt of hydrogen) is dissolved in water,
there are good reasons for believing that part of the molecules become
dissociated — the extent of dissociation increasing with the dilution of
the solution. That is, instead of NaCl, we have Na and Cl; and
instead of H2SO4, we have H, H, and SO4. These particles are not
in their ordinary state, and do not display their ordinary chemical
properties ; for each carries an electric charge — the metals and
hydrogen (in simple salts) always carrying a positive charge, and the
acid Component a negative charge. Sometimes one kind of atom
carries more charge than another kind, but the greater charge is
always some simple multiple of the smaller one. It is probably the
electrical forces due to these charges which constitute chemical affinity.

Such free charged atoms or groups of atoms are known generally
as ions. If the charge be removed by any means, the particle or
group is no longer an ion, and it at once displays its ordinary
chemical behaviour.

According to this theory, dilute sulphuric acid contains a number
of free hydrogen ions, each carrying a certain positive charge, and

204 VOLTAIC ELECTRICITY

half that number of SO4 groups, each carrying a double negative
charge. When a plate of zinc and a plate of copper (or carbon,
platinum, <fec.) are placed in this solution, there is immediately a
tendency (due to chemical affinity) for the zinc to form zinc sulphate,
which really means that zinc tends to form ions passing into the solu-
tion with a positive charge, and thereby leaving the zinc plate, as a
whole, slightly negatively charged. This negative charge will cause
any further action of the kind to cease almost immediately, and thus
we say that pure zinc is insoluble in dilute acid. During this action,
the copper plate has been practically inert, having less tendency to
ionise into the solution, but when the zinc and copper plates are con-
nected by a wire, the negative charge on the zinc can spread over the
wire and copper. This produces two effects : (a) by reducing the
charge on the zinc, it permits further chemical action to take place ;
(&) the presence of a negative charge on the copper now makes it
attract the wandering hydrogen ions in its vicinity. These move up
to it, and (1) neutralise some of the negative with their own positive,
(2) cease to be ions, immediately acquiring their own chemical
properties and coming off as a gas. Evidently these various actions
may go on continuously, the nett result being the replacement of
hydrogen ions by zinc ions. The effect known as polarisation is partly
a consequence of a certain difficulty experienced by the hydrogen ions
in transferring their charge to the copper plate. This difficulty would
be removed if they could combine chemically with the copper.

Let us now apply this theory to a Daniell's cell. Here, we have
exactly similar actions occurring at the zinc plate, but the copper
plate, when it is joined up, attracts copper ions, there being no
hydrogen ions in its vicinity. The copper ions move up to it and give
up their charge ; polarisation, moreover (in its ordinary sense), ceases
because the copper plate merely becomes coated with fresh copper.1

If the action goes on long enough without renewing the copper
sulphate, evidently hydrogen ions may be liberated as before, for
there is a steady passage of zinc ions into solution, and of hydrogen
ions through the porous pot towards the copper plate.

When such a cell is required to work for a long time without
attention, the dilute sulphuric acid may be replaced by a saturated
solution of zinc sulphate, which completely prevents local action. At
first sight, it seems absurd to think that the zinc will dissolve under
such circumstances, but the previous explanations show that there is
no real difficulty in the process, because as the zinc ions and copper
ions in solution both move slowly towards the copper plate, new zinc
ions will pass into solution from the zinc plate, the nett result being

1 It is usual to express these results by writing : —
Zn + H2S04 = ZnS04 + H2
H2 + CuS04 = H2S04 + Cu.
But such equations are apt to be misleading as to the real nature of the actions.

VOLTAIC CELLS

205

FIG. 156.

the replacement of copper ions by zinc ions in what was originally
copper sulphate solution.

Gravitation Cells. — It is worthy of mention that the use of
porous vessels has several objections, so that cells have been devised
in which the liquids are separated merely by a
difference in density. Two such cells will now
be described, both of which have an action
similar to that of the Daniell's cell just
described.

Callaud's Cell (which appears to be
chiefly used in America) is represented in
section in Fig. 156. V is a glass or earthen-
ware vessel ; C, a copper plate soldered to a
guttapercha-covered wire; Z, a zinc cylinder.
A layer of crystals of copper sulphate (CuSO4)
is placed on the copper plate, and the cell is
then filled up with water. These crystals dis-
solve in the water, forming a solution which
gradually diffuses, and, as it comes in contact with the zinc, forms
zinc sulphate (ZnSO4), which, owing to its lower density, floats on the
solution of copper sulphate.

Minotto's Cell (used in India) has a layer
of sand or sawdust instead of a porous cell.
The arrangement of the cell (Fig. 157) is as
follows : At the bottom of an earthenware
vessel, V, a layer of crystals of copper sulphate,
aft, is placed ; and on this a copper plate, C,
having attached to it an insulated wire, i.
Above this is placed a layer of sand or sawdust,
be, having a cylinder of zinc, Z, resting upon it.
The vessel is filled with water, the same
action occurring as we have described in
Callaud's cell.

The Electric Circuit. — We have seen that the continuous
flow of a current necessarily takes place in a circuit, i.e. a conducting
path extending from one terminal of the current generator to the
other, arid completed through the generator itself. The magnitude
of the current depends upon two factors : (1) the nature of the con-
ducting path, and (2) the electromotive force of the generator. Its
strength is found to be the same at every part of the circuit.

Electromotive Force. — This term, which has been already
referred to, is best defined as being the P.D. between the terminals
of any current generator when no current is allowed to flow.

The P.D.'s we are concerned with in voltaic experiments, although
they are usually of much smaller magnitude, are identical in nature
with those discussed previously, and all that has been said in

FIG. 157.

I

206 VOLTAIC ELECTRICITY

Chapter IV. still applies. Hence, it seems natural to measure E.M.F.
in terms of the same unit (see p. 34), but the student will find that
it is customary to use one set of units in dealing with problems in
statics — known as static units — and another wholly distinct set in
voltaic work — known as electromagnetic units. Why these units
came into existence, and how they are related to each other, will be
discussed in Chapter XXXIV. The fundamental or absolute electro-
magnetic unit of potential cannot be defined at the present stage
(see p. 350) ; it is, however, much too small for ordinary purposes,
and a multiple of it, called the volt, is known as the practical unit.
It is one hundred million times larger than the absolute unit — i.e.
1 volt = 108 absolute units of potential — and is very nearly equal to
the E.M.F. of a Daniell's cell.

To give some idea of relative magnitudes, it may be remarked
that the static unit of potential, defined on p. 34, is equal to 300 volts.
A proof will be found on p. 582.

Resistance. — The student must, however, remember that the
strength of the current does not depend merely upon the E.M.F. of
the cell or battery. Another important factor has to be brought into
consideration, viz. the resistance which the current has to overcome
both in the cells themselves and in the external circuit. The greater
the resistance, the smaller the current which a given E.M.F. will
produce; e.g. if the cells have a dense porous partition, or if the
liquids are separated by sand or sawdust (as in the Minotto cell), and
if the external wires be very long and very thin, the strength of the
current is small, although the E.M.F. may be large.

The practical unit of resistance is called the ohm. It is 109 times
the absolute unit of resistance (see p. 587), and is the resistance of
a mercury column, 1 square millimetre in section, and 106 '3 centi-
metres long, at 0° C. To give a more definite idea of the value of
an ohm, we may mention that about 50 yards of 20 gauge copper
wire has 1 ohm resistance, and the same size and length of iron
wire would have about 6 ohms resistance. The fuller study of resist-
ance and its laws will be found in Chapter XVIII., where it is
shown that the resistance of a conductor of uniform section is
directly proportional to its length, and inversely proportional to its
sectional area.

Current Strength. — The meaning of this term has already
been explained. We applied it to the flow of charge, which takes
place when two points, between which a P.D. exists, are connected by
a conductor. Such a flow always involves a passage of positive
electrification in one direction — which we arbitrarily call the direction
of the current — and an equal flow of negative electrification in the
opposite direction. Hence, current strength is naturally measured by
the quantity of charge which flows past any one point in the circuit in
unit time. Although strictly correct, this is not a very helpful defini-

f

VOLTAIC CELLS 207

tion, and we can define it in a more convenient way by the aid of
Ohm's law.

Ohm's Law. — Ohm discovered, long before any electrical units
existed, that the strength of current in any circuit varied directly as
the E.M.F. in that circuit, and inversely as its total resistance. This
statement is known as Ohm's law. It is also true of any portion of
a circuit, the current in that portion being directly proportional to
the P.D. between its extremities, and inversely proportional to its

TE
resistance. Hence, we have Ccx: — , which, by the choice of suitable

K

units, may be expressed as an equality, and it becomes natural to
define the absolute unit of curpent as the current which flows through
one absolute unit of resistance when the P.D. between the ends of
that resistance is one absolute unit of E.M.F.

The practical unit of current — called the ampere — is that which
flows through a resistance of 1 ohm when the P.D. across it is 1 volt.
As 1 volt=108 absolute units of E.M.F., and 1 ohm=109 absolute
units of resistance, it follows that 1 ampere is y^ the absolute unit of
current.

It may be pointed out that the relation

E.M.F.

Current = — — —,

Resistance

holds good numerically for either practical units or absolute units,
but care must be taken not to confuse them in the same equation.
Hence, it is desirable to have some means of indicating which kind
of unit is being used, and for this purpose we shall in future reserve
the letter C for current strength in amperes, and the letter i for
current strength in absolute units. The occurrence of i in any expres-
sion will then indicate that absolute units are being used throughout,
whereas C will mean that practical units are being employed.

Two other important practical units will be required later, and
for convenience we may now mention them. These are (1) the
"coulomb," the practical unit of quantity, which is defined as the
quantity conveyed by 1 ampere flowing for 1 second ; (2) the
"farad," the practical unit of capacity, which will be understood
if we remember that the fundamental equation used in connection
with statics, Q = VC, applies generally, and that a condenser is said
to have a capacity of 1 farad when a charge of 1 coulomb produces
a P. D. of 1 volt between its coatings.

Verification of Ohm's Law.— Most of the simple experiments
performed by students with the object of verifying Ohm's law, really
beg the question at issue. In fact, it is scarcely a suitable task for
a beginner. Perhaps the best method of procedure is to maintain
a current of constant strength (which need not be known) in a long
uniform wire of fairly high resistance, and then to measure the P.D.

I

208

VOLTAIC ELECTKICITY

between points at different distances apart on the wire. It will be
found that the P.D. is directly proportional to the distance between
the points in question, and, as we know that the resistance of a
uniform wire is proportional to its length, it follows that E ocR, when
C is constant, which partially verifies the law. But it is necessary
to measure the differences of potential by means of some form of
electrostatic instrument, such as a quadrant electrometer, which does
not take any current and whose theory is quite independent of the
matter under consideration. Unless this precaution is taken, it is
extremely probable that the reasoning will be invalidated by the
fact that the results obtained depend upon the use of instruments
and methods based on the assumed correctness of the law.

Other Types of Voltaic Cells. — Although, at the present
time, there are many forms of voltaic cells, the only ones of real
importance — except the special forms used as standard cells — are

(1) the DanielPs cell and its modifications just described; and

(2) the Leclanche cell and its modification, the so-called dry cell.
The first group was until recently used exclusively in telegraphy,

and, although now largely superseded by accumulators, many thousands

of such cells are still in operation for this purpose. ;

The Leclanche' Cells, possessing as they do the power of

giving out small and intermittent currents for years without attention,

can be applied to- many purposes for
which accumulators are unsuitable,
and they are, therefore, being em-
ployed in enormous and increasing
quantities.

The ordinary type, shown in Fig.
158, consists of a rod of gas car-
bon, C, placed in a porous cell, P,
which is then tightly packed round
with small pieces of gas carbon
and powdered manganese dioxide.
These fragments of carbon are then
covered by a layer of pitch, M. A
piece of lead, L, is soldered to the
top of the carbon, to which a binding-
screw is fixed. A rod of zinc, Z, is
immersed in a solution of ammonium
chloride (NH^Cl),1 which, when the
cell is working, forms zinc chloride
(ZnCl2) and liberates hydrogen, which
F 150 is slowly oxidised by the manganese

dioxide.

1 It is worthy of mention that a cell, which is to work without attention for a
long time, must be a single fluid cell, and must be practically free from local action.

VOLTAIC CELLS

209

The chemical reactions may be expressed thus —
2NH4C1 + Zn = Zn012 + 2NH3 + H2.

Ammonia (NH8) comes off from the cell, and the hydrogen acts
on the peroxide of manganese, forming the lower oxide (Mn2O3),
as shown by —

2MnO9

Mn2O3 + H2O.

The depolariser acts too slowly to eliminate polarisation com-
pletely, and, therefore, the cell is not suitable for purposes requiring
a constant E.M.F. If, however, it is allowed to rest, it regains
its original strength, and on this account it is well adapted for
ringing electric bells.

The porous pot is used merely to keep the solid depolariser
around the carbon, and is a disadvantage, inasmuch as it increases
the internal resistance. In the " agglomerate " type its use is
avoided, the depolariser being made up into compressed blocks,
which are kept in position around the carbon by rubber bands ;
while in yet another type, the pot is replaced by a canvas bag.

It is a common error to assume that this cell is totally free from
local action. This is not the case, as from what has been already
said, it is evident that some action will be produced by the impurities
in the zinc. In comparison, however, with the amount produced
^when dilute sulphuric acid is used, it is extremely small; indeed
when the zinc is amalgamated, it is practically negligible.

The E.M.F. of a Leclanche cell, when new, is nearly 1*5 volts.
The internal resistance depends, of course, upon the size and type.
In the ordinary form, it is fairly high and increases with use.

Dry Cell. — The dry cell dates practically from 1884 (although
many attempts had previously been made to secure portability by
the use of various absorbent materials). In
construction and action, it is essentially the
same as the Leclanche cell, except that the
exciting substance is in the form of paste —
sometimes jelly — instead of a solution. Fig.
159 indicates the essential parts of its con-
struction. A zinc cylinder, Z, forms the outer
vessel, enclosed in a paper or cardboard case.
The carbon rod, C, is surrounded by the
depolariser, B — a black paste, composed chiefly
of manganese dioxide, powdered carbon, and
graphite mixed with glycerine and ammonium
chloride solution. This paste is often con-
tained in a porous bag. The zinc cylinder
is lined with a white paste, W, made by
mixing plaster of Paris with a solution of

ammonium chloride • certain other materials, e.g. flour and glycerine,

Q

2 Tfi

FIG. 159.

210

VOLTAIC ELECTRICITY

being added to improve the consistency. Sand, S, is placed above
the paste, and upon this is a layer of sawdust, D, into which a glass
tube, G, to carry off the gases, is inserted. The top is then sealed
up with bitumen or pitch, P. A wire, R, is connected with the zinc,
and the carbon carries the usual binding-screw, T.

Grove and Bunsen Cells. — When a fairly large current for
experimental purposes is required for some minutes at a time, the
Leclanche cell and its modifications are unsuitable. Wherever
possible, accumulators are now employed, but if these are not avail-
able, some form of Grove, Bunsen, or chromic acid cell may be used.
The first two are chiefly of historical importance. In the Grove's
cell the outer vessel, is generally a flat cell of glass or earthenware,
A (Fig. 160), which contains a strip of amalgamated zinc, B, im-
mersed in dilute sulphuric acid. The zinc is bent round a porous
B cell, C, in which is placed a piece of platinum

foil, D, immersed in strong nitric acid.

I The chemical reactions in this cell are re-

^^< D -n A presented by the following equations —

FIG. 160.

This hydrogen is liberated at the platinum
surface in the presence of nitric acid, upon
which it acts to form water and nitrogen
peroxide, thus —

H2 + 2HNO3 = 2H2O + 2NO2.

The nitrogen peroxide is a dark red gas,
which, unlike hydrogen, is incapable of pro-
ducing polarisation of the platinum plate.

The E.M.F. is from 1-8 to 2 volts, and as
the internal resistance is low (on account of the
excellent conductivity of nitric acid and the
comparatively small distance between the plates), it is capable of
giving a very large current on short circuit, but, apart from the fact
that — like all double-fluid cells — it must be specially set up and
taken to pieces again every time it is used, the initial cost of platinum
is prohibitive, nitric acid is somewhat expensive, and the nitrogen
peroxide fumes are injurious to adjacent apparatus. Bunsen's cell
is similar to a Grove's in principle and action, and differs in con-
struction only by having a rod of carbon in place of a platinum
plate.

If we replace the nitric acid in Bunsen's form by a solution of
chromic acid, we obtain a cell which is useful, convenient, and free
from fumes.

Chromic Acid or Bichromate Cell. — For many experi-
mental purposes, a single-fluid type of cell is more convenient than
a double-fluid one. In the chromic acid (or bichromate) cell, plates

VOLTAIC CELLS

211

of zinc and carbon, attached to a suitable framework, dip 'into a
mixed solution of chromic acid and sulphuric acid, and are lifted out
when not in use.

The \necessity of setting up the cells every time they are used is
thus avoided. The E.M.F., however, remains constant for a short
time only, as there is more polarisation than in the double-fluid
arrangement. The well-known " bottle bichromate" cell (Fig. 161)
is of this type, and consists of a zinc plate, Z, attached to a brass
rod, which slides up and down a brass tube passing, through an
ebonite cover, and by means of which the zinc, plate may be removed
from the liquid when not in use. The two carbon plates, CO, one
on each side of the zinc, are in this case attached to the cover, and
remain in the liquid.

The absence of porous pots and the closeness of the plates greatly
diminish the internal resistance, and thus enable this cell to give
out a relatively large current, when the external
resistance is low. The E.M.F. is nearly 2 volts.

When the circuit is completed (a) the sul-
phuric acid acts on the zinc, forming zinc sulphate,
(b) the hydrogen reduces the chromic anhydride,
according to the equation

3H2 + 2CrO3 = Cr2O3 + 30H2.

The chromic oxide thus formed then dissolves in
the sulphuric acid.

Cr2O3 + 3H2S04 = Cr2(SO4)3 + 3OH~.

There is no reason to suppose that the reaction
actually occurs in two stages, and the two equations
may be combined by uniting FIG. 161.-

2CrO3 + 3H2 + 3H2S04 = Cr2(SO4)3 + 6OH2.

The student will notice from these equations that the effective
depolariser is chromic anhydride (CrO8). Years ago, this substance
was not on the market, and it is very troublesome to prepare on a
small scale. A solution in water (known as chromic acid) is, however,
readily obtained by adding sulphuric acid to a solution of potassium
bichromate —

K2Cr2Or + H2SO4 + H2O = 2H2CrO4 + K2SO4.

For a long time this method was employed, and hence such cells are
termed bichromate cells. The presence of potassium sulphate is
immaterial.

Now that chromic anhydride can be bought cheaply, its solution

212 VOLTAIC ELECTRICITY

is much better for use, and less expensive to make, than that of the
bichromate.1

Edison- Lalande Cell. — This cell is largely used in America
for special purposes (e.g. operating railway signals), which require
fairly strong currents without risk of failure. It is a single-fluid
cell consisting of plates of compressed copper oxide and of zinc
in a solution of sodium hydrate. It is free from local action and
polarisation ; also, as no porous pot is required, the internal
resistance is low. Hence, it is capable of giving out relatively
large currents for a long period without attention, and it is very
reliable in operation. Its greatest disadvantage is the low ness of
the E.M.F., which is about -7 volt per cell.

The reactions may be written —

Zn + 2NaOH = ZnNa202 + H2.

The hydrogen combines with the oxygen of the copper-oxide
plate, and does not appear as gas.

When the copper-oxide is exhausted, it is readily replaced by a
new plate.

We take the following data from information supplied by the
Edison Manufacturing Co. : A medium size of cell is contained in
an outer jar 7|- in. in diameter and 8| in. high. The internal
resistance is '043 ohm, and hence on short circuit the maximum
current is 15*5 amperes, although such a current could not be
long maintained. It is rated to give a continuous current of
4 amperes and to possess a capacity of 300 ampere-hours, i.e. it
can yield a current of 4 amperes for 75 hours, 2 amperes for
150 hours, and so on, before the plates require renewal.

The Benkb Cell. — This is the most recent innovation in
primary batteries, and it possesses certain novel features, which
may cause it to become very advantageous in practice. It is a
carbon-zinc single-fluid cell (using some form of chromic acid solu-
tion), the chief point of interest being the method adopted for
eliminating polarisation. This consists in keeping up a slow flow

1 Single-fluid Solution —

1 Ib. Cr03 (red powder).
6 Ibs. water (nearly 5 pints).

1 Ib. strong H2S04 (added last with constant stirring).
Allow the solution to cool before using.

Depolariser for Double-fluid Cells —

3 Ibs. Cr03.

4 Ibs. water.

3 Ibs. strong H2S04.

VOLTAIC CELLS 213

of exciting liquid, so that the layers in contact with the carbon are
being continually removed The carbon, which is specially prepared
to make it porous, is in the form of a hollow, flattened cylinder with
open ends. The lower end of the carbon is closed by a leaden plate,
through which passes a leaden pipe (carried up outside and bent
over like a siphon to form the waste-pipe mentioned below), and the
upper end is fitted with a leaden ring, thus making a vessel open at
the top. Inside this vessel is placed the zinc plate. Surrounding
the carbon is an outer jacket of sheet-lead, fused to the lead plate
below and to the lead ring above, so as to form a narrow, closed
chamber outside the carbon. This chamber is also provided with a
pipe, through which the exciting fluid, contained in a vessel placed
about five feet above the cell, is delivered into the chamber, under
the pressure due to this height It then slowly percolates through
the walls of the carbon to the zinc, whence it is led off by the waste-
pipe, either to be thrown away or, if not exhausted, to be pumped
up again to the supply vessel.

Evidently the arrangement is not convenient for single cells ; at
present the standard form is a seven-cell battery.

Excellent results have been obtained, and it seems possible that
this battery may to some extent answer the purpose of accumulators
in places where there are no conveniences for charging the latter.

Standard Cells. — The simplest method of measuring a differ-
ence of potential or an E.M.F. (and thereby indirectly a current) is
to compare it with a known E.M.F. ; and such an E.M.F. is best
obtained by means of a voltaic cell.

A cell, which is to serve as a standard of E.M.F., must satisfy
very stringent conditions. It must be made up from materials
readily obtainable and easily purified, so that cells prepared by
various observers in all parts of the world may not differ in E.M.F.
by any appreciable amount, and it must last indefinitely, without
attention, when once set up. It is also very desirable that the
E.M.F. should not vary to any material extent with change of tem-
perature, i.e. it should have a small temperature coefficient. On the
other hand, as it is not required to give a current, its size may be
very small (and consequently its internal resistance very large) with-
out affecting its usefulness.

For many years the legal standard of E.M.F. in this country has
been Clark's cell, but the International Conference on Electric Units
(held in London in 1908) recommended the adoption of the Weston
cadmium cell in its stead, and hence it is desirable to describe both
these cells. It will be understood that their actual E.M.F. at various
temperatures has been determined with the utmost care by the most
skilled observers, who have made absolute measurements by methods
so refined as to be unsuitable for ordinary purposes.

Clark's Standard Cell.— The metals used are zinc and

214

VOLTAIC ELECTRICITY

— seal of pitch.

/xgi — -cork.

.glass tube,
platinum wire

zinc rod

mercury.

mercury ; the exciting fluid is a solution of zinc sulphate saturated
at 30° C. ; and the depolariser, mercurous sulphate (which is an
insoluble powder). This substance is reduced to metallic mercury
by the zinc ions travelling towards the negative plate (i.e. the
mercury) — an action which may be written Zn + ZnSO4 + Hg2SO4

= 2ZnS04+2Hg. The materials
are purified, and the cell is set up
in accordance with a very elaborate
specification drawn up by the
Board of Trade. One of its forms
is shown — full size — in Fig. 162.
The containing vessel is a small
glass tube, the mercury being at
the bottom. Connection is made
to it either by means of a platinum
wire sealed through the glass, or,
as in the figure, by a platinum
wire protected by a glass tube.

'gj?L~~mercuraits sulphate. Tne E.M.F. of this cell is usually
taken as 1-434 volts at 15° C.,
but at the 1908 International
Conference this was altered to
1-4326 volts at 15° C. The cell
has the disadvantage of possessing
a rather high temperature co-
efficient, its E.M.F. falling by '00115 volt for each centigrade degree
rise of temperature above 15° C. The practical disadvantage is not
so much due to any uncertainty as to the value of the E.M.F. at any
given temperature, as to the difficulty of ascertaining the exact-tem-
perature of the cell and of keeping it constant.

West on Cadmium Cell. — In this cell, cadmium in the form
of amalgam is used instead of zinc ; and cadmium sulphate instead
of zinc sulphate.

The pattern as improved and adopted by the National Physical
Laboratory is shown in Fig. 163, for which we are indebted to
Mr. H. Tinsley. The tubes are hermetically sealed, and the
slight constriction at their lower parts results in the formation
of a taper plug of crystals, which holds everything in place,
and which makes the cell much more portable and safe during
transit.

The E.M.F. was formerly taken as 1*0195 volts at 17° C., but
very careful measurements, carried out recently, have shown that its
true value is 1'0183 (International)1 volts at 17° C. Its great
advantage is due to its small temperature coefficient, which is about

1 The meaning of the term " International " will be understood after reading
Chapter XXXIV.

FIG. 162.

VOLTAIC CELLS 215

•000036 volt per degree centigrade. The vertical tubes are about

Of.

Crystals

Amalgct'
Cd. + Hg.

Cd. 804
Solution

Hg.

FIG. 163.

3 inches long by J inch diameter, and the internal resistance (for this
size) is about 900 ohms.

EXERCISE X

1. "When zinc and copper are placed in contact in dilute sulphuric acid,
hydrogen bubbles (produced by the action of the sulphuric acid on the zinc) are
given off at the copper. How is this explained ?

2. What do you understand by the term electric current ?

3. What is meant by polarisation of the electro-negative plate ? How can
you show its effects on the current ?

4. Each terminal of a battery of, say, 100 cells is connected with a separate
delicate gold-leaf electroscope. State and explain the effect on each electroscope
produced by connecting with the earth : (i. ) the middle of the battery ; (ii.) the
electroscope connected with the platinum end o'f the battery; (iii.) the electro-
scope connected with the zinc end of the battery.

5. A plate of zinc and a plate of copper are in contact at one end. Their
other ends are connected (i.) by a piece of platinum, (ii.) by a drop of dilute
acid. Why does a current flow round the circuit in the second case, and not in
the first ?

6. A Leclanche cell is connected by long thin wires to a galvanometer, the
needle of which is deflected. The poles of the cell are then bridged across for a
short time by a piece of thick copper wire. After the removal of the thick wire,
the galvanometer deflection is much less than before, but gradually rises to its
former value. Explain this. (B. of E., 1898.)

7. Describe one form of primary battery suitable for electric bells, and give
a general explanation of its action.

CHAPTER XV

OHM'S LAW APPLIED TO BATTERY CIRCUITS

LET us consider a Daniell's cell having its terminals joined by a
wire. The E.M.F. is about Tl volts, and is independent of its size
(see p. 199). Let this be denoted by E. Then, if R be 'the total

Tjl

resistance of the circuit, we have C = —

It is important to notice that R is made up of two distinct parts :
(1) the external resistance, which can be varied at will; and (2) the
internal resistance, which depends upon the size of the plates, their
distance apart, and the nature of the liquids used, and which cannot
be altered after the cell is set up.

The internal resistance of a Daniell's cell of about one pint
capacity will usually be from J to 1 ohm.

It is convenient to retain R for the total resistance, and to indi-
cate its components by small letters. Thus, if rx* indicate the
external resistance, and rb the internal or battery resistance, we have

andC= E

The greatest current the cell can give out will be obtained when
rx = 0, i.e. when the terminals are joined by a thick wire of negligible

Tjl

resistance, and its value is evidently — , which (with the cell in

rb

question) is probably less than about 2 amperes. In this case the
cell is said to be " short-circuited."

Grouping of Cells. — (a) In series. In this arrangement, the
copper of one cell is connected with the zinc of the next, and so oh
(Fig. 164).

When similar cells are arranged "in series," the difference of
potential, or as we may call it, the E.M.F., of n cells is n times that
of one cell.

Let us consider two Daniell's cells, A and B.

There is a certain difference of potential between the zinc and

* In this notation, the symbol x is a suffix representing external ; the form x
should not be used, as it is apt to be regarded as an algebraic quantity.

216

OHM'S LAW APPLIED TO BATTERY CIRCUITS 217

FIG. 164.

.the copper of A, and an equal difference between the zinc and the
"copper of B, but when the zinc
of A and the copper of B.are
joined, their potentials are equal-
ised ; whence the difference of
potential between two cells ar-
ranged " in series " is twice that
of one.

The resistance, however, of
n cells is n times that of one, since the length of the liquid traversed
is n times that of one cell. Ohm's law, therefore, becomes, with n
cells arranged "in series,"

(i.)

(1) If the external resistance rx is very large compared with nr^
this equation becomes C = — , i.e. the current is n times that of one

rx

cell, acting through the same resistance.

(2) If the external resistance is very small compared with rb, we

have C = — = -s- , hence, in this case, the current can never be greater
nrb rb

than that given by one cell on short circuit, and can only reach that
value when the battery is short circuited.

(b) In parallel, i.e. all the copper terminals areN connected with
one another, and all the zinc terminals with one another (Fig.

165). In this arrange-
ment the zincs, being
connected together, form,
as it were, one large zinc
plate ; similarly the cop-
pers (in a Daniell's bat-
tery) form one large
copper plafe. Now, as
FIG. 165. we have pointed out

(and as Volta demonstrated long ago), the difference ' of potential
between two metals in a given liquid does not depend upon their size
but merely upon the kind of metal employed. Whence, in parallel,
the E.M.F. is the same as that of a single cell. The resistance, how-
ever, of n cells is - that of one cell, because the sectional area of the
n

column of liquid traversed is n times that of one cell. Whence, for
this arrangement with n cells, Ohm's law becomes

c_ E riE

~

B

218

VOLTAIC ELECTRICITY

(1) If, therefore, the external resistance becomes very large, so
that rb is very small compared with nrx, we have

nrx rx

i.e. the current is the same as that of one cell.

(2) If the external resistance is very small, so that rx becomes

practically zero, we have $ = —

i.e. the current is n times that of one cell. ,

We thus learn that, if we wish to obtain a current strength
directly proportional to the number of cells, we must use the in series
arrangement when we have a large external resistance, and the in
parallel arrangement when we have a small external resistance.

(c) Mixed circuit is a combination of (a) and (&). In Fig. 166,

FIG. 166.

each set of four cells is joined in series, while the four at each end
are joined in parallel.

Suppose that we have m rows-of n J3ells arranged in series. Each
row of n cells will have an E.M.F. of wE, and a resistance of nrb, but
as we have m rows arranged in parallel, the internal resistance will

be — of one row, i.e. — of n?\, or — 6, whence, the current strength
m m m

is given by

n riE

(iu.)

Best Grouping of Cells. — Sometimes it is required to find how
a given number of cells is to be arranged in order to send the
maximum current through a given external resistance (although the
problem has little practical value). ' We have, therefore, to find the

OHM'S LAW APPLIED TO BATTEPxY CIRCUITS 219

values of n and m which will make C a maximum, when mn — o,
constant = the total number of cells.

Dividing equation (iii.) by n, we obtain

Now, the denominator is the sum of two terms, whose product is
constant for all values of n and m, and by a well-known theorem,
this sum is least when the terms are equal.

Therefore, the denominator will be least, and the current greatest,
when

n m

nr-h
i.e. rx= — -

m

Thus, it is evident that we must choose n and m so that the internal
resistance of the battery is as nearly as possible equal to the external
resistance. (See remark on p. 234.)

P.D. between the Terminals of a Cell— In applying Ohm's
law to actual circuits, students are liable to some confusion of thought,
and as we have previously mentioned, it is advantageous to use a fixed
notation, especially at first. To that given on p. 216, we may now
add : ex to represent the P.D. between terminals when a current is
flowing ; eb, that portion of the E.M.F. required to send the current
through the internal resistance of the battery ; where E is the E.M.F.
of the battery (or cell).

Here, we regard E as a definite quantity for a given cell, which
is not altered by the passage of a current, i.e. we assume freedom
from polarisation. On open circuit, i.e. when no current is flowing,
the P.D. between the terminals is identical with E, hence E may be
measured by connecting a suitable voltmeter a to the terminals of the
cell. When a current flmvs, we may regard E as split up into two
components, one of which (ex) can be measured by a voltmeter, and
denotes that portion of E which is actually useful in sending the
current through the external resistance, rx ; while the other (eb), which
apparently disappears and cannot be directly measured, represents the
portion of E required to send the current through the cell itself. There
is, of course, only one current, which is the same all round the circuit.

1 For the time being, we shall assume that certain well-known instruments
are available, e.g. (a) voltmeters, which measure directly, with moderate accuracy,
the P.D. between two points to which they are connected ; (b) ampere meters or
ammeters, which record directly the current flowing through them when intro-
duced into a circuit. In Chapter XXXIIL, their principle and construction will
be more fully discussed.

220 VOLTAIC ELECTRICITY

Hence, although ex and eb vary with the current flowing, we have
under all circumstances

also e x = Or x and eb = Crb
Whence, we have ex + eb = C(rx + rb)

or =

These results may be demonstrated as follows, assuming that we have
(1) a voltmeter of fairly high resistance and of suitable range, (2) a
suitable ammeter of very low resistance, and (3) a variable external
resistance, whose value need not be known.

Exp. 137. Connect up, as shown in Fig. 167, aDaniell's cell, a voltmeter, V,
an ammeter, A, a plug key, K, and an adjustable re-
sistance, rx. (The voltmeter connections are shown by
dotted lines to help the student remember that the
voltmeter should be regarded, not as forming a parallel
circuit, but as a mere accessory, which neither disturbs
nor alters the state of affairs in the real or current
circuit. )

The cell should be set up a short time before use,
or it should be short-circuited for two or three minutes,1
in order to bring it into a steady state, otherwise, as
the liquids soak through the porous pot, the internal
.p 7 resistance may alter during the experiment. A plug

IG< key is used — not a tapper key, as this is apt to intro-

duce a variable resistance (depending on the pressure
applied) which would make the readings unsteady.

(1) Read the voltmeter with key open, i.e. while no current is flowing.
Suppose that the reading is 1'05 volts. Then 1'05 volts is the P.D. between
the terminals on open circuit, and is, therefore, identical with the E.M.F.

Whence E = 1'05 volts.

Now close the key, having first introduced the maximum resistance, so that
only a small current flows. Notice that the voltmeter reads less than before.
Open the key, and notice that the voltmeter reading instantly returns to 1 "05
volts. Close the key and gradually reduce the resistance. Notice that as the
current increases, the voltmeter reading steadily decreases, but that it always
returns to its old value when the circuit is opened.

This effect is not to be confused with that of polarisation (which
would reveal itself if the experiment were performed with a Leclanche
cell by the reading not returning instantly to its old value when the
circuit is opened), for it is inherent in generators of all kinds, and
depends upon their possession of an internal resistance.

Let us suppose that the voltmeter reads '75 volt when the
ammeter reads '6 ampere. Then the P.D. at the terminals is '75
volt, which is the portion of the E.M.F. sending the current through
the external resistance, rx.

-1 See the note at the end of Experiment 138 on short-circuiting a Daniell's
cell.

OHM'S LAW APPLIED TO BATTERY CIRCUITS 221

i.e. in our notation, ex = '75 volt when C = -6 ampere,

but ex = Crx

•75

rx = -— = 1*25 ohms
•6

Hence this measurement gives us the value of the external resistance
(including, of course, that of the ammeter and of the connecting
wires).

Again, 1*05 — '75 volt or '3 volt has apparently disappeared.
This portion of the E.M.F. has been lost through the internal resist-
ance, and is denoted by eb.

Now eb = Crb

.

whence rb = — = '5 ohm
*6

Whence we see that by taking readings (a) on open circuit and (b)
when the cell is sending a current, we obtain data, which determine
both the internal and the external resistance.

(2) Alter the current by varying rx, and take another pair of
readings. Suppose the voltmeter reads '15 volt when ammeter reads
1*8 amperes. Then we know that in this case

ex = -15 volts, and e&-E-ex = 1-05--15 = '9 Volt
but ex = C?*x and eb = Crb

i.e. -15 = l-8rx '9 = l'8r6

*•• r==>5 ohm

Here we notice that the internal resistance, r6, is a property of the
cell, which is not affected by the passage of a current. Strictly, this
is not quite true, for the resistance of the liquids in the cell depends
on the strength of the solutions and the extent to which they are
mixed, and these factors are slightly altered by a current ; but within
reasonable limits our statement is correct.

Simple values have been taken in the above example for the sake
of clearness ; in practice, the accuracy of the results depends upon
the suitability of the instruments, and, for reasons which will be
apparent later, the voltmeter reading will be usually slightly less
than the true E.M.F.

It follows from our results that the cell in question cannot, under

F 1*05
any circumstances, give out a larger current than — or — — - = 2'1

amperes, a fact that may be shown by experiment.

222 VOLTAIC ELECTRICITY

Exp. 138. Short-circuit the Darnell's cell, by connecting it up with thick
wires to an ammeter of very low resistance, and observe that the current is
approximately of the above value.

(A Daniel 1's cell is one that may be short-circuited without injury, but this
experiment should not be done with other cells without due consideration, as
both cell and instrument may be seriously damaged. )

Exp. 139. Repeat Experiment 137, using a cell of low internal resistance—
an accumulator answers well, or failing that, a double-fluid chromic acid cell.

If an accumulator is used we may 'obtain, on open circuit, a reading of
2 volts (or slightly less),

.'„ E = 2 volts.

Insert the maximum resistance, and notice, on closing the key, that a much
larger current is obtained. (The experimenter must be careful to insert the
resistance before completing the circuit, or he may easily injure the ammeter.
With a cell of fairly high internal resistance, such as that used in the previous
experiments, there is little danger of this kind. ) Gradually reduce the resist-
ance, and notice particularly that the voltmeter reading decreases remarkably
little, even for quite large currents.

For instance, it may be found that ex = l'9 volts when 0 = 5 amperes
This means that ei= '1 volt, and as

. •. r& = -fa ohm.

Again, by altering rx, we may find that ex = 1 '8 volts when 0 = 10 amperes,
which again gives r& = fa ohm.

An accumulator has such a remarkably small internal resistance
because its plates are large in area and are placed close together, and
we see, from the above values, that low internal resistance is a very
desirable feature when large currents are required.

We also learn that on short circuit, the current should be
2 ^--^o = l 00 amperes. No doubt it might be something like this
for the first instant, but such a heavy discharge would, by chemical
changes, immediately increase rb; in any case, the cell would be
ruined, and students must be especially careful to avoid short circuits
when working with accumulators.

On account of the importance of this subject, we will now work
a few numerical examples suggested by the above-mentioned facts.

Example I. — Six cells are arranged in series and connected to
an external resistance of 3 '6 ohms. Each cell has an E.M.F. of
1*8 volts and an internal resistance of '3 ohm. Find the P.D.
between the terminals of any one of the cells.

Now C = — = -r— = = 2 amperes.

rx + Qrb 3-6+1-8 5-4

We may arrive at the required result by several independent
lines of thought, and it is useful for a student to check his work
in this way. For instance, (a) we may argue that if 2 amperes flow
through an external resistance of 3-6 ohms, then 2x 3'6 = 7'2 volts

OHM'S LAW APPLIED TO BATTERY CIRCUITS 223

must be the P.D. across rx, i.e. across the terminals of the battery,
or in other words, ex = 7*2 volts.

7*2

Now, if the P.D. across 6 cells is 7'2 volts, it is — =1-2 volts

o

across one cell, which is the answer required.

(b) We may consider each cell individually instead of the whole
battery. We know that 2 amperes flow through each cell, in which
the resistance is '3 ohm. Therefore, 2 x '3 or -6 volt will be re-
quired inside the cell, and this portion apparently disappears, whence
each cell will lose -6 volt out of the 1'8 volts it possesses, and the
P.D. across it will be 1-2 volts.

Example 2. — Using the same data, repeat the calculation when
the six cells are in parallel.

TJ1 -I .Q 1 -ft

In this case C = - = - = - = '49 ampere.
r - 3>«

x+± 3-6 +

(a) Now if '49 ampere flows through 3*6 ohms, the P.D. required
is '49 x 3 '6 volts or 1'76 volts, and this must be the P.D. between
the terminals' of the battery. As the cells are in parallel, this must
also be the P.D. between the terminals of any one of the cells.

(b) Considering one cell only, we notice that it must give out

•49

—- = •081 ampere, and if '081 ampere flows through an internal re-

sistance of -3 ohm, the volts lost will be -081 x -3 = -0243 volt, and
the cell will have left l-8--0243 = l-77 volts, which will be the P.D.
at the terminals.

Example 3. — The P.D. between the terminals of a battery of
twelve cells in series is found to be 16 volts when the current is
4 amperes. When the external resistance is reduced until the current
is 6 amperes, it falls to 12 volts. Find the E.M.F. and the internal
resistance of each cell, and the two values of the external resistance.

The external resistances are found at once, for as ex = Ox, we
have in the first case 16 = 4rx, i.e. rx = 4 ohms; and in the second
case 12 = 6rx, i.e. rx = 2 ohms.

Again, let E = E.M.F. of the whole battery, and

rb = internal resistance of the whole battery,
then we have E - ex = eb, where eb = Crb

••• E-«x = 06

so that we obtain the simultaneous equations

E-16 = 4r6
E- 12 = 6^

from which rb ••= 2 ohms (for the twelve cells in series)
and E - 24 volts.

224

VOLTAIC ELECTRICITY

C Z

Hence, E.M.F. of each cell is 2 volts, and its internal resistance
is T% = J ohm.

Example 4. — Three cells, each of E.M.F. 1 volt and internal
resistance 1 ohm, are connected in series by wires of negligible resist-
ance, but one cell is reversed so that it acts in opposition to the
others (Fig. 168). Find the P.D. between the
terminals of this cell.

The current in the circuit is evidently

r. 2E-E t

C = — 5 = f ampera

orb

Now, to send J ampere through any one of the
FIG 168 ce^s recluires i x 1 = J volt. But the current has

to flow through the reversed cell against its own
E.M.F., and it will be necessary to apply an equal and opposite
P.D. of 1 volt to balance that : hence the P.D. across the cell is
1 J volts.

We can check this result by the following method. As the con-
necting wires are of negligible resistance, the P.D. across this cell
will be identical with the P.D. between the terminals of the other
two cells. Now each of these cells loses J volt through internal re-
sistance, and has f volt remaining. The P.D. across them will, there-
fore, be |- volt, a reading which a voltmeter connected across the
reversed cell would give.

Example 5. — A battery of E.M.F. 16 volts and internal resist-
ance 3 ohms is joined up in parallel with another battery of E.M.F.
12 volts and internal resistance 2 ohms, the circuit being completed
by a wire of 6 ohms resistance (Fig. 169). Determine what occurs.
Let E = E.M.F. of larger battery

= 16 volts.
b = its internal resistance = 3

ohms.

C = current given out by it.
Ej = E.M.F. of smaller battery

= 12 volts.
bl = its internal resistance = 2

ohms.

Cl = current through it.
and rx = external resistance = 6

ohms.

(We have written b and 6X instead of rb and rbl to simplify the
notation.)

Notice that, although the more powerful battery must give out a
current, there are three possible cases as regards the other.

(1) It may give out some current C1? so that the total current
through rx is C + Cx amperes.

OHM'S LAW APPLIED TO BATTERY CIRCUITS 225

, (2) The cm-rent C may divide and a portion Ct may flow through
the weaker battery in opposition to its E.M.F. (Ej.

(3) It may neither give out nor receive a current, i.e. Cj may be
zero, and then it might be disconnected without in any way altering
the current through the other battery or through rx.

The determining condition is the PoD. between the points A and
B. This must necessarily be less than E. If it is greater than Ep
then a current flows backwards against ^E^ if it is less than' Ej,
both batteries generate current ; if it is equal to Ex , then no current
flows in either direction through the smaller battery.

Let ex = I^.D. between the points A and B.

Now the volts lost in larger battery = 0x6,

.-. E-C6 = ex (1)

Considering the smaller battery, if it generates a current Cx we
must have

EL-C&-6X

If CJL flows backwards through it, we have (see last example)

Kj+OiV-fx
or generally, El q= 0^ = ex (2)

Again, if C2 = current through rx, we have

C2?-x = ex
or(C±01)rx = 6x (3)

In applying these equations to numerical calculations, remember
that either the two upper signs or the two lower hold good in (2)
and (3). The beginner will usually find it best to take one pair of
values and work out the result. If one of the currents becomes a
minus quantity, it is evident that he has chosen the wrong case, and
the calculation can be repeated with the signs changed. For instance,
in our problem we have, using the upper signs —

E-C6 = EJ-C16L and E-C6 = (C + C1><X
16-30=12-20! I6-3C = (C + 01)6

or 30-2(^ = 4 or 90 + 60! = 16

whence 0 = \4- amperes
and Cj = J ampere
.*. C2 = Y- amperes.

If the lower signs had been selected, we should have obtained
Cj = — J, which is absurd.

We can check the result by noticing that ex = \7~ x 6 = %4-= 11 J
volts, which is less than EL, and therefore this battery acts as a
generator.

P

226 VOLTAIC ELECTRICITY

To find the condition, mentioned in (3) above, which makes Cj = 0,
we put

16-30=12

.'. C = J ampere.

.•. ?-x must be such that a P.D. of 12 volts sends f ampere
through it,

or rx = 9 ohms.

For this value of rx there is only one current, which is the same
whether the smaller battery is present or not. If we make rx greater
than 9, then ex is greater than Ej^ and a current flows backwards
against E1.

Example 6. — The terminals of a battery of E.M.F. 4 volts
and resistance 1 J ohms are connected to those of a battery of E.M.F.

3 volts and resistance j ohm
4E=4 volts. E=3volts by wires of resistances 1

— |U--vvwvvvyvvvwvv\~-||- |U. ohm and 6 ohms respec-

\>=J'/*ohms *~5oTims$5oKnii.>' J)t=y4ohm.\ tively, so that both batteries

act in the same direction.
Show that, if a third wire
be placed so as to join the
FIG ~170 middle points of these Avires,

no current passes through it.

We have to show that the points A and B (Fig. 170) are at the
same potential, and we notice that when that condition is fulfilled,
there is only one current in the circuit whether AB is joined by a
wire or not.

Let E and Ex = the two E.M.F.'s,

b and b± = the two internal resistances,
and e = the P.D. between A and B.

Suppose that A and B are not connected, and let C be the current
in the circuit. Then considering the portion of the diagram to the
left of AB, we see that e can be obtained by deducting from 4 volts
the volts lost in the battery and the volts required to send C through
3 '5 ohms. Exactly the same argument can be applied to the portion
of the diagram to the right of AB, but whereas the effect of the
former part is to make the potential of A higher than that of B, the
effect of the latter part is to make the potential of B higher than
that of A.

To satisfy the required condition, e must be zero.

OHM'S LAW APPLIED TO BATTERY CIRCUITS 227
.-. e = E -06-3-50 = 0
and e = E1-C61- 3-50 = 0

.-. E -0(6 + 3-5)
and Ej = C(^ + 3'5)

ie E- = b +3'5

On substituting values, it will be found that this condition is
fulfilled, and hence the P.D. between A and B is zero.

This result has several useful applications ; see pp. 306 and 457.

EXERCISE XI

1. What is Ohm's Law?

A cell of electromotive force of 2 volts and internal resistance £ ohm is used
to send a current through a wire of 11 1 ohms resistance. Find the strength of
the current. (Oxford Local, Senior, 1899 )

2. The terminals of a battery, of E.M.F. 4 volts and resistance 3 ohms, are
connected by a wire of resistance 9 ohms. By how much is the difference of
potential altered thereby ? (B. of E., 1900.)

3. Describe the action of a Daniell cell.

The E.M.F. between the poles of a battery is 12 volts when the circuit is
"open," and 10 volts when it is closed by a resistance such that a current
of 6 amperes is passing. Find the resistance of the battery.

(Camb. Local, Senior, 1895.)

4. State the relation between the E.M.F. of a battery, the resistance of a
circuit, and the strength of the current produced. The E.M.F. of a battery
being 12, and its resistance 8, find the strength of the current generated by it
when its poles are connected (1) by a wire whose resistance is 16, and (2) by
a wire whose resistance is 40.

5. If an increase of the resistance of a circuit by 10 ohms causes the strength
of the current to decrease from 5 to 2, find the total resistance of the circuit
after the change.

6. Ten voltaic cells, each of internal resistance 3 and E.M.F. 2, are con-
nected—

(a) in a single series ;

(b) in two series of five each, the similar ends of each series being connected
together.

If the terminals are in each case connected by a wire of resistance 20, show what
is the strength of the current in each case.

7. You have two voltaic cells each having a resistance of 3 units (ohms) and
an E.M.F. of I'l unit (volt). Show what is the strength of the current which
would be produced in a wire of resistance 9-5 units (ohms) —

(i.) By one of the cells alone.
(ii.) By both cells connected in series,
(iii. ) By both cells connected abreast.

8. The terminals of a battery of five Grove's cells, the total E.M.F. of which
is 9 volts, are connected by three wires, the resistance of each of which is 9 ohms.
The current through each wire is f of an ampere. What is the internal resist-
ance of each cell ?

9. Five cells are arranged in series, with a line of 100 ohms resistance. The
resistance of each cell being 7"3 ohms, and its E.M.F. being T5 volts, find the
current strength.

10. How would you connect two equal constant cells of internal resistance

228 VOLTAIC ELECTRICITY

5 ohms each, if you wished to deposit copper as rapidly as possible in a voltameter
of 7 ohms resistance ? (B. of E., 1906.)

11. Four cells, each of 1 ohm internal resistance, are coupled up in series
with an external resistance of 8 ohms. Two similar cells are coupled up in
parallel with an external resistance of 5*5 ohms. Compare the currents in the
two circuits. (Oxford Local, 1901.)

12. A circuit is formed of six similar cells in series and a wire of 10 ohms
resistance. The E.M.F. of each cell is 1 volt, and its internal resistance
5 ohms. Determine the difference of potential between the positive and
negative poles of any one of the cells. (B. of E., 1894.)

13. A circuit is formed of five similar cells in series and a coil of 80 ohms
resistance. The resistance of the connecting wires is 1 ohm. The E.M.F. of
each cell is two volts and its internal resistance '4 ohm. Calculate the differ-
ence of potential between the positive and negative poles of any one of the cells.

(Camb. Local, 1906.)

14. Six cells arranged in series, each having an internal resistance of *4
ohm, are connected by a wire of 1*8 ohms. If each cell has an electromotive
force of 1 volt, what is the potential difference between the positive pole of
the battery and the point of junction of the third and fourth cells ?

(B. of E., 1899.)

15. A metal ball, connected by a fine wire with the copper terminal of a
battery of 120 cells in series, the zinc terminal of which is connected to earth,
acquires a charge of 100. Show what charge the ball will acqiiire if the twenty-
fifth, sixty-first, or ninety-first zinc plate, always counting from the zinc end of
the battery, be connected to earth.

16. Show that if the external resistance is equal to the resistance of a cell,
the same current strength is produced whether the battery is coupled in series
or in parallel. (Camb. Local, 1907.)

17. Define electromotive force between two points of a circuit.

The two poles of a battery are connected to the terminals of an electrometer,
and an E.M.F. of 4 volts is indicated. The poles of the battery are then con-
nected through a resistance of 10 ohms, and the electrometer indicates an E.M.F.
of 3*5 volts. Find (a) the current flowing round the closed circuit, (b) the
E.M.F. of the battery, (c) the resistance of the battery.

(Oxford Local, 1900.)

18. A zinc-and-copper couple is joined up with another couple similar in all
respects except that the plates are twice as large. Describe the effect when they
are joined up in series with a galvanometer ; and, secondly, joined in series so
that their action opposes each other. (Coll. of Preceptors, 1897.)

19. (a) What is meant by "polarisation" in a cell? Explain how the
difficulty is met with in the arrangement of a Daniell cell. (6) Three Leclanche
cells are joined in series, forming a battery. Three more form a precisely similar
battery. The two batteries are joined in parallel circuit, forming a composite
battery. The resistance of each cell is 2 ohms, its E.M.F. is 1'4 volt. Find
what current will flow round a wire of 20 ohms resistance, which joins the poles
of the composite battery. (Oxford Local, 1903.)

20. A battery of 10 volts and internal resistance *5 ohm is connected in
parallel with one of 12 volts and internal resistance '8 ohm. The poles are
connected to an external resistance of 20 ohms. Find the current in each
branch. (London Univ. B.Sc., Internal, 1907.)

21. The electrodes of a quadrant electrometer are joined to the terminals of
a battery of five cells in series. In what ratio will the deflection of the needle
be altered if the electrodes are also joined to the terminals of a battery of three
cells in series similarly arranged, all the cells being alike, and the connecting
wires thick ? (B. of E., 1899.)

22. The terminals of a battery formed of seven Daniell's cells in series are
joined by a wire 35 feet long. One binding-screw of a galvanometer is joined by

OHM'S LAW APPLIED TO BATTERY CIRCUITS 229

a wire to the copper of the third cell (reckoned from the copper end). With
what point of the 35 feet wire can the other screw of the galvanometer be
connected so that the needle shall not be deflected ?

23. Six similar cells are arranged in series, and the circuit completed through
a coil of wire and a galvanometer. The resistances of the battery, coil, and
galvanometer are 10, 50, and 20 ohms respectively. If the difference of potential
between the terminals of the galvanometer be 2 volts, what is the E.M.F. of
each cell of the battery ?

24. You have a battery of twelve similar cells connected in series ; each has
an E.M.F. force = 1'1, and an internal resistance = 3. If the poles of the battery
are connected by a wire whose resistance = 240, what will be the strength of the
current ? What will be the effect on the strength of the current of removing
from the battery three of the cells, and replacing them with their poles inverted ?

25. Two cells, A and B (E.M.F. and internal resistance of each are 1 volt
and 1 ohm respectively), are arranged in series. The positive and negative poles
of this battery are connected with the positive and negative poles respectively of
a third cell, 0, exactly like A and B, the connecting wires having negligible
resistance. What is the current in the circuit, and what is the potential differ-
ence between the positive and the negative poles of the cell C ?

26. Find the best arrangements of twenty-four cells having an external re-
sistance of 3 ohms, and each cell having an internal resistance of 2 ohms.

27. Four points, A, B, C, D, are connected together as follows : A to B,
B to C, C to D, D to A, each by a wire of 1 ohm resistance ; A to C, B to D,
each by a cell of 1 volt E.M.F. and 2 ohms resistance. Determine the current
flowing through each of the cells. (B. of K, 1900.)

28. A battery is formed of four Grove's cells in series, and its poles are joined
by a wire. If one electrode of a Thomson's quadrant electrometer is connected
to the middle point of this wire, and the other electrode to the platinum of each
cell in turn, describe the indications of the electrometer.

29. A battery of twelve equal cells, in series, screwed up in a box, being
suspected of having some of the cells wrongly connected, is put into circuit with
a galvanometer and two cells similar to the others. Currents in the ratio of
3 to 2 are obtained according as the introduced cells are arranged, so as to work
with or against the battery. What is the state of the battery ? Give reasons
for your answer. (B. of E., 1895.)

30. How would you arrange a given number of cells, which are to form
a battery sending a current through a given external resistance, so that this
current shall be greatest ?

Prove that an electric lamp, whose resistance is 20 ohms, and which requires
a current of '6 ampere, cannot be lighted with a battery of fifty cells each
having a resistance of 2 '5 ohms and a voltage of 1'4, if the cells are placed in
series, but that it can if they are arranged in two parallel rows of twenty-five.

(Oxford Local, 1908.)

31. Give full practical definitions of the ampere, the ohm, and the volt.

A battery of fifty-four cells, each of E.M.F. 2 volts and resistance '005 ohm,
is employed with a number of 100- volt glow lamps all in parallel, each lamp
requiring '6 ampere ; what is the maximum number of lamps which can be
used so that the voltage at their terminals shall not fall below 100 volts ?

(Oxford Local, 1907.)

32. An electric light installation consists of a group of lamps in parallel arc
between the ends of leads. The leads have total resistance '4 ohm, and bring
current from sixty accumulators, each with E.M.F. 2 volts and resistance '01
ohm. When twenty-five lamps are switched on, each takes *4 ampere. Find
the resistance of a lamp, and the watts used in each part of the circuit.

(London Univ., Inter. Hon., 1894.)

CHAPTER XVI

ENERGY EXPENDED IN A CURRENT CIRCUIT-
HEATING EFFECT

IT has been shown in Chapter V. that if a quantity of charge Q flows
from one point to another, between which the difference of potential
is V (its. being understood that V is not altered by the passage of the
charge), the work done on, or done by the charge is QV ergs, where
Q and V are expressed in static units.

In a voltaic circuit the facts are essentially similar, although the
units employed are different. Let E be the E.M.F. in absolute units
(see p. 206), and let a current of i absolute units flow round the
circuit for t seconds. Then a "quantity" Q, such that Q = ^.£ has
passed through a difference of potential E, and the units have been so
chosen that the energy expended is EQ, or E^ ergs. When the circuit
contains merely ordinary conductors, e.g. those dealt with in all the
cases up to the present, this energy is wholly converted into heat,
part of which appears in the external resistance, and part in the
generator itself.

Let us now suppose that practical units are used, and that a
current of C amperes flows for t seconds in a circuit in which the
E.M.F. is E volts. The quantity passing round the circuit is Ct
ampere-seconds, and, as one ampere-second is the practical unit of
quantity, known as a "coulomb," the quantity is, therefore, Ct cou-
lombs. From the above statement, the energy expended is evidently
measured by ECf, and will be in ergs, if we express the volts and
amperes in absolute units. Now it has already been mentioned
(see p. 206) that 1 volt = 108 absolute units of E.M.F., and 1 ampere-
T*F of an absolute unit of current, and hence the energy expended as
heat is E x 10s x C x ^ x t ergs

= EC£ x 107 ergs, or EC x 107 ergs per second.

From this, it follows that, if a current of 1 ampere flows through
a P.D. of 1 volt, energy is expended at the rate of 107 ergs per
second during the whole time of flow.

Electrical Power. — The rate at which work is done is the
measure of the " power " expended in the circuit ; and we see that
this rate is proportional to the product of current and E.M.F., and is
independent of the time during which the current flows.

230

ENERGY EXPENDED IN A CURRENT CIRCUIT 231

Definition. — The power corresponding to the product of 1 volt x
1 ampere is called a wait. Hence, 1 watt= 10r ergs per second.

Relation between Horse-power and Watts.— On p. 577,
it is shown that 1 H.P. = 746 x 107 ergs per second. Therefore,

1 watt = -L of an H.P.

746

Again, we know that 1 H.P. =33,000 foot-pounds per minute,

33 000
from which it appears that 1 watt = ' ~. . = 44J foot-pounds per

minute.

Example. — Twelve cells are connected in series, and joined up
to an external resistance of 5 ohms. If the E.M.F. of each cell is

2 volts, and its internal resistance is '25 ohm, find the total power
expended in the circuit.

Let C = current in amperes,

then C = *2x2 ^ = M = 3 amperes

5 + (12x'25) 8

The total power expended in the circuit is EC watts, or 24 x 3 =

72 watts, which is ~-?- H.P.
746

Now the P.D. between the terminals of the battery is ex,
where ex = Crx

.'. ex = 3 x 5 = 15 volts
and e6 = 24-15 = 9 volts

.'. 15 x 3, or 45 watts, is expended in the external resistance, and
9x3, or 27 watts, in the battery itself. It will be observed that
these values do not depend upon the time the current flows. On the
other hand, if it were required to nud the energy expended, it would
be necessary to take into account the time of flow.

It should be noticed that the argument may be applied to the
whole circuit, or to any part of that circuit.

Example. — Find the horse-power required to send 9 amperes
through 10 ohms.

In this case, the 10 ohms may be regarded as a portion of a
circuit through which 9 amperes are flowing. By Ohm's law, the
P.D. across the 10 ohms must be 90 volts, and the power expended

is 90 x 9 = 810 watts = 8-10 = 1 -09 H.P.
746

Equivalent Expressions for Power. —

We have Power = EC watts (1)

also E = CR, and 0 = ?, hence

232 VOLTAIC ELECTRICITY

Power = CR x C = C2R watts (2)

or Power = E x 5 = ~ watts (3)

K K

The three expressions are numerically identical, and we may use
the form which is most convenient. For instance, in the last example,
as current and resistance are given it would have been simpler to
write

Power = C2R watts

= 92x 10 = 810 watts

In saying that the three expressions are numerically identical, it
must be understood that E is the voltage which sends the current C
through the resistance R. There is no difficulty in such cases as we
have already considered, but it may be remarked that if a current C
flows between two points in a circuit, the P.D. between these being
e volts, then in all cases, the power expended in that portion of the
circuit is eC watts. On the other hand, if r is the resistance between
the points in question, we cannot say that the power expended is
C2r watts. That will be the case if the points are joined by an
ordinary resistance, but if, for example, a motor is included between
them, the product C2r merely gives the power which is converted
into heat, the difference being the power which is converted into work
of some kind. These matters will become clearer as we proceed ; in
the meantime we may sum up by saying that eC always gives the
power supplied to a given portion of a circuit, and C2r ahvays gives
that portion of the power which is converted into heat. Very fre-
quently the latter expression is a measure of the wasted power,
although this is not always the case, e.g. in an electric lamp or oven
it is useful power.

Efficiency of a Cell or Battery.— The term "efficiency,"
when applied to any mechanism whatever, denotes the value of

the ratio — e ^ P°" er . and is very frequently expressed as a per-
total power

centage.

In the case of a cell or battery, some convention must be made
regarding the meaning of the expression "useful power." Evidently
the power expended inside the cell is totally wasted, whereas that
expended outside may be usefully applied. Hence, it is convenient
to regard the power expended in the external circuit as useful power.
From this it follows that, if ex is the P.D. at the terminals when a
current C is flowing, we have

Efficiency = useful watts _ !*° _ t* ; (or ?* x 100 per cent.)
J total watts EC E v E

ENERGY EXPENDED IN A CURRENT CIRCUIT 233

It is instructive to write this in another form : —

Putting ex = Crx and E = C(rx + rb), we have

(V V

Efficiency = ,, x = ?x .

C(rx + rb) rx + rb

This expression tells us that the smaller the internal resistance, the
greater the efficiency, and that it can only reach 100 per cent, when
rb = 0. It also tells us that, if we consider any given cell for which
rb has a constant value — large or small as the case may be — the
efficiency may be made as nearly 100 per cent, as we please, provided
we make rx sufficiently large in comparison. On the other hand, if
rx be small, the efficiency is low, and becomes zero on short circuit.

Maximum Rate of Working. — It is, therefore, evident that
maximum efficiency is not the same thing as maximum power.
When, for example, rx is great, the current is small, and the power
expended is small also.

In fact, it will be seen that the efficiency reaches 100 per cent, in
the limiting case when rx = oc , that is, when no work is done.

The power expended in the external circuit is, therefore, zero
when the current is zero, and also zero when the current is greatest,
i.e. on a short circuit. Hence, for some intermediate strength of
current, it will have a maximum value. The same inference may be
drawn by noticing that the useful power is exC, from which it will
be seen that ex decreases as C increases (see Experiment 137). We
have, therefore, to find the condition for which exC shall have its
greatest value.

Now, we know that eb = Crb, or C = —

P
useful power may be written -2

Now, the denominator of this expression is constant, and the
numerator is the product of two quantities whose sum is a constant
(for ex + e& = E). Hence, the numerator (and, therefore, the fraction
itself) will be greatest when ex = eb, i.e. when the external resistance
is equal to the internal resistance.

From the relation already given, it follows that the efficiency in
this case is J, or 50 per cent.

The preceding argument is applicable to a dynamo or any form
of current generator. The main point to grasp is the vital distinc-
tion between "maximum efficiency" and "maximum output." In
practice, the distinction is important, for a dynamo is never worked
with a view to maximum possible output (and, as a rule, it would
be seriously injured if the attempt were made), but under conditions
which enable its efficiency to reach a reasonably high value.

234 VOLTAIC ELECTRICITY

It may be mentioned that on p. 219, we found that the maximum
current through a given resistance, using a given number of cells,
was obtained by grouping them, so that the internal resistance of the
battery became equal to the external resistance. In this case also
the efficiency is only 50 per cent., and from an economical standpoint,
it is in no sense a best grouping of the cells.

Electrical Energy. — The energy corresponding to a power of
1 watt acting for 1 second is called a "Joule." Evidently 1 joule
= 107 ergs. It may be called a " watt-second."

For commercial purposes, the unit of energy is the kilowatt-hour,
also known as the "Board of Trade unit." It will be seen that
1 kilowatt-hour = 1000 watt- hours = 1000 x 60 x 60 watt-seconds or
joules.

Example. — Two hundred lamps, each taking '5 ampere at a
pressure of 100 volts, are run for 24 hours. What is the cost, at 4d.
per Board of Trade unit ?

The current will be 200 x *5 = 100 amperes, and

Watts = EC = 100 x 100 = 10,000
Watt-hours = 10,000 x 24 = 240,000

Kilowatt-hours = - 240

.-. cost = 240 x4d.=<£4

Heating Effect of a Current. — The fact that a conductor
carrying a current is heated thereby, is now a matter of common
observation, on account of its application in electric lamps. It is,
however, necessary to examine experimentally the laws of such heat
production.

Exp. 140. Connect a piece of thin platinum wire in circuit with a battery of
a few chromic acid cells. Probably it will become sensibly hot to the touch, but
not sufficiently hot to be luminous. Gradually shorten the wire, and notice that
it becomes red hot, then white hot, and finally fuses.

This experiment demonstrates the existence of a heating effect,
but it is also apt to suggest that the effect depends in some way on
the length of the wire. This will be shown later to be a misconcep-
tion ; in fact, shortening the wire in the above experiment was
merely a convenient method of reducing the resistance of the circuit,
and thereby increasing the current strength.

The following well-known experiment may be used to show that,
when the current is constant, the heating effect depends upon the
resistance of the conductor.

Exp. 141. Make a chain of alternate links of platinum and silver wire (32
gauge) by cutting the wire into inch lengths, and then bending them into loops.
Pass a current from a 6-cell chromic acid battery through the chain. Notice

ENERGY EXPENDED IN A CURRENT CIRCUIT 235

that the platinum links become white hot, while the silver links remain com-
paratively cool.

This result, however, does not depend only upon the difference in
the resisting powers of the two metals, but also upon their specific
heats. Although the specific resistance of platinum is nearly six times
that of silver, its specific heat is about half as great ; so that the
increase in temperature would be nearly twelve times as great in
platinum as in silver, if the thickness, current strength, and loss by
radiation remained constant. As a matter of fact, the platinum wire
will lose much more heat by radiation than the silver, and hence the
rise in temperature will be correspondingly reduced.

Exp. 142. Connect up an ammeter and an adjustable resistance in circuit with
a battery capable of giving out a fairly strong current (or to electric light mains).
Include in the circuit a pair of terminals, between which lengths of tine iron (or
tin) wire may be conveniently connected (platinum wire is too expensive for use
in such experiments).

(a) Keeping the diameter of the Avire constant, determine the fusing current
for various lengths, between say \ inch and 6 inches. It will be found that the
fusing current is abnormally great for very short lengths, but that it falls rapidly .
as the length is increased, and soon becomes constant for all lengths greater than
a certain value. The large fusing current required for short lengths is due to
the cooling action of the terminals, which rapidly carry off heat by conduction.
When the wire is sufficiently long, this action is negligible, and then we find
that the fusing current for a wire of given diameter is {independent of the length.

(b) Select a length great enough to give the true fusing current, and keep
this constant whilst using wires of the same material but of different diameters.
Determine the fusing current for as many gauges as possible. We shall then
find, as would naturally be expected, that the fusing current increases with the
diameter, and we might reasonably infer that it would be proportional to the
area of cross section, i.e. to the square of the diameter, for evidently two similar
wires side by side would require tioice the fusing current taken by one of them.

Plot a graph, taking current strengths as ordinates, and the squares
of the diameters as abscissas. Notice that the graph is not straight,
but that it bends in a direction which indicates that the fusing
current for the thicker wires is rather less than would be anticipated.
This is because the surface area of a thick wire does not increase at
so great a rate as its cross section, and hence it does not lose propor-
tionally the same amount of heat by radiation. For example, if we
arrange, say, 6 separate wires side by side, the total fusing current
will be 6 times that required for a single wire, but if we imagine that
they are rolled up into a single circular wire of 6 times the sectional
area, then the surface of the single wire is considerably less than the
total surface of the separate wires, consequently it loses less heat by
radiation, and its temperature is higher for a given current. When,
however, we are dealing with wires of nearly the same area, we may
sum up our two experimental results by saying that the fusing
current is independent of the length, and practically proportional to
the area of its cross section.

To give 'some idea of the magnitude of a current to fuse a size of

236

VOLTAIC ELECTRICITY

wire in ordinary use, we take the following figures from a student's
note-book : —

20 gauge copper wire became red hot with 55 amperes and fused at 68 amperes.
20 „ iron ,, , „ ,, 15 ,, „ 25

These metals oxidise rapidly at high temperatures, which also
influences the result. It may also be pointed out that these figures
do not help in determining the greatest current which may be carried
under ordinary conditions, because that depends almost entirely upon
the risk of injury to the insulation. For instance, the " safe " current
to be carried indefinitely by a large coil of double cotton-covered
20 gauge copper wire would be from 1 to 2 amperes.

Relation between Heat Production, Current, and Re-
sistance. Joule's Law. — Exp. 143. Support three stout copper wires
on a light frame. Wind exactly equal lengths (two or three yards) of silk-covered
platinoid wire or manganin wire (30 gauge) into two open
spirals, and solder them to the copper wires as shown at A, B,
and C, Fig. 171. Immerse them in distilled water contained
in a glass beaker (or any other convenient vessel), which may
with advantage be surrounded by cotton wool. Join up to
a battery through a plug key or switch, and include in the
circuit an ammeter and an adjustable resistance.

(a) To find the relation between heat production and
resistance, when the current is constant.

Connect up one of the coils, adjust the current to some
convenient value, switch off the current, stir up the water,
and take its temperature by means of a thermometer, and
then at some noted instant of time, again switch on the
current and let it flow steadily for, say, exactly one minute
(or more). Keep stirring gently, and at the end of the
time, break the circuit, stir up wrell, and again note the
temperature, thus determining its rise in a given time.

Next, place loth coils in series in the circuit, alter the adjustable resistance
until the current is the same as before, then switch off, and take the tempera-
ture. Switch on again at a given instant, and let the current flow for exactly
the same time as before. Then note temperature and determine its rise.

It will be found that the rise in the second case is approximately twice as
great as at first, thus showing that when the current is constant, doubling the
resistance doubles the heat produced.'

(b) To find the relation between heat production and current, when the resist-
ance is constant.

Use either of the two resistances, or put them in parallel, which is sometimes
convenient, i.e. connect A and C together, and make AC and B the terminals.
In either case do not alter the arrangement during the experiment. Adjust the
current by means of the resistance to a convenient value, and determine the rise
of temperature in a given time exactly as before. Then alter the current until
it is exactly twice its previous strength, and repeat the operations.

It will be found that the rise in temperature is now approximately four
times as great as before, thus showing that when the resistance is constant,
doubling the current gives four times the amount of heat.

A series of values may be taken if desired, but the results are
sufficient to verify the law that when resistance and current both

^

^_~_

JT-I-

"Z.

-~

- -

~-

-A

~ (

3" "

C -

FIG. 171.

ENERGY EXPENDED IN A CURRENT CIRCUIT 237

vary, the amount of heat produced -is proportional to the square of
the current, to the resistance, and to the time for which the current
flows,

or Heat <x

This law was established experimentally by Joule, and is known by
his name.

From what has been already said (see p. 232), it follows that the
rate of heat production in any given conductor is proportional to the
power in watts expended in it, and that instead of CV we may write

e2
cG or — as may be most convenient.

In order to evaluate the amount of heat in terms of some heat
unit, we require a knowledge of the mechanical equivalent of heat.
For instance, it is known that 1 gram-centigrade-degree heat unit
(the heat required to raise 1 gram of water through 1°C.) is produced
by 4'18 x 107 ergs, and we also know that 1 watt represents 107 ergs
per second. Therefore, if W = power in watts, and £ = the time in
seconds during which the current flows, we have energy converted
into heat = Wx£x!07 ergs, and as 4'18 x 107 ergs produce 1 heat
unit, we have

This may be written, Heat units = '24W£, from which it appears that
a "watt-second" or "joule" corresponds to '24 of a heat unit.

It is necessary to distinguish carefully between the amount of heat
produced in a conductor and its rise of temperature. The former is a
definite quantity, and as we have seen, depends only upon the watts
expended on it. The latter, however, can be estimated only in simple
cases. It depends upon the specific heat and upon the mass of
the conductor, and also upon the nature of its surface and the sur-
rounding conditions, for obviously, the temperature will rise until the
heat lost by radiation, &c., per second is equal to the number of heat
units produced per second.

Evidently, if we measure both the watts expended in a conductor
and the number of units of heat produced, we can determine the value
of the mechanical equivalent of heat. Such a determination can be
made with the arrangement used in Experiment 143, but it is
better to use a continuous flow of water, as the " water equivalent "
of the apparatus is thereby eliminated. For example, in a certain
experiment a spiral of platinoid wire was enclosed in a glass tube,
and a steady flow of water maintained through it. A current was
sent through the spiral, the voltage across it and the current through
it being measured by a very accurate voltmeter and ammeter respec-
tively. Two thermometers indicated the temperatures at which the
water entered and left the tube, and the amount of water passing per

238 VOLTAIC ELECTRICITY

second was found by weighing. It was arranged that the water
should enter at a temperature a little less than that of the room
and leave at a temperature somewhat higher, thus, to a great extent,
automatically compensating for loss of heat by radiation. The fol-
lowing readings were obtained : —

Current = 3 '37 amperes; P.D. across spiral = 1O60 volts.

The current was allowed to pass for 3 minutes ; during which
time, 182 grams of water passed through the apparatus, entering at a
temperature of 16° C., and leaving at 24-3° C.

.-. Power expended in coil = eC = 10'6 x 3'37 = 35'7 watts.
Energy converted into heat in 3 minutes = 35'7 x 180 x 107 ergs.

Heat units produced = 182 x (24'3— 16) = 1511 gram- centigrade-
degrees.

/. 1 heat unit is produced by — — — ~ = 4c25xl07 ergs.

EXERCISE XII

1. Two Grove's cells, alike in all respects except that in one the plates are
twice as far apart as in the other, are arranged in series, and the poles of the
battery so constituted are united by a copper wire. The liquid in both cells
becomes heated. In which is the rise in temperature the greater, and why ?

2. A thick copper wire and a thin copper wire, of such lengths as to have
the same resistances, are joined end to end and used to connect the terminals of
a battery, so that the same current Hows through them both. Explain why the
thin wire becomes hotter than the thick one.

3. The poles of a cell are joined by two wires similar in all respects, except
that one is longer than the other. In which is the greatest amoimt of heat
produced, and why ?

4. The E.M.F. of a battery is 18 volts, and its internal resistance 3 ohms.
The difference of potential between its poles, when they are connected by a wire
A, is 15 volts, and falls to 12 volts when A is replaced by another wire B.
Compare the amounts of heat developed in A and B in equal times.

5. A current of 1 ampere passes through a coil whose resistance is 2 ohms.
What amount of heat is developed in the coil in 5 seconds ?

6. A current of 10 amperes passes through a wire whose resistance is '9 ohm
for 5 seconds. What amount of heat is developed ?

7. The resistance of two wires made of the same metal, a and b, are as 2 : 3.
What are the relative amounts of heat developed in the wires — (1) when they
are fastened end to end, and the same current passes through them ; (2) when
they are arranged in "multiple arc," that is, when each connects the ends of
the same battery, so that the battery current is divided between them ?

8. State the laws relating to the production of heat by an electric current.
What will be the ratio of the currents which will produce in 1 second the
same amount of heat in two wires of the same material and length, if the radius
of one wire is twice that of the other ? (B. of E., 1902. )

9. A current of 5 amperes flows for 3 minutes through a wire whose resis-
tance is 2 ohms ; given that 1 water-gram-degree = 4 '2 joules, find the amount of
heat, in water-gram-degrees, generated in the wire.

(Oxford Local, Senior, 1908.)

10. A current of 1 ampere, flowing for 1 second through a resistance of

ENERGY EXPENDED IN A CURRENT CIRCUIT 239

1 ohm, produces '239 gram-centigrade units of heat. What current would
have to flow for an hour through a resistance of 41*84 ohms in order that the
heat produced might suffice to raise a kilogram of water from 0° C. to the
boiling-point? (B. of K, 1896.)

11. An electric battery of constant E.M.F., having an internal resistance of
5 ohms, is connected to resistance coils of 10 ohms and 20 ohms respectively,
arranged (1) in series, (2) in parallel. Neglecting the resistance of the connect-
ing wires, compare the amounts of heat produced in the two cases (a) in the
whole circuit, (6) in the two coils. (B. of E., 1904.)

12. Two circuits, whose resistances are respectively 1 ohm and 10 ohms, are
arranged in parallel. Compare the amount of current passing through each of
these circuits with that through the battery. Compare also the amount of
heat developed in the same time in the two circuits. (B. of E., 1901.)

13. Explain how the mechanical equivalent of heat may be determined by
measuring the electric energy spent in heating a resistance. What instrument
would you require, and how would you perform the experiment ?

(B. of E., 1906.)

14. A current of 10 amperes is sent through a platinum wire, the resistance
of which is • 2 ohms. Find the mechanical equivalent in ergs of the heat
generated per second. (B. of E., 1905.)

15. It is required to generate 10 kilograms of steam per hour with power
developed from a 110- volt circuit. What resistance should the heating coil have
in order to do this, supposing loss from radiation negligible ?

(B. ofE., 1907.)

16. A current of 5, amperes is passed through a wire, and therein produces
500 calories per second. If the current were increased to 7 amperes, what
number of grams of wrater would it heat in 1 hour to 100° C. ? Assume that the
resistance of the wire does not change, and that the initial temperature of the
water is 15° C. (Lond. Univ. Matric., 1901.)

17. Describe some method you have employed for measuring the strength of
a current. A lamp, the voltage between the terminals of which is 100, is placed
in a calorimeter, which 'is immersed in 400 grams of water, the water-equivalent
of the calorimeter, &c., being 40 ; the temperature is found to rise 2*5° C. per
minute. Find the current in the lamp. (Camb. Local, Senior, 1901.)

18. The reactions within a cell generate electrical energy at the rate of
1 watt per ampere ; a current of 10 amperes is being generated, with the result
that energy is dissipated within the cell in the form of heat at the rate of 1 watt.
What is the difference of potential between the terminals of the cell ; also what
is the internal resistance of the cell ? (Lond. Univ., B.Sc. Internal, 1909.)

19. What would it cost to raise 10 kilograms of water from 20° to 100° C.,
with an electric supply at 5d. per unit? The mechanical equivalent of heat
may be taken as 427 kilogram -metres per kilogram-calorie.

20. A coil of platinum wire of 9 '45 ohms resistance is immersed in paraffin-
oil in a vessel surrounded by non-conductors of heat ; paraffin-oil is kept
flowing through the vessel at the rate of 8'1 grams per minute ; and a current
is kept flowing in the wire of such strength that the paraffin-oil leaving the
vessel is 5° C. warmer than it is on entering. Calculate the strength of current
in the wire. Specific heat of paraffin-oil = '474.

The above current is to be maintained by Grove's cells, each of electromotive
force 2 volts and resistance "12 ohm ; show how many cells will be required,
and what must be the resistance of the connecting wires.

CHAPTER XVII

MAGNETIC PROPERTIES OF A CURRENT
Magnetic Field due to a Current— Exp. 144. Send a strong

current — say from a 6-cell battery — through a copper wire. Immerse the wire
in iron filings, and on withdrawal, notice that they cling to it. On breaking
contact, the filings fall off.

This experiment suggests that the wire has acquired magnetic
properties, but as we know that copper is not capable of being mag-
netised, we are led to explore the region surrounding the wire.

Exp. 145. Pass a copper wire vertically
through a hole in a sheet of cardboard (Fig. 172).
Connect the ends with a 6-cell chromic-acid
battery. While the current is flowing (a) shake
iron filings from a muslin bag upon the card-
board, and at the same time tap it gently. Ob-
serve that the filings tend to arrange themselves
in concentric circles round the wire, (b) Place
a few small toy compasses on the cardboard, and
notice that they tend to set themselves cross-
wise to the conductor, thus making a similar
172. circular arrangement (except in so far as they

are disturbed by the magnetic field of the earth).

We learn from this experiment that lines of magnetic force,
identical to those of a steel magnet, surround a conductor whilst it
is carrying a current, and that, when the conductor is straight and at
a distance from other conductors, these lines of forces are practically
circles. This constitutes a profound difference between the flow of
a current and the flow of a liquid like water ; and although we can
draw many useful analogies between them, the flow of water produces
no effect whatever outside the pipe containing it, whereas the flow of
a current disturbs the whole of the surrounding space. This dis-
turbance is due to the fact that, when a current is started, it creates
a magnetic field, which dies away when the current is stopiped. The
field thus formed extends without definite limit throughout space,
although it gradually becomes weaker and weaker as the distance
from the conductor increases.

We may also mention that these lines of force differ from those
of a steel magnet, inasmuch as they exist entirely in air, or in what-
ever medium may be outside the conductor,1 i.e. there are no poles

1 We need not consider here what occurs inside the conductor.
240

MAGNETIC PROPERTIES OF A CURRENT 241

in the ordinary sense of the term, for a magnetic pole is the region
where lines of force emerge from a magnet into air. If, for example,
we thread a ring of soft iron on the wire, some of the lines would
pass through the ring, and it would be magnetised, although we
should find no trace of that fact, because in no part of it are lines of
force emerging. The condition would, however, become apparent if
a piece were cut out of the ring, for we should find at the surfaces
of the gap well-defined poles like those of. a horse-shoe magnet.

We are now in a position to understand why the conductor attracts
iron filings. It is a special case of property 5, p. 131 — the filings
tending to place themselves tangential to the lines of force in the
strongest part of the field, i.e. at right angles to the conductor and as
near to it as possible.

If we regard a compass-needle as a large pivoted filing, we can
partly anticipate the results of the following experiments : —

Oersted's Experiment. — Exp. 146. (a) Hold one of the wires from
a voltaic cell immediately above a compass-needle, in the direction of its length,
and complete the circuit by touching the end of this wire with the wire attached
to the other terminal of the cell. Observe that the needle is deflected, tending
to set itself at right angles to the current.

(6) Repeat this with the wire below the needle. Observe that the deflection
is in the opposite direction.

(c) Repeat (a) and (6) Math the direction of the current reversed, and notice
that the direction of deflection is reversed.

This action of a current on a magnetic needle was noticed by
Oersted in 1820, and is historically interesting as being the first
discovered relation between electricity and magnetism.

The positions taken up by a small magnet placed near a vertical
wire carrying a current are
shown in Fig. 173. In these
diagrams, we adopt the
useful convention, intro-
duced by Professor Silvanus
Thompson, that a current
flowing outwards is signified
by a dot,* and a current
flowing inwards by a cross.
These may be remembered
by thinking of them as arrows used to denote current direction, the
dot being the point coming outwards and the cross representing the
tail-feathers going inwards. In accordance with p. 131, we shall also
denote the direction of the magnetic field by arrows, but it must be
very carefully borne in mind that such marks on a line of force are
not intended to suggest motion of any kind ; they simply indicate
the direction in which a free north pole would tend to move.

In the diagrams (Fig. 173) only two lines of force are shown, and
the action of the earth's magnetic field is neglected.

Q

242

VOLTAIC ELECTRICITY

Ampere's Rule. — These experimental results are conveniently
expressed by Ampere's rule. Let the observer imagine that he is
swimming in the wire in the direction of the current with his face
turned towards the particular magnet under consideration and icith
arms outstretched, then the N pole will be on his left hand.

These diagrams are very important, and will be frequently used.
Evidently, if we know the direction of the
current, we can mark the direction of the lines
of force, and vice versa.

Again, these figures suggest that there is
a force acting on the N pole, which tends to
make it rotate continuously around the wire
in a certain direction, and an equal force
tending to make the S pole rotate in an
opposite direction. It is evident that, although
the magnet can place itself only tangentially to the line of force, these
two forces (Fig. 174) must have a resultant directed towards the
conductor, and it is this resultant force which causes the iron filings
to cling to the wire in Experiment 144.

It is not difficult to show by experiment that the suggestion

mentioned above is a real and un-
doubted fact. It is, of course,
impossible to make a magnet with
only one pole, but it is possible to
arrange that one of its poles becomes
inoperative. A simple form of ap-
paratus for this purpose is shown in
Fig. 175.

Exp. 147. A magnet, A, is bent, and
then suspended on a finely pointed wire,
the current being brought by a wire, C, to
a mercury cup, D, supported on the mag-
net, and carried away by a bent wire
which dips into an annular cup of mer-
cury, B. The other end of the battery

FlG. 175.

FIG. 176.

wire is attached to the binding-screw in B, which is in metallic connection with
the mercury. The current from six chromic acid cells, arranged in series, will
cause the magnet pole to rotate, If the current passes down the wire, as shown

MAGNETIC PROPERTIES OF A CURRENT 243

in the diagram, and the uppermost pole be N, it moves in a clockwise direc-
tion. If the S pole is uppermost, it rotates in an anti-clockwise direction. If
the current passes up the wire, the direction is reversed.

The movements will be easily understood by reference to
Fig. 176, which represents the state of affairs when the apparatus
is looked at from above.

Again, from Fig. 175, we infer that if the magnet were flexible,
it would tend to wrap itself round the conductor. A sufficiently
flexible magnet is unattainable, but the con-
verse experiment can easily be performed.

Exp. 148. Select a rather long steel bar magnet
and clamp it in a vertical position (Fig. 177).
Suspend loosely by the side of the magnet, with
a fair amount of slack, a length of tinsel strip (such
as is obtained from toyshops) and connect this to
a current reverser (see p. 253) and a battery. Ob-
serve that, when a current is sent through the
strip, it wraps itself round the magnet, and when
the current is reversed, it unwinds itself and wraps
around again in the opposite direction. It will be
found that the current always arranges itself so
that its influence strengthens the existing magnetism
of the bar. An explanation of this action will be
found on p. 486.

Attractions and Repulsions be-
tween Conductors carrying Currents. — As these forces are
somewhat feeble with currents of ordinary strength, it is necessary,
in order to demonstrate their existence, to devise some simple form

of apparatus, in which a portion
of the circuit may be freely mov-
able throughout a small range.
Such an arrangement is shown
in Fig. 178.

Exp. 149. Bend a piece of copper
or brass wire into a rectangle, R, about
5 inches wide and 9 inches long.
Shape the ends as shown in the figure,
so that one is vertically above the
other. Suspend it by a cotton thread,
C, and connect it by two tinsel strips,
TT, to fixed conductors, which may
conveniently be fastened to a wooden
stand, S. (These strips are shown as
spirals for clearness, but straight pieces
are really used. It is desirable to
place three or four strips in parallel
and twist the ends together, as a
single connection may become too hot
FIG. 178. and break.) Send a current through

the rectangle by means of a battery,
and intercalcate in the circuit a convenient length of wire, L, which can be held

M

244

VOLTAIC ELECTRICITY

close to the side, M, of the rectangle. It is really to be held in front or
behind the side, M, so as to produce a rotation, and not as shown in the figure.
It will be found that when the currents in L and M are in opposite directions,
there is a distinct though feeble repulsion between them, which becomes an
attraction when L is turned round so that the currents are in the same direction.1

The actions will be understood by drawing figures showing the
magnetic fields around the wires. When the currents are in the
same direction (Fig. 179) the two sets of lines are either both clock-

FIG. 179.

FIG. 180.

wise or both anti- clock wise, and there is a tendency for them to merge
into one resultant field surrounding both conductors.

When the currents are in opposite directions (Fig. 180) we see
that the lines of force between them are in the same direction, and,
as we already know, repulsion will occur.

Adjacent conductors, which carry currents and which are free to
move, will always tend to set themselves so that the currents are
in the same direction and as close together as possible. It is, how-
ever, important to bear in mind that the forces acting are between
two sets of lines of force and not between the conductors directly.
Such forces are usefully applied to purposes of measurement in the
various forms of electro-dynamometers, described in Chapter XXXIII.

Magnetic Field due to a Current in a Circular Wire.—

Exp. 150. Pass the two ends of a
piece of stout copper wire through
two holes (about ten inches apart)
in a piece of cardboard. Make the
wire into circiilar form (Fig. 181),
one half being above, and the other
half below the cardboard. Bend
the free ends, as shown in the
diagram, and connect them with
the terminals of a 6-cell chromic
acid battery. While the current
is flowing through the wire, scatter
iron filings over the paper. From
this graphic representation, observe
(1) that the lines of force are cir-
cular near the wires, (2) that at the
centre, they are normal to the plane
of the circle.

FIG. 181.

1 It is quite easy to get greater sensitiveness by making a rectangle of, say,
four to twelve turns of insulated copper wire bound together here and there with
cotton thread.

MAGNETIC PROPERTIES OF A CURRENT 245

This is an important result in studying the tangent galvanometer
(see p. 284).

Magnetic Field due to a Solenoid.— The word solenoid
means "tube-like," and is conveniently used to denote a hollow helix
of wire, when its length is considerably greater than its diameter.

Exp. 151. Place a solenoid in a piece of cardboard, so that its axis is in the
plane of the board (Fig. 182). This
is best done by cutting three sides
of a rectangle in the middle of the
board, and then passing the free end
of the strip through the solenoid.
Attach the free ends of the solenoid
to a battery, and then sprinkle iron
filings over the cardboard, gently tap-
ping it as they fall. Observe that

(1) the lines of force outside are
similar to those of a bar magnet, and

(2) inside they lie crowded together

in a direction parallel to the length of the solenoid.

It follows from this experiment that a solenoid carrying a current
behaves in many ways exactly like a weak bar magnet.

Exp. 152. Hold a solenoid in the hand, and whilst a current is passing,
bring each end in turn near an ordinary compass-needle. Notice that it pos-

sesses polarity. Determine
the N and S poles, and ob-
serve that they are reversed

N * 444 *~4 A i * 4 is Sf i t t f f t t f 4 t* by reversing the direction of

.the current. Ascertain the
relation between polarity and
current direction, and verify
the fact that it is in accord-
ance with Fig. 183.

It will be noticed that the polarity can be ascertained by using
Ampere's rule, if we modify it by saying that when it is applied to
a coil of any kind, the observer must face towards the inside of
the coil.

These conclusions are not limited to a solenoid. They apply to a
coil of any shape or size, e.g. a short, ring-shaped coil, or to a single
turn of wire, which behaves as if the whole of one face were a N
pole, and the whole of the other, a S pole. A very old method of
illustrating these facts is given in Experiment 153, by means of a
"floating battery," as the apparatus is commonly called.

Exp. 153. Fit a beaker in a cork or in a wooden tray (Fig. 184). Fasten
strips of copper and zinc (C and Z) side by side to a cross piece of wood, A.
Bend silk-covered copper wire into a coil (say, about 20 turns) and solder the
ends to the strips. On filling the beaker with dilute sulphuric acid, a current
will pass round the coil. Bring the poles of a bar magnet up to each face of the
coil in turn. It will be found that the coil acts like a very short magnet, the
whole of one side having N polarity and the other side, S polarity. Notice the

FiG. 183.

246

VOLTAIC ELECTRICITY

dirrection of the current in the coil, and apply the swimming rule to determine
its polarity. The results will agree with those found in the experiment.

It will be noticed that the coil tends to thread itself on the bar
magnet, and to move up to the middle of the latter. If the magnet
be put through the coil in the wrong direction, the coil will

FIG. 184.

move away, turn round, and then return to thread itself on the
magnet in the right direction.

Determine the relation between the polarity of the magnet and
the polarity of the coil in the latter position.

A solenoid differs from a bar magnet in having a penetrable
interior, and thus it can produce effects which cannot be obtained
from a bar magnet.

Exp. 154. While a current is flowing, hold two or three knitting-needles, or
straight pieces of iron wire, with their ends just inside the solenoid. Xotice
that they will be sucked into the coil, owing to the tendency to move into the
strongest part of the field (see p. 131). This effect has numerous practical appli-
cations in arc lamps and in other mechanisms.

At first sight there appears to be no relation between the dis-
tribution of the field around a solenoid and the circular distribution

around a straight conductor. To
show that such a relation really
does exist, let us consider Fig.
185, which shows a section of
a s°len°id of one layer deep. If
we think of the lines originating

YIG igs in circles as shown, and then con-

sider their mutual influence, it
will be evident that between any two adjacent conductors, they will

MAGNETIC PROPERTIES OF A CURRENT 247

neutralise each other, for they are in opposite directions, whereas
above and below they will merge into continuous lines, as shown in
Fig. 186. If we also remem-
ber that the lines are very x * > s.

numerous, and that they repel ^ <8> <8> <8> ® <8> <8>J

each other laterally, we shall see > — < < /

that, outside the solenoid, this x * < ^

repulsion must make them bulge T0 © 0 © © © \

out into the shapes found experi- ^- * > X

mentally. The student should IriG lgg

also notice that, as the lines

tend to contract in length, they must tend to squeeze the turns

together like a compressed spring.

As the current in any one turn is in the same direction as that
in the adjacent turn, we may express the same action by saying that
the attraction is due to the currents flowing in the same direction.
This action is very easily verified by means of Roget's vibrating
spiral, which may be made of twenty or
thirty turns of moderately thin copper
wire, suspended from a suitable support.
The lower end just dips into a mercury
cup cut in the base of the support (Fig.
187).

Exp. 155. Connect one wire from a battery of
six oells to the top of the coil, and place the other
wire in the mercury. A current, therefore, flows
through the coil. Observe that attraction takes
place between the wires, so that the lower end of
the coil is lifted out of the mercury. By this action
the circuit is broken, and the spiral drops back
FlG. 187. to its first position. Attraction again takes place,

and so on, thus giving an up-and-down motion

to the coil. Fix a soft iron rod inside the spiral, and observe that the effect is

intensified.

Effect Of an Iron Core. — Exp. 156. Place a knitting-needle inside

the solenoid used in Experiments 151, 152, and after passing a current, show

that the needle is magnetised. Verify the fact that its polarity is the same as

that of the solenoid itself.

Exp. 157. Wind 20 or 22 gauge cotton-covered copper wire on a glass

tube about a quarter-inch diameter, to form a solenoid. A single layer of

wire will be sufficient, but instead of

winding it uniformly in the ordinary

way, reverse the direction of winding .^rrrrn^^^^rrrrt.

in two or more places, as shown I 1 I £ 1 1 1 \ \ \ \ 1 1 Li 1 I

in Fig. 188. Place within it an I J J J J J \ \ \ M 1'J J J J I

unmagnetised knitting - needle, and 5 N N Sv S N

after passing a current, remove the

needle and roll it in iron filings. Also FlG. 188.

test it by means of a compass-needle.

It will be found to possess consecutive poles, which were produced where the

direction of the winding was changed.

248 VOLTAIC ELECTRICITY

Exp 158. Repeat Experiments 156, 157, using a rod or wire of soft iron.
Show that it behaves like the knitting-needle, except that the effect is temporary
— only a slight residual magnetism being left when the current is stopped.

These experiments show that the presence of an iron core in-
tensifies enormously the magnetic effect of the coil, without altering
its polarity, the difference in the behaviour of soft iron and steel being
exactly what would be expected from experiments in Chapter X.

It is evident that each particle of iron in the core tends to behave
like an iron filing placed in the solenoid as in Experiment 151.

Electromagnets. — A coil of wire on a soft iron core constitutes
an electromagnet. Their shapes and applications are innumerable.
Their enormous practical importance depends on the fact that a
mechanical force, completely under control, can be produced at any
distance from the operator.

The general theory of the magnetisation of iron is discussed in
Chapter XXV. ; here, it is only necessary to remark that the magne-
tising power of a coil of given dimensions is proportional to the
number of ampere-turns, i.e. to the product of the current flowing
and the number of turns. Hence, exactly the same effect may be
produced by a small current through many turns of fine wire as by
a large current flowing through fewer turns of thick wire. On the
other hand, the magnetism produced in a soft iron core by a given
number of ampere-turns depends on the shape and quality of the
iron, the more nearly the former approaches a closed ring the better.
Hence, a U-shaped core is more strongly magnetised than a straight
one, and the effect is again greatly increased by putting a soft iron
keeper across the poles (see Experiment 238). It must also be pointed
out that the magnetisation of the iron is not proportional to the
ampere-turns (except, roughly, when weakly magnetised), for the iron
eventually reaches a condition when any further increase in the ampere-
turns has no practical effect. The iron is then said to be saturated.

Ampere's Theory of Magnetism.— From the fact that a
solenoid acts in every respect like a magnet, Ampere propounded a
theory that magnetism is due to current circulation. He considered
that every molecule of a magnet has closed currents circulating round

it. Before magnetisation the
~T~ 7 7"\ molecules, and hence the currents,
/ / I s I move irregularly ; during mag-

T I netisation they assume parallel

\ X, V, V yy directions, and the more perfect

FlG 189 the magnetisation, the more par-

allel they become. The separate

currents, moving round the various molecules, may be considered as
equivalent to one resultant current flowing round the whole magnet
(Fig. 189). It will be seen that the direction of the current therein
agrees with the results found experimentally and also with the

MAGNETIC PROPERTIES OE A CURRENT 249

swimming rule. Looking at a S pole, it is in a clock- wise direction,
and vice versa.

Recent discoveries have thrown considerable light upon the true
nature of these molecular currents, and at the same time strongly
support the essential correctness of Ampere's hypothesis (see p. 422).

A very useful form of electromagnet, which will be frequently
referred to in connection with subsequent experiments, is shown in
Fig. 190 (for which we are indebted to Messrs. Pye & Co., of
Cambridge). The soft iron cores should be about 1J inches in

FIG. 190.

diameter and 8 inches long, screwing into the massive iron base
and readily removable. They are provided with soft iron pole-pieces,
which may be placed in any position, and which are useful in certain
experiments for concentrating the field. The coils are wound on
wooden bobbins, and are also removable.

Exp. 159. Connect the electromagnet to a battery of a few cells in series.
Place a piece of cardboard on the poles and pour iron tilings upon it. Notice
how their arrangement indicates the distribution of the lines of force between
the poles.

Exp. 160. Place a piece of iron on either pole, switch on a current, and test
the pull. Then break the circuit and test the pull again. Notice that the
magnetism has almost completely and instantly disappeared, leaving merely a
feeble "residual" magnetism.

250 VOLTAIC ELECTRICITY

Place a flat iron bar, with an ordinary rough surface, across the poles like
a keeper. Switch the current on and off, Whilst the current is flowing the
bar is, of course, held on firmly. When it is interrupted, measure, by means
of a spring balance, the pull required to detach the bar. It will be found that
the magnetism has not completely disappeared, a fairly strong pull being
required to detach the bar, but when once detached, it is not held on again when
replaced.

Repeat the experiment, using a bar which has a very smooth surface. It
will be found to adhere very strongly after the circuit is broken, most probably
too firmly to be pulled off with the balance. But when it is pulled off, this
residual magnetism instantly disappears.

Repeat the experiment with the same bar, placing between it and the poles
sheets of paper or cards of varying thickness. It will be found that even a
very thin sheet of non-magnetic 'material greatly reduces the amount of residual
magnetism when the current is switched off.

These experiments illustrate what has been said on p. 117, with
reference to the relative power of open and closed chains of particles
to maintain their orderly arrangement. They, also, show that when,
in any mechanism (e.g. an electric bell), an electromagnet is
required to gain or lose its power rapidly, the armature it may
have to attract should never actually touch the poles, or there
will be a strong tendency for it to adhere after the circuit is
broken.

Winding" Electromagnets. — Any given electromagnet may
be regarded as requiring a certain number of ampere- turns to develop
its desired strength. The gauge of wire to be selected for the coils
then depends upon the working conditions. If it is known that a
current of fair strength is available, then evidently comparatively few
turns of thick wire may be used, but if it is to be placed in a circuit
of high resistance in which the current will necessarily be small, it
must be wound with many turns of fine wire.

It often occurs in practical applications that an electromagnet is
to be excited by a constant voltage maintained between the terminals
of the winding. In this case, it is interesting to notice that the
number of ampere-turns produced depends only upon the diameter
of the wire used, and (neglecting the effect of the thickness of
insulation) is independent of the actual number of turns.

Let E be the constant voltage, r the resistance of one turn of
mean length, and N the number of turns. Then the resistance of
the winding is Nr.

TT TT

Hence, C = — - or CN = ampere-turns = — = a constant.
Nr r

-pi

For instance, if there is one turn, C = —

r

TT TP

and ampere-turn = — x 1 = —
r r

MAGNETIC PROPERTIES OF A CURRENT 251

T,1

If there are 1000 turns, C1 = T-XTC-.

TT TT

and ampere- turns = V-KTCTT- x 1000 = — as before.
lOOOr r

The question therefore arises : How many turns shall be used 1
In answering it, there are two points to be kept in mind —

(1) The particular gauge of wire in question can carry only a
certain current without getting hot enough to damage the insulation,
and therefore sufficient wire must be used to bring the current down
to a safe value.

(2) The more wire put on, the less becomes the expenditure of
energy in the coil when in use (although the greater becomes the
first cost of winding), and in practice, therefore, sufficient wire is
used to effect a fair compromise between these two conditions.

These matters will be understood better, if we work out the
values in a particular case.

Example. — An electromagnet requires 4300 ampere-turns, and
is to be excited at a constant pressure of 50 volts. What gauge of
wire must be used, if the average length of one turn is 18 inches 1

We have from above 4300 = - = —

We have, therefore, to find a size of wire such that a length of
18 inches has a resistance of 'Oil 6 ohm. For this purpose we must
anticipate a result arrived at in Chapter XVIII., where it is shown
that

_ Length in inches *65
Area in square inches 106

-A18^65
Area x 106

18 x -65
Area = .=- 001 sq.m.

On referring to wire tables, this is found to correspond to 20 gauge,
and it is also stated that this particular size of wire will carry a maxi-
mum current of 1*5 amperes without undue heating. (It would carry
a larger current if freely exposed to the air, but we have to remember
that the heat cannot readily escape from the inner turns of the
winding.) Hence, the total resistance of the winding must be such

that 50 volts send 1*5 amperes through it, or R = — =33*3 ohms.

252

VOLTAIC ELECTRICITY

.*-• IT]

•-

As each turn has an average resistance of -0116 ohm, we shall

oo.o

need at least ——-, or 2870 turns.
•Olio

The only loss of energy in the coil will be that due to heat pro-
duction, and this will be at the rate of C2R watts. In this case, it

will be more convenient to write it in the form — , which gives — — = 75

11 33 -3

watts (or about ^ H.P.).

From this expression, we see that, as e is constant, the greater we
make E the smaller will become the waste of energy, as previously
stated. Of course, if we push the argument to its logical conclusion,
it follows that it is theoretically possible to obtain an electromagnet
of any strength whatever, without appreciable expenditure of energy,
and it will be seen later that this result embodies a very important
truth.

Electric Bell. — The simplest and most familiar electro-magnetic
device is found in the ordinary electric bell. It consists of an electro-
magnet, whose base is prolonged to carry a steel spring, S (Fig. 191),

to which is screwed a soft iron arma-
ture, A, and which is finally bent out-
wards, as shown, to make contact with
an adjustable screw carried by an insu-
lated brass pillar, P. At the point of
contact, a small disc of platinum is
attached to the spring, and a short
piece of platinum wire to the screw,
in order to prevent oxidation due to
sparking, which would soon cause
bad contact and stop the action. One
of the ends of the electromagnet-
winding is connected to one of the
terminals, T, and the other end is con-
nected to the iron frame by means of
a screw, F ; the second terminal of the
bell being joined up to the insulated
pillar, P. The two wires from the
battery (which usually consists of one
or two Leclanche cells) are connected
to the terminals TT. In one of these
wires is placed a key — commonly
called the " push " — by means of
which the circuit is made or broken.

When the push is pressed, the circuit is completed, and a current
flows round the electromagnet, causing the armature to be attracted.
When this occurs, the hammer, H, being carried by the armature,
strikes a gong, G ; and at the same time, owing to contact being broken

/ 1

t

fffffff in

" / 1-

fffffft i

jjj.j.. y

FlG 191

MAGNETIC PROPERTIES OF A CURRENT 253

between A and the screw in P, the current ceases, and the iron core
loses its magnetism. The spring, S, now comes into play, bringing
the armature back to the screw in P, thus completing the circuit
again. This cycle of operations is repeated as long as the push is
held down.

Current Reverser. — A very convenient form of current reverser,
suitable for use in Experiment 148, and in many later experiments,
is shown in Figs. 192, 193, for the former of which we are indebted
to Messrs. J. J. Griffin & Co. A wooden or ebonite base contains six
mercury cups, arranged as shown, each of which is provided with
a binding-screw, making contact with the mercury. C is joined to F,

FIG. 192.

FIG. 193.

and D to E, by means of copper strips (which should be readily
removable).

The "rocker" consists of an ebonite handle carrying at each end
three bent copper rods in metallic connection, but insulated by the
handle from the similar group at the other end. This enables A and
B to be connected with either CD or EF as required.

For use as a current reverser, only four terminals are required,
which must be AB and either CD or EF. If the galvanometer (or
portion of the circuit in which the current is to be reversed) is joined
up to, say, CD, and the battery to AB, it will be seen from the figure
that the current is reversed when the rocker is moved over, and
evidently the galvanometer and battery connections may be inter-
changed without affecting the result.

It is, however, usual to provide six terminals, because then by
removing the diagonal connections CF and DE, the arrangement is
converted into a useful " change-over " switch, referred to in various
subsequent experiments, e.g. see Experiments 178, 198.

254 VOLTAIC ELECTRICITY

EXERCISE XIII

1. What would be the magnetic effect produced on a straight steel tube by
the passage of a strong current through a straight wire placed along the axis of
the tube ? And how would you prove your statement ? (B. of E., 1898.)

2. A wire is stretched from east to west (magnetic). How, without breaking
it, can you test whether, and in what direction, an electric current is passing
through it? (B. of E.-, 1896.)

3. A road in the northern hemisphere runs magnetic north and south. At
one point an insulated conductor passes beneath it in which an electric current
flows from east to west. How will the indications of a dip circle be affected at
points near to the conductor ? (B. of E., 1894.)

4. A metal ring, through which a current circulates, can move horizontally,
its plane remaining always vertical. Describe and explain what happens when
one pole or the other of a bar magnet is presented to the ring.

5. Describe and explain any arrangement for causing a magnet to rotate
continuously about a wire through which a current is passing, and show the
relation between the direction of the current and the direction of rotation.

6. Two parallel covered wires are traversed by equal currents in the same
direction : what is the joint effect of the currents upon a bar of soft iron (a) laid
across the two wires, on the same side of both ; (6) held between the wires at the
same distance from each.

7. A square of guttapercha-covered copper wire is suspended from one arm
of a balance, so that an electric current can be passed through it, and it is
counterpoised when the lower side of the square is immersed in a trough con-
taining copper sulphate. If an independent current is passed through the liquid,
what effect will be produced on the equilibrium of the balance ?

8. A wire, through which a current is passing, is stretched vertically from
floor to ceiling of a room. A small magnet, suspended by a silk fibre so that it
can turn freely in a horizontal plane, but cannot turn vertically, is brought near
the wire on the east (magnetic) side, and the time is measured in which the
magnet makes, say, 20 oscillations. The magnet is then placed at the same
distance west (magnetic) of the wire, and the time in which it makes the same
number of oscillations is observed. How could you tell from these experiments
whether the current in the wire is flowing upwards or downwards ?

CHAPTER XVIII

RESISTANCE AND ITS MEASUREMENT

Exp. 161. If electric lighting mains are available, connect them to an ordinary
incandescent lamp, putting in circuit with it a suitable ammeter to measure the
current flowing. Observe the value of the current. Add another similar lamp
in series with the first, and observe that the current is half its previous value.
If a third lamp be added, the current is one-third of its original value, and so on.

As the voltage across the mains is constant, this experiment
shows that the resistance of a conductor is directly proportional to its
length.

Exp. 162. Join up two lamps in parallel, and notice that the total current
is twice as great as that with one lamp. With three lamps in parallel, it will be
three times as great, and so on.

Hence, the resistance of a conductor varies inversely as its area of
cross section.

We may, therefore, write, for any conductor —

Resistance oc — s. — or R oc _
area A

Specific Resistance. — We may express the above relation as
an equality by multiplying by a constant, thus —

R = . - x s, where s is a number which is constant for any given
A.

material, but which varies for different materials.

If 1= 1 and A = 1, then R = s, i.e. s is the resistance of a portion
of the material of unit length and unit sectional area. It is called
the specific resistance of that material, and its numerical value will
depend upon the units of length and of resistance chosen. If, for
example, we take the centimetre as the unit of length, and conse-
quently the square centimetre as the unit of area, we might state the
value of s for any substance in ohms per centimetre- cube, i.e. as the
resistance in ohms of a portion 1 centimetre in length and 1 square
centimetre in area. For the majority of metals, this would be a very
small fraction, and it is more conveniently expressed in absolute
units of resistance per centimetre-cube by multiplying by 109.

For many practical purposes, the value of s is expressed in microhms
per inch-cube, a microhm being the one-millionth of an ohm.

The specific resistance of a very poor conductor may be so enor-

255

256

VOLTAIC ELECTRICITY

mous that its value is better expressed in megohms, a megohm being
one million ohms.

The following table shows the specific resistance of a few
materials : —

Absolute Units

Microhms

per

per

Microhms per

Centimetre-

Centimetre-

Inch-Cube.

Cube.

Cube.

Silver annealed

1,468

1-468

0-578

Copper annealed

1,562

1-562

• 0-614

Aluminium ....

2,665

2-665

1-049

Zinc ....

5,751

5'751

2-225

9,065

9-065

3-569

Platinum .....

10,917

10-917

4-298

Platinoid (alloy)

41,000

41-0

16-14

German silver (alloy) l

20,243

20-243

7-97

Mercury .....

94,070

94-07

37-03

Manganin (alloy)

46,000

46-0

18-54

An examination of the above table shows that the specific resist-
ance of alloys is very much greater than that of the pure metals
(excepting mercury). This is a characteristic property of alloys,
which is taken advantage of in the preparation of wires of high
specific resistance. Even a slight trace of another metal, which by itself
may be a good conductor, has an enormous effect on the resistance,
and hence copper used for electrical purposes has to be exceptionally
pure.

Example. — A copper wire 150 centimetres long and -04 centi-
metre in diameter is found to have a resistance of '196 ohm. Find
the specific resistance of copper, (a) in ohms per centimetre-cube,
(b) in absolute units of resistance per centimetre-cube, (c) in microhms
per inch-cube.

As R
we have s

in case (a)

A

RxA
I

_ -196 x TT x (-02)2

150
= '000001643 ohms per centimetre-cube.

For (b), put R in absolute units, i.e. multiply by 10° ; then s= 1643
absolute units per centimetre-cube.

1 German silver contains copper, nickel, and zinc. Platinoid is German
silver with the addition of tungsten. Manganin contains copper, manganese,
and nickel.

or

RESISTANCE AND ITS MEASUREMENT 257

For (c), we must first express the dimensions in inches, knowing
that 2P54 centimetres == 1 inch.

.'. length of wire = inches

2-54

v '02 . ,

and radius of wire « — - inch

'196 X 7T X

2-54
_-02\2
2-54J

150
2-54

•196 x TT x (-02)2 A/wwwxACP
i.e. s = — — - — v ' = -00000065 ohms per inch-cube

or s= '65 microhms per inch-cube x
Effect of Change of Temperature on Resistance. —

Exp. 163. Wind a yard of iron wire (about 20 gauge) into a rough open spiral.
Connect it in series with an ammeter and a single cell of fairly low internal
resistance, e.g. a double-fluid chromic acid cell. Read the current. Now heat
the spiral in a Bunsen flame, and notice that the current falls in strength, but
that it returns practically to its old value when the flame is removed.

This experiment shows that the resistance of iron increases with
the temperature.

Exp. 164. Connect two small spirals of thin platinum wire in series with a
battery of a few cells, so that both spirals are heated to dull redness. Roll up a
paper tube, and blow through it on one spiral, and notice that, whilst this spiral
becomes cooled, the other glows brighter. Again, heat one spiral in a Bunsen
flame, and notice that the other becomes cooler.

This is a very old experiment, but it illustrates in a striking
manner that the resistance of platinunT: increases as the temperature
rises, and vice versa; for as one of the spirals has increased in
temperature, this can be due only to the fact that the current through
it (and therefore through the whole circuit) has increased. Such an
increase in current means that the resistance of the circuit has
decreased, which must have been caused by the cooling of the other
spiral.

It must be pointed out that in order to perform Experiments 163
and 164 successfully, it is necessary for the iron wire in the former
and for the platinum in the latter to be practically the only resistance
in the circuit, so that slight variations in their resistances may alter
the currents by a perceptible amount. Hence, regulating resistances
should, if possible, be avoided, and the cell or battery should have a
fairly low internal resistance.

1 Notice that the expression "cubic centimetre," or cubic inch," though
occasionally used, would be absurd, for the resistance of a cubic centimetre of
any material may have any value between 0 and °<?.

B

258 VOLTAIC ELECTRICITY

The resistance of all bodies varies with temperature. If they
vary directly, i.e. if the resistance increases as the temperature
increases, the bodies are said to have a positive temperature coefficient.
This is characteristic of all pure metals and of most alloys.

If they vary inversely, i.e. if the resistance decreases as the
temperature increases, the bodies are said to have a negative tempera-
ture coefficient. In the latter class, we have the non-metallic
conductors in general — carbon, conducting solutions and liquids ; and
insulators like glass, marble, slate, &c., which become partial con-
ductors at high temperatures.

It has just been discovered that the non-metallic element boron
possesses an abnormally high negative temperature coefficient.

Until recently, it was known only in the form of loose powder,
but Dr. Weintraub has now succeeded in preparing it in the fused
state. In this condition, a certain portion had a resistance of 5,620,000
ohms at 27° C., which fell to 5 ohms at a dull red heat.

We have said that pure metals and most alloys are characterised
by having a positive temperature coefficient, but it is important to
remember that this coefficient is much smaller in alloys than in pure
metals, a property which gives alloys their great value in the prepara-
tion of standard resistances. The majority of pure metals behave
with curious uniformity with regard to their changes in resistance
during variations in temperature, e.g. the resistance of wires made
from them increases by nearly 40 per cent, between 0° C. and 100° C.,
and, moreover, this increase is nearly uniform through quite a wide
range of temperature. To this general rule, iron is a marked excep-
tion : not only is the increase about 60 per cent, between these
temperatures,. but there is a remarkable and unusual increase near a
red heat — a property of iron advantageously utilised in the steadying
resistance of a Nernst lamp (see Chapter XXXII.).

The effect of a change of temperature on the resistance of carbon
and metals may be illustrated as follows : —

Exp. 165. (a) Connect an ordinary incandescent lamp with carbon filament to
supply mains, and put an ammeter in series with it and a voltmeter across its
terminals. Also arrange in the circuit an adjustable resistance, or other con-
venient method of varying the voltage on the lamp, and take a series of
readings spread over as wide a range as possible. From these readings, calculate
the resistance of the lamp filament in each case. It will be found that the
resistance decreases as the current increases, i.e. as the temperature of the
filament rises.

(6) Repeat the observations, using a metallic filament lamp, and show that
the converse is the case.

The following numbers are given as an illustration. They were
obtained in an ordinary class experiment, and have no special value
in themselves. Each lamp was made for working at 100 volts; the
filaments were, therefore, non-luminous at the lowest voltage, and
abnormally bright at the highest : —

RESISTANCE AND ITS MEASUREMENT 259

Carbon Lamp.

Tantalum Lamp.

Volts.

Current

Resistance

Current

Resistance

in

in Ohms

in

in Ohms

Amperes.

(Calculated).

Amperes.

(Calculated).

22

•08

275

•14

157

49

•22

223

•23

213

73-5

•37

198

•32

229

102-5

•53

191

•40

256

124

•67

185

•46

269

152

•89

170

•53

286

A more exact method of measuring the change of resistance with
temperature is described on p. 264, where the subject is further
discussed.

Measurement of Resistance: By using a Voltmeter

and Ammeter. — This method is correct in principle, and is the
only one available under certain conditions (e.g. for measuring the
resistance of a lamp whilst working, as used in the last experiment).
The results, however, are for many purposes not sufficiently exact;
indeed, the method must be regarded as a rough and ready one,
which is very useful when only approximate accuracy is required.

By comparison with a known Resistance. — The inde-
pendent or absolute measurement of a resistance is a matter of
considerable difficulty, and is dealt with later (p. 587). On the
other hand, it is easy to compare two resistances. All ordinary
measurements are of this nature, and involve the use of a standard
resistance, or more conveniently, an adjustable set of such standards,
known as a resistance box.

Method of Substitution. — One of the earliest and most
obvious methods of comparison is that of substitution, which is of
such limited application that it requires only a brief explanation. It
assumes that resistances of known value (e.g. a resistance box) are
available.

Exp. 166. Connect a Daniell's cell in series with the wire or coil whose
resistance is to be measured and with a galvanometer, adjusting the sensitive-
ness of the latter until a convenient deflection is obtained. Read this deflection,
and then remove the wire, substituting for it the adjustable known resistance.
Vary the latter until the deflection of the galvanometer is the same as before.
The known resistance required for this purpose is evidently equal to that of the
wire to be measured.

This method, under ordinary conditions, gives only approximate
values, and is, as a rule, quite inapplicable for practical purposes,
although the principle will be applied later to a special method of
dealing with very high resistances.

260

VOLTAIC ELECTRICITY

Construction of Resistance Boxes.— Fig. 194 shows the
arrangement of a resistance box of the ordinary pattern. Other varieties
are used, but it is unnecessary to describe them here. A thick plate
of ebonite forms the top of the box, and on the upper surface of
this is screwed a series of brass blocks, which can be connected
together by well-fitting brass plugs. It is usual to cut away the
blocks a little at the base in order to facilitate cleaning. The various
coils are wound on wooden bobbins, and are supported by means of
brass pins screwed into the ebonite. The coils are wound " non-
inductively," i.e. the wire is doubled back upon itself before winding.
If the coils were wound in the usual way, each, when carrying a
current, would act like a weak magnet, and there would be other
difficulties due to " self-induction " (see p. 360), which would greatly
limit the usefulness of the box. The two ends of each coil are
soldered to thick copper bars firmly screwed into the brass blocks,

Brass Uocks^piugs.
\ \ jL-Plale of ebonite.

Wooden bobbin trail.

-I Copper wire screwed
[ into brass black.

| Brass pin to
[ support bobbin

FIG. 194.

the final adjustment of value being made with great care when in
position.

It will be seen that the removal of a plug throws the correspond-
ing coil into the circuit. Of course the brass blocks and plugs have
some resistance also, but for many purposes this is small enough to
be negligible.

As the amount of space available is the same for each coil, what-
ever its resistance may be, it is necessary to use finer wires for the
larger resistances. The wire, which is always silk-covered in order
to save space, should be of some material having a high specific
resistance, so that any given value can be made up without using
either an abnormally great length or a dangerously thin section, and
it should also have as small a temperature coefficient as possible.
These requirements are best satisfied by alloys — platinoid or
manganin being largely employed.

The values of the coils are determined by the condition that
it should be possible to make up any integral value within the range

RESISTANCE AND ITS MEASUREMENT 261

of the box with the fewest number of coils. For instance, any value
from 1 to 10,000 ohms can be obtained from 1, 2, 2, 5, 10, 10 20
50, 100, 100, 200, 500, 1000, 1000, 2000, 5000. One of the gaps,
marked " Infinity," has no coil attached to it, and therefore its plug
serves as a key for opening and closing the circuit.

The general appearance of such a box is shown in Fig. 200.

Comparison of Resistances by the Wheatstone Bridge.
— This is the most usual method of comparing resistances. The
principle involved is very simple, and will be easily understood
from Fig. 195, which shows
a current dividing into two
branches at the points A and
B. If we take any point, P,
in one branch, the potential at
that point will be less than
the potential at A, and greater
than the potential at B (as-
suming the current to flow

from A to B), and there is pIG 195

necessarily some point in the

other branch, which is at the same potential. This can be found
experimentally by connecting one terminal of a galvanometer to P,
and moving the wire attached to the other terminal along the branch
AQB, until some point Q is found for which there is no deflection.
We then know that

P.D. between A and P-P.D. between A and Q, and that
PD. „ PandB = PD. „ Q and B.

If i\, r2, r3, r4, be the resistances of the four arms of the bridge
when the above condition is satisfied, and if c^ and c2 be the currents
in the upper and lower branches respectively, as marked in the figure,

then P.D. between A and P = c1r1

A O — c v

J) ,) •£»- 11 *V. — ^9*9

P

>J J? X

» Q

. ' . C-lT-t — C-qTri

B =

and

i.e.

-2' 3

*4

This is the law of the Wheatstone bridge. It will be seen that
if we know the value of one resistance and the ratio of two others,
the fourth can be found by calculation. It is also obvious that the

above expression may be written in the form — = - , which indicates

262

VOLTAIC ELECTRICITY

that if the positions of the galvanometer and of the battery were
interchanged, there would still be no deflection.

The Slide Wire Bridge. — One of the best known forms of
bridge, used in practice, is known as the slide wire bridge, or, as the
wire is generally 1 metre long, as the metre bridge, which is shown
diagrammatically in Fig. 196. A series of thick brass or copper

bars, fitted with the
necessary terminals,
is carried by a
wooden base. Gaps
between the bars
occur at P, x, R, Q.
In the simple forms
of the instrument,
the gaps at P and Q
are often omitted, and even when present they are fitted with stout
copper wires or bars until required. Some of their uses will be
explained later : for the present the bridge may be considered as
having only two gaps at x and R. A uniform wire of platinoid or
other high resistance alloy, L, is soldered to the two end pieces.
Near this wire is attached a scale, usually divided into millimetres,
on which the position of a movable contact piece may be read off.
When connected up as shown, R and x being the known and the
unknown resistance respectively, it will be seen that the current
divides, as indicated by arrows, part flowing through x and R and the
other part through the slide wire. The position of the galvanometer
contact is then adjusted on this wire, until there is no deflection, say
at d divisions from the left-hand side ; then if L is the total length
in scale divisions, the other portion of the wire is of length L— d,
and we have

x^ R
d L-rf
from which x is found at once.

In practice, a number of precautions are necessary. In the first
place, however carefully the slide wire may be soldered to the end
pieces, it is difficult to avoid introducing some unknown resistance
at these points, which of course is equivalent to a slight extra length
of wire at each end. If n and n^ represent these extra lengths in
scale divisions, the true proportion is

x R

d+n it — d + nl

and in exact work, the values of n and wx are found experimentally
to begin with. This can be easily done, but here it is merely
necessary to point out the existence of such a source of error, which
for the present may be taken as negligible.

RESISTANCE AND ITS MEASUREMENT 263

Secondly, it is impossible to be certain that the scale is accurately
placed with respect to the bridge wire, but it is easy to eliminate the
errors due to this source by taking double readings. The scale is
always numbered both ways, and after taking one position of balance
as explained above, it is usual to interchange x and R, and again
obtain balance. The reading is now taken from the other end of
the scale, and the number obtained should be in close agreement with
the first. If the discrepancy is considerable, the first observations
must be repeated. Let d and dl be the two readings thus obtained,

then the mean value — — -1 represents the true reading corrected for

2i

scale error, and we have

x R

Again, it is important to acquire the habit of closing the battery
circuit before depressing the galvanometer key, so that the current is
established before the galvanometer is in circuit. Otherwise, if the
resistance to be measured is inductive, e.g. that of a coil of wire or of
an electromagnet, a momentary kick or throw will be produced even
if exact balance exists, and, in some cases, it may be great enough to
injure the galvanometer.

As it is always possible to find a position of balance, whatever be
the relative values of x and R, it may be thought that a single known
resistance is sufficient for all purposes. In. one sense this is true, but,
if we consider the influence of unavoidable experimental errors on
the final result, it appears that these have the least effect when the
position of balance is near the middle of the wire, i.e. when x and R
are approximately equal. When balance is obtained near one end
of the wire, the effect of such experimental errors becomes very
great, and renders the results unreliable. It is unnecessary here to
discuss the question in detail, but its nature may be illustrated by
taking an extreme case. Let us suppose that the wire is divided into
500 equal parts, and that the position of balance, as found by experi-
ment, is liable to a possible error of one division either way. Let
R = 1 ohm, and let x be such that the true position of balance is at
246 divisions (measured from the end under x),

x I . 246 ,

then = — — . i.e. x = — — ohms.

246 254' 254

If the reading, as found by the experiment, is 245 divisions or

r divisions, we obtain x= —

AD

very much from the true value.

247 divisions, we obtain x = —= or x — , neither of which differs

264 VOLTAIC ELECTRICITY

But if x be such that the balance ought to be obtained at 499
divisions, and working with the same accuracy as before, we actually
get 498 divisions, then the true value is

x = — — ohms, whereas we find it to be

498
x = ohms, the error being 100 per cent.

Hence, it is desirable to have a range of known resistance, i.e. a
resistance box, so that we may always choose a suitable value
for R.

Evidently, this is scarcely possible with extremely large resistances,
and, on the other hand, there is some uncertainty in dealing with
extremely small resistances, because the resistance of the bars of the
bridge is not always negligible in such measurements. Special
methods, which are described later, are required for both cases.

Experiments with Slide Wire Bridge. — Exp. m, to measure

the specific resistance of a material in the form ofivire. It will be convenient at
first to choose one of the high resistance alloy wires (such as platinoid), because
a suitable resistance can be obtained with a moderate length : afterwards the
method may be extended to wires of other metals and alloys.

Assuming that a standard 1-ohm coil is available, cut off a length of the
given wire estimated to have a resistance of about 1 ohm. This can be found
from wire tables, or by means of a rough preliminary measurement. Very care-
fully solder thick copper wires to the ends so that the specimen has a definite
length between these ends. Measure its resistance, taking double readings, as
already explained. Measure with great accuracy the length of the wire, up to
the commencement of the soldered portion, straightening it if necessary. Also
measure the diameter in several places by means of a micrometer gauge, and take
the mean value. If such a gauge is not available, the area of section can be
found from the formula for the volume of a cylinder —

Volume = length x area of section,

the volume being determined by weighing a measured length in air and in water.
Express the results in absolute units per centimetre-cube, in microhms per
centimetre-cube, and in microhms per inch-cube.

This experiment should be repeated with other metals.

Exp. 168, to find the temperature coefficient of a metal. Copper will be the
most convenient metal to use. Measure off roughly a length of cotton-covered
wire having a resistance of about 1 ohm. To avoid an inconvenient length,
a small size should be selected — say 30 gauge, of which 5 yards will be required.
Select a test-tube of about 1 inch in diameter and 6 or 7 inches long. Fit it
with a good cork having two holes bored through it, one to carry a thermometer,
and the other a piece of narrow glass tubing, 4 or 5 inches long, which must fit
tightly and must only just pass through the cork. This tube is used merely to
allow for the expansion of the oil (see below). Another small cork may con-
veniently be placed at the top of this tube to afford a firm grip for a clamp by
which the whole arrangement can be supported. Wind the specimen of wire on
a short piece of glass tubing, tie it down with thread, and solder the ends to two
pieces of 20 gauge copper wire. Fill the tube with colza oil, and then insert
the coil, bringing out the ends of the connecting wires through small cuts in the
cork. Then place the whole arrangement in a large beaker.

RESISTANCE AND ITS MEASUREMENT 265

Fill the beaker with crushed ice or snow, and wait until the temperature
of the oil and wire has become uniform. Measure the resistance very carefully
by the slide wire bridge. Now melt the ice, or, more conveniently, substitute
cold, water, and raise the temperature, by means of a Bunsen llame, a few
degrees above 0° C. Remove the burner, stir up the bath, and again measure
the resistance, noting also the temperature as given by the thermometer in the
oil. Again raise the temperature and repeat the observations, taking ten or
twelve measurements between 0° C. and 100° C.

Plot the values thus obtained on squared paper, taking resistances as
ordinates and temperatures as abscissre. The graph should be a straight line,
showing that the rate of increase of resistance with temperature is uniform over
the range of temperature in question.

Calculate the value of the temperature coefficient from the definition

Temperature coefficient = Increase °f resistance foiarjgeofl^C.

Resistance at 0° C.

Notice that, if we put R100,Ro for resistances at 0° C. and 100° C. respectively,
and a for the temperature coefficient, this becomes

R0 100 x R0

For copper, a= '00428

Again, if we write R; for the resistance at any temperature t° C., we have

generally

or R

From which the resistance at any temperature, t° C., may be calculated, if a is
known ; or conversely, t° may be calculated if R< is measured by experiment. This
is the basis of a very exact method of measuring temperatures, using platinum
instead of copper.

But although the simple expression given above holds good with
considerable exactness between 0° and 100° C., it breaks down when
extremely great ranges of temperature are dealt with. In fact the
graph is not really a straight line, and is better represented by a
parabolic curve, i.e. by an equation of the form Rj = R0 (1+at + bt2),
in which b is an additional constant, small compared with a.

Effect of very Low Temperatures.— It is evident that metals
become better conductors as the temperature is lowered, and it has
been a question of great interest as to whether their resistance would
actually vanish at the absolute zero of temperature ( — 273° C.) or not.
The researches of Dewar and others, first at the temperature of
liquid air, and later at that t>f liquid hydrogen, have proved that the
law changes rapidly near absolute zero.

This investigation has been carried still further by Professor
Kamerlingh Onnes, who has just made known preliminary results
of the highest importance. Using boiling helium, he reached a
temperature of 1'5° absolute, or — 271 '5° C. (as measured on a krypton
thermometer), and found that the resistance of gold and of mercury
dropped rather suddenly to a value too small to be measurable.

266 VOLTAIC ELECTRICITY

For instance, in a certain case the mercury employed had a resistance
of 1727 ohms as a liquid at 0° C. (The change from liquid to solid
produces an effect of its own, which may be allowed for by saying
that if solid at 0° C., the resistance would have been 397 ohms). At
4-3° absolute, this had fallen to -084 ohm, and at 3° absolute it was
below 3 x 10~6 ohm, i.e. less than one ten-millionth of its value at
the freezing-point of water.1 When the temperature was raised, the
resistance first became measurable at about 4*2° absolute. In striking
contrast to this was the behaviour of an alloy (Eureka), whose
resistance remained nearly constant throughout the same range of
temperature.

If it should ever become possible to apply these facts to practical
purposes, far-reaching consequences of the greatest importance may
be anticipated, for conductors of very small section will be able to
carry heavy currents without serious heating and consequent loss of
energy.

Another Method of applying the Bridge Principle.—
Suppose that we arrange the apparatus
as shown in Fig. 197, in which A, B, C,
D are thick metal plates of negligible
resistance fitted with terminals to which
the various wires can be attached, x is
the resistance to be measured, R is a
resistance box, and r, r are two exactly
equal resistances, preferably of the same
order of magnitude as x, but whose
values need not be known. We begin
by adjusting the resistance in R until
FIG. 197 a balance is obtained. Then obviously

x = R ; but as it is impossible to change
H by less than 1 ohm steps, exact balance will rarely be obtained,
and we can say only that x is between, say, 24 and 25 ohms, and
nearer 24 than 25. Such a measurement is quickly made, and is
sufficiently exact for many purposes ; on the other hand, resistances
greater than the total value of R cannot be estimated, and it is
rather difficult to be certain that the resistances r, r are exactly equal.
It is evidently possible to combine the resistance box and the
bridge connections to form one compact piece of apparatus. This
is done in the Post Office pattern, which we have now to describe, but
before doing so, it may be pointed out that a simple arrangement
of the kind shown in Fig. 197 is especially useful, when it is required
to copy a standard coil with great exactness, or to adjust finally the
coils in a resistance box. For suppose that we put a standard 1-ohm
coil in place of R, and that we balance it against any convenient
temporary resistance in place of x. The 1-ohm coil is now removed
and another coil (previously wound and adjusted to a value slightly

1 Quite recently, Onnes has shown that a current once started (preferably
by induction) in a ring of metal at this low temperature persists for hours after
the E.M.F. is removed, owing to the fact that the C-R loss of energy is negligible.

FIG. 198.

RESISTANCE AND ITS MEASUREMENT 267

greater than 1 ohm) is substituted for it. It is then finally adjusted
until exact balance is again obtained. Obviously it must be exactly
equal to the coil it replaced, whether the two resistances, r, r are
equal or not.

Post Office Pattern of Wheatstone Bridge.— This will

be easily understood from the explanatory diagram, Fig. 198. It

differs from the simple arrangement just
described only in the fact that, instead
of a single pair of coils, r, r, it is provided
with three pairs of 1000, 100, and 10
ohms, any or all of which can be thrown
into circuit by taking out the corre-
sponding plugs. These are known as
the "proportional coils." One advantage,
therefore, consists in our ability to choose
the pair nearest in value to the resistance
to be measured, and another advantage
is the extension of range. For, if we put a 10 ohm in the arm CD
and 1000 ohm in the arm BC, balance is obtained when II is T^
of aj. Hence, if the total value of R be 10,000 ohms, resistances up
to 1,000,000 ohms can be measured. The proportional coils can
also be used to estimate fractions of ohms. For example, if, when
using two 10-ohm coils, a resistance is found to be more than 23 and
less than 24 ohms, we may then alter to 100 ohms in DC and 10
ohms in BC, and again obtain balance. Suppose that this requires
R to be between 234 and 235, then we know that the true value
of x is between 2 3 '4 and 23*5. Obviously a second decimal may be
obtained by using the 1000 : 10 ratio, although when such accuracy
is required, it is better to use the slide wire bridge, as it is usually
impossible to be certain of the exactness of the ratio.

The usual arrangement of the box is shown in Fig. 199, lettered
to correspond with Fig. 198. The
galvanometer and battery connections
pass from D and C to the studs under
the keys, P and Q, and when these
are depressed, it will be seen that
the proper connections are made.
The actual appearance is shown in
Fig. 200 (for which we are indebted
to Messrs. Philip Harris & Co). . It
is a compact and useful instrument
for the majority of ordinarv purposes.

Many other forms of bridge have been devised and are in use for
special purposes. We have merely described one of the best known.
Carey Foster's Method of comparing: Two nearly
equal Resistances. — The bridge methods already described are

268'

VOLTAIC ELECTRICITY

not sufficiently delicate when it is necessary to detect very small
differences in resistance, e.g. when comparing standard coils with one

FIG. 200.

another. Many modifications of these methods have been devised,
of which the following may be taken as an example : —

Exp. 169. An ordinary slide wire bridge may be used, provided with the

two additional gaps, mentioned
on p. 262, which are closed \)y
thick copper strips when not in
use. Two auxiliary adjustable
resistances are required, whose
values need not be known. Let
R, Rj be the two resistances to be
compared ; and P, Q the two
auxiliary resistances. Connect
up the apparatus as shown in
Fig. 201.

FIG. 201. (1) Give P and Q any values,

which may for convenience be

approximately equal to R and Rlf and adjust until balance is obtained at some
point reading d divisions on the scale, as in the figure.

(2) Now interchange R and RI, and again obtain balance at some point
reading c?i divisions from the same end of the scale. Let L be the total length
of the bridge wire ; />, the resistance of one scale division of the wire ; and
n, «i, the unknown resistance of the joints and metal at each end respectively
between the end of the bridge wire and the commencement of R and RI.
(These resistances were assumed to be negligible in the previous experi-
ments.)

RESISTANCE AND ITS MEASUREMENT 269

Then, noticing that R and RI are now really extensions of the bridge wire,
we have, in accordance with the simple law of the bridge,

from(l) P .
Q

(2) l

RI + «i + (L - d)p R + nj + (L - dj)p

whence, retaining each numerator, and taking the sum of the numerator and
denominator of each fraction for a new denominator, AVC obtain

_ R + n + dp _ _ _ R! + n + drf _
R + R! -t n f M! + dp + (L - d)p
and as the denominators are identical, we have

or R - RJ = (rfj - d)p

The "end" corrections have cancelled out, and hence we learn that one of the
advantages of this method is that such unknown factors have no influence on
the result. From the equation, we also see that the length of the wire, between
the two divisions read on the scale, has the same resistance as the difference
between the two resistances to be compared, from which it follows that, if this
difference is greater than the total resistance of the bridge wire, the method fails,
because two points of balance cannot be obtained. By adjusting P and Q, the
initial point of balance can be brought to any part of the wire we please.

In order to evaluate the result, p must be determined. This may be done
by measuring the total resistance of the bridge wire in the ordinary way, using
a second bridge ; or it can be done more conveniently by a simple application of
the preceding method as follows : —

Exp. 170. For R, use a known resistance of ,1 value slightly less than that of
the whole length of the bridge wire ; and for R15 use a thick copper strip, the
resistance of which may be taken as zero. Repeat the two operations mentioned
in Experiment 169, adjusting P and Q until the first point of balance lies near
the end of the bridge wire, then the second point will lie near the other end.
This gives

n-0 = (dl-d}p

In our treatment of the subject, we have always assumed the uniformity of the
bridge wire. As a matter of fact, in all exact work, the wire is first carefully
calibrated by experiment, but for information concerning these methods, the
student must consult some work devoted to electrical measurements.

We may also mention that the previous method is capable of inversion in
a way that makes it available for the measurement of very low resistances. For
example, it would be practically impossible to measure the resistance of a thick
copper wire or bar, say a yard long, by the ordinary use of the bridge. But if
the details of the latter are modified, so that the bar can be temporarily but
firmly clamped to become the bridge wire itself, wre may — using for R and R\ two
known resistances differing very slightly in value — mark off' on the bar a length,
whose resistance is equal to this difference. Further, by measuring the cross
section of the bar, we can deduce the specific resistance of its material.

270 VOLTAIC ELECTRICITY

Measurement of very Low Resistances.— As already
indicated, ordinary bridge methods are not well adapted to measure
resistances much less than 1 ohm in magnitude, as the resistance of
the bridge parts and connections are then no longer negligible, and are
difficult to allow for with the necessary accuracy.

Special methods of comparison are then preferable, it being under-
stood that a known resistance of small value is available for the

purpose. Such methods usually depend

A X B C R D on *ke measurement or comparison of
- - - - differences of potential. For example, let

R be tne known, and x the unknown
resistance, and let them be arranged in
FIG 202 series with (say) a Daniell's cell, as shown

in Fig. 202. If e, el be the P.D.'s between

AB and CD respectively, then, as the current is necessarily the same
in both, we have

0 = - = ^, from which - = -
x R' R et

Hence, we see that we have to determine the ratio — , which can be

e\
done in several ways. The most accurate method is to use a potentio-

meter (as explained in the next chapter), and by its means to compare
the two P.D.'s exactly as if they were due to two cells.

A somewhat simpler plan is to use a sensitive reflecting galvano-
meter l of very high resistance compared with either II or x, and to
connect it in turn across AB and CD. It then becomes practically
a voltmeter. Let Clt C2, be the currents through the galvanometer,
producing steady deflections d-^ and d2 in the two cases, and let y be
the resistance of the galvanometer. On account of the high relative
resistance of the instrument, Ct and C2 are negligible compared with
C, the current through R and z, and the values of e and ex are not
appreciably altered.

Now Ct = - oc d1

e i , oo e

.*. — = -* , and we nave — = —

% d2' R ^

whence x = R x — I

''•i

1 It is convenient at this stage to assume that such instruments are avail-
able. They are discussed in Chapter XIX.

RESISTANCE AND ITS MEASUREMENT 271

FIG. 203.

In order to obtain deflections of convenient magnitude, it is desirable
to place a resistance box in the circuit, the actual arrangements being
made as shown in Fig. 203. A mercury
cup switch (see p. 253) is used to enable
the galvanometer connections to be
readily interchanged. The connections
from C and D are crossed so that both
deflections may be on the same side of
the scale. In order to avoid possible
injury to the galvanometer, a large re-
sistance should, at first, be unplugged
in the box (RB), and should . be gradu-
ally reduced until the deflection reaches a suitable value.

Parallel Circuits. — When very large resistances (say, a million
ohms or more) have to be measured, the Wheatstone bridge again
fails, in this case through lack of sensitiveness, and special methods

are required. In order to under-
stand them, it is necessary to study
the laws of parallel circuits.

Let AB (Fig. 204) be a portion
of a circuit, in which the main
current C divides up among a
number of branches of resistance
r]? rv r^&c., arranged in parallel.
It is required to find the equi-
valent resistance R, i.e. the value
of a single resistance, which might replace the branches without
altering the current. Let e be the P.D. between A and B. This is
common to all the branches, and therefore we have

FIG. 204.

e n e n

— , C9 = — , &c.

r 1*

i ' o

AlsoC = C1

R

or = + +
R r r r

rl r2 r3

Observe that, in the special case when n equal resistances are arranged
in parallel, this becomes

1 1 r

272 VOLTAIC ELECTRICITY

The case of two resistances in parallel occurs most frequently in
practice ; this becomes

R^l^*2
= 1 ^ = :

ri T2

To ascertain how the current divides up among the various paths, we
notice that

c =CR c =CR

from which we learn that the currents are inversely proportional to
the resistances.

Case of a Shunted Galvanometer.— The easiest way of
varying the sensitiveness of a galvanometer is to provide it with a
by-pass or " shunt," as shown in Fig. 205.
For instance, when a delicate reflecting
galvanometer is used in connection with
"null" methods, e.g. in potentiometer or
Wheatstone bridge work, it is liable to
be seriously damaged by excessive deflec-
tions during the initial stages of the
measurement whilst the approximate posi-
tion of balance is being found. It is usual
in such cases to connect a resistance box in
parallel with the galvanometer, and to begin
by unplugging a resistance small in com-
parison with that of the galvanometer. If the galvanometer has, say,
500 ohms resistance, and if we unplug, say, 1 ohm in the box, then
by the above argument only -^T of the total current will pass through
the galvanometer, and the deflection will be correspondingly reduced.
By unplugging more and more resistance, the galvanometer can be
made more and more sensitive, until for the final exact adjustment of
the position of balance, the infinity plug can be removed, which is
equivalent to disconnecting the shunt and giving the galvanometer
its maximum sensitiveness.

On account of the practical importance of this case, it is advisable
to discuss it in detail.

Let s be the resistance of the shunt, and g the resistance of the
galvanometer, then, by the previous proof, the equivalent resistance of
the shunted galvanometer is

R = ^L

i.e. the resistance is of its old value.

RESISTANCE AND ITS MEASUREMENT 273

Again, if Cg = current through the galvanometer, Cs = current
through the shunt, C = total current, and e = P.D. between galvano-
meter terminals, then we have

e = Cg x g = Cs x s

ie C'-s
~

• ' and

but
whence equation (i.) becomes Cg = — — .0

and „ (ii.) „

Thus, if we wish to make -^ of the total current pass through the
galvanometer, then we must give s the value determined by

i.e. s = \g ^

Similarly, if we wish yj^ or T^Vo °^ *^e current to pass through the
galvanometer, then the resistance of the shunt must be -^ or ¥Jg-
respectively of the galvanometer resistance.

Sometimes, galvanometers are provided with special " shunt
boxes," containing resistances which are usually J, ¥\, and -^^ of
that of the instrument itself. Their great advantage is that of facili-
tating calculations — a matter of importance when many measurements
have to be made for commercial purposes. On the other hand, there
is the disadvantage that such a box can be used only with the instru-
ment for which it was made. Professor T. Mather devised and, in
conjunction with the late Professor Ayrton, brought out a form of
shunt box which will serve for any galvanometer, but its action is
more complicated, and for the present it will be desirable to limit the
argument to the simple form given above.

(Some additional remarks on the effect of shunts are given on
p. 299.)

Measurement of very High Resistance.— In practice,
such measurements are frequently required, more especially in con-
nection with "insulation resistances." For example, when a building
is to be wired for lighting or power, it is necessary to measure the
resistance between the positive and negative network of wires before

S

274 VOLTAIC ELECTRICITY

any lamps are connected across them. The measurement is, therefore,
a test of the goodness of the insulation, and the value will usually
be something in megohms, although it should be remembered that
the larger and more extensive the system, the lower the insulation
resistance naturally becomes. For this particular purpose, a direct
reading instrument — known as an Ohmmeter — is used, as only
approximate accuracy is required, and the method must be simple and
easily applied.

Makers of the cable used for wiring buildings must also test its
insulation resistance before it is sent out from the works, as they
guarantee that the insulation resistance shall have 'a certain number
of megohms per mile — from about 300 to 1200, according to quality.
This may be regarded as the resistance existing between the copper
core and some conductor in contact with the outer surface of the
insulation, the second conductor being usually provided by immersing
the coil of cable in water, with both ends clear. It will be evident
that the resistance, under these circumstances, is inversely as the
length of the cable, so that, if the value per mile is 600 megohms
and a length of T\ mile is under test, the actual resistance to be
measured is 16 x 600 =- 9600 megohms.

Again, the construction of most electrical machinery involves the
insulation of copper conductors from a metal framework or core, and
it often becomes necessary to test the quality of the insulation, by
measuring the resistance between the copper and the metal upon
which it is wound.

For such measurements as these, the Wheatstone bridge again fails
through lack of sensitiveness, and the most generally useful method
for laboratory work is as follows. It involves the comparison of the
unknown resistance with a single standard resistance of high value —
say 10,000 to 20,000 ohms. As the student is not likely to have a
suitably high resistance for measurement, one, having a value of from
J to 1 megohm, must be extemporised. This may be done by having
a glass tube, which is fitted with corks, carrying copper wires, and
which is filled with distilled water ; or, more conveniently, by making
a conducting surface upon a strip of dry wood or ground glass by
means of an ordinary lead pencil.

Exp. 171. Select a strip of dry wood, 4 or 5 inches long, about 1£ inches
wide, and of such thickness that brass terminal-clamps (such as are used
on carbon plates) may be attached at each end. Make a lead pencil track —
smooth and uniform, of about ^ to f inch wide — from end to end ; at each
end lay on a bit of clean tinfoil and screw the clamp on it. The soft tin
is used merely to assist in making good contact between the graphite and the
brass.

Connect this up to a delicate reflecting galvanometer, and to a single cell,
Fig. 206 (1), taking care to insulate it and the connecting wires from the bench,
because even wood conducts well enough to vitiate the result. (The necessity
for care in this respect will be realised if the student tries the effect of holding
the terminals of the specimen in his hands, thus providing a shunt path through

p >-~- \.

- ( ]

, v X ^J

^t- rJ"""

RESISTANCE AND ITS MEASUREMENT 275

his body.) If the deflection is inconveniently small, add another cell (or more)
ill series ; if too large, reduce the width of the graphite by scraping the edges
with a knife. Carefully note steady
deflection (d divisions), then without ^\ (T\

altering the cells or disturbing the
galvanometer, replace the high resist-
ance by a resistance box and connect up
another box as a shunt to the galvano-
meter, Fig. 206 (2). (In such operations,
always first remove the infinity plug,
which then answers the purpose of a ( ' )

key.) Unplug as high a resistance as FlG. 206.

possible, using a simple number such as

10,000 or 20,000 ohms, in the box R. See that all the plugs, including the
infinity plug, are in their places in the box S, and then insert infinity plug in
box R. The result will probably be a very small, perhaps scarcely perceptible,
deflection (if the shunt were not used, the deflection would be so violent as
probably to injure the galvanometer). Gradually unplug resistances in S until a
convenient deflection is obtained. Then note the values of S, R, and the deflection
of d divisions. The resistance, #, of the galvanometer is also required — if not
known it must be measured as a separate experiment, using a Wheatstone's bridge
method and another galvanometer as an indicator of balance.

Let E = E.M.F of cell or battery ; and b, its internal resistance. Then total

resistance is 6 + R+ — - — (see p. 272). Now, even if g is great, s is not likely to
s+g

be very large, and — £- must be less than s, and is probably negligible com-

s+g
pared with R. Hence, we may take R as the effective resistance.

If, in the first experiment, C is the current, d tho deflection, and x the
unknown resistance ; and in the second experiment, Ca is the total current, C(J
the current through the galvanometer, and d1 the deflection, we have

0 = 5* d (i.)

X

n _ _E_

ft R (practically)

Also from p. 273, Cg= — s—
s+g

and C00C dl

.

s + g R d

It will be noticed that the factor *S£3 is constant for the given setting

S

up of the apparatus ; after it is determined, various unknown resistances can be
found without repeating the second portion of the experiment.

Exp. 172. Estimate the resistance of your body, by noting roughly the deflec-
tions (probably unsteady) which are obtained by holding in the hands the, wires
previously attached to x. Observe also the great decrease in resistance when
the hands are moistened, or when larger pieces of metal are used as handles.

276

VOLTAIC ELECTRICITY

Exp. 173. Adjust the graphite resistance until it is as near as possible
1 megohm. If the measured value is less than this, calculate the deflection
corresponding to 1 megohm, and whilst it is connected in circuit, gently scrape
off the graphite at the sides until the deflection has the required value.

Such a resistance is often useful, and should be retained.
Kirchhoff's Laws. — The equivalent resistance, and the currents in
the various branches of a divided circuit can be readily calculated by the simple
methods already given when there are no cross connections in the network ; it can
be applied, for instance, in Fig. 207a, but the method fails in Fig. 207&, which is
especially interesting on account of its application to the Wheatstone bridge.

a

FIG. 207.

Equations for the complete solution of such problems can easily be written
down by using Kirchhoffs two laws. These are not "laws" in the same sense
as Ohm's law ; they contain nothing new to the student, and are merely useful
rules which show how Ohm's law is to be applied.

They are : (1) In any branching network of ^v^res, the algebraical sum of the
currents in all the wires meeting at a point is zero.

The term algebraical sum means that if the currents are flowing from the
point they are taken with a negative sign, and those flowing to the point with
a positive sign, or rice versa.

(2) In any closed path in the network, the algebraic sum of the C x r products
is equal to the E.M.F. acting in that path.

Here clockwise currents are to be reckoned positive and anti-clockwise
currents negative, or vice versd.

The first law is obvious. The second will be more easily understood by taking

a particular case — as a matter of
general interest, that of a Wheat-
stone bridge.

Let a cell of E.M.F. =E, and
of internal resistance = b, be con-
nected by wires of negligible
resistance to the Wheatstone
network at H and L, the resist-
ances of whose arms are as
marked in Fig. 208.

Then, if we write as usual,
C for the total current through
the cell, 0,7 for that in the gal-
vanometer, g for the galvano-
meter resistance, and if we take
Cj as the current in one of the
arms, say HK, we may deduce
the currents in the other arms
by applying the first law. These
values are marked in the figure. Now, consider the closed path HKM. The
currents are clockwise in HK and KM, and anti-clockwise in HM ; also there is
no source of E.M.F. in this path.
. ' . We have by the second law

Cir1 + CflJ7-(C-C1)r8 = 0, (i.)

RESISTANCE AND ITS MEASUREMENT 277

(This result is more obvious, if we notice that the P.D. between H and K
must be C^ , and that between K and M must be Cg.y. These are in the same
direction, and, therefore, the P.D. between H and M must be G^ + Cg.g. But
this latter P.D. must also be (C-C1)r3, from which the above equation follows.)

Again, considering the closed path KLM, we obtain

In the closed path EHML, the currents are all clockwise, and as there is an
E.M.F. (E) in this path, we have :—

(C - Cj)rs + (C - G! + C,,)r4 + Cb = E

(iii. )

and similarly for the path EHKL,

Taking three of these equations (i. and ii. and either iii. or iv.), we can solve
simultaneously for three unknown currents.

The student will perceive that the chief difficulty is not in obtaining the
necessary equations, but in solving them afterwards. This is a purely mathe-
matical question, dealt with in more advanced works, and need not be discussed
here.

If it is required to find the equivalent resistance of a complex circuit, this-
"really means that we have to determine the P.D. between the common points,
as, for instance, H and L in Fig. 208. If this be e, and R the equivalent resist-
ance, then by the definition of R, e = CR.

Now, e=Cjr| + (O, + G^)»v; or, (C-C1)r3 + (C-C1 + Ci,)r4, and can be
evaluated when the currents have been determined.

In certain special cases, the value of e can be found much more readily by
considerations of symmetry.

Examples. — (1) Find the resistance of a skeleton cube, taken between
opposite diagonal corners, if each edge has a
resistance of r ohms.

Let C be the total current, and e the
P.D. between A and B (Fig. 209). Then by
symmetry, the current divides into three
equal parts at A, and the three equal parts
reunite at B. Hence, currents in AH and

M

KB are each^.
o

Also at H there are two

symmetrical paths for the current - , and

3

r\

hence, the current in HK must be - .

6

Now, P.D. between A and B is the sum
of the P.D. along AH, HK, and KB.

FIG. 209.

but, if R=the equivalent resistance, e = (

or

278

VOLTAIC ELECTRICITY

(2) Find the equivalent resistance of the same cube taken between the ends

of one of the edges.

Let C be the total current, which divides
at A into a component Ca along AB and into
two equal components, each of which may be

written ~& along AH and A? (Fig. 210).

These currents, by symmetry, must reunite
at B. Then, putting C2 for the current
along HL, all the other components can be
written down from the symmetry of the
figure, using Kirchhoffs first law.

Again, it is evident that the P.D. (e)
between A and B is independent of the
path traversed, and hence we have : —

(i.)

FIG. 210.
Path AB,
Path AHKB,

C-C,

o

putting Cjr = e , we obtain e — - Cr - -(

Path AHLMKB, e = 2(C~2CAr + 4C2r

which becomes 2<e = O + 4C2r
From (ii.) and (iii.), C + 4C2r = - C - - C2

/i 1 /~i

(iv.)

Substitute (iv.) in (iii.), then

or
But

(3) Find the resistance of a skeleton tetrahedron between any two corners, if
each edge has resistance r.

The' student will have no difficulty in proving that this is - if> °n drawing

the figure, he notices that the two corners are points, which, by the symmetry of
the figure, must be at the same potential. Hence, the resistance is not altered if
the points are directly con-
nected, and the figure then
reduces to a very simple case-
The method suggested in
this case may also be applied
to Examples 1 and 2. For
instance, in Fig. 209 it is

-.,, 9, ,

obvious that L, H, and N must all be at the same potential, and may therefore
be directly connected together ; the same remark also applies to the points M,
H, and K. It is convenient to draw a key diagram as shown in Fig. 211, on
which all the letters and sides are indicated.

Regarding LHN and MPK as single points, we join up all the sides in accord-

RESISTANCE AND ITS MEASUREMENT 279

ance with the lettering of the original figure, and thus obtain an equivalent
figure, which can be treated by the ordinary laws of parallel circuits, from which,
obviously,

'~3 + 6 + 3~6r

The second example is a little more complicated. H and P are at the same
potential, and so also are K
and N. We, therefore, begin
with the four points, A, HP,
KN, and B, and join up in
accordance with the lettering.
The result is as sliown in Fig.
212, which, by a repeated ap-
plication of the laws of parallel
circuits, gives

7

P N

FIG. 212.

FIG. 213.

Lord Kelvin's Method of Measuring the Resistance

Of a Galvanometer. — Exp. 174. Connect up the galvanometer whose
resistance (y] is to be measured in one arm of any form of Wheatstone bridge,
omitting the galvanometer ordinarily used to obtain,
balance, its position being occupied by a simple
key, KU as shown in Fig. 213.

If another galvanometer is available, we can, of
course, measure the resistance of the first one in
the ordinary way, in which case its own moving
part will be kept clamped. Let us first suppose
that such a second instrument is placed in the arm
occupied by K1? and that we watch the indications
of both galvanometers, whilst going through the
operations of obtaining balance. It will be noticed
that the first galvanometer is always deflected
when the cell circuit is closed, and that, as a rule,
closing the key Kj alters that deflection, but that,
when balance is obtained, as indicated by the non -deflection of the auxiliary
instrument, there is no alteration in the deflection when the key Kj is opened or
closed. The reason of this is obvious, and it follows that we may dispense with
the auxiliary instrument altogether, using merely the arrangement shown in the
diagram. Then, if we adjust the resistance in the other arms until the deflection
is unaltered by opening or closing K1? we know that the ordinary law of the
bridge is satisfied, i.e. gr2= r-^r^ whence g is obtained.

This method is useful when a second galvano-
meter is not available, but the ordinary method is
the more satisfactory. .

Mance's Method of Measuring
the Internal Resistance of a Cell. —

Exp. 175. Place the cell, whose internal resistance
(b) is to be measured, in the position occupied by g,
and a galvanometer in that occupied by KI in the
last experiment. Notice that a deflection of the gal-
vanometer is produced, and that the closing of the
battery key, K, superposes on this current another
due to the testing battery. But when the condition
ftr-g — rjTg is satisfied, the testing battery does not send any current through the
galvanometer when K is closed, and the deflection will be unaltered. Remove

FIG. 214.

280 VOLTAIC ELECTRICITY '

the testing battery, and notice that there is still no deflection on opening or
closing K. If, however, the above condition is ij§t satisfied^ the deflection will
be altered. Hence, we have only to arrange the apparatus as skown in Fig. 214,
and adjust the resistances until the deflection is unaffected by opening or closing
K. Then 6»*2 = r1r3, whence & is obtained. In this experiment, the galvanometer
should not be too delicate. It is a very good plan (if a moving magnet gateano-
meter is used) to place a magnet in such a position as to bring the^-needlQiear
its ordinary zero, because then we obtain the greatest change of deflectienTor &
given change of current, and the test is correspondingly more sensitive. /"

I
EXEKCISE XIV 1%

1. What is the resistance of a column of mercury 2 metres long and '(Lof a
square millimetre in cross section at 0° C. ? r . < |

2. What length of copper wire, having a diameter of 3 millimetres, has the
same resistance as 10 metres of copper wire, having a diameter of 2 millimetres^

3. If the resistance of a yard of iron wire, "03 inch in diameter, be '197 ohpis,
what is the resistance of 15 miles of iron wire, '3 inch in diameter ?

4. What must be the thickness of copper wire which, taking equal lengths,
gives the same resistance as iron wire 6 '5 millimetres in diameter, the specific
resistance of iron being six times that of copper ?

5. If the resistance at 0° C. of an. iron wire 1 foot long and weighing 1 grain
be 1'03 ohms, find the resistance at 0° C. of 1 mile of iron wire weighing 300 Ibs.

6. A piece of copper wrire 100 yards long weighs 1 Ib. ; another piece of copper
wire 500 yards long weighs £ Ib. Show what are the relative resistances of the
two wires.

7. Two exactly equal pieces of copper are drawn into wire ; one into a wire
10 feet long, and the other into a wire 20 feet long. If the resistance of the
shorter wire is '5 ohm, what is the resistance of the longer wire ?

8. Two copper wires, one of which is 4 metres long and weighs 7 '5 grams,
and the other 5 metres long and weighs 4 '2 grams, are joined in "multiple
arc " (that is, so that both wires connect the same two points) in a circuit in
which a current of total strength 8 '09 (amperes) is passing. The current will
divide so that part passes through each wire ; what will be the strength of the
current conveyed by each ?

9. Two wires are joined in parallel circuit, their resistances being 10 and 20
ohms : find the resistance of the conductor thus formed.

10. Three wires are joined in parallel circuit, their resistances being 40, 15,
and 55 ohms : find the resultant resistance.

11. The resistance between two points A and B of a circuit is 30 ohms, but
on adding a wire between A and B the resistance becomes 20 ohms. What is
the resistance of the added wire ?

12. A galvanometer of 1000 ohms resistance is shunted with a shunt of
1 ohm resistance. Find the resistance of the shunted galvanometer.

13. A coil concealed in a box, but with its terminals exposed, is found to
have a resistance of 126 ohms. A piece of wire, 10 metres in length, of the same
material as that of the coil, but of twice its cross -sectional area, is found to have
a resistance of 12 ohms. What is the length of the Avire in the coil ?

(Lond. Univ. Matric., 1906.)

14. A wire, the total resistance of which is 4 ohms, is bent into the form of a
square, ABCD, the loose ends being soldered together. Find the resistance of
the system when a current enters at B and leaves at D. Will it be modified if
the corners A and C are connected by another wire ?

15. If a cell has an E.M.F. of 1'08 volts and '5 ohm internal resistance,
and if the terminals are connected by two wires in parallel of 1 ohm and 2 ohms
resistance respectively, what is the current in each, and what is the ratio of the
heats developed in each. (Lond. Univ. Inter. B.Sc., 1902.)

16. A circuit is made up of (1) a battery with terminals A, B, its resistance

RESISTANCE AND ITS MEASUREMENT 281

being 3 ohms, and its E.M.F. 2*7 volts ; (2) a wire BC, of resistance 1'5 ohms ;
(3) two wires in parallel circuit, CDF, CEF, with respective resistances 3 and 7
ohms ; (4) a wire FA, of resistance 1'5 ohms. The middle point of the last wire
is put to earth. Find the potential at the points A, B, C, F.

(B. of E., 1897.)

17. A wire of resistance r connects A and B, two points in a circuit the
resistance of the remainder of which is R. If, without any other change being
made, A and B are also connected by n-1 other wires, the resistance of each of
which is r, show that the heat produced in the n wires together will be greater
or less than that produced originally in the first wire according as r is greater or
less than R Jn. (B. of E., 1900. )

18. Describe carefully the "Wheatstone's bridge method for comparing the
resistance of two coils. If the two coils are of nearly equal resistance, how would
you arrange the experiment so as to obtain great accuracy ? (B. of E., 1908.)

19. Explain the terms specific resistance and temperature-coefficient of resistance
of a material. What material would you employ for constructing a standard
resistance, and how would you wind the wire ? (B. of E., 1906.)

20. Explain the action of a shunt used with a galvanometer. The resistance
of a galvanometer is 100 ohms ; it is connected in circiiit with a battery, whose
E.M.F. is 15 volts ; the resistance of the battery and wires is 5 ohms ; the
galvanometer is shunted so that one-tenth of the total current passes through it.
Find the resistance of the shunt, and the current through the battery.

(Camb. Local, Senior, 1894.)

21. A closed circuit contains a battery of 1 ohm resistance, a reflecting
galvanometer of 4 ohms resistance, and other conductors of 2 ohms resistance.
The deflection of the galvanometer is 100 divisions of the scale. What will the
deflection be (assuming it to be proportional to the strength of the current) when
the terminals of the galvanometer are connected by a wire of 4 ohms resistance ?

22. A galvanometer, the resistance of which is % ohm, being joined up in
circuit with a cell by "thick copper wires, the resultant current is noted ; and it
is found that the current in the galvanometer is halved if, without any other
change being made, the terminals of the galvanometer are joined by a wire of
resistance '1 ohm. What is the resistance of the cell ?

23. A network is arranged as in Mance's experiment. The resistance of the
battery, of the galvanometer, and of each of the other three arms, is 4 ohms, and
the*EM.F. of the battery is 2 '2 volts; find the current in the galvanometer
(1) with the key open, (2) when it is closed. If the resistance of the arm opposite
the battery is altered to 5 ohms, find the new values of the current.

(Lond. Univ. B.Sc., 1898.)

^ 24. On passinga current of 1 ampere through a piece of platinum wire, it is
found that its temperature rises 10° C. above that of the surrounding objects,
whicfi are at 0° C.^ Assuming that the rate of loss of heat is proportional to the
difference of temperature, calculate the temperature of the wire when a current
of 2 amperes is passed through it. The temperature coefficient of the resistance
of the wire may be taken to be '004 of the resistance at 0° C.

(Lond. Univ. Inter. B.Sc., 1907.)

25. What difficulties are met with in measuring a very small resistance by
the Wheatstone's bridge method ? Describe a method of comparing, low
resistances. (Lond. Univ., B.Sc. Internal, 1909.)

26. A certain electromagnet has to be excited from mains at a definite volt-
age. Show that if any prescribed number of ampere-turns of excitation are to be
given, the coil must be such as to have the resistance per turn of a definite value,
independent of the actual number of turns. A coil, of which the mean length
of turn is 42 inches, is required to produce an excitation of 6300 ampere-turns,
when supplied at 35 volts. Calculate the cross-section of the wire, assuming
that a copper rod 1 foot long and 1 square inch in cross-section has a resistance
of 9-2 microhms. (City and Guilds, 1908.)

CHAPTER XIX

GALVANOMETERS— MEASUREMENT OF CURRENT

THE word "galvanometer" is a term of very general meaning, covering
a wide range of instruments of different types and applications. It
may be regarded as signifying an instrument whose primary function
is to indicate the existence of a current, more especially of weak
currents, and which may, under certain circumstances, be capable
of measuring it.

The simplest and oldest form consists of a coil of wire, within
which is pivoted or suspended a small magnet, usually termed the
needle (see p. 198). Normally, the needle is at rest in the magnetic
meridian, and the position of the instrument is adjusted until it is
at right angles to the axis of the coil. On passing a current through
the coil, a magnetic field is produced, which at the centre of the coil

.--— -glass shade
— bobbin.

/T

j

r

' rSSi ^

^ lupf— ^

r«

br

sti

ass rod.
k suspension,
ick glass pointer,
rdboard scale of degrees,
coil in section.
\\mica carrying
| pointer £ needle .

terminal.

r

—to

loo ooooooooooo 1 jjk] 1 ooo ooooooooooo j

— -ca

VMlrr%S?&+-

~\ leooooooooooool [ ooooooooooooool

n

H

UU

1— I /

FIG. 215.

is at right angles to the earth's field. The two magnetic fields
combine to produce a resultant field, inclined in direction to both,
along which the needle points, its deflection being read off on a
scale of degrees. It will be noticed that the effect of the earth's
field is to produce a restoring or controlling force, which brings the
needle back to its original position when the current is interrupted.

A very convenient form of simple galvanometer is shown diagram-
matically in Fig. 215. The glass shade is an ordinary flat-bottomed
evaporating dish, such as is used in chemical laboratories, about
6 inches in diameter. The rectangular coil (shown in section) is
divided into two parts, separated by about ^ inch, so that the needle

GALVANOMETERS 283

may be suspended between them. This is a short piece of knitting-
needle carried by a strip of mica (shown, in the diagram, as deflected
through 90°) which has an extension upwards to which the silk fibre
and a black glass po'.nter are attached. The mica is made nearly as
large as the inside of the coils, in order that the oscillations may be
checked by air friction. On the coils rests a circular card graduated
into degrees, and the needle is raised or lowered by means of a little
roller carried by a brass rod. This pattern has been used by the
writers for many years, and has proved very serviceable for teaching
purposes.

The greater the number of turns in the coil, the closer it is to
the needle, and the more delicately the needle is pivoted or sus-
pended, the more sensitive will be the instrument ; although as a
rule there is no simple law connecting current and deflection, which
can be found by calculation from the dimensions. For instance, if
one current gives a deflection of 40°, and another a deflection of
20°, it is certain that the first current is the greater, but it is
impossible to say how much greater, except that it is probably more
than twice as great. It is, however, possible to find the relation
between current and deflection ~by simple experimental methods, and
it will be useful to explain one such method here. It assumes that
the resistance of the galvanometer and that of the cell used have
been previously measured.

Exp. 176. Connect up the galvanometer in series with a Daniell's cell and
a resistance box. Adjust the resistance in the- box until a small deflection,
say 5°, is obtained. Call this current unity. Add up the total resistance in
circuit — resistance box, galvanometer, and cell (although, if the cell has a low
internal resistance, this may be negligible). If this total resistance is now
halved, the current will be doubled. Adjust the value in the box until this
is the case. For instance, if r, g, and 6 are the resistances of the box, the
galvanometer and the cell respectively, the new resistance to be unplugged in

the box would be - — | — • - (g + b) ohms. Take the new reading, which will be

nearly twice the first. Then make the total resistance £ of its first value, i.e. unplug

- (y + b) ohms. Proceed in this way as far )0

as possible, thus obtaining a series of deflections due

to currents of unknown value, but which are in a 8

known ratio, 1, 2, 3, 4, &c. Plot a graph with 7

these numbers for ordinates and degrees for |

abscissae. The graph obtained will be of the

form shown in Fig. 216, and from it can be 3

obtained the relative value of the currents pro- 2

ducing any given deflections. It will be observed (

that for a short distance from the origin the curve 0 . . . ^ , . . g. . , g

is nearly straight, which means that for very small 0 30° 60° 90

deflections the current may be taken as approxi- pIG 216

mately proportional to the deflection without

sensible error. It is more nearly proportional to the tangent of the angle of

deflection, as will be shown later, but for very small angles the two are

Deflection.

284

VOLTAIC ELECTRICITY

practically identical. It will also be noticed that the increase of deflection for
a given increase of current diminishes very rapidly above 40° or so (the exact
shape depends on the details of construction of the galvanometer), and that an
infinitely strong current would be required to produce a deflection of 90°.

For many purposes we require only the relative values of current,
but it is evident that if, in addition, we can determine the actual
current corresponding to a given deflection in any one case, then
by using the curve we can find the actual value of the current cor-
responding to any deflection throughout the scale. For instance,
in an experiment with an instrument of the type shown in Fig. 215,
it was found that a current -of '005 ampere produced a deflection
of 44°, and tha£ on the calibration curve 44° corresponded to a
reading of 1*22 divisions on the scale of current. Suppose that we
require to find the value of a current C giving any other deflection
d°, then if D be the scale reading corresponding to d°, we have

C D

•005 1-22

•005

or C = -^ x D = -0041 x D ampere.

The Tangent Galvanometer. — The difficulty encountered
in attempting to deal mathematically with the previous type of
instrument is due to the fact that the field produced by the current
is not uniform throughout the space traversed by the needle, and
hence the direction of the resultant field depends not only on the
current strength, but also on the position of the needle. When
the field at the needle is uniform, like that of the earth (i.e. when

it can be represented by parallel
equidistant straight lines), the diffi-
culty vanishes, and this condition
can be fairly satisfied by a simple
modification of the construction. It
is only necessary to use a circular
coil, whose diameter is at least ten
or twelve times as great as the
length of the needle. Fig. 217
shows one form of the instrument
(made by F. E. Becker & Co.), in
which two coils are wound on a
brass ring — the number of turns
of each being known, and the
radii carefully measured. A short
needle, carrying a long aluminium
pointer at right angles to it, is

suspended by unspun silk. The graduated scale is provided with
a mirror to avoid parallax.

FIG. 217.

GALVANOMETERS 285

In this instrument the field is approximately uniform over a
small region at its centre, as may be inferred by examining Fig. 181,
p. 244, and therefore we can apply the argument given on p. 142,
where it i^ shown that, if F be the strength of the deflecting field,
and H that of the controlling field (both supposed to be uniform),
then the Direction of the resultant field along which the needle

F

points is determined by the expression = = tan 0. In this case, F is

evidently the strength of the field due to the current in the coil,
and we have now to find the relation between the two.

Simple experiments show that the field at the centre of the coil
is directly proportional to the current strength, and it is customary
to write G for its value when the current is unity. Hence, when the
current is i absolute units, the field strength is iG, and the resultant
field will have the direction determined by

iG

jj = tan0

TT

i.e. i = ~tB,nO

Of course the same expression holds if we use amperes, but it is
then necessary to regard G as being the field due to 1 ampere.

The above formula is important, because (1) for very small
deflections it holds good for any galvanometer, in which the con-
trolling couple is due to the earth's field (although it may not be
possible to evaluate G by calculation), and (2) an expression of
similar form may be applied — but only for small deflections — to any
type of galvanometer, whatever be the nature of the controlling force.
These results follow from the fact that, for any very small range of
motion, the field due to the current is practically constant in direction.

We have now to find a value for G, in the special case of the
tangent galvanometer. It will, however, be convenient to generalise
the problem by finding
an expression for the field
strength at any point on
the axis of a circular, coil
carrying a current. Let

AB (Fig. 218) represent V p H

the section of a coil of
radius r and of n turns,
carrying a current of i' JL
absolute units. It is re- & S-J
quired to find the field FIG. 218.

strength at any point P

on the axis. Let d be the slant distance of P from the coil. (If
there is only one turn, then d is measured from the centre of the

286 VOLTAIC ELECTRICITY

wire ; if there are n turns, we may then, with sufficient accuracy for
our purpose, regard d as being the mean distance.)

Now, the field at P is due to the joint effect of every part of the
wire, and it is convenient to consider first the influence of a very
small portion of length s. Some difficulty arises here, as there is no
method by which such a portion can be isolated in order to study its
action experimentally. All that can be done is to observe the effect
of the coil as a whole, and experiment then shows that the field at
P is directly proportional to the current strength, to the number of
turns in the coil, and to the square of its radius (i.e. to its area).

Again, when these quantities are kept constant and the distance
OP varies, it is found that the field varies inversely as (OP)3, as
long as the point P is not too close to the coil.

Laplace pointed out that, if it be assumed that the magnetic force
due to a very small length s is directly proportional to its length
and to the current strength, and inversely proportional to the square
of its distance from P, and that the field thereby produced is at right
angles to the line joining s and P, then the above result follows
mathematically. That is, writing Hs for the force, Fs for the field
produced by the current in s, we have in C.G.S. units

TT i.s T, i.8

(Although p is unity, -we have inserted it in this expression to avoid
difficulties that might arise later.)

It must be clearly understood that this equation is assumed to
begin with, and that it .is not capable of direct verification. The
justification for using it lies in the fact that in this and in many
other cases, it leads to results for complete circuits which are found
experimentally to be correct.

In the present instance, the field ¥s at P acts along PK, the
angle APK being 90°.

Let L KPH = 0, then L. GAP = L KPH = 0

Resolve the field along PK into components PH, PV along the
axis and perpendicularly to it respectively
The component of Fs along the axis

TP n i-s n i*S V i.S.T

= F.cos0=_/* cos 0 = ^/1 .---^.p

If the whole length of the wire be similarly treated, by dividing
it into small portions and considering the effect of each separately, it
is evident that all the components along the axis are in the same
direction, and that their values have simply to be added together ;
whereas the components at right angles to the axis are radial from P
and mutually destructive, their sum vanishing. Hence, the only

GALVANOMETERS 287

field at a point exactly on the axis is along the axis, and its value can
be written down by summing the previous expression for every part
of the wire.

It then becomes

Field at P - F - *' x total length °f wire x r x ;.i

d?

i x 1-rcrn x r

d3

It must be remembered that d is the slant distance, whereas, in
practical applications it is more convenient to measure the distance
along the axis. Obviously, if P is fairly distant, there will be no
serious error if we write OP = AP = d ; but, for short distances we
have

*, and

=_

(r2 + 0P2)i
when P is at the centre of the coil d = r, and we have

'-n 2-Trin
F = -- x /A

And as F = iG, and /x is unity, we find that G = -

This is the case of the tangent galvanometer, for which, there-
fore, we may write (see p. 285)

H tan0

'"H tan 0

If C is the number expressing the strength of the current in amperes,

n

then i = —

lOr.H A

or C = - - tan 0
lirn

It will be noticed that this expression contains a factor H, which
is not, strictly speaking, a constant, for its value will vary in different

288 VOLTAIC ELECTRICITY

parts of the earth ; hence, the portion — - is sometimes called the

27rn

constant of the instrument, and - its reduction factor.

2-rrn

For all practical purposes, however, H has a constant value at
any given place, and it is more convenient to put C = K tan 0, where

lOrH

An experimental method of finding the value of K is given on
p. 335. The theoretical importance of the tangent galvanometer
depends upon the fact that H can be evaluated (see p. 160) at any
given place by a quite independent method, and hence it is possible
to measure a current absolutely, if the construction be such that r
and n are accurately known.

If, as is often the case, r and n are unknown, the relative values
of any two currents can be found, and this makes the galvanometer
at once available for many purposes (as in Experiments 180 and 186).

It may be pointed out that, from its construction, a tangent
galvanometer is not a sensitive instrument, and it is, therefore, not
suitable for use in "null" methods (e.g. Wheatstone bridge and
potentiometer). For such purposes the coil should surround the
needle as closely as possible, and the nature of the relation between
current and deflection is immaterial.

Nor should it be assumed that the tangent law is accurately
satisfied in any instrument over the whole range of scale. It is
instructive to check it by an independent method, and thus to ascertain
the extent of its departure from the ideal law. A number of methods
are available, from which we select the following as a useful exercise.1

Exp. 177. Place a constant cell (e.g. a Daniell's) in circuit with a tangent

galvanometer and a resistance box. It is
better to provide the galvanometer with a
reversing key and to take double readings
(Fig. 219). Let b be the internal resistance
of the cell, g the resistance of the galvano-
meter, and r the resistance unplugged in the
box. Adjust the resistance in box until a
small deflection, say 5° to 8°, is produced.
Note deflection, reverse current, and again
note deflection, entering both values on paper,
and also the mean value. Alter the resistance
until the deflection is 10° or 12°, and repeat
FIG. 219. observations. Proceed in this way until the

deflection is as great as possible, say 70° to

80°. A table of values of r and 0 is thus obtained. For any one of these

we have

1 This method is due to Carhart and Patterson ; see their Electrical Measure-
ments,

GALVANOMETERS

289

o=

b+y + r
also C = K tan B

E

whence tan 0=

Now 6 + g* is constant during the experiment ; let this be rl}

then cot 0 = 5r + -r1

If we put cot 0 = y, and r = x, this is of the form

y = mx + c, where m and c are constants,

i.e. it is the equation of a straight line. Hence (Fig. 220), if we plot a graph

with values of cot Q for ordinates, and r

for abscissae, the result will be a straight gQ£ n

line if the tangent law holds good for the

instrument, and if it does not hold good,

or only so for a portion of the scale, then '•

such deviation will be indicated by the

departure of the graph from a straight

line. We may note, in passing, that the ^,

graph will cut the cot 6 axis in some .s'

point A. Let it be produced to meet the ^^^ —

r axis at B, and let a be the included

angle. Then, by the well-known property jr>IG 220

of the straight line,

K AO

Values of r

(ii.)

from which it appears that BO^

The Sine Galvanometer. — This is not so much a distinct
instrument as a particular way of using a tangent galvanometer, or
indeed any galvanometer having a magnet directed by a uniform
field. Suppose that instead of reading the ordinary deflection 6 on
a tangent galvanometer, the whole instrument is rotated in the
direction of deflection ; at first the deflection will increase, but unless
it is too large, the coil will eventually catch up the needle. The
actual deflection is now, say, a, which is greater than 0, and is evi-
dently equal to the angle through which the coil has been turned.
Hence, in specially made instruments, the coil has an independent
motion, and is provided with another scale, on which this angle may
be read off; but the reading may be taken on any instrument by
simply interrupting the current, when a is the angle through which
the needle swings back.

290

VOLTAIC ELECTRICITY

In this method it is evident that, when the reading is taken,
the field iG is at right angles to the needle, and the resultant
field is along the direction of the needle.
H The construction will be understood

from Fig. 221.

- = sin a

but^? = ^

tG

H

sn a

H

or i = -f^ sm a

A special advantage of this method lies
iG in the fact that it may be applied, as we
FIG. 221. have previously mentioned, to the simple

forms of galvanometer, and does not merely

hold good for the tangent type of instrument. The only difference
will be that, when thus applied, the value of the constant G cannot
be found by calculation, and we must write i = K sin a where K
is unknown, but for all relative measurements of current this is
immaterial.

Helmholtz or Double-Coil Tangent Galvanometer. —
It can be shown mathematically that the best way of producing a
uniform field is to use two equal circular coils, placed at a distance
apart equal to the radius of either. The coils are connected in series
and the needle is placed midway between them. This arrangement
is very useful when it is required to produce a uniform ^magnetic
field of moderate strength and known value for any purpose whatever
(e.g. for deflecting cathode rays in the measurements described on
p. 487).

If each coil has n turns of radius r, the field due to either, at a

point midway between them, is found by putting OP = « ^n tne eclua'
tion given on p. 287, so that

F (due to either) = —

;)'}'

As both fields act in the same direction, the actual field is

f , which becomes

327m?

GALVANOMETERS

291

Galvanometers of Greater Sensitiveness. — It is as ex-
tremely sensitive instruments that galvanometers are more especially
useful in electrical measurements. The tangent galvanometer is of
great theoretical interest, but of comparatively little practical use,
whereas very sensitive instruments are indispensable for many different
purposes.

Speaking generally, there are three methods by which we may
increase the deflection due to a current of given strength: (1) by
increasing the moment of the deflecting couple due to the current
(e.g. by increasing the number of turns, by disposing them to get
the best effect, &c.) ; (2) by decreasing the moment of the controlling
couple ; (3) by increasing the length of the pointer.

The first method may be regarded as obvious ; in any case it will
be carried out as far as is possible.

The second was first applied practically by using an astatic
needle.

Astatic Galvanometer. — As already explained, various systems
of magnets may be arranged on which the earth's magnetic field has
no restoring couple. The simplest case, as applied to a galvanometer,
is shown in Fig. 222. It should be noticed
that, if the needles are perfectly astatic
and there is no friction on a pivot or tor-
sion on a suspension, then all currents,
weak or strong, would produce a deflection
of 90°, and there would be no return of
the needle to zero. It is, however, very
difficult to make needles perfectly astatic,
and there is an advantage in making one

slightly stronger than the other, so that a

O

FIG. 222.

controlling couple may be retained in order
to bring the system to zero. As a rule, if
special pains are taken to make such a pair

of needles astatic, it will be found that the system, when delicately
suspended, has a distinct tendency to set itself east and west (see
p. 164). In any case, the uncertainty as to the zero position makes
comparisons of deflections somewhat difficult, and the instrument is
better adapted for indicating the existence of very feeble currents
than for measuring them.

The astatic galvanometer in its original form is now seldom used,
although the principle is frequently applied in various ways. Speak-
ing generally, we may remark that it is always advantageous for the
moving system of a sensitive galvanometer to be " astatic," if that
term be extended to mean freedom from disturbance due to the
earth's field (or to stray fields due to currents or magnets in the
vicinity of the galvanometer). But this property may be obtained in
different ways, and it does not necessarily imply a feeble control.

292 VOLTAIC ELECTRICITY

For instance, the suspended coil galvanometers to be subsequently
described are ideally "astatic" — the moving part being non-magnetic
— but the control due to the suspension may be as strong as we
please. When the moving part is magnetic, it is best to make it as
astatic as possible, and to work with an independent and adjustable
control due to bar magnets attached to the instrument.

As a good example of this construction, we may mention the
Broca galvanometer. The needle (Fig. 223) consists of two steel
wires, placed vertically and very close together, each
having consequent poles. A very small mirror is
attached below, and the whole is suspended by a fine
quartz fibre. This form makes it possible to use
comparatively powerful magnets, whilst keeping the
moment of inertia small, and at the same time it is
very astatic. The coil, wound in two halves for con-
venience, is arranged as shown with respect to the
needle. It will be noticed that all the poles con-
tribute to the turning moment. A movable bar
magnet at the back provides the necessary control.

Thomson Galvanometer. — The earliest mirror
galvanometer was devised by the late Lord Kelvin
(when Professor Thomson) for receiving signals through
the first Atlantic cable ; and a special class of instru-
ments are still known as " Thomson " galvanometers.

An early pattern is shown in Fig. 224.
IG' ' The magnet is very short and light — so small and

light, indeed, that when attached to' the back of a mirror, A, about
a centimetre in diameter, the total weight is not more than one or
two grains. The mirror is suspended by a single fibre of unspun
silk in a small cylinder, round which the wire is coiled. The length
of wire employed for this purpose depends upon the sensitiveness
required (as explained on p. 291). Through an aperture in a screen,
a beam of light is sent from a lamp upon the mirror, which reflects
it on a scale. A permanent magnet is placed on a vertical support
above the coil, which controls the magnet in the galvanometer, so
that the spot of light is readily brought to the zero on the scale. It
may also be used to weaken the earth's field if necessary, which still
further increases the sensitiveness.

In the most modern forms of this galvanometer the parts are
doubled and the moving system thereby made astatic. The principle
is indicated in Fig. 225.

Two sets of three or four very small magnets (their size is ex-
aggerated in the figure) are attached to a light aluminium wire,
which also carries a mirror, m, and the whole is suspended by a fine
quartz or silk fibre, F. Two pairs of coils surround the needles.
The mirror is sometimes placed between the coils (as shown) and

GALVANOMETERS

293

sometimes under the lower coil on an extension of the aluminium
support. In order to obtain a more delicate adjustment, it is now
usual to supply two controlling magnets, MM, which may be adjusted
either to weaken or to strengthen each other's effect.

The only disadvantage possessed by this type of galvanometer is
due to the fact that the deflections are readily disturbed by the
presence of iron or of magnets in its vicinity, which makes it very
troublesome for ordinary experimental work.

It may be pointed out that, when using a mirror instrument, the
angular deflections of the moving system are necessarily very small

FIG. 224.

FIG. 225.

(or otherwise the spot of light would move off the scale) ; and, for
such small deflections (as already stated on p. 285), the relation,
C oc tan 0, is true to a close approximation with almost any type of
instrument. Again, it has been shown (p. 142) that the deflection
in any arbitrary scale divisions is proportional to tan 0. Hence, for
any mirror galvanometer we may take the current as proportional to
the deflection it produces as long as we avoid unreasonably large
deflections. This is a very great advantage in practical work.

Moving Coil Galvanometers. — These instruments are modi-
fications of the "syphon recorder" devised by Lord Kelvin for use
on submarine cables. They are generally known as D'Arsonval
galvanometers, although improved and developed by many later
inventors. Their great value is due to the fact that they are

294

VOLTAIC ELECTRICITY

practically unaffected by neighbouring currents or magnets, and
hence, they are more generally used both for commercial and for
laboratory work than any other type.

The principle involved in the earliest form will be understood
from Fig. 226. A narrow rectangular coil of many turns of fine
silk-covered copper wire (of which one or two strands are shown
in the figure) is suspended by a fine metal wire, so that it can move
freely in a narrow annular space between the poles, NS, of a strong
steel magnet. Within the coil is a fixed cylinder of soft iron, C,
which serves to concentrate the lines of force in the gap. The ceil
is wound on a light framework (which is either of metal or of non-
conducting material, according as a " dead-beat" or a "ballistic"
action is required, see p. 296), and which is steadied below by another
wire attached to a spring strip to keep it under slight tension. A

FIG. 227.

mirror, M, is fixed just above the coil. The current is passed through
the instrument (as shown) by means of this strip and the metal
suspension. There are several ways of regarding the action of this
instrument ; at present it will be easiest to think of the coil as
becoming, whilst carrying a current, the equivalent of a weak
magnet. For instance, by applying the swimming rule to Fig. 227
(which represents Fig. 226 looked at from above), it is found that
the polarity of the coil is N above and S below, and hence there
will be a couple acting upon it, which tends to rotate it through 90°
in a clockwise direction. If the current is reversed, the polarity of
the coil and the direction of rotation will also be reversed. The
controlling couple opposing rotation is provided by the torsion of
the suspending wire, and the final position will be acquired when
the moments of these two couples are equal.

Evidently, the sensitiveness is increased by making the magnetic
field in the gap very strong and by giving the moving coil many

GALVANOMETERS

295

turns. The effect of the soft iron core is to make the field in the
gap very uniform (except just at the pole tips), and, as a result,
the current is proportional to the deflection through a much wider
range of motion than is usual in a reflecting instrument. For this
reason, the construction just described will be found applied to a very
useful class of commercial ammeters and voltmeters (see p. 560).

In 1892, Professors Ayrton and Mather pointed out the great
advantage of using a thin strip of metal for the suspension, instead
of a wire of circular section — the torsion for a given deflecting couple
being much greater, and the stress on the material much less with
a strip than with a wire. They
also introduced the use of phosphor
bronze for this purpose, and modi-
fied £he shape of the coil and
other details in such a way that
fin entirely new type of instrument
was created.

The Ayrton-Mather form is
shown diagrammatically in Fig. 228.
The magnet, NS, is usually made of
cast steel, circular in section, with
a very narrow air gap between the
poles. These are usually rounded
off as shown in Fig. 228 (b), in
order to concentrate the field; a
construction which is advantageous
for many purposes (e.g. for " null "
methods), but which must be modi-
fied if the proportionality between
current and deflection is required over any large range of scale.
There is no soft iron core, and the coil is long and narrow, the
current being taken in and out by means of the phosphor bronze
strip suspension and a delicate spiral of the same material at the
bottom. If the instrument is to be "dead beat," the coil is placed
inside a thin silver tube, which moves with it ; if it is to be
"ballistic," a light non-conducting tube is used, which in this case
is merely a mechanical support and protection. It is usual to
provide the instrument with a coil of each kind, so arranged that
they can be readily exchanged.

Fig. 229 shows the general appearance of a pattern made by
R. W. Paul, and Fig. 230 shows one of the coils removed from the
instrument. The upper end of the suspension is connected to the
brass carrier, which makes contact through the framework to one
of the terminals. The spiral at the bottom is attached to a pin
passing through an insulating plug of ebonite, and this pin makes
contact with a metal strip connected to the other terminal, which

296

VOLTAIC ELECTRICITY

is insulated from the framework. A simple form of clip is provided,
which serves to clainp the coil when not in use.

Meaning of terms "Dead-beat" and " Ballistic."— As
these terms are so frequently used, it is desirable to explain their
meaning at some length. It will be understood that the moving
part of any instrument, which is controlled by some couple whose
moment increases with the angle of deflection, must, when disturbed,
oscillate about its final position like a pendulum — the amplitude of
the vibrations gradually diminishing until it comes to rest. Hence,

FIG. 229.

FIG. 230.

when a current passes through a galvanometer, the needle does not
at once indicate the corresponding deflection, but oscillates about it,
so that some time elapses before a reading can be taken ; and
similarly, when the current is interrupted, time is again lost before
the needle comes to rest.

It is evident that the needle cannot come to rest until the energy
given to it by the impulse has been converted into heat — in the
simplest case, chiefly by air friction. If some additional means of
converting the energy into heat are provided, the amplitudes of
successive vibrations will diminish more rapidly, and when this

GALVANOMETERS 297

effect is marked, the motion is said to be " damped." It must be
remembered, however, that such damping does not alter the final
position of the needle, but merely enables it to take up that position
in a shorter time.

Again, there is a critical amount of damping, for which the
vibrations are almost completely stopped, the moving part almost
instantly reaching its steady position without overshooting it, and
with equal rapidity returning to zero. In this state, the motion is
said to be " dead-beat/'' If, however, the damping is still further
increased, the motion becomes inconveniently sluggish.

A dead-beat motion is desirable for all instruments with which
readings of steady deflections have to be taken, and it is especially
useful in null methods.

There are various methods of obtaining the necessary damping,
one of the oldest being the attachment of some light vane to the
moving part, thus increasing the air friction, or if this be insufficient
the vane may be immersed in some liquid. In certain voltmeters
and ammeters, a " pneumatic damping " is used, the principle being
similar to that employed in the contrivances fixed on doors to make
them close noiselessly, but the most interesting method to the
student, and one which is very widely used, depends on the pro-
duction of induced currents. This method and its application to
the moving coil galvanometer are explained on p. 356.

The term " ballistic " may be taken at present to mean " not dead-
beat," i.e. the moving part is designed so that the damping is very
small. It is, however, more convenient to defer the consideration of
ballistic galvanometers until the principles of induction currents have
been discussed.

" Period," or " Periodic Time " of a Galvanometer.—

This is the time required by the moving part to make one complete
oscillation, when disturbed from its position of rest. The stronger
the control, the shorter will be the period, and vice versa. With null
methods, a fairly short period, say 3 to 5 seconds, is desirable, for
then the spot of light returns more quickly to its position of rest.
For ballistic purposes (as will be shown later), a longer period, say
from 5 to 20 seconds, is necessary.

Zero-keeping" Property. — This again depends upon the con-
trol ; when the latter is strong the moving part will return exactly to
the same position of rest after being disturbed, whereas when it is
weak there is apt to be some uncertainty about the zero, although the
instrument is then much more sensitive. Whenever a deflection has
to be measured, stability of zero is essential, but for null methods it
is relatively unimportant.

Einthoven " String " Galvanometer. — This is an instrument
of the moving coil type, but the coil is reduced to a single fine wire
or string stretched in a very narrow air gap between the poles of

298

VOLTAIC ELECTRICITY

a powerful electromagnet. In the more sensitive forms, this string
is frequently a quartz fibre, silvered to make it sufficiently conducting.
When a current passes through the string, the latter is deflected
sideways, i.e. at right angles to the lines of force, an effect similar to
that demonstrated in Experiment 204, p. 344. Holes are bored in the
pole pieces, through which this movement is observed by means of a
microscope, which forms part of the instrument — no mirror being
used. The moment of inertia of the moving part is very small (the
mirror is dispensed with because it would increase this quantity), and
hence the natural time of vibration is also very small. This instru-
ment is extremely sensitive and very dead beat. It was designed
more especially for use in physiological research, and, when provided
with a photographic recording apparatus, it fulfils, for such purposes,
the functions of an oscillograph (see p. 571).

Duddell's Thermo-Galvanometer. — Although this instru-
ment depends upon the properties of thermo-electric currents, dealt
with in Chapter XXVIII., it may be alluded to here for the sake of
convenience. It is intended to measure ex-
tremely small alternating or varying currents,
e.g. it may be used to measure telephone cur-
rents or the currents flowing in the receiving
circuits of wireless telegraphy. The galvano-
meters already mentioned will work only with
direct currents, and the need of sensitive in-
struments for rapidly varying currents has long
been felt. A diagrammatic view is given in
Fig. 231. A single loop of silver wire, P,
is suspended by means of a quartz fibre, Q,
between the poles of a permanent magnet.
H is a small glass stem attached to the loop,
which carries a mirror, M. The lower ends
of the loop are connected to a small bismuth-
antimony thermo-couple (Bi, Sb). Immedi-
ately below this is the " heater," which is
a fine filament of high specific resistance
(usually a quartz fibre platinised to enable
it to conduct), 3 or 4 millimetres in length.
The current to be measured passes through
the "heater," producing 02R heat, part of
which is radiated and carried by convec-
tion currents to the thermo-couple, thus
warming it and producing another current in the loop, which then
turns like the coil of a D'Arsonval galvanometer. The instrument is
an adaptation of a principle first applied by C. V. Boys in his radio-
micrometer.

Figure Of Merit. — This is a term occasionally met with

1 MIA?** I

FIG. 231.

GALVANOMETERS 299

in connection with galvanometers. As originally applied to non-
reflecting instruments, it meant the current in amperes required
to produce a deflection of 1°. In the case of reflecting instruments,
it is often taken to mean the current producing a deflection of
one scale division under the particular circumstances of the experi-
ment, but evidently such information is useless unless the distance
of the scale from the galvanometer and the length of a scale
division are also stated. On the whole it seems best to define it
as the current required to produce I millimetre defection on a scale
1 metre from the galvanometer mirror. Using this definition, the
values obtained in practice may be summarised as follows : —

Type.

Figure of Merit in Amperes.

Ay rton- Mather moving coil . .
Thomson

10-6 to 10-9
IO-9 to 10-11

Broca

io-10

Emthoveii .

io-12

Thermo -galvanometer ....

2xlO-6

Effect of Shunting a Galvanometer. — The laws of
parallel circuits and their application to the case of a galvanometer
have already been dealt with (see pp. 27^273). It must, however, be
remembered that the effect produced^by~t&esEunt will depend upon
the resistance in circuit, and may vary within wide limits. Let us
consider two extreme cases : (1) when the resistance in the circuit is
very small, and (2) when it is very large, compared with that of the
shunted galvanometer.

(1) Suppose the galvanometer is connected directly to a cell of low
internal resistance, and a shunt is inserted having i of the resistance
of the galvanometer. Then the joint resistance will be y1^ of the
former value, and as that was practically the only resistance in the
circuit, the effect of reducing it to y1^ will be to increase the total
current to ten times its original amount, and it will be y1^ of this
increased current which passes through the galvanometer, and so the
deflection will be but little affected by using the shunt. Thus, when
the resistance of the shunted galvanometer, i.e. the value of the

fraction — — , is large compared with the rest of the circuit, the

current passing through it will be practically the same whether
shunted or not.

(2) If the resistance in the circuit is very large compared with
that of the .galvanometer, introducing a shunt will only slightly affect
the total resistance, and consequently the total current will be only
slightly increased. In such a case the introduction of a \ or ^ shunt

300 VOLTAIC ELECTRICITY

may reduce the current passing through the galvanometer to nearly
TO" or 1^0 °f ifca original value. For intermediate cases the current
passing through the galvanometer will not be reduced so much.

It is instructive to look at the matter from another point of view. If

e be the P.D. between the galvanometer terminals, then Cg=-, and is

constant if e is constant whether there is a shunt or not. Hence, if
the application of a shunt alters the galvanometer current, it is due
to the fact that it has altered the P.D. between the terminals. Now,
when the galvanometer is practically the only resistance in circuit,
this is also the P.D. between the cell terminals, and it will be found
to alter but little when the total current increases due to the
addition of a shunt. When there are large resistances in the circuit,
e is only a small fraction of ex, and alters very much with change of
current.

The student can easily verify this by taking a particular case.
For example, consider a cell of 1 volt E.M.F. and 1 ohm internal
resistance, connected up to a galvanometer of 504 ohms resistance.
It will be found that the P.D. across the galvanometer terminals is
practically 1 volt, and that this value is scarcely altered by shunting
it with 56 ohms. But when 5000 ohms is also in circuit, the P.D.
across the galvanometer is about ^ volt in the first case, and falls to
about TJ^- volt when shunted with 56 ohms.

Measurement Of Current. — It will be noticed that the tan-
gent galvanometer is the only instrument thus far described which
is capable of measuring a current " absolutely," i.e. without requiring
previous calibration by a method involving, directly or indirectly, the
use of a Imovm current. It may also be pointed out that we can
measure a current only by measuring some property of that current,
having first arrived at some understanding, based upon experiment,
as to the nature of the relation between the two.

Three such properties are available : (1) the magnetic effect, (2)
the heating effect, and (3) the chemical effect produced in certain
solutions.

The first effect we have seen applied to the tangent galvanometer.
It is also applied in various forms of " current-balance " (see p. 567),
which depend upon the measurement of the forces of attraction and
repulsion between conductors carrying a current, and which (like the
tangent galvanometer, only with greater exactness) afford a means of
measuring a current absolutely, because the magnitude of the forces
in question can be calculated from the dimensions of the apparatus.

The second effect is also available, if Joule's equivalent is accu-
rately known, but it is inconvenient in practice ; while the third
cannot be applied unless we begin by defining unit current in terms
of chemical action.

We are therefore restricted to the use of the first property, but

GALVANOMETERS 301

neither current balances nor tangent galvanometers arc convenient
instruments for the rapid measurement of current under commercial
conditions (e.g. for standardising ammeters).

Hence, the problem has been solved as follows: The E.M.F.'s of
various standard cells (already described) have been determined by
very careful and laborious experiments, involving the exact measure-
ment of current by means of a current balance. Having a cell of
known E.M.F. and a set of resistances of known value, it becomes
possible to measure currents of any magnitude very conveniently and
rapidly by the potentiometer method described in the following
chapter. Again, by experiments also involving the use of a current
balance, the weight of silver deposited by 1 ampere in 1 second
from a solution of its salts has been found with great exactness (see
p. 590), and consequently the third effect becomes available for the
general measurement of current. But, although very exact, the
method is so slow and troublesome that its use is confined to special
purposes.

EXERCISE XV

1. A very short magnetic needle is suspended at the centre of a hoop of wire
fixed vertically in the magnetic meridian. One current passing through the wire
causes a permanent deflection of the needle amounting to 30° ; another current
causes a deflection of 45°. What are the relative strengths of the two currents ?

2. Two tangent galvanometers, alike in all respects except that the hoop of
one has twice the radius of that of the other, are employed to measure the
strengths of electric currents. If the galvanometers give equal deflections, show
what are the relative strengths of the currents passing through them.

3. In a tangent galvanometer a current of strength A causes a deflection of
25°, another of strength B causes a deflection of 20°. What is the relation of
A to B? [tan. 25° ='4663, tan. 20° =-3640].

4. Discuss the several forces or moments which act on the needle of a tangent
galvanometer when deflected by the action of a current passing through the coil
of the galvanometer, and deduce the law of action of the instrument.

(B. of E., 1900.)

5. Describe the tangent galvanometer, and the method of using it. The coil
of such a galvanometer consists of eight turns of wire, and has a mean radius of
20 centimetres. Find what current will produce a deflection of 45° if the hori-
zontal intensity of the earth's magnetic field is '18 C.G.S. units.

(Loud. Univ. Matric., 1903.)

6. Calculate the strength of the current in C.G.S. units, and also in amperes,
from the following data : Radius of coil, 12 centimetres ; number of turns in
coil, 10 ; deflection of needle, 45° ; value of earth's horizontal force, *18.

(B. of E., 1902.)

7. Explain how the sensitiveness of a tangent galvanometer can be (1) de-
creased, (2) increased by the aid of a bar magnet.

A Daniell cell connected to a tangent galvanometer of J ohm resistance pro-
duces a deflection of 60°. On interposing a resistance of 2 ohms the deflection
falls to 30°. What is the internal resistance of the cell ?

(Lond. Univ. Matric., 1906.)

8. A current of | ampere passes through a tangent galvanometer, whose
resistance is 10'5 ohms. After the terminals of the galvanometer have been
joined by a wire, the total current in the circuit remains unaltered, but the

302 VOLTAIC ELECTRICITY

current in the galvanometer is reduced to ^V ampere. If the resistance of the
wire is 14 ohms per metre, what length of wire has been used as a shunt ?

(Oxford Local, Senior, 1902.)

9. A current flows through two tangent galvanometers in series, each of
which consists of a single ring of copper, the radius of one ring being three times
that of the other. In which of the galvanometers will the deflection of the
needle be greater ? If the greater deflection be 60°, what will the smaller be ?

(B. of K, 1897.)

10. An electric current of 1 ampere flows round a circular metal ring, the
radius of which is 10 centimetres. Determine the strength and direction of the
magnetic field at a point on the line drawn through the centre of the ring per-
pendicular to its plane and 10 centimetres distant from the plane of the ring.

( B. of E., 1908.)

11. A tangent galvanometer having a coil of one turn of 34 centimetres radius
gives a deflection of 45° with a current of 10 amperes. Calculate the strength
of the earth's magnetic field at the centre of the coil.

(Lond. Univ. Inter. B.Sc., 1907.)

12. A compass-needle is placed at the centre of two concentric circles which
are in the same vertical plane, and are made of wires similar in all respects,
except that the outer is copper, the inner German silver. The wires are con-
nected in multiple arc, but so that the currents which flow through them circulate
in opposite directions. What must be the ratio of the diameters of the circles
so that no effect may be produced on the needle ? [N.B. — Assume the conduc-
tivity of copper to be twelve times that of German silver.]

13. A tangent galvanometer is placed with its coil perpendicular to the
magnetic meridian. When no current is passing through it, the needle when
set in vibration oscillates 10 times in 15 seconds. Will the rate of vibration be
altered when a current is passing through the coil, and if so, will it be increased
or diminished ?

14. What is the relative strengths of two currents passing through the coil
of a sine galvanometer, when the angles through which the coil has been turned,
before the needle stands at zero, are 30° and 45° respectively ?

15. Why is an astatic galvanometer better adapted for the measurement of
weak currents than a galvanometer with a single needle ?

16. Describe some type of galvanometer in which the magnet is fixed and the
coil is the movable part. How is this type of galvanometer rendered dead-beat ?
How can such a galvanometer be arranged so as to be used to measure heavy
currents? (Oxford Local, Senior, 1907.)

17. Describe and explain the mode of action of some form of sensitive gal-
vanometer suitable for use in a place where the earth's field is much disturbed
by the presence of variable electric currents. (B. of E., 1906.)

18. Describe the construction and adjustment of a sensitive mirror galvano-
meter, and explain how you would measure its sensitiveness.

CHAPTER XX

MEASUREMENT OF ELECTROMOTIVE FORCE

IT is by no means an easy matter to measure an electromotive force
directly, although the methods available will be indicated later. It
is, however, quite simple to compare two electromotive forces with
great accuracy, and hence all practical methods depend upon the
comparison of the unknown E.M.F. with that of a standard cell.1

Potentiometer Method. — This is undoubtedly the most accu-
rate method of comparing E.M.F. 's. The principle involved should
be thoroughly understood, as the potentiometer, in its more refined
forms, not only is of the utmost practical and commercial importance,
but is the most generally useful measuring instrument we possess.

In its simplest form, a potentiometer is merely a long piece of
uniform wire of fairly high resistance provided with a scale of equal
divisions. For the use of a student, a length of 400 to 500 centi-
metres of 30-gauge platinoid wire is probably the most useful. Such
a length of straight wire would naturally be inconvenient, and hence
it is usual to zigzag it -

backwards and forwards on ^^_- — ]|

a wooden base; but in doing
this, special care must be
taken not to introduce errors
into the measurement of its
length.

Let AB (Fig. 232) re-
present a potentiometer wire,
the numbers on its scale
running from A to B. Con- FIG

nect an auxiliary battery,
as shown in the figure, to the points A and B, and, for convenience
in calculation, always attach the positive terminal to A. This battery
has nothing to do with the E.M.F. 's to be compared ; its function is
to maintain a perfectly steady P.D. between A and B, which must be
greater than either of the E.M.F. 's under examination. Hence, it
must be capable of giving out a small but steady current for some

1 Students should use a Daniell's cell as a standard until they have gained
some experience, otherwise they may seriously injure an expensive standard cell.
Set up with care, its E.M.F. is exceedingly constant within narrow limits.

303

304 VOLTAIC ELECTRICITY

time. For many purposes two Daniell's cells in series, or one accumu
lator, is convenient. If r be the resistance of the potentiometer wire,
and C the current through it, then the RD. between A and B is Cr
volts.

Let us, for a moment, think of the current as analogous to a
stream of wrater flowing from A to B in consequence of a difference
of level between A and B. Then, if any kind of side channel were
to be joined on, say to the points A and D, it is evident that the
stream would divide at A into two parts both flowing towards D,
and the nearer the point D was to B, the greater would be the differ-
ence of level tending to urge the water along the new path. We
may now suppose that something of the nature of a force pump,
driven steadily by some power, is placed in this path. This may
evidently be arranged to work in either direction. In one, it merely
assists the stream to flow from A to D ; in the other, it opposes the
flow along the new path, and, according to its power, it may diminish
or even reverse the flow. Hence, there will be one case in which
it will just stop the flow, in which case, its effect is exactly balanced
by the difference of level between A and D. If another more power-
ful pump be substituted for the first, the side channel might have to
reach F in order to satisfy the condition of no flow, and thus we see
that the lengths AD and AF might be made a measure of the power
of the two pumps.

The electrical case is exactly similar.

Exp. 178. Coimect up, in turn (as shown in Fig. 232), the two cells whose
E.M.F.'s are to be compared, and alter the point of contact until the galvano-
meter shows no deflection. Let this be at D in the first case, and at F in the
second. Then, as no current is flowing through the cells to be compared, the
P.D. between their terminals is identical with the E.M.F., and we have

E = P.D. between A and D = C x resistance of AD
E!= „ ,, A and F — Cx resistance of AF.

Here, C is the steady current in the wire and is the same in each case.

E _ resistance of length AD
E! ~ resistance of length AF

Xow, the resistance of a uniform wire is proportional to its length, and

JS = length AD = d_ wben rf aud ^ arg th(j gcale readings
Ej length AF rfj

It will be noticed that the argument fails unless the current in the potentiometer
wire is strictly constant during the two readings, and as slight changes may
occur, it is desirable to take the second reading as soon as possible after the first.

Fig 233 shows a convenient arrangement for the apparatus, where a simple
switch is formed by removing the cross wires from a reversing commutator (see
p. 253), and one cell is exchanged for the other by moving over the rocker.

Accuracy in obtaining balance evidently depends upon the sensitiveness of
tlu- galvanometer. If a very delicate instrument is used, it should be shunted
until the approximate point of balance is obtained.

MEASUREMENT OF ELECTROMOTIVE FORCE 305

If a Daniell's cell be used as a standard, no special precautions need be
taken to protect it, but if a Clark or cadmium cell be used, it is necessary to
introduce a high resistance, say
20, 000 ohms, in the galvanometer
circuit, in order to ensure the cell
against injury. This resistance
does not aifect the position of the
point of balance, but lessens the
sensitiveness of the test, and it
may be cut out during the final
adjustment.

The great advantage of
the method is due to the
fact that the cells, whose
E.M.F.'s are to be compared,
are not giving out current
during the measurement
(except very small currents during adjustment), and hence, it gives
accurate results even with cells which polarise readily. A second
advantage is due to its being a "null" method, i.e. it depends on
reducing a deflection to zero, a process which can always be carried
out with greater exactness than the reading of a deflection.

Measurement of Current by means of a Potentio-
meter.— This is the most generally convenient method of measuring
current for such purposes as checking ammeters, &c. It depends
upon the use of standard resistances of known values. For instance,
let such a resistance R be placed in series with the ammeter,
and a steady current sent through both. Also let the cell Ex
(Fig. 233) be removed, and the wires leading to it connected in-
stead to the extremities of R (taking care that the connections are
made so that the current flows through R in the right direction).
Then the P.D. across R may be measured by comparing it with the
E.M.F. of the standard cell E. If the result is e volts, the current

through R and the ammeter will be ^ .

XV

The standard resistances used for such purposes are of a special
type, as they must be sufficiently massive to carry the current without
getting hot enough to be injured or to have their resistance seriously
affected by the rise in temperature. Hence, ordinary resistance
boxes are as a rule useless. Again, it is desirable that the P.D. to
be measured should be of the order of a volt or so, in order to facili-
tate accurate comparison with the standard cell. Roughly speaking,
the standard resistances would be about T^ ohm to measure currents
up to 10 amperes, and y^y ohm and 1U100 ohm to measure currents
up to 100 and 1000 amperes respectively. It is unnecessary to enter
into the details of construction here, but it may be remarked that
they are usually provided with massive terminals for leading the

U

306

VOLTAIC ELECTRICITY

FIG. 234.

with cells

current in and out, and a pair of '* potential leads " for connecting
to the potentiometer. These are small flexible connections soldered
to the conductor at definite points, between which the resistance is
adjusted to have the required value. In this way the varying effects
of bad contact at the ends are eliminated ; a matter of great impor-
tance in the case of such small resistances.

Lord Rayleigh's form of Potentiometer.— A modified form
of potentiometer due to Lord Rayleigh is worthy of notice. In this
case the potentiometer wrire is replaced by
two resistance boxes, Rt and R, arranged as
shown in Fig. 234. A steady current is
obtained as usual from auxiliary cells, and
each of the cells to be compared is in turn
connected up to one of the boxes (Rj in
figure) in such a direction as to oppose the
P.D. across it. The resistances are then
adjusted until the galvanometer shows no
deflection, but in sncli a way that the total
resistance of both boxes is Itept constant. If
R! and R2 are the values giving balance
\ and E2, then, evidently,

E1_R1
E2 R2

(A convenient value for the constant total resistance is usually about
10,000 ohms.)

Other Methods of comparing E.M.F.'s. — Many other
methods of comparing E.M.F.'s might be given, which are now seldom
or never used in practice. As a rule, they require the cells to give out
a current during measurement, and hence are subject to serious errors
owing to polarisation, and to possible changes in internal resistance
due to the passage of a current. A few7 such methods will be briefly
described, mainly as affording simple exercises on Ohm's law.

Lumsden's Method. — Example 6,
given on p. 226, illustrates the principle
involved in Lumsden's or Bosscha's method.
For if the two cells, whose E.M.F.'s are
to be compared, are arranged as shown
in Fig. 235 (also compare with Fig. 170),
and the adjustable known resistances r
and rv are altered until the galvanometer
shows no deflection when the key is
pressed down, we have, from the reasoning giving in Example 6,

E

FIG. 235.

MEASUREMENT OF ELECTROMOTIVE FORCE 307

where 1> and b^ are the internal resistances of the two cells respec-
tively. These are not known, but if we make r and 1\ very large
compared with b and b{, we have practically

A much better plan, however, is, after taking the first readings, to
alter r and i\ considerably, and again obtain a balance. If the new
values be R and R1? then

E

By a well-known theorem in algebra, we can eliminate b and b19 thus

r)= R-r

Wheatstone's Method. — This is the oldest of all methods,
and is noteworthy as not requiring a galvanometer of known law, i.e.
one for which the relation between current and deflection is known.

Exp. 179. (1) Connect up one of the cells in series with a galvanometer and
resistance box, and adjust box until a convenient deflection of d° (about 50°) is
obtained. If R be the total resistance of circuit (which is unknown), and C the
current, we have

-ri

C = — and gives a deflection d°
K

(2) Increase the resistance by a known amount r so that the deflection is
reduced by 10° or 12°, then

Tl

and gives a deflection d^

(3) Remove the cell of E.M.F. = E, and replace it with the other cell of
E.M.F. =Ej, and alter the resistance until the original deflection d° is obtained.
Then the current must be the same as in (1). The total resistance is unknown ;
let it be Rl5

then C = |i

(4) Increase the resistance by a known amount rt until the deflection is again
df. Then the current must be Cj as in (2), and

Now from (1) and (3)

308 VOLTAIC ELECTRICITY

And from (2) and (4)

whence — - = -

i.e. the E.M.F.'s to be compared are in the ratio of the added resistances.

In a particular experiment, with a chromic acid cell, the resistance
required to obtain a deflection of 52° was 3O5 ohms. This resistance
was then increased to 50 '5 ohms, when the deflection was 40°. The
added resistance of 20 ohms thus diminished the deflection by 12°.

A Daniell's cell was then substituted for the chromic acid cell.
To obtain a deflection of 52° a resistance of 10 ohms was required,
while to bring the deflection down to 40° the resistance was increased
to 21*5 ohms; the extra resistance, therefore, was 11*5 ohms. Now,

1-07

'1

11-5
20

20 x 1-07
11-5

-1-86 volts.

We can now perceive the practical defects of this and similar methods.
In the first place, it would be absurd to apply it to cells which polarise
readily, and it is usually such cells whose E.M.F. is required. For
the same reason, a standard cell of the Clark or cadmium type
cannot be used. Again, it is difficult, under the usual conditions of
work, to exactly reproduce a certain deflection with great accuracy.

Sum and Difference Method.— The preceding remarks also
apply largely to this method. It differs from the last method in
requiring a galvanometer of known law, e.g. a tangent galvanometer.

Exp. 180. Connect the two cells in series with a tangent galvanometer

(Fig. 236), if necessary including a
resistance box to adjust the deflection
to a convenient value. Let this be
6°. Without altering the resistance,
reverse one of the cells (Fig. 237),
and again note the deflection.
Let this be ^°. Then if E and El
are the respective E.M.F.'s, and R the
total resistance of the circuit (which

FIG. 236.

FIG. 237.

is the same in both cases), we have

MEASUREMENT OF ELECTROMOTIVE FORCE 309

_

^ E _ tan 0 + tan 6l

E~tan 0-tan 0

In a particular experiment, when the two cells were arranged in series the
deflection was 36°; when they were in opposition it was 11°. Now, tan 36° =
•7265 and tan 11° = '1944.

Whence, substituting, we have s,

_-9209
•5321

Again, taking the E.M.F. of the Daniell's cell as 1'07 volt, we have

E_ -9209x1-07
•5321

= 1-85 volts.

Electrometer Method. — If an electrometer of the quadrant
type is available, electromotive forces may be compared with great
ease, once the instrument is set up. It is only necessary to connect
each of the cells in turn to the terminals of the instrument, and to
read the steady deflections thereby produced. If these are d and d^
divisions respectively, then

Any kind of standard cell may be used, and, as the cells are always
on open circuit, the method is ideally perfect as regards freedom from
polarisation. It is inferior to the potentiometer method, because it
is more difficult to read a certain deflection accurately than it is
to determine zero deflection accurately ; and further, a quadrant
electrometer is a much more troublesome instrument to use than
a potentiometer.

Voltmeter and Allied Methods.— The quickest method of
measuring the E.M.F. of a given cell is evidently to connect it up to
a voltmeter of suitable range, and to take the reading. Where no
great accuracy is required, this is usually done, but the method must
be applied with caution. It is not merely that the calibration may
be incorrect, for, even if we assume that it is perfect in this respect,
the readings may be seriously wrong unless the instrument has a very
high resistance, and this, with certain types of voltmeters, is not
always possible. As an example (a) let us consider a certain instru-
ment reading up to 2'5 volts, and having a resistance of 12 ohms.
Suppose that we connect it up to a cell, otherwise known to have an
E.M.F. of 1'5 volts, an4 an internal resistance of 3 ohms, The

310 VOLTAIC ELECTRICITY

1*5
current flowing will be C = .. ^ » *= yV ampere. Now, the "drop"

due to the internal resistance will be Cr6 = T1u-x3 = -3 volt, so that
the P.D. at the terminals of the cell will be 1*5 — -3 = 1-2 volts, and,
if the instrument is correctly graduated, this will be the reading — an
error of 20 per cent. The student should notice that this source of
error is not due to polarisation. It is entirely different in nature.
(b) If, however, the resistance of the voltmeter had been 120

1-5
ohms, the current would have been y^K = -yV ampere nearly, and the

"drop" would have been -^ x 3 = -0375 volt, which would give
a reading of 1*5 — -0375=1*46 volt — a fair approximation to the
correct value.

It would, however, be unusual for a voltmeter of this range to
have so high a resistance, and hence the readings, when applied to a
cell, are always slightly too low, although they give quite correctly
the P.D. between the terminals under the circumstances.

Method of using a Reflecting- Galvanometer as a
Voltmeter. — The student has just learnt that the higher the resist-
ance of a voltmeter, the more nearly the P.D. at the terminals
becomes equal fo the E.M.F., the limit being reached only when that
resistance is infinite, as in the quadrant electrometer, which is really
only a special form of electrostatic voltmeter.

Now, suppose that a reflecting galvanometer — preferably a dead-
beat suspended-coil instrument — of average sensitiveness is (a) put in
circuit vith a resistance of, say, 20,000 ohms at least, and (b) shunted
with another and variable resistance. Neither of these resistances
need be known, but ordinary resistance boxes are ^convenient for the
purpose. The arrangement is shown in Fig. 238.
The two resistances are to be regarded as forming a
constituent part of the instrument, whose terminals
are the ends of the connecting wires TT. Then,
whatever the actual resistance -of the 'galvanometer
may be, the presence of 20,000 ohms insseries with
it makes the working resistance sufficiently high.
If, however, this were used alone, a cell connected
to TT would give too large a deflection — perhaps
enough to damage the instrument — but the presence
of the shunt-box enables the sensitiveness to be
adjusted with great nicety, so that a convenient
deflection is obtained. (Perhaps it may be pointed
FIG. 238. out that all such adjustments are to be made with

the shunt-box ; the other resistance not being
altered.) The arrangement is equivalent to an uncalibrated voltmeter
of high resistance, and the readings will be proportional to the P.D.
between the points to which it is connected.

MEASUREMENT OF ELECTROMOTIVE FORCE 311

Exp. 181. Let each of. the cells under examination be connected to TT,
in turn, giving deflections d and d^ divisions respectively, then if R = resistance
of the instrument between the two terminals, we have

<>=£*-*
and C'=ft7nrrf'

where 6 and 6X are negligible compared with R

This method is especially convenient when it is desired to demon-
strate rapidly to a class the fact that different combinations of metals
give different electromotive forces when used in simple cells.

Exp. 182. After Arranging the apparatus, as shown in Fig. 238, standardise
the scale by connecting a Clark's cell to the terminals TT, and adjusting the
deflection by varying the shunt until it is 143'4 scale divisions (as nearly as
possible). As the E.M.F. of a Clark's cell is 1-434 volts, a. deflection of 100
scale divisions means 1 volt, and so the values corresponding to any deflection
can be read off at once, e.g. a deflection of 95 divisions means '95 volt. Now,
read off the E.M.F.'s due to zinc and copper, zinc and lead, zinc and iron, or
any other combination of metals in dilute sulphuric acid or in other liquid.

Results are obtained by this means with very fair accuracy for
a class experiment. Moreover, any small pieces or wires of th§
various materials may be used. It is also excellent for showing that
the E.M.F. of a cell does not depend upon the size of the plates —
the plates of a cell may be raised out of the solution, but the reading
does not alter until they break contact.

Methods of measuring the Internal Resistance of
Cells. — Several of the previous arrangements may be readily
modified to measure internal resistance, and hence it is convenient
to consider them here.

The most generally applicable method depends on measuring the
ratio of the P.D. at the terminals on open circuit to the P.D. at
the terminals when a current is flowing through a known external
resistance. For example,

Exp. 183. Using the last arrangement, but without taking the trouble to
standardise the scale, join up the cell whose internal resistance is to be measured,
and obtain a steady deflection d divisions.

Now connect, across the terminals of the cell, a known external resistance
of from 2 to 4 ohms, and again read the deflection. Let this be d± divisions.
It will be less than before (the external resistance should be chosen so that
dl is about f of d. )

Then, if E==the E.M.F. of the cell, r& = the internal resistance, rx = the
known resistance, and ex = the P.D. between the terminals when rx is in circuit,
we have

Eo=c£ (i.), and ex«=«&i ("•)

whence, by subtraction, E-ex = rf-(Z1 (iii.) .

312 VOLTAIC ELECTRICITY

and dividing (ii.) by (iii.). we obtain

E — €x d — d-^
but ex = Ox, and E-ex = ^ = l

rx c?j
r^^d^d\

/,7

whence r& =

It must be pointed out that, if polarisation occurs, a serious error
may be introduced into the second reading (taken while the current
is flowing), hence the method is strictly applicable only to constant
cells. But if the galvanometer used is sufficiently dead-beat, the
second reading may be taken almost at the instant the circuit is
closed, i.e. before the cell has had time to polarise seriously.

The same measurements may obviously be carried out whatever
be the particular method adopted for obtaining the ratio E to ex.

A potentiometer is very suitable for the purpose, an experimental
arrangement being shown in Fig. 239. A balance is obtained with

the key open, and then with the
key closed. In practice, a re-
" ^>^ sistance box is used for rx, and

i i^N R ^s iu^ity plug f°r the key.

If d and dl are the two
positions of balance, the argu-
ment is the same as before.

It is, however, not very easy
to obtain the second reading

FIG. 239. quickly, and, therefore, the

method is suitable only for
constant cells. To obviate this difficulty completely, the condenser
method may be employed.

Use of Condenser. — A condenser suitable for such purposes
is made by interleaving many sheets of tinfoil with larger sheets of
paraffined paper, or better still, of mica, as dielectric.

As explained on p. 57, the result is essentially the same as a
Leyden jar, but, as the coatings are larger and the dielectric thinner,
the capacity of even a small condenser is much greater than that of
a large Leyden jar ; on the other hand, it is intended to stand only
comparatively small differences of potential. . To charge it from an
electrical machine would probably break it down at once, owing to
the sparks piercing the insulation.

Let this condenser be connected up to a cell and a ballistic 1

1 Practically any galvanometer that is not too " dead-beat" maybe used,
provided it is sufficiently sensitive.

MEASUREMENT OF ELECTROMOTIVE FORCE 313

galvanometer, as shown in Fig. 240. The normal position of the key
is as shown ; when depressed the condenser is charged by the cell until
the P.D. between its coatings is equal to
the E.M.F. of the cell. On releasing
the key, the cell circuit is opened and
the condenser discharged through the
galvanometer, producing a swing or
" throw " of the needle. For the present
purpose, it is sufficient to say that, with
a reflecting galvanometer, the first swing
of the needle in scale divisions is pro-
portional to the "quantity" discharged
through the galvanometer.

Condenser Method of comparing two E.M.F.'s. —

Exp. 184. Connect up each cell in turn (as explained above) and obtain throws
of d and dl divisions. If K be the capacity of the condenser,

and Q,i = .

EI Ctj

The equation Q = EK is identical with the equation Q = VC discussed on p. 50,
but, when dealing with voltaic measurements, the units of potential, quantity,
and capacity are different, and in order to suggest the change, as well as to
avoid confusion between the use of C for capacity and for current, we shall
write Q = EK, instead of Q = VC.

This method of comparing E.M.F.'s is absolutely perfect with
respect to freedom from polarisation. Its only disadvantage lies
in the fact that it is necessary to read a "throw" in scale divisions,
which means that the possible accuracy is less than that obtainable
by using the potentiometer method, which is a null method.

Comparison of Two Capacities by the Condenser
Method. — If two capacities are to be compared — not differing too
widely in magnitude — it is only necessary to use each in turn with
the same cell, obtaining throws of d and dl , then

and Q^EK^ocdj

K d

or — = — -

Measurement of Internal Resistance by Condenser.—

Evidently, we may use a condenser to obtain the ratio of the quantities
E and ex (as in Experiment 183), and the peculiar advantage of the
method lies in the fact that it is possible to obtain the throw corre-
sponding to ex at the instant the circuit is completed through the
resistance ?*x, i.e. before the cell has had time to polarise, and hence

314

VOLTAIC ELECTRICITY

the method is applicable to all kinds of cells, being especially advan-
tageous with those which polarise quickly.

Exp. 185. The connections may conveniently be made as shown in Fig. 241.
In this case the throw observed is that due to the charging and not to the dis-
charging of the condenser (although for our purposes
K either may be used indifferently in this and in previous

_G experiments). When obtaining the first throw, remove
the infinity plug of the resistance box. We get, as before,

FIG. 241.

Then unplug 2 or 3 ohms in the box, insert the
infinity plug, and again take the throw. This will give
Cxccd1 , and the argument is then identical with that
already given on pp. 311, 312.

It will be noticed that by closing the key, we simul-
taneously close the cell circuit and charere the condenser.

Condenser Key. — In experiments with
condensers, it is convenient to use a special type of key, known as
a " condenser key," of which a simple form is shown in Fig 242. A
strip of brass, A, is fixed at B, and when depressed it makes contact
with a metal block carrying
a terminal, C. At the same
time it breaks contact with
the screw, D, which is carried
by a metal bridge-piece, and
which is adjustable to any
height. When A is allowed
to fly up again, it breaks con-
tact at C and makes it again
at D. The working parts are
supported by ebonite pillars,
in order to provide the high
insulation required for certain
purposes. Such a key can be FIG. 242.

connected up in a number of

different ways according to the purpose for which it is to be used. Fig.
243 shows diagrammatically the connections required in Experiment

FIG. 243.

FIG. 244.

184, and Fig. 244 those required for the measurement of internal resist-
ance in Experiment 185. A, B, and C have the same meaning as in
Fig. 242, b is the cell, G is the galvanometer, and K is the condenser.

MEASUREMENT OF ELECTROMOTIVE FORCE 315

Reduced Deflection Method. — For the sake of comparison,
we may mention an older method of measuring internal resistances,
sometimes called the "reduced deflection method."

Exp. 186. Connect up the cell, whose internal resistance is to be measured,
in series with a tangent galvanometer and a resistance box. Adjust the resist-
ance until a deflection of about 50° is produced. Let this be 0°, and r the
resistance unplugged in the box. Increase the resistance until the deflection is
reduced by 10° to 12°, and let 0l be the new deflection and ^ the resistance in the
box. Then, if y be the resistance of the galvanometer and connecting wires,
and b the internal resistance of the cell, we have —

i-r + g tan 0X
or (& + f+30 tan 6 = (b + r1 + g) tan 0j

J(tan 6- tan ^i) = (»'i + <7) tan ^-(r + r/) tan 6
j. (ri + </) tan d-i -(r + q) tan 6

Or 0= 2 — * Si i 1_ — i:

tan 6 - tan 0X

In a particular experiment a deflection of 53° was produced with 10 ohms,
and a deflection of 40° with a resistance of 20 ohms. The resistance of the
galvanometer and connecting wires was ascertained to be 1*25 ohms. Now,
tan 40°= *839 and tan 53° = 1*327, whence from the above formula —

, _ -839(20 + 1*25) - 1*327(10 + 1*25)
1*327 -'839

_ 17 -82875 -14 -92875

-488

= 5*9 ohms nearly.

Not only is this method tedious and solely applicable to constant
cells, but it is also liable to very large errors if the total resistance of
the circuit is much greater than b. If, for instance (using a galvano-
meter of low resistance), the resistance of the same Darnell's cell be
measured repeatedly, first with small and then with large values
of r and rlt the results will be found to be very variable. If b is
about 1 ohm, and if r = 0 and rt = 1 ohm, or if r = 1 ohm and rt = 2
ohms, the result will be fairly correct. If, however, we make
r == 60 ohms and i\ = 100 ohms, the result will probably be absurd.

This is because the effect of b on the deflections is negligible
compared with that of r and rlt and is less than the unavoidable
errors of reading.

When the galvanometer has several coils to choose from, as is
usually the case, these considerations will indicate the one to be
selected.

316 VOLTAIC ELECTRICITY

EXERCISE XVI

1. Describe and explain a method of comparing an E.M.F. of two voltaic
batteries.

2. Two batteries, known to have different electromotive forces, are found to
give equal deflections when joined up, one at a time, with the same tangent
galvanometer. Explain how you could find out by experiment which battery
has the greater E.M.F., and what proportion the E.M.F. of one bears to that
of the other.

3. A Daniell's cell and a Grove's cell were arranged in series with a tangent
galvanometer. When their electromotive forces \vere acting in the same direc-
tion, the deflection was 42° ; when in opposition, 14°. Taking the E.M.F. of the
Daniell's cell as 1'07 volts, find that of the Grove's cell. Tan 42°= '9004 ; tan
14° ='2493.

4. Compare the electromotive forces of two batteries, A and B, if when they
are successively connected in series with a tangent galvanometer and a resistance
box, the deflection is 20°, and that to reduce the deflection to 10°, 3 ohms must
be added when A is in circuit, and 5 ohms when B is in circuit.

5. A cell was connected in series with a tangent galvanometer and a resist-
ance box. When the resistance of the external circuit and galvanometer was
1 '4 ohm a deflection of 45° 30' wras obtained, but when an ohm was added to the
external resistance the deflection was 32° 20'. Find the internal resistance of
the cell. Tan 45° 30' = 1 '0176 ; tan 32° 20' = -6331.

6. A cell was arranged in series with a tangent galvanometer and a resistance
box. A deflection of 40° was obtained with a resistance of 8 ohms, and a deflec-
tion of 35° with 10 ohms. The resistance of galvanometer and connecting wire
was ascertained to be 1 '05 ohms. Find the internal resistance of the cell. Tan
35° = 7002; tan 40° ='8391.

7. Two cells, the E.M.F. 's of which are 2 : 1. are joined up in series, with their
E.M.F/S acting in the same direction, and the circuit is completed through
a tangent galvanometer, the needle of which is deflected through 60°. If one of
the cells is reversed, no other change being made, what will be the deflection
of the galvanometer ? (B. of E., 1896.)

8. A wire AB, of *33 ohm resistance, forms part of a circuit through Avhich
an electric current flows in the direction from A to B. The points A and B are
also connected by another conducting path in which is included a cell of E.M.F.
1'287 volts and a galvanometer, the positive pole of the cell being that joined to
A. If the galvanometer is not deflected, what is the strength of the current in
wireAB? (B. of E., 1897.)

9. The zinc poles of two batteries, A and B, are connected by a wire, and
likewise the platinum poles by another wire. When the poles of the battery A
are also connected with each other by a wire whose resistance is to that of the
battery A itself as 7:3, there is no current in the battery B. Show what
relation the E.M.F. of the battery A bears to that of the battery B.

CHAPTER XXI

ELECTROLYSIS

CHEMICAL actions, similar to those already described as taking place
in the cells themselves, are also produced outside the cells when a
current is passed through certain liquids. Neglecting mercury and
molten metals, which behave exactly like solid metals and require
no special notice, we may divide liquids into two classes : —

!1) Those incapable of conducting a current.
2) Those which conduct, but in a manner which differs from
metallic conduction.

In the first category must be placed the more complex liquids
such as oils, alcohol, &c., some of which are amongst' the most perfect
insulators. Pure water must also be included in this group.

In the second category we are chiefly concerned with solutions
and fused salts. If, for example, one of the large class of bodies,
known as chemical salts (themselves non-conductors in a dry state),
be dissolved in a non-conducting solvent,, such as water or alcohol,
the result is a more or less conducting solution or electrolyte.
Such a liquid solution behaves quite differently from a metallic
conductor, for it can only conduct by being decomposed during the
process. Ordinary acids, which may be regarded as hydrogen salts,
behave similarly. Such decomposition by the electric current is
called electrolysis. The conductors, by which the current is
conveyed in and out of the solution, are known as electrodes ;
the electrode by which the current enters being called the anode,
that by which it leaves the cathode.

Exp. 187. Join up a battery in series with a galvanometer and dip the ends
of the connecting wires in some oil or turpentine. Observe that no deflection
is obtained.

Exp. 188. Connect the ends to platinum electrodes dipping into the purest
distilled water obtainable. There should be no visible effect at the electrodes,
and although a sensitive galvanometer may show that some current is passing,
it will be extremely small.

Exp. 189. Add to the water a drop of sulphuric acid, and stir. Notice that
bubbles of gas now appear at both electrodes, and that the galvanometer shows
that a current of considerable strength is flowing round the circuit.

Exp. 190. Instead of using the acid, show that similar results are obtained
when various salts are dissolved in the water. Try common salt, potassium or
sodium sulphate, potassium iodide, &c. Notice that no action is visible any-
where except at the electrodes.

317

318

VOLTAIC ELECTRICITY

When the apparatus used in such experiments is designed to
collect the substances liberated, so that they can be weighed or
measured, it is known as a voltameter. A lecture -table form,
especially usoful with acid solutions, is shown
in Fig. 245. An alternative form of apparatus
can be easily made as follows : —

Obtain a glass funnel, five or six inches
in diameter across the top. File off the stem
at a point about half an inch from the bottom
of the funnel. Solder strips of platinum foil
to two copper wires, and then pass the wires
from the inside of the vessel through the
stem. Arrange the platinum strips parallel
to each other, and then fill the stem and part
of the funnel with plaster of Paris, so that
the pieces of platinum project above it. If
the wires- are uninsulated, take care that they
are not in contact. In order to make the
apparatus water-tight, melt some parafiin-wax
and pour it over the plaster of Paris.

FIG. 245.

Electrolysis of Dilute Sulphuric

Acid. — Exp. 191. Partially fill the vessel with water
acidulated with sulphuric acid. Fill two test-tubes
of equal size with acidulated water, and invert them
over the electrodes. Connect the free ends of the

wires to a battery of three to six cells in series. Observe that bubbles of gas
rise from the electrodes. After the action has proceeded for a short time, it
will be found that the volume of gas (H) in the tube over the cathode is nearly
double that (0) in the tube over the anode.

If the apparatus is filled with a solution of sodium sulphate or of potassium
sulphate, exactly the same effect is obtained as regards the evolution of gas.

Outline of Theory. — When discussing the actions which occur
in voltaic cells, we explained that simple salts may be regarded as
made up of a metallic portion and a non-metallic portion held
together by electric forces, and that, in solutions, these portions to
some extent appear to dissociate and to wander about as free charged
atoms, or groups of atoms, known as ions.

This process of ionisation seems to be essential to conduction in
liquids (and gases) — if no free ions are present, they cannot conduct.
The extent to which a substance ionises in solution depends partly
upon the nature of the solvent; and of all known sol vents, "water is
by far the most effective.

Suppose that two inert electrodes, say of platinum, are placed
in such a solution, and that an electric field is created between them
by connecting them to the terminals of a cell or battery, then
evidently any free charged particle, or ion, will tend to move one

ELECTROLYSIS 319

way or the other along the field according to the sign of its charge.1
Such motion is slow, owing to the frictional resistance of the water,
but there must evidently be a tendency to set up two opposite
processions and to cause an accumulation of ions around the electrodes,
as far as the counter tendency of diffusion permits. There can,
however, be no current in the circuit, and the process must soon
stop, unless the ions in contact with an electrode can give up their
charge to it, or receive one from it. "V\Jhen this occurs, the particle
or group ceases to be an ion (this term, as already explained, being
strictly limited to a charged particle or group) and displays at once
its ordinary chemical properties. What happens next depends upon
circumstances. If it has no chemical action either on the electrode
or on the solvent, it appears in a free state ; but in many cases such
chemical affinity does exist, and then it combines with one or the
other, or both, the product actually liberated in a free state depending
upon the nature of the reaction. It is therefore convenient to
distinguish between primary and secondary effects in electrolysis.
The former depend only on the nature of the substance ; the latter,
also upon the nature of the electrodes and upon the experimental
conditions, as will be seen when we apply these ideas to the results
of experiments. Let us consider the case of dilute sulphuric acid
with platinum electrodes, as given in Experiment 191. Then we
have : —

Primary action H2SO4 becomes HH (at cathode) and SO4 (at anode).

+ +

Secondary action. There is no secondary action at the cathode,
because hydrogen does not combine with either platinum or water,
and hence it comes off as a gas. At the anode the (SO4) group or
" sulphion " is unable to attack the platinum, and may be regarded
as acting on the water present, according to the equation

and hence, sulphuric acid is re-formed in solution and oxygen gas is
evolved. (This is the simplest way of considering the matter,
although we might think of the SO4 group as obtaining the necessary
negative charge to keep it in solution as an ion by removing the
charge from the oxygen in a molecule of water — the oxygen then
becoming free and the two hydrogen atoms becoming charged ions.)

These statements (and others made in this chapter) must be
regarded merely as a simple outline of the more important facts
and not as an exhaustive treatment. Experiment shows that more
complex ions may be present in small quantity, sometimes formed
by simple ions uniting with undissociated or neutral molecules — a

1 The products of electrolysis, which appear at the anode, are called anions ;
those, at the cathode, are called cations.

320 VOLTAIC ELECTRICITY

tendency especially marked in strong solutions — and sometimes re-
presenting a partial stage in the process of ionisation. It is possible,
for instance, that H2SO4 ionises at first into H and (HSO4), the

+ + - -

latter group then breaking up into H and (SO4) ; for a few HSO4

ions appear to be usually present.

The behaviour of water itself is peculiar, on account of its solvent
action on all ordinary containing vessels. The early experimenters
tried to electrolyse the purest distilled water they could obtain, and
they invariably found that acid and alkaline substances were formed
in the liquid around the electrodes (acid at anode, alkali at cathode),
and for some time it was thought that a current had some power of
creating such substances. Sir Humphry Davy finally cleared up
the matter by a series of laborious experiments. He found that, when
a marble vessel was used to hold the water, the alkali was sodium
hydrate and the acid HC1, both due to a trace of NaCl present in
the marble ; when agate cups were used, the effect was less, but
silica was obtained ; with gold vessels, nitric acid and ammonia
appeared (traced to the surrounding air) ; and by working in a
vacuum, the effect was finally reduced to a minimum, but then the
liquid was almost totally non-conducting, although he used fifty cells
in series. Hence, all ordinary water is slightly conducting, although
it appears that water itself is not an electrolyte.

Exp. 192. Put a current reverser in circuit with a battery, attach copper
strips to the free ends, and insert them into a beaker of dilute sulphuric acid.
Notice that a gas is evolved as before at the cathode — if collected, it proves
to be hydrogen — whilst no gas is evolved at the anode, but observe that the
liquid immediately around it gradually becomes blue. Reverse the current and
notice that the actions are also reversed.

The primary action is unaltered. As before, there is no secondary action
at the cathode, but at the anode the S04 group combines with the copper to
form copper sulphate, which dissolves in the surrounding water. (If the current
strength is great compared with the area of the anode, part of the S04 may
attack the water as in Experiment 191, and some oxygen gas will be given oh*'
in addition to the formation of the sulphate).

Exp. 193. Electrolyse a saturated solution of copper sulphate with platinum
electrodes (as before, two strips dipping into a beaker will do).

Primary action CuS04 becomes On (at cathode) and S04 (at anode). Again

+ +

there is no secondary action at the cathode, copper being deposited. (If the
cathode surface is sufficiently great compared with the current strength, the
copper will be in the form of a firm coherent coating of bright metal ; witli
greater current strength it may be deposited partly as a loose powder.) At
the anode, the secondary action will be the same as in experiment 191 — oxygen
gas being given off and sulphuric acid re-formed in solution. Reverse the current,
and notice that the deposited copper returns into solution.

Exp. 194. Repeat the last experiment, using copper electrodes. No apparent
effect will be noticed at the anode, but the cathode will become coated with a
deposit of copper.

The primary action is the same as in Experiment 193, and as before there is
no secondary action at the cathode, copper being deposited upon it ; in fact the

ELECTROLYSIS

321

nature of the cathode is immaterial ; if it is a conductor, copper will be deposited

upon it. At the anode, the secondary elfect will be the same as in Experi-

ment 192, copper sulphate being formed and
entering into solution, which is unaltered
in strength.

Exp. 195. Make a saturated solution of
sodium or potassium sulphate, and electro-
lyse it, iising platinum electrodes, in a
U-tube, as shown in Fig. 246 (from Carey
Foster and Porter's Electricity and Maynet-
ism). Gas will be evolved at both electrodes,
and we know from Experiment 191 that it
consists of oxygen and hydrogen in the pro-
portions to form water. Pour blue litmus
solution in each limb. At the cathode, it
will remain blue, and at the anode it will
turn red. Reverse the current, and notice
that these colours are gradually reversed also.
These operations may be repeated several
times. Hence, an alkaline substance is
formed at the cathode, and an acid substance
at the anode.

Assuming that potassium sulphate is
used, the primary action is evidently
K2S04 becomes K2 (at cathode) and S04 (at
anode). ++

Secondary actions now occur at both
electrodes. At the cathode the liberated
metal attacks the water, forming strongly

alkaline potassium hydrate, and liberating hydrogen gas, according to the

equation

FlG. 240.

At the anode, the action is the same as in Experiment 191, and we have (see p. 319)

Exp. 196. Electrolyse a solution of lead acetate witn platinum electrodes.
Gas will be evolved at the anode, and feathery crystals of metallic lead will form
on the cathode (Fig. 247). Reverse the current, and notice that little or no
gas is evolved at the new anode until the previously deposited lead has either
redissolved or dropped off.

This illustrates the tendency of many metals to
assume a crystalline form. There is no secondary
action at the cathode, but at the anode the radicle
of acetic acid behaves like the S04 group, taking
hydrogen from the water and liberating oxygen,
which may again react on the substance to produce a
dark red powder, — lead peroxide, Pb02.

Exp. 197. Repeat the observations, using copper
electrodes. Again we find that the nature of the
cathode makes no difference, lead crystals being
formed as before, but at the anode the solution soon
turns greenish blue, owing to the formation of copper
acetate. FlG. 247.

These experiments sufficiently illustrate the general nature of
electrolytic actions. It will have been noticed that the metal (or

X

322 VOLTAIC ELECTRICITY

hydrogen) is invariably liberated at the cathode ; this is due to
the fact that in simple salts it always carries a positive charge.
If an ammonium salt is electrolysed, the group NH4 behaves
like a metal and is set free at the cathode, where it breaks up
into ammonia and hydrogen. Sometimes the secondary actions are
very complex. If, for example, a not very strong cold solution
of ammonium chloride be electrolysed with platinum electrodes,
the above action takes place at the cathode, and chlorine gas is
evolved at the anode, where it partly dissolves in the liquid and
then comes off as gas. But if the solution is warm and saturated,
and the current density fairly great, the very dangerous explosive,
chloride of nitrogen, is formed. The experiment can be performed
safely in an apparatus of the kind previously described and used in
Experiment 191, provided a thin layer of turpentine is poured on
the top of the liquid (the test-tubes are not required), for, as the
chloride is lighter than water, it rises to the surface and instantly
explodes on coming into contact with the turpentine. Hence it
is never allowed to accumulate, and so results which might be
disastrous are avoided.

Practical Applications. — In 1808, Sir Humphry Davy
isolated the metals potassium and sodium by electrolysing their
hydrates, which were then regarded as elements. The chief difficulty
was to exclude secondary actions, and many experiments with solu-
tions in water as well as with the fused hydrate were failures. He
succeeded eventually by using a slightly moist piece of hydrate
placed on a platinum-plate anode and touched with a platinum-wire
cathode. This illustrates the general principle to be adopted in such
cases. When secondary actions are unavoidable, on account of the
chemical activity of the body required, the details of the experiment
should be arranged so that the substance is liberated at a greater rate
than secondary actions can remove it. This usually means a small
cathode and a relatively large current.

In this way, many years later, Bunsen succeeded in depositing the
metals calcium, strontium, and barium from hot aqueous solutions of
their chlorides — a result which might well have been deemed impossible.
Davy succeeded in obtaining these metals only as amalgams, from
which the mercury was afterwards removed by distillation out of
contact with oxygen. Such amalgams can be easily obtained by
using mercury as a cathode in a solution of any convenient salt
of the metal required.

At the present time many substances are produced on a com-
mercial scale by electrolysis, and its applications are being extended
daily. Amongst these, we may instance the following : —

Metallic sodium is now almost entirely produced by electrolysing
fused sodium hydrate with iron electrodes (Castner process).

Practically all the copper used for electrical purposes is refined

ELECTROLYSIS 323

by making it the anode in copper sulphate solutions and depositing it
on a suitable cathode.

Gold is very largely obtained from poor ores and residues by
treating the finely ground material with potassium cyanide. This
dissolves the gold, forming a double cyanide of potassium and gold,
which is then electrolysed.

Caustic soda and chlorine are made in quantity by the Castner-
Kellner method of electrolysing a solution of common salt.

Metallic calcium was obtained many years ago in small quantities
and with considerable difficulty by the electrolysis of the fused
chloride, but recent improvements in details have enabled it to be put
on the market on a commercial scale at a relatively low price. The
chloride is contained in a graphite crucible, which also serves as
anode, the bottom being kept cool by water. The cathode is an
iron rod rather more than an inch in diameter, also water-cooled.
At the commencement of the process, the cold chloride is partly
fused by forming a temporary arc from the side of the crucible, and,
when it begins to conduct, an alternating current is applied until
the chloride is completely fused by the heat thereby produced. The
alternating current is then cut off, a direct current switched on, and
the process begins (the chloride being kept in a state of fusion by the
heating effect of the current). It is necessary to keep the end of
the cathode only just below the surface, because then the spongy
metal first formed melts almost at once, and its surface tension draws
it into a compact globule, which is almost instantly solidified by the
cooling effect of the cathorle. In this way, by raising the cathode
very steadily, a solid stick of calcium is gradually built up. If the
cathode dips too far below the surface, the spongy metal is apt to be
swept away by convection currents, and therefore lost.

Aluminium is obtained by electrolysing the oxide (alumina),
dissolved in fused cryolite (a naturally occurring double fluoride of
aluminium and sodium). This metal is remarkable for several
reasons. It has never been successfully deposited from an aqueous
solution, so that aluminium plating is unknown ; and when used as
an anode in many electrolytes, it offers a very considerable resistance
to the passage of a current. If, for example, electrodes of aluminium
and lead (or iron) are placed in a solution of ammonium phosphate,
the current passes readily when the aluminium is the cathode, but
when it is made the anode, the cell is almost non-conducting. It,
therefore, acts like a valve, and may be used to obtain a direct
current from alternating mains. When used for this purpose, the
device is known as an aluminium rectifier. The best plan is to
use four such cells, arranged as shown in Fig. 248, for then both
directions of the alternating current are made available. A and L
denote aluminium and lead respectively ; MM are the alternating
mains ; TT the points from which the direct current is taken off to

324 VOLTAIC ELECTRICITY

nome load (generally accumulators, which cannot be charged with an
alternating current). Remembering that A acts only as a cathode,
i.e. the current through the cells must go from L to A, it will be
found, on tracing out the path of the current, that its direction in
< .- ... the load circuit is the same,

^

A .L f i . A \ whichever way it may be

\. LOAD flowing from the mains.

In practice, an adjustable
resistance would be placed
in series with the mains,
in order to prevent an ex-

FlG 248 cessive current at starting,

for the action appears to

depend on the formation of a thin non-conducting layer of oxide at
the aluminium anodes, and this takes a few minutes to form. Then
the resistance is cut out. A four-cell rectifier will easily stand alter-
nating pressures of from 100 to 150 volts, and greater voltages may
be dealt with by increasing the number of cells ; in fact, in this way
one of the writers has for several years worked up to 2000 volts.
When the rectifier is in good order, the current strength is about the
same on both alternating and direct current sides, but the voltage
on the latter is only about half that on the former. This means
that there is only 50 per cent, efficiency at the best, but in practice,
it is not likely to average more than 30 to 40 per cent. One of
the most troublesome defects is due to the heating of the cells,
for as the temperature rises, the rectifying property diminishes.
It is, therefore, necessary to make the cells rather large for a given
current output. In any case, they are somewhat troublesome to
keep in order, but they are often convenient when no other way of
obtaining a direct current is available.

Two other practical applications may be mentioned : —

1. Electrotyping, by means of which impressions of coins, wood
engravings, &c., are obtained.

2. Electroplating, by which the surface of a base metal— e.g. German
silver or copper — is covered with a superior metal — e.g. silver or gold
— either for the purpose of protecting the article from oxidation, or to
give it the appearance of being wholly composed of the superior metal.

Electrotyping". — In order to obtain a copper electrotype of any
object, a mould must first be made, on which the layer of metal is to be
deposited. For objects, such as medals, which can be submitted to
pressure, guttapercha may be advantageously used for this purpose.
It is softened in hot water, pressed upon the object, and then allowed
to cool. For wood blocks or type, wax moulds are commonly used,
which are made by pouring a mixture of wax, tallow, and Venice
turpentine into a shallow vessel, and, before it completely sets, press-
ing the block or type upon it. Moulds of plaster of Paris, or of a

ELECTROLYSIS 325

fusible alloy, are sometimes used. It is very important that the face
and edges of these moulds should be covered very carefully and
thoroughly with graphite, so as to make the surfaces conduct. The
mould is placed in a saturated solution of copper sulphate, and then
made the cathode of a Daniell's cell or battery, while a copper plate
forms the anode, which, gradually dissolving in the solution, keeps it
at a constant strength.

Electroplating1. — This term includes : —

(a) Electro- gilding, the process by which gold is deposited on a
baser metal. The bath is a solution of the double cyanide of gold
and potassium. In order that the gilding bath may be of constant
strength, the anode consists of a gold plate, which dissolves at a rate
equal to that at which the gold is deposited on the cathode.

(b) Electro-silvering. — In this process, the bath consists of the
double cyanide of silver and potassium, and the anode is a silver plate.

(c) Electro-nickeling. — The bath in this case consists of a solution
of the double sulphate of nickel and ammonium (made slightly acid).

The principle of these processes will be understood from the
following method of coating a German silver spoon with silver. The
spoon must be thoroughly cleansed by (1) boiling it in a weak solu-
tion of caustic soda to remove grease, (2) washing with water, (3) im-
mersing it for a moment in dilute nitric acid to remove any film of
oxide, (4) brushing it with a hard brush, and (5) plunging it in clean
water. Two metal rods are placed across the vessel containing the
solution, which should be gently warmed while the deposit is being
made. The spoon, hung from one of the rods by means of a wire
(waxed all over), is made the cathode, while a silver plate, suspended
from the other, is the anode.

Faraday's Laws of Electrolysis. — By a most laborious and
extensive research, Faraday established the following important
generalisations : —

I. The mass of a substance liberated during electrolysis is propoi'-
tional to the " quantity " passing through the solution, i.e. to the product
of the current strength and the time during which the current flows.

II. The mass of a substance liberated l>y a given " quantity" is
proportional to its "chemical equivalent" (where chemical equivalent

_ atomic weight\ l
valency )

1 The chemical equivalent of an element is obtained by dividing its atomic
weight by its valency.

The atomic weight of an element is the weight of an atom of the element
compared with the weight of an atom of hydrogen. (At the present time atomic
weights are usually referred to that of oxygen, taken as 16, bnt this introduces
a source of confusion, and the older definition is used in the table on next page.)

The valency of an element is the atom-fixing or atom-replacing power of the
element; e.y. when zinc is acted on by sulphuric acid, the chemical action is
represented by the equation Zn + H2S04 = ZnS04 + H2, in which it is seen that
one atom of zinc replaces two atoms of hydrogen.

VOLTAIC ELECTRICITY

The first law may be written

WocCU

where W is the weight in grams of the substance liberated by a
current of C amperes flowing for t seconds,

or W = (Us

where z is a number which is constant for a given substance, but
which varies for different substances. It is called the "electro-
chemical equivalent " of the substance.

If we put C= 1, t = 1, it is evident that z is the weight in grams
deposited by 1 coulomb.

From the second law we learn that the quantity which will

deposit W grams of hydrogen, will also deposit W x — grams of

silver, or W x

64-85

1
grams of zinc, i.e. if we know the value of z for

hydrogen, its value for any other substance will be the product of
this value for hydrogen and the chemical equivalent of the substance.

If a substance has two valencies, like iron in ferrous and in ferric
salts, it will be deposited by the same current at different rates from
the two classes of salts, in the ratio of its chemical equivalents of each.

The following table gives the atomic weight, the valency, the
chemical equivalent, and the electro-chemical equivalent of various
elements : —

Atomic
Weight.

Valency.

Chemical
Equivalent.

Electro-chemical
Equivalent (given in
Grams per
Coulomb).

Hydrogen
Potassium

1

38-86

1

1

1

38-86

•0000104
•0004065

Sodium

23

1

23

•0002403

Silver .

107-03

1

107-03

•0011183

Gold .

195-7

3

65-23

•0006816

Copper (cupric) . '

63-06

2

31-53

•0003294

, , (cuprous) .

63-06

1

63-06

•0006588

Mercury (mercuric)

198-5

2

99-25

•0010370

,, (mercurous)

198-5

1

198 5

•0020750

Tin (stannic)

118-1

4

29-52

•0003084

,, (stannous)

118-1

2

59-05

•0006169

Iron (ferric)

55-4

3

18-5

•0001930

, , (ferrous)

55-4

2

27-7

•0002894

Nickel

58-3

2

29-15

•0003046

Zinc .

64-85

2

32-42

•0003388

Lead .

205-45

2

102-72

•0010732

Oxygen

15-88

2

7-94

•0000830

Chlorine

35-18

1

35-18

•0003676

Iodine

126

\

126

•0013166

Bromine

79-36

1

79-36

•0008292

ELECTROLYSIS 32?

It is instructive to combine the two laws into one general equa-
tion, which then becomes

\*r n j. atomic weight
W = CU x — — - - ^— x a constant
valency

It is evident that this constant is the same for all substances, and
hence, it is a number of great importance.

Putting unity for C and t we have 1 coulomb, and as the atomic
weight and the valency of hydrogen are both unity, it will be seen
that this constant is the mass (in grams) of hydrogen deposited by
1 coulomb, i.e. it is the electro- chemical equivalent of hydrogen. Its
numerical value is '0000104 very nearly. For our purpose, it is

more convenient to write it in the form x-x--» thus

YVT _ CU x atomic weight
96,000 x valency1

It follows from this equation that 96,000 coulombs must pass
through a solution in order to liberate 1 gram of hydrogen (i.e.
96,000 coulombs of positive in one direction, and the same quantity
of negative in the opposite direction), and from our previous state-
ments as to the nature of the process, this must also be the amount
of positive charge carried by 1 gram of hydrogen. Similarly it will

take- — '—- coulombs (-4174 coulombs) to deposit 1 gram of
2o

sodium, and this must be the charge carried by 1 gram of sodium ;
but, as the atom of sodium weighs 23 times as much as the atom of

hydrogen, 1 gram of sodium contains only — as many atoms as

1 gram of hydrogen, which means that the monad atom of sodium
carries the same charge as the monad atom of hydrogen.

The argument applies also to ions (like chlorine), which carry a
negative charge, and thus we arrive at the very important conclusion
that every monooalent ion in an electrolytic solution carries exactly the
same amount of charge — negative or positive as the case may be. It
also follows that every divalent ion, like oxygen, or copper in cupric
salts, carries exactly twice this charge, and so on. These facts suggest
that an electric charge is essentially atomic in nature, being made up
of small charges, which are incapable of division by any known
process, and this conclusion is strongly supported by other evidence
derived from the passage of a current through gases.

The charge carried by one monad ion is herce a natural unit of
quantity, being the smallest portion of electricity known to exist.

1 It must be understood that this number (96,000) is given in round figures,
as the value of the last three significant figures is somewhat uncertain. The value
of z for hydrogen is probably '00001045, which corresponds to 95,690.

328 VOLTAIC ELECTRICITY

Now the number of atoms in 1 gram of hydrogen has been
determined by many different methods, and is known to be about
6-16 xlO23.

96 000

Hence, the natural "atom" of electricity is .=

o'lo x 1023

1*57 x 10~19 coulombs, or 1-57 x 10~20 absolute units of quantity in
the magnetic system of units. This is 4'68 x 10~10 static units of
quantity.

(It will be seen later that free negative charges of this magnitude —
known as electrons — are present in vacuum tubes, and the above values
agree with those determined by direct experiment in various ways.)

Ionic Velocities. — If an electrolytic cell be divided into two parts by
a porous partition (to prevent diffusion as much as possible) and a current passed
for some time, the solution is no longer of the same strength in each compart-
ment. For instance, if a solution of copper sulphate is used with copper
electrodes, after a time it is stronger in the compartment containing the anode,
although the solution as a whole contains as much copper sulphate as before.

This fact was carefully inves-
tigated many years ago by
Hittorf, who showed that it
could be explained by assum-

(Anode)*_ , , , _ (Cathode) ijl£ tliat difterent ions moved

' with different velocities, under
the influence of the same
P.D., and that the ratio of
FlG 249. ^'ne vel°cities could be ascer-

tained by measuring the relative

strengths of the solution in the two compartments. This will be understood
from Fig. 249, which is a modification of a diagram due to Ostwald.

P is a porous partition, and the two upper rows represent the state of the
solution before the current flows, each compartment containing eight molecules
of some simple salt, e.g. NaCl.

Let v = velocity of positive ions towards the cathode, and u — velocity of
negative ions towards the anode ; and assume that v=3 and u — 2. When the
current has passed for a certain time, the state of the solution will he as shown
in the two lower rows. Now, the electrical conditions determine that charges
must be given up to the two electrodes at exactly the same rate, otherwise there
would be an accumulation of charge at one or the other. Suppose that five ions
are discharged at each electrode ; then five molecules out of the original solution
have disappeared — three from the anode compartment and two from the cathode
compartment.

If the strength of the solution in each compartment is determined experi-
mentally and compared with the original strength, it will be found that
anode loss _3_v
cathode loss~2~w

or anode loss x anion velocity = cathode loss x cation velocity.
Which may be written

anode loss v cathode loss u

total loss ~v + u' total loss

If another equation can be found connecting v and u, it will evidently be
possible to determine their absolute values. Such an equation was ultimately
provided by Kohlrausch ; his argument being briefly as follows : —

ELECTROLYSIS *329

First let us suppose, for the sake of clearness, that only one kind of ion
moves — say the positive with velocity v. This would he the equivalent of a
current, because an equal number of negatively charged ions would be simul-
taneously liberated at the other electrode, but as will be seen, by drawing
another diagram similar to Fig. 249, all loss of electrolyte must necessarily
occur at the electrode from which the ions are moving, which in this case is
the anode.

Let there be n charges of each sign per cubic centimetre, each of magnitude
c coulombs, where e is the charge on one monad ion. (This does not mean n
ions unless the valency of each ion is unity ; in fact the number of ions will be

-VM

valency /

Consider a column of liquid 1 square centimetre in section between the
electrodes. Then the quantity of charge given up to the cathode per second
will be + ncv units ; an equal charge, - nev units, being simultaneously given up
to the anode.

But current strength = quantity passing per second,
. •. current = n ev amperes.

Secondly, if the other kind of ion moves also in the opposite direction with
velocity u, it will, by similar reasoning, also correspond to a definite current in the
same direction as the previous current, and the diagram will show that if u = r, the
loss will be the same at each electrode ; but if the velocities are unequal, the
loss will be greatest near the electrode at which the slower-moving ions are
discharged.

The actual current will be the sum of the two components, hence

C = ncv + neu — ne(v + u) amperes.

Let Z Jbe the distance between the electrodes, and E the PD between them in
volts. The resistance of the column in question will be - x is, where s = specific
resistance of the solution. It is, however, usual .to employ specific conductivity

in this connection, where specific conductivity =-=K,

s

.'. resistance of column =--- ohms
Jv

"F1 1\F

Now C = — = — = ne(v + u) amperes
I I

K
If E0 be the P.D. per centimetre of length of solution, then

E T?

T=EO

whence KE0 = ne(v + u)

Now nc is the total charge per cubic centimetre of solution. Let us
suppose its strength is N gram-equivalents per cubic centimetre (one gram-
equivalent means a number of grams of each ion equal to its chemical equivalent,
i.e. to its Atomic weight \ ^ Then W() , the game total cjiarge per cubic centi,

valency /
metre, whatever the nature of the dissolved substance may be ;

i.e. each cubic centimetre will contain N x 96,000 coulombs of each sign ;

.-. we = NX 96, 000
whence KE0= N x 96,000(y+w)
K _ 96,000(-t> + M)

N" "E

330 VOLTAIC ELECTRICITY

The ratio A is called the nwlecular conductivity. It varies with the strength

of the solution, but, for many substances, it approaches a limiting value for
very dilute solutions, and it is this value which is to be used in the above
equation, for we have, so far, really assumed that all the molecules are ionised,
which is not the case except in infinitely dilute solutions.

It follows, from the above equation, that v and u are directly proportional to

the P.D. per centimetre of length ; also, if the limiting value of = can be found

by measuring the specific resistance of solutions of various strengths (Eo being
easily measured by means of a suitable voltmeter), we can calculate v and u
independently by combining it with the ratio determined by Hittorf s method.
It is found that the velocity of a given ion in different dissolved salts is always
the same for a given P.D. for infinitely dilute solutions.

Of all ions, hydrogen moves the fastest, but even its velocity is very small.

The following table, taken from Carey Foster and Porter's Electricity and
Magnetism, gives the velocities of a few ions in centimetres per second due to
1 volt per centimetre of length : —

( + )H 320xlO-5

(-)OH 182x10-5
(-)Cl 69x10-5

(-)I 69x10-5

( + )K 66x10-5

66x10-5

( + )Ag 57x10-5

( + )Na 45x10-5

36x10-5

It has since been found possible to measure these velocities directly by several
methods, the first of which was due to Sir Oliver Lodge.

He made a solution of common salt and added some phenolphthalein,1 which
he made slightly alkaline with sodium hydrate to bring out its red colour.
This solution was made into a semi-solid mass with agar-agar jelly and placed
in a horizontal glass-tube, which connected two vessels containing dilute
sulphuric acid, into which the electrodes were dipped. When a current passed,
the hydrogen ions travelled along the tube, forming hydrochloric acid, which
decolorised the phenolphthalein. Hence their velocity was found by measur-
ing the rate at which the decolorisation progressed along the tube. The results
obtained by this method, which has been greatly improved in detail by later
experimenters, afford a strong confirmation of the general correctness of the
theory.

Back E.M.F. in Electrolytes, — Exp. 198. Remove the cross
connections from a current reverser, R, and connect up (as shown in Fig. 250) a
battery of a few cells ; a galvanometer, G, and any form of voltameter, V,
having platinum electrodes in dilute sulphuric acid. It will be seen that in
one position of the rocker the battery is connected to the voltameter, and in
the other the voltameter is connected to the galvanometer. After passing
the current for a few minutes, throw over the rocker, and notice that the
galvanometer is deflected, and that the deflection rapidly decreases. Find
the direction of the current through the galvanometer, and notice that the
voltameter is giving out a current in the opposite direction to that previously
passed through it.

1 The indicator pheuolphthalein becomes red in the presence of an alkali,
but becomes colourless in presence of an acid.

ELECTROLYSIS 331

Exp. 199. Repeat the experiment, using copper electrodes of lair size in a
saturated solution of copper sulphate. No effect of the kind will be noticed.
Replace the copper anode by platinum, and repeat, and notice
that there will be a deflection as in Experiment 198.

Exp. 200. Repeat the experiment, using two lead plates in
dilute sulphuric acid and replacing the galvanometer by an
electric bell. On passing the current through the voltameter,
gas will be evolved at each plate, and after a short time the
anode will be found to be covered with a dark- brown coating of
lead peroxide (Pb02). "When the rocker is reversed, the bell will
ring for a few minutes.

These experiments show that the passage of a cur-
rent through a voltameter is, in certain cases, able to
make it act temporarily as a voltaic cell, the anode of
the voltameter becoming equivalent to the copper or
the carbon of a cell. That is, there must be a P.D.
produced by the passage of a current, which persists
after that current has ceased, and which is in a direc-
tion opposite to that of the applied E.M.F. Hence, FIG. 250
the equation of a current through an electrolyte must

be of the form C = , where r is the true ohmic resistance and e

r

is a back electromotive force.

This is another instance of the fundamental law that a current
can do work only by flowing against a back E.M.F. In the case we
are discussing, the work done is the breaking up of a chemical com-
pound, and the back E.M.F. is a definite quantity depending on the
nature of that compound. In fact, we can say generally that, when
chemical action occurs in a circuit due to the passage of a current (i.e.
excluding " local actions ") the formation of a compound means the
production of a certain E.M.F. in the direction of the current flowing,
and the breaking up of a compound means the production of a lack
E.M.F. of the same value.

Consider Experiment 199, in which a solution of copper sulphate
was electrolysed with copper electrodes. Copper is going into solution
at the anode with the formation of CuSO4, and producing a forward
E.M.F., and going out of solution at the cathode and producing an
equal back E.M.F. Hence there is an exact compensation, which
explains the absence of any effect in that experiment. When, how-
ever, we used a platinum anode, this balance is upset, and a back
E.M.F. was found to exist. We may mention that this will be the
case to some extent even with a copper anode, if the current density
is so great that the SO4 ion acts on the water as well as on the copper
and evolves oxygen.

Exp. 201. Try to electrolyse dilute sulphuric acid, using platinum electrodes,
with one Daniell's cell. Put a suitable ammeter or galvanometer in circuit. A
current will be indicated at first, but it will rapidly decrease and become practi-
cally zero.

332 VOLTAIC ELECTRICITY

Add a second Darnell's cell in series with the first. Observe that there is
now a current and evolution of gas, but that the action is slow.

Add a third cell, and notice a marked increase in current strength, and
consequently a much more 7~apid evolution of gas.

These results depend upon the fact that the back E.M.F. is, in this case,
about 1'48 volts. This is greater than the E.M.F. of a Daniell's cell (about 11
volt), and hence the action must stop as soon as the back E M.F. lises to an
equality with that of the cell. With two cells in series, the current is given by

2'2 — 1'48 "72

C= = — , where r is the resistance of the voltameter.

r r

q.q _ 1 .Aii 1 .09

With three cells, C = * L™ = L^, which is about 2£ times as large as
with two cells.

Measurement of Resistance in Electrolytes.— It follows

that the apparent resistance of an electrolyte is greater than its true
value, and that the ordinary methods of measurements do not apply.
Hence, special methods, in which the back E.M.F. is eliminated,

must be used in order to determine the specific conductivities re-
jr

quired to evaluate — .

The method most generally used is due to Kohlrausch. It
depends upon the fact that, when alternating currents of a sufficiently
high frequency are passed through a solution, the chemical actions
at the electrodes are reversed so rapidly as to be practically negligible.
Increasing the surface area of the electrodes is a further advantage,
which is secured by platinising the platinum electrodes, i.e. deposit-
ing a rough coating of platinum upon them. Kohlrausch employed
ordinary Wheatstone bridge methods, using alternating currents
obtained from a small induction coil, and substituting a telephone for
the galvanometer — the position of balance being determined by
adjusting the resistances until the telephone is silent. An induction
coil of suitable design is better than alternating currents derived from
supply mains, as the abrupt changes of current give more distinct
sounds in the telephone. Using alternating currents of frequency 50,
it is almost impossible to obtain an exact balance, for the point of
contact can be varied considerably without perceptibly affecting the
telephone. With a frequency of 100, better results can be obtained,
but even they are not really satisfactory.

A telephone is less convenient than a galvanometer (1) in re-
quiring a fairly quiet room, and (2) in not indicating (as the latter
does by the direction of its deflection) which way a resistance is to be
altered to obtain a balance. These difficulties are avoided by using a
dynamometer of suitable form, although such instruments cannot
easily be given sufficient sensitiveness (see p. 562). At present a
dynamometer may be regarded as a kind of galvanometer, which will
work with alternating currents.

Another excellent, but more elaborate, plan, is to use the ordinary

ELECTROLYSIS 333

bridge method with a galvanometer to detect the position of balance,
and to employ a double form of rotating contact-breaker to reverse
rapidly the current supplied to the bridge, the galvanometer con-
nections being simultaneously reversed, so that the current is always
in the same direction through the galvanometer. This involves
the use of a motor of some kind to drive the contact-breaker.

Calculation of E.M.F. — We have been gradually led to the
conclusion that any chemical action corresponds to a definite E.M.F. ;
for instance, the formation of zinc sulphate under circuit conditions,
sets up an E.M.F. which is exactly the same in value as the back
E.M.F. produced when that compound is electrolysed.

In practice more than one reaction is usually going on simul-
taneously, and the observed value is a resultant.

Suppose that it is required to find the E.M.F. represented by the
formation of a given compound, say, ZnSO4. Let a solution of this
salt be electrolysed by a current of strength i flowing for t seconds
against a back E.M.F. e, and depositing W grams of the metal.

Then the work done against the back E.M.F. is e.i.t ergs, which
must be the potential energy of the liberated metal with respect to
that compound. Let H heat units be produced when 1 gram of the
metal re-forms the combination, then W grams will give WH heat
units, and, as 1 heat unit = 41 '8 x 10° ergs, the potential energy of the
metal is WH x 41-8 x 10° ergs.

/. e.i.t =WHx41-8x 106

but if z = absolute electro-chemical equivalent of the substance

w=».fe

.: e.i.t = i.tz'Kx 4:1-8 x 106

whence e = 2xHx41-8x!06 absolute units.

Now, as H is known for many reactions, e can be calculated.

The above expression can be simplified as follows : We know
that z for any substance is equal to z for hydrogen multiplied by the
chemical equivalent of that substance.
/. if zh = value for hydrogen,

atomic weight TT , -, 0 1A« , , , ..
e = zhx — - x H x 41-8 x 106 absolute unit,

valency

Now 1 coulomb liberates '0000104 gram of hydrogen, but, in our
equation, we have used absolute units, and the absolute unit of
quantity is 10 coulombs.

Hence zh = "000104 grams.
Again, we can divide by 108 to express the result in volts ;

. x -000104 x H x chemical equivalent x 41-8 x 106
t.e. e (volts) = " 108

/ ,, x H x chemical equivalent
whence e (volts) = _ — _TL_

334 VOLTAIC ELECTRICITY

The numerator is the heat given out by the equivalent weight in
grams, i.e. by 1 gram- equivalent of the substance, when forming the
given compound.

.-. we have generally, e (volts) = Heat produced bg 1 gram-gquivalenl

2o,000

Let us now apply this to a DanielPs cell, in which there are two
distinct reactions going on : (1) the formation of ZnSO4, producing a
forward E.M.F., and (2) the decomposition of CuSO4, producing a
back E.M.F. From chemical data, we find that 1 gram of zinc gives
out 1670 heat units in forming ZnSO4, and 1 gram of copper gives
out 909*5 heat units in forming CuSO4 ; we also know that the

£t t*

equivalent weight of zinc (in round figures) is — - = 32'5, and that of

2

for the first reaction, e = 167^* 32'5 = 2'36 volts

'

copper —

and for the second reaction, e = — — - - = 1*25 volts

23,000

V E.M.F. of a Daniell's cell - 2-37 -1-25 = M volts.

The argument is, however, incomplete, for it does not take into
account the fact that the E.M.F. of a cell depends, to a certain
extent, upon its temperature. The result just given happens to be
correct, because the temperature coefficient of a Daniell's cell is so
small as to be negligible. To deduce the necessary correction requires
an application of the second law of thermo-dynamics, which must be
omitted here. What it means is, that a cell at work may either
absorb heat energy from its surroundings, or give out such energy
to them, and this effect must be considered in order to obtain the
correct E.M.F. It is related to the phenomena discussed in Chapter
XXVIII. on thermo-electric currents.

In connection with voltaic cells, it may be pointed out that the
number of coulombs obtained by the consumption of a given amount
of active material is independent of the nature of the reaction, where-
as the E.M.F. obtained, and, therefore, the output of energy, does
depend upon the nature of the reaction. For instance, the electro-
chemical equivalent of zinc is '000338 gram per coulomb, which
means that, if no further loss occurs through local action, the con-
sumption of '000338 gram of zinc will give out 1 coulomb, i.e.
1 gram of zinc will give out 2967 coulombs ; and this is true whether
the zinc is forming ZnSO4 (as in a Daniell's cell), or ZnCl2 (as in a
Leclanche cell. But the energy value of the reaction, and therefore
the E.M.F., may be widely different, e.g. 1 gram of zinc in forming

ELECTROLYSIS

335

ZnCl2 gives out 1559 heat units as against 1670 for ZnS04. In this
example the numbers do not differ much, but if (for the sake of
illustration) we imagine a cell in which zinc iodide is formed instead
.of zinc chloride, then, although the number of coulombs would be
unaltered, the heat units evolved per gram of zinc would be 821, and

the E.M.F, could not exceed 821QXA^'5 = M6 volts.

2o,000

Measurement of Current by Chemical Action.— One of

the most exact methods of measuring the strength of a current
depends upon the use of the equation W = U.t.z, but obviously an
exact knowledge of the value of z for at least one substance is re-
quired, and it is equally obvious that the selected substance should
be one whose liberation is least affected by secondary reactions.

For very accurate work, a solution of pure silver nitrate is electro-
lysed in a special form of voltameter (which often consists of a
platinum bowl as the cathode and a silver plate as the anode), the
electro-chemical equivalent of silver having been determined, with the
utmost care, by Lord Rayleigh and others.

For ordinary purposes a copper voltameter may be used. A very
convenient form for the use of a student is shown in section in Fig.
251. The anode is a U-shaped sheet of fairly stout copper, 2J to 3
inches wide, the sides being about 5
inches long and 1J inches apart. A
portion of one side is bent as shown
to carry a terminal. The anode is
screwed to a strip of wood (about 4J
in. x 1 in. x 1 in.), which acts as a
support in the containing vessel and
which carries the cathode through a
3 in. slot cut in it. The cathode is a
strip of very thin copper (4^ in. x 2J
in.) to which is fastened a screw ter-
minal. This terminal not only serves to
take the connecting wire, but also prevents the cathode from slipping
through the slot. The arrangement is placed in any convenient vessel
to contain the copper sulphate solution ; although it is preferable to
use a glass vessel, because the position of the cathode can then
be seen.

Exp. 202, to find the constant of a tangent galvanometer ly means of a
copper voltameter. Arrange the apparatus as shown in Fig. 252. The tangent
galvanometer, T, is provided with a reversing key, R, and is in series with a
Daniell's cell, b, the voltameter, V, and a plug key, K. The cell should be set
up a little time in advance, and it may with advantage be short-circuited for
a minute or two before being used.

The first step is to adjust the resistance until a convenient deflection is
obtained. For this purpose a dummy cathode of the same size as the real
cathode should be employed, and a short length of 30-gauge platinoid wire

-Removable terminal,
Wood,.

Thick copper anode.
Thin, capper cathode.

x^y

Glass vessel.

pIGi 251.

336 VOLTAIC ELECTRICITY

intercalated in the circuit at P. Alter the length of this wire until a
deflection of about 45° is obtained. Then substitute the real cathode,
previously cleaned and weighed (taking care that it is connected up the right

way, and also that it does not touch the
anode beneath the surface of the liquid), and
at a noted instant insert the plug key. The
current must be allowed to flow format least
20 or 30 minutes, and during the time its
strength ought to remain constant, but in
practice it is almost certain either to increase,
or to decrease slowly. In this case, the value
deduced from the amount of copper deposited
is evidently the avcraye value during that
time, and it is necessary to ascertain the cor-
responding value of the deflection. Again,
any inaccuracy in the initial adjustment of
the pointer to zero will make the deflections
vary with the direction of the current. Both
sources of error can be eliminated by reversing
the current and noting the deflection at definite
periods, say every three minutes, thus getting
FIG. 252. ten readings of the deflection in the course of

30 minutes.

Let 6 be the average value of these deflections. When time is up, break the
circuit at a definite instant, carefully remove the cathode, wash it with distilled
water, dry — -partly with clean blotting-paper, and finally in a current of warm
air (remembering that the deposited copper will readily oxidise if unduly heated)
— and then weigh it.

Let W be the increase in weight in grams during time t seconds,

then W = C. t.z, where z = '0003294

W

or C = — amperes

also C = K tan 6

• K- W

t.z. tan 8

Notes on Method. — A tangent galvanometer is usually pro-
vided with several coils, and with some of them it is probable that
a current, which will produce a suitable deflection, will be too small
to deposit a weighable amount of copper in a reasonable time. In
such a case, the resistance of the coil may be measured, and it can
then be shunted with a suitable resistance. The total current in the
circuit is obtained as above, and the current in the galvanometer is

- of. that value.

In order to obtain a satisfactory deposit, at least 30 square centi-
metres of cathode surface should be allowed per ampere ; if the current
density is greater than this, some copper may fall off and be lost.
The solution should be • saturated, and may with advantage contain
about 1 per cent, (by volume) of strong sulphuric acid.

Accumulators or Secondary Batteries.— The principle
involved in Experiments 198-200 was first employed by Hitter, in

ELECTROLYSIS

337

1803, in the construction of secondary batteries. He used large
plates of platinum, having pieces of moistened cloth between each
pair. Each end of the pile was then connected with the poles of a
battery. After receiving a charge it was separated from the battery,
when it was found to be capable of producing all the effects of an
ordinary voltaic battery.

Later, Sir William Groves constructed his gas battery (Fig. 253).

FIG. 253.

A cell, M, consists of two glass tubes, each containing a platinum
plate, to which platinum wires are attached, and then connected out-
side with binding-screws, A, B. The tubes were filled with dilute
sulphuric acid and inverted over a similar solution. On passing the
current, oxygen and hydrogen are liberated and collected. When the
charging battery is removed, and AB joined by a wire, the arrange-
ment is found to give out a current in the opposite direction to the
charging current, the liquid meanwhile slowly rising in the tubes

as the gas disappears.
The E.M.F. of such a
cell is about 1*4 volts.
It is best to platinise
the platinum plates, i.e.
coat them with finely
divided metallic plati-
num in order to increase
the effective surface.

Plante,inl860, con-
FIG. 254. structed a secondary cell

by using two sheets of

lead, each provided with a tongue (Fig. 254). The sheets were
then rolled up — narrow strips of felt being put between them to

Y

338 VOLTAIC ELECTRICITY

prevent contact — and then immersed in dilute sulphuric acid.
The terminals of a battery were attached to the two tongues, so
that a current passed through the cell. By this means the liquid is
decomposed, oxygen being liberated at the anode, which combining
with the lead forms peroxide of lead (PbO2), while hydrogen is
liberated at the cathode. The current is then reversed until the PbO.,
is reduced to spongy lead by the action of the hydrogen, while the
other plate (the first cathode, now the anode) is in its turn oxidised.
Thus by sending repeated currents in alternate directions, the plate
which served last as the anode is left deeply coated with PbO2,
while that which served last as the cathode is deeply coated with
spongy lead. Plante's cells were vastly more efficient and practical
than any previously produced, and he must be regarded as the real
inventor of the modern accumulator.

In order to obviate the lengthy process of the " formation of the
cells," Faure, in 1881, improved the construction by coating the two
plates with minium or red lead (Pb3O4). When the current is passed
through the cell to charge it, the red lead at the anode is oxidised to
PbO9, while at the cathode it is reduced by the hydrogen — first to
PbO, and then to the spongy metallic state.

Plates formed by Plante's method are superior to Faure's pasted
plates in durability, because the peroxide is more firmly attached \
on the other hand, a greater capacity for a given weight is obtained
more readily with pasted plates. It is now a common practice to
make the positive plates by some modification of Plante's method
and to use pasted plates for the negative. Various devices are em-
ployed to keep the paste from falling off, one of the most recent
being the " box " type of negative, in which the plate is made of two
perforated grids, fitting together to form a narrow box, which is
filled with the oxide (litharge). As the grids are merely supports
for the active material, they are usually made of an alloy of antimony
and lead, which is stiffer and stronger than pure lead.

A modern accumulator consists of a number of positive and
negative plates sandwiched together (precautions being taken to
avoid contact), all the positives being connected together and simi-
larly all the negatives. The outside plates are always negative, and
hence there is always one more negative than positive. The electro-
lyte is dilute sulphuric acid.

In consequence of the large area of surface and the closeness of
the plates, the internal resistance is very low, and as the effective
E.M.F. is about 2 volts, the possible current on short circuit is very
great ; in fact great enough to seriously injure the cell by causing
the plates to bend and the active material to drop off.

It should be borne in mind that there is no essential difference
between a primary cell and a secondary cell. In the former, it is
more convenient to renew the active materials themselves when the

ELECTROLYSIS 339

cell is exhausted, whereas the latter is designed to permit of the
materials being brought back to their original state by electrolysis.
For this purpose, it is essential that no product formed during
normal work (or discharge} should be lost. A little consideration
will show, for instance, that an exhausted Daniell's cell might be
restored to its original state by sending a reverse current through
it, but this could not be done with a Leclanch4 cell on account of
the ammonia lost as gas.

A lead accumulator is essentially a single -fluid voltaic cell in
which the zinc is replaced by a lead plate of relatively enormous
surface area (spongy lead) and the copper by lead peroxide. It is
free from "local action" and "polarisation."

The acid produces hydrogen ions and "sulphion" (SO4)ions, and
during discharge (1) the sulphion gives up its negative charge to
the lead plate, and forms PbSO4, which, being insoluble, remains on
the plate as a white film ; (2) the hydrogen has its positive charge
neutralised at the peroxide plate, and combines with the oxygen to
form water ; thus

(1) Pb-fS04 =

(2) Pb02 + H2

The PbO thus formed is acted on by the acid, and also forms
PbSO4.

PbO + H2SO4 = PbSO4 + OH2

Thus, after discharge both plates are coated with a thin layer of
insoluble lead sulphate.

To recharge, a current in the opposite direction is passed through
it from an external source. The peroxide plate now becomes the
anode, and the lead plate becomes the cathode. When the acid is
electrolysed (1) the hydrogen at the cathode reduces the PbSO4
to Pb, thus

(2) the oxygen at the anode acts on the PbSO4 and water, re-forming
the peroxide

PbS04 + O + OH2 = Pb02 + H2S04

The equations given above are neither exact nor complete. They
are given merely as useful aids by which the nature of the principal
reactions may be indicated.

From what has been said previously, it will be evident that the
voltage required to send a given charging current through a battery
of accumulators is very much greater than the value calculated from
the internal resistance. For instance, suppose that the battery con-
sists of ten cells, each having an internal resistance of ^ ohm, and
that the charging current is to be 12 amperes. The total internal

340 VOLTAIC ELECTRICITY

resistance is i ohm, and to send 12 amperes through -J- ohm requires
12x^:=2'4 volts. But each cell, during chart /in<j, will develop a
back E.M.F. of at least 2'5 volts, or 25 volts in all, and therefore
the charging E.M.F. must be at least 27 -4 volts. That is, the equa-
tion of current must be written, as already explained, in the form

where e is a back EJVI.F. and r is the ohmic resistance.

It will be seen later (see p. 372) that an equation of this form
indicates that work of some kind is being done by the current, and
it will be found to admit of very simple interpretation — EC being
the power in watts supplied to the accumulator, eC the power ex-
pended in doing work (in this case chemical action), and the differ-
ence being the power wasted as C'2r heat on account of internal
resistance.

The Edison Accumulator. — This is the only practical
accumulator which does not involve the use of lead. Although
little known at present in this country, it seems likely to become of
great commercial importance in the near future.

The positive plate is a grid of steel, supporting closely packed
vertical rows of perforated steel tubes, made by rolling steel strip
into a long spiral. These are filled with thin alternate layers of
metallic nickel (in the form of excessively thin flakes) and of nickel
hydrate. The negative grids are also of steel and carry rectangular
pockets filled with an oxide of iron. The electrolyte is a 21 %
solution of potassium hydrate. The containing vessel is also of steel,
and all the steel parts, grids, &c., are nickel-plated.

The chemical reactions are said to be as follows : Beginning with
iron oxide on the negatives, green nickel hydrate on the positives,
and potassium hydrate in solution, the first charging of the cell
reduces the iron oxide to metallic iron, and converts the nickel
hydrate to a higher oxide, black in colour. During discharge, the
iron becomes oxide again, and the nickel is reduced to a lower oxide,
but does not again return to its original form of green hydrate. The
operation of charging, therefore, amounts to a transference of oxygen
from the iron to the nickel, and that of discharge to a transference
back again. When fully charged, the plates are no longer acted
upon, and hydrogen and oxygen gases are given off as in ordinary
cells. The amount of potassium hydrate in solution never changes,
and hence the density remains constant. The nickel flakes in the
positives and a small amount of mercury in the negatives do not
take part in the reactions, and are used merely to ensure good
electrical contact between the active material and its supports.

The cell is remarkable for its mechanical strength and perfection
of design. The absence of a corrosive acid is a distinct advantage,

ELECTROLYSIS 341

but even more important is its freedom from the most troublesome
defects of lead cells — the plates do not "buckle" or "sulphate,"
neither do they deteriorate if the cell is left uncharged.

Again, the cell is remarkably light for a given capacity in ampere-
hours, and only requires filling up occasionally with distilled water.

Its only disadvantage appears to be its low E.M.F., which is
about 1*3 volts per cell.

EXERCISE XVII

1. How many amperes would deposit 2 grams of copper in 15 minutes, the
current being supposed constant ?

2. How many grains of copper would be deposited by a constant current of
12 amperes acting for 1 hour ?

3. What would be the strength of a constant current which would deposit
36'36 grams of copper in 5 hours t

4. What would be the strength of a constant current which liberates 50
cubic centimetres of hydrogen in 5 minutes ?

5. How many amperes would liberate 250 cubic centimetres of hydrogen in
15 min. 32 sec., the current beim* constant?

6. What is the result of passing a current through a solution of sulphate of
sodium (Na2S04), by means of platinum electrodes separated from one another
by a porous partition ?

7. Explain the term electro-chemical equivalent. If 3 amperes deposit 4
grams of silver in 20 minutes, what is the electro-chemical equivalent of silver ?

(B. of E., 1899.)

8. If the electro-chemical equivalent of silver is '01118, what is the electro-
chemical equivalent of oxygen ? (B. of E., 1901.)

9. State Faraday's laws of electrolysis. A current of 1 ampere is passed for
two hours through an electrolyte and decomposes 2'4 grams. Find the electro-
chemical equivalent of the electrolyte. (B. of E., 1905.)

10. A current is passed through a voltameter and through a coil of wire in
series with it. If the current is altered in such a way that the heat produced
in the coil is doubled, show what change will be produced in the rate at which
chemical action takes place in the voltameter. (B. of E., 1903.)

11. Distinguish between the chemical and electro-chemical equivalents of
an element. What weight of hydrogen is separated from water by the passage
of 1000 coulombs of electricity, given that the chemical equivalent of copper
is 31 '5, and its electro-chemical equivalent 0 '000328 per coulomb ?

(B. of E., 1902.)

12. It is stated that, in order to separate 1 gram of hydrogen from acidu-
lated water by electrolysis, 96,500 coulombs oi^ electricity must pass: how
would you proceed to verify the statement ? (B. of E., 1907.)

13. Six Grove's cells, connected (a) in a single series, and (b) in two series
of three each, are used to decompose water in a voltameter. If there is no local
action in the battery, show how much zinc is dissolved in each case (a) and (b)
in the whole battery while one grain of hydrogen is evolved in the voltameter.

Take sulphate of hydrogen (sulphuric acid) as H2S04 (98) and sulphate of
zinc as ZnS04 (161).

14. A battery of eight cells, connected in series, is used to decompose water
in a voltameter ; the chemical equivalent of zinc being 32'5 times that of
hydrogen, show how much zinc is consumed in the whole battery while one
grain of hydrogen is liberated in the voltameter.

15. A current is passed through a coil of wire and then through a galvano-
meter arranged in series with it. If the strength of the current is altered so

342 VOLTAIC ELECTRICITY

that the heat produced per minute in the coil of wire is doubled, show what
change will be produced in the rate at which chemical action takes place in a
voltameter.

16. After acidulated water is electrolysed, find the electromotive force of
hydrogen tending to recombine with oxygen (34,000 units of heat are produced
by the combination of 1 gram of hydrogen with oxygen).

17. Find the E.M.F. of zinc dissolving in sulphuric acid, where z ='00338,
H = 1670.

18. An electromotive force of 3 volts is required to force a current of
1 ampere through a voltameter containing acidulated water. If the work re-
quired to separate one gram of hydrogen is 142,000 watt-seconds, and the
electro-chemical equivalent of hydrogen is '00001035, find the resistance of the
voltameter. (B. of E., 1904.)

19. Given that the electro-chemical equivalent of copper is '00033, calculate
what must be the output in watts of a dynamo to deposit 10 kilograms of copper
per hour, the voltage of the dynamo being 10 volts ?

20. What do you understand by the polarisation of the platinum plates
employed as electrodes in a voltameter ? How would you show experimentally
in which direction the polarisation acts ?

21. A current flows through two troughs, which are connected in multiple
arc. and contains a solution of copper sulphate. If all the circumstances of the
two paths which are thus open to the current are the same, except that the
metal plates by which it enters and leaves the liquid are, in the one case copper,
and in the other platinum, will the currents be equally strong in the two
troughs ? Give reasons for your answer.

22. A number of cells formed of plates of zinc and platinum, immersed in
dilute sulphuric acid, are to be connected in a circuit, so that the platinum of each
cell is in contact with the zinc of the next. What effect, if any, would be pro-
duced on the current if, by mistake, one cell was made up with two platinums,
instead of with one platinum and one zinc plate ?

23. A single Grove's cell is joined up in circuit with a voltameter, in which
acidulated water is decomposed between platinum electrodes, and the strength
of the current is noted. On connecting a second Grove's cell in series with the
first, the strength of the current becomes considerably more than twice as great
as at first. Explain this.

24. A current of 20 amperes passes through a resistance of 2 ohms and is
also sent through a battery of 10 accumulator cells in series to charge them.
If each accumulator cell has a counter electromotive force of 2 '4 volts and an
internal resistance of T£7 ohm, how many volts must be applied to furnish this
charging current? (Lond. Univ. Matric., 1899.)

25. Describe a method of investigating the relation between the strength
of an electric current and the rate of chemical change produced by it.

(B. of E., 1897.)

26. Is the strength of the current that passes through a simple circuit the
same at all points of the circuit, however its parts differ in resistance ? How
would you "justify your answer by experiment ? (B. of E., 1894.)

27. Two liquid resistances, A and B, of 5 and 10 ohms respectively, are
connected in parallel, and a battery of electromotive force of 8 volts and 2 ohms
internal resistance is used to send a current through them. Find the currents
in the two liquids, being given that the electromotive force of polarisation is
•1 volt in A and 1'8 volts in B. (Lond. Univ. B.Sc., 1908.)

28. Describe the construction and method of charging a secondary battery.

CHAPTER XXII

ELECTRO-MAGNETISM
Mutual Actions between Magnetic Fields and Currents.--

We have already learnt that a current always produces a magnetic
field in the space around it. Hence, if a current is made to flow in
another magnetic field of independent origin, the mutual action of the
two sets of lines of force will, as a rule (but not in all cases), set up
mechanical forces which react upon the conductor carrying the current.
In the case of a straight conductor, the nature of the effect will
be easily understood if we draw a simple diagram and then apply the
laws already established ; e.g. in Fig. 255, the con-
ductor, seen in section, is at right angles to the
independent magnetic field, which is not neces-
sarily produced between the poles of a magnet,
although for the sake of clearness such poles are
shown. For convenience also, only a few lines of
force are drawn, and these, according to the general
rule, are marked in the direction in which a free
N pole would tend to move.

We now see that, at A, the two sets of lines are

in the same direction, and therefore they repel one another, whereas,
at B, they are in opposite directions, and so tend to neutralise each
other. The result is a force, which tends to drive either the conductor
in the direction shown by the arrow or the field in
the opposite direction.

It must be clearly kept in mind (1) that such
forces always act between the invisible lines of the field,
although we can perceive the effect only on the bodies
with which the fields are associated ; (2) that, in reality,
we should not obtain the distribution of lines shown in
the diagram, the two fields really producing a resultant
FIG. 256. field in which no two lines intersect (Fig. 256). But
in order to understand the actions taking place, it is
better to draw the two components superposed, rather than the
resultant field itself.

The diagram also informs us that reversing either the direction of
the current or the polarity of the magnet would reverse the direction
of the force, whereas reversing loth would leave the latter unaltered.

343

344

VOLTAIC ELECTRICITY

Many experimental illustrations are possible, and as the principle
is important — all practical electric motors depending upon it — we
shall investigate the matter rather fully.

Exp. 203. Bend a wire, AB, as shown in Fig. 257. Flatten one end, A, and
drill a bole through it. Fix AB so that it is in connection with the binding-
screw, T. Pass a wire, C, through the hole, keeping it in position by making
a small loop at the top, and of sufficient length to just touch some mercury con-
tained in the hollow, D, made in the
base. The mercury is connected to the
binding-screw T' by means of a wire
passing underneath the base (indicated
by the dotted line). Attach a wire
from the battery to one end of the coil
of an electromagnet, E, the other end
being connected to the binding-screw
T'. Fasten the other wire from the
battery to T, and observe that the wire
moves out of the mercury across the
lines of force of the magnet, and that,
at the moment contact is broken, it
falls back again. These motions are
repeated while the current lasts. If
FlG. 257. the direction of the current is altered,

the direction of motion of the wire will

remain the same as before, because the current is reversed in both wire and magnet.'
(This experiment also works satisfactorily if a fairly strong steel horse-shoe
magnet be used instead of an electro-
magnet.) O-

The student will find that the
direction of motion can be deduced
by applying the method indicated in
Fig. 255.

A similar experiment can be
performed in an interesting
manner as follows : —

Exp. 204. Suspend a straight
brass or copper wire so that the lower
end swings freely between the poles
of the electromagnet shown in Fig.
258 (a). (To do this it is only neces-
sary to make a loop of copper wire and
hang a straight piece loosely on it.)
Join it up to a battery of one or
two cells through a current reverser,
making connection to the lower end
by means of a strip of flexible tinsel.
Excite the electromagnet from some
independent source (not shown in the
diagram) and notice that, when a
current is sent through the vertical
wire, it is driven sideways out of the
field. Reverse the current in it, and

(a)

FIG. 258.

notice that the direction of motion is also reversed. Determine the polarity of
the electromagnet and the direction of the current, draw a diagram to represent

ELECTRO-MAGNETISM

345

the facts (as in 6), and observe that the direction of motion agrees with that
deduced from the diagram. (It will be seen that (6) indicates the state of affairs
when (a] is looked at from above.)

AVhen alternating currents are used for lighting purposes, a very simple
demonstration of the existence of the force in question may be given by holding
an ordinary bar magnet (or horse-shoe magnet) near an incandescent lamp pro-
vided with a carbon filament. The force on the filament is quite small — indeed,
if it were carrying a continuous current, the displacement would probably
be imperceptible — but as it is reversed many times per second, a vibration
occurs, which may be so violent as to break the filament unless the magnet is
removed.

These experiments can easily be modified to produce a continuous
rotation. One of the oldest devices for this purpose is known as
Barlow's wheel, which usually consists of a brass or copper wheel

As it rotates, the points just

FIG. 259.

cut into the shape of a star (Fig. 259).
dip into a mercury cup.

Exp. 205. Connect the binding-screws, B and C, with a battery. The current
then passes from one binding-screw, B, up the support to the axis A, thence
down a vertical radius of the wheel to the mercury, and so back to C and the
battery. When the wheel is placed between poles of a strong magnet, as shown
in the diagram, observe that it begins to rotate, and, one point of the wheel after
another passing out of the mercury, that the rotation is kept up while the current
lasts. As in the last experiment, the direction of rotation can be reversed, by
either changing the direction of the current or the polarity of the magnet.

In this experiment a steel magnet may be used, and again the student should
deduce the direction of motion by aid of Fig. 255.

Another method of obtaining rotation — due to Faraday — is shown
in Fig. 260. If desired, a steel magnet may be used instead of the
electromagnet therein indicated.

Exp. 206. Fit two corks, AB (Fig. 260), tightly into a lamp chimney.
Remove the cork B, and bore a hole to admit a round piece of soft iron. Pass
the iron a short distance through the cork, and then coil insulated copper wire
round the outer part, so as to form an electromagnet. Now fix the cork in its
place, so that one end of the coil passes between the cork and the glass and
the other end is free. Pour mercury in the tube, so that the end of

projects slightly above the surface.

the iron
Through the cork A pass a copper wire,

346

VOLTAIC ELECTRICITY

making a loop at the end which is to be inside the tube. In this loop hang a
piece of copper wire, D, of such a length that its lower end just rests in the

mercury. Connect the wires from a chromic
acid battery to the free ends of the wires out-
side the apparatus, and observe that the wire
D revolves round the pole, C, of the magnet.

The result can be explained, and the
direction of rotation predicted, in exactly
the same way as before. Looking down
upon the apparatus, the conductor and
the magnet will be seen in section, and
some lines of force will spread laterally
from the upper pole towards the lower
pole. On a flat surface these must be
shown as a series of radial lines. Hence
we deduce Fig. 261. The upper end of
the magnet is assumed to be a N pole,
and the current is supposed to be flowing
doim the wire. Two positions, A and
B, of the wire are shown, and only one
line of force is drawn as a circle around
it to represent the field due to the
current. When the direction of the two
fields are marked according to the usual
rule, it becomes apparent that the two
fields are in. the same direction on one
side of the wire, and hence there is a
tangential force tending to drive the lower end of the conductor
sideways, the final result being a continuous rotation in a clockwise
direction.

As usual, reversing
either the polarity of
the magnet, or the
direction of the cur-
rent, will reverse the
direction of rotation.

Exp. 207. A simple
modification of the last
experiment can be made
by pivoting a wire, bent
twice at right angles, on
the pole of a magnet
(Fig. 262), and letting the FIG. 261. FIG. 262.

ends dip into an annular

trough containing mercury, fixed about the middle of the magnet, so that a
current can be sent through the magnet and into the wire, returning by wray of
the mercury trough, or vice versa.

The student should draw an explanatory diagram as in the

Fig. 260,

ELECTRO-MAGnSTETISM 347

last experiment, except that the two conductors must be shown 180°
apart. He will then see why the current must flow in the same
direction in both sides of the wire — for if it were to flow up one
and down the other, the conductors would tend to rotate in opposite
directions.

Similarly, a liquid conductor may be made to rotate.

Exp. 208. Fix a fairly powerful bar electromagnet vertically, e.g. one of the
two uprights of the horse-shoe electromagnet described on p. 249. On the top
of the magnet, support a flat-bottomed circular glass dish having a diameter of
5 or 6 inches. Bend a long strip of sheet copper into a circular hoop, which fits
just inside the dish. Make a conducting liquid by acidifying water with sulphuric
acid, and pour it into the dish.1 Support another strip of copper, so that it dips
into the liquid at the centre. Connect the two pieces of copper to a battery, so
that a current may be sent from the centre to the circumference of the liquid, or
vice versa. Notice that, when the magnet is excited, the liquid Avill begin to
rotate, as may be made evident by floating small pieces of wood or cork in it.

As before, reversing either the current or the polarity will reverse
the direction.

In this experiment, it is the approximately vertical V
lines of force that are operative, and the diagram I *
(Fig. 263) may be drawn by taking any radius of •—
the liquid to represent a conductor. Supposing
that the current flows from the centre outwards,
Ave obtain a figure which shows that the liquid,
as seen from above, is rotating in a clockwise

. , , , J IV*. ^ '-'•'.

direction.

Exp 209. Perhaps the most interesting variation of Experiment 207 is to make
a magnet rotate under the influence of its own field. For this purpose, prepare
a permanent magnet of round steel about f to \ inch
diameter, and 6 or 7 inches long, with both ends carefully
pointed, so that it may be pivoted to rotate freely about
its axis. Twist a copper wire round the middle of the
magnet, and bend it so that its end dips into an annular
mercury trough. The arrangement is shown in Fig. 264.
. , Notice that the magnet begins to rotate when a fairly

M i •! i |l|ljhx strong current is sent through one half of the magnet.
| | A better effect Avill be produced if the two ends are

connected in parallel ; the current then enters at the
middle and flows towards each end, or vice versa.

The results of such experiments may be ex-
FIG. 264. pressed in the following terms : —

If a conductor carries a current at right angles

to the lines of force in a magnetic field, a mechanical force is exerted
upon it which tends to move it laterally, i.e. in a direction which
is at nght angles both to the direction of the current and to that of
the field.

If the conductor is not at right angles to the field, but is inclined

1 Mercury or any sufficiently conducting liquid may be used.

348 VOLTAIC ELECTRICITY

to it at some angle 0 — as in Fig. 265, where AB represents the

portion of the conductor under consideration — then the force in it is

^ the same as if it were of length AC. This means

—fa- that we can write force =p sin 0, where p i« the

j\ value when 6 = 90°.

j— r- When the conductor is parallel to the field,

IL | 0 = o and the force vanishes.
// Q-^+- The subject is further dealt with on p. 372.

Q Q An exactly similar effect is produced when the

conductor is gaseous. For instance, the luminous
discharge in a vacuum tube may be made to rotate
around a magnet pole very much as the wire rotates in Experiment 206,
and the electric arc in air at ordinary pressure may be driven sideways
until it is blown out, by means of a magnetic field at right angles to
its direction. This action is usefully applied in the " controllers " used
for driving electric cars, in order to prevent arcs forming when the
circuit is broken. It is also applied in a modified form in the " flame "
arc lamps now largely used, as explained in Chapter XXXII.

Production of Electromotive Force by the Motion of
a Conductor in a Magnetic Field. — Let a straight conductor
be placed in a magnetic field at right angles to the lines of force.
We have just shown that, if a current be sent through it, a
mechanical force is produced which tends to move it sideways.
If, however, instead of sending a current through it, the conductor
itself, by some external means, is moved sideways so that it cuts
the lines of force, we find that, during such motion, an E.M.F. is
produced in it, i.e. a difference of potential is produced between its
ends. If a circuit is formed, say by joining the two ends of the
conductor by a wire outside the field, then a current flows. Such
currents are commonly known as induced currents, but it is important
to remember that an E.M.F. is produced whether there is a circuit or
not. This action was discovered by Faraday in 1831, and upon it is
based the construction of all dynamos for generating current on a
large scale, just as all electric motors depend on the mechanical force
produced when a current is sent through a conductor.

To demonstrate this fact, it is advisable to employ a reflecting
galvanometer of the suspended coil type (preferably fairly dead-beat),
as it is not affected by the magnets used during the experiment.

Exp. 210. Connect the terminals of such a galvanometer by means of a long
copper wire. Move a portion of the wire rapidly in, say, an upward direction,
between the poles of a steel horse-shoe magnet, and notice a very slight move-
ment or "throw " of the spot of light. Notice that, when the wire is moved in
the opposite direction, say downwards, the direction of the deflection is reversed.
Also notice that the deflection is increased with the speed of motion, and that a
slow movement produces no perceptible effect. After trying the experiment with
an ordinary horse-shoe magnet, the student should repeat these operations with
the electromagnet described on p. 249.

ELECTRO-MAGNETISM

349

FIG. 266.

In Fig. 266, AB represents the portion of the wire which is moved through
the field.

It is important to mention that the best effect will be obtained when AB
is moved at right angles to the iield, and that no
effect will be produced when it is laid across the
poles and moved upwards, i.e. when it is moved
without cutting the lines of force.

Exp. 211. Wind the wire into a few loose turns,
and slip them on and oil' one of the poles of a steel
horse-shoe magnet. Notice that larger deflections
are now easily obtained.

The student will readily understand that the
loose coil is equivalent to the joining together of a
number of conductors in series, an arrangement
which causes their separate E.M.F.'s to be added
together.

Exp. 212. After trying the effect of moving the coil on and off at A (Fig. 267),
move the coil from B to C, and notice that little or no
effect is produced. Hence, we infer that the action
depends upon the turns of the coil cutting through the
lines of force in such a way that the number of lines
threading the coil must either increase or decrease.

Exp. 213. Place the coil anywhere between B and C,
and try the effect of putting a soft iron keeper on the
magnet, and then suddenly pulling it off'. In this
position, no induced current will be obtained. Repeat
the experiment with the coil at A. The result will
be a deflection in one direction when the keeper is
applied, and another in the opposite direction when
it is removed.

Evidently this is due to the fact that more lines of force pass through A when
the keeper is in position than when it is absent ; in the latter case many of the
lines emerge from the magnet pole below A. But the flickering of the lines of
force is confined to the neighbourhood of the poles and does not affect a coil near
the bend. This experiment illustrates the action of telephones (described in
Chapter XXX.).

Similar results may, of course, be obtained by using a bar magnet.
Exp. 214. Connect a hollow coil of wire to the galvanometer and then insert
a bar magnet. (Try this gently at first, as, if the coil has many turns, the
deflection may be violent enough to injure the galvanometer.) Notice that a
"throw" is produced when the magnet pole enters the coil, and that another
occurs in the opposite direction when it is removed, whereas no effect whatever
is obtained by merely leaving the magnet in the coil. If the other end of the
magnet is inserted, similar actions are produced, but the directions of the throws
are reversed.

Exp. 215. Reinove one of the coils of the large electromagnet (p. 249) and
connect it to the galvanometer, using wires long enough to permit of free move-
ment without jerking the instrument. Hold the coil in such a position that as
many lines as possible of the earth's field pass through it. This will occur when
the axis of the coil is in the line of dip. Turn the coil suddenly through 90°, i.e.
until it reaches a position where no lines of force pass through it. Notice that a
small throw will be produced due to the cutting of the earth's field. Contimie
the process, turning it through 90° at each step, until the coil has made a
complete revolution, and notice the directions of the throws obtained. It will
be found that they are in the same direction during one half revolution, and in
the opposite direction during the second half., This is due to the fact that
during the first quarter revolution, the lines threading the coil are decreasing in

FIG. 267.

350 VOLTAIC ELECTRICITY

number, and during the next quarter are increasing again. *\Ye might, therefore,
expect a reverse throw (and this would be the case if we turned the coil back
again), but, when motion is continued for the secoild quarter, the lines as they
increase in number are threading the coil in the opposite direction, which again
reverses the E.M.F., leaving the direction of throw as at tirst. Thus the rotation
of the coil produces an alternating E-M.F., which will be more fully understood
after performing Experiment 217.

The existence of an induced current may be demonstrated without
using a galvanometer by a simple method due to Faraday.

Exp. 216. From the electromagnet, remove one coil and its iron core, so that
the remaining coil and core form a vertical bar electromagnet. Excite this as
strongly as possible by means of a current. Attach two loose bare copper wires
to the terminals of the other coil, and bend them until their free ends just touch
crossways to complete the circuit. Hold this coil, with its iron core inside,
above the bar electromagnet, and bring it suddenly down upon the latter. An
induced current will flow during the process, and the shock of contact of the two
iron cores will jerk the loose wires apart at the right moment, a distinctly visible
spark being produced at the gap.

(It should be noticed that all the experiments given to demon-
strate the existence of a force on a conductor carrying a current in
a magnetic field are necessarily reversible, i.e. if, instead of sending a
current through it, the moving part is moved by some external means,
then an induced E.M.F. is produced.)

Magnitude Of Induced E.M.F. — Faraday established, by
experiment, the laws underlying the production of induced currents,
and we may conveniently summarise the facts in the following way : —

When a conductor is moved so that it cuts the lines of force of a
magnetic field, an E.M.F. is induced in it, ivhich depends only upon
the rate at which the lines are cut, i.e. upon the number of lines cut
per second.

Hence, steady motion in a uniform field will induce a steady
E.M.F. during that motion ; if, however, the field is not uniform,
the E.M.F. will vary from moment to moment according to the rate
of cutting at any particular instant.

It is important to notice that the magnitude of an induced E.M.F.
is absolutely independent of (1) the material of the conductor, (2) its
shape, (3) its size, or even (4) the existence of a circuit. If a circuit
does exist, a current will flow therein, the magnitude of which
depends, of course, upon the resistance of the circuit, and therefore
upon the nature, size, and shape of the conductor.

The conception of an induced E.M.F. is thus simpler than that
of an induced current, and should precede it.

From what has been said, the student will see that we may have
an induced E.M.F. without an induced current, but that we cannot
have the converse.

Absolute Unit Of E.M.F.-— The fact that an induced E.M.F.
is independent of the material and dimensions of the conductor,
enables us to define the unit of E.M.F. in the following simple way: —

ELECTRO-MAGNETISM 351

The al>*olute unit of E.M.F. {A the E.M.F. produced when n, con-
ductor moves at such a rate that it cuts one line of iim<ji/<'ii<- force
per second.

(Here, it must be mentioned that the essential point is relative
motion. The conductor may be stationary and the field may move,
but the result is the same if the rate of cutting is unaltered.)

As this unit is too small for practical convenience, a multiple is
used for most purposes, which is already known to us as a "volt."
A volt is one hundred million (or 108) absolute units of E.M.F.

It follows from the definition that, if Z lines of force cut N
conductors in t seconds (the rate of cutting being uniform during
that time and the conductors being connected in series), then

ZxN ZxN

e = — — absolute units = n .Qt volts.
t

If the rate of cutting is not uniform, this is the average value during
that time.

Sometimes we are concerned with the motion of straight con-
ductors (as in dynamo and motor armatures), and sometimes with
the motion of coils — either circular or rectangular in section — and
there is sonic danger of confusion when applying the above formula.
Now a single turn of a coil is really the equivalent of two conductors
in series, but it is convenient to adopt a method which enables us
to apply the formula to both cases, and which, therefore, obviates
the necessity of using two special equations. Suppose that Z lines
of force pass through a coil of N turns, and that the coil be rotated
through 90° so that no lines of force then pass through it; if we
agree to define this as meaning that Z lines have cut N turns, the
above equation applies without ambiguity, whether N" stands for the
number of turns or for the number of conductors, and we then have

lines cut x turns , , , , ., .
e = — - j - -. — (absolute units)
time ot cutting N

This definition of "cutting" as applied to a coil means that if
the coil makes one complete revolution about a diameter in a magnetic
field — its axis of revolution being at right angles to the field — it cuts
the lines threading it four times.

Example I. — A solid fly-wheel, 1 metre in diameter, makes 250
revolutions per minute in a plane at right angles to the
magnetic meridian at a place where the horizontal component
of the earth's magnetic field is '18. Find the P.D. between
the centre and the circumference of the wheel.

The rotating wheel cuts the horizontal component of the
earth's field, and is really equivalent, as far as the production
of an E.M.F is concerned, to a single conductor OP (Fig. 268) FiG. 268.
equal in length to its radius r. For if OP rotates round the
centre 0, a P.I), will be produced between 0 and P, and this will be true for
any number of radial spokes such as OP. Merging them together to form

352 VOLTAIC ELECTRICITY

a disc will not affect matters, and, hence, we see that every point on the cir-
cumference is at the same potential, and that the P.D. between the centre and the
circumference is the same as between 0 and P. No current will ilow unless a
connection is made between the centre and some other point on the disc (best
at the circumference).

•v lines cut x number of conductors in series , , , ,

JNow e = —. jr- — rr- (absolute units)

time of cutting

Consider the time of one revolution, which is -^j- seconds. In that time OP
has swept through an area irr-, and has cut H x irr- lines ; hence

_Hx-7rr2xl_ -18 x 3-14 x(50)2x 1x250
i>Yo 60

= 5890 absolute units

= -y^g- = '0000589 volts.

It will be noticed (1) that this is a uniform and unidirectional
E.M.F., and (2) that e varies as r2, hence the P.D. between 0 and
any point on the disc midway between the centre and the circum-
ference is only \ of the value just given.

Exp. 217. Make a small rectangular coil by winding a number of turns
of 30-gauge copper wire around a match-box ; dip it
into melted paraffin-wax to hold the turns together,
and then trim off the projecting ends of the box.
Connect the two free ends of the wire to a dead-beat

^__ reflecting galvanometer, and then hold the coil be-

•fin Ilfc A, tween the poles of an electromagnet, in the position
AB (Fig. 269), where the horizontal lines- indicate
the general direction of the lines of magnetic force.

Turn it through a quarter of a revolution and

B notice the direction of throw ; turn it through

FIG 269. another quarter of a revolution in the same direction,

and observe that the throw produced is in the same

direction as before ; continue the motion and notice that the throw is reversed
in direction during the last two quarters of a revolution.

Hence we see that in such a rotating coil, the E.M.F. is reversed
in direction once in each half revolution. Each turn may be regarded
as made up of two active portions, which are really cutting the lines
of force, and of two inactive portions which are sliding past the lines
without cutting them, and which serve merely as conductors necessary
to complete the circuit. The active portions will be the parts at
A and B which are perpendicular to the plane of the paper; the
inactive portions will be the parts actually shown in the figure from
A to B and the corresponding portions on the further side. Consider
one active conductor at A. It will be seen that, in passing from
A to B, it is moving downwards through the field, and an E.M F.
in the same direction will be produced during the whole of the half
revolution, which will have a maximum value at Alf where the
conductor is cutting the lines at the greatest rate, and which will

ELECTRO-MAGNETISM 353

be zero at A and B, where it is sliding along the field, and, for an
instant, is cutting no lines. In passing back from B to A, it is
moving upwards through the field,
and hence the E.M.F. will be re-
versed in direction, again having a
maximum value at Br This may
be represented as in Fig. 270, where
the angle of rotation is taken for
abscissae, and the E.M.F.'s at any
instant are plotted as ordinates.
It will be seen that the alternating
K M.F. passes through one com-
plete cycle in each revolution. The " .
number of complete cycles passed
through per second is called the pIG 270

frequency. It may be remarked

that the symmetrical curve shown in this figure implies that the
coil is rotating with uniform velocity, and also that the field is
uniform. With the same proviso, a similar curve would be obtained
in Experiment 215. Again, it is quite immaterial as to whether the
coil is circular or rectangular in shape.

Example 2. — A coil of 50 turns, and mean diameter 40 centimetres,'!?;^^ i~
rotates about a vertical axis at the rate of 12 revolutions per second, at a place
where H — '18. Find the average value of the E.M.F. produced. Generally, we
have

. . lines cut x number of turns , , , , . , >

e (average) = —. „ — — -. — (absolute units)

time of cutting

The lines threading the coil, when it is at right angles to the field, are given
by H x irr\ and these lines are cut four times during one complete revolution.
Whence

e (average) = ^1^-7r^2Q)2x50 = 542,592 absolute units

s= -00543 volt

The meaning of this result will be understood from Fig. 270.
If the curved line represents the actual E.M.F. at any instant, then
OA is the average value, where OABD is a rectangle having the
same area as the space enclosed by the actual curve.

If, for the sake of illustration, we solve the problem from the
point of view of active conductors, the argument is as follows : —

1 active conductor in one complete revolution cuts all the lines
threading the coil twice — once moving down, once moving up.

.*. 1 active conductor in one revolution cuts 2 x H x Trr2 lines,

i.e. 1 active conductor in one second cuts 12 x 2 x H x Trr2 lines.

Whence, by definition, e (average) =12x2xHx Trr2 absolute units
per active conductor.

Now, as each turn consists of two such active conductors in series,

Z

354 VOLTAIC ELECTRICITY

and as there are 50 turns, i.e. 100 active conductors in series, we
have o- (average) for whole coil = 100 x!2x2xHx 7r/>2 absolute units,
which is identical with the value previously obtained. We are not
in a position to calculate the maximum or instantaneous values at
present ; for this see p. 427.

To Find the Direction of an Induced Electromotive
Force. — An induced current evidently possesses energy, for it may
be made to do work in various ways, and in any case heat is pro-
duced in the conductor carrying it. This energy must be supplied
in some way, and, in the experiments just described, it can have
been supplied only at the expense of mechanical work. Whilst an
induced current was being produced, the moving part must have
experienced a resistance to motion greater than that required to
overcome friction. Our experiments were on too small a scale for
this resistance to be directly perceptible, but there is no difficulty
in demonstrating its existence, which, on a larger scale, is at once
obvious. For instance, a dynamo generator requires very little power
to drive it when it is not producing a current, but when it is allowed
to give out a current, it requires power in proportion to its output
of electrical energy, although the moving part is rotating as freely
in air as before. It must be understood that no such resistance to
motion is experienced when only an E.M.F. is produced. The circuit
must be complete, so that a current can flow before any energy is
expended in this way.

Exp. 218. Set up a suspended coil galvanometer, using a coil supplied for
ballistic work, and produce a deflection by putting the ends of the two wires
proceeding from it on the tongue (without touching) for an instant. If necessary,
hold a silver coin in contact with one wire, and place this and the end of the
other wire on the tongue. Notice that the spot of light oscillates backwards
and forwards for a long time before coming to rest. Again set it swinging, and
then bring the ends of the wires in contact with each other, thus completing
the circuit through the coil. Notice that the result is very marked — the
oscillations being powerfully " damped " and the coil speedily coming to rest.

The explanation is very simple. Whilst the coil was swinging
in the magnetic field of the permanent magnet, an (alternating)
E.M.F. was produced in the coil ; but this, as stated above, does not
demand any expenditure of energy, and therefore the coil oscillated
until its energy of motion became gradually converted into heat
(principally by air friction), and this took some time. When the
circuit was completed, the induced E.M.F. was able to produce an
induced current, the energy of which was necessarily derived from
the energy of motion. Hence, this became rapidly used up, the coil
moving as if against a resistance, or as if immersed in a viscous
medium. Under these circumstances, the energy of motion is not
merely converted into heat by air friction, but much of it appears
as C2r heat in the wire of the coil itself, and the process of dissipation
is, therefore, more rapid.

ELECTRO-MAGNETISM 355

This resistance to motion is due to the action of the magnetic
field produced by the induced current upon the field produced by the
permanent magnet, and we will now apply it to determine the
direction of the induced E.M.F. in any given case.

Consider Fig. 271, which is almost identical with Fig. 255,
although the method employed in drawing it is different. A con-
ductor,' shown in section in the figure, is placed in a magnetic field.
It is then moved, say in the direction P, and
it is required to find the direction of the induced
E.M.F. Assume that an induced current flows,
then at once a new field appears around the
conductor, which we represent by drawing a
circle. If we can find the direction of this field,
that of the current at once follows. Now, we
cannot mark it clockwise, because then the two
fields would be in the same direction on the
right-hand side, and the forces between them 7

would tend to drive the conductor in the way
it is being moved. This is evidently wrong, as it would lead to
something like perpetual motion, and hence we mark the circle
anti-clockwise, which means that a force comes into play tending
to oppose the motion, and against which work must be done in
moving the conductor, as is found to be the case by experiment.
Hence, by applying the results given on p. 241, we find that, in the
case shown in the diagram, the current flows outwards from the
plane of the paper (and even if there is no current, that must be the
direction of the E.M.F.).

The direction can be obtained in all cases by applying the above
argument to a suitable figure, although sometimes it is more con-
venient to adopt the following method. Suppose it is required to
find the direction of the induced E.M.F. when one
rv"i pole of a magnet is inserted into a coil of wire

I (Fig. 272).

If an induced current flows, the circuit being
completed, work must be done against some opposing
force in inserting the pole, and, therefore, the induced
current must produce a polarity in the coil oppositely
directed to that of the magnet. For example, if a
S pole is approaching the coil, the current in the coil
must give it the polarity shown in the figure, from
FIG. 272. which the direction at once follows. Similarly, work
must be done in removing the pole ; this requires an
opposite polarity, and, therefore, the induced current, during removal,
is reversed in direction.

Lenz's Law. — All such results can be conveniently summarised,
as was done by Lenz many years ago, in the statement : —

356 VOLTAIC ELECTRICITY

When induced currents are produced by motion, they are always in
such a direction that they tend to stop the motion whicli produces
them. Illustrations and applications of this fact, which is commonly
known as Lenz's law, are often met with. The damping of a
galvanometer, due to closing its circuit, described on p. 354, is an
instance of it. It is particularly noticeable in the suspended-coil
type, although it occurs to a smaller extent in any form of instrument.
Again, it has been stated that, in the original form of suspended-
coil galvanometer, the coil is wound on a metal frame, and that in
Ayrton and Mather's form it is enclosed in a silver tube, in order
to make it "dead-beat." The explanation is now obvious, for
induced currents are produced in the frame or in the tube as they
swing in the field, and the reactions must damp the motion.

The same principle, for steadying the motion of some moving
part, is applied in many commercial instruments. The only essential
point is the attachment of a suitably shaped piece of metal (preferably
silver or copper, as they are tho best conductors) to the moving part,
which then is arranged to move in a narrow gap between the poles
of a magnet.

Eddy Currents or Foucault Currents. — That induced
currents are produced in any piece of metal — and not merely in
wires — may be easily demonstrated. Such currents are often known
as "eddy currents," or, although they were discovered by Joule,
" Foucault " currents. If an exceptionally powerful electromagnet
is available, any sheet of metal will show a perceptible resistance
to motion when moved rapidly between its poles, and if the motion
is continued the metal becomes heated.

If such a magnet is available, the following experiment may be
made.

Exp. 219. Attach a thread to a lump of copper (an old twopenny piece
answers excellently). Spin it until there is a good twist on the thread, and
then let it untwist ; whilst rotating rapidly, bring it between the poles of the
electromagnet, described on p. 249. Notice that it will at once slow down
perceptibly, and that it will speed up again when the magnet circuit is broken.

If an alternating current is available, a simple method of demonstrating the
heating effect, due to eddy currents in masses of metal, may be employed.

Exp. 220. Connect one of the electromagnet coils — having first removed
its iron core— to alternating supply mains, taking care that the current does
not exceed 3 or 4 amperes. Place an iron bar, about f in. diameter, inside the
coil. Notice that, in a short time, the part inside the coil will become too hot
to touch. Repeat the experiment with a bundle of iron wires of about the
same section. Notice that this will remain cool, the eddy currents being
greatly reduced by thus laminating the iron. (Part of the heat produced in
the bar, especially if not very soft iron, is due to hysteresis ; see Chapter XXV.).

Arago's Rotations. — Arago discovered (some years before
Faraday's discovery of induced currents supplied the explanation)
that if a compass-needle, or a pivoted or suspended magnet, be
oscillated very close to a horizontal metal disc, it came to rest much

ELECTRO-MAGNETISM 357

more rapidly than usual, i.e. its motions are damped. He also found
that, the better the conductivity of the metal, the greater the effect.
Evidently, induced currents are produced in the disc by the moving
lines of force of the magnet, and these must tend to stop the motion
which produces them. The forces concerned are, however, mutual,
and hence we might infer that a moving disc would have some effect
on a stationary magnet. Arago rotated a copper disc very rapidly
close to a magnetic needle, and found that the needle was also set
in rotation in the same direction. The reaction between the field due
to the induced currents and that due to the magnet again tends to
stop the motion, but as the magnet is now movable, it is dragged
after the disc. Conversely, if the magnet is rotated rapidly close
to a pivoted disc, the latter is set in motion. This action has
become of great practical importance in connection with "induction
motors " working with alternating currents.

Actions due to Starting and Stopping of Currents.—
Up to the present, we have moved either a magnet or a conductor
in order to obtain an induced E.M.F. It is, however, possible to
obtain a motion of lines of force by another method.

Exp. 221. Procure a long piece of "twin" bell wire (i.e. two insulated
conductors bound together). Con-
nect the ends of one wire to a
sensitive galvanometer, G (Fig. 273),
and those of the other to a cell (b)
and key (&). Close the circuit and
notice the direction of the " throw."
Release the key and notice that the
throw is in the opposite direction.
In such experiments, it is usual to
speak of the wire (AB) connected
to the cell as the primary, and the **
one in which the induced current
is produced (CD) as the secondary.
Evidently, the effect obtained in
the experiment is due to the lines
of force produced in the primary
cutting the secondary as they FIG. 273.

appear and disappear.

To find the direction of the induced E.M.F. at "make" and
" break " respectively, we can apply Lenz's law. We may regard
the starting of the primary current as equivalent to bringing the
current suddenly from an infinite distance up to its actual position,
and the stopping of it as equivalent to taking it away again. In
the first case, the secondary current must produce a field which tends
to repel the primary field, and hence, at " make," it must be in the
opposite direction to that in the primary. At " break," it must
oppose the withdrawal of the primary, and must therefore be in the
name direction as the primary current.

358 VOLTAIC ELECTRICITY

The actual existence of such attractions and repulsions can easily
be shown by experiment.

Exp. 222. Drill a small hole through a wooden lath (a metre scale answers

well), put a pin through the hole to serve
as a pivot, P, (Fig. 274), and support it
on a wooden stand slotted at the top.
To tme end of the lath attach, with wax
or any suitable cement, a copper disc, D,
and balance the arrangement by means
of a weight, AV — a small piece of sheet
lead is convenient. Bring the disc im-
mediately above one pole of the electro-
magnet (Fig. 190) — the other pole may
be temporarily removed. Excite the
FlG. 274. electromagnet and notice that the disc

is momentarily repelled. Demagnetise it
by breaking its circuit, and notice that the disc is momentarily attracted.

Remembering that the magnet may be regarded as a whirl of
molecular currents in the same direction as the current in the coil
surrounding it, it is evident that the experiment is equivalent to
(1) bringing such a current suddenly up to the disc and (2) suddenly
removing it. In the first case, the induced current in the disc is
in the opposite direction to the molecular currents, and thus we get
repulsion ; in the second case, the induced current is in the same
direction, and attraction takes place.

In Experiment 221, two straight wires were used for the saky of
simplicity, but that method is not the most convenient, as a consider-
able length is needed to obtain a well-marked effect. It would be an
improvement to first wind the double bell wire into a coil, but the
most satisfactory arrangement is obtained by using two separate coils,
one of which will slip inside the other. In the following experi-
ments, it will be found convenient to use one of the coils of the
electromagnet shown in Fig. 190, and another of smaller diameter
made to fit easily inside it. (This small auxiliary coil is very useful,
and will be required for a number of experiments ; it may consist of
a few layers of 20 -gauge wire, wound on a wooden bobbin, with a
hole through it in which an iron rod may be placed.)

Exp. 223. Place this small coil inside the larger and connect one of the coils
in circuit with a cell and tapper key, and join the other to a galvanometer (if a
sensitive instrument is used, protect it by means of a shunt). Notice that a
"throw" is produced when the primary circuit is closed, and another in the
opposite direction when it is broken. Insert an iron rod, and notice that the
effect is very much greater. Interchange the connections, so that the coil
previously used as primary is now the secondary, and notice that the general
result is unaltered.

Remembering that the primary coil, when carrying a current,
behaves as a weak magnet, we see that this experiment is equivalent
to inserting such a magnet into, and to withdrawing it from, the

ELECTRO-MAGNETISM 359

secondary coil. It is generally convenient to use the inner coil as
primary, although the experiment shows that it is not absolutely
necessary ; what is really essential is that the field produced by one
coil (the primary) should cut the other coil (the secondary) as it
appears and disappears.

Now the E.M.F. produced in the secondary depends only upon
three factors : (1) the number of lines of force that cut it, (2) the time
they take to appear and disappear, and (3) the number of turns in
the secondary (for an E.M.F. is produced in each turn and these
E.M.F.'s are added together). Hence, we learn that the secondary
E.M.F. can be made as large or as small as we please, by using
many or few turns, and that it is quite independent of the E.M.F.
of the cell used in the primary circuit, except in so far that
increasing the primary E.M.F. increases the primary current, and,
therefore, the primary field. It is to this property that induction
coils and commercial transformers owe their usefulness — they
enable any desired E.M.F. to be obtained in a simple manner.
The ordinary induction coil is more especially a device for obtain-
ing a very high E.M.F., and its actions may be illustrated as
follows : —

Exp. 224. Connect the large coil to two pieces of metal of sufficient size to
form convenient handles, and join up the smaller inner coil to a tapper key and a
battery of from three to six cells in series. Hold the metal handles while contact
is made and broken. Nothing will probably be felt when the primary circuit is
made, but notice that a distinct, though feeble, shock is felt when the circuit is
broken. Place an iron rod inside the coils ancl repeat the operations. Now
notice that a shock may be felt at "make " and also at "break," but that the
latter is much the stronger, and that it is very much more powerful than it was
without the iron core.

As rather a high E.M.F. is necessary to give a perceptible shock,
the experiment shows that the induced E.M.F. in the secondary
must have been greater than that of the battery ; and it also brings
out two other facts : (1) the importance of the iron core, and (2) the
difference between the induced E.M.F. at "make" and that at
" break." The effect of the iron core is readily understood — we
know that the core increases enormously the strength of the primary
field ; but it is not so obvious why the two E.M.F.'s should differ in
strength, seeing that they are due to the same lines cutting the
same number of turns. The best way to investigate the matter
would be to use an oscillograph (see p. 571) to indicate the rate of
rise and fall of the current, but such an instrument is not likely to
be at the disposal of the student. If it were, he would learn that
the primary" current, and hence the field due to it, takes longer to
grow to its full strength when the circuit is made, than it does to
die away when the circuit is broken, and, as the induced E.M.F.
depends upon its rate of cutting, it is necessarily weaker at " make "
than at " break."

360 VOLTAIC ELECTRICITY

This difference in time is due to an effect generally known as
self-induction,

Self-induction. — Exp. 225. Break the circuit of a battery when its
terminals are joined by a straight piece of wire. Only a faint weak spark is
obtained, although the current strength may be considerable. Put a large coil
of wire in circuit, e.g. the electromagnet coil used in the previous experiment.
The spark is now much brighter, although the current itself is much less on
account of the increased resistance in the circuit. Insert the iron core of the
electromagnet ; the spark is still more intense.

(A more striking method of carrying out these experiments is to connect one
wire to a file and to draw the other over its surface. Contact is thus rapidly
made and broken.)

Such effects will often be noticed during experimental work.
Whenever a circuit contains coils of wires with iron cores, i.e. when-
ever it is linked with strong magnetic fields, a bright spark is
produced when the circuit is broken. It is easy to show that this
is due to a new E.M.F. of considerable magnitude and not to the
E.M.F. acting in the circuit before it was interrupted.

Exp. 226. Connect the large electromagnet coil to a few cells in series and a
tapper key. Also connect metal handle's to the coil (Fig. 275), so that the body
of the person holding them may act as a shunt. When
the circuit is " made," nothing is felt, whether there is an
iron core or not ; but, when the circuit is broken, there
is a shock, which is greatly increased in intensity by the
presence of the core.

We infer from this experiment that the P.D.
FIG 275 between the ends of the coil rises suddenly to

a high value when the circuit is broken, and it
is not difficult to understand that this effect must be due to an
E.M.F. generated by the lines of force cutting the turns of the coil
itself as they disappear.

Such actions are of great importance, and must occur to some
extent in any circuit whatever. The student should keep clearly in
view the general fact that the existence of a current in a circuit
implies the existence of both electric and magnetic lines of force in
the space surrounding it. The electric lines must have been present
before the current began to flow, whereas the magnetic field did not
come into existence until the current started. At present we are
considering more especially the magnetic field, but in doing so we
must not forget the existence of the other, or unnecessary difficulties
will be created at a later stage of progress.

It is extremely important to grasp the following fact : Whfin_a
m^aneJic_field appears andjjjisappears, it producf « a-fL "TC-M JK1f"trnflti
conductors cut by it, and it is immaterial (a) whether these con-
ductors belong to afTTintependent circuit, or (b) whether they
constitute the circuit in which the magnetising current is flowing.

We shall speak of the E.M.F., which we apply to any circuit to

ELECTRO-MAGNETISM 361

produce a current therein, as the "impressed" or ''applied" E.M.F.,
and we shall denote it by E. We know that Ayhen the current is
starting, an independent induced E.M.F. — which is termed the E.M.F.
oj self-induction and which we shall denote by ex — must exjsJL for a
certain time (usually a small fraction of a second) due to the magnetic
field, as it forms in the surrounding space, cutting the circuit. This
E.M.F. is in the opposite direction to the impressed E.M.F.' and
hence the current rises gradually and does not reach its full value
until es has disappeared. On the other hand, when the circuiLjs
broken _j.od the current is stopping, the process is reversed ; the
"Held, as it disappears, cutting the circuit in the opposite direction,
and another self-induced E.M.F. is produced, which is now in the
urmic. direction as the impressedJE. M. F. , and which tends, therefore,
to prolong the duration of the current:-- , But whereas the value
of es Tat " make" cannot possibly be greater than the applied E.M.F.
(otherwise the current could not rise in value), its value at " break "
is not limited by any such condition, and when the break is sudden,
its instantaneous value may be very great. This explains the results
obtained in Experiments 224 and 226. Certain forms of motor
ignition — known as the " low-tension " system — depend on this
principle. A battery current is sent through a coil of wire wound on
an iron core, and the circuit is suddenly broken inside the chamber
containing the gaseous mixture to be ignited.

Inductive and Non-inductive Circuits. — It is usual and
convenient to speak of circuits (or parts of circuits) as being inductive
or non-inductive, according to the power they have of showing
these effects in a marked degree or not, e.g. a circuit containing
electromagnets is typically inductive, and one consisting of short
straight wires is practically non-inductive. It must, however, be
borne in mind that no circuit is absolutely non-inductive, for, even
with a straight wire, some magnetic field must exist around it, and
the term non-inductive must therefore be taken to mean that the
effects due to self-induction are so small as to be negligible. On the
other hand, it is possible, and often necessary, to make some portion
of a circuit almost completely non-inductive, as for example in the
coils of resistance boxes.

Again, the magnetic field around a steady current represents a
certain amount of energy stored up in the surrounding space. This
energy was supplied at the expense of the circuit while the current is
starting, and it is paid back to the circuit when the current is
stopped (not necessarily in the form of a discharge at a very high
voltage, for the applied E.M.F. may be cut off without actually
breaking the circuit). Hence, the starting of a current in a circuit
resembles, in a marked degree, the starting of some heavy mass, such
as a fly-wheel. It cannot be started suddenly ; nor does it naturally
stop suddenly, and if it be made to do so compulsorily, the result is

362 VOLTAIC ELECTRICITY

disastrous. The amount of energy associated with it, when in steady
motion, and the magnitude of all the effects depending on that
energy, will vary with its mass, so that an inductive circuit is the
analogue of a body of great mass, and vice versa.

Such actions are of special importance in connection with alter-
nating currents.

Ezp. 227. Measure the resistance of one of the electromagnet coils l>y
Wheatstone's bridge method. Connect it to alternating mains, having put an
ammeter in series with it and a voltmeter across it. (Take any necessary pre-
cautions to avoid too large a current.) Calculate the current from the measured
resistance, and then observe that the ammeter reading is very much less than
this value. Hence, when alternating currents aro» used, an inductive resistance
cannot be measured by the method indicated in Experiment 165, p. 258, although
correct results would be obtained with continuous currents.

Exp. 228. Insert an iron bar in the electromagnet coil, and notice that the
current is reduced. Add more iron, and notice that there is a further reduction
of current, the smallest value being obtained when the iron core belonging to
the coil is inserted. (This should soon be removed, or, for reasons already given,
it will get hot.)

This effect is really due to the presence of an E.M.F. of self-
induction of variable value, although the experiment seems to indicate
that the coil has a variable resistance, and consequently such a coil
with a movable iron core may be used, instead of a resistance, to
regulate the strength of an alternating current. A coil, specially made
for the purpose, is known as a "choking coil." As such coils may
be given a very low resistance, this method of regulation wastes much
less energy in the form of heat (for the number of heat units varies as
The subject is further discussed in Chapter XXVI.

Induction Coils. — Induction coils are devices for obtaining very
high secondary voltages. During recent years they have acquired
considerable commercial importance, owing to their use in wireless
telegraphy and in connection with X-ray work. The principle is
fully illustrated in Experiment 223, and the practical form merely
represents improvements in details. As the secondary E.M.F. is
produced only at make and break — chiefly at break— it is necessary
to have some device for automatically breaking the circuit many times
per second. With large coils, an independent arrangement is now
always used for this purpose, but the simplest and best-known
contact-breaker (shown in Fig. 276, taken from Brooks and James'
Electric Light ami Power) is a modification of the electric bell
principle. Its construction will be sufficiently evident from the
diagram. A soft iron armature nearly touching the core is fixed at
the top of an elastic strip of brass, which on its other side is fitted
with a short piece of thick platinum wire, normally in contact with
another similar piece of platinum attached to a screw head. It will
be seen from the connections that the primary current passes across
this contact until the circuit is broken by the pull of the core on the

ELECTRO-MAGNETISM

363

armature. Then the core loses its magnetism and the strip springs
back to remake contact. The lower screw, moving through the insu-
lating washer, I, is merely a device for adjusting the tension of the
spring, thereby affording a means of regulating the strength of current
required to break contact. It is desirable to insert a current reverser,
11, in the primary circuit.

The Coil Windings. — In constructing induction coils the point
of supreme importance is the effective insulation of the secondary.

The primary usually consists of two or three layers of fairly

Or Or

FIG. 276.

S, Space occupied l>y secondary winding.

V, Vertical insulation.

V1} Vertical insulation, drawn further apart to show wire

more clearly.

"VV, "Wooden base.

C, Condenser.

R, Primary current reverser (actually mounted on base W).

I, Insulating washer.

T, Primary terminals.

stout copper wire (which may be cotton-covered, but is preferably silk
covered, especially in large coils), wound upon a core of iron wire.
This is placed inside a thick ebonite tube, which may with advantage
be somewhat longer than the secondary.

In winding the secondary, S (shown in section in the diagram),
it is necessary to remember (1) that an E.M.F. is produced in each
turn and that the turns are in series, so that the difference of
potential between any point of the secondary and the earth will be
greatest towards the ends, where it will, therefore, be essential to

364 VOLTAIC ELECTRICITY

have the most effective insulation ; and (2) if the secondary is wound
backwards and forwards in the usual way, there will be an enormous
difference of potential between the first turn of one layer and the last
turn of the next layer which lies immediately above it, and conse-
quently there arises a great danger of sparking through the insulation.
To avoid such danger, horizontal layers of insulation are practically
useless, and thus it becomes necessary to ensure that there is only a
small difference of potential between any turn and those immediately
adjacent to it. We can do this best by winding the wire for a short
distance (say y1^ to T\ of an inch) in a horizontal direction, then
returning the wire on itself, and so on. It is convenient, for this
purpose, to use a special winding device, consisting of two smoothly
turned brass checks, which can be separated by a thin disc of wood to
form a narrow deep bobbin.

The wire, which should be fine (say 36-gauge), and must be silk
covered, is run through melted paraffin-wax, and then wound on the
bobbin until a sufficient depth is obtained. The narrow coil thus
formed is held together by the wax, and should be carefully removed,
placed between thin discs of paraffined cardboard (or of ebonite), and
the whole sweated together with a warm flat-iron. This is paired off
with another similar coil — the inside ends being soldered together
and pushed out of the way — and the two coils then sweated together
as before.

The result forms one element of the winding, and both the free ends
arc on the outside. Perhaps 50 to 100 such elements are required for
a large coil, and each should be tested separately to stand a voltage
much greater than they will have to produce during use. It is usual,
as will be seen from the diagram, to wind less depth on those coils
which are to bs situated near the ends of the instrument (where the
need of insulation is greatest), and to gradually increase the depth of
the winding towards the middle. They are then placed in the proper
order on the ebonite tube containing the primary, the necessary con-
nections being made by soldering. The interstices between the
vertical discs of insulation are next filled in with melted paraffin- wax,
until the whole forms a solid cylinder (shown in the figure by the
shaded portions surrounding the secondary, &), which is then covered
with a thin sheet of ebonite.

Coils may be made in this way, which are capable of giving very
long sparks without risk of piercing the insulation.

A most ingenious and valuable improvement has recently been
patented by Miller, who has devised a method of winding the sections
continuously without breaking joint, each section containing only one
conductor horizontally — like a watch-spring — so that there are about
1000 separately insulated sections in a coil of moderate size.

The Condenser and Its Functions. — The most important
detail yet to be mentioned is the condenser, C, connected as a shunt

ELECTRO-MAGNETISM 365

to the spark-gap of the contact-breaker, and generally placed inside
the base, W, although for the sake of clearness it is shown diagram-
matically in the figure.

From the result obtained in Experiment 225, we should expect
to get considerable sparking at break, due to self-induction, and this
would mean serious heating and wear of the platinum contacts, as
as well as much waste of energy. This would actually happen if
no condenser were used, or one of too small capacity. With a
suitable condenser, however, the sparking at the contact-breaker is
greatly diminished, and the length of the secondary spark is in-
creased. This action of suppressing or of neutralising the effects of
self-induction will not be completely understood until Chapter XXIII.
has been read. However, we know from Experiment 226 that, when
the primary circuit is broken, the lines of force, in disappearing, set
up not only an induced E.M.F. in the secondary but also another in
the primary, which is in the same direction as the current previously
flowing, and which, therefore, tends to prolong its flow. This, in
itself, is very undesirable, for the magnitude of the E.M.F. depends
on the rapidity with which the field, and, therefore, the current dis-
appear. As the capacity of the terminals forming the spark-gap is
small, the P.D. between them rises suddenly to a value sufficiently
high to spark across the gap, thus prolonging the current and
diminishing the secondary E.M.F. The presence of the condenser
greatly increases the effective capacity of the terminals, and the P.D.
across them can only rise at the rate the condenser charges up, and
may never reach such a high value as before.

Again, it will be noticed that the two coatings of the condenser
are always in conducting communication through the battery and the
primary, and hence, immediately after charging, it discharges itself
through this path. This discharge means a rush of electricity round
the circuit in the opposite direction to the dying-away current, and
this rush itself tends to produce an induced E.M.F. in the secondary
in the right direction, and thus to strengthen the secondary spark at
break.

Such a condenser is usually made of sheets of tinfoil interleaved
with paraffined paper.

When the primary current is alternating, a contact-breaker and
a condenser are not required, but the secondary spark, with an
ordinary induction coil, will be shorter and of an entirely different
character. The changes in the field strength are now comparatively
gradual, and the induced E.M.F. is, therefore, less. A direct current,
interrupted by a contact-breaker, is, in fact, desirable when very high
voltages are required, in order to obtain the necessary rapid changes
in field strength.

Transformers. — For many commercial purposes, it is useful to
have a method of obtaining any required E.M.F., high or low, from

366 VOLTAIC ELECTRICITY

a given primary E.M.F. ; and induction coils made for this purpose
and working with alternating currents are known as transformers.
Their construction and theory are given in Chapter XXVI. ; at pre-
sent, it is sufficient to show that a contact-breaker may be dispensed
with when alternating currents are used.

Exp. 229. Connect one of the electromagnet coils to alternating mains,
taking the necessary precautions. (Probably it will stand 100 volts pressure for
a short time, the current rising only to 3 or 4 amperes on account of the choking
effect.) Place inside it the small coil, already used in Experiment 223, to act as
secondary, having joined its terminals by a short piece of fine iron wire (e.g. that
used for bouquets). If the wire is of a suitable thickness, it may become very
hot without being luminous. Insert gradually an iron bar. Notice that the
wire now becomes dull red, and that the temperature increases as the bar is
lowered, until it becomes white hot, and finally breaks.

EXERCISE XVIII

1. If a current is flowing through a coil, what effect is produced by inserting
into the coil and then withdrawing rapidly, (a) a bar of wood, (b) a bar of iron ?

(Oxford Local, Senior, 1908.)

2. State generally under wrhat circumstances a current is induced in a coil.
Describe some method by which these circumstances can be realised in practice.

(Canib. Local, Senior, 1907.)

3. Describe a machine which is set in rotation by passing an electric current
through some of its movable parts. What is the effect on the current if, while
it is still flowing, the movement of the machine is stopped ? (B. of E., 1895.)

4. The binding-screws of two astatic galvanometers, a considerable distance
apart, are connected by wires, so that their coils form a continuous circuit. If
tne needles of one galvanometer are moved, those of the other are disturbed.
Explain fully this effect.

5. Two equal magnets, each bent into a semicircle, are fastened together with
like poles in contact, so as to form a complete circle, and a copper ring, through
which one of the magnets had been thrust before the two were fastened together,
is carried round and round the circle at a uniform speed. Show how the cur-
rents induced in the copper ring by the magnets vary in strength and direction
as the ring passes different parts of the magnetic circle.

6. A metal ring is put round the end of a bar magnet. Upon bringing a
mass of soft iron near to this end of the magnet, a current is produced in the
ring. Show what motions (a) of the magnet and (b) of the ring will produce a
similar current in the absence of the soft iron.

7. A piece of wire is bent into the form of a rectangle, and the ends are
joined. It is laid upon a horizontal table with two sides pointing magnetic
north and south. If the rectangle be turned about the east side as a hinge so
as to lie on the table to the east instead of to the west of it, what will be the
direction of the current which circulates in the wire during its motion, in con-
sequence of the earth's magnetism ?

8. When a circular metallic hoop is rotated in the earth's magnetic field,
electric currents are generally produced in it. In what positions of the axis of
rotation will the induced currents be the greatest and least respectively ? Give
reasons for your answer.

9. A circular hoop of wire is suddenly twisted half round about a vertical
axis. What is its electrical condition during this movement ? Determine in
what position of the hoop as it moves the E.M.F. is the largest.

(B. of E., 1897.)

10. A vertical hoop of wire, at right angles to the magnetic meridian, is

ELECTRO-MAGNETISM 367

quickly Imt with uniform speed turned through 180° about a vertical axis, its
originally eastern half moving northward at first. State the direction in which
the induced current passes round the wire, and determine the position of the
hoop in which the induced E.M.F. is the greatest. (B. of E., 1898.)

11. A coil of wire, whose ends are joined to the terminals of a galvanometer,
is continuously and rapidly rotated about a given axis. Explain the effect upon
the needle. (B. of E., 1900.)

12. A light metal ring is suspended over the end of a solenoid : if a large
current is suddenly sent through the solenoid, show that the ring will be re-
pelled. (B. of E., 1903.)

13. An iron hoop is held in the magnetic meridian, and is allowed to fall
over towards the east. Explain why an electric current traverses the hoop, and
state whether the current would flow north or south in the part of the hoop
which touches the ground if the experiment were performed in England.

(B. of E., 1894.)

14. State the law of the induction of currents ; illustrate your statement by
describing the behaviour of two parallel coils, A and B, placed side by side,
when currents are started and stopped through A ; and when A, while convey-
ing a current, is moved towards and from B. (B. of E., 1901.)

15. A bar magnet is suspended on a stirrup by a string, and oscillates in a
horizontal plane. How are the oscillations affected (if at all) when a thick non-
magnetic metal plate is placed horizontally beneath the needle so as to be close
to without touching it ? (B. of E., 1896.)

16. Explain how it is that a disc of copper, revolving in a horizontal plane
below a magnetic needle, causes the needle to turn in the same direction as the
disc.

17. Describe an induction coil, and explain why the iron core is made of
wire. (B. of E., 1902.)

18. In an induction coil, show how the condenser is connected, and explain
its function.

19. A copper ring is hung by a torsionless thread between the vertical
parallel and flat opposed poles of an electromagnet. The plane of the ring is
vertical, and is inclined at 45° to that of either pole face. If a current is started
round the magnet, the ring turns through a moderate angle, but quickly comes
to rest. If it is replaced in its former position and the current is stopped, it
starts twisting in the opposite direction and keeps on spinning. Account for
these actions. (Lond. Univ. B.Sc., 1902.)

CHAPTER XXIII

0

ELEMENTARY THEORY OF INDUCTION

SUPPOSE that Z lines of force cut N turns in t seconds, then, from
the definition of unit E.M.F. (p. 351), we have

(average)

ZxN , ,

— - — absolute units.

If the circuit has a total resistance of R absolute units, then by
Ohm's law the average current is given by

(average) ^ * R

Now in most cases the actual E.M.F. and current vary rapidly in
strength during their brief duration, and their average values are of
little use to us. If, however, we rewrite the last expression in the
form : —

ZN

i x t = -—-

(average)

the product (average current x time) is a definite " quantity " (Q),
and we have : —

ZN

Q = absolute units. (i.)

R

Hence we learn (a) that the quantify, which flows round a circuit
due to induction occurring therein, is independent of the time taken
by the process ; and (b) that it varies directly as " the amount of
cutting " 2 which takes place, and inversely as the resistance of the
circuit.

In the above expression, the absolute unit of quantity is the
quantity conveyed by 1 absolute unit of current in 1 second. This
is 10 coulombs, where 1 coulomb = 1 ampere-second. Hence, if we

1 As already stated, where i occurs in an equation, it indicates that absolute
units are to be used ; and when C occurs, that practical units are to be employed.

2 When Z lines cut N turns, we shall call the product ZN "the amount of
cutting." It is often said to be ZN "Maxwells," but this term has scarcely
passed into general u.°>o.

368

ELEMENTARY THEORY OF INDUCTION 369

write Q= — ^ , the result is in coulombs, where R is expressed in

absolute units.

If we wish to use practical units throughout, we must write

Q

(average)

C = -^ — amperes, where R is in ohms

(average)
'. CU - Q = y-g-.p coulombs. (ii.)

In equation ii., if we put R in ohms, Q is in coulombs; and
obviously if we put R in absolute units and delete the 108, Q is in
absolute units. Equations i. and ii. are important, because Q can be
measured by means of a ballistic galvanometer (p. 296). Various
applications will be met with later.

Rxample I. — A coil of inductive area, 15,000 square centi-
metres, is connected to a galvanometer. The coil lies flat on the
table, and when it is turned over, the galvanometer is momentarily
deflected. Discharging a condenser of 1 microfarad capacity, charged
to T5 volts through the same galvanometer, produces the same throw.
If the vertical component of the earth's field is *4 C.G.S. units, what
is the resistance of the coil and galvanometer together with the con-
necting leads 1

By the term "inductive area" is meant the product of the area
a'nd the number of turns, i.e.

Area x number of turns= 15,000

(such a coil, which is usually of rectangular or of circular shape —
although it may be of any shape whatever — is often called an " earth
inductor ").

If A = area of surface, N = the number of turns, and V = the
vertical component, then, as it lies on the table, V x A lines of force
ss through it, and when it is turned over these lines are cut twice
see p. 351).

From equation ii. we have

Q = coulombs, where R is in ohms

... Q = 2 x V x^A x N coulombg) where A x N = 1 5,000

108R

2A

370 VOLTAIC ELECTRICITY

For the condenser discharge, we have

Q = EK, where, if E is in volts and K in farads,
Q will be in coulombs

.*. Q = 1'5 x — ; coulombs.

Now these two quantities are equal by. the terms of the question,
2x -4 x 15,000 1-5

whence R = 80 ohms

It should be noticed that the question as set ignores the influence
of " damping," considered later in Chapter XXIV., the solution being
only exact, when we assume that no damping exists. Neither is it
correct to say that the damping factor will be the same in both
experiments, and, therefore, will cancel out. The conditions are
different, and so will be the damping. In practice, serious errors
may arise when these facts are ignored.

It should also be noticed that, when R is known, the experiment
suggests an instructive method of measuring V.

Example 2. — Find the strength of the field between the flat-
ended poles of a large electromagnet from the results of the following
experiment. A small coil of fine wire, having 30 turns and a mean
area of 2 square centimetres, is connected in series with a ballistic
galvanometer and an earth inductor coil, having 500 turns and
a .mean area of 480 square centimetres. The small coil is placed
between the poles, and suddenly pulled out of the field, the resulting
throw being 153 divisions. The inductor coil is then quickly turned
over (as in Example 1), and the throw is 102 divisions. Neglect any
corrections due to damping, and assume that V= '4.

If F be the strength of the field due to the electromagnet, then,
when the small coil of area A square centimetres is placed in the field,
it is threaded by F x A lines, and when it is removed, these are cut
once.

Let Q be the quantity producing the throw of d divisions.

Now, as we have seen, Q = — ^ — absolute units oc d

±\i

~ F x A x N ,
^.e. Q = - - ccd

XV

whence Q = F X.^X 3° oc 153. (i.)

ELEMENTARY THEORY OF INDUCTION 371

In the second case, let QL be the quantity producing the throw of
dl divisions,

9V v A v "NT
then Q! = - :Jp£U absolute units cc dl

XV

whence Q1 = 2 x "4 * 48° x B«> x 102
R

We, therefore, have * 2 x 30

2 x -4 x 480 x 500 102
whence F = 4800 lines per square centimetre.

(In this example the resistance of the circuit is the same for both
throws, and the damping factors will be equal and cancel out.)

Back Electromotive Force.— Consider any one of the ex-
periments mentioned in Chapter XXII. in which motion is obtained
by sending a current through a conductor in a magnetic field. The
current is produced by creating a P.D. between the ends of the con-
ductor, and we will assume for convenience that this impressed P.D.,
which we will denote by E, is kept constant. If r be the resistance
of the conductor, then, as long as it is at rest, the current through it

will be - ; but, as soon as it moves, an induced E.M.F., e, is produced

in it, due to its motion in a magnetic field, just as it would be if it
were moved by any other means. This is in the opposite direction to
the impressed E.M.F., and is usually called a "back E.M.F." The

current through the conductor is now C = _ Z£ , and is, therefore, less

than before. Again, e is proportional to the speed, and, as we have
assumed E to be constant, the faster the conductor moves, the smaller
will be the current through it.

The student has been accustomed to consider only one E.M.F. in
a circuit, the current flowing naturally with it. It must now be
pointed out that whenever this occurs, all the energy expended in the
circuit is converted into heat, and is necessarily wasted, unless it is
required in that form for lighting or heating purposes.

A current flowing in the direction of an E.M.F., i.e. down a slope
of potential, resembles a stream of water flowing downhill, and
dissipating its energy of motion as heat by friction. In order to
make the stream do some useful work, e.g. to drive a mill, it must
turn a wheel or turbine, which resists being turned, i.e. sets up a
back electromotive force, and the greater this resistance to motion the
slower will be the flow of water, which, it should be noticed, is still
flowing down a slope of potential. We see that a maximum current
of water does not mean that a maximum amount of work is being
done ; it merely indicates that there is the least resistance to its flow,

372 VOLTAIC ELECTRICITY

and that occurs when no work is being done. Exactly the same
reasoning holds good with regard to a falling body. Falling freely, it
does no work, and its energy is ultimately converted into heat. If it
is to do work, it must act against some resistance to motion (other than
friction), which is analogous to the back E.M.F. we are now con-
sidering. (Further remarks on this point are made on p. 465.)

An electric current can do no work (other than heat production)
except by flowing against an opposing E.M.F. — being, of course,
impelled "by a greater E.M.F. This is expressed by the equation

C = — , already given in one important case in the chapter on

electrolysis. Various other applications will be met with later.

We may interpret it by noticing that the watts given to the circuit
(or the part of the circuit we are considering) must be EC, and the
portion expended as heat must be C2r, so that EC — C2f watts are
left to be accounted for.

T> + T?/^ f^2i* /~l^ tf^iAf^ of^

i.e. the product of the back E.M.F. and the current represents the
watts converted into some foryi of work. We shall now use this prin-
ciple to obtain certain important formulae in a simple way.

To Find the Force on a Conductor carrying a Current
in a Magnetic Field. — Let a straight conductor of length / centi-
metres carry a current, i, at right angles to the lines of force of a uni-
form magnetic field of strength B. We know that a force, tending
to drive it sideways across the lines, acts upon it. Let
». ^TT^.TT^,. this force be p dynes. If we allow it to move under
T. •.'•'." :'.'; '/•':!•. the influence of this force, a back E.M.F. will be pro-
duced in it, and the current does work in moving it.
(If, by mechanical means, we move it in the opposite
direction, we do work against the force, and the induced
E.M.F. is now assisting the impressed E.M.F., so that
the energy we expend is thereby converted into elec-
trical energy.) As the current, when the conductor is
at rest, is not the same in magnitude as when the conductor is in
motion, it must be postulated that i is the current flowing while the
conductor is moving with uniform velocity.

Let it so move in the direction of the force through d centimetres
in t seconds. The work done in moving it will be force x distance,
p x d ergs.

Again, if e be the back E.M.F. during motion, the work done by
the current = e.i.t ergs.

.'. p.d. = e.i.t. (i.)

Now the lines cut by the conductor in its motion are those lying
within the area I x d (Fig. 277), i.e. B x I x d lines, and as

*

FIG. 277.

ELEMENTARY THEORY OF INDUCTION 373

amount of cutting
e = — r — — ; — ", we nave
time or cutting

(unity is inserted in this expression because we are dealing with one
conductor),

B x I x d . , T> 7 7 • /•• \

i (11.)

.. .. .

t

Whence we have from equations i. and ii.

p.d. = B.Z.<i x i
or p = B.e'.Z dynes.

That is, the force per centimetre is B.i dynes.

If the conductor makes an angle 0 with the field, then the force is
given by_2> = B.fc.Z sin 0 dynes, as already explained on p. 348.

Rxample. — A conductor carries a current of 50 amperes at
right angles to the lines of force in a field of 10, 000 'lines per square
centimetre. Find the force in Ibs. -weight per foot of length.

By calculation, we first obtain 1 ft. = 30 '48 centimetres, and 50
amperes = 5 absolute units of current.
Now p = J$.i.l dynes

= 10,000x5x30-48 dynes
= 1,524,000 dynes

But 1 lb.- 453-6 grams, and 1 lb.-weight = 981 x 453'6 dynes

= 445,000 dynes (approxi-
mately)

= 3-4 Ibs. -weight per foot-length.

To Find the Field Strength at any Point P, at
Distance d from an Infinitely Long Straight Wire
carrying Current i. — Let B be the field strength at P, Fig.
278 (as the lines of force are circles, it will be the same all round a
circle of radius d). Imagine that an ideal unit magnetic pole is placed
at P, then, by definition, the force c
acting on this pole is H dynes, xT^X- _______ d _____ -->

where B = /*H or H = ?. As /x is Yjy~

P1 FIG 278

unity in air, the two quantities

are numerically equal, but as in many previous instances, we
must be careful not to confuse them.

This force will tend to urge the pole in a circle around the con-

374 VOLTAIC ELECTRICITY

ductor. Let it do so, making one revolution iu t seconds (i being the
current strength during the motion).

Then work done = force x distance = H x ^ird ergs.

Also work done = e.i.t ergs, where e is the back E.M.F., produced
by the lines due to the pole cutting the conductor.

Now the pole has 4vr lines (see p. 136), and in one revolution
these have cut the conductor once,

^4?rx 1
t

whence e.i.t = — x i.t = fari ergs
.-. H x 2-n-d

But B = /xH

So that B = -^.
<i

This result shows that the field strength varies inversely as the
distance from the wire, and not as the square of the distance. This
was originally discovered by direct experiment. (The field due to
a very small part of the conductor does vary as the square, as already
mentioned (see p. 286), but when the effects of all such parts are
added together, the result is as stated above.)

To Find the Force of Attraction or Repulsion between
Two Infinitely Long Straight Conductors carrying Cur-
rents i and iv respectively at Distance d Centimetres
apart. — Consider the conductor carrying the current /P By the

2/M,

preceding proposition, it is in a field of strength -^- , and the force
upon a portion of it of length I centimetres is given by

2*1, 7

i.e. p = — -i x L x /z.
ct

This force is mutual, and hence each conductor exerts a force on the
other, whose value per centimetre of length is JV dynes.

To Find the Field Strength inside a Solenoid having
N turns, Length I Centimetres, and carrying a Current i
Absolute Units. — Let B be the field strength within the solenoid
(assumed to be uniform throughout). Suppose that a unit pole is

ELEMENTARY THEORY OF INDUCTION 375

placed at one end and that it is driven by the force H through the
solenoid in t seconds.

Now, work done = force x distance = H x I ergs

Also, work done = e.i.t ergs, where e = — -—

t

or til = 47T.2.JN
Whence H = — ^— dynes, and as B =

B = — i-l — '£ lines per square centimetre,

where /x is unity for air.

It is important to notice that the assumption, made respecting
the uniformity of the field throughout the solenoid, is not strictly true.
Evidently it must change, at any rate near the ends, and a more
exact investigation would lead to the result that the field at any
point P on the axis, subtending angles a and ax at the end of the

coil, as shown in Fig. 279, is
given by

B= -— /u,(cos a + cos ax) lines
L

per square centimetre.
FIG. 279. Hence, we see that our value

is correct for the field strength at

the centre of an infinitely long solenoid, and that the field exactly at an
end is just half as strong. We may, however, apply our result, without
serious error, to any solenoid of reasonable length, as the cosines
of very small angles do not differ appreciably from unity. It should
also be noticed that B is independent of the diameter of the solenoid.

Coefficient of Self-induction.— For any circuit (or portion of
a circuit), the coefficient of self-induction may be defined as "the
amount of cutting " which takes place in that circuit when unit
current is stopped or started therein. It is usually denoted by the
letter L, and is often called the inductance (or the self-inductance}
of the circuit. In the case of a long solenoid, its approximate value
can be easily calculated as follows : —

When the solenoid carries a current of i absolute units, the fteld
within it is B, where

B = ^'^ x \L lines per square centimetre.

1 It follows from the definition on p. 168 that 4ir.*.N is the difference of
magnetic potential between the ends of the solenoid.

376 VOLTAIC ELECTRICITY

When the current is unity, B = _^— x //, and the total number of
lines passing through the solenoid = B x area of section.
" .-. Total number of lines inside solenoid = — — - x p, where A

I

= area of section.

When the current is stopped or started, these lines cut N turns.

and /. "amount of cutting " = L = - - — x a.1

It will be seen from this expression that when the length of the
solenoid is constant, L varies as the square of the number of turns.

When no iron core is present, L is a constant for any given coil
or circuit, although it is not always easy to calculate its value ; but
when iron is present, p is usually much greater than unity, and, as
will be shown later, its value depends upon the state of the iron,
hence, L is no longer a constant (unless, indeed, the circumstances
are such that /A may be regarded as constant).

It will therefore be perceived that the presence of iron enormously
increases the value of L.

Subject to these restrictions, some of the equations, previously
obtained, may be conveniently expressed in terms of L. For instance,
if the current flowing is not unity, but i absolute units, then the
amount of cutting which takes place, when that current is stopped or
started, is lui. Xow, we have shown that when Z lines cut N turns
in t seconds, the average induced E.M.F. is given by

e (average) = 5_

If this is a self-induced E.M.F., then Z x N - Li, and

es (average.) = -^
t

This does not apply merely to the stopping or starting of a
current : it is simply necessary for the current to change in value by
i units in t seconds. Hence, we have an alternative method of
defining L, for suppose that the current is increasing (or diminishing)
at a uniform rate of 1 absolute unit per second, then the induced E.M.F.
is constant in value, and is numerically equal to L, so that a circuit
may be said to have unit inductance when a current changing at the
above rate produces a steady induced E.M.F. of 1 absolute unit.

If we employ volts and amperes, the expression given above becomes

G

L|"Q J^Q

es (average) = _ = _ volts,

1 For a straight solenoid this expression is approximately correct. For an endless
solenoid (i.e. for a circular coil like that shown in Fig. 289, wound on a non-magnetic
ring), it is strictly correct, and hence such a coil may be used as a "standard in-
ductance, although only small inductances can be readily obtained in this way.

ELEMENTARY THEORY OF INDUCTION 377

which may be interpreted by saying that, when the current is chang-
ing uniformly at the rate of 1 ampere per second in a circuit for
which L— 109 absolute units, the induced E.M.F. will be 1 volt.

In this case, the circuit is said to have an inductance of 1 henry,
or of one practical unit of self-induction, where 1 henry— 10° absolute
units of self-induction.

In a similar manner, equation (i.), p. 368, may be written in the
form

^ = R~

If L and R are expressed in absolute units, Q is in absolute units.
If the current is expressed in amperes, L in henries, and R in ohms
then Q is in coulombs, or

Q = -|£-

Energy Stored up in a Circuit. — When a current is flowing
steadily in a circuit, a certain amount of energy becomes latent.
This energy is set free when the current ceases, and it produces,
among other effects, the self-induction spark when the circuit is
suddenly broken. As already mentioned, it is analogous to the
kinetic energy of a mass in motion, but it must be carefully borne
in mind that the energy is not stored up in the circuit itself, but in
the magnetic field linked with the circuit.

Now, the equation Q ••= . l shows that when a current i is stopped
R

in a circuit of inductance, L, and resistance, R, a quantity, Q, flows
round the circuit, i.e. the circuit behaves as though a quantity, Q, was
stored up in it. We know that this is not really the case, but the
effect is very similar to that obtained when, by the application of
a steady E.M.F. to a circuit containing a condenser, we charge that
condenser to an electromotive force, E, and store therein a quantity,
Q. As a matter of fact, in the one case the energy is stored up in
the form of a magnetic field, and in the other in the form of an
electric field ; in both cases, however, the amount of energy so stored
will be JQ x E ergs.

Now, E = iU by Ohm's law,
and Q =

.*. Energy stored up in the circuit = J-— x ^'R

= JL*2 ergs.

Hence, when L is great, as in circuits with many turns and much
iron, a relatively enormous amount of energy may be stored up, and,

378 VOLTAIC ELECTRICITY

in practice, special precautions may have to be taken to avoid break-
ing such circuits too suddenly ; otherwise, serious injury to the
insulation, due to the sudden rise in voltage, may occur.

Coefficient of Mutual Induction between Two Circuits
pr Parts of a Circuit. — Let Fig. 273, p. 357, represent two
independent circuits of any shape and size, at any distance apart.
When a current is started in one of them, the field thereby produced
spreads outwards through space, and some of its lines must cut the
other, although the effect, except at short distances, may be negligibly
small. Then, M, the coefficient of mutual induction — often called
the " mutual inductance " — between them (in any fixed relative posi-
tion) may be defined as being the "amount of cutting" icltich takes
place in one of them, when unit current is stopped or started in the
other.

As before, M may only be regarded as constant when iron is
absent, and, in that case, if the current in one circuit is i absolute
units, the amount of cutting in the other is M.i.

Hence, when current i is stopped or started in the primary
circuit, the quantity Q passing round the secondary is given by

Q = -— , where R is the resistance of the secondary. The relation
K

is mutual, i.e. either circuit may be used as the primary without
altering the value of M.

The simplest case occurs with two solenoids, one inside the other,
as in Experiment 223. Suppose that the primary has Nx turns, and
the secondary N2. Then if I be the length of the primary, we have,

by a repetition of the argument given on p. 376, a total of -

lines within it when the current is unity.

Assuming that all these lines cut all the turns of the secondary,
when the primary circuit is made or broken, we have, by definition,

As a matter of fact, some of the lines would miss the secondary
unless very special precautions were taken, and hence, this must be
regarded as the limiting value.

Methods of measuring L and M are given in Chap. XXVII.

Magnetic Moment of a Coil carrying a Current—

It has been shown on p. 287 that a circular coil carrying a current produces
a magnetic field, which for a point on its axis has the value

-r, _ 27TTuV2/A

d* '

where d is the distance of the point, and r is the radius of the coil (its being
understood that d is great compared with r).

ELEMENTARY THEORY OF INDUCTION 379

Now, a short magnet of moment M, placed along the axis with its centre at
the centre of the coil, would produce at the same point a Held of strength
2M
d?'

If, therefore, we agree to define the moment of the coil as being numerically
equal to the moment of the equivalent magnet, we have

2M

or M = i.n x irrz x /* = i.n.A.p

where A is the area of the coil.

This result has been obtained by simple reasoning for a circular coil, but if
we investigate the matter more fully, it Avill be found that it is quite independent
of the shape of the coil.

Couple Acting on a Coil carrying a Current in a

Uniform Magnetic Field. —Let a coil of n turns and area A square
centimetres carry a current i in a uniform field of strength B lines per square
centimetre. We have already had occasion to consider such a coil, under some-
what similar circumstances, in connection with suspended coil galvanometers
(see p. 294), and we know that it may be regarded as a magnet, which tends to
place itself along the lines of force, i.e. the coil will tend to turn until its axis
is parallel to the field.

If its axis be inclined at a direction 6 to the field, we know that, in the case
of an actual magnet, the moment of the restoring couple is MH sin 0, and we

T>

can at once apply this result to the coil, if we put M = *'.n.A./x, and H = — •

Then moment of couple becomes i.n.A.E sin 8, and is independent of the
shape of the coil.

It is a useful exercise in fundamental principles to verify this result in the

particular case of a fiat rectangular coil by applying the equation p=}$.i.l
rce on a straight conductor in a magnetic field.

obtained on p. 373 for the force

Let the sides of the coil be of lengths I and b centimetres respectively, and,

for simplicity, let it be placed with
its axis at right angles to the field,
as in Fig. 280, with the sides of
length I perpendicular to the plane
of the paper. The sides of length I
are acted upon by a force p, whereas
the sides of length 6, being parallel
to the field, are inactive. Evi-
dently the moment of the couple
is p x b, and p — B.i.lxn (for n con-
ductors).

. '. moment of couple = TS.i.l.n x b}
FlG. 280. but l.b is the area of the coil,

.'. moment of couple = B.t'.n. A,
which agrees with the above result, as sin 0 = 1.

Magnetic Shell. — The magnetic effect of a flat coil is often more
conveniently treated by regarding the coil as producing a "magnetic shell."
For instance, we may regard the whole of one side of the coil as a north pole,
and the whole of the other side as a south pole, and thus it resembles a thin
plate transversely magnetised, i.e. the area of the polar surfaces is relatively
large, and the distance between the poles very small. Such a transverse slice
of magnetisation is known as a magnetic shell.

380 VOLTAIC ELECTRICITY

Let the area of each end .-surface be A, the thickness t. and let m be its pole
strength. Then the intensity of magnetisation (I) will be given as usual by

I=s-j=pole strength per unit area,

A

. and the "strength" of the shell is defined as being the product Ixt or from
above

Strength of shell = ™ x t.
A.

Now, mt is the magnetic moment, and, therefore,

Strength of shell = S = — = magnetic moment per unit area,

which is independent of the shape of the shell.

Now, it has been shown (see p. 169) that the magnetic potential at any
point P near a short magnet is —L^- , where d is the distance of the point

from the centre of the magnet, and 0 is the angle which this direction makes
with the axis.

Applying this to the shell, for which M = SA,

and writing VOT for the magnetic potential at the point P, we have

•CT- S.A. cos 9
v m --- ->-« - •
fj.d2

But ~dz ~ is the solid anSle subtended by the shell at the point P, and writing
w for this solid angle, we have

__Sw

\,n--.

Or, the potential at any point due to a magnetic shell is equal to the strength of
the shell multiplied by the solid angle it subtends at that point, and is quite
independent of the size and shape of the shell.

Now, we have shown that in the case of a circular coil of n turns, area A,
and carrying a current i,

M — i.n.A..fj,

but S =

Hence, for one turn, the absolute unit of current is numerically equal to the
strength of the magnetic shell it produces.

Again, the magnetic potential at any point due to the coil will be

Quite close to the surface of the coil, and on its axis, w = 2-ir. Hence, for a
single turn the potential is 2iri. At a similar point on the other side of the
turn, the potential has the same value, but with the sign reversed. Hence, the
magnetic P.D. between the two sides is 4iri. The argument may be extended
to any number of turns, and thus we deduce the fact that the P.D. between the
two ends of a solenoid is 4irin (as stated on p. 375).

EXERCISE XIX

1. A copper disc having a diameter of 40 centimetres is rotated about a
-horizontal axis perpendicular to the disc and parallel to the magnetic meridian.

ELEMENTARY THEORY OF INDUCTION 381

Two brushes make contact with the disc, one at the centre and the other at the
edge. If the value of the horizontal component of the earth's field is 0'2 C.G.S.,
find the potential difference in volts between the two brushes when the disc
makes 3000 revolutions per minute. (B. of E., 1906.)

2. What is the magnitude and direction of the force acting on a straight
conductor, 10 centimetres long, placed at right angles to a magnetic field of 50
lines per square centimetres, the current through the conductor being 5 amperes ?
In what unit is your result expressed ? (B. of E., 1905.)

3. A square conducting frame cut through at one place rotates in the earth's
magnetic field about a vertical axis passing through the middle point of opposite
sides. Describe the variation in E.M.F. between the two sides of the break which
consequently occurs, and calculate its maximum amount when there are 120 re-
volutions per minute, if the edge of the square is 25 centimetres, and the inten-
sity of the earth's horizontal force is '18. (Loud. Univ. Inter. B.Sc., 1904.)

4. A circular coil traversed by a current is placed horizontally in the earth's
field. What is the nature of the force acting on the coil, and how does it vary
with (1) the strength of the current, (2) the strength of the field, (3) the number
of turns of wire in the coil, (4) the size of the wire, (5) the radius of the coil ?

(B. of E., 1904.)

5. Two points in the circuit of a voltaic battery are connected by two long
covered wires arranged in multiple arc (that is, each wire would complete the
circuit by itself if the other were removed). The resistances of the wires are in
the proportion of 3 to 4. The one is now bent into a zigzag, the other is
wrapped in a continuous coil round a soft iron core. Show in what proportion
the battery current is divided between the wires (1) when the battery contact
is made continuously, (2) when it is made momentarily.

6. How may the intensity of the magnetic force inside a solenoid be approxi-
mately calculated? What is it in one of 300 turns, 15 centimetres long, which
carries a current of '2 ampere ? What effect has the diameter of the solenoid ?

(Lond. Univ. Inter. B.Sc., 1904.)

7. When a conductor I centimetres in length,. carrying a current c (in C.d.S.
electro-magnetic units), is placed at right angles to a magnetic field of strength H,
the force acting on the conductor is equal to H/c dynes. Use this result to
determine the value of the electromotive force generated when a conductor is moved
with velocity v c.ms/sec. in a direction perpendicular both to its length and
to a magnetic field of strength H. Deduce the result that the E.M.F. generated is
equal to the rate at which the magnetic lines of force are cut by the conductor.

(Lond. Univ. B.Sc. Internal., 1909.)

8. A solenoidal coil 70 centimetres in length, wound with 30 turns of wire per
centimetre, has a radius of 4*5 centimetre. A second coil of 750 turns is wound
upon the middle part of the solenoid. Calculate the coefficient of self-induction
of the solenoid, and the coefficient of mutual induction of the two coils. Will
the inductance of the solenoid be affected by short-circuiting the ends of the
secondary coil ? (Lond. Univ. B.Sc. Internal., 1909.)

9. In what way may a circuit carrying a current be considered equivalent to
a magnetic shell ? Find the work done in taking a magnetic pole round a closed
curve which threads an electric circuit once. (B. of E., Stage III., 1909.)

CHAPTER XXIV

BALLISTIC GALVANOMETERS

GALVANOMETERS are not required merely as indicators in null
methods, or as instruments for reading steady deflections. In dealing
with condenser discharges, as in Experiment 184, p. 313, and with
measurements involving induced currents, we are concerned with a
sudden rush of current through the instrument, which is completed
in a small fraction of a second, and thus the term current is no
longer applicable. We have then to measure a "quantity," not as
a continuous flow but as a sudden discharge, the difference being
somewhat analogous to that involved in measuring (a) a steady flow
of water, and (/>) a sudden rush due to emptying, say, a single
bucket of water through some apparatus. Such a discharge of
electricity will produce a swing or "throw" of the needle of the
galvanometer, and then it will settle down to rest again. We can
no longer obtain a relation between current and deflection by equating
the moments of opposing couples when the needle is in equilibrium,
but we may estimate the kinetic energy given to the needle by the
quantity passing through the instrument, and then equate this to the
potential energy of position at the instant it reaches the end of its
first swing. The method implies

(1) that the discharge through the instrument is completed before
the needle has moved sensibly from its zero position ;

(2) that no energy has been lost during the swing, in other
words, that the deflection is entirely undamped.

Condition (1) is necessary, because it enables the energy due to
the impulse to be easily estimated, and it is satisfied by making the
natural time of vibration of the moving part rather great, so that
it starts slowly from rest. About five seconds for one complete
vibration is a convenient value; it may be greater than this with
advantage, but should not be much less.

Condition (2) cannot be completely satisfied, neither is it always
desirable to approximate too closely to it ; it is sufficient if the
damping is comparatively small in amount. The effect of departure
from this condition is evidently to make the throw, as measured on
the scale, less than it would be if there were absolutely no damping,
but as the undamped value is required by the theory, it is necessary
to multiply the observed throw by a "damping factor," which is

BALLISTIC GALVANOMETERS 383

calculated from data obtained during the experiment. When only
comparative values are required and the throws are not very different
in magnitude, the factor cancels out, and hence it is not introduced
in the experiment mentioned in the worked Example 2, Chapter
XXIII. For further details, the student is referred to the Appendix
to this chapter.

It must be distinctly understood that the term "ballistic," when
applied to a galvanometer, does not mean any special type or con-
struction. Any pattern may be used, if only conditions (1) and (2)
are satisfied, and these do not involve any serious structural changes.
In practice, instruments of both the moving magnet and the moving
coil types are employed, but, on the whole, the former are the more
generally useful, for although the latter possess the great advantage
of being unaffected by external fields, in some cases the damping effect
due to induced currents in the moving coil is apt to be inconveniently
great, as will be seen in some of the following experiments.

Hence, we shall first consider the simple type in which the
moving system is a permanent magnet controlled by the horizontal
component (H) of the earth's magnetic field.

Let M — the magnetic moment of the needle system,

I = its moment of inertia,1
and to = its angular velocity.

Now, the " impulse " due to the discharge is equivalent to the
action of some couple, C, acting for a very short time t, and we have,
by a well-known theorem in mechanics,

CJ = Io> (1)

Let G be the strength of the magnetic field at the needle, due to
unit current flowing through the instrument. Then, if i be the
average current during the time t, the average field at the needle
is Gi. This field is at right angles to the undisturbed position of
the needle, and from p. 152, the moment of the couple acting on the
latter is Gi x M.

.-. from(l) G*M# = Ico
but i.t = Q

.-. GMQ = Iw (2)

Again, the kinetic energy given to the moving system is JIt»2,
and if 0 be the angle of the first swing, the potential energy at the
end of that swing is (from p. 168) MH(1 - cos 6).

-cos (9)

= MHx2sin2f (3)

2

1 The letter "I" had a different meaning in the last chapter, and will
shortly be used for yet another purpose. Experience shows that there is no
real danger of confusion.

384 VOLTAIC ELECTRICITY

Finally, if T be the time of one complete vibration (see p. 155)

We have, therefore, equations (2), (3), and (4) from which to obtain
an expression for Q.

G2M2Q2

From (2) o>2 =
from (3) &>2 =

I2

4MH sin2 ~
_

4IHsin2-
*'• Q'r MG2

Also from (4) ^ = -7- -?
x y M 4?r2

T2H2 sin2 ?

n - /~\

or Q^-^TQ-nn-g

Therefore, the " quantity " passing through the galvanometer is pro-
portional to the sine of half the angle of the first throw.

In practice, H is usually some unknown field due to the joint
effect of the earth and a controlling magnet. This does not affect
the argument, although it means that in order to obtain an expression
suitable for general use, the factors H and G must be evaluated in
some way by experiment. This can be done by noticing (from p. 285)
that if a steady current i passes through the instrument producing
a steady deflection a°, then if a is very small,

i = — tan a
or

tan a G

Hence, Q = — - .sin ^ (6)

Tr.tan a 2

Here, Q will be in absolute units or in coulombs according as i is
expressed in absolute units or in amperes.

In actual practice, ballistic galvanometers are always reflecting
instruments, which means that 0 is always fairly small, for under

BALLISTIC GALVANOMETERS 385

ordinary circumstances, a deflection of 6° or so would send the spot
of light off the scale. Hence, for many practical purposes, we may
regard the sines and tangents as numerically equal to the angles
themselves,1 and these can be expressed in terms of the observed
deflection and the distance of the mirror from the scale. Thus, if
d and d1 are the scale readings corresponding to deflections 6 and a,
and / the distance of the scale from the mirror, we have (remembering
that the angular motion of the spot of light is twice that of the
mirror)

sin 20 = 20 = ^, and tan2a = 2a = ^

. e

SU12 6 d

tana 2a
and equation (6) becomes

There is some risk of confusing the i and d^ in this expression
with similar letters necessarily used in subsequent work, and hence
we shall write I and D respectively for them. Again, to indicate
when practical units are being used, i.e. when i is in amperes, we
shall write A instead of I ; so that equation (7) becomes

T.I
Q = n—fz x d (absolute units)

or Q = E x d (coulombs) (8)

If we examine the case of a suspended coil galvanometer with
metallic strip suspension, we shall find that Q is proportional to 0

n

and not to sin -, although the resulting expression, in its final form,

a

will be identical with (8). In fact, we may say that this equation,
which is the most convenient form for ordinary use, holds generally
for all types of galvanometer, provided that the deflections are
sufficiently small, so that we have

Q = m x d,

where m is a constant depending on the instrument.

This result has been obtained by assuming that no energy is lost
during the swing, i.e. that the oscillations would continue for ever,

1 These approximations are sufficiently exact for most purposes, but when
necessary a correction can easily be applied.

2B

386 VOLTAIC ELECTRICITY

but we know that this is not true, and that the observed throw d is
less than it would be if there was no damping. Hence, we have to
multiply the observed throw by a " damping factor " before we can
use it in the above expression (unless we are dealing with "comparative
values under similar conditions, as in Experiment 184, p. 313, in
which case this . factor may cancel out), and it is shown (see
p. 393) that this factor is (i+-|-), where I is the "logarithmic
decrement." Hence, the complete expression is

Q = wkJ(i+i). (9)

One word of warning may be given — the use of shunts should
be avoided as much as possible. When it is necessary to reduce the
sensitiveness of a ballistic galvanometer some other method must be
adopted (unless, of course, it is being used with steady currents, as
in the experiment given below). This is because a momentary rush
does not divide up between two possible paths inversely as their
resistances, but inversely as their impedances, and the galvanometer
behaves as if its resistance were temporarily increased. Hence, the
results given on p. 273 cannot always be applied.

Practical Applications. — Exp. 230, to find the constant of a
ballistic galvanometer. First method. — We have Q — md (l+^), where m is
the constant to be determined.

Tx A

Now m=- — , and T can be readily found by setting the needle swinging

£TT X \j

and observing the time in seconds taken to execute a certain number of complete
oscillations.

To find a value for — we may proceed as follows : Insert the galvanometer

(Fig. 281) in a circuit containing a cell of known E.M.F. and a resistance box, R.
Connect up another resistance box, S, as a shunt to the
galvanometer (in order to keep the spot of light on the

C

©

scale, and also to avoid injuring the instrument). A
Daniell's cell is suitable, and if necessary its E.M.F. may

the method described in Experiment 178, p. 304. For

1'07

be determined by comparison with a standard cell, using

scribed in I

ses, it may

tness. Ma

20,000 ohms— and adjust the shunt^until a convenient
FIG 281 deflection is obtained. Let this be D divisions, then we

l ^ rg-|_>' ordinary purposes, it may be taken as 1'07 volts with

f \ sufficient exactness. Make R large— say 10,000 to

HJLr 20,000 ohms— and adjust the shunt until a convenient

deflection is obtained. Let this be D divisions, the

have

=— [ because —— is usually negligible compared with R )
K\ s+g J

also C =A= _i_ xC= -f-x?

s+g s+g R

. A s E 1
' D s+g R D

BALLISTIC GALVANOMETERS 387

and.'. TO=T^A = — TxsxE

2?r x D 2?r x (s +g) x R x D

The following values (taken from a student's note-book) were obtained in an
actual experiment : —

Galvanometer resistance = 960 ohms
Shunt =15 ohms

R = 20, 000 ohms

D =158 divisions

T = 4'1 seconds

Daniell's cell was used taken as 1*07 volts.

4-1x15x1-07 "..xlo-9 coulombs,

2 x TT x (15 + 960) x 20, 000 x 158

which is the quantity that would give a "throw" of one scale division, if no
damping existed.

Second Method. — This method requires a condenser of known capacity, and
a cell of known E.M.F., but as the latter is not needed to give a current, some
form of standard cell may be used.

Exp. 231. Arrange the apparatus as in Experiment 184, p. 313, using a con-
denser key. Observe several throws, due to the discharge of the condenser
through the galvanometer, and take the mean. Let this be d divisions. Also
determine I, the logarithmic decrement (see Appendix to Chapter).

Then Q = EK =
EK

or m — — — j- .

The following result was obtained in an actual experiment with the same
galvanometer as before : —

E = 1-0183 (a standard cadmium cell)
K=-5 micro farads = '^ farads

d=l50 scale divisions
£=•06

1-0183 x — „

*• =15o'x01l8033XX'lO»=3-3><10-9-'-b-

Third Method (by using a standard solenoid).1— In this case we require an
accurate ammeter of suitable range, a current reverser, an adjustable resistance,
a battery .of a few cells capable of maintaining a small steady current for a short
time, and (possibly) a resistance box.

Exp. 232. The apparatus should be set up as shown in Fig. 282. It will
be seen that the battery is arranged to send a steady current through the
solenoid, whilst the secondary coil within it is connected to the galvanometer.
The resistance box, r, is inserted to diminish the damping, and with some
instruments may not be required.

Adjust the current to some convenient value. Then observe the throw when

1 See Appendix to this Chapter, Section II,

388

VOLTAIC ELECTRICITY

that current is suddenly reversed. If the throw is inconveniently large, either
reduce the current or suddenly interrupt instead of reversing it ; this is done by
lifting up the rocker, instead of moving it across, and it means that the lines
of force cut the secondary coil once instead of twice, so that the throw is half as
great. (Increasing the resistance r will also diminish the throw.) Then deter-
mine the logarithmic decrement, Z, taking care not to alter the conditions, i.e. circuit

remains closed through the same resistance.

( *t^3

V/wwwww \ f \ A I

R

/ — EX}-

Let C — current in amperes, as read on ammeter,
d —throw produced when this current is

reversed.
R = total resistance in the galvanometer

circuit.
Zg = number of lines threading secondary

coil when the steady current of

Qs amperes flows through the

solenoid.

NS= number of turns in the solenoid.
Zs=length of solenoid in centimetres.
t = number of turns in the secondary coil.
A — Area of section of secondary coil = irr

Then, if Q = quantity passing through the galvanometer,

9 7 f

also Q = — — (for the lines Zg cut tg twice. If current is merely broken, the

2 disappears).

•'• m~ ~~77 1~\ (in which all the quantities are known except Zs).

It will reduce the risk of error in working, if we express R in absolute units ;
then m will also be in absolute units.

To find a value for Za> we proceed as foyows : —

Number of lines per square centimetre inside solenoid = — s(see p. 401)

/. Number of lines threading secondary coil = Zs=

T257CN

x -A-

1/257C N
/. Zs= —^ — x irr2, where Cs is in amperes, and r is the radius

ls

of the section of secondary coil.

Substituting this value in the expression for m, we obtain
2x1-2570 xN XTrr2xt

(This is in absolute units ; for coulombs, the value will be 10 times as great.)

BALLISTIC GALVANOMETERS 389

In an experiment, using the same galvanometer as before, the values obtained
were —

Cg= '504 amperes
3sTj,= 869 turns
r— 2 '53 centimetres
^=1000 turns
^=136 centimetres
R=1957 ohms
d — 194 scale divisions

• m = 2 x 1 '25? x '504 x 869 x TT x (2'53)2 x 1000
136 x 1957 x!09x 194x1 -3

= 3-2xlO-10
or m=3'2 x 1Q-10 absolute units=3-2 x 10~9 coulombs.

In this experiment, the damping was excessive, and should have been reduced by
putting more resistance in the circuit. For it will be seen in the Appendix that
the expression 1 + 4- does n°t no^ g°0(l when I is large.

The preceding experiments should be performed on account of the
useful training in manipulation and theory thereby obtained, but, in
other respects, they are of no particular importance. For instance,
-any alteration in the distance of the scale or in the nature of the
working conditions will alter the value of m, and, for this reason, it
is customary, as will be seen in most of the experiments subsequently
described, to either eliminate or evaluate m in the course of the work.
The following experiment is given as an application of the previous
results : —

Exp. 233, to determine the vertical component of the earth's magnetic field.
Connect up to a galvanometer an " earth -inductor " coil. This may be simply a
rectangular or circular coil of known area and number of turns. Insert a resis-
tance box in the circuit if necessary in order to diminish the damping. Lay the
coil flat upon the bench, and then turn it quickly over. Note the throw, and
repeat several times, and then obtain the mean. In this operation the turns of
the coil cut the vertical component twice (see p. 351). Also determine the
value of I.

Let V= vertical component

R= total resistance of the galvanometer circuit
te,— number of turns in earth inductor
A = area of inductor
d= throw obtained.

Then Q = md(^ + -|-) (absolute units)
Also Q =

2A x te

390

VOLTAIC ELECTRICITY

The results in an actual experiment were —
#=960 ohms

R=3041 ohms (being the resistance in box, 2000 ohms, resistance
of inductor coil, 81 ohms, and of galvanometer, 960 ohms)
d=17l'5 divisions

1 + 1=1-025

A=415 square centimetres

m taken as 3'3 x 10-10 absolute units

As V=

y_

2Axte
3041 x 109x 3-3 xlO~10x 171-5 x 1°025

2x415x500
= *43 nearly.

Exp. 234, to measure the capacity of a condenser. Connect up a constant

cell, e.g. an accumulator or a Daniell's cell,
to a potentiometer ; charge the condenser
by means of the P.D. between the ends of
the potentiometer wire, and then discharge
it through the galvanometer. For this pur-
pose a condenser key may be used, the
general arrangement being as shown in
Fig. 283. It will be seen that depressing'
the key charges the condenser, which is
discharged through the galvanometer when
the key is released. Also determine the
damping factor.

FIG. 283.

Let e=P.D. between the ends of the potentiometer wire, AB, in

volts,

and K=the capacity of the condenser in farads.
Then Q=eK (1)

.-. K=:

(2)

We can determine T and — substantially as in Experiment 230, and the

potentiometer wire is merely used because it
affords a ready means of eliminating e. Let the
apparatus be now arranged as in Fig. 284. Place
a large resistance, r, in series with the galvano-
meter, and connect it across a portion of the
potentiometer wire, AP, of such a length that
a steady deflection of a convenient amount is
obtained. Let this be D divisions. Then if
A is the current through the galvanometer
in amperes, and e1 the P.D. across AP, we
have

FlG. 284.

BALLISTIC GALVANOMETERS 391

*$

1 AB
. A_AP
D~ABX

Substituting this value in (2), we obtain

K=TJ^(lj__l/_l I farads.*

Sometimes it is desirable also to shunt the galvanometer in order to avoid
making AP inconveniently small.

The results of an actual experiment (using another galvanometer) were —
T=3 seconds
c£=61 scale divisions

#=250 ohms

r=20,000 ohms

s=40 ohms
AP=550 centimetres
AB = 1000 centimetres

D = 113 divisions.

In this case, the student also shunted the galvanometer in the second part of
the experiment, which makes a slight difference in the calculation, for

A will be ^ x -f-

r+
s + g

and as — — is negligible compared with r,

-= 6l v S
D rD s + g

Again, e^xe
.'. substituting in (2), we have

K-_Tx^(i+ j)xAPxs
2ir x r x D x AB x (s+g

3x61x1-07x550x40

~2 x IT x 20,000 x 113 x 1000 x (40 + 250)
= 1 '04 xlO-6 farads
= 1'04 microfarads.

The condenser used in the experiment had not been finally adjusted ; it was
known to be nearly 1 microfarad.

1 It must be remembered that such measurements of capacity may be seriously
affected by "absorption" in the dielectric (see p. 68).

392

VOLTAIC ELECTRICITY

APPENDIX TO CHAPTER XXIV

(1) DAMPING CORRECTION

LET us suppose that the needle is set in motion by the passage of a discharge
through the instrument, and that the scale readings are observed as the oscilla-
tions die away. Let these be plotted as ordinates against time, as shown in
Fig. 285, where + d merely means a deflection to the right, and - d a deflection
to the left. We are really only able to observe the turning points of the spot
of light, but the complete curve is of the type OCFG, the amplitudes of suc-
cessive swings gradually decreasing. If, however, the assumption underlying
our formula held good, the curve would be that shown in OAH, the amplitudes
remaining constant. Hence, in using the instrument we actually read the first

271

FIG. 285.

throw BC, whereas we require the value of the undamped throw AB, and we
have therefore to show how this value can be found.

For this purpose, it is necessary to assume an elementary knowledge of
simple harmonic motion. Students possessing such knowledge will know that
the ideal undamped curve can be represented by an equation of the form
d=a sin 6 where a is the maximum amplitude AB, and positions on the hori-
zontal axis are denned in terms of angle by regarding one complete oscillation as
equivalent to 2?r radians or 360°.

Similarly, if we take into account the influence of a frictional retarding force
proportional to the velocity, we find that the actual curve is represented by

d=ae~bt sin 6.

Where a has the same meaning as before, e is the base of the natural logarithms,
6 is a constant depending on the instrument, and t is the time that has elapsed
since the needle started from rest.

Now, the successive maximum deflections (i.e. the scale readings we can actually

observe) will correspond to sin 0=1, and will occur when 0=o' o*"» o71"' &c-

T

Let T be the time of one complete oscillation, then when 0=o>

4'

and when

3T

••-. -, and so on.
4

DAMPING CORRECTION 393

Hence, we can tabulate tho successive readings as under : —
To right. To left.

ae—Y (BC in figure) ac—'i~ (DF in figure)

. .,, T, WT

(GIv ,, ) ae-~*-

&c. &c.

Let A15 A2, As be the successive amplitudes of swing (i.e. if the first throw is
d divisions, and then the spot of light swings back to dl on the other side of the
zero, the amplitude Al is d + d^ and similarly A2 is
dj +d2 divisions, and so on), then

»3T iST WT/_

A2 = oe— r+«-T = ae-~^\\ +e

boT WT &5T/,

A3 = ae—r + ae—i- = oe— i"\l-|-«

From the symmetry of these results, it will be
seen that if m, w represent any two amplitudes (not jTIG 286

necessarily consecutive),

A

= oe- 6(2»-Di(l + e-J

_ _6(2m-l)J-f&(2n-l)

Or j!*= e(n-mT
An

As 6 and T are constant, this ratio also remains constant as the oscillations die
away.

Let b . | = J, then A^ e(»-*»)«

Aw

or/ = _L_loge^ (1)

n~m An

I is known as the " logarithmic decrement " of the swing.

In order to apply this result, we notice that the first throw of d divisions,
which is the throw actually observed in practice, may be written

Let d0 be the corresponding undamped throw, then d0=a, and this is the
value we- really need.

•2+ ,L +&c.j

And when I is small (as it always should be in practice) we may neglect I2 and
higher powers, and, therefore, we have

d0=d(

l +

The value of I must always be obtained under the working conditions, for it will
depend upon the nature of the circuit, e.g. upon whether it is open or closed and
upon the amount of resistance in it. The needle is set in vibration in any
convenient way, the extent of its excursions being approximately the same as

394 VOLTAIC ELECTRICITY

the observed deflection, and, as the oscillations gradually die away, thro
readings corresponding to three, successive swings are taken. For example, if th
readings are 128, 122, 117 divisions, then

An 122 + 117 139
and as n - m=l, we have from (1)

When the damping is small, it may not be easy to determine with accuracy th
difference between three successive swings, and then it is convenient to mak
n - m greater than unity.

In practice, students should remember that the numerical value of 1+4
is to be only slightly greater than unity. If it is large, the working condition
should be altered. Little trouble will be met with when using galvanometers
whose moving part is a steel magnet, for, even on short circuit, the damping wil
be small, but it is very different with instruments of the moving-coil type
In this case (as shown in Experiment 21 8, p. 354), on a closed circuit the motioi
is powerfully damped by induced current produced in the coil, and unless tin
effect is kept in check by introducing a sufficiently high resistance into th
galvanometer circuit, our results will only hold good approximately.

It will help the student to realise the fact that the amount of damping s
depends upon the working conditions, if he performs the following experiment : —

Exp. 235, to find the resistance of a, galvanometer by observing the "damping"
under different conditions: 1. When the galvanometer is on open circuit (Zj)
2. When the galvanometer is on short circuit (Z2) ; 3. When a resistance or i
ohms is in circuit with it (Z3).

Now, the damping is partly due to air friction, and partly to induced
currents whose magnitude is inversely proportional to the resistance in th
galvanometer circuit, and it may therefore be represented by an equation o
the form,

T>

£=A + — , where A and B are constants.
it

Hence, we have

*!=A + ^ (i.)

^=A + 5. (iL)

y *

*>=A + JL

If we eliminate A and B between these equations, we obtain

and hence the resistance of the galvanometer may be found, although, of course
it is not suggested that this is a method of determining g with exactness.

The results of an actual experiment with a suspended coil galvanomote:
(taken from a student's note-book) were: Z1='0162; 12='36Q; /3='174 whei
r=1000 ohms. This gives #=820 ohms. The real value was about 900 ohms.

It will be noticed that the damping on short circuit is excessive, and henn
the result is unreliable. Under the circumstances it would have been better t(

DAMPING CORRECTION 395

ii,iv<> inserted a resistance in (2) as well as in (3), say 1000 ohms in (2) and
2000 ohms in (3).

Simplified Expressions for Damping-. — We have treated the

question generally, on account of its importance in other branches of the subject,
but simplified forms of the expressions obtained are frequently given in text-
books which obviate the necessity of determining the logarithmic decrement and
at the same time are sufficiently exact for most purposes when the dampitig is
small.

We have, from the results given on p. 393, and using the notation indicated
in Fig. 286,

, MT

d2 = ac — ¥"

or 21 = \oge~ (1)

Again, we have shown that

.'. tf = log,2 (2)

From (1) and (2) we obtain log ^iloa.A (3.)

d ft.,

This may bo written in the form

log d0 - log d = i (log d - log dg)

and when the damping is sufficiently small, this becomes approximately
d0-d = \(d- d2)

or d0 = d + 1^2 (4)

4

For instance (taking numbers at random), let d, d^ d2, be 90, 85, and 81
respectively. Then, if we use the expression

d0 =d(i+|)

it will be found that d0 = 92 '37.
If we use equation (4), we obtain d0 = 92*25.

(2) STANDARD SOLENOID

It is shown on p. 375 that the field strength at the centre of a long solenoid
— without an iron core — is given by the expression : —

B47T.Z.N ,

— j — x jw, where p. = 1 for air,

and hence the field strength can be readily calculated when *, N, and I are
known. (It will be noticed that this result is independent of the area of cross
section.)

Such a solenoid is very convenient when a field of known value, but not of

396 VOLTAIC ELECTRICITY

great strength, is required for standardising a ballistic galvanometer, as in
Experiments 232 and 236.

A secondary coil is necessary, which is sometimes merely wound outside the
middle part of the solenoid. It is, however, distinctly preferable to wind it on
an accurately turned wooden bobbin, which can be placed inside (as indicated in
Fig. 282).

The solenoid used in Experiment 232 (and in several subsequent experiments)
was made by winding one layer of 18-gauge cotton- covered wire on a pasteboard
tube about 8 centimetres in diameter, fitted with wooden ends to which the
terminals are attached. It has 869 turns, and is 136 centimetres long. The
secondary coil sliding within it has 1000 turns of 30-gauge silk-covered wire
(although 500 turns would have been amply sufficient), and is wound in one
layer on a truly turned boxwood bobbin. This is done because its area of
section must be accurately known (when its position is inside the solenoid),
whereas such information is not required for the solenoid itself.

The above dimensions are given as an illustration, but they are largely
immaterial, provided that the solenoid is long compared with its diameter, and
that the construction is such that N and I can be readily evaluated.

CHAPTER XXV

THEORY OF MAGNETISATION

WE have already indicated in Chapter XII. the importance of the
ideas embodied in the fundamental equation B = /xH, and it now
becomes necessary to discuss them more fully. In the first place,
we must remember that both B and H may be represented by a
system of " lines of force " ; and by many writers, diagrams such as
those shown in Chap. XI. (and the term " line of force ") are understood
to represent H. In air, these diagrams may be regarded as repre-
senting either B or H, but inside magnetisable bodies, B and H
have different numerical values, and then it becomes necessary to
distinguish between "lines of force" (H)," lines of magnetic induc-
tion " (B), and (sometimes) " lines of magnetisation," which may be
regarded as the difference of the two quantities.

There is, however, some risk of confusion of thought in dealing
with two or three distinct kinds of "line," and we prefer (unless
otherwise stated) to use the term " line " in one sense only, viz. in
a manner analogous to that already adopted in connection with " lines
of electric force."

Let a magnetising force (H) be produced in any region of space.
(We are not at present concerned with its origin ; it may be produced
in various ways, e.g. by a current in a wire or coil, or by a permanent
magnet.) At any point therein, H is measured by the force in dynes
exerted upon a unit pole placed at that point. This magnetising
force produces a magnetic field, which we shall represent by "lines
of force," its strength being measured by the number of lines per
square centimetre. This number is denoted by B, and is often called
the "magnetic induction." By a perfectly arbitrary convention, we
then agree to regard B as being numerically equal to H in air (more
precisely, in a vacuum, but the difference is very slight), which means
that the permeability (p) of air is taken as unity.

Now, suppose that a substance other than air — e.g. iron — is
placed in this field. If we can regard the substance as filling up the
whole space so completely that we need not consider its boundaries,
then the magnetising force acting upon it will be the same as it was
before the substance was introduced. But it is quite otherwise when
we have to deal with substances which are not unlimited in extent.
In such a case, if the body is more strongly magnetic than air, from

397

398 VOLTAIC ELECTRICITY

one point of view we may think of it as concentrating the lines of
force, which will tend to pass through it in preference to the sur-
rounding air.

The distribution of the field inside the body depends upon the
shape of the latter ; it is not as a rule uniform, but it can be shown
to be uniform in two cases, (a) when the body is a sphere, (b) when
it is an ellipsoid with its major axis parallel to the field.

The state of affairs when the body is a sphere of soft iron is
shown in Fig. 287, which is taken from Carey Foster and Porter's
Electricity and Magnetism. There is no particular difficulty in
measuring the value of B inside a sphere (by methods subse-
quently to be described), but it is important to notice that B is
much less than it would be under similar conditions inside a mass
of iron of unlimited extent. This is due to the self -demagnetising
effect of short chains of iron particles already explained on p. 11 7. It
is an effect which is greatest at the poles, i.e. where the lines pass

FIG. 287.

from iron to air, and is least at the middle of a long chain of
particles, and hence it is particularly marked in the case of a sphere.
These facts are usually expressed by saying that the poles developed
in the iron exert a " demagnetising force," and hence the effective
value of H inside the iron, on which B depends, is much less than
it is outside in air. Now, the value of H inside the body cannot
be found experimentally, and we therefore see that it would lead
to grossly incorrect results if we were to measure the values of B
produced in an iron sphere by various magnetising forces, and to
regard them as being solely due to those magnetising forces. If
we use a bar, instead of a sphere, the difficulty still exists, but we
can easily see that it will become of less importance as the length
of the bar increases relatively to its cross section, and it can be
shown that if the length is from 500 to 1000 times as great as the
diameter, there will be no serious error if we regard the magnetising
force at the centre of the bar as being identical with its value before
the bar was introduced into the field.

THEORY OF MAGNETISATION 399

It follows that long bars or wires should be used in such
measurements, when we are dealing with strongly magnetic substances.
It also follows that, when the substance is only feebly magnetic, this
condition is not so important, for the disturbing influence due to
the induced magnetisation will be small, and then good results may
be obtained with relatively short specimens.

Induced Polarity. — In order to attain clearness of ideas, it
is necessary to regard the phenomena of magnetisation from several
points of view. For instance, we may think of a substance, like
iron, as becoming magnetised and developing poles, from which
emanate a field of its own, superposed upon the field due to the
original magnetising force. These induced poles may be treated
mathematically as points, and in the case of a sphere they will be
very near together on each side of its centre. Let m be the strength
of each induced pole, then, as stated on p. 136, each may be regarded
as a source producing a radial magnetising force, whose value at

T , 7 • in

distance d is -— .
/xcr

In order to show what these statements mean, it will be con-
venient to represent the three magnetising forces (not the field) by
a line diagram. Let NS, Fig. 288, be the induced poles, from which
are drawn a series of radiating lines to represent their " magnetising
force." Let these lines be superposed upon r x v / ^

a set of parallel lines to indicate the original •> — \y V/ *

magnetising force H. Taking the resultant, ~ ^A, ~ /*v
we obtain the effective value of H at each ^ 4 ^x \ >

point, and it can be shown that this would
lead to the state of affairs shown in Fig. 287.

Now it will be seen from Fig. 288 that, in the space between the induced
poles, the direction of the magnetising force due to the poles them-
selves is opposite to the direction of the original magnetising force ;
from which it follows that in the body itself the poles give rise to a
"demagnetising force," in consequence of which the effective value
of H will be less than its value outside. Again, as the effect of these
poles varies inversely as the square of the distance, the longer we
make the body (thus separating its poles) the smaller will become
this demagnetising force at its centre, until at last it becomes small
enough to be negligible, although by this means it can never be
completely eliminated.

Method of completely Eliminating Effect of Induced
Poles. — In practice, however, such complete elimination can be
obtained in a very simple way. For most purposes, the original
magnetising force is produced by a coil of wire carrying a current,
and if, instead of experimenting on a long bar inside a long solenoid,
we suppose the ends of the bar (and also the winding) to be bent
round to form a closed ring, then there a.re no free poles, and the

400

VOLTAIC ELECTRICITY

lines of the field are wholly inside the iron, so that the theory be-
comes as simple as in the case of a medium of unlimited extent.

We have now to apply these ideas to the more important experi-
mental methods adopted in practice.

Exp. 236. Procure a ring : as nearly circular as possible, having a diameter
of about 6 inches, and made out of J inch soft iron bar. Anneal it as care-
fully as possible. Measure the diameter of the iron in several places, and
then calculate its cross section. Also measure the mean diameter of the ring.
Cover the iron with a thin layer of silk or of paraffined paper, and then wind
one layer of 18 or 20 gauge double cotton-covered copper wire uniformly all
round it. Where the ends meet, wrap a strip of silk around each of them to
obtain good insulation, twist them together, and bring them out to serve as
terminals. Finally varnish or dip in melted paraffin-wax. Count the number
of turns N.

Now wind upon the ring a small " test coil," or secondary, of say 10 or 20
turns. This is conveniently made of fine wire, say 30 gauge, and is one layer
deep, so that the number of turns can be easily counted. " (The best number
to use will depend upon the sensitiveness of the galvanometer, and is unknown
at present, but this coil is a very simple affair, readily removed and altered.)

Connect up the arrangement as shown in Fig. 289, in which E is an ad-
justable resistance ; A, an ammeter ; ab, a current reverser ; and B, a battery.

By these means, any required
current up to about 5 am-
peres may be sent through
the magnetising coil, N, in
either direction. The test
coil, t, is connected to a re-
flecting ballistic galvano-
meter, G, preferably of the
suspended coil type, and in
the circuit is included some
device for calibrating the
instrument.

Many such devices are
possible ; for instance, if the
vertical component of the

earth's field is known with sufficient accuracy, an earth inductor coil may
be used, as explained in Experiment 233 ; but perhaps the most convenient
and accurate method is to use a standard solenoid, described on p. 395. Its
secondary, t8t is included in the galvanometer circuit during the experiment,
as shown in the figure.

(1) Adjust the resistance and the number of cells until the maximum
current intended to be used is obtained — say 5 amperes. Suddenly reverse
the current, and notice the throw obtained. If the spot of light moves off
the scale, the number of turns in the test coil, t, must be reduced until it
remains on the scale ; if the throw is not fairly large, increase the number of
turns in the test coil.

(2) Then adjust the current to some small value until the throw on re-
versal is just large enough to be measurable — say 5 to 10 scale divisions.

FIG. 289.

1 Such a ring is readily procured, and is useful for teaching purposes.
When the ring is made with a view to exact measurements, it is better to give
it the form of a hollow cylinder, which can be trued up in a lathe. A con-
venient size will be about 1 inch long, 4 inches mean diameter, and thickness
of iron J inch. Before winding, these dimensions must be very exactly
measured in centimetres.

THEORY OF MAGNETISATION 401

Note current and throw, repeating the readings once or twice to avoid mis-
take. Slightly increase the current, note its value, then suddenly reverse and
note the throw. Proceed in this way (taking all the readings on the same
side of the scale), until the maximum current is reached.

(3) The next step is to calibrate the galvanometer. Without, in any other
way, disturbing the apparatus, disconnect the magnetising coil from the
reverser at the points a and 6, and, in its place, join up the ends alf bt of the
solenoid. Adjust the current very exactly to some definite value, and note
the throw on reversal. This is due to the lines of the solenoid cutting the
turns of the secondary coil, ts. If the throw is too small to be read accurately,
either increase the current or increase the number of turns in the secondary.
If possible, take two or three readings with different currents, and notice if
the readings are— as they should be— exactly proportional to the current. By
doing this, accidental errors are avoided.

We are now in a position, from the various data obtained in this experi-
ment, to work out the values for both H and B from equations already
obtained.

To obtain Values Of H. — In the case of H, this is a simple
matter, for H = 7rz , where N is the number of turns in the mag-

netising coil of the ring, and I is its mean circumference, i.e. the mean
length of the path of the lines of force inside the iron expressed in
centimetres.

As, in the experiment, the current was measured in amperes, it'
is convenient to write the equation in the form

10Z
where C is in amperes ; this therefore becomes

I

1*257N
Work out the value of the constant - - — : let this be m. Then

t

for each reading obtained in the experiment, H = m x C.

To obtain Values Of B. — Let t = number of turns in the
test coil on iron ring; ts = number of turns in the secondary coil
inside the solenoid (as time does not enter into the calculation, we can
use t for turns in this case without risk of error); Ns = number of
turns in the standard solenoid ; and la = length of standard solenoid
in centimetres.

Consider any one of the " throws " obtained in (2) of Experiment
236. Before reversing the current, the iron was magnetised up to a
certain value of B ; on reversal, it was suddenly demagnetised and
then remagnetised in the opposite direction up to the same value of
B. The throws due to each operation are in the same direction, and
the observed throw is, therefore, due to all the lines present in the
iron cutting the turns of the test coil twice.

2c

402 VOLTAIC ELECTRICITY

Let Z = number of lines present in the iron, and Q the quantity
passing through the galvanometer, which produces the throw of d
divisions,

then Q = «d (1)

K

where R is the total resistance in the galvanometer circuit.

Again, in the calibration experiment (part (3) above), let Zs = the
number of lines present inside the secondary, and Qs the quantity
passing through the galvanometer, which produces a throw ds, when
the current in the solenoid is Cs amperes,

then qs = 2x^xtsocds (2)

-TV

Dividing equation (1) by equation (2), we obtain
Z# _d_
RterM*

.-. Z = Z'x*'xd (3)

t X Clg

Now all the values on the right-hand side of this equation are
known except Z,,, which we can find as follows : —

Applying the general formula for H to the standard solenoid (it
is convenient to use the affix s to denote this application), we have

TT_1-25.7C.N.
Hs IT

Now let Bs be the strength of the field inside the solenoid, then
Bs = /*HS, but ft = 1 for air.

1-257C.N.

•• Bs= nr

Let As = area of cross section of the secondary coil ; then the
total number or " flux " of lines inside it is Bs x As, and we have

Zs = Bs x As = - s (all of which are known)
ls

whence, substituting this value in (3), we obtain

ls x t x ds

Now as Z is the actual number of lines inside the iron, and as
the area At of cross section of the iron in square centimetres is
known, we have

'

££

_

A^ lsxt xdsx At
or B = mx x d, where mx is a constant.

THEORY OF MAGNETISATION

403

Work out its value, and calculate the various values of B by multi-
plying each throw by it.

T»

Finally work out the various values for /A, which equals ^.. The

« xl

results may be tabulated as under : —

Current in
Amperes
CC)

Throw
(=d)

H

= in, x C

B

=mixd

"B
= H

Magnetisation and Permeability Curves. — Plot tho
"magnetisation curve," taking values of H for abscissae, and of B
for ordinates.

20.000

5OOO

Also plot the permeability curve, taking values of B for abscissae,
and of /A for ordinates.

Figs. 290 and 291 show such curves for various materials. If

the graphs were straight lines, the ratio ^ i.e. the permeability,

404

VOLTAIC ELECTRICITY

would be constant. It will be observed that the permeability in-
creases at first and then decreases, and it is for this reason that
those expressions which contain p as a factor (e.g. the coefficient
of self-induction), become indefinite when iron is present, unless
information is available as to the state of the iron.

Three well-marked stages are indicated by a complete magnetisa-
tion curve : —

(1) A very short initial stage (just perceptible near the origin
in the curve for soft steel, Fig. 290, and perhaps better seen in
Fig. 294). During this stage, the iron is under the influence of a
very weak, but increasing, magnetising force, and the magnetic
induction B is increasing relatively slowly — a given change in H

3.OOO

5OOO JOPOO /5.00O.

MAGNETIC INDUCTION (B)

FIG. 291.

2.G.OOO

producing only a small change in B. In other words, the per-
meability is small.

(2) A stage indicated by the nearly straight and vertical portions
of the curve. During this stage, a very slight change in H produces
a relatively enormous change in B, and the permeability increases
rapidly, passing through a maximum value.

(3) A stage indicated by the portion of the curve beyond the
bend. Here, the iron has reached " saturation," and further increases
in H produce relatively little effect. The curve slowly rises, however,
because B consists of two parts, (a) the field due to the iron, and (b)
the field due to the coil itself (i.e. without the iron), and the latter
part increases without limit. Meanwhile the permeability falls
steadily to quite low values.

In practice, such curves are absolutely necessary to furnish data
for calculating dimensions and windings in designing motors, dynamos,
transformers, &c. As the curves vary appreciably for different

THEORY OF MAGNETISATION 405

specimens of soft iron, it is important to be quite sure that the curve
used refers to the particular variety in question.

It will be seen that, in the case of soft iron of average quality,
saturation commences at about 15,000 lines per square centimetre,
and may be regarded as complete at about 18,000, whereas cast
iron is saturated when B = 5000, and nickel at a much lower
value.

The method of measuring B, H, and /x, described in Experiment
236, is theoretically perfect, but is unfortunately very tedious in
practice, because a ring has to be specially made and wound for each
kind of iron that has to be investigated. It is therefore often con-
venient to modify it in such a way that the same magnetising coil
may always be used. Now, we have shown that if the iron is in
the form of a long rod, magnetised by a long solenoid, the value
of H at the middle of the rod will be very nearly the same as if
the iron were in the form of a ring.

Exp. 237. Make a magnetising solenoid by winding three or four layers
of 20-gauge double cotton-covered wire on a paper or cardboard tube, not
less than 50 centimetres in length and about 1 centimetre internal diameter.
Cut oft' about five lengths of soft iron wire, say 20-gauge, each about twice as
long as the solenoid, and tie them together in several places with thread.-
Cover about 3 inches at the middle of the bundle with thin paraffined paper,
and wind on it about 50 turns of 30-gauge cotton-covered wire in one layer.
Insert the iron into the solenoid, so that the test coil is in the middle, and
bring out the ends carefully. This arrangement replaces the ring used in the
previous experiments, and is to be connected up exactly in the same way, i.e.
the ends of the magnetising solenoid will be connected to the reverser at ab ;
one end of the test coil wound on the iron wire will be joined up to the
secondary of the standard solenoid and the other to the galvanometer. The
method of experiment and the details of calculation are unaltered, except
that the total area of cross section of iron must be found by calculating the
section of each wire, and adding the results together, the diameter being
measured with a micrometer gauge.

(Lt may be pointed out that if a specially designed standard solenoid is
not available for calibrating the galvanometer, the magnetising solenoid itself
may be used for that purpose, first removing the iron core, and winding a
small secondary around it at the middle.)

The Magnetic Circuit. — In order to appreciate the usefulness
of such curves as those drawn on pp. 403 and 404, it is necessary
to look at the subject from another point of view.

As lines of magnetic force always form close curves, however
much they branch out in some part of their path, we may regard
them as forming complete circuits. It is not of the same nature as
a current circuit, for there is no suggestion of a flow round the
circuit ; yet similar ideas may be applied to it, especially when the
path of the lines is entirely, orjjSmost entirely, inside iron.

Consider an iron ring inside a uniformly wound magnetising coil,
similar to that mentioned in Experiment 236. The shape of the ring
need not be circular, although that is the simplest case.

406 VOLTAIC ELECTRICITY

From p. 401 we have

„ 1-257CN

H = - = - , and

L

w lj257CN.fi
_B = ftH = -- = —

(/

Let A be the area of section of the iron, then total lines in iron
(which we shall call the " Flux ") is given by

1 "257CN//A
Total lines = Flux = BxA = - — -- , which may be written

in the form

^ 1-257CN
Flux = -- —

0
Now the numerator is really 47ry~N = 47n'N, and we have already

found this to be the magnetic P.D. between the ends of the solenoid
(see pp. 375 and 380).

Also, the denominator has the same form as a resistance, if we re-
gard ft as being a kind of specific conductivity for lines of force. Now,

ft is not a conductivity, and the expression — is not a resistance,

A/A

but there is a mathematical similarity of form, which makes it
possible to apply the ideas embodied in Ohm's law to a magnetic
circuit. To avoid suggesting that the denominator is a resistance,
it is called the magnetic reluctance of the circuit. The numerator
is often termed "the magnetomotive force," and hence we have
generally

Magnetomotive force

Magnetic reluctance

If the magnetic circuit is not the same at all points, the magnetic
reluctances of each part may be calculated separately, and the results
added together to obtain the total reluctance, just as resistances are
summed in current circuits. It is, however, usually more convenient
to apply the expression as a whole to each part of the circuit in
turn, as we apply Ohm's law to a part of a current circuit as well as
to the whole.

Kxample I. — The mean length of an iron ring is 62 centimetres.
Find the ampere- turns required to produce 12,000 lines per square
centimetre within it — assuming the permeability curve to be avail-
able.

Generally, B =

THEORY OF MAGNETISATION 407

and on reference to the curve, we find, say, that when B = 12,000,
ju.= 1500. It is probably more convenient in the majority of
problems to write this equation as follows : —

= l-257//-

,OT» 7

i.e. CN = — — as the reciprocal of 1 • 257 = -8 nearly

Whence CN = ampere turns = '8 x 12'QQO x 62 = 397 nearly.

loUu

Hence, the required degree of -magnetisation will be produced
with a uniform winding of any number of turns provided that the
product of the current and the turns is 397. It will be noticed that
the thickness of the iron is immaterial. Whatever the area of section
may be, the value of B is unaltered. The Jfux, however, does depend
upon the area, and equals B x area of section.

Example 2. — Find the number of ampere-turns required to
produce the same value of B when a gap 1 millimetre in width
is made in the iron ring mentioned in Example 1.

This gap is so small that to a first approximation we may con-
sider the field uniform within it, and neglect the fringing out of the
lines which must occur at the edges.

It is now best to think of the total ampere-turns required as
being divisible into two parts, one part being required to produce
the flux in the iron, and the other in the gap.

From Example 1 we have, therefore,

nxr -8B.Z,. x -8B.Z,. .

CJN = (in iron) -\ (in air gap).

fJj (J*.

In the iron ring this is practically what it was before ; for the
only difference will be the putting of 61 -9 for I instead of 62. It is,
therefore, unnecesary to repeat that calculation.

Applying it to the air gap, for which ju, = 1 , we have

-8x1 2,000 x'-l

V^JM = - = you

.-. total CN - total ampere-turns « 397 + 960 - 1357.

This example brings out, in a striking way, the enormous im-
portance of permeability, and it also serves to illustrate the meaning
of the statements made at the beginning of this chapter from yet
another point of view.

For suppose we used this ring in Experiment 236. ^We
should measure the values of B quite correctly, but if we obtained

408 VOLTAIC ELECTRICITY

the values of H in the same way, then, -when B=12,000, we
L-257CN 1-
~~l~
1-257x397

1-257CN 1-257x1357
should have H = = whereas its true value for

the iron would be

We can also understand why approximately correct results are
obtained when very long rods are used, as in Experiment 237.
The expression

/^IXT . /f x . /. . v

CN= - (for iron) + - (for air gap)

still applies, but now the " air gap " is the whole return path of the
lines in air, and the longer the bar the farther away the lines will
be thrown out into the surrounding space, and the greater will be
the effective cross section available. As a result, in spite of the
extra length of the air path, the average value of B in it is so small
that the second term becomes almost negligible.

It will be seen that this line of thought leads to exactly the
same conclusions as those based upon the conception of the de-
magnetising effect of free poles, and serves to illustrate the statement
that the idea of poles is mainly a convenient mathematical conven-
tion. In the majority of practical applications, the magnetic circuit
point of view is almost exclusively employed, but, for purposes of
research, it often happens that a theory based on the properties of
poles is most useful.

We have now to develop such a theory, but before doing so
another experiment may be given to illustrate the properties of a
magnetic circuit.

Exp. 238. — Connect an electromagnet to a current reverser and a single
cell, if necessary putting a resistance in the circuit so that the magnet is

only weakly excited. Wrap one or two turns
of wire around the iron base and connect to
a galvanometer as shown in Fig. 292. Shunt
the galvanometer at first to avoid possible
injury. Keverse the current and note the
throw ; divide this by 2, as it is due to
all the lines in the iron cutting the test coil
twice. Now put on an iron keeper, and
FIG. 292. notice the large throw produced. This

means that the number of lines of force

passing through the iron must have increased. Confirm this by again
reversing the current. Taking the throws as proportional to the flux in the
iron, ife will be found that the flux is much greater — perhaps ten times
greater — when the keeper is on than when it is off. As the ampere-turns
have remained constant, this must be due to the improvement in the magnetic
circuit, i.e. to the decrease in reluctance obtained by putting on the keeper.

Intensity of Magnetisation and Susceptibility. — It has

already been stated that, for any magnet, the term moment represents

THEORY OF MAGNETISATION 409

a perfectly definite quantity, directly measurable without ambiguity,
although we cannot say as much for the term magnet pole. Now
instead of thinking about the lines of force a magnet possesses,
and measuring its strength in terms of their number, we may think
of its magnetic moment per unit of volume. This is known as the
intensity of magnetisation, which we shall denote by I. Hence, by
definition,

Moment
= Volume

For our purpose it will be sufficient to consider the special case
of a long magnet of length I and sectional area A the poles being
regarded as located at the ends.

M m x I m strength of pole
=^=-r — 7 = ~r = - -V-
V Ax I A Area

or I is the number of unit poles per square centimetre of end
surface.

Now suppose that the rod is of soft iron, magnetised by a long
solenoid as in Experiment 237. For each value of H there will be
some value of I, which we may measure experimentally by means of a
magnetometer (see p. 146), and then a curve may be plotted for
I and H. It will be almost identical in general shape with the
B and II curve, and, for some purposes, will represent the facts
equally well. We may write the relation between the two quantities
in the form

where 7c is known as the susceptibility of the iron. It is, therefore,
a measure of the magnetic quality of a substance from the "polar"
point of view, just as ^ is a similar measure from the " lines of force "
point of view.

To find a relation between these quantities, we may refer to
Fig. 288, from which it will be seen that the total flux inside the
iron must consist of two distinct parts — (a) a uniform field due to the
original magnetising force H, which would exist whether the iron were
present or not, and which is numerically equal to H ; (b) a very
much stronger field, not necessarily uniform, contributed by the
iron itself, which may be regarded as passing through the iron
and emerging at the poles. The "magnetic induction" (B) inside
the iron is the sum of the two (i.e. the total flux) divided by the area
of section.

Now, the number of lines of force per square centimetre due to the
iron is 4?rl (for there are I unit poles per square centimetre of section,
and each unit pole has 4?r lines). By some writers these are termed
"lines of magnetisation."

410 VOLTAIC ELECTRICITY

.-. B =

Also I =

.-. B =

=

But by definition B =

If our reasoning is correct, it follows that, when an iron rod is
subjected to a magnetising force, which is raised to an extremely high
value, B ought to gradually increase without limit because the term
^H increases without limit. On the other hand, I is the contribution
of the iron itself, and should reach a limit when the iron is " saturated.''
Recent measurements with very intense magnetising forces have shown
that this is really the case.

Magnetometer Method of obtaining Magnetisation
Curves. — We give this method not only because it is especially
useful for investigating the effects of residual magnetism, but also
because it is an instance of the application of ideas previously
outlined.

The solenoid already referred to may be used, but it will be better
to make one specially for the purpose by winding from 6 to 10 layers
of 22-gauge copper wire on a brass tube of about J inch internal
diameter and 70 centimetres in length. The number of turns and the
length must be accurately known. An iron wire can then be
magnetised beyond the point of saturation with a current of less
than 1 ampere, which, if a suitable ammeter is not available, is
readily measurable on a tangent galvanometer. A new piece of
apparatus is required, viz. a coil of wire of any convenient size on a
hollow circular frame (like a tangent galvanometer coil) and known
as a "compensating coil."

Exp. 239. Set up the reflecting magnetometer described on p. 146. About
6 inches behind it, clamp the solenoid firmly by some non-magnetic support 2
in a vertical position, with its lower end 2 or 3 inches belmv the level of the
magnetometer needle. About 10 or 12 inches in front of the magnetometer,
place the compensating coil with its plane at right angles to that containing
the solenoid and mirror, so that the light passes through it to the scale. The
coil should be arranged to permit of adjustment by moving it slightly back-
wards or forwards. Connect up, as shown in Fig. 293, so that the same
current flows round both coils. The compensating coil is used to produce a

1 Although this expression is usually written in the form given, it will
be seen later (Chapter XXXIV.) that' it is physically incorrect, because
B and H are quantities of different dimensions. In fact, it should be written
B=/<^H + 47rI, where nv is the permeability of a vacuum, but it would be
pedantic to do so.

2 Obviously all iron fittings must be avoided in any experiments with a
magnetometer.

THEORY OF MAGNETISATION

411

1*-*\ ^

III \A/WW\/WVW\A-

FIG. 293.

field at the magnetometer needle equal and opposite to that produced there
by the solenoid, so that for all strengths of current, the effect of the latter
field is neutralised and the deflections obtained in the experiment are due to
the iron only. The first step is to obtain this
adjustment. Switch on a small current, and
make sure that the deflection due to the com-
pensating coil alone is in the opposite direction
to that due to the solenoid alone. If not,
reverse the connections of one of them. Then
move the compensating coil backwards or
forwards until the deflection is zero. It is
very convenient to complete the adjustment
by shunting this coil with a suitable resis-
tance (usually a piece of platinoid wire).
Increase the current to the maximum value
intended to be used, and thus make sure that
the adjustment is exact.

There is one difficulty still to be overcome.
When the iron is placed in position, it will be
subjected to the magnetising influence of the
vertical component of the earth's field, as well
as to that of the solenoid, and the softer the
iron used, the more conspicuous and trouble-
some is the effect. There are various ways of
dealing with it. One of the best is to have
another independent winding on the solenoid,
in which, during the experiment, is main-
tained a steady current from an independent source of a strength sufficient to
produce a field equal and opposite to the vertical component. As our purpose
is to demonstrate the method rather than to secure great accuracy in detail,
we shall refer the student to works specially devoted to electrical measure-
ments, and shall neglect the correction.

Let us suppose that the object of the experiment is to investigate the
magnetic properties of steel. Cut off a piece of steel piano wire, about
50 centimetres long, i.e. rather shorter than the solenoid, and solder to one
end about 15 centimetres of a stouter brass wire, springy enough to move
easily up or down in the solenoid and yet to retain its position without
slipping. Insert this in the solenoid. If it is not already accidently magne-
tised, there will be practically no deflection. Switch on a very small current,
just enough to produce a readable deflection. Move the wire up or down
until the deflection is greatest. Then the pole in the wire (which is not
necessarily at the end) is in the correct position with reference to the needle,
and the wire must not be moved again during the experiment. Increase the
current by successive small steps, at each step reading the current and the
deflection, until the maximum current to be used is reached. It will be
known when saturation is obtained, because after that point is reached the
deflection will scarcely be. altered by any increase of current. Do not disturb
the apparatus — it will be required for the next experiment — but measure the
distance of the scale from the mirror, and also the distance of each pole in
the iron from the mirror, marked D and D15 in Fig. 293. The position of the
lower pole is known, for it is on the same level as the mirror, and hence its
distance from the top of the brass wire can be measured. Having located
this pole, the position of the other can be found by symmetry.

If it is merely desired to show the shape of the magnetisation curve,
without knowing actual values, plot a curve with current "(which is propor-
tional to H) for abscissae, and the observed deflections (proportional to I) for
ordinates.

412 VOLTAIC ELECTRICITY

If absolute values are required, the reasoning is as follows (although,
as we are concerned both with H, our symbol for the magnetising force, and
also with H, the symbol for the horizontal component of the earth's field, we
shall denote the latter by He) :—

Let He = Horizontal component of earth's field.

A = Area of section of steel wire.

D, D^ Distance of poles from magnetometer needle (marked in Fig. 293).
a = Angle between lines marked D and Dx.
s= Distance of mirror from scale.
m=Pole strength of wire for any given current.
F= Field produced by wire at needle.
6= Angle of deflection of mirror.
d= Deflection in scale divisions (for any given current).

Then F =^-^ cos a, but cos a=^-

Also F-

And as m=IA

TT

we have 1= — — — ^-? — x tan 6= (a constant) x tan 8

Again, if the deflection d and the scale distance s are measured in the
same unit, then tan 26 = — > and as 0 is small, we may put tan 6=~

approximately.

Hence, I can be calculated in terms of a constant multiplied by the
observed deflection d, and as the values of the magnetising force H can be
found in the form of another constant multiplied by the current, exactly as
was done in the previous experiment, we obtain data from which a curve
can be plotted with I for ordinates and H for abscissae, and another curve
with k for ordinates and I for abscissae.

Although it would scarcely be the object of the experiment, it is also
possible to obtain values of B from the data, for we have

(p. 410)
Also I = (a constant) x tan d
:. B= H + 4irx (a constant )x tan 0
= H + (a constant) tan 0.

Cycles Of Magnetisation.— Exp. 240. Repeat Experiment 239
until the maximum current is reached. Then, without interrupting it,
gradually decrease it step by step, at each step reading the current and the
deflection, until the current is zero. There will still be a large deflection, due
to the residual magnetism. Eeverse the current, again take the readings as
it is gradually increased to its former maximum value. Then decrease again,
still taking the readings, until the current is zero. Reverse, and continue
readings until the maximum value is again reached. The whole cycle of
operations may be repeated with advantage so that greater accuracy is secured.

THEORY OF MAGNETISATION

413

To find the shape of the curve, plot deflections as ordinates
against current as abscissae. If the result is fairly satisfactory,
the values of I and H may be
calculated and plotted in the
same way. Fig. 294 gives the
result obtained in a certain ex-
periment with steel wire, and
the dotted line curve indicates,
for the sake of comparison, the
shape that would be obtained
with soft iron. It will be seen
that the values of H are plotted
to the right and left of the origin
respectively, according to the
direction of the current, and the
corresponding values of B above
and below.

The branch OP is the first
part obtained in the experiment
just described. From P to Q
the current is decreasing, and
OQ is a measure of the residual
magnetism when the current is
reduced to zero. From P to R
the current is increasing in the
opposite direction, and the iron
is not demagnetised until this
has reached the value correspond-
ing to R. Still increasing the
current, the point V is reached, and by again decreasing and re-
versing the portion of the curve, VWXYP is obtained. The initial
path OA is never again traversed unless the iron is first completely
demagnetised. This is best done by heating it to redness, or (if that
is undesirable, as affecting its state), by placing it in a solenoid carry-
ing an alternating current which is gradually decreased to zero.

Such curves may also be obtained by ballistic methods — modifi-
cations of those already given — but as the details and precautions arc
rather more complicated, space does not permit of their description
here.

It will be noticed that there is more than one value of B corre-
sponding to any given value of H. In order to bring this out more
clearly, when the point S was reached, the current was gradually
reduced nearly to zero, thus tracing the lower part of the loop ST.
Then it was increased again to its old value, thus tracing the
upper part of the loop and again reaching S, the normal sequence
of operations already described then being resumed. Exactly

414 VOLTAIC ELECTRICITY

the same thing was done at the stage corresponding to Y, in
this way obtaining the loop YZ, and evidently any number of
loops could have been superposed on the main curve. Consider
the points a, &, c, d, e, for each of which H has the same
value. There are five values of //. corresponding to this value of
H, and obviously there might be many more. It may, therefore,
be asked — What meaning is to be attached to this term if its
numerical value is so indefinite ? Without giving a complete answer,
it may be remarked that the use of p in calculations is entirely in
connection with soft iron, which readily loses its residual magnetism,
so that if a soft iron wire be tapped or vibrated while we are carrying
out the cycle of operations referred to above, the two branches of the
curve tend to coalesce into one definite curve, which, under most
practical conditions, is the magnetisation curve. The ballistic
methods mentioned previously naturally give this mean curve. It
is, however, necessary to keep in view the fact that the numerical
value of p does depend upon the way the iron is treated, owing to
the tendency of the iron to persist in the state in which it happens to
be. To this tendency, the term hysteresis has been applied, and the
closed curves in Fig. 294 are known as hysteresis curves.

Loss of Energy due to Hysteresis.— If no such lag existed,

i.e. if the curves of increasing and decreasing magnetisation coincided,
so that for any value of H there was only one value of B, the energy
expended in magnetising the iron would be completely returned to
the circuit during demagnetisation. But the energy is not wholly
returned, and the difference appears as heat in the iron.1

It can be shown that the energy lost in this way per complete
cycle is proportional to the area of the figure enclosed by the curves.
In the course of a B, H curve, if the area is reckoned in' terms of
B and H and divided by 47r, the result is the loss in ergs per cubic
centimetre per cycle.

If the I, H curve be plotted, then the area expressed in terms of
I and H will give the loss in ergs without dividing by 47r. In other
words, the area included by the BH curve is 4vr times the area
included by the IH curve. This loss may be as much as 200,000
ergs per cubic centimetre in magnet steel. It is very important in
connection with alternating currents (see Chapter XXVI.).

"Coercive Force and Retentivity." — Hopkinson first gave
these terms a real meaning by defining OR, Fig. 294, as coercive
force, and OQ as retentivity. Hence, retentivity is measured by
the "residual" or "remanent" magnetism, which persists when the
magnetising force is removed. The figure shows that it may be

1 The heat is never sufficiently great to produce any marked rise of tem-
perature. In Experiment 220 there is a certain amount of heat produced by
the hysteresis, but the effect is mainly due to eddy currents, as stated.

THEORY OF MAGNETISATION

415

^ Scale

greater in soft iron than in steel, but it is understood that the
substance is not to be exposed to shock or vibration. Coercive force
or coercivity is measured by the inverse magnetising force required
to destroy this residual magnetism, and is much greater in steel than
in soft iron.

Practical Measurement of Hysteresis Loss. — The deter-
mination of hysteresis losses can be very satisfactorily accomplished by the
magnetometer (or ballistic) method, but only at the expense of much time
and labour. It is, however, necessary to know the amount of such loss
(under given conditions) in the sheet iron used for armatures and for trans-
former cores, so that, in practice, more rapid methods are required.

Ewing's Hysteresis Tester. — One of the earliest and most con-
venient arrangements for this purpose is due to Professor Ewing, and is
shown diagrammatically in Fig. 295. A permanent steel magnet, M, is
pivoted at P, so that it oscillates in a plane at right angles to the plane of the
paper. At the top is fixed a long pointer mov-
ing over a divided scale. At the bottom is
a weight, W, which can be raised or lowered
to adjust the restoring force acting when the
magnet is slightly deviated from its position
of rest ; and also a vane, D, moving in oil to
make the motion more " dead -beat." Strips are
cut from the sheet iron to be tested and
made up into a bundle about 3 inches long and
f inch square. This bundle is clamped together
and fixed to an arm, B, so that it can be rotated
between the poles of the magnet by means
of a hand wheel. When this is rotated
at sufficient speed (the exact speed is imma-
terial), it will tend to drag the magnet after
it, and as a result the latter will take up a
definite position, and the corresponding deflec-
tion can be read off on the scale. The greater
the tendency of the induced magnetism to
persist, the greater will be this deflection, and
it is, therefore, a measure of the hysteresis loss.
The makers supply with each instrument a
standard specimen of iron, for which the loss
is known for a given value of B. This sample
is placed in the holder and rotated in the same way — the reading thus ob-
tained being used to* calibrate the scale. Each bundle to be tested is made up
as nearly as possible to the size and weight of the standard specimen.

The magnetic cycle due to the rotation of iron in a constant magnetic
field is exactly similar to the cycle occurring in an armature core, but is not
strictly similar to the magnetic cycle performed when an iron core is mag-
netised by means of an alternating current, as in the case of a transformer.
In the latter case, the actual iron loss is better measured by means of a
wattmeter, and the principle of such methods will be indicated in connection
with that instrument (see p. 566).

Relation between Hysteresis Loss and Magnetic

Induction. — It is obvious, from Fig. 294, that the loss during a cycle
depends on the maximum value of B attained during that cycle. A graph
drawn connecting these quantities shows that they are not proportional, the
loss increasing at a greater rate than B. No theoretical connection has been

FIG. 295.

416 VOLTAIC ELECTRICITY

obtained between them, but Steinmitz has shown that if W be the loss
in ergs per cubic centimetre per cycle, the experimental results are repre-
sented by an equation of the form

W = axB1>6

(max.)

where a is a number, which is constant for a given sample of material. For
good soft iron a is about "001 ; for hard steel about '025 ; and for the new
silicon and aluminium alloys (see p. 423) it may be as low as -0007.

E wing's Theory of Magnetisation. — The loss of energy
due to hysteresis has often been vaguely assigned to a kind of molecu-
lar friction, such that work must be done in moving the particles
of the iron into new positions. Ewing experimented on a large
model, which contained a great number of small pivoted magnets
placed very close together and surrounded by a magnetising coil.
The results obtained strongly support the conclusion, that it is quite
unnecessary to postulate anything akin to friction between the
particles in order to explain hysteresis, and his theory may be briefly
outlined as follows. The ultimate magnetic particles in a piece of
unmagnetised iron, left to itself, tend to arrange themselves in groups
— not necessarily alike in configuration — each group being stable for
small disturbances and producing no external magnetic field. When
the iron is subjected to a magnetising force of gradually increasing
strength, the first effect is to modify the configuration of each group,
although its stability or general form, as a whole, is not destroyed.
This corresponds to the short initial portion of the magnetisation
curve, where B is slowly increasing and p small.

As H increases, the individual members of each group tend to
point along the field, except in so far as their directions are still
modified by their influence on each other, and as a result some groups
become unstable and break up. All the groups will not break up for
the same value of H, but when instability begins, this action will go
on somewhat rapidly. This corresponds to the steep portion of the
BH curve, where a small increase in H produces a large increase in
B. When most of the groups have been broken up, the third stage
is reached, in which the iron is said to be "saturated." In this stage
an increase of H merely tends to weaken the effect of their mutual
influence, and produces more and more perfect alignment with the
field. Suppose that H is now gradually diminished. Some groups of
particles will tend to persist in their arrangement, being more or less
in equilibrium under the action of their mutual forces. Therefore,
when H is reduced to zero, there will be some residual magnetism,
and H will have to be reversed in order to break up the new group-
ings. It is also evident that reducing H does not make the iron pass
through each of its previous states for a given value of H.

The loss of energy is explained as being due to the fact that,
whenever a particle becomes unstable and suddenly swings round into

THEORY OF MAGNETISATION 417

some new position of equilibrium, it acquires energy of motion, and
it oscillates about its new position of equilibrium until that energy is
gradually converted into heat.

Hence, the hysteresis loss is greatest in the second stage of
magnetisation, and relatively small in the third.

Up to this point, we have assumed that the field varies in strength
in order to produce the changes in magnetisation. There is, however,
another method of procedure, for a piece of iron may be rotated in a
field of uniform strength. This case occurs in practice in dynamo
armatures. Evidently, during one complete revolution, the iron
passes through one complete cycle of magnetisation. Let us suppose
the strength of the field to be very great, so great, indeed, that the
mutual magnetic forces between the particles becomes comparatively
negligible in comparison. Then each particle always points along
the field in the right direction, irrespective of the rotation of the
iron as a mass, and no energy will be lost in its taking up new
positions. Thus the theory leads to the conclusion that there should
be little hysteresis loss, when iron is rotated in very strong fields — a
conclusion which has been confirmed by direct experiment.

Attraction between a Magnet Pole and its Armature. —
Let a magnetic field exist between two iron surfaces, each of area
A, and distance d apart, and let us assume that the field is perfectly
uniform, and of strength B, over this area. It will not be uniform,
especially at the edges, but the departure from uniformity becomes
smaller as d, compared with A, becomes smaller. (It will be remem-
bered that similar assumptions were made in discussing a kindred
theorem in statics ; see p. 52.)

A unit pole, placed anywhere in the gap, will experience a force

T>

of II dynes, where H = — and u is unity. Consider one of the iron
M

surfaces. Its pole strength is — , and if such a pole were placed in

4?r

T)TT A

the field, the force on it would be - — dynes. /

But, in this case, the pole forms the boundary of the field, and
the force is, therefore, only half as great (for the reasoning given on
p. 93 for electrostatic fields applies to magnetic fields also).

.'. Attraction = -5 — dynes.

O7T
T>

As H = — , we have
P

Attractions?2^ dynes ( or 5^ dynes^

OTT/A \ STT /

where /A is unity in all practical cases.

2D

418 VOLTAIC ELECTRICITY

The result is independent of the distance between the surfaces,
and, therefore, the above expression gives, very approximately, the
pull between two iron surfaces in good contact.

Potential Energy of Uniform Field. — We may, also, apply
it to find the energy of a uniform magnetic field. For, suppose
that the surfaces are in contact, and are then separated through a
distance, d, the Held still remaining uniform.

The work done = force x distance = — - — x d ergs, and as a result,

8?r

a magnetic field has been formed in space, whose volume is A x d cubic
centimetres,

-r>TT T>2

.'. the energy of the field per unit volume = — - ergs per

O7T O7T/X

cubic centimetre.

Example. — A horse-shoe electromagnet with a core and keeper
forged from 1 inch square iron is excited by 300 ampere-turns. Let
the joints between the pole faces and the keeper be scraped so as
to make a perfect fit, and assume that the permeability of the metal
at the induction produced by 300 ampere-turns is 1500. The length
of the magnetic circuit is 16 inches. Find the total flux through core
and keeper, and the force required to tear the keeper off.

(City and Guilds of London Institute, 1893.)

(The chief difficulty in this example is due to the fact that all the
dimensions are given in inches, whereas it is necessary to use centi-
metres in the calculation.)

-™ _ Magnetomotive force _ 1 '257 CN
Magnetic reluctance I

A/,
_l°257x 300 x(2.54)2x 1500

16 x 2-54
= 8*9 x 104 lines of magnetic force.

Also B = - o = 1 '2 x 104 lines per square centimetre.

Area (2'54)2

B2A

Now, attraction for each pole is - dynes,

,. Total pull . 2 x 1^ _ 2 x (1-2 xl(yx (2-54)*

STT Sir

= 74 x 106 dynes =160 Ibs. weight nearly.

Various methods based on this principle have been devised for
measuring B in terms of the pull required to detach two surfaces
of iron, j The uncertainty as to the actual area of contact and other
difficulties detract from their accuracy, and it does not seem necessary
to describe them.

THEORY OF MAGNETISATION

419

Refraction of Lines of Magnetic Force. — The argument
given on p. 99 for lines of electric force applies equally well to lines
of magnetic force, provided that we write "unit pole" in place of
"unit charge," and B, ju, H in place of F, K, U respectively.

Consider a magnetic field passing from a medium of permeability
ju,, into another of permeability p2. Fig. 67, p. 100, will represent this
case, if we suppose that the above changes are made in the lettering.

Then, we have, as before : —

(1) The normal components of the field (B) must be the same in
each medium.

(2) The tangential components of the force (H) must be the same
in each medium.

The first condition requires that

s 02 (1)

cos =

and the second condition, that

Ht sin Ol T H2 sin 6»2
But B = ^ and B =

(2)

• ± JL

cos Ol
(?)" "Itan*,

or '—*- =

cos 02
- tan 6>2

tan ^t
tan 6L

Hence, lines passing (say) from air into iron are bent away from
the normal, and when passing into a diamagnetic substance (for which
ft< 1 ) they are bent towards the normal.

If ^ refers to air, and fJL2 to soft iron of good quality, as, for
instance, in Fig. 287, p. 398, then p2 is so large compared with /^
that the ratio is practically zero, in which case 0X = 0, i.e. the line is
incident normally.

Diamagnetism. — Faraday, in 1845, by aid of powerful electro-
magnets, demonstrated that all bodies are acted on by magnetic
influence — some being attracted, others repelled.
When experimenting with solids, he suspended
small bars of various substances, w, between the
poles (Fig. 296), and he found that some of them
set themselves axialfy, i.e. in a line joining the
poles. These substances were, therefore, attracted
by the poles of the magnets, and to these he gave
the name paramagnetic. Others, however, were
repelled by the poles of the magnet into a position
at right angles to the line joining the poles, i.e.
equatorially, and to these he gave the name
diamagnetic.

When liquids, contained in thin glass tubes, were suspended

FIG. 296.

420

VOLTAIC ELECTRICITY

between the poles, they behaved similarly. Nearly all liquids are para-
magnetic, and the tubes, therefore, set themselves axially ; a few, how-
ever, are diamagnetic — notably blood, water,
and alcohol — and the tubes, therefore, set them-
selves equatorially. The action on liquids may
be observed in the following manner with very
powerful electromagnets. The liquid is placed
in a watch-glass, and rests on the poles S, Q
(Fig. 297). If the liquid is diamagnetic, it is
repelled from the poles and forms a little heap
between them, A. If it is paramagnetic, it
rises in a little heap over each pole, B. These
changes, however, are so exceedingly small that
it is very difficult to detect their existence.
In experimenting with gases, Faraday caused it to be mixed with
a small quantity of a visible gas or vapour, and then to ascend
between the two poles of the magnet. He found that if the gas was
paramagnetic, it spread out like a flame from an ordinary gas-burner
between the poles ; while, if it was diamagnetic, it spread out across
them.

The following table gives the chief substances arranged in two
classes : —

FIG. 297.

Paramagnetic.

Iron.
Nickel.
Cobalt.
Manganese.
Chromium.
Cerium.
Platinum.
Oxygen.
Titanium.
Palladium.
Osmium.

Many salts and ores of the
above metals.

Diamagnetic.

Bismuth.

Phosphorus.

Antimony.

Mercury.

Zinc.

Lead.

Tin.

Copper.

Silver.

Gold.

Sulphur.

Selenium.

Water.

Alcohol

Air.

Hydrogen.

It must be kept in mind that the behaviour of a feebly magnetic
substance depends to some extent on the magnetic properties of the
surrounding medium. Just in the same way as a balloon filled with
a light gas rises through the air, because the force of gravity attracts
the gas less than it attracts the atmosphere, so, if we suspend a para-
magnetic body in a medium which is more strongly paramagnetic than
the body, it behaves as though it were diamagnetic. Thus, if we
suspend a weak solution of ferric chloride in air, it is paramagnetic ;

THEORY OF MAGNETISATION 421

if it be suspended in a strong solution of the same substance, it
appears to be diamagnetic.

Oxygen is remarkable as a paramagnetic gas, and this property is
still more evident in the liquid state. An experiment, first performed
many years ago by Professor Dewar, shows this fact in a striking way.

He placed a vessel, containing oxygen in the liquid form, between
the poles of an electromagnet (which happened to be the identical
magnet with which Faraday first discovered the magnetic properties of
oxygen gas). Upon exciting the magnet, the liquid oxygen sprang
upwards, and adhered to the sides of the glass nearest the poles,
until, owing to evaporation, it gradually disappeared.

The magnetic properties of oxygen and other feebly paramagnetic
substances are, however, almost infinitesimally small in comparison
with those of iron. For instance, the permeability of liquid oxygen
is only 1*0041. Diamagnetic properties are relatively even feebler.

Nature of Diamagnetism. — A diamagnetic substance may
be defined as a body, which, when placed in a magnetic field, acquires
an induced polarity of the opposite kind to that induced in iron under
similar conditions. Hence, while a rod of a paramagnetic substance
tends to move into the strongest part of a magnetic field, and to set
itself axially so that as many lines of force as possible may pass
through it ; a diamagnetic rod behaves in just the opposite way — it
tends to move into the weakest part of the fieid, and to set itself so
that as few lines of force as possible may pass through it.

We can express these facts in terms of ft and k by means of the
relation /z=l-f47rA; (already given on p. 410). For air, ft =1, and,
therefore, k = 0.

For a paramagnetic body, ft is greater than unity, and k is greater
than zero, both being positive.

For a diamagnetic body, the reversal of the induced polarity
makes k negative, and, therefore, ft is less than unity, but as k is
always very small, ft never becomes negative. For instance, the
susceptibility of bismuth (the most powerfully diamagnetic substance
known) is only- Ab(^uuo.

Without attempting any complete discussion of the subject, it may
be pointed out that diamagnetism is a property which all bodies
might naturally be expected to possess. For consider Experiment
222, p. 358, in which we found that a copper disc suddenly
brought into a magnetic field develops a polarity opposite to that of
the field in virtue of an induced current produced in it. This current
dies away almost instantaneously, because its energy is converted into
heat by ohmic resistance ; but suppose that a similar induced current
is produced in the atoms of copper themselves, and that each atom
may be regarded as a circuit of perfect conductivity ; then the induced
currents in the atoms would persist as long a^ tiie body remained in
the field,1 and consequently it would behave like a diamagnetic sub-
1 For experimental confirmation, see footnote, p. 266.

422 VOLTAIC ELECTRICITY

stance. When removed from the field, an equal and opposite induced
E.M.F. would destroy those currents, and leave the body in its
original state. (According to modern ideas, we should not regard the
current as something flowing inside the atom, but rather as being
due to the orbital revolution of electrons outside it, much as planets
rotate round the sun. But this does not affect the argument given
above).

Considerable light has been thrown on the whole subject of
magnetism by some comparatively recent researches of the late M.
Curie (better known by his work on radium).

Langevin has embodied Curie's results in a general theory of great
importance, in which the old classification (paramagnetic and diamag-
netic) is departed from, and all bodies are ranked in one of three
groups : —

(1) Diamagnetic. — These bodies develop a polarity opposite to
that produced in iron. The susceptibility, k, is a constant, inde-
pendent of the magnetising force H, and also independent of the
temperature.1

(2) Paramagnetic.- — Under this head are placed a number of
very feebly magnetic substances, whose polarity is the same as that
of iron. The susceptibility, &, is still independent of H, but is in-
versely proportional to the absolute temperature.

(3) Ferromagnetic. — This class includes a limited number of
bodies (practically iron, cobalt, and nickel), of which iron may be
taken as the type. Their polarity is the same as, but enormously
stronger than, that of group 2, and k is not a constant. It varies
largely with both the magnetising force and the temperature,
although the variation does not follow any simple law.

As regards the effect of temperature, it has already been stated
that iron becomes non-magnetic at a temperature — known as the
critical temperature — a little above redness (about 750° C.). Curie,
however, showed that in reality the iron at that temperature began to
change into the paramagnetic state as defined in (2), k being very
small, independent of H, and varying inversely as the absolute
temperature.

According to Langevin's theory, all atoms are regarded as being
surrounded by electrons in motion. If the orbits of these electrons
are not substantially in the same plane, the magnetic moment may
be zero, in which case the body is diamagnetic. If the atom has
a magnetic moment, but the conditions are such that it is not
appreciably affected by neighbouring atoms, then it is paramagnetic.
If, however, the atoms do exert directive influences upon each other,
the body is ferromagnetic.

1 Bismuth shows exceptional properties. In the solid state it is the most
strongly diamagnetic substance known, whereas after fusion it becomes the
weakest.

THEORY OF MAGNETISATION 423

Magnetic Alloys. — Heusler has recently discovered that certain
alloys of manganese, aluminium, and copper are ferromagnetic to an extent
almost equal to that of nickel or cobalt or cast iron of poor quality. These
alloys differ from iron in not showing the " Kerr " effect (p. 498).

This property is also shown more or less strongly in quite a large number
of alloys, in which the aluminium or the copper is replaced by another
element, but they all contain manganese.

Manganese itself, as usually obtained, is only feebly paramagnetic, but
when fused in an electric furnace it becomes ferromagnetic, having enor-
mous coercive force and small retentivity, and not becoming " saturated "
even under the influence of an extremely high magnetising force. When
alloyed with iron in certain proportions it yields the well-known non-magnetic
manganese steels. Its properties are somewhat peculiar, and have not yet
been fully investigated.

Iron Alloys. — Barrett discovered that the addition of aluminium or
silicon to iron in certain proportions yielded a material whose permeability for
fairly small magnetising forces exceeded that of pure iron. For strong mag-
netising forces, however, it drops below that of pure iron. Such alloys are
now very largely used for armature and transformer stampings, for they give
increased permeability with small eddy current losses. This is due to the
fact that, being alloys, they have naturally a specific resistance higher than
that of pure soft iron, and therefore the eddy currents are smaller for a
given E.M.F.

Professor B. Hopkinson has recently completed a very extensive examina-
tion of allo}'S under exceedingly strong magnetising forces. Under such
conditions, no alloy has been found to be more magnetic than pure iron, but
some were found to be more magnetic than the value corresponding to the
sum of the effects of their constituents. Such a result, taken in conjunction
with the extraordinary properties of Heusler alloys, is very suggestive. If it
should be found that the property of magnetism depends to some extent upon
the arrangement of the atoms, and not entirely upon their nature, then there
remains the possibility of producing a body more magnetic than iron from
non-magnetic materials. The opinion has, in fact, been recently expressed, that,
whereas diarnagnetism and paramagnetism are both purely atomic properties,
ferromagnetism is not so dependent upon atomic properties as upon crystal-
line structure. The effect of temperature upon the three states may be
regarded as supporting this view. In the first two states, the laws connecting
temperature and magnetism are very simple ; whereas in the third, the
relation is extremely complex. For instance, under weak magnetising forces,
the permeability of soft iron increases with a rise in temperature— at first
slowly, but at about 700° C. it suddenly rises to a very high value, and then
at the critical temperature of about 750° C., even more suddenly falls again
practically to unity. Under stronger magnetising forces, the permeability
merely falls with a rise of temperature, until it becomes unity at 750° C., .
without any such sudden increase in value just below that temperature.

Note. — Professor Kamerlingh Chines has just published the results of
important researches made at temperatures closely approaching absolute
zero (already alluded to on p. 265), and he has found that at these extremely
low temperatures, paramagnetic and diamagnetic substances deviate con-
siderably from Curie's law (which states that their susceptibilities vary
inversely as the absolute temperature).

EXERCISE XX

1. What is meant by magnetic hysteresis ? How would you determine it
for a specimen of iron given in the form of a rod ?

(B. of E., Stage III., 1910.)

424 VOLTAIC ELECTRICITY

2. Describe one good method of determining the permeability of a speci-
men of iron. (Lond. Univ. B.Sc., 1903.)

3. Define magnetic induction B and magnetising force H, and give an
account of an experimental method of determining their relation for a
specimen of soft iron. (Lond. Univ. B.Sc. Internal, 1909.)

4. Show in what features a magnetic circuit is analogous to an electric
circuit. In what respects does the analogy fail ?

(Lond. Univ. B.Sc. Internal, 1907.)

5. Calculate the number of ampere-turns of excitation required to mag-
netise, up to 14,000 lines per square centimetre, a soft iron ring, 28 inches in
mean diameter, made of round iron 1 inch thick. [Assume permeability
= 800.] (C. and G., 1894.)

6. A ring-shaped electromagnet has an air-gap of 6 millimetres long and
20 square centimetres in area, the mean length of the core being 50 centi-
metres, and its cross-section 10 square centimetres. Calculate, approximately,
the ampere-turns required to produce a field of strength, H = 5000, in the
air-gap. [Assume permeability of iron as 1800.] (C. and G., 1908.)

7. Two bars, each 5 square centimetres in cross-section, have their faced
ends touching within a solenoid excited so as to produce a magnetic induction
of 15,000 lines per square centimetre through the joint. Give, in pounds, the
force requisite to pull them apart. (C. and G., 1894.)

8. Explain what is meant by residual magnetism, coercive force, perme-
ability.

Draw a curve showing the manner in which the magnetism induced in a
soft iron rod varies as the magnetising field is taken through a cycle, and
state in a general way how from this diagram you would obtain the residual
magnetism, coercive force, and permeability of the iron. (B. of E., 1911.)

CHAPTER XXVI*

INTRODUCTION TO THE THEORY OF ALTERNATING
CURRENTS

THE conception of an alternating E.M.F. or current has already been
introduced (p. 353).

Such E.M.F.'s and currents as occur in practice may be regarded
as "strictly pi'viodic, i.e. the maximum and minimum values recur
after equal intervals of time, and all the half cycles are exactly alike.
The wave form, however, is not necessarily of the simple and sym-
metrical shape shown in Fig. 270, p. 353. Now, the only shape
of curve jwhich can be readily treated mathematically is that known
as the sine curve,1 and, as a rule, the results obtained in this chapter
apply only to curves of this kind. This is not such an absurd pro- .
cedure as at first sight it may appear to be, partly because the wave
form of many alternating generators approximates closely enough to
the ideal curve to make our calculations practically useful, and partly
because, in more advanced work, there are other ways of meeting
the difficulty. At the same time, the fact that we are making certain
initial assumptions must be kept in mind when comparing theory
with practice.

What is implied by the statement, that an alternating E.M.F. or
current follows a sine law, will be best understood by taking a
particular case.

Consider a narrow coil, which may be either circular or rect-
angular in shape, rotating in a perfectly uniform field. (If this field
is due to the earth, the above condition is satisfied. We are not,
however, concerned with its origin, but unless it is perfectly uniform,
the induced E.M.F. will not follow a sine law.)

For the sake of definiteness, we may think of the coil as being
wound on the outside of a non-magnetic cylinder (serving merely as
a carrier), which rotates between the poles of a magnet as shown in
Fig. 298, it being assumed that the field within the space swept
through by the rotating coil can be represented by equidistant and
parallel straight lines. As already stated elsewhere, each turn of the
coil is equivalent to two active conductors — perpendicular to the
plane of the paper — joined in series by two inactive conductors, which
do not cut the lines of force ; and, as all the active conductors

1 It may be advisable for the student to consult some work which treats
of harmonic motion.

425

426

VOLTAIC ELECTRICITY

FIG. 298.

behave alike (provided that the coil is narrow enough to enable us
to regard them as being sensibly in the same
position), we may simplify the argument by
considering only one of them, i.e. one half of
a single turn.

This case is shown in Fig. 299, where P
and Q represent two consecutive positions of
the rotating conductor, seen end-on, in a uni-

form field parallel to the plane of the paper and perpendicular to AB.
We have now to find the value of the E.M.F. induced in the

conductor at any given instant. A simple application of the calculus

would give us the solution in a few lines, but experience shows that

students eventually acquire a more thorough grasp of the subject if

at the outset they deduce their results from first principles.

Let the conductor rotate (say) clockwise, making n revolutions

per second in a circle of radius r. Its linear velocity is, therefore,

2irnr centimetres per second, and its posi-

tion at any given instant may be con-

veniently defined in terms of the angle 6,

measured from A, one of the positions of

zero E.M.F. Let Z be the number of

lines of force passing through a cross-

section of the cylinder taken through AB.

Then the conductor cuts Z lines of force

in moving from A to B, or from B to A.

(Its length, perpendicular to the plane of

the paper, need not be considered ; it is

really taken into account in defining Z.)
Suppose that it moves from P to a

new position, Q, very close to P, in t

seconds. We shall find (1) the average E.M.F. during this time,

and then (2) deduce its value when t is so small that P and Q coin-

cide. This will be the instantaneous E.M.F. at P.

(1) Let e be the overage E.M.F. during the motion from P to Q.

Then e= meS CU * , and the lines cut are those lying between

T

C and D, so that we have

lines cut = -^ x Z = -^— x Z

CDxZ

2rxt

(absolute units).

Now, as t is the time required to move through the arc PQ with
velocity Zirrn centimetres per second,

_ space _ arc PQ
velocity

THE THEORY OF ALTERNATING CURRENTS 427

and as CD = PK,

PKxZ PK

"~arcPQ"" arcPQ

2-irrn

As the arc PQ is by assumption exceedingly small, it will not
differ appreciably from the straight line PQ, so that we may write

e = irnZ sin PQK.

(2) We have now to find what this becomes as t approaches
zero, i.e. when, in the limit, P and Q coincide.
Evidently the straight line PQ becomes
the tangent at P, and the angle PQK (Fig.
299) becomes the angle between this tangent
and CP.

This is shown in Fig. 300, where HK is
the tangent at P, and the angle PQK becomes
the angle HPO. But by the geometry of the
figure, angle HPC = 0,

e = 7rnZ sin 0 (absolute units),

(instantaneous)

If there is another conductor, diametrically opposite, connected
up to form one turn of a coil, then

e = 2'trn'Zi sin 0,
(tost.)

and if there be N turns in a coil so narrow that 0 may be taken as
defining its mean position, then

e = 2vmZN sin V.1
(mat.)

We see that the maximum value of this expression is reached
when 6 - 90°, for then sin 0=1,

e = 27raZN.

(maximum)

Writing em for this value, we have

e = em sin 0.
(inst.)

PC1

Again, in Fig. 300, "^--sin 0, or PC = r sin 0, hence, if the

length of T be taken to represent the maximum value of the E.M.F.,

1 It will be noticed that the coil (or conductor) moves through 2irn radians
in 1 second, and therefore through 2irnt radians in t seconds. Hence, if t is
the time required to move from A to P, we have 6=.2irnt. This is often
written pt, where p =

428 VOLTAIC ELECTRICITY

then PC represents, to the same scale, the instantaneous value. We
see that at A and B, PC = 0, and becomes equal to r when 0 = 90°
or 270°. In one complete revolution, the induced E.M.F. will pass
through one complete cycle, and the frequency is (in this case) equal
to the number of revolutions per second.

As in Fig. 270, mark off along a horizontal line any convenient
length to represent 2vr radians or 360°. On this plot as ordinates
to any convenient scale the values of PC during one complete revolu-
tion — above the horizontal axis during the motion from A to B, and
below it (to indicate the change in the direction of the induced
E.M.F.) during the motion from B to A. The continuous curve
drawn to connect the summits of these ordinates is known as a sine
curve, and we shall assume that such a curve may be taken to repre-
sent any alternating E.M.F. or current. If the maximum value be
known, the curve can be plotted by using a table of sines. For in-
stance, when 0 = 30°, the instantaneous value is em sin 30°, i.e. \em\

when 0 = 45°, it is -^ ; for 6> = 60C, it is^ew; these values being

evidently repeated for angles TT + 30, TT + 45, and TT + 60.

Virtual Values. — We have now to define precisely what is
meant by the terms " volt " and " ampere " when applied to alternat-
ing circuits. If we could deal with instantaneous values, no difficulty
would arise ; but in practice, we are obliged to read steady deflections
on instruments calibrated by using direct currents, such deflection^
the rean]|,a,ntf ftf impulses wjrjcjb vary in value throughout each

^alternation, It would seem natural to suppose that these steady read-
ings represent average values, but such is not the case, for, although
the average value of a current (or of an E.M.F.), whose maximum is
100, works out at fffi^^jt reads 70*7 on a suite-hlp. foatmipftnt ;
and this, moreover, represents its actual value as regards power.

As already stated on p. 353, the average value is the height oj
OA, Fig. 270, of a rectangle of the "same area as the figure included
by llreTSurve]

1 f T e f *

or, e = - 1 em sin Odd = — 1 1 sin OdO

VJo ^Jo

(average)

9

= ~x««

It is not difficult to understand why this is not the value given
by the instruments, for, taking the case of a hot-wire voltmeter or

E2

ammeter, the heating effect is given by C2R or by -p-, and is, there-
fore, proportional to the square of the current or of the E.M.F. at
any instant, and if the construction is such that the deflection is
proportional to the heating effect, equal distances along the scale will

THE THEORY OF ALTERNATING CURRENTS 429

not represent equal increments of average current or E.M.F., but
equal increments of the average value of their squares. The scale
is, however, graduated by means of direct currents, and so its reading
indicates the value of the direct current (or E.M.F.), which pro-
duces the same heating effect, and this amounts to taking the square
root of the average or mean value of the squares. It is these " square
root of mean square " values which are implied by the term volt or
ampere, when used in connection with alternating currents ; they are
also known as " virtual " volts or amperes, in order to distinguish them
from the true averages} with which, as a rule, wre are not concerned.
We may show this mathematically as follows : —

e2 = - f'ej sin* OdB

(average value of square)

e = vaverage value of square = */ -£-
(virtual)

For instance, if the pressure of alternating mains supplying a
town ia 100 volts, it implies that the maximum value of the E.M.F.
is v/2* 100=141-4 volts.

For the sake of definiteness, it will be convenient to consider
what occurs when an alternating E.M.F. is applied to a coil of wire,
without an iron core, such as the electromagnet coil already used in
many experiments.

Let us suppose that the resistance of the coil is 10 ohms, and
that an impressed E.M.F. of 100 volts sends 4 amperes through it,
In this simple case, the only way in which power can be expended is
as heat in the copper, which is C2R = 42 x 10-160 watts. Thus the
current and the power correspond to an effective E.M.F. of 40 volts.

We shall call the voltage of the mains connected to the coil
(i.e. the reading of a voltmeter connected across its terminals), the
impressed E.M.F., and denote it by E, and the volts required by
Ohm's law, the resultant E.M.F., and denote it by er. Hence,
apparent power = EC, and true power — erC. \

The ratio true Power is known as the «power factor," which
apparent power •

evidently, in the case before us, is ^ = ^x = *4.

JL 1UU

430

VOLTAIC ELECTRICITY

•90

360

FIG. 301.

Now, we know that as_the current in the noil rises in strength,
the magnetic field, formed in the space around it, also increases in
strength, and we may regard the field as being "in step" or "in
phase " with the current, i.e. they reach their maximum and mimimum
values respectively at the same instants.1 This field produces a self-
induced E.M.F., already denoted by est which is greatest when the
rate of cutting is greatest, and rice w*d. A little consideration will
show that this means it will be zero when the field .(and the current)
is at a maximum, and will be greatest when the field (and current)

is passing through zero
to reverse its direction.
Thus, if the full .line
curve in. Fig. 301 re-
presents the current,
cs will pass through
zero at .the points A
and B. Now, there
are two possible ways
of drawing the curve
through these points,
and in^ordfir -to.. decide
whether the maximum

value of f.K lies above or below AB, we must remember that whilst a
current is decreasing in strength, the induced E.M.F. is in the santr
direction. Hence, the curve of es must be on the same side of the
axis of abscissae as the decreasing current from M to L, and, there-
fore, it has the position shown by the dotted liqe. It will be under-
stood that no meaning is to be attached to the amplitude of these
curves ; this diagram merely shows their phase relation~Weach other,
which is expressed by saying that ?s always lags ^ of a period, or 90°
in phase behind the current wliich produces it.

It is, however, important to remember that it is not jreally the_
rate of change of current which produces ea but^Jie^ate_jpJL.ciiaiige-
PfJJUJi ?- fythfi ra.f.n pf f-bfl-ngft in the number of lines of force passing
through the interior of the coil. It will be found later that, whilst
es always lags 90° behind the flux, it may lag more than 90° behind
the current in the circuit. This is due to the fact that a part only
of that current may be effective in producing the flux.

Applying these ideas to the case in question, let us consider what
occurs when an alternating E.M.F. of E volts is applied to the coil.
(It may be remarked that we are not concerned here with what
occurs in the first fraction of a second after switching on.)

We know that an alternating current and flux will be established,
and that the latter will call into existence a certain E.M.F. of self-

1 As these terms "in step" and "in phase" are frequently used, the
student should pay particular attention to this definition.

THE THEORY OF ALTERNATING CURRENTS 431

-•-^ et

90

Xe.

ISO

360

induction, ea. If, for the sake of argument, we assume the current
to be in step with E at starting, es must be 90° behind C, and
therefore 90° behind E. Hence, there are two distinct E.M.F.'s
acting on the circuit, and the current will be determined by, and
will be in step with, the resultant E.M.F. (er). Now, the resultant
of two curves, such as those drawn in Fig. 302, may be obtained
graphically by taking
the algebraic sum of
the ordinateSj which
evidently is represented
by a third curve lying
between the two com-
ponent curves. But
the current depends on
this third curve, and
is in step with it, and
therefore immediately
begins to lag behind

E, also as es must keep pIG 3Q2

90° behind C, that

lags with it until some steady state is reached, for the current cannot
lag more than 90° (for reasons that will be seen later).

Hence, we have finally (1) the self-induced E.M.F., es, lagging
more than 90° behind E; (2) the resultant E.M.F., en lagging less
than 90° behind E, and smaller than E in magnitude ; and (3) a

current C in step with er and determined by Ohm's law as ~.

Hence, the effect of self-induction has been to make the current
smaller than it would otherwise be, and also to make
it lag behind the impressed E.M.F. in phase, so that
the true power in watts is no longer given by the
rn product EC.

Vector Diagrams. — Currents and electromo-
tive forces are not really vectors, but, when they
follow a sine law, the properties of a sine curve
make it possible to represent them graphically, and
such methods are exceedingly useful both in theory
and in practice.

Let OP (Fig. 303) be ajme, whose length _re-_
presents the maximum value of an impressed E.M.F., and let it rotate
clockwise 1 about O, making one revolution in one complete alterna-

FIG. 303.

tion. Then, in any position, its projection

^OP sin a which

1 The International Electro-Technical Commission (Turin, 1910) has recom-
mended the general adoption of the anti-clockwise direction for vector diagrams.
We should have accepted this recommendation but for the fact that the
blocks used in this chapter were engraved before the Commission met.

432

VOLTAIC ELECTRICITY

therefore represents the instantaneous value of the E.M,F. at the

instant it reaches that position.

Similarly, the maximum value of the current can be represented
by another rotating line making an angle with OP equal to the
angle ot phase -difference ; and the E.M.F.
P of self-induction by a third line, which must
lag behind the current line by 90°. Then the
projections of these three rotating lines will
indicate the instantaneous values of the three
quantities in their correct phase-relation. If
the diagram is drawn to scale, it will be
found, by completing the parallelogram, that
E and fls have a resultant er along the line of
current, of such a magnitude that ef = CR.

We thus obtain Fig. 304, in which a
length OA to represent E is marked off
along OP. The current acts along OC,
lagging behind E by an angle' 0. Then
es must act along OQ, which makes an angle
of 90° with OC, and if OB represents its
value on the same scale as E, then OD is the
resultant E.M.F., er.

Hence, the maximum values of E, es, and
er can be represented by the three sides of
a right-angled triangle, derived from the above figure,
and this triangle is so frequently required, that we
draw it in any convenient position without always
thinking of the method by which it was obtained.

From Fig. 305, we perceive that jf the impressed
E.M.F. is a constant (as is often the case), then the
greater es becomes, the greater will be the angle of
lag, and the smaller will be the value of er (and,
therefore, of the current).

When es = 0, then er = E, and there is no lag, as **s

is the case in non-inductive circuits. FlG° 305-

Evidently E2 = er2 + es2, and although this relation has been ob-
tained for maximum values, it also holds good for virtual values,
because we can draw another triangle with the same angles and with

its sides -?-L -^=, -jL to represent such virtual values.

\/2

FIG. 304.

-0'

Of course this relation does not apply to instantaneous values ;
for these we have simply er = E ± es

In the example given on p. 429, we have E
= 40 volts. Now

100 volts, and er

^e?= Vl002-402 =

91 '6 volts.

THE THEORY OF ALTERNATING CURRENTS 433

Also from Fig. 305, cos 0 = J^- =-4, which, from mathemati-

Jii 1UU

cal tables, gives 0 = 66° 25' nearly.

It will be noticed that er = E cos 0, and as C = — , we have

R

R R

Again, it will be seen that : —

True power = er x C = EC cos 6.

This result is especially important, because (although we have
deduced these results from the consideration of a special case, and
may consequently have to extend their meaning at times) under all
circumstances the power supplied to an alternating circuit is EC cos 0.

Value of Self-induced Electromotive Force in Terms

/E2 — e 2

of Current. — The expression C = -^— = — - is inconvenient in its

K

present form, because es is also a function of the current. To find a
value for ea we make use of the equation

e = 27TWZN sin 0

(instantaneous)

This was obtained for a moving coil, which cut a stationary field
according to a sine law, but it holds good for a stationary coil cut by
a moving field, provided that the rate of cutting follows a sine law ;
and it therefore applies in the present instance, n being now the
frequency of the alternating E.M.F.

es
(max.)

. and

e

(virtual' volts) v2 x 10*
Again, on p. 376 we obtained the equation, in absolute units,

L;=ZN,

which referred to a steady current of strength i being stopped or
started, and therefore when applied to an alternating current, i must
be the maximum value. For our purpose we must express this result
in henries and virtual amperes.

Now L ' = L x 10°
(absolute) (henries)

and i = ix ^2 = TK x \/2
(max.) (virtual)

2E

434 VOLTAIC ELECTRICITY

.*. ZN = L x i L x!09x-=4x ^/2

(abs.) (max.) (henries)

2™ x L y 1Q9 x c,

(virtual volts) "v
Therefore, we have in practical units : —

Substituting this value in
C:

R

and simplifying, we have

E

C

from which we see that, when L = 0, C = — as usual.

tv

This expression is often convenient for use, but it is misleading
unless the real nature of the facts is kept in mind. It suggests, for
instance, that the decrease in current strength in an inductive circuit
is due to a kind of increased resistance, and disguises the fact that it
is really due to the presence of another E.M.F. Neither does it indi-
cate that the current is out of step with the impressed E.M.F. It
does, however, make clear the importance of frequency. When n is
small, the second term in the denominator will also be comparatively
small ; when n is very great indeed, this term may become very large ;
and thus we find that, for currents of extremely high frequency, even
straight copper bars behave as if they possessed considerable resistance,
and a few turns of thick copper wire will " choke " powerfully.

The denominator of the last equation, V/R2 -f- (2?raL)2, is generally
known as the impedance of the circuit, where impedance is simply a
useful term to express the apparent resistance of the circuit, i.e. the

Tf
ratio — , when E is in volts and C in amperes. This ratio in the case

Vy

of steady currents becomes, of course, the ohmic resistance.

The quantity lirriL is often termed the reactance.

The relation between the resistance, the reactance, and the im-
pedance of a circuit is easily understood by deriving a new triangle
from Fig. 305. If each side be divided by current strength, C, we
shall evidently obtain a similar triangle with sides

~ X/
(Fig. 306), and we, therefore, have

where 9 is, as before, the angle of lag.

2rrnL

THE THEORY OF ALTERNATING CURRENTS 435

This tells us that, when L is constant, & depends upon the fre-
quency of the impressed E.M.F., but not at all on its numerical'value,
nor on the current strength.

But we know that, when iron is
present, L is not constant, because it
depends on the permeability, /x, which
in its turn varies with the state of
magnetisation. At present, we are
assuming the absence of iron, in " R

which case L has a definite meaning, FlG 306

and can be approximately measured
in a very simple way by means of a voltmeter and ammeter.

For instance, in our typical example, we have

n- ^ E
or 4 = J°£

from which 27mL -=23 nearly,

23

whence L = — = °073 henry.

27T X 50

Example I. — A coil of wire, Fig, 307, having inductance

= "01 henry and negligible resist-
ance, is placed in series with a
non-inductive resistance of 4 ohms.
-£ — > If the -impressed E.M.F. is 120

^'01 henry Wohms* voltSj find (^ tne current ; (2) the

FIG. 307. P.D.'s across the coil and across

the non-inductive resistance; (3)

the phase-relation between the current and the E.M.F. ; (4) the power
expended. (Frequency = 50.)

(1) Applying the previous equation to the coil and the resistance
taken together, we have

24 amperes

^R2 + (2imL)a V42 + (2T x 50 x -01)2 V25'6 nearly
(2) Applying the equation to the coil only, we have

24= I =
27mL 3-1

whence Et - 24 x 3-1 = 74-4 volts.
(Notice that, in this case, Ej is numerically equal to es.)

436 VOLTAIC ELECTRICITY

Again, for the non-inductive resistance,

CandE2

.-. E2 = 96 volts.

Hence we see that E is not the sum of Ex and E2, as would be
the case with a direct current (and as it actually is for the instanta-
neous values of alternating E.M.F.'s).

(3) The current lags behind the impressed E.M.F. of 120 volts

by an angle 0, such that tan 0 = —^ — = = -77,

i.e. (9 = 37-7°.
Also, it lags behind Ex by 90°, for the tangent of the lag angle is

—^, and R = 0 for the coil.1
n

Evidently this is a limiting value, for an actual coil must have
some resistance (although it can be made as small as we please), and
hence 6 will never quite reach 90°.

Finally, the current must be in step with E^
Hence we can represent the state of affairs iby drawing a line OE
(Fig. 308), on which we mark off a length to represent 120 volts.
The current and es will be marked off
along a line OC, making an angle of 3 7 '7°
with OE.

Ej must be set off 90° in advance of
the current, and es is exactly equal and
opposite to it.

It will be seen that the three E.M.F.'s
bear the same numerical relation to each
other as they would have done had we
considered the coil to have both induc-
tance ='01 henry and resistance = 4 ohms.
(4) By the conditions of the question,
the power expended can only be the C2r
heating effect in the non-inductive resistance, which is 24 x 24 x 4
= 2304 watts. It will be found that this is identical with EC cos 0,
which in all cases gives the power supplied to the circuit.

Example 2. — What would be the result if the resistance of
4 ohms was equally divided between the coil and the non-inductive
resistance ?

In the first place, such a change would not affect the equation
for current, which would, therefore, be 24 amperes ; nor would it

1 Or we may say — because R = 0, no energy is converted into heat, and
there is no other source of loss; hence, no power is expended in the coil.
Therefore, EiC cos 0 = 0, or 0 = 90°.

FIG. 308.

THE THEORY OF ALTERNATING CURRENTS 437

affect the angle between the current and the impressed E.M.F. of
120 volts.

EL would be altered in value, for we should have

' ° = ~

i.e. Ej = 24 x 3'7 volts = 88-8 volts
and the current would lag behind this by #r where

tan

3-1
= — 2— =-3- = 1-55

.e.

l = 57° nearly.

Again, E2 would be 2 x 24 = 48 volts, in step with- the current, whilst
the power is evidently unaltered.

The figure can now be drawn as before, except that Ej will be
greater than es, and the angle between them will be less than 180°.
Or the following method may be adopted, if we notice that there
is now a resultant E.M.F. (er) of 48 volts in the coil itself, which

(b)

FlG. 309.

(C)

must be in step with the current, and, therefore, also in step with
E2. Hence, the resultant voltage is 48 volts for the coil alone, and
96 volts for the coil and resistance taken together.

Considering the coil alone, we obtain (a), Fig. 309. Considering
coil and resistance together, we obtain (b). Superposing the two
diagrams, we obtain (c).

Effect of Iron. — The results we have already obtained hold
good only when iron is absent. When it is present, care must be
taken in using them, and, without attempting an exhaustive discussion
of the subject, it is desirable to point out certain effects caused by
its presence.

In the first place, /x is a variable depending on the current
strength, and hence L is some complicated function of the current.
The result is that, even if the impressed E.M.F. follows a sine law,

438

VOLTAIC ELECTRICITY

the current (and the self-induced E.M.F.) will not do so. Secondly,
some energy is expended in the iron on account of hysteresis and
eddy currents. Now, this energy is expended outside the circuit,
although it must have been derived from the circuit, and we have
repeatedly found that this only happens when a current is flowing
against an E.M.F. As there are only two E.M.F.'s acting in the
circuit — the impressed E.M.F., E, and the self-induced E.M.F., es —
we infer that es must have a component in opposition to the
current.

To a first approximation, the actual state of affairs is very simple,
and is shown in Fig. 310. The current, C, lags behind E by some
amount 0, and the total power taken from the
circuit is, as always, EC cos 9.

But es now lags more than 90° behind the
current, and, as shown by the dotted lines,
this means that it can be resolved into a
component in direct opposition to the current
and another at right angles to it. In fact,
if a is the angle between es and the current,
esC cos a represents the power expended in the
iron, and if we obtain the resultant E.M.F.,
e^ by completing the parallelogram, then er x C
is the power expended as C2r heat, which is
equal to the difference of the quantities EC cos 9
and esC cos a. (Of course, cos a is negative
in sign, but this need only be taken to indicate
the distinction between power taken in and
power expended as work.)

Now es must lag 90° behind the alternating
flux which produces it, and hence it follows
that this flux is no longer in step with the current. This apparently
paradoxical result may be at first somewhat difficult to understand.
It will become clearer when the student has met with other instances
of the kind, and he will then find that the actual current may
conveniently be regarded as the vector sum of two components, viz.

(1) the magnetising current, C7n, as required by the reluctance of
the magnetic circuit, and calculable, in simple cases, by the methods
already given on p. 407. The magnetising current is alone effective
in producing the flux, with which it is strictly in step ; although as
it makes an angle of 90° with es, it does not represent external power ;

(2) an iron loss current, CA, in step with the applied E.M.F., such
that ECA represents the power expended in the iron in hysteresis
and eddy currents.

These matters are so important in their practical applications,
that it is desirable to consider a few special cases.

CASE I. — Let an E.M.F., E, be impressed upon a circuit (or part

FIG. 310.

THE THEORY OF ALTERNATING CURRENTS 439

of a circuit), which has self-induction, but has neither resistance
nor iron losses. (This is approximately the case of a coil of wire
of exceedingly low resistance, without an iron core).

The assumptions made imply that no power whatever is expended,
even when the current or the flux is exceedingly great. Hence, the
current must make an angle of 90° with both E and es,
and, as there is no resistance, there is also no resultant
E.M.F.

We, therefore, draw es (Fig. 311) equal arid opposite
to E, and put C at right angles to both, but lagging

behind E and in advance of es. The magnitude of C, flux \0
which is all magnetising current, is determined by the
relation es = 2irriLC.

CASE II. — Let the circuit have iron losses and self-
induction, but no resistance. FlG 311

There is still no resultant voltage, but power is
taken from the source, and expended outside the circuit. Hence we
obtain Fig. 312, where E and es are still equal and opposite, but C
lags less than 90° behind E. Then EC cos 0 is the
power taken from the source, and esC cos a is the
power expended in the iron, and these are obviously
equal. But the flux is out of step with the actual
current, which is resolvable into magnetising and

90°

Flux • \ power components as already shown.

This is approximately the case of an ordinary
choking coil, which has an iron core, but which has
as small a resistance as possible. Such a coil is often
said to afford a means of regulating current strength
FIG. 312. without wasting power, but we see that this statement
ignores the iron losses. It is, however, much less wasteful than a
mere resistance would be.

CASE III. — Let the circuit have self-induction and resistance,
but no iron losses.

This was the case first considered in obtaining various formulae,
but the student will see, from the preceding argument, that it
necessarily implies the existence of a resultant voltage and an angle
of 90° between es and C.

We have also dealt in Fig. 310 with the case of the circuit having
iron losses in addition. We see that it involves modifying the figure
in Case II. to give a resultant E.M.F.

Transformers. — We have already shown that the work a
current is capable of doing is measured by the product of its E.M.F.
and its current-strength (i.e. using practical units, the product of its
volts and amperes). We may have one current of 50 .amperes at
1000 volts, and another current of 1000 amperes at 50 volts, and
in both cases the energy expended to produce them and the work

440 VOLTAIC ELECTRICITY

they are capable of doing will be the same, although one form may
be more convenient for a particular purpose than the other. For
instance, if we wish to transmit electrical energy from one place to
another at some distance from it — say, to light a town five miles
off — although it is more economical as regards transmission to keep
the current small and the E.M.F. as high as possible (because a
thinner and less expensive wire may be used to convey it, and because
the loss of energy in a wire increases with the square of the current),
it cannot, in that form, be used with advantage, as for distribution to
consumers it is more convenient to employ as many amperes as possible
at an E.M.F. rarely exceeding 100 or 200 volts.

Now, just as we can exchange a sovereign for smaller coins, or
small coins for a sovereign, the money in each form having the same
value, so we can exchange one current for another of different
amperage and voltage, both having the same energy, with just a
slight unavoidable loss in the transformation. An instrument to do
this is known as a transformer, which is really a modified induction
coil, with a closed magnetic circuit, i.e. the path of the lines is wholly
within iron ; the primary and secondary coils being arranged to ensure
that, as far as possible, all the lines produced by the primary shall
cut the secondary— an ideal state never actually reached in practice.

The first transformer was invented by Faraday in 1831, and
consisted of two separate coils wound on a soft iron ring (Fig. 313).
He wound 72 feet of bare copper wire (of about
18 B.W.G.) in three helices, one over the other,
and insulated by layers of calico. The ends of
each helix were brought out so that the coils
could be used either together or separately.
On another part of the ring, he wound 60 feet of
wire in two helices, which were then connected
. in series with a galvanometer. When the three

helices were joined in series, and connected with

a battery so that they formed the primary, there was a sudden
deflection of the galvanometer needle in the secondary. The needle
soon came to rest, but when the circuit was broken, it was suddenly
deflected in the opposite direction.

If a steady current be sent through one coil, the inductive effect
on the second coil will be at making and breaking only ; but if an
alternating current be used, a current is induced in the other at
each alternation, without using a contact-breaker.

An ordinary induction coil is a special type of transformer
designed to give a very high secondary voltage. Hence the secondary
must have many turns. Commercial transformers may be required
either to raise or lower the primary voltage, i.e. either to transform
"upwards" or "downwards," and in the latter case, as will be
shown, it is the primary which has the greater number of turns.

THE THEORY OF ALTERNATING CURRENTS 441

For details of construction and descriptions of the various types
of commercial transformers, we must refer the student to works on
electrical engineering ; in these pages it will be sufficient to regard
them as affording instructive object-lessons to which the principles
we have already outlined may be applied.

For the sake of definiteness, it will be convenient to take as an
example an actual (but very small) transformer, made to work on
100 volt alternating supply mains at frequency 50, and to give out
about 50 volts and a proportionately greater current. (This trans-
former was made by some students, and all necessary dimensions
were taken during the process.) It consists of two long, almost
rectangular coils, separately wound and insulated, placed side by
side,1 and then interlaced with iron stampings. Fig. 314 shows its
general appearance; Fig. 315 its section in which the path of the
lines of force is indicated.

One coil consists of 107 turns of 16-gauge copper wire, and has a

FIG. 314.

FIG. 315.

resistance of '66 ohm. The other has 53 turns of 13-gauge wire,
with a resistance of '16 ohm.

The mean length of the magnetic circuit, i.e. the length marked
with arrows in Fig. 315, is 25'4 centimetres; and its sectional area,
i.e. the nett section of iron core at right angles to the path of the
lines, is 122 square centimetres.

In a particular experiment, (1) the coil of 107 turns was used
as the primary and connected to mains at a pressure of 104 volts,
the secondary being on open circuit. It was then found that a
current of '27 ampere passed through the primary, and that a volt-
meter connected across the secondary read 51 volts.

This illustrates the powerful " choking " effect under such cir-
cumstances ; for we know, by Ohm's law, that a steady E.M.F. of
104 volts acting on a resistance of '66 ohm would produce a current

of -- = 156 amperes, which would have at once burnt out the

1 It is more usual to place one coil inside the other, as such an arrange-
ment tends to diminish " magnetic leakage."

442 VOLTAIC ELECTRICITY

insulation. We infer that the current must, therefore, lag consider-
ably, and that ea must be nearly equal to E.

(2) When the secondary circuit was closed through an adjustable
resistance, and a current taken from it, the primary current auto-
matically increased — always keeping at roughly half the strength of
the secondary current — the P.D. at the secondary terminals mean-
while slightly decreasing. The practical limit of the current which
may be taken out is determined by the rise in temperature of the
conductors, i.e. upon their thickness. In this case it was about
5 or 6 amperes for the primary and twice as much for the secondary.

The power obtained from the secondary will always be less than
the power supplied to the primary, the difference representing the
waste in C2r heat in both coils, and the loss due to eddy currents
and hysteresis in the core ; but in well-designed transformers of
fair size, these losses can be made relatively small and an efficiency
of considerably over 90 per cent, easily obtained.

Theory of Transformer. — Let us now consider what occurs
when an E.M.F. of E volts is applied to a primary of Nj turns, the
secondary of N2 turns being on open circuit.

Evidently an alternating flux is produced, which, as it appears
and disappears, cuts the turns of both coils at the same instant.
(We shall assume, as before, that none of the lines miss the secondary.)
In the primary it produces a self-induced E.M.F., which we have
hitherto denoted by ea but, for convenience in this case, we shall
write ev where

or

j absolute units.
(max.)

21TOZN, 4-45WZNJ . . . u

6 = = Vlrtual volts'

From Fig. 312, p. 439, we know that this E.M.F. must be nearly
equal in magnitude to E and in almost exactly opposite phase.

In the secondary, the same flux produces an induced E.M.F., e2,
such that

4-45raZN9 . A . ^ /0.

e2 = - — — — - virtual volts. (2)

This must be in exactly the same direction and phase as elt for
the secondary conductors are cut by the same flux in the same
direction at the same instant.

e N
From equations (1) and (2) we notice that -1 = ^4, and as el is

E2 N
very nearly equal to E, we have approximately — = ^i, a very

e-z ™a
important result. (The student can verify this statement by noticing

that, in the example given on p. 441, — - - _ nearly.)

0 1 Oo

THE THEORY OF ALTERNATING CURRENTS 443

Value of B in the Iron Core. — In the secondary, we have
seen that e.2 = 51 volts, and that the frequency = 50.

4-45 x n x ZN9
Now e., = —

4-45 x 50 x Z x 53
or 51= -^-

whence Z = 43 x 1 04 lines of force.

Again, Z = B x area of section of iron ; and area =-122 sq. cm.

whence B = — =^9 — = 3500 lines per sq. cm. nearly.

It will be noticed, from the manner in which equation (2) was
originally obtained, that this is the maximum value of B.

Explanation Of the term " Load." — When the secondary of
a transformer is on open circuit, there is " no load " upon it. When it
is closed, it is said to be " loaded." This does not mean that it is short-
circuited, but closed through, say, a load of lamps. If the external portion
of the secondary circuit is such that no magnetism is produced therein, the
load is said to be " non-inductive." As a matter of fact, however, a real load
is never absolutely non-inductive, although when it consists of, say, a number
of incandescent lamps, it may be treated as non-inductive in ordinary calcula-
tions. If the secondary circuit contains coils of wire with iron, cores, e.g. if
alternating motors are being driven by a current from a transformer, then
the load is said to be " inductive."

In all cases, the number of watts expended in the secondary circuit is the
measure of the load. Hence, when this is inductive, the load may be small,
although the current output is large.

Calculation of "No Load" Current, i.e. the current in
the primary when the secondary is in open circuit. — This consists of
two components — (1) the magnetising current, Cm, and (2) the iron
loss current, C7i. The former may be obtained from the expression
given on p. 407 for direct currents —

Hence, if we apply it to an alternating magnetic circuit, using
the maximum value of B, we shall obtain the maximum value of C,
but we require the virtual value, or

-8BZ

, m,= f^N,
(max.)

(virtSal

\ /

In this expression, B = 3500; Z = 25'4 centimetres; Nj = l07;

444 VOLTAIC ELECTRICITY

and by reference to the permeability curve for this kind of iron, it
is found that when B - 3500, p = 3200.

•8 x 3500 x 25-4

" 1-414 x 3200 x 107

This current is small because the magnetic reluctance is low. If
part of the path of the lines were in air, as in a spark coil, the
magnetising current would be enormously greater.

To determine C7l, the actual power supplied to the primary (with
secondary on open circuit) was measured by a wattmeter (see p. 565),
and was found to amount to 18 watts. This loss must be due
entirely to hysteresis and eddy currents, for the copper loss (C'2r) is
(•27)2 x -66 - -048 watt, and is negligible.

This component of the current is in step with the impressed
E.M.F., and therefore EC7l= 18,

18

i'e' C/l = 104 ^ ' 1 7 amPere'

Now Cm is almost exactly 90° behind the impressed E.M.F., and
/. no load current = \/Cm2 + Ch2 = >/'1472 + -172 = '23 ampere,

which, for a rough calculation, agrees, within experimental error,
with the value actually read on an ammeter.

The angle of lag is found by putting EC cos 0=18, and using
the observed value of the current,

i.e. 0 = 50°.

Were it not for the iron losses, this would Lave been practically 90°.

Effect Of a Secondary Load. — Suppose the secondary circuit
to be closed on a non-inductive resistance, such that the secondary
current is 12 amperes. The iron core is now subjected to the
magnetising influence of 12x53 = 636 secondary ampere- turns, and
(as we know that the primary current must be at least 6 amperes)
about the same number of primary ampere-turns, whereas previously
the apparent ampere-turns were '27 x 107 = 29. We know that the
secondary current must be nearly in exact opposition to the primary
current, and hence the question arises as to what is the resultant
effect of these nearly equal and opposite sets of ampere-turns. The
answer is indicated by noticing that, as the secondary current in-
creases, the secondary volts (the secondary load being non-inductive)
remain nearly constant, falling very little more than is naturally due
to internal resistance. As the secondary resistance is '16 ohm, and
as 1'92 volts are required to send 12 amperes through it, even if er
remained constant, we should expect the P.D. at the secondary

THE THEORY OF ALTERNATING CURRENTS 445

terminals to drop by that amount (i.e. 1'92 volts). (Ileally there is
another cause of drop due to the fact that some of the flux misses the
secondary winding, an effect known as " magnetic leakage.")

Hence, as the P.D. at the secondary terminals does not fall more
than we should expect from the above causes, we infer that the flux
remains constant at all loads — a most important result, from which
it follows that the magnetising current and the iron losses are constant
at all loads.

We can now predict, from principles of energy, what the value of
the new primary current must be. Suppose that the P.D. at the
secondary terminals is 48 volts, when the current is 12 amperes.
This current is practically in step with the watts, when the secondary
load is non-inductive. Then we have

Watts given out by secondary = 48 x 12 = 576

„ lost as heat in „ = 122 x '16 = 23

„ „ ,, primary = 62 x -66 (approx.) = 24

623

In the last line, we have estimated the primary current at 6
amperes — it cannot be far from that value, and any slight error is
negligible. To find its actual value, we notice that 623 watts must
be supplied in addition to those required at no load, and this means
a current in step with E, such that 104 x C - 623, or C = 6 amperes.
To this must be added the iron loss current, shown above to be
unaltered, of '17 ampere, or 6-17 amperes in all.

Finally, we have to compound this with the magnetising
current of -147 ampere, the actual primary current being, there-
fore, /(6- 1 7)2 + (' 1 47)2. We see that the second term is negligible,
and that, therefore, the current must, for all practical purposes, be in
step with the impressed E.M.F.

Efficiency of the Transformer at this particular load will be
given by

•prffl • _ Watts given out _ Output

Watts supplied Output + losses

- 48x12 _ - -9, or 90 per cent.

(48x12) + 24+ 23 +18

It will be noticed that the efficiency is less at smaller loads. This is
important, because, in practice, transformers usually have to be
running continuously for most of the time on light loads. Now, the
copper loss increases as the load increases, whereas the iron losses do
not, and hence it is all important to keep these losses small, by exer-
cising care in the selection of the iron and in the details of the design.
Automatic Adjustment of Primary Current to Load.—
Our argument, up to the present, has depended largely upon the

446 VOLTAIC ELECTRICITY

axiom that power given out must be balanced by power supplied, but
it has not explained the manner in which the primary current adjusts
itself to the load.

Space does not permit us to explain this fully. It may, however,
be remarked that our previous results show that when an alternating
E.M.F. is applied to an inductive circuit of very small resistance, the
current rises in strength until the flux produced is great enough to
establish an E.M.F. of self-induction, very nearly equal and opposite
to the impressed E.M.F. This is the case in the primary when the
secondary is on open circuit. When the latter circuit is closed (the
load being non-inductive, and magnetic leakage being practically
negligible), the first effect of the secondary ampere-turns is to weaken
the flux, and therefore es in the primary. Hence, the primary current
immediately rises until es is again practically equal to E, until the
flux reaches its old value. In fact, the two opposed sets of primary
and secondary ampere-turns are not quite in exactly opposite phases,
and they adjust themselves until there is a small resultant, which
represents their effective magnetising value, and which is practically
the same at all loads.

Effect of Condensers. — CASE I. — Let a condenser of capacity
K be connected, by wires of negligible resistance, to alternating mains
of E.M.F., E, and frequency n. It will behave, to some extent, like
a tank fed by a pipe connected to a large reservoir, in which the level
of the water is alternately raised and lowered. If the level in the
reservoir were constant, it would correspond to the case of direct
currents, the tank (or condenser) merely filling (or charging) up until
the water-level (or P.D. across it) became equal to that of the
reservoir.

As it is, the water surges in and out of the tank, which acts,
therefore, as if it had an impedance instead of an infinite resistance.
We see that the P.D. between the condenser terminals must rise
during the charging like a back E.M.F., until it becomes equal and
opposite to the impressed E.M.F. at the instant the latter reaches its
maximum value (the current being then zero). As the impressed
E.M.F. falls in strength, this condenser E.M.F. becomes operative in
producing a current against it, thus discharging the condenser. This
current reaches its maximum value at the instant the impressed E.M.F.
is zero, and then, when the latter reverses its direction, again begins to
charge up the condenser, but with the opposite polarity.

Evidently the current is out of step with the impressed volts, and,
if we assume a negligible resistance in the connections and no loss in
the condenser, there can be no expenditure of energy, and we can
predict that the phase-difference must be 90°. We can also regard
the condenser as possessing an E.M.F. of its own, measured by the
P.D. between its terminals, just as a coil possesses a self-induced
E.M.F. of its own. We shall denote this condenser E.M.F. by ek.

,THE THEORY OF ALTERNATING CURRENTS 447

In Fig. 316, let the curve E represent the impressed E.M.F. In
this case, the condenser E.M.F., ek, is equal and opposite to it.

During the first quarter period, while E is rising, the condenser is
charging up, and therefore ek rises in
opposition to E, meanwhile the current
impelled by E must be diminishing in v ' /6K

strength, and reaches zero at the instant
the condenser is fully charged, i.e. when
E is a maximum. Hence, the current
curve must pass through the points A
and B. As there are two ways of draw-
ing it through these points, we must
select the right one by noticing that C JPIG

is in the same direction as E, whilst

E is increasing, and against E whilst E is decreasing (for then the
condenser is discharging), the current being then impelled by ek in its
direction. We now see that the current lags 90° behind e^ and leads
90° in advance of E. This latter fact is most important, for just as
introducing self-induction into a current makes the current lag with
respect to the impressed E.M.F., so the introduction of capacity makes
the current lead with respect to it. (Really some loss occurs in the
dielectric, for it becomes warm. This is an effect analogous to
hysteresis, and means that the lead is actually less than 90°).

We have now to find an expression for the current strength. As
in previous cases, this is most readily obtained by means of the
calculus, but it is instructive to deduce it from first principles.

Let T be the time of one complete alternation, so that T = -, where

n

n is the frequency. The condenser charges up in J of a period, i.e. in

T

time - , and the quantity in it is Qm when ek is a maximum (which we
4

shall denote by em),

.-, Qm = emK (i.)

T

Now Qm may be written Crt x — , where Ca is the average current, and

4

it has already been stated (p. 428), that for a sine curve : —

o
Average = — x maximum.

7T

From(i.) Cflx? = eTOK

or CTO = 27rnKem

448

VOLTAIC ELECTRICITY

This obviously holds good for virtual values, for to obtain them it i
only necessary to divide each side by ^2,
and, therefore, we have generally

C = 27rnKek
or, as E = ek, we may write

which shows that a condenser may be regarded as having an impedance
1

of

27TOK'

CASE II. — Let the condenser be in series with a non-inductive

resistance, R, Fig. 317.
M*

M

FIG. 317.

We know ek is 90° in advance of the current,
and that the current is in advance of E
by some angle 6. Also that there are
two E.M.F.'s in the circuit, viz. E and

which must have
a resultant er in step
with the current, such

that er = CR. Hence, we obtain Fig. 318, from
which we learn that

R R

but C = 27rnKek

whence, eliminating ek, we obtain
E

FIG. 318.

The figure shows that this current is in advance of the impressed
E.M.F. by an angle <9, such that tan 6= - --.

CASE III. — Let the resistance, R, be inductive, having a self-
inductance, L. In this case, the effect of L is to introduce a
lag (flj), and that of K to produce a lead (#2) of current
with respect to E. The two triangles of impedance are shown in
Fig. 319. Combining the two, we obtain Fig. 320, and whether that

is a lag or a lead depends on which is the greater, 2-rrnL or

2ir»K*

Hence C =

Z-rrnL

,__L)2

From which we see that when 27rriL = -, the impedance reduces

THE THEORY OF ALTERNATING CURRENTS 449

to R, and there is no lag. If R is small, this may mean an enormous
current, although the resistance of the circuit is really infinite. Here
we have an example of the general rule that inductance and capacity
produce opposite effects, so that one of them can be used to neutralise
the influence of the other.

The series arrangement just given is by no means the only method
available, but it is impossible for us to discuss the question in
detail.

It is, however, very desirable to obtain some idea of the real
meaning of these results, and it may be pointed out that self-
induction implies that energy is being stored up in the form of a
magnetic field whilst the current is rising in strength, and is being
paid back into the circuit whilst it is falling in strength. On the
other hand, the presence of capacity implies that energy is being
stored up in the form of an electrostatic, field in the dielectric whilst
the current is falling, and is being paid back whilst it is rising.

FIG. 319.

FIG. 320.

Hence, when both are present, energy is being stored up in one
form whilst it is being paid back in the other, and this see-saw
'action may go on without making any further demand on the
circuit (except to make good small losses due to heating, &c.) after
the first few moments during which this state of affairs is being
established. All actual circuits possess some self-induction and
some capacity, and hence the ordinary form of Ohm's law is a
limiting case, always applicable to 'direct current circuits when the
current has reached a steady value, but only to alternating circuits
when the above quantities are small enough to be negligible.

Example. — What E.M.F. will be required to drive 10 virtual
amperes through a condenser, of which the resistance is 1200 megohms,
and the capacity 22 microfarads, the frequency of supply being 80 ?

(City and Guilds, London.)

(Hitherto a condenser has been treated as having an infinite
resistance, whereas it is obvious that it really has an extremely high
one, which, although practically negligible in this question, must
be taken into account. But it must not be supposed that the
resistance of the condenser itself can be treated as if it were a

2F

450 VOLTAIC ELECTRICITY

non-inductive resistance in series with the condenser, and therefore
the equation we have given in Case II. above does not apply.)

In this example, the actual current of 10 amperes is made .up of
two components —

(1) A current (Cr) in step with the impressed E.M.F., E, and of

-p

magnitude — , where R = 1200 x 106 ohms.
K

(2) A current (Ck) , 90° in advance of E, and of magnitude 2irnKE.

As these currents differ in phase by 90°, we must have

-+ + amperes
or Cr2 + CA2 - 1 00 amperes

orE= 10 10

\ R2 + (2™K)2 V \ 1 200 x 10 V

The first term in the denominator is obviously negligible, and we
have, therefore,

9QB volts nearly. * ;

Power in an Alternating Current. — The measurement of
alternating power by means of a wattmeter is discussed on p. 565,
here it is merely necessary to point out how it varies during a
complete cycle, when the current lags (or leads) with reference to the
impressed E.M.F.

In Fig. 321, let E and C represent the curves of E.M.F. and
current respectively for one complete cycle. Draw vertical dotted
lines through the points where either of them is zero. We notice that
at the instants corresponding to a, b, c, d, e, the instantaneous power
is zero (because it is EC, and at these instants one of the factors
is zero). Hence the power rises and falls periodically, its frequency
being twice that of the current. Also, during the period of time
a bt the current is flowing in the opposite direction to the impressed
E.M.F. (of course, being impelled by a temporarily stronger E.M.F.,
es, not shown in the figure), which means that energy, supplied
by the energy of the disappearing magnetic field, is being returned
to the circuit.

During the interval 6c, the current and the E.M.F. are in the
same direction, which means that energy is being taken from the

THE THEORY OF ALTERNATING CURRENTS 451

source, and, in this case, partly expended in storing up energy in
the form of a magnetic field, i.e. the circuit loses energy.

The period cd is similar to ab, and de to be, as far as energy is
concerned. Hence, the effective power, EC cos 0, depends on the
nett energy expended in the circuit during the cycle. If we draw
a second curve to indicate the rise and fall of power, it will
take the form shown at the bottom of Fig. 321. If there is no
lag, the power expended in the circuit is EC ; if the lag is 90°,
evidently the periods marked G and L are equal, and on the whole
no power is expended. If, on the other hand, the lag is greater than

FIG. 321.

90°, evidently more power is gained by the circuit than is supplied
from it, which is absurd in this case, although such a figure would
correctly represent the state of affairs when E is the E.M.F. of an
alternator working as a motor ; the interpretation being that it is npt
now acting as a generator, and on the whole giving out energy to
the circuit ; but that it is acting as a motor, and on the whole receiving
energy from the circuit, which it expends as mechanical work.

EXERCISE XXI

1. Find the impressed voltage required to send a virtual current of 50
amperes through a resistance of 2 ohms, if the self-inductance is '005 henry
and the periodicity is 63.

2. A 100 candle-power glow-lamp requires an alternating current of 6
amperes at 60 volts. The supply is at a pressure of 100 volts, at a frequency
of 75 complete cycles per second. To give the lamp the right voltage, a

452 VOLTAIC ELECTRICITY

choking-coil is to be inserted. Calculate how much self-induction, in henries,
this choking coil must have. (C. and G., 1893.)

3. Define impedance and reactance as applied to alternate current circuits,
and draw a clock diagram showing the phase-relation between E.M.F. and
current in a circuit whose impedance is 5 ohms and reactance 4 ohms.
What is the resistance of the circuit ? (C. and G., 1905.)

4. Calculate the number of alternating volts needed to drive an alter-
nating current of 10 ampei-es through a choking-coil having a coefficient of
self-induction of T^¥ henry, the frequency being taken at 80 periods per
second. (C. and G., 1896.)

5. A certain choking-coil of negligible resistance takes a current of 8
amperes if supplied at 100 volts, at ^50 periods per second. A certain non-
inductive resistance, under the same conditions, carries 10 amperes. If the
two are transf erred to a supply system working at 150 volts, at 40 periods per
second, what total current will they take, (a) if joined in series, (b) if joined
in parallel ?

6. The core of a transformer contains 100 square centimetres nett sectional
area of iron. Assume that it is to be used on a circuit in which the frequency
is 75 complete periods a second, and that the primary currents supplied will be
such as to raise the magnetic induction at each alternation to 5000 lines per
square centimetre. Find how many turns there must be in the secondary
coil if it is to produce 100 volts on open circuit. (C. and G., 1893.)

7. An alternate current transformer has 200 turns in its secondary wind?
ing, the induced E.M.F. in which is 102 volts, the frequency being 100
periods per second. Assume that B will not exceed 4000 lines per square
centimetre. What must be the sectional area of the core ?

(C. and G., 1894.)

8. Point out in what way the pressure at the secondary terminals of a
good transformer depends on that at the primary terminals, and on the
respective number of windings and their resistance. A transformer with
a well-closed magnetic circuit is taking 10 amperes at 2050 volts. Its
primary coil consists of 860 turns of wire having a total resistance of 5
ohms. Its secondary consists of 43 turns, with a resistance of ¥V of an ohm.
Assuming that magnetic leakage^ eddy currents, and hysteresis are negligibly
small, calculate the pressure at the secondary terminals when 200 amperes
are being supplied to the lamps. (C. and G., 1896.)

9. What is meant by the magnetising current of a transformer, and what
is its phase-relation to the supply voltage ? Find the value of the magnetis-
ing current of a transformer with closed iron circuit, and of which the data
are as follows : —

Primary voltage 2200, primary turns 320, area of core 130 square inches,
mean length of core 40 inches. /t=2000, frequency =50. (C. and G., 1908.)

10. An alternating pressure of 1000 volts, frequency 50, is applied to a
circuit containing a condenser of 4 microfarads capacity in series with a
resistance of 600 ohms. Find the current in the circuit, and the P.D. across
the condenser and the resistance respectively. On what assumptions does
your result depend, and how far do you consider them to be justified ?

CHAPTER XXVII*
MEASUREMENT OF INDUCTANCE

UNTIL recently, the measurement of the coefficients of self and mutual
induction was of little practical importance. Electrical engineers seldom or
never required such measurements, as they were mostly concerned with
circuits containing iron, for which L and M have no definite values. The
rapid development of wireless telegraphy has, however, altered this state of
affairs. In its theory and practice, L and M are extremely important
quantities, and it is becoming necessary for commercial purposes to measure
their values with rapidity and accuracy.

We, therefore, describe a few well-known methods based upon the use of
a ballistic galvanometer. As such methods depend upon the properties of
varying currents, their theory cannot be readily expressed in terms of equa-
tions relating to steady currents, and the descriptions given must be regarded
as explanations rather than proofs. (See also remarks on p. 460.)

Measurement Of Self-Inductance.— (1) By means of an alternat-
ing current of known frequency. This method has already been rnent ioned (p. 435).
Besides having the great advantages of convenience and readiness of appli-
cation, it is often exti-emely useful when only moderate accuracy is required.
For instance, the self-induction of the standard solenoid used in several
previous experiments was measured in this way, and it was found that a
P.D. of 2() virtual volts sent 7 amperes through it, the frequency being 50.
The resistance of the solenoid was 3'54 ohms, and, therefore, we have

o=_ E '

7_ 26

v'(3-54)24-(27rx50xL)2

or 27r x 50L- f 26 Y - (3'54)2

from which L= '003 he my.

The next significant figure cannot be relied upon, for it would be altered
by an extremely small change in the value of current or of E.M.F., and, more-
over, the frequency is seldom exactly the nominal value when public supply
mains are used. Again, the formula assumes that the current obeys a sine
law, a condition which may be only approximately satisfied.

(2) By comparison with a known capacity. — The coil, or portion of a circuit
whose self -inductance is to be measured, is placed in one arm of a Wheat-
stone bridge, as shown at P(L) in Fig. 322 (1), where P stands for the resis-
tance in that arm. It is understood that the resistances in the other arms
are non-inductive. Exact balance for steady currents is first obtained in the

453

454

VOLTAIC ELECTRICITY

usual way, i.e. there is no deflection when the galvanometer circuit is closed
after the battery circuit is closed, which means that Px RX = S x K. (1)

Then the battery circuit is opened while the galvanometer circuit remains

closed, the result being a momentary
throw due to self-induction L. Let
this be d scale divisions. Also de-
termine the damping factor as usual.
The next step is to calibrate the gal-
vanometer by some method which
will facilitate calculation. There
are several ways of doing this ; in
the present case, the connections
are altered as shown in Fig. 322 (2),
a condenser of known capacity K
and the galvanometer being con-
nected between the points A and B.
When the battery key is closed, a
P.D. will exist between A and B,
and then, on closing the galvano-
meter key, this becomes the P.D.
between the condenser coatings, the
charging throw being rfj divisions.
The damping factor should again be
determined, because the circuit ar-
rangements have been altered, and
it cannot be regarded as necessarily
the same in both cases.
FlG 3" Theory of Method. — Let i be

the steady current through the arm

P when balance is obtained in the ordinary way. Then, the effect of open-
ing the battery key (whilst the galvanometer key is closed)- is equivalent
to setting free a quantity Q in the arm P, such that

Resistance of circuit

This resistance is easily written down (being P + S + resistance of shunted
galvanometer), hence, we have

O— k*

P i Q i ( •" + -IV] )fj
•t ~i~ O ~\~ .p- —

Of this quantity a portion q passes through the galvanometer, such that

R + RI f\

q = s — x y

R + RI + Q

also q oc d( 1 + - ) , where d is the throw produced.

R+:

ocrfl 1 +

1)

but from equation (1) S = ±£± , and by substitution and simplification we obtain
R

LiR

then

(2)

In the second part of the experiment, let c be the P.D. between A atid B,

e=i (P + R), by Ohm's law.

MEASUREMENT OF INDUCTANCE 455

Also, if ql be the quantity charging the condenser and producing a throw dl,
we have

ql =

whence, dividing (2) by (3), we obtain

UR

(3)

from which i cancels out, and we have finally

<H)

The self-inductance of an electromagnet coil (without iron) was measured
in this way. R and Rx were each 10 ohms, and in order to obtain exact
balance for steady currents (which is the chief difficulty in such methods),
an adjustable non-inductive resistance was placed in series with the coil and
altered until balance was obtained, when S=12 ohms. Hence, P=:12 ohms.
The throw d was 242 divisions, and the logarithmic decrement / was found to
be '051. In the second part of the experiment, using a condenser of 1 microfarad "
capacity, the throw was 208 divisions, and I was '059. The galvanometer
resistance was 957 ohms, and hence,

{12(10 + 10 + 957) + 10x957}(12 + 10)xiL 242xl'025

10

208 x 1-029

- -054 henry.
(The alternating current method gave '058 henry.)

(3) Anderson's Method. — This is a very good method, and, in practice,
it would often be adopted in preference to any other.

The theoretical diagram is shown in Fig. 323, and the actual arrangement
in Fig. 324.

The Wheatstone bridge method, slightly modified, is again used, but in
this case the throw due to self-in-
due tion is neutralised by an equal
and opposite effect due to a known
capacity, and hence it has the ad-
vantage of being a null method.

First obtain exact balance for
steady currents. During this opera-
tion the resistance r can be cut out.
The galvanometer should be shunted
at first to avoid injury, and it will
be necessary (as in the previous ex-
periment) to introduce a small
variable resistance in the arm P
(such as a straight piece of bare
wire indicated by 6; Fig. 323), in
order to obtain balance for steady
currents. It will also be advisable to make R large and

FIG. 323.

small.

This balance will not be disturbed by introducing the resistance r into the

450

VOLTAIC ELECTRICITY

galvanometer circuit. Adjust its value until there is no throw when the
galvanometer key is closed first. It may be necessary to find by trial the

'best value' of K, if
several condensers are
available. As a rule,
select a small capa-
city when measuring
a small inductance,
and vice versa.

The theory of the
method cannot be
worked out in any
simple way, but it
leads to the following
expression :

FIG. 324. When this method

was used to measure

the self-induction of the standard solenoid (p. 395) the following values were
obtained : R=1000 ohms, R, = 10 ohms, S = 359 ohms, therefore P (including
&) = 3*59 ohms, r=28 ohnis, K = J microfarad (obtained by putting two
| microfarad condensers in series).

. • . L = l 6{(359 x 28) + (3-59 x 1000) + (3'59 x 10) }

= *0034 henry.

Value of L by Calculation. — It has already been shown that for a lony
solenoid,

L= -*n £ henries (approximately).

In this case, n = 369 turns, Z=136 centimetres, diameter=8*4 centimetres.
.'. A=7r(4'2)2.

As the value arrived at from this formula is known to be an approximation
(see p. 376), the general agreement is satisfactory.

Meaning of Result. — From the explanations already given (see p. 376), it
will be seen that the statement
that L=*0034 henry for a certain
coil means that when unit current
(i.e. 10 amperes) is passing through
it, a magnetic flux is linked with it
such that the product of the flux x
number of turns is '0034 x 109.

Measurement of
Mutual Induction. —The

coefficient of mutual induction has
already been defined (p. 378). In
practical electrical engineering, it

is a quantity of little importance,
except in connection with wireless
telegraphy. Hence, we merely
indicate two or three methods of general application.

(1) Comparison of Two Mutual Inductances. — The two mutual inductances,
Mj and Ma, are arranged as shown in Fig. 325. The respective primary coils

MEASUREMENT OF INDUCTANCE 457

are connected through a reversing key to a cell or battery ; and the secondaries
are placed in series with each other and two resistance boxes, R: and R2, a
galvanometer bridging across this circuit at AB. R1 and R2 are then ad-
justed until there is no galvanometer throw when the primary current is
reversed. To obtain this result, it is necessary to connect up the secondaries
the right way, i.e. so that the two induced E.M.F.'s are in series, a condition
easily found by trial. Let i be the steady primary current, el and e2 the
average value of the E.M.F.'s produced in the secondaries when this current
is reversed. Then

_ 2Mji ^ The coefficient 2 would disappear

Cl~time of reversal if the circuit was merely broken.

. V Reversal means a double cutting of

c 2= & the lines, and increases the sensitive-

1 time of reversalj ness Of the test.

•'• |=M2

The arrangement of the secondary circuit is exactly similar to that of the
case discussed on p. 226, where it was shown that the condition of A and B
being at the same potential is

c2 r2 + R2

Hence, ^=
M2

This method was adopted to compare the mutual inductance between the
standard solenoid and its secondary, used in Experiment 232 (Mj), with
that between one of the electromagnet coils and a small coil slipped inside
it, used in Experiment 223 (M2).

It is a matter of convenience as to which coil of each pair is made the
primary — in this particular experiment, the coil of 1000 turns inside the
solenoid was used in the one case, and the small coil inside the magnet coil
in the other. Their resistances were 46'2 and 1*75 ohms respectively, there-
fore r1 = 46'2 and r2=l'75. Balance was obtained when R1=30 ohms and
B^=2tJO ohms.

. M1_r1 + R1_ 46-2 + 30 ' 76-2 __ 1_
M2~ r2 + R2~ 1-75 + 290" 291-75 ~ 3^8

(2) Carey Foster's Method of measuring a Mutual Inductance. — In this method
the discharge due to mutual induction is balanced by an opposite discharge
from a condenser of known capacity, the arrangement being shown in Fig. 326,
where P and S are the primary
and secondary coils of the mutual

inductance to ba measured, and * p

R and R! are resistance boxes. If
- the infinity plug be taken out of
the box Bj, thus rendering that
part of the circuit inoperative, it
will be found that a throw is pro-
duced when the battery key is
open or closed, due to the con-
denser ; and if the plug is replaced
and the condenser circuit broken
between A and D, another throw

is produced under similar con- FIG. 326.

ditions, due to the mutual induc-

tance. Hence, if the connections are such that these throws are in opposite
directions, it is possible to adjust R and Rx until they balance each

458 VOLTAIC ELECTRICITY

other, i.e. until no throw is produced when the key is opened or closed. A
current reverser may be employed to increase the sensitiveness of the
arrangement. When this condition is attained, A and B are at the same
potential, and, therefore (if i be the steady current in R and P), the P.D.
across the condenser=P.D. across DB = iR, and the quantity discharged from
the condenser =i.Rx K (when the circuit is broken).

Similarly the quantity produced by the mutual inductance is

and the condition for balance is

where M will be in henries, if K is in farads and the resistance in ohms.

The mutual inductance between the electromagnet coil and its secondary
(referred to as M2 on p. 457) was measured by this method. For convenience,
in this case, the small inner coil was used as the secondary, and the values
obtained were : —

K= 1 microfarad = —— absolute units of capacity, S = 1'75 ohms,

R=108 ohms, R^SO ohms.
NowM=KR

=__ x 108 x 109{(50 + 1'75)109}

= 5,589,000 absolute units
or -0056 henry.

The mutual inductance between the standard solenoid and its secondary
may be found approximately by calculation as explained on p. 378, from the
equation : —

M= ™i^^, where M=l

In this case, M = ** x 869 x 1000 x T x (2-68)»

136

= 1,600,000 absolute units
or '0016 henry.

The ratio of the mutual inductances is, therefore,

•0016 __ 1_
•0056 ~ 3-5

which is a rough confirmation of the result obtained in the previous case.

Note on Choice of Galvanometer. — The two preceding methods are nidi
methods, and it might be thought that a dead-beat galvanometer would be
the most suitable — • as it is certainly the most convenient — instrument to
employ. But the state of " balance " does not necessarily exist at each
instant during the process of " charge " or " discharge," and hence, it is
desirable to employ an ordinary ballistic instrument of fairly long period.

Relation between M and L. — There is another very instruc-
tive method of measuring a coefficient of mutual inductance, which may
be mentioned, although it is not recommended in practice. Each of the
two circuits or ooils, which together form the mutual inductance, has,

MEASUREMENT OF INDUCTANCE 459

when removed from the other, a definite self-inductance. Let these
be L! and L2 respectively. Let the two coils be arranged so as to
form the mutual inductance in question and connected in series in such
a way that their respective magnetic effects are in the same direction.
Then the two coils constitute a single winding having also a definite
self-inductance, which can be measured by any of the methods previously
given. Let this be L3. Next reverse the connections of one of the
coils, so that their ampere-turns are in opposition, and again measure
the joint self -inductance, L4. The relation between these quantities will
be most easily understood, if we consider the special case of two solenoids,
one wound over the other, having the same length and practically the same
diameter, the turns being n-^ and n.2 respectively.

Then Lo=

ButL1 = 47mi2A^,andL,:
L

and M = ^

and L4= Lx + L2 - 2M
whence, by subtraction, 4M=L3 - L4

Hence, M can be found from two measurements of an inductance, and although
this result has been deduced from the particular case of two solenoidal coils,
a little consideration will show that it holds generally, whatever may be the
shape or arrangement of the circuit, because in the first arrangement, the
inductances of the coils will become L1 + M,.L2 + M; and in the second,
Lx - M, La - M respectively.

.*. L3 — Li + M + La + M, and

LI = L ^ — M + L i — M,
which are identical with equations already obtained.

Variable Standards of Inductance. — The previous results

may evidently be applied to the construction of standards of self-induction,
although for this purpose the solenoidal form is not convenient. If two coils
(of the tangent galvanometer type, for example) are connected in series, and
one is arranged to rotate inside the other, the nett value of L will vary with
their relative positions, and may be read off on a fixed scale by means of a
pointer attached to the movable coil. The actual values are usually obtained
by calculation from the dimensions of the system ; an operation of some
difficulty when great accuracy is required. Such standards were first intro-
duced by Ayrton and Perry many years ago. At the present time, standards
based on mutual induction are preferred, as their working range is greater
and the absolute values are more readily calculated. In practice they are
equally useful, for M and L are expressed in the same units, and it is as easy
to compare L with M as to compare two L's or two M's.

Vibration Galvanometers.— A student, who works through the

previous methods (and possibly others given in more advanced treatises), will
find them not only troublesome in many respects, but also particularly un-

460 VOLTAIC ELECTRICITY

satisfactory in the case of small inductances. In the problems of wireless
telegraphy, the accurate measurement of small inductances is of great impor-
tance, so that other methods, depending on the properties of alternating
currents, are rapidly being developed, which, in all probability, will soon
render obsolete those already described. The chief difficulty, hitherto met
with, has been the want of a suitable galvanometer for alternating currents.
Ayrton and Perry long ago made the first step in the right direction by intro-
ducing their sccohmmcter (practically a rotating commutator, which made an
ordinary galvanometer serviceable with alternating currents produced by
rapidly reversing a direct current), but this instrument never came into
general use. At the present time, the problem appears to be in a fair way
of being solved by the use of vibration galvanometers. Wien and Eubens
first made use of the principle, and, a few years ago, a very convenient form
was brought out in this country by Campbell. This particular instrument is
of the D'Arsonval type ; a very narrow coil, carried by a bifilar suspension,
being supported under adjustable tension between the poles of a permanent
magnet.

When an alternating current is passed through the coil, the latter tends
to oscillate with it, but the effect is almost negligibly small under ordinary
conditions, and hence the chief feature of the instrument is an application
of the principle of resonance. By van-ing the tension of the suspension, the
moving system is " tuned," until its own natural period of vibration is equal
to that of the alternating current in use at the moment, and then the spot oi?
light on the scale widens out into a continuous band, the length of which is
a measure of the current strength as long as the frequency is unchanged. When
thus " tuned," the instrument is suitable for use in all kinds of null methods,
such as those already described. It also lends itself to a very ready and exact
comparison of inductances and capacities with a suitable variable standard
of inductance, and thus the chief difficulties met with in ballistic methods
disappear.

As vibration galvanometers a»e not yet in general use, space does not per-
mit us to go further into details. It may be remarked, however, that the
operation of " tuning " is equivalent to ensuring that the moving system is
exactly at the end of its swing when the current changes its direction. Then,
evidently the successive impulses act together, and a veiy feeble alternating
current may produce a large cumulative effect. It might be thought that the
amplitude would steadily increase without limit, but, as in other cases of the
kind, it only increases until the energy dissipated in each swing becomes
equal to the energy imparted to it during that swing.

CHAPTER XXVIII

THERMO-ELECTKICITY

The Seebeck Effect- — Exp. 241. Connect two copper wires to the
terminals of a reflecting galvanometer, and complete the circuit by a piece of
iron wire, twisting the junctions tightly together. Hold a lighted match to
one junction, observe that the galvanometer is deflected, and note the
direction. On removing the match, the spot of light gradually comes to rest.
Now apply a match to the other junction, and notice that the deflection is in
the opposite direction.

Compare these directions with that produced by putting the ends
of the galvanometer wires on the tongue, as in Experiment 218,
and thus show that the current passes from the copper to the iron
through the liot junction.

Such currents — discovered by Seebeck in 1822, and called thermo-
electric currents — are produced in any circuit formed of two dissimilar
conductors, the only necessary condition being a difference in tempera-
ture between the two junctions. It is noteworthy that the effect is
usually greater for poor conductors than for good conducting metals
like copper, silver, or gold. This is clearly shown by the following
list, in which the order is such that, if a circuit be made with any
pair of substances, the current through the hot junction is from the
one above to the one below, its strength increasing with their distance
apart on the list.1

Bismuth

Platinum

Cobalt

German silver

Lead

Copper

Gold

Silver

Zinc

Iron

Antimony

I

Evidently an E.M.F. is produced in the circuit, the magnitude of
which depends both on the nature of the substances and on the
difference of temperature between the junctions. It is usual to

1 It must be understood that this sequence refers to comparatively small
differences of temperature under ordinary circumstances. It is shown later
that for wider ranges of temperature the dii-ection of the current may be
reversed. In fact, such a list has no meaning unless the temperature
conditions are specified.

461

462

VOLTAIC ELECTRICITY

express certain important experimental results in the form of two
laws, which may be stated as follows : —

1. Law of Successive Temperatures. — Suppose that

10E20 is the E.M.F. in the circuit, when the junctions are at 10° C. and
20° C. respectively ; and that 20E30 is the E.M.F., when they are 20°
and 30° respectively. Then the law states that the E.M.F., when the
junctions are at 10° and 30°, will be the algebraic sum of the pre-
ceding values, or 10E30 = 10E20 + 2oE3o> which may be generalised by
writing

2. Law of Intermediate Contacts.— This very important
experimental law states that we may open any junction at a given
temperature — say t° — and insert in the circuit any number of different
conductors in series, without altering the E.M.F. in the circuit, pro-
vided that all these intermediate conductors are kept at the same
temperature, t°, as that of the original junction.

This means, for instance, that we may solder two wires together
to obtain good contact, and the result will be the same as if the bare
wires are twisted together, although a thin layer of solder may every-
where prevent actual contact. It also makes it possible to insert a
galvanometer in the circuit, and to use such an arrangement as that
shown in Fig. 327.

Measurement of Thermo-electromotive Force.— This

offers certain difficulties on account of the very small value of such
E.M.F.s ; for instance, the E.M.F. of a copper-iron circuit, when the
junctions are kept at 0° C. and 100° C. respectively, is only about
•0013 volt.

The following arrangement is given on account of its simplicity.
A more exact method is described later.

Exp. 242, to measure t/ie E.M.F. for copper and iron at different temperatures.

Connect up anordinarypotentiometer
as shown in Fig. 327. The potentio-
meter, AB, is in circuit with a single
Darnell's cell, D, and a resistance
box, E. The thermo-couple is made
Q by cutting off any convenient length,
say 1 yard, of iron wire, and solder-
ing to its ends two other pieces of
copper wire, each of about the same
length. It is then connected up as
shown, one junction being kept in
melting ice, and the other being

FIG. 327.

placed in a vessel, which can be
heated over a Bunsen burner.

If it is intended to carry the measurements above 100° C., colza
oil may be used instead of water up to about 200° C. The galvano-
meter should be a reflecting instrument, and it must be remembered
that, as the E.M.F. 's to be measured are so small, bad contacts will

THERMO-ELECTRICITY 463

lead to troublesome difficulties. Again, if we attempt to employ the
ordinary potentiometer method described on p. 304, the P.D. between
the ends of AB will have to be greater than the E.M.F. of the
standard cell used for purposes of comparison, and then all the points
of balance obtained with the thermo-couple will be so close to A as to
be unreadable. We must, therefore, make the constant P.D. across
AB only a little greater than the E.M.F.'s to be measured, and in
this simple form of experiment, we dispense with a standard cell by
making use of the fact that the E.M.F. of a DanielPs cell is practically
I'l volt. Begin with the hot junction at about 10° C., and then
increase R until the position of balance occurs at a convenient point
on the potentiometer scale. It is desirable to make the first reading
as large as is consistent with getting the last reading on the scale, e.g.
if it is intended to work up to 100° C., the position of balance when
the hot junction is at that temperature will roughly correspond to ten
times as many divisions as for 10° C., hence, if there are 1000 divisions
on the potentiometer scale, it will be convenient to start with about
80 divisions for a difference of temperature of 10°.

This adjustment being made, R remains constant during the
experiment.

Increase the temperature of the hot junction to about 20° C.,
again obtain balance, and so on, obtaining readings for as many
different temperatures as possible.

Let R be the number of ohms unplugged in the resistance box,
r the resistance of one division of the potentiometer wire (found, if
necessary, by a separate experiment), N the total number of divisions,
and d the reading in scale divisions at any temperature t° C.

Then, as the resistance of cell and connecting wires is negligible
compared wi^h R,

Current through potentiometer wire = — amperes.

Xv ~T~ -1^ i

Also, the P.D. across any one division of that wire = O = = — ^ x r

= a constant (provided that the current through potentiometer wire
does not alter during the experiment). Therefore, we have only to
multiply this constant by the observed reading, d, to obtain the
E.M.F. of the thermo-couple for that difference of temperature. It
will be convenient to express the result in absolute units or in micro-
volts (one millionth of a volt).

If the results of this experiment be plotted with the values of
E for ordinates, and the temperatures of the hot junction for abscissae,
the graph obtained will not differ much from a straight line, but when
the measurements are extended through a wider range of temperature,
the E.M.F. passes through a maximum, then decreases in value to zero,
and finally reverses its direction — the resulting curve being sensibly

464 VOLTAIC ELECTRICITY

parabolic (see Fig. 331, p. 468). The temperatures at which these
effects occur depend upon the substances used ; in the case of copper
and iron (one junction being kept at 0° C.), the maximum E.M.F. is
obtained when the other is at 275° 0., and reversal occurs at 550° C.
The reversal of the current is easily shown as follows : —

Exp. 243. Connect two copper wires to the terminals of a dead-beat reflect-
ing galvanometer, and complete the circuit by means of a piece of iron wire.
Gradually bring one of the junctions near a very small gas flame, approaching
the coolest part of the flame at first. The deflection will be observed to
increase to a maximum, then decrease, and finally increase again in the
opposite direction. Remove the junction from the flame, and as it cools,
watch the deflection, which will be found to pass through the same stages in
the inverse order.

Peltier Effect. — About twelve years after Seebeck's discovery,
Peltier discovered that, when a current is passed across the junction
of two dissimilar substances, there is either a small absorption or
a small evolution of heat at the junction. This effect is quite distinct
from, the ordinary Joule effect, which is always an evolution of heat,
and is independent of the direction of the current. Moreover, the
heat appears in all parts of the circuit, and is proportional to C'2R.
On the other hand, the Peltier effect is always confined to the junc-
tions, and is reversible, being either an absorption or an evolution of
heat according to the direction of the current. Again, it is propor-
tional to the current and not to its square, and it does not depend
upon the ohmic resistance at all (except in so far as that modifies the
current strength).

Peltier's Cross. — One of the methods used by Peltier to
demonstrate this effect is shown in Fig. 328. Two rods of antimony
and bismuth, A and B, are soldered together to form a cross, and
connected up as shown to a cell and a galvano-
meter. The current from the cell does not pass
through the galvanometer, and if the bars were of
the same material, nothing would happen. As it
is, the galvanometer shows a deflection, and if the
cell current passes from bismuth to antimony, this
deflection shows that the galvanometer current
passes from antimony to bismuth through the
junction, i.e. it indicates that the junction is
cooled.

Mr. S. G. Starling has suggested another
method of demonstrating the Peltier effect. A bar
of bismuth is placed endwiss between two bars of
FIG. 328. antimony, the surfaces being amalgamated to reduce

the resistance due to bad contact. A coil of 36-
gauge insulated copper wire is wound over each junction, and the
arrangement protected from air currents by a surrounding tube. The
two coils are then connected up in the arms of a Wheatstone bridge,

THERMO-ELECTRICITY 465

and balance obtained. When a current is passed through the com-
pound bar (1 ampere is said to be sufficient), one of the two coils is
heated and the other is cooled by the Peltier effect ; the consequent
change in resistance disturbing the balance and producing a deflection
of the galvanometer. The direction of this deflection indicates the
existence of a cooling effect when the current flows from bismuth to
antimony, and vice versd.

The fact that the Peltier effect is reversible affords a clue to its
nature. Consider a copper-iron circuit, Fig. 329. Energy is always
absorbed when a current flows from copper to iron,
whether that current is a Seebeck effect due to
heating the junction, or whether it is obtained from
some external source, and this indicates that it is
flowing up a slope of potential.1 When a current FlG

flows from iron to copper, energy is evolved in the
form of an additional amount of heat, which means that the current
is flowing down a slope of potential. Hence, we infer the existence
of a P.D. between two unlike substances in contact; in the above
case iron being positive to copper (at ordinary temperatures).

Here there is some danger of confusion of thought, because we
are naturally inclined to think that when a current is flowing flown
a slope of potential, it is flowing in the same direction as an E.M.F. ;
and when it is flowing up such a slope, it is flowing atjainst a back
E.M.F. As a matter of fact, it is easy to see that at a part of the
circuit in which an E.M.F. is beinr/ generated, a current in the same
direction as the generated E.M.F. is flowing up a slope of potential,
and vice versa ; whereas in conductors serving merely to complete the
circuit, it is of necessity in the same direction as the E.M.F., and is
also flowing down a potential slope. Consider an ordinary battery
circuit. In the battery itself the current is flowing in the direction
of the generated E.M.F., and also up a potential slope (as if pumped
up to a higher level) ; a process which requires a supply of energy.
In this case the energy is derived from chemical actions going on in
the cells, such actions liberating less free heat than they would other-
wise do. That is, energy is being absorbed where the current is
flowing up a slope of potential. In the external circuit, the current
is flowing in the direction of the E.M.F. (i.e. down a slope of poten-
tial), and is liberating, as C2R heat, the previously acquired energy,
which in this case cannot be liberated in any other form. Kow in
this part of the circuit, insert a back E.M.F. (say a cell of negligible
internal resistance connected up the wrong way), and we may suppose

1 We might express this in the language of electrostatics by saying that,
when electric charges are moving in an electric field against the electric force,
i.e.. when work must be done upon them, they are moving up a slope of poten-
tial. When they move with the electric force, they are flowing down a slope
of potential.

2G

466 VOLTAIC ELECTRICITY

that the battery power is increased to keep the current unaltered in
strength. A little consideration will show that the opposed cell is
equivalent to a locally steep downward slope of potential, and, as a
consequence, an extra amount of energy is liberated in it over and
above the C2R heat which is the same as before. But where the
current flows downwards through the source of a back E.M.F., the
energy evolved is not necessarily in the form of heat, and in this case
it is expended in reversed chemical action, i.e. in decomposing some
chemical compound, while in other cases — such as that of a motor —
it may take the form of mechanical work.

Applying these considerations to the thermo-electric circuit shown
in Fig. 329, we notice that when the junctions are at the same
temperature, there are two equal and opposite Peltier E.M.F.'s, and
hence no current flows. When this balance is upset by altering the
temperature of one of the junctions (say by raising the temperature
at A), we know that a current flows from copper to iron through the
hot junction, energy being absorbed at A and given out at B, and we
should naturally expect the former to be the greater, the difference
between the two representing the energy available in the circuit.
But experiment shows that, in this particular case, the energy evolved
at B is greater than that absorbed at A, i.e. the Peltier effect at A
is weaker than that at the colder junction B, and yet the current is
flowing in the direction of the weaker E.M.F.

Thomson Effect. — Considerations such as these led the late
Lord Kelvin (then Professor Thomson) to predict the existence of some
other reversible heat effect in a circuit ; a conclusion he soon verified
experimentally by showing that a P.D., or slope of potential, existed
between the hot and cold parts of one and the same substance,
e.g. hot copper is positive to cold copper, and hot iron is negative to
cold iron. Without going into details, the principle of the method
will be understood from Fig. 330. AB is a bar of the metal in
question, which by some external means is kept hot in the middle
and cold at the ends. Intermediate points, P and Q, on the bar can
p Q be found, which are at the same

Aj . . . R temperatures. If there is any

Cold Hot COM\ difference of potential between the

;} hot and cold parts of the bar, a

llilil ~~"^ current flowing through it must,

FlG 33Q in one half, be flowing up the

slope of potential, and, in the

other half, down it. In the former case, energy will be absorbed and
the temperature lowered ; in the latter, energy will be liberated and
the temperature raised. As a matter of fact, when a strong current
is sent through the bar, it is found that the temperatures at P and Q
are no longer equal : in the case of copper, that at Q is greater than
that at P, when the current flows from P to Q ; in the case of iron,

THERMO-ELECTRICITY 467

the higher temperature is at P. Silver, cadmium, zinc, and antimony
behave like copper; platinum, bismuth, cobalt, nickel, and mercury
like iron. In the latter metals, the Thomson effect is said to be
"negative." Lead is noteworthy on account of the entire absence
of the effect — at any rate, it is too small to be measurable.

Mr. S. G. Starling demonstrates these facts by a method similar
in principle to that already given in the case of the Peltier effect.
The two ends of a U-shaped iron rod are bent to dip into vessels of
mercury, thus ensuring good contact and also keeping the ends cool.
Over the middle portion of each limb a coil of fine insulated copper
wire is wound, the two coils being placed in the arms of a Wheat-
stone bridge. The rod is then heated to redness at the bend, and the
bridge adjusted until balance is obtained. When a current is passed
through the bar (by means of the mercury cups), this balance is dis-
turbed, and the direction of the ensuing deflection shows that the
Thomson effect in iron is negative. The parts on which the coils are
wound must be packed in asbestos wool in order to eliminate external
disturbances.

We can now restate the general argument as follows : If, in any
circuit made up of different materials, the temperature is the same
at all points, the algebraic sum of the Peltier and Thomson effects
is zero ; but if such uniformity of temperature does not exist, then
that sum may amount to an absorption of heat, in which case a
thermo-current corresponding to the energy absorbed per second will
flow round the circuit.

Coefficient of the Peltier Effect for Two given Sub-
stances.— This is usually defined as the amount of energy, measured in ergs,
absorbed or evolved tvhen one absolute unit of current flows across the junction for
one second. We shall denote it by P.

Hence, if H units of heat are absorbed or evolved by a current i flowing
for t seconds, then

H x 41-8 x 106= P x i x t ergs.

Now this energy is also equal to exixt ergs, where e is the P.D. between
the substances at the junction, from which we see that P is also the value of
e in absolute units, and there is often a distinct advantage, as regards clear-
ness of thought, in regarding P as an E.M.F. rather than as energy.

Example. — The coefficient of the Peltier effect for copper-iron at 0° C. is
436,000. If a current of 40 amperes flows from copper to iron for one minute
at this temperature, what happens at the junction?

We found by Experiment 241 that a thermo-current flows from copper
to iron through the hot junction, hence we know that, when a battery
current is sent in this direction through the junction, there is an absorption
of heat.

Now energy absorbed =Y.i.t ergs

= 436,000 x |? x 60= 104-64 x 106 ergs.

.-. Heat units absorbed = 104'64 x 10-6=2'5 (gram.C0).
41'8x 106

Of course, this tells us nothing respecting the actual fall of temperature at

468 VOLTAIC ELECTRICITY

the junction. That depends upon the size of the conductors, their specific
heat, and the rate at which heat is carried away by conduction, and it is
also affected by the production of ordinary C2R heat. Obviously it will be
very small.

Coefficient Of the Thomson Effect. — Let one absolute unit of
current flow for one second from a part of a substance at temperature T to
another part at temperature Ta, where T: differs wry little from T (this
stipulation is made necessary by the fact that the effect varies with tempera-
ture). Then if s be the coefficient of the Thomson effect, the amount of
energy absorbed or evolved is s(T-Tj), more conveniently written s.rfT,
s being taken as positive for substances that behave like copper and negative
for those that behave like iron.

Hence, when current i flows for t seconds, the energy absorbed or evolved
= i.t.s.dT ergs, from which it follows, as before, that s.dT is the P.D. between
the points in question.

When wre are dealing with great differences of temperature, s varies from
point to point along the conductor, and the total P.D. must be obtained by a

Pi
summation, expressed by writing 1 s.dT.

JT

Consider again the circuit shown in Fig. 329, and suppose that the
junctions are at T and Tl respectively (Tl being greater than T). A current
is flowing, whose direction is from copper to iron through the hot junction,
due to an E.M.F., which is the resultant of four components, viz. the two
Peltier E.M.F.'s at the junctions, which oppose each other but which are
unequal in magnitude; and the Thomson E.M.F.'s between hot and cold
copper, and between hot and cold iron, which are in the same direction.

It will be noticed that energy is being absorbed both in the wires and in
the hot junction, and is being given out at the cold junction. If E be the
thermo-electromotive force in the circuit, as measured in Experiment 242,
then

IT JT

As the sign of s is negative for iron, the second term is numerically positive.

Neutral Point and Thermo-electric Heights. — As

already stated, the curve obtained by experiments, such as the one
described on p. 462, is sensibly parabolic, and for copper and iron has

the shape shown in Fig. 331. Hence,
when one junction is kept at 0° C., we
^ know from the theory of parabolic curves

that the relation between E.M.F. and
temperature can be represented by an
equation of the form

*

FIG. 331. where a and b are constants for a given

pair of metals, and are opposite in sign. Solving the quadratic
(&T2 + aT - E = 0) we obtain

2b

* T is used for temperature to avoid confusion with t (time). It does not
necessarily mean absolute temperature until so stated (on p. 476).

THERMO-ELECTRICITY 469

From this \vc learn that there are two values of T for any given value
E, unless the quantity under the root sign becomes zero, in which case

T — — ---- . Hence, this is the temperature corresponding to the

summit of the curve, which for copper-iron is 275° C. It is called
the neutral point.

We also learn from the equation that E vanishes when T = 0,

and when IT2 = — «T, or when rT= — - , i.e. when T is twice the

b

temperature of the neutral point.

Thermo-electric Diagram. — These results are more simply
obtained by differentiating the equation E = «T + 6T2, for

— = a + 26T, which vanishes when T = - ~.
a L 2tO

Now, this is a linear equation in temperature, and hence if we plot a

jjji

graph with the values of -~ for ordinates, and temperatures for

a J.

abscissae, the result will be a straight line. With a few exceptions,

JTf

this holds good generally. The values of — are known as thermo-
electric heights, for reasons which will become apparent later.

Because the Thomson effect is zero for lead, this has been taken
as the standard metal, and in obtaining the curves of E.M.F. each
metal has formed a couple with lead. Hence, when a value of the
thermo-electric height for, say, copper at a given temperature is
stated, it must be understood to refer to such a lead-copper couple.1

When the curves of thermo-electric h tight and temperature for
different substances are plotted' on the same sheet, the result is known
as a thermo-electric diagram. Such a diagram is shown in Fig. 332
(taken from Carey Foster and Porter's Electricity and Magnetism},
in which the ordinates represent thermo-electric heights in microvolts
(millionths of a volt) per degree, and the abscissa? temperatures in
centigrade degrees. When the direction of a current is from lead
to the other substance through the hot junction, the E.M.F.'s are
plotted above the base line, and vice versa.

Consider one of the lines, shown separately for convenience in

Fig. 333. As ~ =a + 26T, it follows that a is the intercept on the

1 It is usually assumed that the axis of the parabola in Fig. 331 is
parallel to the axis of E.M.F., but since the above was written, it has been-
shown that the metals platinum, copper, cadmium, nickel, manganese, pal-
ladium, and aluminium give, when combined with lead, parabolas with
inclined axes, the range of temperature being from -200° C. to +100° C.

The graph of d~ and temperature is then no longer a straight line.

470

VOLTAIC ELECTRICITY

axis, and 2b the tangent of the angle of slope. Also it is evident
mathematically that E, the electromotive force in the circuit corre-
sponding to junction-temperatures T and T1? is proportional to the area
of the figure ABTjT.

7-p

This may also be seen by regarding — as the E.M.F. in the
circuit due to a unit difference of temperature of the junctions.

FIG. 332.

Then AT would give the E.M.F. when the junctions were at T - J,
T + J respectively ; and similarly BTj would give the E.M.F. for a
difference of 1° about the mean temperature 1\. Hence, when the
temperatures are T and Tl respectively, we may take the E.M.F.
per degree difference of temperature as being the mean of these values,
represented by the line CD in the figure. This must be multiplied
by Tj - T, the actual difference of temperature, and the result is the

area of the trapezium ABTjT as
before. The value thus obtained is
the E.M.F. when lead forms part of
the couple, but the argument can
easily be extended to other cases.

Suppose we require to find the
E.M.F. due to a copper-iron couple,
at junction -temperatures T and Tp
both being below 275°. Then from
Fig. 334, it follows that the area
ABTjT represents the E.M.F. of
an iron-lead couple, and DCTjT that of a copper-lead couple. These
areas both lie on the same side of the zero line, and the E.M.F.
of the copper-iron couple is given by their difference, or is proportional
to the area ABCD

T D

FIG. 333.

THERMO-ELECTRICITY

471

The two lines intersect at the neutral point 275°, and evidently
when one junction is at 0° C., the E.M.F. is a maximum when the
other is at 275° C.
Any area beyond the d
neutral point corre- <^
sponds to a reversal of
direction, and hence the
circuit E.M.F. will not
only be ze% when one
junction is at 0° C. and
the other at 550° C., as
already stated, Imt also
whenever the mean tem-

Lead,

-275° Temperature. 550°

perature of the junction FIG. 334.

is 275° C.

Hence, the rule for calculating the E.M.F. in a thermo-electric
circuit may be stated as follows : To find the E.M.F., multiply the
difference of tJie thermo-electric heights at the mean temperature of the
junctions by the difference of temperature.

Example. — If the thermo-electric height of iron at any tem-
perature t° C. is given by 1734-4'87/, and that of copper by
136 + -95/; find the E.M.F. of a copper-iron couple when the junc-
tions are at 20° C. and 100° C. respectively.
Difference of thermo-electric heights = (1734 - 4'87£) - (136 + -95£)

= 1598-5-82^

90 ' 1 00
and the mean temperature of the junctions is — "^ — = 60°

.-. Difference of thermo-electric heights = 1598 - (5'82 x 60) * 1249
also difference of temperatures = 100 - 20 = 80

.-. E = 1249 x 80 = 99,920 absolute units = "|^° volts.

By applying the principles of thermo-dynamics (see appendix to
this chapter) it can also be shown that P, the coefficient of the
Peltier effect at any temperature T, is AD x T, where T is the absolute
temperature of the junction. (Thus far, all temperatures have been
expressed in centigrade degrees.) From this it appears that the
Peltier effect vanishes at the neutral point, i.e. at that temperature
two different substances behave as though they are alike, there being
no P.D. across the surfaces in contact.

It is there also shown that the value of sdT for the Thomson
effect, between any two absolute temperatures, T and Tv is equal
to the difference of the thermo-electric heights at T and Tt multiplied
by the mean temperature of the junction.

In fact, the slope of the lines in the thermo-electric diagram

472 VOLTAIC ELECTRICITY

indicates the existence and nature of the Thomson effect. For example,
the line for copper slopes upwards as the temperature rises, which
means that hot copper is positive to cold copper ; whereas the line
for iron falls, telling us that hot iron is negative to cold iron. When
there is no slope, there is no Thomson effect, as in the case of lead.
Also, the method of plotting leads to the result that, when a current
flows upward in the diagram, i.e. either along a rising line, as from
cold to hot copper, or from any metal to another whose line at that
temperature is above that of the first metal, then energy «s absorbed,
because a current is flowing up the slope of potential, and vice versa.
These results are easily remembered as being analogous to the
gravitational case — bodies, when falling, evolving energy ; bodies,
when rising, acquiring energy.

Example. — Using the data given in the last example, again
find the E.M.F. in a copper-iron circuit with the junctions at 20° C.
and 100° C. by calculating the values of the Peltier and Thomson
effects.

As already found, the difference of the thermo-electric heights
for copper and iron is 1598 — 5'82f, where t is the temperature in
centigrade degrees.

Now the Peltier E.M.F. is equal to this difference multiplied by
the absolute temperature of the junction.

.-. Peltier E.M.F. at hot junction = {1598 -(5-82 x 100)} x 373

-378,968

and reverse Peltier E.M.F. at cold junction - (1598 - 5-82 x 20) x 293

-434,109

Again, the Thomson effect in each metal is equal to the difference
of the thermo-electric heights multiplied by the mean absolute tem-
perature of the junctions.

.*. Thomson's E.M.F. in iron

= {(1734 - 4-87 x 20) - (1734 - 4-87 x 10U)}(60 + 273)
-129,737
and Thomson's E.M.F. in copper

-95x 100)-(136 + -95x 20)}(60 + 273)
-25,308

Wo have already found that the first, third, and fourth of these
are in the same direction, the second being in the opposite direction ;
hence

E.M.F. - 378,968 - 434,109 + 129,737 + 25,308

= 99,904 absolute units,
a result which agrees with the former value.

THERMO-ELECTRICITY

473

FIG. 335.

FIG. 336.

Practical Applications. — The earliest of these was the famous

thermopile, invented by Melloni,

much used in the past in connec-
tion with researches in radiant

heat. It consists of a number of

small bismuth antimony couples

connected in series (Fig. 335),

and built up into a rectangular

or a cubical form (Fig. 336).

When the terminals are joined

to a sensitive galvanometer (G,

Fig. 337), a very slight difference

of temperature between the two exposed faces of the pile (A and
the opposite one inside the cone C) will
produce a perceptible deflection, which for
temperatures between 0° C. and 100° C.
may be taken as proportional to the tem-
perature - difference without serious error.
The chief defect of the thermopile, in its
original form, is due to its comparatively
large mass, which means that it does not'
respond quickly to changes in temperature.
This difficulty has recently been overcome in
a form devised by Kubens, now almost ex-
clusively used for purposes of research.
FlG- 337- A thermopile of the ordinary type may

be used to demonstrate the Peltier effect as follows : —

Exp. 244. Connect a battery of a few cells to a thermopile, and pass a
current for a short time. Then quickly disconnect the battery, and join up
the thermopile to a galvanometer. Notice that a deflection is produced, and
that it gradually dies away.

The passage of the current heated one set of junctions and cooled the
other, which thus- left the pile in a state to produce a thermo-current. Had
there been no Peltier effect, both sets of junctions would have been equally
heated according to the C2R law.

Measurement of Temperature.— A single thermo-couple is

frequently used to measure temperature, its great advantage being
due to its extremely small mass, and hence its ready response to
temperature-changes.

It is necessary (a) to use materials that will not oxidise and will
not melt at the highest temperature to be investigated, and (/>) to
avoid the temperature of the neutral point. For temperatures
up to 1200° C., a wire of pure platinum joined to a wire of
platinum-rhodium (or platinum-iridium) alloy, is largely used, and
instruments made on these lines are supplied commercially with a
galvanometer, calibrated so that the temperature of the junction
can be obtained at once from the deflection. In such measurements

474 VOLTAIC ELECTRICITY

the temperature of the cold junction, i.e. the wires connected to the
galvanometer, should be kept constant, although when this is not
the case, a simple correction may be applied, which is exact enough
for most industrial purposes.

The most direct method of calibrating the galvanometer is carried
out by taking a series of readings with the thermo-electric couple at
a number of known temperatures between, say, 0° C. and 1000° C.
It is now possible to obtain known temperatures with sufficient
accuracy for this purpose.

Boys' Radio-Micrometer. — This is probably the most sensi-
tive of all appliances for the detection of radiant heat. It resembles
a galvanometer of the moving-coil type (Chapter XIX.), but the coil
is replaced by a single loop of copper wire, carrying a mirror and
suspended by a quartz fibre. The two ends of the loop below the
polar gap carry a small bismuth-antimony couple soldered to a thin
blackened disc of copper. The couple is heated by radiation falling
upon it, and the current flows round the copper loop, which is then
rotated by the field like the coil of the galvanometer in Fig. 228.
When the junction is placed at the focus of a suitable parabolic
mirror, it is said to detect the radiation from a candle flame at a
distance of three miles.

The principle of the construction will be understood by referring
to Fig. 231, p. 298, which shows the arrangement adopted in
Duddell's thermo-galvanometer. If the "heater" be removed, and
the thermo-couple warmed by radiation falling upon it from a distant
source, the result is Professor Boys' instrument, of which the thermo-
galvanometer is an adaptation.

Callendar's Radio-balance. — This is the latest instrument
for the measurement of radiation. In its simplest form, the radiation
is admitted through an aperture 2 millimetres across, and falls upon
a copper disc having a diameter of 3 millimetres to which two
thermo-j unctions are attached to form a Peltier cross. One couple
is connected to a sensitive galvanometer for indicating changes of
temperature; the other is connected to a battery and an adjust-
able resistance, and included in the circuit is a suitable ammeter or
potentiometer device for measuring accurately the current required
to reduce the galvanometer deflection to zero. When the disc is
heated by radiation, a thermo-current flows, and then the battery
current is adjusted until this is exactly neutralised by the Peltier
cooling effect. Hence, it has the advantages of a "null" method,
and is especially suitable for absolute measurements.

Much greater sensitiveness is obtained in another form of the
instrument, known as the Gup radio-balance, in which the radiation
enters a small copper cylinder — thus absorbing practically the whole
of it — and a pile of several couples in series is used in place of a
single couple. Various devices are employed to eliminate external

THERMO-ELECTRICITY

475

disturbances and the C2R heating effect. This type is especially
suitable for measuring the heat evolved by small quantities of radio-
active bodies.

Thermo-electric Generators. — Many attempts have been
made to apply the principles outlined in this chapter to the construc-
tion of commercially successful current generators. Such a generator,
as compared with other devices for producing electrical energy,
possesses several great advantages, e.g. (1) no consumption of material
takes place in the pile itself ; (2) but little attention is necessary,
the action being easily set up by lighting a gas jet or other
heating arrangement ; (3) it can work continuously for long periods
without detriment. Its disadvantages, which have hitherto proved
fatal to extensive commercial use, are (1) inefficiency, only about
5 per cent, of the energy of the fuel consumed being actually utilised
in producing the current ; (2) the very low difference of potential
produced at a junction, which necessitates the use of a large number
of elements to obtain even a moderate E.M.F. The latter dis-
advantage is partially compensated by the low resistance, and for
plating purposes, in which high E.M.F. is not required, a certain
amount of success has been obtained.

It appears probable, however, taking everything into consideration,
that further research may lead to important practical results.

APPENDIX TO CHAPTER XXVIII

A THERMO-ELECTRIC E.M.F. may be measured more accurately as
follows: AB (Fig. 338) is an ordinary -potentiometer wire, having,
say, 1000 divisions and a total resistance of R ohms, through which
a steady current is maintained by a Daniell's cell or an accumulator,
and in whose circuit two ordi-
nary resistance boxes, P and
R13 are included. Against this
can be balanced either a stan-
dard cell or a thermo-electro-
motive force, by using the
simple switch described in Ex-
periment 178. A reflecting
galvanometer is used to indi-
cate balance, provided with a
shunt to protect it during the
preliminary adjustments, and
another resistance box, K, in
which about 20,000 ohms are
unplugged, is included in the circuit merely to protect the standard
cell. It does not affect in any way the position of balance.

476 VOLTAIC ELECTRICITY

The principle of the method will be easily understood from the details
of an actual experiment. The resistance of AB is 85 ohms ( — 11), and
the E.M.F. of a Clark cell is taken as 1*434 volt; and it is desired
to adjust matters so that the P.D. per scale division is y^VoT) vo^>
i.e. the P.D. across AB has to be yj^- volt. A connection is made
from K to Rt as indicated by the dotted line (1) — that marked (2) is
not wanted yet — so that the Clark cell can be balanced across the
total resistances, R + Rt. The question then arises, What value must
be given to Rj to make the P.D. across AB -^ volt, when the P.D.
across R + R! is 1'434 volt?

Let C be the steady current in AB, then we must have : —

and l

85

or R1 = 12, 104 ohms.

Unplug 12,104 ohms in Rt. Then alter the value of P until the
Clark cell is balanced across R + Rj (removing the galvanometer
shunt and cutting out K for the final adjustment). Rx and P must
not be touched again.

Remove connection (1), and replace by (2), and then begin the
actual measurements of the thermo-electromotive force, by keeping one
junction of T at 0° and gradually raising the temperature of the other.
If balance is obtained at d divisions, then the thermo-E.M.F. is
Tinnnnr x ^ volts = <? x 1000 absolute units. If the limit of the scale
is reached, the arrangement can be readjusted, making, say, the
P.D. across AB = 1 volt, and, therefore, the P.D. per scale division

volfc'

Application of Thermo-dynamics.— Without attempting an

exhaustive treatment of the subject, it may be remarked that, at the junctions
of two different substances, there is a reversible transformation of work into
heat, or heat into work, to which the second law of thermo-dynamics may be
applied. On the other hand, superposed upon this is an irreversible change
of work into heat due to the C2R effect, to which that law does not apply.
Cut as the one effect depends on the first power of the current, and the other
upon its square, if the current is sufficiently small the C2R heat is relatively
a small quantity of the second order, and is, therefore, in comparison, negli-
gible. This condition implies that the difference of temperature between the
junctions must also be very small.

Consider a copper-iron circuit with junctions at temperatures T and
T + eZT, a small current i, therefore, flowing round the circuit. The quantity
of heat, measured in ergs, which is absorbed in the wires in unit time by the
Thomson effect is («cu - *Fe)cTT x i ; and the Peltier effect produces, at one
junction, an evolution of heat P x i, and at the other an absorption Pl x i.

But in any reversible process, the quantity of heat taken in, divided by
the absolute temperature at which it is taken in, is equal to the heat given

out, divided by the absolute temperature at which it is given out, or 7^=^'

THERMO-ELECTRICITY 477

Hence, («c.i - *Fe)dTxj + P^ P^t where T is the absolute tem.

perature. Now, as P: is only very slightly greater than P, it may be written
P + dP, and i cancels out,

u,

T T + dT T

Hen, («Cn-*FB)dT
~~~

T(T + dT)

T -I- dT
or sCu-sFe=_: — — — - , , which in the limit

becomes sou - »Fe = ~ - -r~

Now, on p. 468, we obtained the expression

E-
dE , k , dP

Substituting tlie value of *c« - *Fei we have

^.E = ? = thermo-electric height,
t/T T

or P for any two substances = the difference of T.E.H (eacli given with respect
to lead) x absolute temperature.

TrfP
. . d P d'£ 1/P rfP

Agam' ^•T=~T2-

But SCu-*Fe=?-

= -26T

Although we have taken the case of copper-iron as an illustration, a
similar result obviously holds good when one of the substances is lead, for
which »=0, and hence we see that the absolute values of the Thomson
coefficient can be obtained in terms of 6. The details of such calculations
are purposely omitted on account of the limitations of space. We are,
however, now in a position to understand more fully the meaning of the
thermo-electric diagram. The minus sign in the above expression depends on
our convention as regards direction, and does not affect the magnitude of the
quantity.

478

VOLTAIC ELECTRICITY

In Fig. 339, let AB be any two points on the line for copper, correspond-
ing to absolute temperatures T and T3.

«

T Ti Temp?'

?™ OQO

i:

Then, because = a + 2bT, 26 is
the tangent of the angle of slope, or

26=AD

_ Difference of T.E.H at T and Tx

T!-T

Also, sCu=26T

Diff. of T.E.H ™

. . SC'll — • FS m ^ JL

T1-T

And the Thomson P.D, between
T and Tx will be

1\-T

_Diff. of T.E.H. T]2-Ta
~~~"

= Diff. of T.E.H xTl+T

which, it will be noticed, is represented by the area KBAL.

Now, consider again the diagram for copper and iron, as in Fig. 340.
The Thomson E.M.F. in iron is represented by the area KABL, and in copper
by the area MDCN, and wo
know that these are in the
same direction. Again, the
Peltier effect at the hot junc-
tion is, by the equation on
p. 477, represented by the
area LBDM, and this is
known to be in the same
direction as the Thomson
effect. The sum of these
will be given by the area
KABDCN.

From this must be de-
ducted the Peltier E.M.F. at
the cold junction, known to
be in the opposite direction,
i.e. we must subtract the area
KAON. This leaves the area

ABDC to represent the actual nett E.M.F. in the circuit, as already stated
on p. 470.

Volta Effect. — Volta discovered (by means of his condensing
electroscope) that, when two rods or plates of copper and zinc are
brought into contact, the copper becomes negatively, and the zinc
positively charged. Other metals show similar effects, but in different
degrees. This result implies the existence of a P.D. between copper
and zinc (the copper being positive to the zinc) before contact, a flow

T, Temperature.

FIG. 340.

THERMO-ELECTRICITY 479

then taking place which makes their potentials equal. This contact
electrification was regarded by Volta as the source of the E.M.F. in
a voltaic cell, a hypothesis which excited endless discussion for
nearly one hundred years after his death.

Now, the Volta P.D. between copper and zinc is about -8 volt,
and if such a large difference of potential actually existed between
these metals, reversible heating effects of a distinctly noticeable
amount would be produced whenever a current passed across the
junction of the two metals. But experiments reveal only the exist-
ence of a Peltier E.M.F. of almost negligibly smaller magnitude, and
there is every reason to believe that this is the only true contact
E.M.F. between the metals.

It is now usual to regard the Volta effect as being due to the
immersion of the metals in air (oxygen), incipient chemical action
tending to set up a P.D. between each metal and the surrounding
air, this difference being the Volta E.M.F.

Apparently the question could be easily decided by measuring
the Volta effect in a vacuum, but it is extremely difficult to get rid
of the surface films of occluded gases, which vitiate the results of the
experiments. By taking the most elaborate precautions, it has, how-
ever, been shown that the magnitude of the Volta E.M.F. is largely
reduced in the absence of oxygen, a fact which sufficiently confirms
the general accuracy of the explanation.

EXERCISE XXII

1. Two wires, one of copper, the other of iron, are twisted together at one
end, the other ends being connected to a suitable galvanometer. Describe
and explain the indications of the galvanometer as the iron-copper junction
is gradually heated to bright redness. What becomes of the heat absorbed
at the junction ? (B. of E., 1908.)

2. Write a short account of the construction and practical applications of
the thermo-electric junction. (B. of E., 1904.)

3. Explain how the metals can be arranged in a thermo-electric series, and
the conditions under which such a series has a definite meaning.

(B. of E., 1896.)

4. Two bars of bismuth, A and B, are attached to the extremities of a bar
of antimony, and a current is passed from A to B. Is there any difference,
and if so what, between the effects produced at the two junctions ? How does
the effect in each case depend on the strength of the current ?

5. A ring is made, partly of iron and partly of copper wire, the junctions
being A and B. If A be kept at 0° and B at 100°, a thermo-electric current is
produced in the circuit. Similarly, a current is produced if A be at 100° and
B at 200°. Have the currents in each case the same strength ? Give reasons
for your answer.

6. A thermopile is joined up in series with a Daniell's cell, and the current
allowed to flow for a short time. The thermopile is then removed from the
circuit, and connected to the terminals of a galvanometer, the needle of which
is thereupon considerably deflected but gradually returns to its undisturbed
position. Explain this. (B. of E., 1898.)

480 VOLTAIC ELECTRICITY

7. Under what circumstances can an electric current cool the conductor
through which it passes? What will be the result if the direction of the
current be reversed ? (B. of E., 1894.)

8. Explain how thermo-electric currents can be. utilised for measuring
temperature. Illustrate your answer by describing the arrangements you
would employ for measuring the temperature of the outside of a pipe which
is conveying steam. (B. of E., 1910.)

9. What is meant by thermo-electric power, and how can the data for
a diagram representing it be obtained ? c (Lond. Univ. B.Sc., 1903.)

10. Prove that the coefficient of the Peltier effect at a given junction is
the product of the absolute tempei^ature of the 'junction and the rate of
change of the whole E.M.F. of the circuit with the temperature of that
junction. (Lond. Univ. B.Sc., Honours, Internal, 1907.)

11. If the thermo-electric height of iron is 1734-4'87£, and of silver
214 + I'oOt, where t is the temperature in centigrade degrees, find the E.M.F.
of a silver-iron couple when the junctions are at 16° C. and 180° C.
respectively. Also find the temperature of the " neutral point."

CHAPTER XXIX*

PASSAGE OF A DISCHARGE THROUGH GASES

THE phenomena attending the passage of a discharge through a gas
differ in many important respects from those observed in the case of
metallic or electrolytic conduction ; for instance, one and the same gas
may, according to circumstances, act like an excellent insulator, a
poor insulator, or a true, if feeble, conductor.

At atmospheric pressure, most gases act like good insulators of
small dielectric strength, i.e. they insulate well, but are somewhat
readily pierced by a spark. At greater pressures, they are still better
insulators, and are much less readily pierced by a spark.

If the pressure is reduced, the dielectric strength decreases, pass-
ing through a minimum value (at a pressure which depends upon the
nature of the gas, but which is of the order of a millimetre of
mercury), thence increasing again until, when extremely rarefied, the
gas becomes totally non-conducting.

Again, by the action of certain agencies (X-rays, radium rays, &c.),
a gas at ordinary pressure may be brought into a state in which it
possesses true conductivity, i.e. instead of requiring an applied P.D.
above a certain minimum value to ensure a discharge — which then
occurs more or less suddenly as a spark— a small P.D. can send a
current through it, which, up to a certain point, obeys Ohm's law. In
this state the gas is said to be ionised.

The very existence of the phenomena discussed in the first section
of this book shows that air at ordinary pressure is a non-conductor.1
This fact and the increase of insulating power with increased pressure
indicate that some real difficulty attends the passage of a charge from
a solid body to the gas particles immediately in contact with it. The
same difficulty is strikingly met with in the case of liquids, e.g.
electrified water gives off unelectrified vapour, and so does boiling
and electrified mercury.

On the other hand, it has been shown (see p. 82) that a gas can
be electrified — e.g. by point discharges — and that it can be pierced
by a spark, both effects requiring a certain high strength of the

1 Kecent work connected with radio-activity has shown that air always
possesses a certain (although very feeble) conductivity. This is why a charged
body, no matter how perfectly insulated, in time loses its charge ; an effect
which, until the discovery of radio-activity, was always ascribed to imperfect
insulation. See p. 105.

481 2H

482 VOLTAIC ELECTRICITY

electric field at the junction of solid and gas, and suggesting some
disruptive process quite distinct from ordinary conduction. (It should
be noticed that when a spark passes through a gas, it produces a
temporary path of very small resistance, and is, in fact, equivalent to
suddenly joining the spark terminals by a good conductor. As
Faraday first observed, it may be difficult to get the first spark to
pass, but it is then easier for others to follow.)

Spark Discharge at Ordinary Pressure.— The earliest
measurements of the relation between sparking voltage and spark
length .were made many years ago by Lord Kelvin by means of the
(then newly invented) absolute electrometer, and his work brought
out the "fact that the dielectric strength of air was relatively much
greater for very short sparks than for longer ones. To a considerable
extent, however, the sparking voltage depends upon the experimental
conditions (e.g. upon the size and shape of the terminals), and thus
the question is somewhat complicated.

The order of magnitude of these quantities is indicated in the
following table, which refers to brass balls two centimetres in
diameter in air at ordinary pressure. It will be seen that the voltage
does not increase proportionally to the length of spark : — •

Spark Length in p n . Vnlta

Centimetres. FJL m Volts'
•1 4,700

•5 17,500

1-0 31,300

2-0 47,400

3-0 57,500

4-0 64,200

5-0 69,800

The chief facts may be briefly summarised as follows : —

(1) In gases at ordinary pressures, the relation between P.D. and

spark length (except in the case of very short sparks) is practically

linear, and may be expressed in the form

where a and b are constants depending upon the nature of the gas.
and I is the spark length.

(2) As the spark length decreases, this expression holds good
until (in the case of air), I is about 4010o inch, and E is about 350
volts. This is known as the critical spark length, and for it E has
its minimum value.

(3) For still shorter sparks, E increases up to a certain point, and
then the law again abruptly changes, and for extremely short sparks
of microscopic length, the expression becomes E = ml, where m is a
constant.1

1 The peculiar properties of microscopic spark gaps have been applied
recently by S. G. Brown in the construction of a very remarkable telephone
relay.

PASSAGE OF A DISCHARGE THROUGH GASES 483

(4) The minimum sparking potential is the same for the same gas
at all pressures, but the corresponding sparking distance is nearly
inversely proportional to the pressure, and hence arises the greatly
increased length of spark in vacuum tubes.

(5) For extremely low pressures, however, this relation ceases to
hold good, and the discharge tube finally becomes nori-conducting.

Discharge in Rarefied Gases. — The phenomena met with
in gases under reduced pressure are usually demonstrated by means
of exhausted glass tubes, of extremely varied shapes and sizes, pro-
vided with metal electrodes (generally of aluminium, because this
metal disintegrates least under the action of the discharge). The
electrodes are joined to platinum wires sealed through the glass.
Such a tube, suitable for examining the effects that occur in the
initial stages of an exhaustion, is shown in Fig. 341.

If the tube, before exhaustion, is connected to the secondary of
an induction coil in action, no effect is produced until the pressure
falls to about half an inch of mercury, and then a faint glow becomes
visible near the terminals. As exhaustion proceeds, this luminosity

mawm

FIG. 341.

gradually increases, and long thin undulatory threads of light extend
from terminal to terminal, until gradually expanding and combining,
they form a straight luminous column — red in the case of air — which
begins at the anode and reaches to within a short distance of the
cathode. This is known as the positive column. The cathode is
surrounded by a very striking bluish glow — known as the negative
glow or the cathode glow — separated from the positive column by a
comparatively non-luminous space, known as the Faraday dark space.
With increasing exhaustion, this space gradually lengthens, the
negative glow increases in size and brilliancy up to a certain point,
and the positive column begins to split up into thin distinct discs or
slices, known as strice (always having their concave sides towards the
anode) which gradually increase in size and move further apart.
Meanwhile, the negative glow has separated from the surface of the
cathode, leaving a sharply defined space, known as the Crookes' dark
space. The colour of the positive column depends on the gas used,
and to some extent on the nature of the discharge. In air or
nitrogen, it is red ; in • hydrogen, blue or red ; in carbon dioxide,
white ; but all these colours decrease in brilliancy as the exhaustion
proceeds. They are best seen in narrow tubes, such as are used for
spectrum analysis.

484 VOLTAIC ELECTRICITY

The distribution of potential along the tube is far from uniform.
It behaves as if nearly the whole of the resistance was concentrated
in the comparatively small region between the cathode and the
negative glow. This "cathode fall of potential" is about 300 volts,
and appears to be closely related to the minimum sparking potential
already referred to.

With further exhaustion, the tube begins to decrease in brilliancy,
both dark spaces increasing in size, the negative glow becoming more
nebulous and ill-defined, and the column of striae receding until,
unless the tube is very long, only a few indistinct traces are left near
the positive terminal, and these at last disappear.

Meanwhile new phenomena have been gradually coming into
existence. When the Crookes' dark space is large enough to reach
the walls of the tube, the glass becomes brilliantly luminous near
the cathode with a phosphorescent light, which is yellowish-green
if the tube is made of German (or soda) glass, and blue if made of
English (or lead) glass.

At an exhaustion of something like — — of an atmosphere, the

Crookes' dark space practically fills the whole of a tube of moderate
size, and within this dark space, substances like lime, chalk, alumina,
diamond, ruby, and many salts and minerals shine with a phos-
phorescent glow of various colours, the emitted light in certain cases
giving a characteristic spectrum.

If the exhaustion is still continued, the discharge has an increasing
difficulty in passing, until at last the tube refuses to conduct.

Cathode Rays. — The phosphorescence produced on the glass
near the cathode, at rather high exhaustion, appears to have been
first noticed by Pliicker, and afterwards studied by Hittorf and
Goldstein. The last two observers found that the phosphorescence
was due to something emitted from the cathode, which could be stopped
by obstacles. They attributed the effect to some peculiar kind of
wave motion emanating from the cathode, which Goldstein termed
kathodenstrahlen or cathode rays, and the term still survives, although
the implied meaning has been modified. The subject was then very
thoroughly investigated by Crookes, who not only systematised the
results obtained by previous observers, but also largely extended our
knowledge concerning the facts and properties of these rays.

We may briefly summarise his results, and those of later workers,
as follows : —

(1) The cathode rays always leave the cathode normally to its
surface and are negatively electrified.

(2) They behave as if possessed of irfertia, being capable of
deflecting obstacles and of turning light wheels.

(3) The substances struck by them are heated, and the temperature
may be sufficiently high to melt glass or platinum.

PASSAGE OF A DISCHARGE THROUGH GASES 485

(4) A very large number of crystals, minerals, and salts phos-
phoresce brilliantly under their impact. In the case of salts, this
effect is profoundly influenced by the presence of small quantities of
an impurity. Specimens prepared with ordinary care may phosphoresce
brilliantly, although when extreme pains are taken to ensure purity,
they may refuse to phosphoresce at all.

(5) In certain cases, e.cj. the haloid salts of tli3 alkalies, a change
of colour is produced, which is apparently due to a change in
chemical composition.

(6) They are deflected both by a magnetic field and by an
electric field.

(7) Their production and properties are quite independent of the
nature of the gas used in the tube.

(8) A substance struck by them emits an entirely new kind of
radiation, generally known as the X-rays. This effect was discovered
by Rontgen in 1895, many years after the discovery of the preceding
properties.

Nature of Cathode Rays. — Properties (2) and (5) are
especially suggestive. They are not of the kind found to be associated
with wave motion, but are exactly such as might be expected to
belong to a stream of electrified particles projected from the cathode.
Crookes strongly advocated the latter view, and as the state of the
particles could not well be defined as solid, liquid, or gaseous, he
used the term radiant matter to denote the peculiar conditions
holding good in the cathode stream. On the other hand, the con-
tinental school of physicists almost universally supported Goldstein's
theory of wave motion, and the question remained undecided for
many years. The remarkable independence of the nature of the gas,
mentioned in result (7), was particularly difficult to account for on
the particle theory, for one would naturally expect to find some
difference in the behaviour of atoms of varying mass. Certain
experiments by Lenard, which showed that the cathode stream could
pass through a very thin aluminium Avindow into the open air, were
also regarded as strongly supporting the hypothesis of wave motion.

Magnetic Effect of a Moving Charge.— According to the

ideas of Faraday and Maxwell, a moving charge should be equivalent to a
current, and should produce by its motion a moving field. This fact was
first demonstrated by Rowland in 1876. By rotating a charged disc with
great speed, he obtained a deflection of a compass-needle, similar to that
produced by a current moving in a circular path. In such an experiment,
various disturbing influences have to be eliminated, and in recent years its
accuracy has been questioned. Further investigation, however, has completely
established its validity.

It follows that a stream of electrified particles moving from the cathode
should behave like a flexible current and should be acted upon by a magnetic
field exactly like any other current, i.e. the particles should move at right
angles to the field in order to obtain the maximum effect, and should then
be acted upon by a force at right angles to the field and to their own direction

486 VOLTAIC ELECTRICITY

of motion. The latter condition implies that the actual direction of the force
changes as the deflection changes, being always directed to the centre of
curvature of the path. Hence, that path is approximately circular for small
deflections.

The nature of this effect has been illustrated by means of Experiment 148,
p. 243, in which a flexible conductor carrying a current was found to wrap
itself round a bar magnet, i.e. round the magnetic field inside the magnet.
Similarly, an electrified particle, moving at right angles to a magnetic field,
tends to wrap itself round the lines of force, and its path may become a
portion of a circle or a spiral according to circumstances.

Velocity Of Cathode RayS. — The elementary theory of the
subject may be outlined as follows : —

Let a mass, m, carrying a charge, c, move with a velocity, r, at right
angles to the lines of force in a magnetic field of strength B. Then the
quantity e is conveyed through a distance I in t seconds, where l=rt. Now,

by definition, the equivalent current strength, i, is given by i= - = - = -v-
Again, Force = B x i x I (see p. 373)

.'. Force = BxeJx£= B.C. v dynes. (1)

Let r be the radius of curvature of the path at any instant, then the
effect of this force will be to produce an acceleration, a, directed towards
the centre.

Now we know by ordinary mechanics that for motion in a circle

i-*

r
and force = mass x acceleration

or ?.«=! ,2)

v m r

This equation contains three unknown quantities, e, v, and m (B and r
being measurable experimentally). Various devices have been employed to
obtain other relations between these quantities, but here it will be sufficient
to indicate the principle of what is probably the most satisfactory method.

Action of an Electrostatic Field.— Let us suppose that the

moving particle is subjected to the action of an electric field, the direction of
motion being at right angles to the field. The particle then experiences an
electrostatic force tending to move it along the field. This force differs from
the preceding one in several respects: (1) its direction is invariable, and is
independent of the direction of the moving particle ; (2) it does not depend
upon the velocity of the particle, being merely U x es, where U is the electric
force on unit charge, and eK the strength of the charge in static units. As a
result of (1), the path is not circular or spiral around the lines of force. In
fact, it resembles that of a body moving horizontally with uniform velocity
under the influence of gravity, in both cases the path being parabolic. But
if we restrict the argument to quite small curvatures, no serious error will be
made if we treat the path as a circular arc.

Then we obtain, by a repetition of the former argument,
force = mass x acceleration

(3)

PASSAGE OF A DISCHARGE THROUGH GASES 487

This deflection, due to an electric field, is much more difficult to obtain
experimentally than the magnetic deflection, for the magnetic lines freely
penetrate the tube, and are thus readily brought to bear upon the cathode
stream, whereas the electric lines are apt to be diverted or partially cut off
on account of the residual gas acting as a semi-conductor. Sir J. J.
Thomson succeeded, where other experimenters had failed, by using an
extremely high vacuum. His method, carried out in 1897, consisted in sub-
jecting a narrow pencil of the cathode stream to the simultaneous action of
the two kinds of field, so arranged that the deflection produced by the one
tended to neutralise that produced by the other.

For balance, i.e. no deflection, we have

B.0 (magnetic) _ U<? (static)
V.DI mv2

from which « = J?ifi__
B.e (magnetic)

In this equation the same quantity of charge must be expressed in static
units in the numerator and in magnetic units in the denominator. Now the
static unit of quantity is very much the smaller of the two units, and hence
the numerical value of the charge is greater in the numerator than in the
denominator. Again, on p. 581 it is shown that 3 x 1010 static units are equal
to 1 magnetic unit of quantity ; and hence, in the above case,

e (static) = e (magnetic) x 3 x 1010.
The final result is therefore

1Q10 centimetres per seconcl. (4)

13 13

Knowing v, the ratio — can be found from the curvature produced when

m
one field acts alone.

The results obtained by Sir J. J. Thomson were

-y-from 2'8 to 3'6 x 109 centimetres per second ;

— — 1'Tx 106 (where e is in magnetic units) *

m
and these values were found to be independent of the nature of the gas.

Hence, the velocity of the cathode particles is about TVth the velocity of
light — enormously greater than any other velocity known at the time.

Value of the Ratio of Charge to Mass.— The value of the

ratio — can be determined by several quite independent methods. For in-

iii

stance, it appears in the theory of the Zeeman effect (p. 499), and it can also
be measured by experiments on radio-active bodies, which are essentially of
the same kind as those just described. All methods agree in assigning to

— a value of about 107 (e being in magnetic units).

Now, in electrolysis, we are also dealing with charged particles, but the
ratio — is not constant. For example, 1 gram of hydrogen carries 96,000
coulombs, or 9600 magnetic units,

for hydrogen - = - = 104 in round figures.

Later experiments have given

— — I'll x 107 magnetic units.
m

488 VOLTAIC ELECTRICITY

HI.* nf)f)

Again, 1 gram of sodium carries — — coulombs,

2o

... for sodium -i =

m~ 23 23

The greatest value is evidently obtained in the case of hydrogen, but even
this is very much less than 107.

Hence, if the particles which constitute the cathode rays are as large as
the atoms of hydrogen, the charge they carry must be much larger than the
charge carried by those atoms during electrolysis. On the other hand, the
fact that all monad atoms; during electrolysis, carry the same charge, is strong
presumptive evidence that some natural electrical unit is involved. But, if a
particle in the cathode stream be assumed to carry this typical monad charge,
its mass must be much smaller than that of the hydrogen atom — a somewhat
startling conclusion.

The question could only be decided by measuring either e or m indepen-
dently, and this was eventually accomplished by Sir J. J. Thomson in 1898.
Space does not permit us to give the details of this extremely difficult and
brilliant research. His results, since confirmed in various ways, show that
the charge carried by each particle is really identical with that carried by
the monad atom in electrolysis, its value being, as already stated on p. 328,
about 1'57 x 10"20 magnetic units, or 4'68 x 10~10 static units.

Hence, if — = 1*77 x 107 for cathode particle,
hydrogen atom,

and JL=104 in electrolyis, where mh is the mass of the
ftift

10* 1

_

mh ~ 1-7 x!07~ 1770

i.e. the mass of the cathode ray particle is about --— of the mass of the hydro-
gen atom.

Electrons. — It appears, therefore, that in the cathode stream
we are not dealing with ordinary mutter. Moreover, the same charge,
associated with the same mass, is found to be concerned in many
other phenomena, and there is good experimental evidence for be-
lieving that the apparent mass is purely electrical in origin. In fact,
the cathode rays appear to consist of independent particles of negative
electricity, now known as electrons.

Negative electricity, therefore, is a real entity, i.e. it exists inde-
pendently of matter, and what we call a current must in all cases
involve the flow of electrons round the circuit.

The questions at once arise : What is the nature of positive elec-
tricity? Do positive electrons also exist, or does a positive charge
consist of ordinary atoms minus electrons? As yet it is impossible
to give a definite answer, and until that can be done many other
extremely important questions must remain in abeyance.

Positively charged particles can certainly be obtained in many

different ways, and the ratio — has often been determined for them.

m

This value always approximates to 104, a result which indicates that

PASSAGE OF A DISCHARGE THROUGH GASES 489

either the charge is smaller than that met with in electrolysis — an
improbable supposition — or else that the particles themselves are of
atomic dimensions. There is no experimental evidence, which even
suggests the existence of positively charged bodies as small as the
negative electrons.

Canal Rays. — This term is due to Goldstein, who discovered
that when holes were made in the cathode, luminous beams were
formed extending from the holes on the side remote from the anode.
These were ultimately found to consist chiefly of positively charged

particles, for which — = 104, moving from the cathode with a velocity

of about 2 x 10s centimetres per second.

During recent years, these rays have been very carefully studied
by Sir J. J. Thomson, and his work, although yet unfinished, shows
that several kinds of positively charged particles exist in vacuum tubes.

Some of these possess — ratios corresponding to the kind of ions

??&

actually present, and, therefore, depending upon the gas used in the
tube, but there are also others for which — = 104, and ivhich are

?/£

quite independent of the nature of the gas.

This last result is extremely suggestive, and at the time of writing
it seems possible, but by no means certain, that positively charged
bodies may exist, which are not merely electrified particles of ordinary
matter. As to their actual nature, nothing can be surmised at
present. It may, however, be remarked, that positively charged
particles are thrown off by radium (a particles). These, also, have the

ratio — = 104 (nearly), and have proved to be atoms of helium,
m

At the same time, there are no reasons for thinking that these are
identical with, the bodies found in the canal rays.

Rb'ntgen or X-RayS. — Ilontgen's epoch-making discovery
deserves special notice. In 1895, he discovered that any substance
struck by the cathode rays — in the first instance, the glass walls of a
highly exhausted tube — gave off a totally new kind of radiation,
invisible to the eye, but capable of affecting photographic plates
and of exciting phosphorescence in certain bodies, of which the
various platinocyanides, and scheelite (native calcium tungstate) are
among the most important. The new radiation was especially remark-
able for its power of penetrating ordinary matter more or less readily ;
the relative transparency of different materials being roughly pro-
portional to their density, but being otherwise quite independent of
their physical nature. It could not be refracted, it did not yield
interference phenomena, and it was incapable of regular reflection.
Neither was it deflected by either a magnetic or an electric field.

490 VOLTAIC ELECTRICITY

Its nature has given rise to much discussion, and although it is
generally regarded as being some form of wave motion in the ether,
it is by no means certain that this is the case, in fact recent research
appears to disprove it.

The best form of tube for producing the X-raj7s is one in which
the cathode stream is brought to a point focus by means of a cup-
shaped cathode, and then made to impinge on the anode, which is
a metal plate inclined to their direction at an angle of 45°. The
X-rays then emanate from practically a point source, and consequently
yield sharp shadows on photographic plates or on phosphorescent
screens. Their application in surgery is too well known to require
attention here.

Ionising Power Of X-RayS. — Perhaps the most important
and characteristic property of the Rontgen rays is their power of
producing temporary conductivity in gases ; a property most readily
demonstrated by the fact that an electroscope will not retain a
charge near a tube in action.

According to modern ideas, such conductivity depends upon the
presence of charged particles or ions, and is essentially of the same
character as that met with in electrolysis.

Under ordinary circumstances, gases contain very few free ions,
and hence are excellent non-conductors, but they may be ' ' ionised "
by various agencies (see p. 494), and they then acquire a relatively
feeble but true conductivity. Until the discovery of the electron,
there was some difficulty in conceiving the nature of the process, for,
although ordinary gaseous molecules might be regarded as being
resolved into their constituent atoms — charged positively and nega-
tively respectively — this explanation could not be applied to mon-
atomic gases, which are also readily capable of being ionised. If,
however, we regard ionisation as due to the detachment of one or
more electrons, the difficulty largely disappears.

Suppose that two metal plates are placed a short distance apart
and connected in series with a battery and a very sensitive galvano-
meter.1 Let the radiation from an X-ray tube traverse the space
between the plates. If now a gradually increasing P.D. be applied
to the plates, the current at first obeys Ohm's law, i.e. is proportional
to the P.D., but its rate of increase gradually diminishes, and after
a time the current remains constant in strength, although the
P.D. is still increasing. In this stage, the current is said to be
"saturated." When the applied P.D. reaches a very high value
(which for air at ordinary pressure is about 30,000 volts per centi-
metre), another stage is reached in which the current somewhat
suddenly begins to increase again with very great rapidity.

i Ordinary galvanometers are scarcely sensitive enough for this purpose,
and another method of measuring these extremely small currents is described
later (see p. 491).

PASSAGE OF A DISCHARGE THROUGH GASES 491

Again, when the potential difference is sufficient to produce
saturation, the current (although the P.D. remains constant) is
increased by increasing the distance between the plates — a behaviour
exactly opposite to that shown by liquid or metallic conductors.

These results are easily explained, if we suppose that the X-rays
are producing a certain number of ions in the gas, which tend to
recombine. The number of ions, therefore, increases until a state
of balance is reached, exactly equal quantities then being formed and
being recombined per second.

If now a P.D. be applied to the gas, oppositely-directed pro-
cessions of charged particles will move towards the plates, the
actions going on being similar to those already explained in the
case of electrolysis. Increasing the P.D. will increase the number
of ions arriving at each plate per second, and as long as there is an
abundance of such ions available, Ohm's law is satisfied. It is,
however, evident that no more ions can arrive at a plate per second
than are produced by the ionising agency in the same time, and
when this state of affairs is reached, further increases in the applied
P.D. cannot increase the current, and we get the stage of saturation.
If, however, the P.D. be raised to some high value, depending on
the pressure, the velocities acquired by the moving ions may become
sufficiently great to enable them to produce fresh ions by collisions
with neutral atoms or molecules, and then the current rapidly
increases. (An effect of this kind is probably operative in ordinary
sparks in air. There are always a few ions present, and when the
electric field becomes great enough, they acquire sufficient velocity to
produce new ions by collision, the process being then rapidly cumula-
tive and ending in the spark.)

It appears, therefore, that the magnitude of the current depends
not only upon the applied P.D., but also upon the number of available
ions present. Now, increasing the distance between the plates
increases the volume of gas exposed to the ionising agency, which,
therefore, increases also the number of ions produced per second,
and consequently the saturation current.

The absence of the saturation state in the case of conduction in
solids and electrolytes is an indication that relatively inexhaustible
supplies of ions are present.

Measurement of Extremely Small Currents. — A sus-
pended coil galvanometer of ordinary sensitiveness will easily measure
currents of the order of 10~6 ampere, but it is often required to
measure much smaller currents in experiments of the kind described
above. For this purpose, a quadrant electrometer, or a gold-leaf
electroscope of suitable design, is used — previously calibrated by
means of a standard cell, so that the P.D. corresponding to any
given deflection is known. The apparatus is then arranged as shown
in Fig. 342. A battery, ZC, is connected in series with the plates

492 VOLTAIC ELECTRICITY

A and B and a quadrant electrometer, E, one terminal of the battery
and one of the electrometer being earthed. Hence, there is a P.D.
between A and B, but the plate B and both sets of quadrants are
initially at zero potential, and there is no deflection. An ionising
agency, such as an X-ray tube, is then made to act upon the gas
between A and B, and as a result the potential of B gradually rises
and a deflection is produced.

Let this deflection increase from d to d± divisions in time t
seconds. Then dl — d measures the increase of P.D. between the
quadrants of the electrometer, and its value is known from the

Earth,

Eartfv

FIG. 342.

previous calibration. Let this value be e, and let K be the capacity
of the electrometer (also previously determined by experiment).

Then, the quantity Q passing into the electrometer (and also
between A and B) is given by

Q = eK.

Let i be the average current during the time t (or the true current
if it happens to increase uniformly with time),

then Q = it . eK

•, 77- or i =

i.e. u = eK i

From this equation we see that the smaller the value of K, the
smaller will be the value of i corresponding to a given change in
deflection. This shows us why gold-leaf electroscopes are even
superior to the electrometer for the purpose of measuring small
currents. Suitably designed, an electroscope can be given a much
smaller capacity than any electrometer, and will, therefore, indicate
correspondingly smaller currents.

Nature of Discharge in Sparks and in Vacuum Tubes.—
We are now in a position to indicate the general nature of such
discharges, regarding them as dependent upon the presence of
free ions. There is good reason to believe that gases at ordinary
pressure contain a few ions, and in a spark-gap these tend to move
along the electric field, acquiring velocity exactly as a falling body
does in a gravitational field. If this velocity reaches a sufficiently
high value, they may produce other ions by impact with neutral
particles, and then, as already stated, the process is rapidly cumu-
lative and ends in a brush or spark. The velocity acquired depends
partly upon the P.D. per unit of length, and partly upon the length

PASSAGE OF A DISCHARGE THROUGH GASES 493

of path available, although, as this path is small at ordinary pressures,
a very high voltage is necessary to initiate ionisation by shock. At
low pressures, the mean free path of an ion is much longer, and it
can acquire sufficient velocity under the action of smaller voltages.
Again, the velocity acquired depends upon the mass of the particle,
and hence, if electrons are present, they are much more effective in
producing ionisation than positive ions, for, owing to their small
mass, they can acquire the necessary velocity in moving through a
smaller difference of potential. In a highly exhausted tube, the
potential gradient is, as already stated, particularly steep between
the cathode and the negative glow, and there the greatest ionic
velocities are acquired. In all probability, it is the impact of
positive ions which facilitates the escape of electrons from the
cathode, and when there is an aperture in it, some of these positive
ions may pass through to form the canal rays. The liberated
electrons similarly acquire their enormous velocity in moving along
the potential gradient away from the cathode, and become active in
ionising the neutral particles in the cathode glow. When, however,
they have lost some of their velocity by collisions, they also lose their
ionising power, and it is necessary for them to travel a certain dis-
tance in order to again acquire sufficient velocity for that purpose.
Hence, under certain conditions, stratifications are produced, the dark
spaces between them representing convective motion, and the striae
themselves the places where ionisation is going on by collisions.

In the electric arc the action is somewhat different. In this case,
the voltage is much too low to produce ions by impact, and they
must be supplied by the electrodes themselves. Consequently, the
formation of an arc depends very largely upon the nature of the
electrodes (see p. 553).

Certain substances, chiefly metallic oxides (of which lime is a good
example), when heated possess in a conspicuous degree the property
of readily emitting electrons. If a metallic cathode in a highly
exhausted tube is coated with lime, and arranged in the form of a
loop so that it can be readily heated by an independent current, the
discharge passes readily at quite low voltages. This discovery is due
to Wehnelt.

Metallic Conduction. — Good conductors must be regarded as
bodies containing enormous numbers of free electrons.

A piece of metal is a region in which they can move about
freely, although they cannot readily pass through its surface into the
surrounding space. If their number were not exceedingly great, the
phenomenon of saturation would occur as it does in gases, and Ohm's
law would fail for very large current densities.1

1 This remark applies also to electrolytes. It has, however, been shown
recently that certain very badly conducting solutions (e.g. lead oleate in
hexane or light, petroleum) do give a saturation current.

494 VOLTAIC ELECTRICITY

Experimental evidence supporting these views is to be found it
the Hall effect (see p. 498).

Sources of lonisation. — Gases may be ionised, i.e. brought
into a conducting state, in many ways, of which the most important
are —

(1) By raising the temperature to a very high value.

The effect of high temperatures on a gas is somewhat complicated,
depending largely upon the proximity of the gas to heated surfaces.
Hot metals, in particular, have a strong tendency to give off positively
charged particles. In some gases the effect is small; when large, it
can always be traced to chemical decomposition. Hence, gases drawn
from the proximity of flames or arcs, or from contact with glowing
metals, are found to conduct.

(2) By passing X-rays, radium rays, or Lenard rays through the
gas.

The effect of radium and other similar bodies is referred to
separately. Lenard rays are produced outside a vacuum tube when
cathode rays are made to impinge upon a small aluminium window.
Their influence is confined to the immediate neighbourhood of the
window, and they are relatively unimportant as a source of ionisation.

(3) Air is ionised by passing it over phosphorus, or by bubbling
it through water.

(4) By ultra-violet light.

Hertz found that the incidence of ultra-violet light on a spark-
gap facilitated the passage of a spark. This observation formed the
starting-point of a large amount of research. It was discovered that
a clean zinc surface charged negatively, rapidly loses its charge when
ultra-violet light falls upon it. If uncharged to begin with, it be-
comes positively charged, i.e. it emits electrons. If positively charged,
there is no loss. Other metals also show this effect in varying
degrees. The more electro-positive elements are the most efficient,
the order of efficiency being identical with Volta's electro-chemical
series. With copper, iron, lead, &c., the effect is negligible; with
magnesium it is stronger than with zinc ; sodium and potassium are
sensitive to ordinary daylight; whilst rubidium, the most electro-
positive of all metals, is sensitive to a glass rod just heated to
redness.

(The student may find some difficulty in understanding why we
have given the latter statements under the heading "ultra-violet
light." The point to be brought out is that waves of short length,
i.e. ultra-violet, are the most effective. The alkaline metals are,
however, sensitive to longer waves, and rubidium is sensitive even
to the extreme red rays.)

Becquerel Rays, Radio-active Bodies. — Immediately
after Rb'ntgen's discovery, Becquerel made experiments to ascertain
whether ordinary phosphorescent bodies emitted penetrating radia-

PASSAGE OF A DISCHARGE THROUGH GASES 495

tions. Amongst other substances he tried uranium salts, which had
been made to phosphoresce by exposure to daylight, and found that
they gave out a radiation similar to X-rays ; further investigation,
however, showed that the exposure to sunlight was unnecessary, the
action being just as intense when the salts were kept in the dark.
Shortly afterwards, Schmidt discovered that thorium possessed similar
properties. Crookes then discovered that uranium salts could be
separated by ordinary chemical means into two portions — one of
which was radio-active and the other was not. He called the active
portion Uranium X. Becquerel found that, if these two portions
were kept for some months, the non-active part gradually regained
its activity, whilst the active part gradually lost it.

M. and Mine. Curie discovered that certain minerals containing
uranium were more radio-active than uranium itself, and hence they
inferred that other active bodies were present. After some years of
laborious research, they isolated two new substances, polonium and
radium, enormously more active than uranium. Debierne discovered
yet another intensely active substance associated with thorium in
extremely small quantity, which he called actinium.

These new substances possess, in a conspicuous degree, the power
of ionising surrounding gases. They have been found to emit three
distinct kinds of radiation 1 —

(1) Streams of positively charged particles — known as the a-rays
— of atomic mass, moving with a velocity something like T^th that
of light. These have proved to be atoms of helium.

(2) Streams of negatively electrified particles — known as the
/2-rays — moving with a velocity approaching that of light itself.
These seem to be identical with the electrons present in the cathode
rays, although they move with a much greater velocity.

(3) An unelectrified radiation — the y-rays — which closely re-
semble X-rays, and may be identical with them.

Apparently these radio-active elements are gradually breaking up
into constituents of smaller atomic weight ; the disintegration being
accompanied with the liberation of relatively enormous amounts of
energy. These newly formed bodies, after a life of very varied
duration, may again break up ; and the final result may be the
ordinary stable elements as we know them.

The experiments of Rutherford have shown that the rate of dis-
integration follows a perfectly definite law, and is unaffected by any
change of experimental conditions; it is always proportional to the
amount of substance present.

This condition can be stated in a simple form. If M0 is the
mass of active substance initially present, and M^ the mass after a
time*, then Nt = M0e~at

1 This term usually implies wave motion, but it is now generally used to
denote the emissions from radium and other radio-active bodies.

496

VOLTAIC ELECTRICITY

where e is the base of natural logarithms and a is a constant,
whose value depends on the substance, and which can be determined
by experiment. For example, the time required for half the sub-
stance to break up is given by

or eflt = 2

It will be noticed that the time is independent of the amount of
material present, and hence it has become usual to employ it as a
measure of the rate of change. It is often called the "period" of
the body. Only products of long period are likely to accumulate
naturally in any large quantity.

The following table, corrected to June 1911, by the kindness of
Professor Rutherford, represents the state of our knowledge : —

Substance.

Period.

Emitted Radiation.

Uranium . . .
Uranium X . .
Ionium .... . .

Radium . . ; ' • f .
Radium Emanation
Radium A . .
Radium B . . . -v

5 x 109 years
23 days
?
2000 years
3-85 days
3 minutes
26 minutes
19 minutes

(Two) a
p and 7
a
a
a
a
)8 and y
o, 8 and y

(Radio-lead) Radium D .
Radium E
(Polonium) Radium F

16 years
5 days
140 days

Slow£
/3 and y
a

3 x 1010 yeai-s

Mesothorium (1)
Mesothorium (2) ...
Radio-thorium . . , ' .
Thorium X . . . ;V
Thorium Emanation . . :> :. ;
Thorium A . . • :.
Thorium B • . "* ." ' ". .
Thorium C -, . . . .. V:
Thorium D . . .

5'5 years
6 hours
2 years
3'6 days
54 seconds
10-6 hours
55 minutes
?
3*1 minutes

None
£ and y
a
a
a
Slow£
ct
a
/3 and y

Actinium
Radio-actinium ....
Actinium X ....

Actinium Emanation . - *
Actinium A . . .
Actinium B
Actinium C

?
19-5 days
10-5 days
3-9 seconds
36 minutes
2*1 minutes
4'7 minutes

None
a
a
a
None
a
(3 and 7

PASSAGE OF A DISCHARGE THROUGH GASES 497

It will be noticed that the y-rays are never produced without
/?-rays, and as they resemble the X-rays and cathode rays respectively,
it seems probable that the y-rays represent a disturbance due to the
sudden starting of an electron, just as the X-rays are produced by
its sudden stoppage. The various products are all solid except the
"emanations." These are true gases, inactive chemically and closely
allied to the argon group. They can be liquefied by extreme cold,
and appear to possess characteristic spectra, but their life is so short
that only the radium emanation has been studied at all carefully,
and even that has been done in the face of very great experimental
difficulties.

Faraday, Kerr, Hall, and Zeeman Effects.— Certain out-
lying phenomena of great theoretical importance may conveniently be
grouped together. We may summarise the more important of these in
historical order as follows : — -

(1) The Faraday Effect. — In 1845 Faraday discovered that when
certain substances were placed in a strong magnetic field, they acquired the
power of rotating the plane of polarisation of a beam of plane polarised light,
if that beam passed througli them in the direction of the lines of magnetic
force. If the two directions are not the same, the effect is weaker, becoming
zero when the path of the light is at right angles to the field. It is now
known that all substances exhibit this effect t6 some extent, but in varying
degrees. It is especially marked in bodies having large refractive indices
for light, like Faraday's " heavy glass," in which it was discovered, and
carbon disulphide. Non-transparent bodies like the metals can be used only
in extremely thin films, and it is noteworthy that an iron film, when very
strongly magnetised, produces a relatively enormous effect.

The experiment is usually performed by 'placing the substance between
the poles of a powerful electromagnet, the pole pieces being perforated in
order to permit the passage of the light. As a rule the plane of polarisa-
tion of the beam is rotated in the same direction as that of the current,
which produces the field, e.g. if we suppose that the field is produced by a
solenoid, inside which the substance is placed, then the direction of rotation
is usually in the same direction as the current in the solenoid, although
there are a number of substances for which the rotation is in the opposite
direction. The amount of rotation is directly proportional to the length of
path in the field (/) and to the strength of the field (B), and is approxi-
mately inversely proportional to the square of the wave length of the light
used ; hence, if 6 be the observed rotation, we may write

6 oc B x I.

From p. 375 we know that, in air, BZ=47rm, and therefore we see that
the rotation for a given length is proportional to the magnetic difference of
potential acting on that length. This is known as Verdet's law, and the
rotation produced in any given substance by unit magnetic P.D. is known
as Verdet's constant for that substance. If the light be reflected backwards
again through the substance, the rotation is increased, just as though the
length of the path had been doubled, i.e. the effect is independent of the
direction in which the light passes along the field — a fact which sharply
differentiates the phenomenon from that observed in the case of naturally
optically active substances like quartz, sugar solutions, &c., for in these the
direction of rotation does depend on the direction of the light in the sub-
stance, and the reflection of the light back through them would simply
annul the rotation altogether.

2i

498 VOLTAIC ELECTRICITY

(2) Kerr Effects. — Ken* discovered that when a beam of plane polarised
light is reflected from the polished surface of the pole of a strongly excited
electromagnet, it becomes elliptically polarised. Apparently, this is a special
instance of the Faraday effect, the light penetrating very slightly, but suffi-
ciently to cause its plane of vibration to be affected by the molecular rotations
which must be present in magnetised iron.

Another effect discovered by Kerr is of a totally different nature. He
found that a transparent dielectric placed in a strong electrostatic field
becomes doubly refracting. This might be regarded as the natural result of
the strain thereby set up in the dielectric, for it is a familiar fact that singly
refracting substances, like glass, become doubly refracting when subjected
to strains produced by purely mechanical means. But Kerr found that it
occurred in liquids also, in which mechanical strains cannot exist, and hence
it must be regarded as a molecular action of some kind.

(3) The Hall Effect.— Let AB, Fig. 343 (taken from Carey Foster
and Porter's Electricity and Magnetism) represent a very thin plate through
which a current is passed from A to B. Let a sensitive galvano-
meter be connected to the points a and
6 at the sides of the strip. If these
points are symmetrically placed, they
will be at the same potential, and there

_ will be no deflection, a result easily ob-

I tained by slightly adjusting the position

l ' of the wires. Hall discovered that, if

-/ 1 a very strong magnetic field is then

\_ produced at right angles to the plane

FiG. 343. of tne striP (*•*• in tbe direction of

the arrow), this state of balance is

disturbed and the galvanometer is deflected, thus showing that a P.D.
has been produced between the points a and 6. The effect is proportional to the
field strength and inversely proportional to the thickness of the plate, and is
much larger in bismuth than in any other substance. Its direction varies — in
gold and nickel it is the same as in bismuth, but the deflection is reversed in
iron, antimony, cobalt, and tellurium.

The Hall' effect has not been observed in liquids, but it is well-marked
in gases. Up to a certain point, its explanation is fairly easy if we regard
the current in a metal as being due to a stream of negatively charged
electrons in one direction and (more doubtfully) a stream of positively
charged particles of some kind in the opposite direction. We know that,
in the magnetic field, a mechanical force acts on the conductor, tending to
drive it sideways (see p. 343), and we must now suppose that this force acts
primarily on the electrons, which, whether positive or negative, will be urged
in the same direction. (For, in the magnetic field, a positively charged
particle moving in a certain direction will be urged in the same way as
a negatively charged particle travelling in the opposite direction). If the
carriers of opposite sign behave in exactly the same way, there will be no
effect, but if there is any difference of velocity between them, so that there
is an excess pf one kind at the side (a or I) of the plate, then there will
exist the transverse P.D. discovered by Hall. If the current consisted
merely of a stream of electrons, the deflection would in all cases be the
same as that observed in bismuth ; but the fact that its sign varies in
different metals is somewhat puzzling, and might be interpreted as indi-
cating the existence of free positively charged bodies, which in these metals
have greater mobility than the electrons. It is extremely improbable that
this is the correct explanation, and the question must be regarded as open
until the true nature of a positive charge is determined. The absence of the
effect in electrolytes is probably due to the fact that the velocities of the

PASSAGE OF A DISCHARGE THROUGH GASES 499

oppositely charged ions do not differ sufficiently, for the negative charge
is not carried by rapidly moving free electrons, but by relatively massive
bodies to which electrons are attached. In gases, the Hall effect is very
easily observed, because free electrons are again met with, which move
much more rapidly than the positively charged atoms or atomic groups
also present.

Various other phenomena, which may be regarded as dependent upon the
Hall effect, have been observed and measured in bismuth. Perhaps the
most important of these is an increase in the resistance of the metal, a fact
that has been applied with considerable success to the measurement of strong
magnetic fields. A flat spiral of bismuth, wound non-inductively, is mounted
on a suitable holder, and, when placed in the field in question, its resistance
is measured in the usual way, a calibration curve having previously been pre-
pared by measurements in fields of known strength. The increase of resist-
ance amounts to about 50 per cent, in a field for which B = 10,000 lines per
square centimetre.

Other secondary actions are (1) a slight decrease in the heat conductivity
of the material, and (2) a transverse difference of temperature between its
sides. These temperature effects can be deduced as necessary consequences
of the electron hypothesis. In fact, a current of heat through the strip is
found to act like an electric current. It is deflected by a magnetic field, and
the result is the production of an electrical P.D, between the sides of the
strip, which can be detected by a galvanometer.

(4) Zeeman Effect. — Zeeman discovered that if a source of light,
giving line-spectra, be placed in a very strong magnetic field and then viewed
through a spectroscope of great dispersive power, some of the lines widened
out and became resolved into doublets, triplets, and sometimes into more
complex systems, although all the lines, even of the same substance, were
not affected alike. In the simplest case, when the source was viewed at
right angles to the direction of the magnetic field, a line became a triplet,
each constituent plane polarised, but the plane of polarisation of the middle
one was at right angles to that of the two outer ones. When viewed along
the direction of the field, the line became a doublet, with constituents circu-
larly and oppositely polarised. This discovery strongly supports our ideas as
to the existence of charged particles, for according to the electro-magnetic
theory of light, the source of a light-wave must be an oscillating electric
charge, and this is probably an " electron " rotating around some central
body as a nucleus. Now, all possible motions of such a particle can be
resolved into two opposite circular paths at right angles to the field and a
rectilinear path along the field, the magnetic force accelerating one circular
component and retarding the other, but leaving the rectilinear component
unaffected. Thus the original line gives rise to three, the middle one being
of the original frequency and in the original position, and the two outer ones
of higher and lower frequency respectively ; and evidently, when viewed
along the direction of the field, two circularly polarised components should
be produced. It is not difficult to calculate the alteration in frequency in
terms of the strength of the field, assuming the original vibration to be
simple harmonic, and thus by measuring the alteration in frequency by spec-

troscopic observations, another value of — is obtained, which agrees very

nearly with the results derived from quite different methods.

Until this discovery of Zeeman in 1896, the only known relation between
magnetism and light was Faraday's rotation of the plane of polarisation in
a magnetic field. In the latter, the effect depends upon the action of the
field on light-vibrations previously produced, i.e. it produces a difference in
the velocity of the two circularly polarised constituents of a plane polarised
beam, but does not alter their frequencies. In the former, the magnetic

500 VOLTAIC ELECTRICITY

field acts directly upon the vibrating particles which are the source of
light- waves, and by influencing their velocities causes them to emit light of
different frequencies.

EXERCISE XXIII

1. Describe the mode of production of Rontgen and Ldnard rays, and dis-
cuss the possible origin of the phenomenon. (B. of E., Stage III., 1909.)

2. Describe the arrangements for generating Rontgen rays, and discuss
briefly their character and the question of how they are produced.

(Lond. Univ. Inter., B.Sc., Honours, 1904.)

3. An electrified particle traverses an electric field, the intensity of the
field being normal to the original direction of motion of the particle. Find
an expression for the deflection of the particle. What other experiments must
be made in order to determine the ratio of the mass of the particle to its
electric charge ? (Lond. Univ. B.Sc., 1906.)

4. What is meant by a saturation current ? Explain the experimental
details you would adopt to investigate the relation between the electromotive
force and current for a gas ionised by Rontgen rays.

(Lond. Univ. B.Sc.', Honours, Internal, 1909.)

5. Give a short account of the effects of transmitting light through a
magnetised material, and of reflecting it from the surface of a magnetised
body. (Lond. Univ. B.Sc., Honours, Internal, 1909.)

6. An electron of mass 10~25 grams rotates 5x 1014 times a second round
an atom supposed to be at rest ; find the force with which the electron must
be attracted to the atom in terms of the radius of its orbit. If the atom has
a positive charge of 10 -J0 electrostatic units and the electron an equal nega-
tive charge, calculate the radius of the orbit, assuming that the attractive
force is purely the force between the charges.

(Lond. Univ. B.Sc., Internal, 1903.)

CHAPTER XXX

TELEGRAPHY AND TELEPHONY

ONE of the most important applications of electro-magnetism is the
electric telegraph, by means of which messages are transmitted from
one place to another at a considerable distance apart. The essential
parts of most telegraphic systems are : —

(1) Line wires connecting the two stations.

(2) Batteries for generating the current.

(3) Instruments for sending and receiving the signals.

Lines. — In the earlier days of the electric telegraph, two wires
were employed, one to carry the current from the sending station to
the receiving station, and the other to convey it back to the battery.
If was soon discovered, however, that if each end of the first wire
was connected to a large plate sunk in the ground, the earth itself
would act as the return wire. The conductivity of the earth appears
to be chiefly due to the moisture contained in it. Although its
specific resistance is high, this is counterbalanced by
the enormous cross-section available ; indeed, the
resistance between two earth plates rarely, exceeds a
few ohms, no matter how far they are apart.

Lines may be divided into three classes : (a)
overhead or aerial ; (b) underground ; (c) submarine.

Overhead lines are constructed chiefly of iron,
but the use of copper for long and important circuits
is increasing. The poles are generally made of fir
or larch, treated with creosote or with a solution of
copper sulphate to prevent decay. As wood is to
some extent a conductor — especially when wet with
rain — the wires are supported on glass, glazed
earthenware, or porcelain insulators, to stop, as far
as possible, the current leaking to the earth. A
modern form of insulator is shown in Fig. 344 (taken
from Slingo and Brooker's Electrical Engineering}.
As most of the leakage takes place, not through
the substance of the insulator, but along a film of
moisture on its surface, it will be seen that the insulator is designed
to expose as long a surface as possible between the wire and the

501

FIG. 344.

502 VOLTAIC ELECTRICITY

supporting bolt, a, and that, moreover, part of the surface is sheltered
from the rain.

Underground wires are generally made in the form of a cable,
containing a large number of copper wires, each usually about 40
mils l in diameter, and insulated from each other by means of a gutta-
percha covering. The cable, generally placed in cast-iron pipes, is
laid a foot or so below the surface of the ground. During recent
years, "dry-core" cables have been much used. The separate
wires of such cables are insulated from each other by wrappings of
manilla paper, the whole being bound together with cotton and
enclosed within a sheathing of lead — the external appearance being
that of a leaden pipe. (All paper-insulated cables are sheathed in
lead to exclude moisture, as paper is very hygroscopic.) These
cables are also laid in cast-iron pipes or in earthenware conduits.
The chief advantages are (1) a lower electrostatic capacity ; (2)
economy of space, owing to the fact that the wires are placed

FIG. 345. FIG. 346.

nearer together than in the guttapercha type ; and (3) material
decrease in cost.

Submarine cables, in which great strength and perfect insulation
are necessary, vary considerably in detail, although the principle
involved is the same in all. Fig. 346 represents a cross section of
the longitudinal piece of an Atlantic cable shown in Fig. 345, which
consists of a core of seven strands of copper wire, each about -^ of an
inch in diameter (forming one conductor), insulated by being covered
with alternate layers of guttapercha, and of a mixture of tar, gutta-
percha, and resin. This is covered with hemp, and then wrapped
round with a coiled sheath of steel wires to take the tensile strain
and to protect the cable from injury.

Batteries. — The cells commonly used are the Daniell, the
Leclanche, the chromic acid, and, for some special purposes, the
so-called dry cells. In the larger offices, accumulators are generally
used. The voltages required for different circuits vary from 5 to 180,
and are obtained by joining up a sufficient number of cells in
series.

1 A " mil " is YoVo inch.

TELEGRAPHY AND TELEPHONY

503

Sending and Receiving Instruments.— The simplest forms

of sending and receiving instru-
ments are the single current key
and the sounder respectively.

The key consists of a brass
lever with its fulcrum at A (Fig.
347), and normally held on the

back contact stop at B by means FIG 347

of a spiral spring. On depressing

the ebonite knob, the front of the lever makes contact with the
contact stop at 0. The lever moves through a distance of ^ inch
only. Its general appearance is shown in Fig. 348.1

FIG. 348.

The Sounder (Fig. 349) consists essentially of a double core
electromagnet, the upper poles of which lie immediately below a soft

iron armature A, fixed to a brass
bar. One end of the bar is sup-
ported on bearings at B, while the
up-and-down motion of its free
end is limited by the screws at
C and D. Normally the bar is
held against the upper screw C,
by tension of a spring (shown in
diagrammatic form at S) ; but when
a current flows through the coils,
the cores are magnetised, and

FIG. 349. therefore attract the armature, so

that the screw D strikes the "anvil"

N, and emits a sharp metallic click, which denotes the commencement
of a signal. When the current ceases, the cores are demagnetised,

1 Many figures in this chapter (e.g. 348, 350, 352, 356, 362, 368, 369, 371,
372, 374) are taken by pel-mission from Preece and Sivewright's Telegraphy,
published by Messrs. Longmans.

504

VOLTAIC ELECTRICITY

and the spring S lifts the bar against the screw C — another sharp
click announcing the end of the signal. Fig. 349 illustrates the
principle of the instrument ; the actual form being shown in
Fig. 350.

FIG. 350.

The Simple Telegraphic Circuit (Fig. 351) shows diagram-
ma tically the scheme of connections. D and D' are the keys ; A and B
the batteries ; E, E, the earth plates ,• the galvanometers are supposed

FIG. 351.

to represent any form of receiving instrument. We may mention that a
special kind of galvanometer, with a vertical needle, was used before the
advent of the sounder ; and in fact is still used, on account of its simpli-
city, on railways. When a sounder takes the place of the galvanometer

TELEGRAPHY AND TELEPHONY

505

shown in the diagram, it is usual to insert a simple form of galvano-
meter at the ends of the line, so that the operator, when " sending,"
may know that his apparatus is working properly. (These galvano-
meters are shown at G and G' in Fig. 352). On depressing the
key, say D, at one station, the sounder there is disconnected, and the
Lattery becomes connected to the line. The current flows along the
line, through the key D' (by means of its back contact), to the
sounder, thence to earth, and so back to the battery at the sending
station. As long as the key is depressed, the current flows through
the sounder — thus, by depressing the key for short or for longer

I

SINGLE

ami1

PRINTING.

NEEDLE.

PBTNTIN'G.

NEEDLE.

A

J

N

A

B

Ax,

o

I/I

C

AA

p

Jk

D

Ax

Q

IIJ

E -

X

R

vA

F

xxA

S

xxx

G

/A

T —

/

H

XXX

u

xx/

I --

\x

V

xxx/

,J

x///

W

x//

K

/x/

X

Ax/

L

x/.

Y

A//

M

//

Z

/Ax

periods, the dots and dashes of the Morse code, shown above, are
signalled. An expert operator can send 25 to 30 words per minute.
In the table, the column for " single needle " refers to the galvano-
meter with a single vertical needle, mentioned above, and indicates
the direction and duration of the deflection.

The direct-sounder circuit just described should be carefully
studied, as it illustrates very clearly the main principles underlying
the more complicated systems.

It can be used, however, only on circuits of a few miles in length.
On longer circuits, difficulties arise from many causes — defective
insulation being, perhaps, the chief. A portion of the current leaks

506

VOLTAIC ELECTRICITY

down each pole to earth, especially in wet weather ; and as a hundred-
mile line requires some 2500 poles, the total leakage becomes so
great, that a current does not reach the distant sounder of sufficient
strength to produce loud, clear, and reliable signals. The higher
resistance of the longer wire, also, of course, weakens the current.
This would be remedied by increasing the number of cells, except for
the fact that such increase of E.M.F. also increases the leakage. The
method adopted in practice is to employ an additional sensitive instru-
ment called a relay, which actuates the sounder.

The Relay is practically a very delicate form of sounder, with
an electromagnet containing many turns of fine wire, and attracting
a very light armature, which plays between contact points only
TJ-o inch apart. This armature closes and opens a circuit containing
the sounder and a local battery of a few cells. The weak line

FIG. 352.

currents need only be of sufficient strength to move the relay arma-
ture, the energy required to produce the loud beat of the heavy
sounder being furnished by the local battery. The connections are
shown in Fig. 352, where RR' are relays ; LB, LB', local batteries ;
MM7, the sounders (or any variety of receiving instrument) ; GG', the
line galvanometers already referred to. Notice that the relays take
the place occupied (in Fig. 351) by the sounders.

There are many forms of relay, but the one almost universally
used at the present time is the Post Office Standard Relay, which is
described on p. 508.

Difficulties with Long Lines. — The introduction of the
relay thus overcame the difficulty of dealing with weakened currents,
but, when it became necessary to employ still longer lines, other

TELEGRAPHY AND TELEPHONY 507

difficulties, which we must now consider, were produced. After the
working current has ceased, some means of bringing the relay arma-
ture back to its normal position must be provided. This could be
done by means of a spiral spring (as in the case of the sounder) or by
the attraction of a magnet. In either case, this controlling force
must be adjustable to suit stronger or weaker line currents. If the
relay is delicately adjusted to respond to very weak currents (by
moving the armature nearer to the poles of the electromagnet, or by
reducing the tension of the spring), the residual magnetism of the
cores may cause the armature to stick during the interval between
two signals, so that, say, two dots become one dash ; if, on the other
hand, this adjustment is reversed, the currents may be too weak to
attract the armature, so that dots may be lost although dashes are
recorded. Again, when the relay is in its most sensitive position,
stray currents from other wires on the same poles may cause false
signals to be received. The chief obstacle to rapid working on long
Hiv is, however, due to their electrostatic capacity. Every telegraph
circuit may be regarded as a condenser — the line wire forming one
"plate," and the earth the other, the air between them being the
dielectric. As we have shown in Chapter V., the capacity of a
condenser is directly proportional to the area of the plates, and
inversely proportional to the distance between them, so that, although
the "plates" are usually some 15 to 20 feet apart, this is counter-
balanced by their great length, which may be 100 miles or more.
When a signal is being transmitted, the "plates" are joined together
through the receiving apparatus at the distant end, while the poles
of the battery are joined, one to each, at the sending station. The
first portion of the signal is, therefore, absorbed as a static charge,
the nearer parts of the line being first raised to the potential of the
sending battery, and the distant parts later ; the maximum current
not flowing through the relay until this effect has been produced.
The nett result is that the first signal is delayed. If now the
key be raised, the battery is disconnected, both ends of the line
becoming connected to earth. The static charge drains out at
each end, thus prolonging the signal on the relay. If another
signal is sent into the line before this discharge is completed, the
returning discharge neutralises the first portion of this second signal,
and the dash may become a dot, or a dot may be completely lost.
In order that undistorted signals may be transmitted, sufficient time
must be allowed to elapse between two successive signals, so that the
accumulated charge may leak out. The longer the line, the greater
its capacity and its consequent charge, which, of course, makes the
time required for discharge longer, and, therefore, the speed of
signalling slower.

The above statements must be regarded only as a very elementary
explanation. The rate of working depends upon the capacity, resist-

508

VOLTAIC ELECTRICITY

ance, inductance, and leakance l of the circuit, the four quantities
being connected by very complicated formulae.

All the difficulties mentioned above were overcome, or at any rate
greatly reduced, by the introduction of double-current working, but
before discussing this in detail, it is advisable to describe the Post
Office Relay and the Double- Current Key.

The Post Office Standard Relay consists essentially of two
chief parts — the permanent magnet combination (Fig. 353) and the
electro-magnetic combination (Fig. 354).

NS, Fig. 353, is a large, semicircular
permanent magnet, near the poles of which
are two soft-iron armatures of rectangular
section, rigidly fixed to the spindle F, which
is pivoted at P1 and P2. Above the arma-
ture is the tongue or contact maker, also
rigidly fixed to the spindle. The free ends,
n and s, of the armatures are seen between
the pole-pieces of the electromagnet in
Fig. 354. These armatures are kept mag-
netised with the polarity shown in the
figure by induction from the permanent
magnet. (It may be necessary to point out
that, if permanent steel magnets were used
in place of the armatures, they would
gradually lose their strength owing to the
vibration. As it is very important that
the strength should remain constant, soft iron magnetised induc-
tively by a fixed permanent magnet is used.) Each of the electro-
magnets shown in Fig. 354 consists of a straight
core of well-annealed iron, about 3 inches long,
upon which is wound a coil of fine silk-covered
copper wire of many turns. At each end of
the cores are soft-iron pole-pieces, S1N1S.>N.,.
In the spaces between the pole-pieces play
the free ends of the armatures n and s of
Fig. 353. The two coils are joined in series,
and a current in the proper direction will
magnetise the pole-pieces with the polarity

shown in the figure. As unlike poles attract aud like poles repel,
it will be seen that all four pole-pieces unite in moving the two
armatures to the left — and the contact maker with them — in which
position the circuit of the local battery is broken and the sounder
is unaffected. If the direction of the current through the relay is

1 This is a term much used in telegraphy. It may be defined as follows : —

Leakance = , ,— . . , — -

Insulation resistance in megohms

0(

dU

FIG. 354.

TELEGRAPHY AND TELEPHONY

509

reversed, the polarity of all four pole-pieces is also reversed — the
polarity of the armatures, of course, remaining unchanged. The
contact maker is then urged to the right and closes the sounder
circuito This reversal of the current is performed by the double
current or reversing key described later.

The metal plate carrying the contact maker and the contact screws
is shown in Fig. 355. The screws, St and S.2, are fixed so that the
contact maker, T, does not move more than TJ<y inch. In order to
adjust for stronger or weaker signals, the plate may be slightly
rotated about its centre by means of a thumb-screw, thus bringing
the armature nearer to or farther from the left or right hand pair of
pole pieces.

— GPTi

FIG. 355.

FIG. 356.

Double-current Key. — There are two forms of double-current
key. "With the first form, a single battery is used, the poles of which
are alternately joined to the line and to earth; with the second,
separate batteries are used for each direction of current. The prin-
ciple of the first type is shown in Fig. 356. The brass bar, EL, is
fixed at the end of the key lever, and moves up and down between
four flat, steel contact springs, ZjCjZC, which are connected to the
battery, as shown in the figure, alternate pairs being also joined
together. The two halves E and L of the bar are insulated from
each other by an ebonite partition. In the position illustrated, the
bar is resting on the two lower springs, the positive pole of the
battery is connected to the line (which, although not shown in the
figure, is permanently connected to E), and the negative pole to
earth (which is permanently connected to L). This direction of
current moves the distant relay tongue to the left, and the sounder
there is not actuated. On depressing the key, the bar rises against
the upper springs. The negative pole is now joined to the line, and
the positive pole to earth. In this direction, the current moves the
distant relay tongue to the right, closes the local circuit, and attracts
the sounder armature. The long and short up-and-down movements
of the key are thus reproduced on the sounder. When the operator

510

VOLTAIC ELECTRICITY

Line.

indicated in Fig.
illustrates the

FIG. 357.

has finished signalling his message, the line is connected to his own
relay by turning a switch, which forms part of the key. Such a

switch is
357, which

second method of double-
current working. When the
key is resting on the back
contact, the positive pole
of the battery A is joined
to the line. On depressing
the key, this is replaced by
the negative pole of the bat-
tery B. It may be remarked
here, that the direction of
current which moves the re-
lay tongue to the right and
actuates the sounder is called
by telegraphists a " marking " current ; the oppositely directed current,
which causes the sounder to rise, is called a " spacing " current. These
terms arise from the fact that, in an obsolete form of instrument,
which preceded the sounder, the dots and dashes of the Morse
alphabet were printed on a moving ribbon of paper. While the
current was flowing in the first direction, a line or " mark " appeared
on the paper. On reversing it, a " space " was left.

Advantages of Double-current Working.— The advan-
tages of double-current working on long lines are as follows : —
(1) The movement of the relay tongue to either side being effected
by the line currents, no controlling or " antagonistic " force is
required to pull it back after each signal. Thus, any cause which
reduces the working currents affects both the "spacing" and the
"marking" currents alike, and the relays can, therefore, be adjusted
to possess great sensitiveness. (2) A current in one direction or the
other is always flowing during transmission. Stray leakage currents
are, therefore, shut out. (3) The reversal of current after each signal
obliterates any residual magnetism in the relay cores. (4) Last, but
not least, the application of a reversed E.M.F. greatly facilitates the
electrostatic discharge.

With modern appliances, and especially with the greatly im-
proved construction and insulation of the lines now attained, no
difficulty is experienced in fair weather in working over lines
200 miles in length with double-current apparatus.

Repeaters or Translators.— When it is desired to work
direct to greater distances, a repeater is used. This instrument is
placed in some office situated somewhere near the middle of the line,
and is a special form of relay in which the local battery currents,
instead of working the sounder at the office where it is placed, are

TELEGRAPHY AND TELEPHONY

511

transmitted to the distant office relay and again actuate the sounder
there. Thus, a message from London to Glasgow is sent over two
lines, each connected to earth at Leeds, where relays repeat the
London signals on to Glasgow.

Duplex Working.— This is a method by means of which it is
possible to send messages simultaneously in both directions, i.e. station
A may be signalling to B, while B is signalling to A.

The two chief methods of working duplex are (1) the Bridge
system, mainly used in submarine cables, and (2) the Differential
system, which we shall now describe.

The Differential Duplex. — The essential feature of this
system depends upon the fact that the cores of the relay electro-
magnets are each wound with two separate and distinct coils of wire

r

r

^ '^j^v^^';;^^^>^:^^^ ^\^sjK$^^ftv!<(9&

™

FIG. 358.

of equal resistance, and of an equal number of turns — the two wires,
indeed, being laid on side by side.

If currents of equal strength are sent in opposite directions
through the two coils, their magnetic fields cancel each other and
the cores are not magnetised. If one current is stronger than the
other, the polarity produced in the cores will be determined by
the difference between the current strengths. Hence, the term
Differential.

The arrangement of the apparatus is shown diagrammatically in
Fig. 358, where mm' represent the cores of the double wound relay
coils; cc', the keys; and KR', two variable resistance coils called
rheostats, by means of which the currents in the inner relay coils
may be made equal to those in the outer relay coils.

A little consideration will show that in order to work duplex, the
following conditions must be fulfilled : —

512 VOLTAIC ELECTRICITY

(a) When both keys are at rest, both relays must be unaffected.
(/;) When the key at station A is depressed, the relay there must

be unaffected, but the relay at B must work.
(c) When the key at B is depressed, the relay there must

remain neutral, but the relay at A must work.
(cZ) When both keys are depressed, both relays must work.

The last case is the only one which may strictly be called duplex
working; the other three cases being those that occur in ordinary
single working.

We have now to show how these conditions are fulfilled.

Case (a) is self-evident. The keys being at rest, no current can
flow to actuate either of the relays.

In case (6), the current leaving the key at A has two paths of
equal resistance open to it: (1) through the inner relay coil of A,
through the rheostat to earth and back to the battery ; (2) through
the outer coil of A's relay, the line, the outer coil of B's relay, and
then through the back contact of B's key to earth. Two currents of
equal strength, therefore, flow through the coils of A's relay, and,
being in opposite directions, that relay remains neutral. At B,
however, the arriving current flows through the outer coil only, and
therefore its armature is attracted, which closes the local circuit of
the sounder (not shown in the figure).

Case (c) is exactly similar to case (1) — the signalling being merely
in the reverse direction.

In case (d) both keys are depressed, and therefore equal and
opposite electromotive forces are applied to the line, the points
a and a being raised to the same potential. Therefore, no current
flows through the outer coils of the relays and the line. At each
station, however, a current can flow from a and a' respectively
through the inner coils and the rheostats, and it is these currents
which actuate the relays and produce the signals.

The description given above refers to the single-current duplex.
In practice, however, the double-current system is used, and although
the principle remains the same, the explanation is somewhat more
complicated, so that if the student wishes to study the subject more
fully, he must consult some technical work on telegraphy, e.g.
Telegraphy, by Preece and Sivewright.

The Central Battery System. — One of the most recent
developments in telegraphy is the Central Battery system, chiefly
used on the circuits which radiate from a large central office to sur-
rounding smaller towns and villages or to branch offices. Instead
of providing a battery at each end of the line, the whole of the current
required is supplied from one large storage battery at the central
office. The only apparatus used at the distant office are a key, a
condenser, and a special form of receiving instrument called a
polarised sounder. The principle of the polarised sounder will be

TELEGRAPHY AND TELEPHONY

513

understood from Fig 359. The lower ends of the electromagnet,
instead of being joined, as in ordinary form, by a soft-iron yoke, rest
upon the poles of a large compound horse-
shoe permanent magnet placed under the
base of the instrument, so that the cores
form its pole-pieces. The path of the lines
of force of the magnet is, therefore, from
the N pole up the left-hand core, across the
armature, and down the right-hand core to
the S pole. The tension of the spiral spring
can be so adjusted that, if the armature is
forced down to the cores, the attraction of
the permanent magnet is sufficient to keep
it down, but if the armature is raised, it
remains up in consequence of the increased

air-gap reducing the magnetic flux. The armature is fixed to the
sounder lever as in the ordinary form of the instrument.

A momentary current, e.g. a condenser discharge, passing in the
right direction to increase the magnetism of the cores, will suffice
to pull down the armature, which will remain down until another
momentary current in the opposite direction weakens the cores and
allows the spring to pull it up again.

The arrangement of the apparatus is shown in Fig. 360, which

FIG. 359.

80V

soy

80V

indicates the connections for three offices on one line. B is the central
battery, S the sounders, k the keys, K the condensers, and E the
various earth-connections. The battery, usually of 80 volts, has one
pole connected to earth and the other to the line, the distant end of
which is insulated, instead of being connected to earth. A resistance
of about 3000 ohms is intercalated between the battery and the line,
indicated by the zigzag line in the figure. This is to avoid short-
circuiting the battery when a key is depressed.

It should be noted that, if the line is perfectly insulated, no

2K

514 VOLTAIC ELECTRICITY

current flows while the apparatus is at rest, but that the condensers
are maintained in a charged condition — the P.D. between their
plates being (say) nearly that between the poles of the battery, viz.
80 volts, in fact, every portion of the circuit between the positive
pole of the battery and the positive plates of the condensers is
80 volts above the potential of the earth.

When a key at any station is depressed, the line is connected to
earth. As all the positive plates of the condensers are connected
through the sounders to the line, they also become earth-connected.
The negative plates of the condensers are permanently joined to
earth, so that, in effect, the plates are connected together and the
condensers therefore discharge. This momentary discharge pulls
down the armatures of the sounders, and they remain down after
the current ceases. When the key is permitted to rise, the earth
connections are broken, and the battery immediately re-establishes the
original P.D. of 80 volts between the plates of the condensers. Then
the transient current flowing through each sounder to recharge the
condenser permits the armature to rise and denotes the completion of
the signal. In the ordinary sounder, the dots and dashes are formed
by long and short currents, while with the polarised sounder they
are due to longer or shorter intervals between two currents of equal
duration but opposite in direction.

It will be noticed that two alternative methods of connecting
up the key are shown in the figure. When arranged as in the first
office, depressing the key operates all the sounders on the line
except the one at the sending station, and when arranged as in
the second or third office, the sender's own sounder is operated also.
As a rule, the latter plan is adopted, but sometimes the former is
convenient.

Telephony. — Before we discuss the electrical instruments —
known as telephones — which reproduce distant sounds, it may be
pointed out that all sounding bodies are in a state of vibratory
motion, which sets up waves in the surrounding air, and that it is
these waves which, striking the ear, cause the sensation of sound.
These sound-waves differ among themselves in three particulars, viz.

(1) the rate of vibration, i.e. the number of vibrations per second;

(2) amplitude, i.e. the amount of displacement ; and (3) shape, due
to the coexistence, with the principal, of 'other secondary vibrations.
These physical conditions determine respectively (1) the pitch, (2)
the loudness, and (3) the quality or timbre of a sound. Given, then,
a sound which has these three characteristics in a fixed and definite
way, the problem of practical telephony is to produce an exactly similar
wave-motion at a distance.

The first person to construct an apparatus for this purpose was
Reis of Friedrichsdorf. His apparatus, although fairly successful
with musical notes, only imperfectly reproduced speech, mainly owing

TELEGRAPHY AND TELEPHONY

515

to its faulty receiver. The sending arrangement of the instrument
depended upon the alteration of a loose contact under the influence
of sonorous vibrations, and was identical in principle with the best
modern transmitters.

The Bell Telephone. — This well-known instrument was in-
vented by Professor Graham Bell in 1876. Its original form is shown
in Fig. 361. A bobbin of fine silk-covered copper wire, B, is fixed

FIG. 361.

on a short soft-iron core, attached to one end of a steel permanent
magnet, M, supported in a wooden case. The ends of the coil pass
through holes in the case, and are attached to binding-screws, CC.
In front of the soft- iron pole-piece, and as close to it as possible
without actual contact, is placed a very thin disc, D, of soft
iron. This disc, or diaphragm, is held in position by the mouth-
piece, E.

FIG. 362.

At the present time, the double pole type, shown in Fig. 362, is
generally used on account of the greater magnetic strength obtained.
The principle is unaltered, the only difference being that a horse-shoe
magnet is employed instead of a bar magnet. The magnet is made

516 VOLTAIC ELECTRICITY

from a long thin bar of steel bent double in the middle, thus bringing
the poles within J inch of each other. The two coils of wire are
joined in series, and the ends brought out by a flexible connection
(shown at the end) consisting of two separate insulated conductors
bound up together. In Fig. 363 two such instruments are shown,
connected to a line wire, the circuit being completed through the
earth, although in practice another wire is generally used. Their
action is easily understood. Considering one of the instruments,
we see that the magnetic circuit is partially completed through the
diaphragm. The distribution of the lines of force is very unstable,
and varies greatly with any alteration in the distance between the
diaphragm and the pole-pieces. The smaller the distance, the greater
the number of lines completing their circuit through the disc, and
vice versa, and any movement of the lines means that some of them
must cut the turns of the coils. (This action is well illustrated by
Experiment 213, p. 349.)

When sound-waves strike the disc, it vibrates in unison with them,

Line

D'

FIG. 363.

and the distribution of the lines through the coils on the bobbins is
disturbed, the opening-out and closing-in of the lines setting up
E.M.F.'s in opposite directions, which generate alternating currents,
when the circuit is completed. The impulses of current thus set up
are more or less wave-like in form, and reproduce with remarkable
fidelity the chief characteristics of the sound-waves producing them.
We may, therefore, regard the instrument as a very delicate alter-
nating current generator, which transforms the energy of sound-waves
into oscillatory currents of electricity. The currents from the trans-
mitting telephone, as they pass through the coils of the receiving
instrument, alternately increase and decrease the strength of the steel
magnet, thus varying the flexure of the diaphragm, the disc moving
inwards when the pull increases, and outwards (by virtue of its own
elasticity) when the attraction decreases. The discs of both instru-
ments, therefore, vibrate in harmony, with the important difference
that the amplitude of the receiver is much less than that of the trans-
mitter, and as a consequence the loudness of the resulting sound is
much reduced.

TELEGRAPHY AND TELEPHONY

517

The Watch Receiver.— This is the typical form of receiver
used in the instruments of the National Telephone Company, in
which the transmitter and receiver are fixed to a common support
held in the hand. For this construction, the Bell type would be
too heavy.

The watch receiver is very compact and light, the essential feature
being the shape of the steel magnet, which in the earlier forms was
a nearly semicircular plate of steel, B (Fig. 364). This is fitted with
soft-iron pole-pieces, AA, on which the coils are wound, as shown in
section in Fig. 365. In later patterns, the permanent magnet is a
complete ring of steel, magnetised so that consequent poles, N and S
respectively, are produced on opposite sides of a diameter, although
Fig. 364 represents this type equally well.

FIG. 364.

FIG. 365.

Transmitters and Receivers.— Any of these forms of Bell
telephone will act either as transmitter or as receiver (without the
use of a battery), if connected up as shown in Fig. 363. As a re-
ceiver, the telephone is unequalled, but as a transmitter it is not so
satisfactory. It will transmit speech over 10 to 15 miles of well-
insulated wire, but as the currents generated by the motion of the
diaphragm are very small and are incapable of augmentation, the sound
becomes more and more inaudible as the distance increases. In order
to be able to converse over long distances, it became necessary to
devise some new form of transmitter, which would permit of a
stronger current being used. This was achieved by the introduction
of a microphone in circuit with a battery.

Carbon Transmitters. Hughes* Microphone. — In 1878,
Professor Hughes showed that any loose or imperfect contact inter-
posed in a conducting circuit could be used as a telephonic trans-
mitter, owing to its action in varying the resistance, and consequently
the current. For example, two ordinary nails were made the terminals
of a battery, in the circuit of which a telephone was included. The

518

VOLTAIC ELECTRICITY

nails were connected by a third nail lying across them, thus forming

an imperfect contact. When
speaking near the nails, the
vibrations of the voice were
communicated to them, and
as differences in pressure pro-
duced differences in resistance,
similar variations were, of
course, produced in the cur-
rent through the telephone.

The most satisfactory re-
sults were, however, obtained
when points of carbon, mounted
on a sounding-board of thin
flexible pine- wood, were used
instead of the nails. In Fig.
366 is shown Hughes' original
FIG. 366. form of microphone, which

merely consists of a carbon

pencil, D, with pointed ends loosely fixed between two carbon blocks,
CC.

If the microphone be connected in circuit with a battery and
telephone (Fig, 367), and any sound
be produced near it, the sound-waves,
falling upon the carbon pencil, will
cause it to vibrate slightly. The
alterations in resistance thus pro-
duced result in a varying current
through the telephone, which repro-
duces the sound in the usual manner.
To transmit very feeble sounds or
vibrations, such as the ticking of a
watch, the carbon pencil must be
delicately balanced ; for spoken words or louder sounds, a coarser
adjustment is necessary to give good results.

Further improvements were made by using a number of pencils
joined in parallel.

Granular Transmitters. — All transmitters in use at the
present time are of this type. The principle is indicated in Fig. 368.
Two discs of carbon, placed very near each other, are connected
respectively to the terminals, the space between being filled loosely
with carbon granules. One of the carbon discs is fixed to the back
of the case, and the other is a thin plate which serves as a diaphragm
and is provided with a mouthpiece. When this diaphragm vibrates
under the influence of the voice, the resistance of the granules varies
, with the compression. The granules may, in fact, be regarded as

FIG. 367.

TELEGRAPHY AND TELEPHONY

519

equivalent to an enormous number of pencils. The outside of the
diaphragm is varnished to exclude moist are from the breath ; and
inside, both the carbon surfaces are made extremely smooth and are
highly polished. The carbon granules are very carefully selected ;
they should be extremely hard and well polished.

The chief difficulty with granular transmitters is due to " packing,"
i.e. a tendency of the granules to cohere into a more or less solid
mass. Various devices are employed to prevent this defect. In
the Deckert type (largely used in the Post Office systems) the surface
of the fixed carbon plate, instead of being smooth, is cut into a
large number of four-sided pyramids (Fig. 369). The summits of
these pyramids are very close to the carbon diaphragm, so that the
granules can only move freely along the zigzag grooves between
the pyramids. The apexes of about fifteen of the central pyramids
are cut off and replaced by little tufts of fluffy material. The tufts

FIG. 308.

FIG. 369.

on the tops of the pyramids assist in delaying the packing of the
granules, prevent short-circuiting between the two carbon electrodes,
and damp the vibrations of the diaphragm, i.e. check any tendency of
the vibrations to continue after the exciting sound-waves have ceased.
The Telephone Induction Coil.— It should be clearly under-
stood that the efficiency of a carbon transmitter depends not only
upon the number of ohms resistance by which it varies the total
resistance in its circuit, but also upon the ratio which that number
bears to the total resistance ; consequently, the longer the circuit
the less the efficiency of the transmitter. A microphone, whose
resistance varies between some small value and 10 ohms, if placed
in a circuit whose resistance is also 10 ohms, would alter the
total resistance from 10 ohms to 20 ohms, and vary the current
by 100 per cent. If it were placed in a circuit of 1000 ohms^it
would vary the current by only one per cent. A transmitter is,
therefore, most efficient when it is practically the only resistance

520

VOLTAIC ELECTRICITY

in the circuit. The longer the line, the less percentage change in
total resistance (and therefore in the current). Hence, it is imperative
that the total resistance should be kept low, but on long lines this
is impossible. Edison ingeniously overcame this difficulty by using
a small induction coil. This is a small and very simple affair,
consisting of two coils on a core of iron wire. The general arrange-
ment is shown in Fig. 370. A battery, B, and microphone, A, are
joined up to the primary of the induction coil, C. By using a special
type of low-resistance battery, the resistance of the microphone
becomes practically the only resistance in the circuit, that of the
battery and primary not exceeding 1 ohm. The secondary coil
of many turns, having a resistance of 25 to 100 ohms, is connected
to the line L and to the receiving telephone D, the circuit being
usually completed by means of a return wire, and not by the earth
connection shown in the figure. Although the resistance of the

FIG. 370.

coils is fairly high, their insertion in the line whose resistance is
already considerable, does not increase the latter by a large percentage.

The action is as follows : The variations of resistance, caused
by the sound-waves striking the transmitter, produce corresponding
changes in the strength of the current flowing through the primary
of the induction coil, and these fluctuations are considerable because
the microphone resistance forms so large a portion of the total
resistance.

The variations of current in the primary induce similar currents
in the secondary, but at a much higher E.M.F., which is an advantage
on account of the high resistance of the line. These induced currents
actuate the receiving telephone, D, which is of the usual Bell type.

The Magneto-Generator.— The magneto-generator is of the
type shown in Fig. 374, p. 526, having an H pattern armature wound
with a coil of fine wire of about 500 ohms resistance, and driven by

TELEGRAPHY AND TELEPHONY

521

a hand- wheel through gearing. No commutator is used, connection
being maintained with the ends of the coil in a very simple manner,
and an automatic device is arranged to cut it out of the line circuit
except when the handle is being turned.

The Magneto-bell. — As the generator produces an alternating
current, the bell must be designed to work efficiently with such
currents. Its principle is indicated in Fig. 371. E is an electro-
magnet, provided with a soft-iron armature rocking on a pivot, A,
and carrying a hammer which strikes the
gongs as it moves to and fro. This armature
is " polarised," being magnetised inductively
by means of a steel magnet (not shown in
the figure), which is bent so that one pole
is just below A, and the other just above the
middle of the base of the electromagnet, E.
The lines of force, entering at A, divide to
pass along the armature to both ends, and
then return to the distant pole through the
cores of the electromagnet. Hence, the arma-
ture is kept magnetised with " consecutive "

poles; both ends being of the same polarity, and this magnetism "is
not affected by its vibrations.

The alternating current from the line passes round the coils of
the electromagnet, and produces an alternating polarity superposed
upon the steady inductive effect due to the fixed magnet. If the
polarity alternates, a little consideration will show that each end
of the armature is in turn attracted and repelled, thus oscillating
rapidly, and if the current is so small that the poles are merely
strengthened and weakened respectively, the resulting inequality in
pull will produce the same effect.

Return Wires. Effect of Induction. — Except on very
short lines, it is not possible to use the earth as a return wire for
telephone circuits. The modern telephone is such a remarkably
sensitive detector of weak currents that it may almost be regarded as
an electrical microscope. When two earth plates, situated some
distance apart, are connected by a wire, a sensitive galvanometer
placed in the circuit will show that there is a continual ebb and
flow of current between the two plates, owing to the fact that their
potentials are constantly changing. This effect is very pronounced
in towns having an electric tramway system. These currents, although
they are^too weak to affect ordinary telegraph instruments, produce
a very disagreeable buzzing in the telephone, which renders conversa-
tion very difficult, if not impossible.

Another trouble is due to the fact that telephone circuits are
especially affected by induced currents from other wires — more
particularly from high-speed telegraph circuits — running parallel to

522

VOLTAIC ELECTRICITY

them. Owing to such induced currents on two telephone circuits, it
is possible for a conversation taking place on one circuit to be over-
heard on the other.

To overcome these difficulties, it is usual to provide
two wires, thus forming a metallic loop and dispens-
ing with the " earth return."

In Fig. 372, the upper wire is part of a telegraph
circuit, and the two lower lines represent a telephone
"line" and "return" wire, carried on the same
poles. The "starting" of the current in the tele-
graph line would tend to induce currents in the
opposite direction (as shown) in both wires of the
telephone circuit. If the two latter wires were equi-
distant from the former, the two induced currents
would be equal in strength and would neutralise
each other, as they tend to flow in opposite direc-
I 1HU tions round the circuit ; but unless precautions were

^ I taken, the telephone wires would not be equidistant,

/ / r J and there would be a resultant induced current, which
would introduce a disturbing influence. Hence, the
wires are arranged as shown, being made to change
places at every fourth or fifth pole. Equal portions
of each wire thus become alternately nearer to, or
farther from, the source of disturbance, so that no
resultant induced current is produced in the circuit.
Attenuation and Distortion of Speech.—
The currents in a long-distance telephone circuit may
be regarded as belonging to a complex series of
electro-magnetic waves propagated through the space
surrounding the conducting wire. Such waves of
current can be recorded by means of an oscillo-
graph ; they have approximately a sine-curve form,
and are usually between ten to twenty miles long.
The speed and mode of propagation are determined
by the four electrical constants of the circuit, viz.
resistance, leakance, inductance, and capacity.

The resistance and leakance, by abstracting
energy from the circuit, reduce the amplitude of the
waves at the distant end of the line to a mere ripple
with scarcely enough energy to vibrate the diaphragm
of the receiver. The sounds reproduced are faint
but distinct. This effect is called attenuation.

The effects of inductance and capacity are more
serious. Inductance tends to cause the wave to lag
in phase behind the E.M.F. producing it, and as waves of higher
frequency are the more retarded, the sounds of the voice — including,

TELEGRAPHY AND TELEPHONY 523

as they do, many different frequencies — become distorted. Capacity
affects the waves in much the same way, with, however, the im-
portant difference that a lead instead of a lag is produced in the
wave-form. This distortion, which is much more pronounced in
cables than in aerial circuits (owing to the relatively enormous
capacity of the former), may so modify the resultant wave-form,
that speech becomes quite unintelligible, although the sound pro-
duced may be loud.

The fact that inductance and capacity displace the wave-form in
opposite directions, indicates the possibility of using one to neutralise
the other — a problem which has been attacked in various ways.

In all cables the capacity greatly exceeds the inductance, and as it
is not possible to reduce the former, the only remedy is to increase the
latter. This is done by placing specially constructed induction coils
(in series) in the cable, their number and distance apart being deter-
mined by the "wave-length constant" of the cable.

In the year 1910, a specially designed cable, "loaded" (as it
is termed) with such coils, was laid between Dover and Cape
Grisnez in France, with, it is said, excellent results. It is hoped
that, by means of this cable, London may hold telephonic communi-
cation with Berlin, Amsterdam, or any other centre in Northern"
Europe.

The investigations of Pupin in America, of Heaviside in England,
and of other workers have provided data such that, knowing the
electrical constants of any type of cable, we can calculate the distance
in miles over which it is possible to conduct a satisfactory conversa-
tion. A unit or standard cable, of the paper-insulation type, has been
constructed having the following dimensions per mile of loop : weight
of conductors, 20 Ibs. ; resistance, 80 ohms ; capacity, '054 microfarad ;
insulation resistance, 200 megohms ; inductance, '001 henry ; and
speech limit, 43 miles. The properties of all other cables can be con-
veniently expressed in terms of this standard cable.

EXEECISE XXIV

1. Describe the construction and explain the use of a relay.

2. What is meant by electrostatic induction in cables ?

3. Describe Bell's telephone, and explain how it transmits and repro-
duces the vibrations.

4. Describe the method of communicating telegraphically between two
stations provided with Morse instruments, relays, and local batteries.

(B. of E., 1904.)

5. Give a general explanation of the action of a telephone, and describe
some form of transmitter. (B. of E., 1905.)

6. Describe the construction of some one form of telephonic receiver.
Why would not soft iron do instead of steel for the electromagnet ?

(B. of E., 1907.)

7. Describe the ordinary form of Morse sender and receiver, and explain

524 VOLTAIC ELECTRICITY

how a line may be used for sending signals simultaneously in opposite
directions. ' (B. of E., 1908.)

8. Describe and explain the reproduction of sound by a telephonic receiver
of the Bell type. Why should the core be of magnetised steel and not of
soft iron ? (B. of E., 1909.)

9. Describe, giving sketches of the instruments and connections, a com-
plete arrangement for telephonic communication between two stations.

(B. of E., 1911.)

CHAPTER XXXI

DYNAMOS AND MOTORS

ALL practical dynamo-generators are applications of the principle
established in Experiment 210, p. 348, i.e. they are devices for
obtaining relative motion between a number of conductors connected
in series, (forming what is termed the armature), and a magnetic field,
the direction of the field being as nearly as possible at right angles
to the conductors, and also to the direction of motion.

When the magnetic field is produced by steel permanent magnets,
the arrangement is known as a magneto -machine. Such machines
are used only in very small sizes for special purposes, e.g. for ringing
bells, for exploding fuses, and the like.

When the field is produced by soft-iron electromagnets, which
are excited by the whole or by a portion of the current generated,
and which are known as field magnets, the machine is called a
dynamo, and may be either (1) a direct current, or D.O. machine, or
(2) an alternating current, or A.C. machine.

In D.C. machines, it is usual to make the armature to rotate in
a stationary field; in A.C. machines, it is now customary to make
the field magnets rotate with reference to a stationary armature,
although in the past both arrangements have been used.

In our treatment of this subject, we shall first consider the simple
case of a coil of wire rotating in a magnetic field. We have already
shown (see p. 352) that an alternating current, making one complete
cycle per revolution, is produced, and we have now to give the con-
trivance a practical form. For this purpose, it is evidently desirable
to concentrate the lines of force upon the coil, which is conveniently
accomplished by winding the latter upon a soft-iron cylinder rotating
with small clearance in the polar gap. Also, as the end portions of
the coil are inactive, it will be advisable to make its length great in
comparison with its diameter. We are thus led to the original
Siemens type of armature, which is still useful for small magneto-
machines, such as are employed in telephony for ringing bells
(see p. 520).

SIEMENS OR H PATTERN ARMATURE

Fig. 373 shows one of the early armatures of this type. The coil
is wound in a deep longitudinal groove in the iron core, and when the

525

526

VOLTAIC ELECTRICITY

machine is intended to give alternating currents, the two free ends
are merely joined up to insulating rings (or other forms of rubbing
contact) so that they remain in connection with the stationary
terminals of the machine as the armature revolves. Fig. 374 shows
the iron core and steel field magnet of a magneto-machine. The

FIG. 373.

magnet, NS, is provided (1) with soft-iron pole-pieces, ns, which
serve to concentrate the lines of force upon the armature ; and (2)
with distance-pieces, bb, of brass (or other non-magnetic material),
which keeps them at the right distance apart.

When the permanent magnets are replaced by electromagnets
excited by the machine itself, the current in the external circuit must
be continuous in direction, and in order that this
condition may be satisfied, some form of commu-
tator must be used. The simplest is that shown
in Fig. 373. It is merely a metal ring split into
halves by two longitudinal and slightly oblique
slits (the obliquity is not, however, essential, being
a detail which tends to reduce sparking) insulated
from each other and from the shaft, to which the
two free ends of the coil are connected and upon
which rest two stationary brushes. These brushes
serve to maintain connection between the rotating
coil and the remainder of the circuit. If their
position is such that the contact changes from
one half ring to the other at the instant the
current reverses its direction, each brush will
always have the same sign, and the current out-
side the armature will be continuous in direction,
although pulsating from zero to some maximum
value. Evidently the current may be led round
the coils of the soft-iron field magnets in order to
excite them, but as there are various ways of
accomplishing this purpose, which do not in the
least affect the consideration of armature details, it will be convenient
to deal with such methods later, and at present to assume that field
magnets of some kind exist.

The E.M.F. of a machine, with a field of a given strength, depends
upon the number of turns on the armature and upon the speed of

DYNAMOS AND MOTORS

527

rotation ; the current then depends chiefly upon the resistance of the
external circuit, and its safe value is determined by the heating of the
armature conductors. At the same time, it must be remembered that,
as the output increases, so does the mechanical power required to
drive the machine.

The first armatures of this type were made (and are still made for
very small machines) with solid iron cores, but then induced currents
were not only produced in the winding, but also in the core itself.
Such "eddy currents" involve a very serious waste of energy, and
the iron core may become hot enough to injure the insulation of the
winding. This difficulty is not a peculiarity of the type of armature
just referred to. It is met with in all armatures, although it is very
greatly reduced (but never completely eliminated) by laminating the
armature core, i.e. by building it up of thin soft iron plates or
stampings, whose planes are at right angles to the direction of the
eddy currents. In large machines these stampings are usually
insulated from each other by a thin coat of varnish ; in small ones,
the natural coating of rust or scale is sufficient. Such simple means
are effective, because the E.M.F. producing eddy currents is very
small.

Defects of Single-coil Armatures with Split-ring Com-
mutator.— Of these defects, the two most important are : (1) the
E.M.F., and therefore the current, falls to zero twice in each revolu-
tion— this makes the machine useless for charging accumulators;
(2) the full P.D. the machine produces, exists between the two
halves of the commutator, and as a consequence there is a serious
danger of an accidental short-circuit. The latter defect prohibits the
use of this winding for high voltages.

The Gramme Ring Armature. — A very great step in ad-
vance was made by the introduction of the Gramme armature.
Although obsolete, it is worth attention, because it leads naturally to
forms of winding now in common
use. The study of its construction
and action is best approached by
considering a number of straight,
insulated conductors, grouped sym-
metrically around a rotating cylindri-
cal armature, as shown in section in
Fig. 375, which indicates the direc-
tion of the E.M.F.'s on the two sides
of the armature for the given polarity
and direction of rotation. The prob-
lem to be solved is the method of
connecting these conductors together and to the external circuit.
Gramme solved it by making the armature hollow and providing
inactive connecting wires inside. The arrangement is equivalent to

FIG- 375.

528

VOLTAIC ELECTRICITY

winding the iron ring with a uniform coil until the ends of the
winding meet, and then joining the ends to form a closed winding.
Of course, this is not the actual method of winding, but the result is,
electrically speaking, the same. The principle is indicated diagram-
matically in Fig. 376.1 It will be seen that all the active conductors
on each side of the armature are in series, but the E.M.F. induced on
one side is opposed in direction to an equal E.M.F. on the other side,
so that yet no current can flow. If, however, we suppose that the
outside surface of the active conductors are free from insulation, and

that two brushes, joined by a wire forming the external circuit, make
contact with them at the top and bottom of the armature respectively,
it is evident that the two halves of the winding are joined up in
parallel to the external circuit, in which a current will flow of twice
the strength of that in any one active conductor. In practice, it
would be inconvenient to collect the current directly from the active
conductors, and so they are connected to a number of insulated copper
bars, forming the commutator, upon which the brushes (marked + and
— in the figure) rest. It is not necessary to provide a bar for every
active conductor, as, in most cases, such an arrangement would make

1 This and several following figures (Figs. 376, 377, 378, 379, 380, 381,
383) are taken from Slingo & Brooker's Electrical Engineering, published by
Messrs, Longmans.

DYNAMOS AND MOTORS 529

the commutator impossibly large ; but it is desirable to have as
many bars as possible, because that helps to keep down sparking
at the brushes. Consequently, the total winding may be regarded
as being split up into a number of coils (known as " sections "),
connected to the commutator bars as shown diagrammatically in
Fig. 377.

Advantages and Disadvantages of Ring Armatures.—
The advantages are : (1) the current in the external circuit is nearly
uniform in strength, instead of falling to zero twice in each revolution ;
(2) the points, between which the P.D. is greatest, are at opposite
ends of a diameter, both on the armature and on the commutator, and
thus there is little difficulty as regards insulation.

On the other hand, there are two great drawbacks : (1) each
section of the winding must be wound in its place by hand ; (2) all

FIG. 377.

the lines of force of the field must pass through the iron of the
ring, and it is difficult to give it a cross section sufficiently great to
keep the magnetic reluctance within reasonable limits.

Drum Armatures. — Practically all modern armatures are of
the drum type, in which these defects are avoided.

It has been stated that the conductors inside the Gramme ring
are merely devices for joining together the external or active con-
ductors in series, and we see that this can be accomplished equally
well by connecting each of the external conductors shown in Fig. 375
to another diametrically opposite one, thus keeping all the winding
on the outside of the armature, which need not then be hollow. For
the actual details of the connections, the student must refer to
special treatises ; the result, however, is a closed winding, which
behaves, from an elementary point of view, very much like the ring
winding already described. In both, there are two parallel paths
through the winding from brush to brush, and in each path half the
external conductors are in series, the difference being mainly in the
method of connection by which this result is obtained.

Early drum armatures were hand- wound, and the windings over-

2L

530 VOLTAIC ELECTRICITY

lapped at each end, so that in case of injury, it was usually necessary
to unwind and rewind the whole. This
was a very great disadvantage, which is
entirely avoided in modern practice — a
damaged part being now easily replaced
without seriously disturbing the remainder
of the winding.

The iron core of both ring and drum
armatures is, of course, built up of thin
stampings, and the conductors are now
invariably imbedded in slots. A stamping

FIG" 378 ^°r a ^mm armatlire is shown in Fig.

378. The apertures around the shaft are

partly to reduce weight where iron is not required, and partly for
ventilation.

E.M.F. induced in Armature. Example.— An armature
has 420 active conductors, and is to run at 900 revolutions per minute
in a two-pole field. How many lines of force must pass through it in
order to produce an E.M.F. of 110 volts?

Let Z lines of force be required. Then one active conductor cuts

2Z lines of force in one revolution, or 2Z x — — lines per second. By

oO -\

definition, this must be the average value of the E.M.F. per conductor

in absolute units. Also - active conductors are in series, therefore
2

the total induced E.M.F. between the brushes is given by

E = 2Z x ?22 x absolute units.
oO 2

But E is to be 110 volts,

2Z x 900 x 420

110

60x2

rj 110x60x2xl08 , *, 1A81. , ,

or Z PTTTTT— — = 1-74 x 106 lines of force.

2 x 900 x 420

In this case, the result will be the same whether the armature
is ring or drum wound.

It will be observed that the E.M.F. induced in any one active
conductor is alternating in direction, whereas the method of collection
ensures that the P.D. between the brushes shall be continuous in
direction. Hence, this P.D. is the average value, as assumed in the
above calculation.

It must not, however, be supposed that this is the reading
actually obtained on a voltmeter connected across the terminals of
the machine. Such may be the case under certain circumstances
when no current is flowing, but the machine has a certain internal

DYNAMOS AND MOTORS

531

resistance, and to it applies all that has been said in Chapter XV.
as regards the distinction between induced E.M.F. and P.D. at the
terminals in current generators. For instance, if, in the worked
example just given, the internal resistance is -5 ohm, then, when the
current is 20 amperes, 10 volts will be expended within the machine,
and the P.D. at the terminals will be only 100 volts. But the case is
not quite so simple as that of a cell or battery discussed in the
chapter referred to, because there are other influences at work,
which may alter the induced E.M.F., and, therefore, modify the P.D.
at the terminals.

The Field- Magnet System. Self-exciting Principle.—
If the field magnets possess initially a slight amount of magnetism,
a definite but very small E.M.F. will be induced in the rotating
armature. If the feeble current thereby produced be passed through
the coils of the field magnets in the right direction, the existing
magnetism will be strengthened, and consequently the induced E.M.F.
and current will be increased. This current, in its turn, will increase
the magnetism of the fields. The action is, therefore, cumulative, but
as the iron tends to become saturated, further increases of current
produce a smaller and smaller effect,, and so the strength of the field
soon reaches a limiting value for a particular speed of rotation.

In order to produce the required magnetic flux, the excitation
must amount to a certain number of ampere-turns, which will depend
upon the details of the machine, and which is calculated by the
method outlined in Chapter XXV. The higher the permeability of
the iron, the less will be the magnetic reluctance, and hence the best
material for field magnets would be wrought iron, but as this cannot
be cast, it is now usual to employ what is known as " mild cast steel,"
which is really a fairly pure form of iron, containing just enough
carbon to make it sufficiently fusible.

Series Winding of Field Mag-
nets.— In the oldest form of winding —
known as series winding — the armature,
the field coils, and the external circuit are
in series with each other, and hence the
fields must be wound with conductors
thick enough to carry the whole output
of current without excessive heating.
However, as the exciting current is large,
only a moderate number of turns is neces-
sary to give the required number of
ampere-turns.

This winding is simple and inexpensive,
and is also very satisfactory when the cur-
rent output is always to be the same, but it is evident that, when
the current alters in strength, the magnetic flux also alters, and with

FIG. 379.

532

VOLTAIC, ELECTRICITY

it the induced E.M.F. As, however, for most purposes, it is neces-
sary that the current should be capable of variation without altering
the E.M.F., the simple series winding is seldom used for generators.
Such a winding is shown diagrammatically in Fig. 379. It will
be noticed that no current can flow, and therefore the machine
cannot excite itself, until the external circuit is closed.

Shunt Winding". — In a shunt winding, the required number
of ampere-turns is produced by a small fraction of the total current
passing round many turns of fine wire, the ends of the winding

being connected directly across the
brushes, and therefore in parallel with
the external circuit. This form is
shown in Fig. 380. Evidently the
machine can excite itself even if the
outer circuit is open ; moreover, the
exciting current is constant if the P.D.
at the terminals is constant, and
is,- therefore, largely independent of
changes in the current output. For
this reason, the machine maintains a
much closer approximation to a con-
stant voltage at all loads, although
it is easy to see that the voltage must
drop to some extent as the load in-
creases, for even if the induced E.M.F. remains constant, the P.D.
at the terminals must fall on account of internal resistance.

One method of keeping the pressure constant is to place an
adjustable resistance in the shunt circuit; then when the voltage
falls, this resistance can be reduced (thereby increasing the field
current and the ampere- turns) until the
voltage rises again to its original value.
Such a method, of course, requires the
presence of an attendant. The same
result can, however, be obtained automati-
cally by using a compound winding.

Compound-wound Machines.—
These may be regarded as shunt- wound
machines provided with an auxiliary semes
winding for regulating purposes, as shown
in Fig. 381. The shunt coil produces the
number of ampere- turns required to give
the desired voltage at no load, i.e. when
the external circuit is open and the series
winding is inoperative. When the circuit
is closed and a current flows, it also flows round the turns of the
series coil, which, therefore, produces an extra number of ampere-

FIG. 380.

FIG. 381.

DYNAMOS AND MOTORS 533

turns proportional to the load, and by adjusting the number of
turns, it can be made to keep the P.D. at the terminals practically
constant throughout the working range of current, or if necessary
to make the voltage rise slightly as the current increases.

Armature Reaction. — It should be noticed that the symmet-
rical distribution of the field assumed to exist in Figs. 375 and 376,
only holds good at no load. When a current flows through the
armature, it must produce another magnetic field of its own, and the
actual flux in the air-gap will be the resultant obtained by superpos-
ing this field upon the original one. For an adequate discussion of
this important subject, the student must consult text-books on
electrical engineering ; but we may state that in the case of a
generator, the final result is very much as if the original field had
been slightly dragged round in the direction of rotation, and
thereby distorted. As this displaces the positions of zero E.M.F., it
necessitates a similar forward displacement, or " lead," of the brushes
in order to bring them into the correct position. Again, as the
strength of the armature field depends upon the strength of the
current, it must vary with the output of the generator, and in con-
sequence it was once necessary to alter the lead of the brushes, when
the load altered, in order to avoid sparking. In modern practice,
machines are made to work with fixed brushes at all loads, a result
obtained partly by improved design, and partly by taking advantage
of the properties of carbon brushes, or of reversing poles, as mentioned
in the next paragraph.

Commutation. — In the simple case shown in Fig. 376 we see
that the current in each armature conductor must be reversed in
direction as the latter passes a brush; whilst Fig. 377 tells us that
if, as is generally the case, the conductors are grouped in sections,
the current in each section as a whole must be reversed as that section
passes a brush. Hence, while the current is stopping, there will be
produced a self-induced E.M.F. in the same direction, and therefore
tending to retard the stoppage ; and another in the opposite direction
while the current is starting again, retarding its rise. These effects
lead to very injurious sparking between the brushes and the com-
mutator bars, wrhich must be eliminated in some way. The difficulty
may be minimised, to begin with, by making the number of turns in
each section as small as possible, because the magnitude of the
induced E.M.F. is proportional to the square of the number of turns.
This means using numerous sections and many commutator bars. It
is then customary to provide what is known as a " reversing field "
at the brushes. According to Fig. 376, the correct position for a
brush is exactly at the neutral point of the field, so that the section
undergoing commutation is not at that moment cutting the lines of
force. If, however, the brush is advanced slightly in the direction
of rotation, the section will be moving in a magnetic field as it passes

534 VOLTAIC ELECTRICITY

the brush, and the E.M.F. induced in it by this motion will be in
the opposite, direction to (and can be made to neutralise) the self-
induced E.M.F. due to the stoppage of the current. Moreover, as
each section passes the brush, there is a brief instant during which
it is removed from the main circuit, and is short-circuited upon
itself through the brush as the latter bridges across two consecutive
commutator bars. Under these circumstances, the reversing field can
induce an independent current in the section, and ideal commutation
is obtained when this local current reaches the same strength as the
working current at the instant the section becomes part of the wind-
ing beyond the brush.

These actions are so important that it is becoming an increasingly
common practice to provide the field magnets with small auxiliary
poles near the brushes, whose sole purpose is to provide the necessary
reversing field, and thus to secure sparkless commutation at all loads.

Carbon brushes, wide enough to bridge over several commutator
bars at once, are now invariably used. The extra width is beneficial
in prolonging the time available for commutation, whilst carbon is
remarkably effective in suppressing slight tendencies to spark, chiefly
on account of (1) its relatively high specific resistance, and (2) the
peculiar way in which the resistance of a carbon-metal contact varies
with the pressure, the practical result being that energy, otherwise
expended in a spark, is harmlessly dissipated as heat in the brush.

Multipolar Machines. — For convenience, we have thus far
considered two-pole or " bipolar " generators, but at the present time
multipolar types are preferred — four poles being used for small
machines, and six, eight, or more poles for larger outputs. The
poles are N and S alternately, and, with the usual form of armature
winding, there are as many brushes as there are poles. For instance,
an eight-pole machine will have four positive brushes all connected
together to form the positive terminal, and four negative brushes
similarly connected ; and instead of there being two paths in parallel
through the armature winding from brush to brush, there will be
eight such paths, so that, when calculating the E.M.F. by the method
given on p. 530, -J- of the total conductors must be reckoned in series.

The parts of a small four-pole generator of modern type (made
by the Lancashire Dynamo and Motor Co., Manchester) are shown
in Fig. 382, which will serve to indicate the usual construction of
such machines. A machine of this size gives out a current of 40 or
50 amperes at 500 volts pressure, but, of course, the winding may
be adapted to any output of about the same number of watts.

The advantage of the multipolar type, apart from increased sym-
metry and better mechanical protection of the windings, is chiefly
due to the fact that large current outputs may be obtained without
using abnormally thick armature conductors. For instance, if the
full load current is to be 400 amperes, each conductor in a bipolar

DYNAMOS AND MOTORS

535

type must carry 200 amperes, but with eight poles each conductor
has only to carry 50 amperes, and can, therefore, be of proportionately

smaller section. This is not only more convenient from the con-
structive point of view, but it also diminishes the loss due to eddy

536 VOLTAIC ELECTRICITY

currents in the copper itself, which always occurs in thick conductors ;
and, what is of much greater importance, it greatly facilitates the
sparkless collection of current, owing to the fact that the self-induced
E.M.F. which impedes commutation is proportional to the current in
the section undergoing reversal, and therefore will be only J as great
as in the previous case.

Efficiency of Direct-current Generators. — Evidently, the
mechanical power supplied to drive a current generator must be
greater than the output of electrical power by an amount equal
to the various sources of loss. These are (1) the C2R loss in the
armature and field windings, (2) the eddy current loss, due to in-
duced currents in the framework, iron cores, &c., (3) the hysteresis
loss in the armature core, and (4) frictional losses, due to bearings,
brushes, air resistance, &c.

Theoretically, the first source of loss can be made as small as we
please (see p. 252) by making the resistance of the armature and
series coils sufficiently small, and that of the shunt coil sufficiently
large. As this increases the size of the machine, and, therefore, the
first cost, there is a practical limit which it does not pay to exceed.
Obviously, these losses are (apart from that in the shunt coil) pro-
portional to the load.

The eddy current loss can be kept dowa by careful attention
to details of construction, but, as we have said, it can never be
eliminated. It is proportional to the square of the speed, because
the E.M.F. producing the eddy currents is directly proportional to
the speed, and, in a circuit of constant resistance, the watts lost are

/ E2\

proportional to the square of the E.M.F. f watts = — ).

The third loss depends upon the fact that the iron in the arma-
ture core is continually passing through cycles of magnetisation. As
there is the same loss in each cycle, it is proportional to the speed,
and can only be reduced by carefully selecting the iron used for the
armature stamping.

The fourth or frictional loss is, of course, unavoidable, although
in well-designed machines it is relatively small. Like the hysteresis
loss, it is proportional to the speed.

Now, efficiency may be defined as the ratio Power given. out, and

Power supplied

if e be the P.D. between the terminals and C the current given out,
this may be written — -

Efficiency =

eC

CO + (C2R loss in windings) + (eddy current loss) + (hysteresis and friction losses)

The student must be referred to technical manuals for the methods
adopted in measuring the various losses, and thereby determining the

DYNAMOS AND MOTORS 537

efficiency. Here, it is sufficient to say that in the case of large
machines at full load it is well above 90 per cent.

Direct-current Motors. — These do not differ in any essential
respect from current generators, although certain details may be
modified to suit the working conditions. For instance, Fig. 382
represents a motor just as well as it represents a generator. In
generation, the force resisting motion is the force on armature con-
ductors carrying a current in a magnetic field. Evidently, if the
driving-power were suddenly removed, and if, at the same time, the
current were kept flowing in the same direction as be/we from some
external source, this force would still act and would drive the
machine backwards as a motor. A reversal either of the field or
of the direction of the armature current would reverse the direction
of the force, and therefore of the rotation ; but if both be simultane-
ously reversed, the direction of the force is unaltered. This is exactly
what is found to occur with a series- wound machine — it runs in one
direction as a generator and in the opposite direction as a motor —
but its direction of running as a motor is not altered by reversing the
current through the machine as a whole.

A shunt-wound machine behaves somewhat differently. If a
current from some external source be sent through it, the student will
see, by drawing a simple diagram, that it can never flow in the same
direction as before in both armature and field. In one of the two
windings it must be reversed, and this reverses the direction of
rotation, so that a shunt- wound machine runs in the same direction
both as generator and as motor. Compound windings are very little
used for motors, and need not be discussed here.

Contrasting the three windings, it will be sufficient to say that
a series-wound motor slows down under a heavy load and races
dangerously under small loads ; a shunt- wound motor runs at nearly
the same speed at all loads within its range, although it slows down
slightly as the load increases • a compound- wound motor can be
made to run at practically constaut speed at all reasonable loads.

Back E.M.F. Of a Motor. — In accordance with the prin-
ciples stated on p. 371, an induced E.M.F. will be produced in the
armature as soon as the motor begins to rotate. This E.M.F. will be
exactly the same in value as it would be if the machine were running
as a generator at the same speed and in a field of the same strength,
Avith the important difference that it will be now in opposition to the
impressed voltage.

It is, as we have already pointed out (see p. 372), this back
E.M.F. which is the essential factor in the performance of external
work. For instance, if E be the impressed voltage (assumed to be
constant) and r the internal resistance of the motor, the current

through it at rest will be — amperes. As r is quite small, perhaps
r

538 VOLTAIC ELECTRICITY

only a fraction of an ohm, this current is excessively large, and
would burn out the armature in a very short time. Hence, the full
working voltage is never applied to a motor at starting ; an adjustable
resistance being included in the circuit, which is gradually cut out as

the speed rises. When running, C = ~gffl, where em is the back

E.M.F. of the motor; the power supplied to the motor is EC watts,
and the power developed is em x Ctt watts (here Ca is the current
through the armature. This is obviously the same as C in a series
motor, but slightly less in a shunt motor). The difference EC — emCa is
equal to C2r, and represents the power wasted as heat in the wi»d-
ings. However, not all the power represented by em x Cu is available
for external work, for from it must be deducted the losses due to
hysteresis, eddy currents, and friction.

Exp. 245. Connect up any small machine, that may be available, to a
battery or to supply mains, including in the circuit an adjustable resistance
and an ammeter. Watch the ammeter as the motor starts up from rest, and
notice that the current tends to decrease as the speed increases. (If the motor
is series wound, care must be taken to keep it from racing dangerously.)
Slow it down by holding something against the pulley, and notice that the
current increases, reaching a maximum when the motor is forcibly prevented
from turning.

When a motor is run without a load, it speeds up, thereby re-
ducing the current, until the power (EC) taken from the source is
just sufficient to balance the total losses occurring at that speed.
With a well-designed machine, the losses are small, and hence the
speed must be great enough to make em nearly equal to E, and the
current correspondingly small. Obviously, if a machine could be
made in which no loss occurred, it would, at no load, speed up until
em was actually equal to E (when the current would be zero), but, of
course, it would not then be doing any work.

When the machine is shunt wound, its fields are always fully
magnetised (because the impressed E.M.F., E, is always acting
directly on the field coils), and in this case a very slight increase of
speed, at no load, is sufficient to raise the back E.M.F. to the
required value. When it is series wound, the field strength decreases
as the current decreases, and this means that, at no load, a very much
higher speed is necessary to raise the back E.M.F. to the required
value.

When a load is applied, it tends to slow down the machine ; and,
if shunt wound, a very slight decrease in speed will lower the back
E.M.F. sufficiently to permit the necessary increased current to flow;
if series wound, the field strength rises as the current increases,
and this raises the back E.M.F., so that a much greater slowing
down must take place before the current can rise to the amount
required.

The series- wound motor is much more useful than the series-wound

DYNAMOS AND MOTORS

539

generator. It is almost exclusively employed for traction purposes,
because the tendency or effort to rotate (known as the torque or
turning-moment) is greatest when the resistance to motion is greatest.
This is because the turning-moment depends entirely upon the force
on a conductor carrying a current in a magnetic field, and thereby
depends, in a given armature, merely upon the strength of the
current and the strength of the field. Now, in a series machine,

• FIG. 383.

these two factors increase simultaneously, being greatest when the
machine is at rest, i.e. forcibly restrained from turning. Thus, the
motor makes its greatest effort when it is most needed.

Fig. 383 is a four-pole traction motor, made by Messrs. Dick,
Kerr, & Co., and serves to show the nature of the modifications
in design necessitated by the special conditions of service. As it
has to be carried beneath a car, it must be dust-proof, take up
little space, and be able to stand large temporary overloads without
injury.

540

VOLTAIC ELECTRICITY

Alternating Current Generators.— These are simpler in
principle than direct- current machines, because, as we have seen, the
induced E.M.F. in a coil rotating in a magnetic field is naturally
alternating. Hence, the commutator is unnecessary, and thus many
possible sources of trouble are eliminated. On the other hand, a
direct current is required to excite the fields — usually supplied by a
separate D.C. machine, known as an " exciter."

On p. 352 we have considered a very simple type of alternator,

FIG. 384.

and have shown that in a two-pole field the frequency is equal to
the number of revolutions per second. To obtain a frequency of
50 cycles per second (a very usual value) from a bipolar machine
would, therefore, mean an excessive speed, and hence the construction
is always multipolar, for a little consideration will show that if there
are n pairs of poles there will be n complete cycles per revolution.

During recent years, however, turbine -driven alternators, running
at very high speeds, have been largely used, and this has enabled the

DYNAMOS AND MOTORS 541

necessary frequency to be obtained with two or four poles (more
usually the latter). It is the invariable practice to make the field-
magnet system rotate inside a stationary armature, because the
former is simpler in structure and it is easier to give it the necessary
strength and rigidity to withstand centrifugal forces.1 The stationary
armature has also the advantage of enabling the current to be
collected from fixed terminals, instead of through rubbing contacts — a
matter of considerable importance in the case of large currents.

In Fig. 384 is shown the armature of an alternator, made by
Messrs. Siemens Bros., to whom we are indebted for the illustration.

FIG. 385.

The active conductors are embedded in slots in the stampings, and
the end connections are held in position by very strong clamps
bolted to the framework. As these end connections are not in
the magnetic field, there is, under ordinary conditions, no force
acting upon them except the very small one due to the currents
in the neighbouring conductors, and the necessity for much care
in supporting them would not naturally be anticipated. However,
the experience of all makers of alternators of this type has shown
that, when an accidental short circuit or other mishap occurs when
working, the end connections are liable to be torn out of place and
broken, thus completely wrecking the armature, although the part
of the winding embedded in the slots is uninjured. This happens
because the current under such circumstances may reach an enor-
mously high value for a brief instant, thus producing momentary
forces of very great magnitude. Hence, all modern armatures are
provided with clamps as shown in the figure.

A four-pole field magnet for a turbine-driven alternator is shown
in Fig. 385. The four polar projections are each wound with an

1 It is a convenient and usual practice to speak of the stationary and
moving parts as the " stator " and " rotor " respectively.

542 VOLTAIC ELECTRICITY

exciting coil, and the whole structure is very firmly braced together.
The exciting current is led into and out of the windings by means
of two insulated rubbing contacts on the shaft, which, at the further
end, is arranged for direct coupling to the engine. This field magnet
was constructed by Messrs. Dick, Kerr, & Co., and is therefore not
the one actually used with the armature shown in Fig. 384. There
are, in fact, two distinct types of field magnets in use at the present
time: (1) the salient pole type, as shown in Fig. 385; (2) the non-
salient pole or cylindrical type, which resembles externally the
ordinary drum armature of a direct-current machine, such as is
shown in Fig. 382, and is built up of slotted stampings in much
the same way. It is wound so that the necessary poles are produced
on the cylindrical surface, although there are no pole-pieces. It is
difficult to say which form will finally survive — probably the latter.

A cylindrical four-pole field magnet of this kind is used with the
Siemens armature described above (Fig. 384), but as an illustration
would merely show a plain cylinder supported on a shaft, it is not
given here. The output of this particular machine is 1000 kilowatts
at a pressure of 2200 volts,, i.e. the full-load current is about 450
amperes. The frequency is 50, and as it is a four-pole machine, there
will be two complete cycles per revolution, and hence the speed must
be 1500 revolutions per minute. The surface speed at the circum-
ference of the rotor is therefore very high, and the greatest care must
be taken to secure perfect balance. The machine is totally enclosed
(excepting for air inlets and outlets), to minimise noise, and it
is usual to provide forced ventilation by means of fans fastened to
the shafto

The Winding of Alternator Armatures. — This is very simple

in principle, as there is only one path through the winding from terminal to
terminal. Two varieties are shown in Fig. 386.1 The machine is supposed to
be of the ordinary slow-speed type, having a large number of pole-pieces, of
which only four are shown for convenience, and for the sake of clearness the
armature conductors are projected on the polar faces. It is evident that as
the poles move over the conductors, the induced E.M.F. will be in one direc-
tion under a North pole and in the opposite direction under a South pole, as
shown, so that it is merely necessary to join the active conductors in series,
leaving two free ends to be connected to the terminals. This condition is
satisfied by either form of winding — electrically they are equivalent — but
whereas the end connections in the upper figure are the shorter, thus reducing
the resistance and the amount of inactive wire, the lower form — sometimes
called a hemitropic winding — has the advantage *of being more readily adapted
to the varying requirements of different designs.

Polyphase Windings. — It will be noticed that only half the avail-
able space is filled up with active conductors. Although in practice rather
more than half is utilised, this is an essential condition in " single-phase "
alternator windings, because there must be an instant at which all the active

1 This figure is taken from Electric Light and Power (Brooks and James),
published by Messrs. Methuen.

DYNAMOS AND MOTOKS

543

conductors are so placed that there is no induced E.M.F. Evidently it would
be possible to put on two independent windings, each occupying half the
available space (thus increasing the output for a given size), and it is also
obvious that one alternating E.M.F. will be a maximum at the instant the
other is zero, i.e. they would differ in phase by |- period, or be " in quadrature."
The result is a " two- phase " alternator.

When the space is filled with three independent and symmetrical wind-
ings, the result is a " three-phase " alternator. Such machines have most
important properties, which, however, cannot be discussed here.

Alternating Motors. — In practice, it is more difficult to construct
satisfactory motors for alternating than for direct currents. Although an

FIG. 386.

ordinary alternator will run as a motor, when its armature is supplied with
an alternating current of suitable frequency, the property is of little value,
because the machine will not start from rest, and it also requires an auxiliary
direct current to excite its field. Moreover, the speed is determined by the
frequency and the number of poles ; it is, therefore, constant at all loads, and
cannot be varied or controlled as in the case of direct-current motors. The
first practical solution was based on the properties of the two and three-phase
currents mentioned above, and such " polyphase " or " induction " motors are
still largely used. These motors are quite distinct in principle from direct-
current motors, being based on the properties of rotating magnetic fields.
Single-phase motors took longer to develop, and several distinct varieties are
in existence. One of these is similar to the polyphase motors, and another is
really a modified series motor of the ordinary type with laminated field
magnets — to avoid eddy currents. Motors of this kind can be made to run

544 VOLTAIC ELECTRICITY

on D.C. or A.C. circuits indifferently, and are successfully used for electric
railways,

Examples. — 1. A motor running light takes 1 ampere at 100
volts. If its resistance is 20 ohms, how much power is spent in
turning the armature, and how is it expended 1

(B. of E., Day, 1907.)

Writing C = 5n^, we have
r

1 = 1°em or em = 80 volts.

The power supplied to the motor is EC = 100x1 = 100 watts;
the power expended in turning the armature is em x C = 80 x 1 = 80
watts, and the CV loss is I2 x 20 = 20 watts, which makes up the
difference. But of the 80 watts expended in turning the armature,
only a portion (of amount unknown) is spent in doing useful work —
the balance being required to 9vercome resistances to motion due to
eddy currents, hysteresis, and mechanical friction.

2. A motor whose resistance is 6 ohms is connected to a supply
at 100 volts, and it is found that 3 amperes flow through the motor.
Neglecting friction, and losses due to hysteresis and eddy currents in
the iron, find the amount of mechanical work done by the motor.

(Lond. Univ. B.Sc., 1908.)

TT o

As before, we have C = — — -

r

0 100 — em on ,,

.-. 3 = - ^_ 2, or em = 82 volts.

Hence, with the above-mentioned limitations, the power expended
in mechanical work is em x C = 82 x 3 = 246 watts. The "amount of
mechanical work" cannot be stated unless the time for which the
motor runs is given — the question in this respect being a little
ambiguous, confusing "work" with "power."

3. A battery of 50 volts and negligible resistance supplies a
current of 5 amperes to a motor at some little distance. The E.M.F.
across the terminals of the motor is found to be 35 volts. Find the
ratio of the heat lost in the leads to the energy supplied to the motor.

(B. of E., Day, 1908.)
The pressure required to send the current through the leads is

50 — 35 = 15 volts, and the power expended in them will be 15x5

= 75 watts.

The power supplied to the motor is EC = 35 x 5 = 175 watts.
Now, the heat produced in the leads per second is proportional to

the power expended therein, and the energy supplied to the motor

per second is measured by the power it takes in watts.

DYNAMOS AND MOTORS 545

, Heat lost in leads (measured as work) _ 75 _ 3

Energy supplied to motor 175 7

The question is, however, worded ambiguously, for in asking for
the ratio of "heat" to work, it should be stated clearly in what units
the "heat" should be expressed.

4. Enumerate the principal sources of waste of power in an electric
motor. Current is supplied to a series motor at 100 volts, the re-
sistance of the circuit being 0'5 ohm. Determine the power expended
in turning the armature when the current is 10 amperes. Determine
also the current when the power thus expended is a maximum. Com-
pare the values of the electric efficiency in the two cases.

(1) We have C = ^^

... 10 = 1UU e™, or em = 95 volts.

'O

Also, Power expended in turning armature = em x C = 950 watts.

Efficiency = Pol^e-IeIoPed . 5-^ _ fl. _ .95 95 cent.
(electrical) Power supphed ExC E 100'

(2) Now C increases as em decreases, and there will be some value
of C for which the product is a maximum. To determine this value,
we may wrrite

Power developed = em x C = ™^ — .

The numerator of this expression is the product of two quantities
whose sum is a constant, and therefore the product will be greatest
when the quantities are equal,

i.e. when em = E - em, or em = -E.

When this condition is satisfied, the motor is developing the
maximum power. In the case given, it evidently corresponds to a
current of 100 amperes. The efficiency, however, is only 50 per cent.

This theorem is often misunderstood. It is not of any practical
importance, for a motor is never worked at such a rate, and it would
probably be burnt out in a few seconds by the excessive current if an
attempt were made to do so.

EXERCISE XXV

1. How do magneto-electric machines differ from dynamo-electric machines?

2. Describe the construction and explain the action of a magneto-electric
machine for the conversion of mechanical work into current energy.

(B. of E., 1899.)
2M

546 VOLTAIC ELECTRICITY

3. Describe the two chief forms of armature in continuous current dynamo:- .

4. What is the meaning of " self-excited machines " ? Describe the various
methods of winding the field magnets.

5. A battery is employed to drive a magneto-electric engine. Does the
rate of consumption of zinc increase or decrease (and why) if the speed of the
engine is increased by lessening the work it has to do ?

6. Describe the general principles of the construction of a simple form of
dynamo. (B. of E., 1897.)

7. State the difference between a series, a shunt, and a compound dynamo.
Why is it not advisable to use a series dynamo for charging storage cells ?

(B. of E., 1905.)

8. Describe a drum armature. What are the advantages of this form of
armature over the Gramme armature. (B. of E.. 1906.)

9. Give a diagrammatic sketch of a shunt-wound motor. How will the
speed of such a motor vary under a given load when the resistance in the
field circuit is altered ? (B. of E., 1907.)

10. Describe, illustrating your answer with a diagram of the winding of
the armature, a motor of about 10 horse-power. If 90 per cent, of the energy
supplied is turned into useful work, what current would the above motor take
at 100 volts ? (B. of E., 1908.)

11. Describe any simple form of electric motor to work on a direct-current
system. (B. of E., 1910.)

12. In some forms of high-tension magneto machines (small machines
with permanent magnets used to produce a spark to fire the charge in internal
combustion machines) the armature has two sets of windings, one consisting
of a few turns of thick wire, and the other of many turns of thin wire, and
the spark occurs in the thin wire circuit when the thick wire circuit is inter-
rupted. Explain the action of such an instrument.

(B. of E., Stage III., 1910.)

13. A Gramme armature, intended for a 2-pole field, has 120 turns of con-
ductor wound upon it. Give the total flux required through armature core if
an electromotive force of 100 volts is to be induced at a speed of 1000 revolu-
tions per minute. (C. and G., 1894.)

14. It is found that a motor (shunt-wound, separately -excited, or series-
wound) that is supplied from mains at constant pressure runs faster if its
field magnet is weakened. Explain (1) the reason for this fact ; (2) what
arrangements you would make to weaken the field magnet in the case of each
of the three sorts of motors mentioned. (C. and G., 1896.)

CHAPTER XXXII

ELECTRIC LAMPS

Electric lamps may be divided into two classes —

(1) those in which the source of light is a solid body heated by
the passage of a current ;

(2) those in which the source of light is a gas or vapour rendered
luminous by a current.

The distinction is important, because in the former class the
efficiency of the lamp (i.e. the ratio of the candle- lower produced to
the watts supplied) is almost entirely dependent upon the tempera-
ture, although it is limited by the fact that, at all practicable working
temperatures, an enormous amount of energy is wasted in the form of
useless heat radiation. In the latter class, the emission of light is
not necessarily a question of temperature, and hence it is possible
to obtain a much higher efficiency.

In certain cases, e.g. in arc lamps, the emitted light is to some
extent due to both sources — chiefly to the first in the ordinary
"open" arc, and to the second in the more recent "flame" arc.

Incandescent or Glow Lamps. — Of all forms of lamp, these
are the most extensively used,, Evidently, they are to be included in
the first category, inasmuch as they are a direct and simple applica-
tion of the C2R or Joulean heating effect. It will be convenient to
summarise as follows the conditions which should be satisfied in
such a lamp in order to obtain the maximum efficiency.

(1) The surface area of the heated substance should be great
compared with its mass, in order that the radiating surface may
be as large as possible. This leads naturally to the use of a thin
filament.

(2) The material of the filament should permit of its being used
at extremely high temperatures, without too rapid disintegration.
The laws of temperature radiation are such that a small increase in
working temperature produces a relatively enormous increase in the
emitted light.

(3) As it is convenient to supply power to such lamps in the form
of a small current at a high voltage, the filament should have a high
resistance. Hence, it is an advantage if the material naturally
possesses a high specific resistance.

(4) The filament should be placed in an exceedingly good vacuum

547

548

VOLTAIC ELECTRICITY

in order to keep it from being destroyed by the impact of 1 he rapidly
moving gas particles. An additional advantage is due to the fact
that the loss of heat by convection is greatly reduced. (A badly
exhausted lamp is much hotter to the touch than a well exhausted
one.)

The first incandescent lamps, developed by Edison between 1878
and 1880, were made with platinum filaments. The chief dis-
advantage of this material arises from its comparatively low fusing
point, which means that it has to be worked dangerously near its
temperature of fusion to give out any useful amount of light.

About the same time, Swan introduced the use of carbon, which
has the advantage of infusibility. At high temperatures, however,
it softens and tends to volatilise, thus gradually blackening the bulb.
But so conspicuous did this advantage of carbon appear, that all
other substances were discarded, and until recently there seemed
little probability of its being superseded.

Fig. 387 shows a recent type of Edison
and Swan carbon filament lamp. The raw
material of the filament consists of cotton wool.
This is usually dissolved in a strong solution
of zinc chloride, and the viscous mass is then
" squirted " through a fine nozzle into a
hardening liquid (e.g. alcohol), which removes
the zinc chloride. The threads thus formed
are wound into shape, and gradually raised to
a very high temperature in crucibles packed
with carbon, to exclude the oxygen of the air,
the result being the well-known thin carbon
filament. During the operation of exhausting
the bulb, both it and the filament are raised
to a high temperature, so that any gases
F G 387 absorbed by the glass and carbon may be

expelled.

Life and Efficiency of Incandescent Lamps.— The candle-
power of a carbon filament lamp usually increases a little during the
early stages of use ; it then falls slowly but continuously, until at
last it becomes desirable to replace the lamp by a new one. The
" useful life " of the lamp is regarded as extending until the candle-
power falls to 80 per cent, of its initial value.

As we have just mentioned, the efficiency of the lamp is really

,, . ,, ,. Candle-power emitted . -,11-11

the value of the ratio TTT F — , i.e. it should be mea-

Watts supplied

sured in candle-power per watt, although as a matter of fact it has
become customary to express it in watts per candle-power.

It must be borne in mind that the normal working conditions are
a matter of compromise between high efficiency and long life. For

ELECTRIC LAMPS 549

example, a certain carbon lamp, marked 100 volts, gave 16 candle-
power, when worked at that voltage. It took -56 ampere, and
would have a useful life of about 800 hours, its efficiency being

12PX '56 = 3'5 watts per candle. When run at 77 volts, it took
16

•42 ampere and gave out 3 candle-power, its efficiency being

77 x *42

- =10°8 watts per candle. Under these conditions it would
o

have a very much longer life than in the first case. On the other
hand, at 112 volts it took '62 ampere and gave out 29 candle-power, its

efficiency being - - — - = 2°4 watts per candle ; but at this voltage

2li)

its life would be considerably shortened.

Under ordinary circumstances, the average efficiency of a carbon
lamp may be taken as about 4 watts per candle.

Resistance of the Filament. — As lamps are worked at a
definite voltage, it follows that the smaller the candle-power, the
higher must be the resistance of the filament, if the efficiency is to be
maintained constant. For example, if a 16 candle-power lamp works
at 100 volts and takes 64 watts, its resistance, when hot, must be

E2 (100^2

given by =64, orr=>- — A = 156 -25 ohms; whereas a 5 candle-
r 64

power lamp of the same efficiency would take 20 watts, so that we

have — = 20, or r = ± — -A. = 500 ohms. This extra resistance can

only be obtained either by reducing the section of the filament or by
increasing its length, and hence it is always more difficult to make
satisfactory lamps of small candle-power. This difficulty is still further
increased when the voltage is raised, e.g. suppose that the 5 candle-
power lamp is to work at 200 volts, then

= 20, or, = = 2000 ohms.

r 10

That is, if the voltage is doubled, the resistance must be quadrupled
in order to obtain the same efficiency.

Metallic Filaments.— Within the last few years, lamps having
metallic filaments have been introduced, which, apart from accidents,
combine a very long useful life with the extremely high efficiency of
from 1 to 2 watts per candle. The increase in efficiency is due to
the fact that, for a given useful life, they can be run at a higher
temperature than carbon. Evidently only metals of extremely high
melting-point can be used.

The first lamp of the kind was introduced by Welsbach in 1902.
Its filament was made of osmium, but owing to various practical
difficulties it never passed into general use. In 1905, Siemens

550

VOLTAIC ELECTRICITY

FIG. 388.

brought out the " tantalum " lamp, shown in Fig. 388, which really
initiated a new era. Since then many other metals and substances

have been tried, of which tungsten
has been found to be most satis-
factory, and, at the present time,
lamps made of either tantalum or
tungsten are almost universally used.
As the specific resistance of metals is
lower than that of carbon, the new
filaments have to be made longer and
finer than carbon filaments in order to
obtain the required resistance, and at
first they were very fragile. Although
much improved in this respect, they are
somewhat easily broken, and hence car-
bon lamps are still used in positions
subjected to much vibration.1 Again,
metal filaments differ from carbon in
having a positive temperature coeffi-
cient, i.e. their resistance increases with
temperature, which on the whole is a
distinct advantage. This fact is clearly
brought out in Experiment 165, p. 258.

Use of Transformers with Metal Lamps.— On account
of the smaller resistance necessary, it is easier to make thoroughly
satisfactory and durable lamps for low voltages (e.g. 25 to 50 volts)
than for the ordinary working pressure of supply mains, which is
usually either 100 or 200 volts. Hence, where the pressure is alter-
nating, small transformers are much used for house lighting, for the
purpose of lowering the voltage to suit such lamps. The efficiency
of the transformer is very high, and the small loss occurring in it is
more than counterbalanced by the increased durability of the lamps.

Nernst Lamp. — -In this lamp, advantage is taken of the fact
that non-metallic substances have negative temperature coefficients,
and hence may conduct fairly well at sufficiently high temperatures.
The appearance of one of the many patterns is shown in Fig. 389,2 and
a key diagram in Fig. 390. The source of light — termed the " glower "
— is a short and relatively thick rod, composed of certain metallic
oxides (chiefly zirconia, with a little yttria). It is non-conducting at
ordinary temperatures, but if connected to the supply mains, and then
heated by some auxiliary means, it at length conducts well enough to

1 Metallic filaments are more brittle when cold than when hot. Many
breakages would be avoided if the lamps were always switched on before
attempting to clean the bulbs or shades.

2 Figs. 389, 391, and 393 are taken from Slingo and Brooker's Electrical
Engineering.

ELECTRIC LAMPS

551

maintain its high temperature by means of the C2R heat developed in
it. For the initial heating a spirit lamp may be employed, but usually
a special heating coil (H) of fine platinum wire embedded in porcelain
is provided, and also a small automatic switch, S, which cuts out this
" heater " when the glower lights up. As, however, the resistance of
the glower continues to fall (and thereby to increase the current), its
condition is unstable, and without some regulating contrivance it would
soon be destroyed by the excessive current. This is provided by intro-

. 389.

FIG. 390.

ducing, in series with the glower, a spiral of very fine iron wire, I,
sealed up in a glass bulb containing hydrogen at low pressure. Now,
the temperature coefficient of iron (which is positive, like metals in
general) is remarkable for its sudden and abnormal increase at a
temperature near redness, and hence, if the conditions are such that
the iron wire is kept by the working current at a temperature just
below redness, it will be in its most sensitive state, and as its be-
haviour is exactly opposite to that of the glower, the resistance of the
lamp as a whole remains constant. In spite of the complicated

552

VOLTAIC ELECTRICITY

details, and of the loss of energy in the iron wire, these lamps are
fairly strong, and have the very high efficiency of about J watt per
candle. Perhaps their greatest disadvantage is due to the fact that
they do not light up instantly when the current is switched on. It
will be noticed that they do not require an exhausted globe, and they
also differ from incandescent lamps in working better, and in being
more efficient at 200 volts than at lower pressures.

The Electric Arc. — If two carbon rods, connected to some
source giving a steady pressure of about 50 volts, are brought into
contact and then separated by a short distance — say y1^- inch — the
current persists across the gap, and the carbon tips become intensely
hot, emitting a very white and brilliant light. In order to avoid too
great a rush of current, when the carbons are in contact, a resistance
of from 1 to 2 ohms should be included in the circuit ; in fact, such
a resistance is necessary for other reasons in order to obtain steadiness
of working. The P.D. between the carbons must not be less than
about 39 volts, and it need not be increased much above that value
unless a longer arc is required. The current strength is largely a
matter of choice, depending on the intensity of the arc required — in
ordinary lamps it is usually from 8 to 10 amperes. The low voltage
and relatively large current are highly characteristic of the arc
discharge, and distinguish it from the spark.

The processes occurring in the gap between the carbons are
analogous to those discussed in Chapter XXIX., the chief difference
being that the necessary supply of charged ions must be derived from
the electrodes themselves. The essential factor is the negative elec-
trode, which in all probability is the chief
source of ions. This electrode must be hot,
or the current cannot pass, whereas it does
not matter whether the positive electrode is
hot or not ; in fact, in some recent lamps,
it is made of copper massive enough to
remain fairly cool, and which works for a
long time without appreciable loss. At the
same time, with carbon electrodes, the posi-
tive carbon actually becomes the hotter, and
hence burns away more rapidly, most of the
light emanating from a small hollow or
" crater," which forms at its tip, whilst the
negative carbon remains pointed and is less
luminous (see Fig. 391). With alternating
currents, both carbons naturally behave alike,
the crater is not formed, and the efficiency
as a source of light is much decreased. The
arc is, in fact, not at its best with alternating
currents ; under such conditions, it dies out at each reversal, and has

FIG. 391.

ELECTRIC LAMPS 553

to start again in the opposite direction before the bridge of vapour
ceases to conduct.

Carbon electrodes are not essential to the formation of the arc
discharge, although they are almost universally employed for lighting
purposes. With direct currents, arcs may readily be formed between
electrodes of iron or of other metals, but not so readily in the case of
metals of lower fusing point and greater affinity for oxygen. With
alternating currents, it is always difficult, and as a rule almost im-
possible, to run metallic arcs.

Theory Of the Arc. — The various actions going on in the arc are

somewhat complicated, and we can only briefly indicate certain general
results as follows : —

1. When the current strength is kept constant, the relation between the
P.D, across the arc (E) and its length (1) is given by E = a + U, where a and b
are constants for electrodes of a given material. Evidently a is the minimum
voltage required to form the arc at all, and, as already stated, this is about
39 volts with carbon electrodes and a direct current.

2. When the length of the arc is constant, the relation between the P.D.

and current is given by E = m + ^-' where m and n are constants under fixed

C

conditions. This peculiar relation between voltage and current means that if
E increases, C decreases, and vice versd. Hence, the arc does not obey the
simple form of Ohm's law, i.e. it does not act as if it possessed a definite
resistance.

Very important methods of obtaining electrical oscillations of high fre-
quency depend upon this property of the arc (see p. 615).

Arc Lamps. — The distinguishing feature of practical lamps lies
in the mechanism employed to control the distance between the carbons
and to "feed" them together as they wear away. For descriptions of
the many forms in actual use, the student must consult special treatises.
Here it is only possible to mention briefly certain distinct types of
lamps.

Open and Enclosed Arcs. — So far, we have referred to
arcs working with free access of air, and, until about 1895, all lamps
were of this kind. Under these conditions, however, there is a rapid
loss due to combustion, and, as a consequence, the carbons must be
renewed frequently. To reduce this loss, they may be enclosed in a
semi-airtight globe, in which case they become surrounded by the
products of combustion. As the ready access of oxygen is thus pre-
vented, a pair of carbons last a much longer time, and the lamps
therefore require less attention. This construction also lends itself
to the working of longer arcs with a larger P.D. between the carbons,
and it was introduced mainly to make it possible to run a single arc
from 100- volt supply mains. The great disadvantage of enclosed
arcs is their relatively low efficiency.

Flame Arcs. — In the types previously mentioned, the light is
almost entirely derived from the white-hot carbon tips, the arc itself
giving merely a feeble bluish light. In the " flame " arc, the carbons

554 VOLTAIC ELECTRICITY

are impregnated with a large proportion of metallic salts (chiefly of
calcium), and the vapours thus produced become intensely luminous,
the colour of the light depending on the composition used. The
carbons themselves are usually of relatively small diameter, in order
that they may become thoroughly and uniformly heated at the tips,
and, in the majority of lamps of this type, they are placed side by
side, like the letter V. As the chief source of light is a luminous gas
and not a solid, the conditions which limit the efficiency of light
- sources of the latter type are avoided, and a greatly increased efficiency
is obtained, although the carbons burn away very rapidly. In round
numbers, the efficiency of an open direct-current arc is about *5 watt
per candle ; of an enclosed arc, about 1 watt per candle ; whilst that

of a flame arc may be ^ watt
per candle. The general ap-
pearances of the various types
of arc are shown in Fig.
392, adapted from Electric
Light and Power (Brooks and
James : Methuen), in which
(1) is an open arc, having as
usual the positive carbon

^ ' uppermost and of greater

diameter than the other.
An open alternating arc is

shown in (2) ; while (3) gives the characteristic appearance of any
form of enclosed arc. It will be noticed that the carbons are farther
apart than in (1) and (2), and as oxygen is excluded, they do not
round off at the ends. In (4) the arrangement adopted with flame
arcs is seen, the carbon tips passing just inside a small shallow
vessel known as an " economiser," which steadies the arc by ex-
cluding draughts, and also serves as a radiator. If left to itself, the
discharge would not remain at the bottom ; it would at once ascend,
and either go out as the gap widened or do mischief in the frame-
work, and hence an important part of the lamp mechanism is a
simple electromagnet, excited by the working current, which creates
a magnetic field at right angles to the discharge in such a direction
as to force the latter downwards, and of such strength as to keep it
well extended without actually blowing it out.

Magnetite and Titanium Arcs. — In America, magnetite
arc lamps have been introduced with some success. In these, the
electrodes are placed vertically, as in ordinary lamps, the lower one
(negative) being a thin tube of soft iron packed with a mixture of
magnetite (Fe3O4) and titanium oxide, and the upper (positive) being
a massive copper rod. This rod is practically unaffected, the loss
occurring entirely at the negative electrode. The light is derived
from the vapour of iron and titanium, the idea being to obtain the

ELECTRIC LAMPS 555

advantages of the flame arc with more durable electrodes. These
lamps are very efficient, and run a long time without attention. They
are, however, inclined to flicker, and chiefly on this account they have
not been much used in' this country,,

Mercury Lamps. — These are vacuum tubes with mercury
cathodes, the discharge really being a very long arc in mercury
vapour. One of the best-known forms is the Cooper-Hewitt lamp,

shown in Fig. 393.
The long glass tube
is fixed to a support,
which holds it in a
slanting position, and
is capable of being
tilted. In the normal
position the mercury
— which forms the
FIG. 393. cathode — is in the

bulb, B, making con-
tact with a platinum wire sealed through the glass. The anode,
A, is iron. To start the lamp, it is tilted until the mercury runs
slowly from B to A, thus momentarily completing the circuit, and,
when the lamp is allowed to return to its original position, the
discharge persists through the vapour. As usual, when the con-
ductor is gaseous, the current does not obey Ohm's law in its
simple form, and hence it is necessary to include a steadying
resistance in the circuit (not shown in the figure). The diameter
of the tube is 1 inch and the length 40 inches for a working
pressure of 100 volts and a current of 3J amperes. For running
at 50 volts, the length is halved, the diameter and the current
remaining unchanged.

Originally the tubes were made of glass, but because glass is
readily fusible, they are now frequently made of quartz. In the
latter case, a strong smell of ozone is produced after the lamp has
been working for a short time. This is due to the fact that the light
is extremely rich in ultra-violet rays, and as quartz (unlike glass) is
very transparent to such rays, they pass through it and produce ozone
in the surrounding air. To avoid this action, as well as to obviate
the injurious effect of ultra-violet light on the eyes, the quartz tube
is surrounded by a larger one of flint glass.

Mercury lamps are, perhaps, the most efficient sources of light
known, although the colour of the light is very unpleasant, owing to
the almost complete absence of red rays. Hence, such lamps have
at present a very restricted application.

The " Moore " Light. — This method of illumination has been
used to some extent in America and elsewhere, although it seems
very unlikely to meet with general acceptance at present. The " lamp "

556 VOLTAIC ELECTRICITY

is really a long vacuum tube, built up in the room to be illuminated
by welding together lengths of glass-tubing. In some cases, the total
length is as much as 70 metres. This tube contains either nitrogen or
carbon dioxide, according to the colour required (the former giving
a reddish, and the latter a fairly white light). It takes about
•3 ampere at a pressure of 12,000 volts, which is supplied from
alternating mains by a transformer of special form. A very complete
system has been worked out, one ingenious device being a valve which
admits a small quantity of gas, as it is required, to keep the pressure
constant (for the vacuum in all tubes tends to improve with long
running). The method may be regarded as an attempt to obtain
light without a simultaneous production of useless heat, and is
undoubtedly excellent in principle. The losses, however, which
necessarily occur in the transformer, choking coils, &c., are sufficiently
great to lower the efficiency to something like 1*5 watts per candle,
while the complicated details required for high voltages and the
difficulties of making repairs in the case of breakage are very serious
objections.

It may be remarked that the nature of the gas used has an
important influence on the efficiency. For the same consumption of
energy, nitrogen gives out more light than carbon dioxide ; and some
recent researches appear to show that neon is superior in this respect
to nitrogen.

Meaning of the term " Candle- Power. "—We have used
this term in the last few pages without comment, as being sufficiently
definite for its purpose. It is, however, quite obvious that light
sources in general do not radiate uniformly in all directions, so that
their candle-power will depend upon the particular direction in which
it is measured. For this reason, no exact meaning can be attached
to the term when used in a general sense, and it has, therefore, become
customary to express the power of a lamp in terms of either its "mean
spherical candle-power," or its " mean hemispherical candle-power."
The former may be defined as the candle-power of an ideal lamp,
which would give out the same total amount of light emitted uniformly
in all directions. In the latter, the light emitted above a horizontal
plane is ignored, and it is, therefore, the candle-power of an ideal
lamp in which the emission is uniform over a hemisphere.

EXERCISE XXVI

1. Write a short essay on the incandescent electric lamp, dealing par-
ticularly with any improvements which have been made within the last few
years. (B. of E., 1910.)

2. State the conditions under which the electric arc can be formed and
maintained between two carbon electrodes, and describe carefully the differ-
ences in shape, temperature, and rate of consumption of the two electrodes.

(B. of E., 1903.)

ELECTRIC LAMPS 557

3. A dynamo feeds 1000 16-candle-power lamps. What current must the
dynamo supply, if the difference of potential at its terminals is 200 volts, and
each lamp absorbs 3'G watts per candle ? (B. of E., 1903.)

4. Find the cost of running 20 16-candle-power lamps for 6 hours, if each
lamp requires 3 '6 watts per candle, and if each Board of Trade unit costs 4d.

5. Describe what arrangements you would make if you wanted to run a
single 50- volt incandescent lamp off a 110-volt circuit. If the lamp takes
*5 ampere, how much power is taken from the mains, and how much power
is absorbed by the lamp 1 (B. of E., 1911.)

CHAPTER XXXIII

MEASURING INSTRUMENTS

IN the preceding chapters, various forms of measuring instruments
have been repeatedly referred to and assumed to be available. We
must now briefly outline the elementary principles on which their
construction is based.

Voltmeters and Ammeters. — A voltmeter is an instrument
with a scale graduated to read in volts the RD. between its terminals,
i.e. the P.D. between any two points to which it is connected by
wires of negligible resistance. The ideal voltmeter should have an
infinite resistance, for, if it takes an appreciable current, it may in
certain cases alter the P.D. previously existing between the points
to which it is connected, and, apart from this, many simple measure-
ments depend upon the assumption that a voltmeter current is small
enough to be negligible (see, for example, pp. 309 and 310).

An ammeter is an instrument graduated to read in amperes the
strength of the current flowing through it. The ideal ammeter
should have no appreciable resistance, otherwise its insertion in
a circuit may alter the strength of the current previously flowing.

Both voltmeters and ammeters should be as dead-beat as possible.

Any property of a current (or charge), which can be made to pro-
duce motion of a pointer may be employed, the most important being—

(1) Electrostatic attraction between a movable and a fixed

conductor.

A coil moving in a magnetic field.
Soft iron moving in a magnetic field.

(4) Expansion of a fine wire, by the heating effect of a current.

This list is by no means exhaustive, but it contains all the types
we need consider here.

(1) Electrostatic Instruments, of which the electroscope
and the quadrant electrometer may be regarded as types, are in
practice limited to voltmeters. They are ideally perfect in having
infinite resistance, and they also possess the very great advantage
of working equally well on both direct and alternating circuits. But
the force of attraction on which they depend is relatively feeble, and
hence there is some difficulty in making instruments sufficiently
sensitive to read low voltages, without being too delicate for ordinary
commercial use. Lord Kelvin's multicellular voltmeters are perhaps

558

MEASURING INSTRUMENTS

559

the best-known instruments of this class. One of the latest pattern,
with part of the outer case removed, is shown in Fig. 394.1 The
moving system is a series of light metal vanes, V, mounted one above
the other on a vertical rod, suspended by a fine wire, W. The
bottom of the rod ends in a loop of wire (visible in the diagram),
which dips into a small vessel containing a suitable liquid for damping
the vibrations. To the top of the rod is attached a pointer, P, moving
over a scale. In some patterns this scale is horizontal, but in the
type illustrated it is vertical, in order to be readily seen from a

FIG. 394.

distance. When the pointer is at zero, the vanes are partly within
a series of horizontal fixed plates or shelves carried by the metal
plate, C. When the moving vanes and the fixed plates are connected
respectively to two points between which a P.D. exists, lines of
electric force pass from one to the other (i.e. they become charged,
respectively positively and negatively). Hence, the attraction causes
the movable vanes to approach the fixed ones, thereby setting up a
rotation controlled by the torsion couple due to the suspension. It is
usual to connect the moving part of the framework and to insulate the

1 Figs. 394, 395, 396, and 398 are taken from Slingo and Brooker's

Electrical Engineering.

560

VOLTAIC ELECTRICITY

fixed plates only. These instruments may be regarded as derived from
the quadrant electrometer by multiplying the parts, and, like it, they
are evidently condensers with one movable coating. Unfortunately
they are not very dead-beat, which is sometimes an inconvenience.

(2) Moving Coil Instruments. — The principle of these instru-
ments is similar to that embodied in the D'Arsonval galvanometer
(p. 293). A steel horse-shoe magnet is provided with cylindrical
soft-iron pole-pieces, and the polar space is very nearly filled up by
a fixed cylindrical core of soft iron. A small coil, carrying a pointer,
is pivoted to move freely in the narrow annular gap, its motions
being controlled by two springs, similar to watch-springs. AVhen

FIG. 395.

the instrument is used as a voltmeter, a large non-inductive resistance
is placed in series with the coil ; while, as an ammeter, the latter
is shunted with a conductor of low resistance and of sufficient section
to carry (without undue heating) the largest current to be measured.
The general arrangement is shown in Fig. 395, in which the soft-
iron pole-pieces have been removed in order to show the coil and the
core more clearly.

Instruments of this type are convenient, very accurate, and the
scale is remarkably uniform. They are unsurpassed for direct-current
work, but as the direction of the deflection depends upon the
direction of the current, they cannot be used on alternating circuits.

(3) Soft-iron Instruments. — A very small piece of thin soft

MEASURING INSTRUMENTS

561

iron, attached to a spindle carrying a pointer, is pivoted inside
a small coil. Motion may be obtained in several ways ; perhaps
the method most frequently employed is to place a fixed piece of iron
in the coil, so that the two pieces become magnetised with the same
polarity, and therefore repel each other. As in the previous case,
a large non-inductive resistance is placed in series with the coil when
it is to be used for a voltmeter. An ammeter is often wound with a
coil of thick wire of low resistance.

Instruments of this type possess several advantages. They will
stand much rough usage without injury, and can be made to work

FIG. 396.

on either alternating or direct-current circuits. But they are natur-
ally less accurate than the moving-coil type, and, in consequence
of hysteresis, they are liable to read too low with gradually increasing
currents, and too high with decreasing ones. Again, the scale is not
uniform, for the force on the soft iron depends upon the square
of the current, and therefore the deflection is also proportional to the
square of the current (although this effect can be partially com-
pensated by the details of construction). A well-known form is
shown in Fig. 396, which represents the moving part of an instru-
ment made by Messrs. Everett, Edgcumbe & Co. A spindle, pivoted
in a brass frame, carries a pointer and also a half cylinder of thin soft

2N

562 VOLTAIC ELECTRICITY

iron (shown in the sectional view), and is balanced by a small
adjustable weight, seen at the side of the pointer in the figure.
Round the outside of the brass frame is fixed another strip of soft
iron, bent round until the edges nearly — but not quite — meet, and
cut away very much on that side. The whole arrangement is re-
markably small and compact, being about 2 inches long and J inch in
diameter. This is surrounded by a coil of wire, which, when
traversed by a current, produces an axial magnetic field passing
longitudinally through both pieces of iron. If we regard them as
becoming magnetised with like polarity, it is evident that the mov-
able half cylinder will tend to set itself so that its ends are as far as
possible from the ends of the fixed piece, i.e. it tends to take up the
position shown in the figure.

(4) Hot- wire Instruments. — In this very important class of
instruments, a fine wire (usually of platinum-silver alloy) under
tension carries a current, and the slight increase in length due to the
consequent rise in temperature (magnified by various devices) is
made to move a pointer over a scale. As usual, voltmeters are
provided with a large non-inductive resistance in series with the hot
wire, and ammeters have the latter in parallel with a suitable shunt.

The great advantage of these instruments is due to the fact that
they read as correctly on alternating as on direct-current circuits.
They are also remarkably accurate. Their chief drawback is a ten-
dency to " burn out," if by some mischance they are slightly over-
loaded. As the heating effect is proportional to the square of the
current or of the voltage, the scale is not uniform.

Ammeters and voltmeters of all types are calibrated by sending
known currents through them, or by applying a known P.D. to their
terminals, the values being thus found by actual trial throughout the
whole of the scale. Such calibrations are readily carried out by
means of a potentiometer, but space does not permit us to describe
the methods used in practice.

Dynamometers and Current- Balances. —In this category
must be placed an extensive range of instruments depending upon
what is known as the " dynamometer principle," i.e. the mutual
forces between conductors carrying currents, when iron is not present.
To some extent, this principle has been applied to the construction
of direct-reading ammeters and voltmeters, but its great importance
is due to the fact that it lends itself readily to the construction of
wattmeters and current-balances, which may be used on either alter-
nating or direct-current circuits.

Siemens' Dynamometer (shown in Fig. 397) was the first
really practical instrument of the kind. It was devised for the
measurement of alternating currents, and although now seldom used
in its original form, it is worth studying as a good illustration of the
principle in question.

MEASURING INSTRUMENTS

563

The construction is shown diagrammatically in Fig. 398, where
ABCD is a fixed coil composed of comparatively few turns of stout
wire, the section depending upon the minimum current to be measured.
At right angles to it is a movable coil, GEFH (which is usually one
turn of thick copper wire), suspended by a silk thread from a
specially made terminal forming the " torsion-head." A spiral spring,
surrounding the thread, is connected below to the movable coil and
above to the torsion-head, so that the thread carries the weight of
the coil whilst the spring controls its position — the arrangement
being such that the upper end of the spring can be rotated without

j t

FIG. 397.

FIG. 398.

twisting the top of the suspension. The lower ends of the movable
coil dip into mercury cups (not shown in the diagram), thus making
good contact and yet allowing freedom of motion, and the two coils
are connected in series. Remembering that currents in the same
direction attract, and in opposite directions repel, it will be evident
that the movable coil tends to move round until it is parallel to the
fixed coil, and it will also be seen that the direction of motion is
unaltered when the direction of the current is reversed. The torsion-
head carries a pointer (shown in Fig. 397) which moves over a scale
of degrees (or any .convenient scale of equal parts). The movable
coil also carries a pointer (also shown in Fig. 397) which passes

564

VOLTAIC ELECTRICITY

upwards and bends just over the same scale, its motion (and, of
course, that of the coil) being restrained by two stops.

Normally, both pointers are at zero, and when a current passes,
the coil moves until its pointer comes in contact with one of the stops.
The torsion-head is then turned in the opposite direction, thus
putting a twist on the spiral and bringing the coil back to its exact
zero position. The angle of rotation required for this purpose is
then read off on the scale. Let this be d divisions. Then d is a
measure of the force between the coils, and this can be shown to
be directly proportional to the product of the currents in them.
Hence, we have, if C is the current in each,

C*~d .'. Goc Jd

which means that we can write C = m, Jd, where m is a constant for
a given instrument. Evidently the value of m can be found by
taking a single reading with a current of known strength (although
it is, of course, better to confirm the result by taking a number of
readings).

The importance of this instrument is due to the fact that when
thus calibrated by means of direct currents, it reads correctly with
alternating currents.1 On the other hand, it is not direct reading
and not very portable.

Application to Wattmeters.— Let the two coils of the dyna-
mometer be fitted with separate terminals. The fixed coil need not

be altered, but the movable coil
should preferably consist of a
few turns of fine wire, and should
be connected in series with a
large -non-inductive resistance.
(More turns are needed to give
sufficient sensitiveness, because
the current in this coil will now

••MIPMM^MMMMMM^HMH^^

B

FIG. 399.

®Ayyyy\ C
3> " " "

J

be very small ; in any case, as
few turns as possible are used.) |
To measure the power ex- j
pended in any part of the circuit, the thick wire coil is connected in j
series with the circuit (like an ammeter), and the movable coil, with i

1 It may be pointed out that any instrument which is to read with equal j
accuracy on either direct or alternating currents must satisfy two conditions :|
(1) The direction of motion must be independent of the direction of current, |
and (2) it must be practically non-inductive. Both conditions are satisfied ini
Siemens' dynamometer: (1) we have already mentioned, and (2) because it
contains few turns and no iron. Hot-wire instruments also satisfy both
conditions. The moving iron instruments satisfy condition (1), but as a rule;
only partially satisfy condition (2), and hence their readings depend upon the
frequency, and they are correct only at the particular frequency for which|
they have been calibrated.

MEASURING INSTRUMENTS 565

its resistance, is connected across that part as if it were a voltmeter.
The arrangement is shown in Fig. 399, where the current is supposed
to be derived from terminals TT, and the part of the circuit in which
the power is to be measured is indicated by the word "load."

In the case of direct currents, the theory is very simple. If E be
the RD. across the mains, C the current through the load and the
rixed coil, and C^ the current through the movable coil and its
resistance, we have

Power = EC watts (expended in load and fixed coil).

Also, as the reading of the torsion -head depends on the currents in
each coil, we have

E

but Cj = — , where R is the resistance between
K

A and B (Fig. 399), i.e. practically the non-inductive resistance.

CE ,
•'• IT0**
or «EC oc Ud
or EC = m. Rd.

Now, m is a constant for the given instrument, and as R is also
constant, we may write m.R = M ; therefore

Power = EC = M x d

This includes the power expended in the fixed coil, which, however,
is usually negligible. The value of M can be found by taking a
reading with known values of E and C.

With direct currents, the same result could be obtained equally
well by using a voltmeter and an ammeter, but with alternating
currents we know that the product of volts and amperes (often called
the " volt- amperes ") does not give the true power.

Application to the Measurement of Power in Alter-

nating" Circuits. — Suppose that the load is inductive, then C lags behind
E by some angle 6, and the true power is EC cos 6. Again, as the shunt-path
between A and B (Fig. 399) is almost entirely made up of the non-inductive
resistance R, the current Cx must be practically in step with E. Therefore, C
lags behind Cl by the same angle 6. Now, it can be shown that, when the
coils of a dynamometer are traversed by currents which are out of phase
with each other, the force between them is proportional to the cosine of the
angle of phase-difference (and therefore vanishes when that angle becomes 90°),
Hence, in the present case, we have, if d is the scale reading,

CCX cos 6 oo d,
•pi

but G!=— , as R is non-inductive.
B

566 VOLTAIC ELECTRICITY

EC cos 6 j

-§—~d

or EC cos 6 or ~Rd

or True power =m x d, where m is some constant
to be determined by experiment (using a known power).

In this simple proof, we have assumed that E and C follow a sine law,
but fuller investigation would show that this is not an essential condition ;
and hence wattmeters may be used on alternating circuits without reference
to the shape of the wave curve.

In practice, it is necessary to take care that no masses of metal are near
the coils, otherwise eddy currents may be induced in them, and large errors
thereby occasioned. Hence, metal covers should be avoided.

In any case, the argument given above is only approximate!}7 true, for the
fine-wire coil must have some self-induction. The nature of the error thereby
introduced can be readily investigated to a certain extent, for we know that
Cj really lags behind E by some very small angle a. Then the phase-
difference between C and Cx is 6 - a.

Hence. CCj cos (0 - a) oa d
and C = Ecsa

.'. EC cos a cos (6- a) oc ~Rd=md.

The power deduced from the readings is therefore EC cos a cos (6 - a),
whereas the true power is EC cos 0.

Power given by reading cos a cos(0*- a) cos a(cos 6 cos a + sin 0 sin a)
True power cos 0 cos 0

= cos2 a + cos a sin a tan 0

If this expression is equal to unity, the instrument will read correctly.

Now, as a is very small, the first term is practically unity. Sin a is
also small, but if 0 be great, tan 0 is very large, and the second term may
not be negligible. Hence, the more inductive the load, the less accurate
become the readings. (It may be pointed out in passing that the above
expression becomes unity, not only when 0=0, but also when 0 = a. This
fact is, however, of no practical importance.)

Our argument has tacitly assumed that the resistance R is not only
non-inductive , but is also destitute of capacity. If R be made non-induc-
tive in the same way as coils in ordinary resistance boxes (i.e. by doubling
the wire upon itself, and then winding it upon a bobbin), it may have
sufficient capacity to introduce an error of considerable magnitude ; for
doubling the wire upon itself has no influence on the capacity, which
depends mainly upon the contiguity of the layers in the winding.
Resistances for use with wattmeters are never wound in that manner,
but in such a way as to open up the winding as much as possible.

Measurement of Iron Losses. — Suppose that the load in
Fig. 399 is a coil of wire, with an iron core, and that it is supplied
with an alternating current from the terminals TT. The wattmeter
reading gives at once the power supplied to the coil, which must
be expended as (1) C2R heat, (2) hysteresis, (3) eddy currents.
The first source of loss is often negligible, and in any case it is easily
estimated by a measurement of current and resistance. The actual
" iron loss " is thus obtained in watts. The eddy-current loss can be

MEASURING INSTRUMENTS

567

made small by laminating the -iron, but it is only possible to separate
(2) and (3) by measuring the loss at different frequencies, and
making use of the fact that (2) varies directly as the frequency,
while (3) varies as the square of the frequency.1

What is required in practice, however, is not merely a knowledge
of the iron loss, but the relation between that loss arid the maximum
value of B in the iron, i.e. we must make simultaneous measurements
of the iron loss and B. Now, in discussing the theory of the trans-
former, it was shown that the value of B in the core can be deduced
by measuring the P.D. at the secondary terminals on open circuit,
when the area of section of the core and the number of secondary
terms are known. Hence, one of the best methods of testing iron
stampings for hysteresis loss is to make up a definite weight of them
as the core of a simple form of transformer which is designed to be
readily fitted with temporary cores, and to measure simultaneously
the iron loss by a wattmeter and the P.D. at the terminals. In this
way, the loss in watts per Ib. of iron at a given induction is obtained.
Current-Balances. — Instruments of this type have been largely
used for the absolute measurement of current ; for instance, by Lord
Rayleigh in his important research on the electro-chemical equivalent
of silver. One of the latest instruments of the kind has recently
been set up at the National Physical Laboratory. Two coils with
their axes vertical are suspended from the arms of a delicate balance
within vertical marble cylinders, on each of which two coils are
wound. The general idea may be gathered from Fig. 400, which

is merely a diagram, and in no
way represents the actual details
of construction. AAX and BBj
are the two pairs of fixed coils,
and CCj the two movable coils
suspended from the arms of a
balance. The current to be
measured passes through all six
coils in series in such directions
that the forces act together to tilt
the beam the same way, and the
total force can be measured with
great accuracy by adding weights
until the balance is again in equilibrium. From what we have already
said, it is evident that this force varies as the square of the current, and
its value in terms of the current can be calculated from the dimensions
of the apparatus without reference to any other electrical quantity.

1 The induced E.M.F. producing eddy currents varies directly as the fre-
quency, and the power wasted as heat in a path of constant resistance varies

"R"2

as the square of E.M.F. (for watts = ^-). Hence, eddy current losses are pro-
portional to (frequency)2

FIG. 400.

568

VOLTAIC ELECTRICITY

As the reversal of the current in all the coils does not alter the
direction of the force, such an instrument can be used in connection
with alternating currents.

Kelvin Balances. — These are instruments capable of measuring
currents with great accuracy, and yet not too delicate in construction
for use as commercial standards. If in Fig. 400 we suppose that
the coils CCj are carried by a horizontal lever suspended at the
middle, and the tilt, due to the passage of a current, balanced by

. 401.

sliding a rider along the lever, the result will embody the principle
of a Kelvin balance. These instruments are made in several forms,
designed to measure currents from T^Q- ampere up to 1000 amperes.
Fig. 401 shows the appearance of the " centi-ampere " balance, whose
effective range extends from T^ ampere to 1 ampere, and also of
the " deci-ampere " balance for currents between J^ ampere and
10 amperes. Fig. 402 (from Aspinall Parr's Practical Electric Testing)

indicates ^can-ange-
ls ment diagrammati-
cally. In the latter
figure, aa is a light
frame, which forms

FlG 402 the lever carrying

the coils BB1, and a

triangular scale-pan, t, in which a weight, W, can be placed. At
each end is a pointer moving over a small fixed vertical scale —
only one of which, however, is shown in the figure at I. AA1

MEASURING INSTRUMENTS 569

and CC1 are the two pairs of fixed coils, and, as in the instrument
just described on p. 567, the current passes in series through the
six coils in such directions that the various forces act together to
tilt the lever (the left-hand side moving downwards). In order
to neutralise the effect of the earth's magnetic field, which would
of itself produce a tilt, the current flows clockwise in one moving
coil and anti-clockwise in the other. To the whole length of
the lever is attached a light L-shaped bar of aluminium, the lower
divisions forming a shelf and the upper edge having a scale of equal
divisions engraved on it. On the shelf slides a carriage, drawn to and
fro by means of threads. To this carriage is attached a pointer,
by which its position on the scale can be read off. When the
pointer is at zero and no current is flowing, the total weight of
carriage and pointer balances the weight placed in the scale-pan at
the other end, so that the arrangement is then equivalent to a
loaded and balanced lever.

In Fig. 401, the scale-pan is visible just below the upper fixed
coil on the right ; the sliding carriage and pointer are seen in front
of the graduated movable scale, i.e. the lower scale. After the
sliding carriage has been placed exactly at zero, the beam is made
horizontal (as indicated by the pointers at the ends) by means of
a rider worked by the handle shown under the base of the instru-
ment. At the back of the moving scale there is a fixed scale (i.e. the
upper scale in the figure) not divided into equal parts. The numbers
marked thereon are the doubled square roots of those coinciding with
them on the lower scale (which are marked 1, 2, 3, &c. for 10, 20,
30, &c.) ; for instance, on the fixed scale 50 is visible near the right-
hand end, which is the doubled square root of 625, coinciding with
it on the movable scale. This doubling of the square roots is
merely a matter of convenience, the essential point being that the
readings on the fixed scale do not give the distance from the zero,
but are proportional to the square root of that distance.

When the beam is tilted by the passage of a current, obviously
equilibrium could be restored by diminishing the weight of the
carriage, W, but it is more convenient merely to slide the carriage
into some new position at distance d from its zero. Then it is
evident that W x d balances the new couple due to the current, and,
as already mentioned, this couple is proportional to C2

Hence, we have C2 oc Wd

or C oc jWd
i.e. C = m ijd, where m is some constant.

Thus the current is proportional to the square root of the distance
through which the carriage is moved to restore equilibrium, i.e. to
the reading on the fixed scale. This reading can be taken at a

570 VOLTAIC ELECTRICITY

glance, and, when multiplied by a constant supplied by the maker,
gives the current with sufficient accuracy. When the greatest
accuracy is required, the reading on the lower scale is taken with
the aid of a lens, and the corresponding number on the upper scale
more exactly determined by reference to a table of doubled square
roots provided for the purpose.

In order to extend the range of these instruments, it is usual
to supply four pairs of weights, each pair consisting of one for the
scale-pan and one for the carriage.1

These weights are in the ratio 1 : 4 : 16 : 64, and from the pre-
ceding argument, it follows that the currents corresponding in each
case to one division of the fixed scale are in the ratio 1:2:4:8.
For instance, in the size known as " the ampere-balance," there are
50 divisions on the fixed scale, and with the smallest pair of weights
each division is equivalent to °25 ampere. Hence, when using this
pair of weights, currents up to 12 -5 amperes can be measured. With
the second pair, each division represents *5 ampere, and the range
is from 0 up to 25 amperes. Similarly, the third pair gives 1 ampere
per division ; and the fourth, 2 amperes per division — the total
effective range being from about 1 ampere to 100 amperes.

The ceuti-ampere balance, with a range of from T^ ampere to
1 ampere, serves equally well for the measurement of voltage. Its re-
sistance is about 60 ohms, and if this is known exactly and multiplied
by the current as measured, the result is the P.D. between the terminals
of the instrument. In practice, special resistances are supplied for
connecting in series with the balance in order to increase its range.

Special instruments are also made for use as wattmeters ; in
these, the fixed coils are of thick wire, and the movable coils of
thin wire, the latter being provided with a separate non-inductive
resistance. The connections and the theory are identical with those
explained on pp. 564 and 565.

One of the most important details in these instruments is the
ingenious way in which currents of any magnitude are taken in and
out of the moving coils without interfering with their freedom of
motion. This is the great difficulty with all dynamometers, mercury
cups being a clumsy and troublesome expedient which is fatal to
high accuracy.. In a Kelvin balance, the frame of the lever is fixed
to two metal bars, each suspended by a large number of very fine
copper wires from two other fixed bars. This provides a soldered
contact which can be made to carry large currents and is yet extremely
flexible throughout the very small range of motion required. It
was, in fact, the invention of this suspension which made these
instruments possible.

1 The carriage itself balances the smallest weight. Hence, there are four
weights made to fit the scale-pan, and three in the form of riders to attach to
the carriage.

MEASURING INSTRUMENTS 571

Oscillographs. — An oscillograph is a variety of galvanometer,
m which the deflection at any instant is proportional to the current
strength at that instant, no matter how rapidly the current may be
varying or alternating. It is used to obtain the actual shape of
alternating current curves ; to trace the rate of rise or fall of a current
under various conditions ; to record the number and amplitude of the
oscillations set up by a spark discharge ; and, in fact, to obtain exact
data regarding the manner in which an E.M.F. or a current changes
in strength under all possible conditions.

In order that the moving part may faithfully and instantaneously
follow such extremely rapid changes, it must satisfy certain conditions :
(1) its own natural time of vibration must be exceedingly small
compared with the periodic time of the current to be investigated
(say gL of the latter) ; (2) its motions must be very exactly dead-beat.
The first condition implies that the moment of inertia of the moving
system is small and the controlling couple relatively great; the
second can be satisfied by a suitable method of damping.

The first instrument of the kind was devised by Blondel in 1892.
More recently, Duddell has applied the principle of the suspended-
coil galvanometer with great success, and it is chiefly to him that
we owe the development of the oscillograph as f

a really practical instrument. In his form, the
permanent magnet shown in Fig. 229 is replaced by
a powerful electromagnet, and the moving coil is
reduced to a single loop kept under tension by a
spring. This is shown diagrammatically in Fig. 403,1
where M represents a very small mirror^ attached to
the loop between the pole-pieces, NS. The necessary
damping is obtained by immersing the whole loop
and mirror in oil.

In order to obtain the wave-form, two methods
may be employed. In one, the spot of light, reflected

from the mirror, is received upon a photographic '" I I I I

film or plate moving vertically with uniform velocity, *^ 1 1_ *£ J |^

thus giving a permanent record. In the other, the

light from the mirror, M, falls first upon another

mirror, which is rocked by a small motor in exact

unison with the motion of M, and in a plane at

right angles to it. Thence, the light passes to a

screen, on which it is seen — on account of the persistence of vision —

as a luminous curve.

Many other forms of oscillograph have been introduced — one of

the most recent, due to Irwin, being of the hot-wire type. Braun

has devised a special kind of highly exhausted vacuum tube, in which

the wave-form is traced out on a phosphorescent screen by means

1 From J. A. Fleming's Principles of Electric Wave Telegraphy.

572 VOLTAIC ELECTRICITY

of a beam of cathode rays deflected by a magnetic field due to
the current under investigation. Gehrcke has also applied, for the
same purpose, certain phenomena, which occur in much less highly
exhausted vacuum tubes; but on the whole, the Duddell type just
described appears to be the most generally useful.

EXERCISE XXVII

1. Describe carefully, stating the precautions necessary, how you would
test the accuracy of an ammeter reading to about 1*5 amperes.

(B. of E., 1906.)

2. Explain why an electro-magnetic voltmeter should have as high a
resistance as is practicable. Given a voltmeter of this kind reading from
0 to 5 volts with a resistance of 500 ohms, how may the same instrument
be adapted to read from 0 to 50 volts with the same scale ?

(B. of E., 1904.)

3. Describe the construction of an electrostatic voltmeter. An electro-
static voltmeter gives deflections of 15, 18, and 21 scale divisions for constant
potentials of 50, 60, and 70 volts respectively ; what deflections will be
produced by an alternating electromotive force E sin pt, (a) when the
amplitude E is 70 volts, and (b) when E is 90 volts ?

(Lond. Univ. B.Sc., 1908.)

CHAPTER XXXIV

UNITS AND DIMENSIONS

EVERY physical quantity consists of the product of two factors —
(1) a measure, and (2) a unit. Thus, if we speak of 10 metres, this
is the product of the measure 10, and the unit 1 metre; and any
one, who knows what a metre is, can form some idea of the length
of 10 metres. If, however, we have no knowledge of the value of
the unit, we can form no idea of the value of any quantity containing
that unit; e.g. if a person, ignorant of electrical quantities, hears of
a current of 10 amperes, no idea of the magnitude of that current
is conveyed to his mind, and, for anything he knows to the contrary,
it may be either an exceedingly small or an exceedingly large
current.

In the case of most physical quantities, e.g. work, power, velocity,
&c., the magnitude of the unit depends upon, and is determined by,
the values we agree to take for the units of length, mass, and time.
Hence, the latter units are known as fundamental units, and the
others as secondary or derived units. There are, however, several
physical quantities (ft and K, for instance), whose real nature is
unknown, and which cannot be expressed in terms of the fundamental
units.

The Centimetre-Gram-Second System.— In scientific

work, the centimetre, the gram, and the second have been adopted,
by common agreement, as the fundamental units.

Derived Units. — The unit of velocity is, therefore, a velocity
of one centimetre per second, and is a derived unit. Other derived
units are —

Acceleration. — By the term acceleration is meant the increase or
decrease in velocity per second.

A body moves with unit acceleration when its velocity increases or
decreases by 1 centimetre per second in every second.

The acceleration (g) produced by gravity on falling bodies is
roughly 981 centimetres per second per second. This value, however,
continually changes as we change our latitude, being greatest at the
poles (983'1), and smallest at the equator (978'1).

Force. — The unit of force is called the dyne, and it is that force
which, acting on a mass of 1 gram for 1 second, gives to it a velocity
of 1 centimetre per second.

573

574 VOLTAIC ELECTRICITY

A force of 1 dyne is nearly equal to the weight of ¥^T gram,
i.e. of 1'02 milligrams.

Weight. — The student must be careful not to confuse the terms
mass and weight. The weight of any mass is the force with which
the earth attracts it, and as the value of this force varies at different
places on the earth's surface, the weight of any mass also varies.
In fact, the weight of a body is equal to the product of the mass
of the body and the earth's acceleration, i.e. W = mg, in C.G.S. units,
this means that a mass of 1 gram has a weight of 981 dynes.

Work. — Work is measured by the product of the magnitude of
a force and the distance through which the point of application moves
in the direction of the force.

The unit of work is called the erg, and is the amount of work
done through a distance of 1 centimetre against a force of 1 dyne.

Energy. — As the energy of a body is its power of doing work, the
erg is also the unit of energy.

Heat. — The unit of heat, a calorie, is the amount of heat required
to raise the temperature of 1 gram of water from 0° C. to 1° C.,
and its dynamical equivalent is 4*18 x 107 ergs.

Dimensions of Units. — It is usual to employ capital letters
in square brackets to represent the abstract ideas of length, mass, &c.,
apart from any numerical values. For instance, if we have v = 6, and
wish to emphasise the fact that v is a velocity, we may write

o = 6[V],

or, because a velocity is measured by the ratio .^ ,

time

where [LT"1] must be understood to mean the unit of velocity in the
particular system we are using ; in the C.G.S. system, for example,
we must read

v = 6 x (one centimetre per second).

Definition. — The powers to which the fundamental units are
raised in order to express the magnitude of any derived unit are
called the dimensions of that derived unit. For instance, the dimen-
sions of velocity are zero with respect to mass, + 1 with respect to
length, and - 1 with respect to time, which is expressed by writing

This is known as a dimensional equation.

It is worthy of mention that, in any physical equation, the terms
connected by + or — signs must necessarily be of the same kind, for
it would be absurd to add or subtract quantities of different kinds.
From this, it follows that all such terms must have the same dimen-
sions. On the other hand, it is possible to multiply numbers ex-
pressing quantities of different kinds, the result being a third quantity

UNITS AND DIMENSIONS 575

different in nature from either, e.g. if we multiply a force by a length,
the product represents work,

Dimensions of Various Derived Units. —

Area = length x length [L2]

Volume = Area x length [L3]

Velocity = [MM]

time

Acceleration ^

time

T-> •/ mass niTT QT

Density = — - [ML-3]

volume

Force = mass x acceleration [MLT~2]

Kinetic energy = J mass x (velocity)2 [ML2T~2]

Work = force x distance [ML2T-2]

Power = rate of doing work = FML2T

time L

Angle = _arc length [LO]

radius length

Elasticity = force Per ™t area

a ratio area L L2

With regard to the meaning of these statements, it may be remarked
that volume, for instance, is always the product of three lengths
(there may be a numerical coefficient), and we may therefore alter
the units of mass and time without, in any way, affecting the unit of
volume ; but, if we alter the unit of length, say by doubling it, the
new unit of volume is then eight times as great as before, the unit
of area would be four times as great, and the unit of velocity would
be only twice as great.

The result for angle shows that a physical quantity may have
zero dimensions, owing to the fact that it is expressed as the ratio of
two similar quantities.

Applications of the Preceding Results. — There are two
special uses to which a knowledge of dimensions may be applied.

The first of these soon becomes almost instinctive. It is to test
the accuracy of any physical equation, which may have been obtained
in some mathematical investigation, or which may, perhaps, have been
written down from memory. This depends upon the fact, already
mentioned, that, in any physical equation, the terms connected by +
or — signs must necessarily be of the same kind, from which it follows
that they must have the same dimensions. For instance, in abstract
mathematics, we may accept the expression a + b + c = d as one merely

576 VOLTAIC ELECTRICITY

stating some relation between numbers ; but in a physical equation,
if a is a length and b is a force, such an expression becomes absurd.
Let us, for example, consider the %familiar equation of accelerated
motion

As s is a length, the other terms must also be lengths.
Now, ut = velocity x time or ^ x T = [L]

In the last term, the numerical coefficient may be ignored, as it does
not affect the dimensions, and we have

ft2 = acceleration x (time)2 or ^ x T2 = [L]

The student should test various equations, with which he is ac-
quainted, in the same way ; for instance,

V = A/- (seep. 583).

The second useful application is to find the change that takes place in
the numerical value of a certain quantity, when the fundamental units
are altered. This depends upon the fact that the number expressing
that quantity is inversely as the size of the unit, e.g. a certain sum of
money is measured by 1, if the unit is a sovereign, and by 960, if the
unit is a farthing. The dimensions of the quantity are, of course,
unaltered by changing the units, and hence, if n} is the number ex-
pressing it when the fundamental units are M15 Lp Tt ; and n2 the
number when these are changed to M2, L2, T2, we have

This will be better understood if we consider an example.
Example. — Find the value of 42 x 106 ergs, when the units are
the pound, the foot, and the minute.

As the dimensions of work are ML2T~2, we have

or

42 x 10.r(gam) (centimetre)*! Rib.) (foot)*]

(sec.)2 J "L (min.)2 J

= 358,790.
Had we not altered the unit of time, this result would have been

UNITS AND DIMENSIONS 577

in foot-poundals.1 But as the dimensions of work are - 2 with respect
to time, increasing the time unit 60-fold has decreased the work
unit 3600-fold, and so the result obtained above is in terms of a
work unit, which is ^JL^ of a foot-poundal.

Example. — What number would represent 33,000 foot-pounds
per minute in the C.G.S. system1?

The first step must be to express the quantity in terms of the
fundamental units in the old system, for in ft.-lb. -second units,
the unit of work is the foot-poundal — not the foot-pound.

Now, 33,000 foot-pounds per minute = > " - foot-poundals

per second = 550 x 32-2 foot-poundals per second.

This quantity is the rate of working, i.e. power or — — , and its

time

dimensions are [ML2T~3]

550 x 32-2{(lb.) (foot)2 (sec.)-3} = ra{(gram) (centimetre)2 (sec.)-3}

^y foot Vj
gram / Vcentimetre/ )

= 550 x 32-2 x 453-6 x (30-48)2

= 746x 107

.-. 33,000 foot-pounds per minute, i.e. 1 H.P. = 746x 10r ergs
per second.

This example indicates the origin of the number 746 in express-
ing the relation between watts and H.P. For it has been shown
on p. 231 that 1 watt=107 ergs per second, and is, therefore,

i TT P
74F "••*•

Dimensions of Electrical Units. — It will be convenient
first to show how some of these dimensions may be obtained, and then
to discuss certain peculiarities exhibited by them.

Static Units.—

Quantity or Charge — By Coulomb's equation,

Force = 1, or QQj = K x force x d*

.-. [QJ = [(K x MLT-* x L2)*] =
Electric Force 2 '(at a point in an electric field} —
Mechanical force = Q x electric force,

or [Electric force] =

1 A poundal is the unit of force in the ft.-lb.-sec. system, and equals _ _ Ib.

o2'2

weight. 2 Denoted by U in this book.

2o

578 VOLTAIC ELECTRICITY

Electric Field = K x electric force (or F = KU)
.-. [Field strength] = [
Static Potential —

Capacity =

[Hence, an electrostatic capacity is correctly measured in terms
of a length, e.g. for a sphere, in terms of the length of its radius in
centimetres.] *•

Electromagnetic Units. —

Pole Strength — By Coulomb's equation,

Force = m X ^1

- [(/* >< MLT"2 >< L2rt = [M*L*T- V*]
Magnetic Moment = pole strength x length,

Magnetising or Magnetic Force —

Mechanical force = pole strength x magnetic force,

Field Strength, or Magnetic Induction (B) —
"Field

Intensity of Magnetisation = ^g"^ moment,

volume

Current — Force = B x i x I.

UNITS AND DIMENSIONS 579

Quantity — QTO = zY,

Electromotive Force (or Potential) —

QWEW = work,
E.M.F.

Resistance = 'x

^ current

r-M^T-2/
[Resistance] = A ^ _1

(Hence, in magnetic units, a resistance can be expressed as a
velocity, and it will be found that all absolute methods depend on
the measurement of a velocity. (See also the dimensions of resist-
ance in the static system, p. 580.)

Inductance. — For a solenoid, Inductance = — — — ^

* - L

or [Inductance] - I £i£ - [L/x].
L L J

From which it appears that in magnetic units a coefficient of self-
induction may be expressed in centimetres of length.

Contrasting these results with those previously obtained for
mechanical quantities, we notice at once the prevalence of fractional
indices, and the occurrence of K and ^ in the static a,nd magnetic
systems respectively.

If we regard K and //, merely as numerical ratios, in terms of air
taken as unity, they disappear from the dimensional equations, but
then we meet with two initial difficulties : (1) that of understanding
the physical nature of quantities whose dimensions are fractional, and
(2) that caused by the dimensions of magnetic pole strength and of
static charge becoming identical, which would lead to the absurd con-
clusion that they are really the same thing.

If, however, we regard K and /x as real physical quantities of
unknown dimensions, both difficulties disappear.

It may be pointed out that the static system of units was. obtained
by ignoring current phenomena (i.e. the phenomena of magnetism),
and the electro-magnetic system by similarly ignoring static pheno-
mena } the two underlying conventions — that K and \i are to be

580 VOLTAIC ELECTRICITY

taken as unity for air — being probably incompatible with one another.
Thus both systems are entirely arbitrary, but it must be clearly
understood that the terms potential, capacity, £c., mean the same
physical quantity in either. Hence, the dimensions assigned to them
in the two systems respectively must be really identical. Let us see
what occurs, when we equate the two forms obtained for quantity
or charge.

Lx/K/

,

x/K/J ,

which indicates that , — has the dimensions of a velocity.
Repeating the process in the case of capacity, we have

which is the previous result squared.

Similar results may be obtained with all the units represented
in both systems. One obvious consequence is that we may change

T
from one system to the other by making use of the relation KJ/x5 = j->

and substituting for either K or p.. For instance, if we know that in
the static system

we can find the dimensions in the magnetic system by putting

T
K^ = - — j , which gives

Example. — Find the dimensions of resistance in the static
system.

[RJ = [LT- V] and [K/x] = [T^], i.e. [P]

From which we see that R9 comes out as the reciprocal of a velocity,
i.e. as a slowness, and, in fact, the idea of static resistance may be
interpreted as signifying the slowness with which a static charge
leaks away through the substance in question.

Ratio of Units in the Two Systems. — It follows that if
we substitute numerical values in the expression — _ ? the result is

a velocity ; and our convention, that K and /A are unity for air, really

UNITS AND DIMENSIONS 581

amounts to making its value for air the unit of velocity. We can,
however, estimate its actual value in C.G.S. units by applying the
method already used for mechanical units.

Let Qs, Qm be the numbers representing the same quantity in the
two systems. Then we must have

In this equation, the ratio -^ is a pure number (for instance, if

Scs»

Qm is one absolute unit (i.e. 10 coulombs), Q6, will be the number ex-
pressing this quantity in static units), and as already stated [LT-1]
stands for the unit of velocity, i.e. I centimetre per second in the

C.G.S. system. Hence, the expression —j= stands for a velocity,

which, when K=l, /*=!, is measured in centimetres per second by
the numerical value of the ratio -^ .

Qm

It is usual to denote this velocity by v,

• Q._e

' Q.

Here, Qs is a large number compared with QBl, which, of course,
indicates that the magnetic unit of quantity is v times as large as the
corresponding static unit.

The value of v must be determined experimentally by actually
measuring the same quantity in both systems of units. For instance,
the capacity of a condenser in static units may be found by calcula-
tion from its dimensions (or by comparison with that of a sphere).
It is then charged, and its potential is measured absolutely in static
units by an electrometer, thus determining Qs in terms of potential
and capacity.

The condenser is then discharged through a ballistic galvanometer
and the throw noted, Qm being calculated from the theory of the
instrument.

In this way, it has been found that v = 3 x 1010 centimetres per
second.

Ratio of Units of Potential. — Obviously, the numerical value
of v may be obtained by comparing any two units. In the case of
potential, a steady current i may be allowed to flow through a resist-
ance R, and the value of the P.D. across it, in magnetic units, found
by measuring the current and the resistance, for Vm = ^'R. The P.D.
may be directly measured at the same time, in static units, by means
of an electrometer.

582 VOLTAIC ELECTRICITY

To ascertain the real meaning of the ratio thus obtained, we have
only to notice that, if a current i flows for time t seconds, a quantity
passes measured by Qs in static units and Qm in magnetic units, and
that in one system the work done is VgQs ergs, and in the other
VmQm ergs.

or ==.

Vs Qro
y

i.e. ~ = v = 3 x 1010 (centimetres per second),
» j

from which we learn that the static unit of potential is v times as
large as the magnetic unit.

Example. — How many volts correspond to 1 electrostatic unit
of potential?

Now, 1 static unit = 3 x 1010 absolute magnetic units of potential,
and 108 absolute magnetic units = 1 volt.

Q v i mo
/. 1 static unit = ^ = 300 volts.

Ratio of Units Of Capacity. — The units of capacity may be
compared by determining Cm as on p. 390, and calculating Cs from
the dimensions of the condenser. To ascertain what this means, we
have only to notice that, for a definite charge and potential, we have

Magnetic Units Of Capacity.— It has already been stated that
the equation —

Quantity ,of charge in condenser = P.D. between its coatings x Capacity,

holds good generally, whatever system of units we may adopt in actual
measurement, and we have written it in the form

Q = VC

to indicate that static units are employed, and

Q = EK
when using magnetic units.

From this it follows that a condenser tvill have 1 absolute unit of capacity
when a charge of 1 absolute unit of quantity produces a P.D. between its terminals
of 1 absolute unit of E.M.F.

Now, the absolute unit of quantity is 10 coulombs, which is fairly large,

and the absolute unit of E.M.F. is -i volt, which is relatively very small.

Hence, a condenser must be of enormous size to have 1 absolute unit of
capacity. It is very much as if we defined a vessel as having unit capacity
for holding water, when a charge of 10 gallons would fill it to the depth of

UNITS AND DIMENSIONS 583

jL inch. Evidently its area of section would be exceedingly great, or other-

wise the 10 gallons of water would fill it to a greater depth than — g inch.

The result obtained above tells us that 1 absolute unit of capacity in the
magnetic system is equal to 9 x 1020 static units, i.e. equal to the capacity of
a sphere 11 x 1015 miles in diameter.

Again, as mentioned on p. 207, a condenser will have a capacity of I practical
unit (known as a "farad ") when a charge of 1 coulomb produces a P.D. of 1 volt
between its coatings.

Obviously, this is a smaller capacity, and we can find its value in absolute
units by writing

If A _ 1 coulomb _ Y1^ absolute unit of quantity
1 volt *" fO8 absolute units of E.M.F.

or 1 farad = — - absolute units of capacity.

This is still too large for ordinary purposes, and the " microfarad " is the
most convenient unit for actual practice, where

1 microfarad — — ^ farad = — ^ absolute unit of capacity,
which is equal to the capacity of a sphere about 11 miles in diameter.

Electro-magnetic Theory of Light. — Many careful experi-
ments of the kind just given have produced results which have a
remarkable agreement to one another, and which give a mean value of
almost exactly 3 x 1010 centimetres per second.

"Within the limits of experimental error, this is also the velocity
of light in space, and Clerk-Maxwell, who gave to the mathematical
theory of electro-magnetism its present form, suggested that light
itself was an electro-magnetic phenomenon. In his time, no other
experimental evidence existed to support- this suggestion, but it has
since been abundantly confirmed by the labours of Hertz and his
successors (see p. 605).

According to Maxwell's theory, the numerical value of — /===

determines the velocity of an electro-magnetic wave (p. 607) in any
given medium. If, for example, for a certain dielectric, K = 4, p = 1,

then ,-— = J, which means that the velocity in that substance is

yK//.
half its value in space, i.e. % x 3 x 1010 centimetres per second.

The student should refer to more advanced treatises for a fuller
treatment of the subject. We may, however, deduce the previous
result by means of a simple analogy, as follows : —

It was shown by Sir Isaac Newton that the velocity of wave
motion in any given medium may be expressed in the form
_ /Elasticity hereelast;ciigdef,ned ag fte ^ stress applied

V Density strain produced'

584 VOLTAIC ELECTRICITY

and we may, therefore, try to evaluate this equation in the case of
electrical quantities.

Let a small body, charged with + Q units, be placed at the centre
of a hollow conducting sphere of radius r. Then +Q units appear
on the outside of the sphere, which we may regard as being due to
the introduced charge displacing an equal amount of charge, very
much in the same way as a body lowered into water displaces its own
volume of water. Now, 4?rQ lines of force pass through the sphere,

and the field strength at its surface is, therefore, -— ^.

Also, F = KU, . •. U = electric stress =. JPj^ .

This stress has produced a displacement of charge per unit area
of — ^, which may be regarded as the measure of the strain produced
by the applied stress. Therefore, we have, by analogy,

_Q_

elasticity of dielectric = stress = Kr2 = 4?r * , 1 .

(electric) strain Q K '

We must next find an expression for density. We see from
p. 377 that the kinetic energy of a current is JLe2. Let us now
apply this to the case of a ring-shaped coil, e.g. an endless solenoid,
which is convenient because the whole of the field is contained within

the turns of the coil. For such a coil we know (p. 376) that L = y — ^

where n is the number of turns, A the area of cross section, and / the
mean length of the axis of the circular coil.

/. Energy of the magnetic field inside coil = \ x — n ^ x i2

I

This energy is contained in a volume, I x A cubic centimetres,

J X 47T/A X

. \ Energy of field per unit volume = — —

(2)

Now, the kinetic energy of a moving body is Jwv2, where m is

1 In this and the following expression, the 4*- is a coefficient depending
on our initial choice of units, and has no physical meaning.

UNITS AND DIMENSIONS 585

its mass, and v its velocity, which is strikingly similar to the expres-
sion (2) for the energy of a current.

If the analogy has any real basis, 47T//, is the electrical mass per
unit volume of the dielectric, i.e. its density.

Whence, we have v = ^J

Density * 47371, *J~Kp>

Examples. — 1. Find the dimensions of specific inductive capa-
city in magnetic units, and of permeability in static units.

and [/*] - [T2L-2K-!]

It follows that the permeability of air (or space) in static units,
and its specific inductive capacity in magnetic units, are each

2. Find the value of a quantity of electricity, of which the electro-
static measure is 250 (cent.^ gram, sec."1) \ (&) in magnetic units,
(b) in coulombs, (c) in magnetic milligram-metre-second units.

(a)

and

and these are identical,

.-. if n represents the value in magnetic units, we have

Now the numerical value of K^ = - , and L and T are each
., v

unity

whence rc = — = 25° 10 = 83-3 x lO"10 magnetic units.

1) O X J-\J

(b) In coulombs, the number will be ten times larger,

or 83-3 x 10~9 coulombs.

(c) We have to express 83*3 x lO"10 magnetic units (C.G.S.) in
another system where the fundamental units are the milligram, the
metre, and the second.

8S-S x 10-iof M" ^ ^~\-J M" L"
bd 3x 10 JU[_(o-ram)(cm.) ~ n[_(mg.) (metre)

586 VOLTAIC ELECTRICITY

iVmilligram/ \ metre / /

2

100
83'3xlO-1°xlOi

Clerk-Maxwell's relation between Dielectric Constant
and Refractive Index for Light. — According to the theory of

light, the refractive index of a substance is the ratio of the velocity of light
in space to its velocity in that substance. Hence, if n be the refractive
index of some transparent dielectric, we should have

but for all such substances, /* does not differ appreciably from unity, and
therefore

n= \/K, or K = n2.

When we examine the experimental values of K and n for various trans-
parent dielectrics, we find that, whilst the above relation is satisfied in
certain cases, it certainly does not hold good generally, and considerable
discussion has taken place as to the cause of the discrepancies. If, however,
we remember that neither n nor K are simple definite numbers for a given
substance — the value of the former depending upon the wave-length for
which it is measured, and that of the latter (as stated on p. 69) upon the
temperature and the frequency — it will be evident that some difficulty must
exist in selecting the values to be used in the comparison. As K is usually
measured by processes equivalent to the use of very long waves, it may be
supposed that we require the refractive index for waves of infinite length.
Our ordinary tabulated values have been determined for waves of very short
length, which lie within the range of vision, and if the refractive index
varied with the wave-length according to some uniform law, it would be
possible to calculate the corresponding values for waves of infinite length.
But the phenomenon termed " anomalous dispersion," which is now known
to be of general occurrence, makes the relation between wave-length and
refractive index extremely complicated, the latter quantity varying between
wide limits for one and the same substance. For instance, in the case of
water, K = 80 (under ordinary conditions), and n for lightwaves is about 1-33.
When, however, n is measured for electric waves having lengths of 5 centi-
metres and upwards, its value is found to be nearly 9, in accordance with
Maxwell's law. Similar results have been obtained in the case of alcohol.

Relation between Opacity and Conductivity.— Maxwell

also pointed out that a perfect conductor should be perfectly opaque, i.e. it
should reflect completely all light falling upon it, and Drude has expressed
the relation between reflecting power and conductivity in a convenient
mathematical form. In Maxwell's time there were certain outstanding
difficulties, e.g. gold-leaf is more transparent than it apparently ought to be.
The discovery of the "electron" has enabled the theory to be extended to
meet such cases, and it has been verified to a remarkable extent by the experi-
mental work of Rubens on infra-red light waves.

The Ether. — In this book, we have implicitly assumed the
existence of a medium, which is the seat of the phenomena denoted

UNITS AND DIMENSIONS 587

by the terms electric and magnetic lines of force. It may, however,
be mentioned that at the present moment, the various questions
associated with the ether give rise to problems of great complexity
and difficulty. The experimental knowledge acquired during the last
twenty years, taken in conjunction with recently acquired knowledge
regarding the " electron " and the constitution of matter, leads to
apparently irreconcilable results, and the real nature of the ether
— if it exists at all in the old sense of the word — must be regarded as
absolutely unknown. For instance, if the ether is incompressible, as
it is usually assumed to be, we are driven, by one line of argument, to
the conclusion that it is 2000 million times denser than lead and
possesses enormous energy of internal motion. On the other hand, if
it is compressible, it may be much rarer than the rarest gas. There
is no intrinsic difficulty in either view, but at present no method is
known by which we may hope to discriminate between them. The
whole subject of the ether is in that state of uncertainty and apparent
confusion, which in other branches of science has usually preceded
some great advance in knowledge.

Practical Standards Of Resistance. — The theoretical defini-
tions of certain units have already been given. For instance, the absolute
unit of E.M.F. has been defined as the E.M.F. produced when one conductor
cuts one line of magnetic force in each second, and the volt has been arbi-
trarily taken as 108 absolute units. With regard to resistance, it will be
seen (p. 579) that, in magnetic units, it has the dimensions of a velocity
multiplied by /*, and hence, in a system of units in which /j. is taken as unity
for air, a resistance can be expressed as a velocity, and the absolute unit of
resistance will, therefore, represent a velocity of 1 centimetre per second.
The practical unit, or ohm, has been arbitrarily defined, perhaps unfor-
tunately, as 109 absolute units, and it is, therefore, equivalent to a velocity
of 109 centimetres per second.

These statements do not imply that a resistance is really the same
physical quantity as a velocity, for obviously that cannot be the case if p. is
regarded as a quantity of unknown dimensions, but they do imply that any
method of measuring directly the resistance of a given conductor must
resolve itself into a measurement of a velocity (or, of a length and a time).

These definitions, taken in connection with Ohm's law, determine the
magnitude of the absolute unit of current, and incidentally it follows that
an ampere is TV of that absolute unit.

It is, however, not sufficient merely to define these units in a consistent
and logical manner ; it is necessary to give them some concrete form, which
may be used as standards for commercial purposes, and this is a problem of
quite a different kind.

Evaluation Of the Ohm. — The first step was to evaluate the ohm,
i.e. to find the resistance of some given conductor in absolute measure. This
task was undertaken in 1863 by. a Committee of the British Association, who
used a method suggested by Lord Kelvin. A circular coil — something like a
tangent galvanometer coil — was rotated about a vertical axis in the field
of the earth, and although we know that an alternating current was thereby
induced in the coil, a small compass-needle suspended at the centre ex-
perienced a steady deflection. The theory of the method is rather too
lengthy to be given here, but the resistance of the coil can be expressed in
terms of the number of revolutions per second and the observed deflection

588

VOLTAIC ELECTRICITY

of the needle (i.e. in terms of a
must also be taken into account.

FIG. 404.

course the two P.D.'s must be in opposition).
Let i = steady current in coil and in K,

velocity). The self-induction of the coil
This method is subject to various sources
of error, difficult to eliminate, and is
not now used.

Of the numerous other methods
proposed or carried out, perhaps the
most important is that of Lorenz,
which seems likely to become gener-
ally adopted. The principle of the
method is indicated in Fig. 404.

A metal disc, D, is rotated inside a
coil carrying a current, so that it moves
at right angles to the field produced
by the coil (indicated by dots). The
conductor, K, whose resistance is to
be measured, is placed in series with
the coil, and the P.D. between its ends
balanced against the induced P.D. be-
tween the centre and the circumference
of the disc. The P.D. induced in the
disc D depends upon its speed, and
hence if it is connected up as shown
there will be some speed at which the
galvanometer will not be deflected (of

then: — .

also

= resistance to be measured,
M = the mutual inductance between coil and disc,
n = number of revolutions of the disc per second,
e = P.D. between centre and circumference of disc,

lines passing through disc = M.^,

lines cut in 1 revolution x number of conductors

time of 1 revolution

Mix I

Also, the P.D. between A and E=iE, and, therefore, if these P D.'s are equal
and opposite,

iR = M.£n, or E = M.n.

Now M can be calculated from the dimensions of the apparatus as a purely
geometrical problem, so that again K is measured as a velocity.

The chief drawback to this method is the fact that only comparatively
small resistances can be measured by it.

The Ohm in Terms of a Column of Mercury. — in order

that such absolute determinations may be practically useful, it is desirable to
express them in terms of a substance readily obtainable in a state of purity,
such as mercury, and the result of numerous determinations by different
methods has been to show that the ohm is the resistance of a "column of
mercury 1 square millimetre in section and about 106'3 centimetres in length.
In 1890, this was adopted as the legal definition of the ohm in this country,
but as the only method of determining the cross section is to obtain the
weight of mercury filling a given length of the tube containing it, the
definition was afterwards amended, and at the present time it runs as
follows : " The international ohm (as recommended by the International
Conference on Electrical Units, held in London in 1908), is the resistance

UNITS AND DIMENSIONS 589

offered to an unvarying current by a column of mercury at the temperature
of melting ice, 14'4521 grams in mass, of constant cross section, and of
length 106-3 centimetres."

To construct a standard resistance embodying this definition, it is not
necessary to use exactly the above dimensions. It would be exceedingly
difficult to do so, and in practice a glass tube of uniform section, conveniently
about 1 millimetre in diameter, is fitted into larger end vessels (to contain
terminals), and the whole filled wifch mercury. The report of the Conference
specifies that these end vessels shall be spherical, of approximately 4 centi-
metres in diameter, connection being made by means of platinum wires
sealed through the glass.

The actual resistance of this standard is then calculated as follows : Let
L = length of tube, r=radius, A=area of section, s = specific resistance,
D = density of mercury, M = mass of mercury filling the tube, and R=its
resistance in ohms. Then (neglecting at first the resistance of the mercury
in the spherical ends) we have : —

=A2X8

But M= A x L x D = TH

2_ M
~LD

and R=-^-

Substituting in this expression the values given in the definition, we have

1 = (106'3)2xD.s

14-4521
4521
(101T3)1

D.^14'4521

1,2 14-4521
"~M"X( 106-3)2*

= -001278982 x — ohms.
M

Hence, we have only to determine experimentally the weight of the mercury
filling the tube, and its length, in order to obtain a concrete standard of
known resistance.

The report of the Conference gives the following formula for evaluating
the slight extra resistance due to the spherical ends, when made in accord-
ance with their instructions :

Additional resistance = — (— J-— ) ohms,
106-37T Vrj rrj

where rx and r2 are the radii in millimetres of the end sections of the bore of
the tube.

Practical Standards of Current and E.M.F.— it is neces-
sary to define one other practical standard (the third then follows from Ohm's
law), but, whereas the ohm has been unanimously adopted by all countries,
there has been some difference of opinion and of practice as to whether
current or E.M.F. shall be selected for the second standard. If the former
be chosen, the ampere must be defined in terms of chemical action, and, of
course, cannot be represented by an actual standard. If the latter, the volt

590 VOLTAIC ELECTRICITY

must be defined in terms of the E.M.F. of some standard cell, and there is a
strong inducement to adopt this course, because, whatever be taken as the
legal standard, practical men will always use a standard cell and a known
resistance as the basis of their measurements.

The International Conference strongly recommended the adoption of the
ampere, as a current can be measured quite independently of the obm ; and
in carrying out practical measurements by means of a silver voltameter it is
only necessary to use one salt (silver nitrate), which is easily procurable in a
pure state. On the other hand, in a standard cell several salts are required,
one of them (mercurous sulphate) is difficult to purifv ; and further, the
absolute determination of its E.M.F. depends on the measurement of a
current and a resistance. They, however, recommended the use of the
Weston cadmium cell as a convenient sub-standard.

In order to define the ampere in terms of chemical action, it is necessary
to measure a current absolutely, and to determine the amount of some sub-
stance liberated by it in a given time. A silver nitrate voltameter has proved
to be the most suitable, and the current has usually been measured by some
form of current- balance (see p. 567), because its absolute strength* can be
calculated in. terms of the dimensions of the instrument. The final result
shows that the ampere is the current which liberates '001118 gram of silver
per second.

The numerical value of the E.M.F. of a standard cell is based upon these
two definitions. For the details of the modified potentiometer method most
generally used for measuring the E.M.F. of such cells, the student should
consult treatises on electrical measurements, but the principle is easily under-
stood, and may be outlined as follows. Assume that a standard resistance of
accurately known value — say 1 ohm — is available, and that it is able to carry
one or two amperes without sensible heating. In any case it would be
arranged so that its temperature could be measured and kept constant. Jjet
this be connected to a suitable battery ; an adjustable resistance and a silver
voltameter being included in the circuit. The cell whose E.M.F. is to be
determined is placed in series with a sensitive galvanometer, and the two
connected across the terminals of the resistance R like the cell Ej in Fig. 234,
p. 306, in such a way that its E.M.F. is in opposition to the P.D. across R.
The current is then adjusted until the galvanometer shows no deflection. If
C is this current, then CR is the P.D. across R, and is equal and opposite to
E. Hence, it is only necessary to measure C, but this must be done with
the greatest possible accuracy. For this purpose the silver voltameter may
be used, precautions being taken to ensure that C does not vary during the
time required to deposit a convenient weight of silver.

For a Weston cadmium cell, set up in accordance with the specification
published by the Conference, it may be taken as 1-0183 volts at 20° C. In
the report of that body, the following expression is given for the value at
other temperatures : —

If E»= E.M.F. at e C. ; E^^RM-F. at 20°, then :—
Et= Eoo - -0000406 (t - 20) - -00000095 (* - 20)2 + -00000001 (t - 20)3

Table Of Electrical Units. — For convenience of reference we
append the following table, indicating the various relations between
the more important units and also the pages on which they are
mentioned in this book.

It may be remarked that the ideas of potential, quantity, and
capacity are met with in considering current phenomena, and also
in electrostatics, and hence each of these possesses two well-known
units.

UNITS AND DIMENSIONS

591

On the other hand, the ideas of current, resistance, and inductance
more naturally belong to the former section, and hence are represented
by only one unit.

It is, of course, quite easy to express these in the static system
(an instance is given on p. 580 in the case of the unit of resistance),
but it would be misleading to insert such numbers in the following
table.

*^*4

o

§

wg

^3 o 'o S3

llfl.-§

ll^'ls"

11

l|<i

"3 2 *a |j

1*|§x

Piig-e.

rt £

feS^i

i£&i

o3 II

&

HI

I"8 *

& O ^ 3 ^

Resistance ....

Ohm

109

206, 587

Current •

Ampere

lo-1

207, 590

Electromotive Force )
or Potential \

Volt

108

ixlO-2

\ I \

206, 351

Quantity or Charge .

Coulomb

10-1

3xl09

V

207, 230

(or Auipere-secoud)

Capacity .....

Farad

io-9

9 x 1011

vn-

207

Inductance . . . .

Henry

109

...

...

377

Power . *',

Watt

= IO7 ergs per sec. = ^ H.P.

231

Work or Energy . .

Joule

= 107 ergs

234

(or W:itt-secoiid)

Each of these is 10'

Multiplcs and Sub-multiples : —

Megohm = 106 ohms.

Microhm, microvolt, microfarad, micro-ampere.

of the corresponding practical unit.
Milli-amperc, millivolt. Each is 10 ~3 of the corresponding unit.
Kilowatt = 1000 watts.
Kilowatt-bow, or "Board of Trade Unit " = 1000 watt-hours

= 1000 x IO7 x 60 x 60 ergs.
A mpere-hour = 3600 coulombs.

EXERCISE XXVIII

1. Define unit magnetic pole and unit electrical current in the electro-
magnetic system, and state the relation of the ampeie to the latter.

(B. of E., 1895.)

2. Draw up a table giving the practical units in terms of the C.G.S.
electro-magnetic units for current, quantity, electromotive force, resistance,
and capacity, detailing the fundamental relations on which the latter system
is based. (B. of E., 1907.)

3. Describe the electrostatic and electro-magnetic systems of units. Find
in each of the systems the dimensions of (1) potential difference, (2) capacity
of a condenser, (3) specific resistance. (B. of E., Hon., 1903.)

592 VOLTAIC ELECTKICITY

4. Define the terms magnetomotive force, magnetic flux, and reluctance
of a magnetic circuit. Find the dimensions of these quantities.

(Lond. Univ., B.Sc. Internal, 1909.)

5. According to the usual definitions, the dimensions of capacity in the
electro- magnetic system are those of the reciprocal of an acceleration, while
in the electrostatic system they are simply a length. Show how these
results are obtained, and explain the apparent discrepancy.

(Lond. Univ., B.Sc. Internal, 1908.)

6. A disc 16 centimetres in diameter is rotated at a speed of 3700 revolu-
tions per minute inside a long spiral coil of wire of 13 turns in each centi-
metre length of coil, the axis of revolution being parallel to the axis of the
coil and perpendicular to the plane of the disc. When 1-5 amperes are passed
through the coil, what difference of potential exists between the axis and the
circumference of the disc. (Lond. Univ., B.Sc. Internal, 1903.)

7. Find the number of watts in 1 horse-power, given I foot=30*48 centi-
metres; 1 lb. = 453'6 grams; ,7 = 981 centimetre/(seconds)2.

Electrical energy is sold at the rate of 4d. per kilowatt-hour. The
mechanical equivalent of the heat given by the burning of coal worth 4d. is
108 foot-lbs. Compare the prices of the two forms of energy. Why is
electrical energy so much dearer than coal energy ?

(Lond. Univ., B.Sc., 1902.)

CHAPTER XXXV

EFFECT OF INDUCTANCE AND CAPACITY AT STARTING
AND STOPPING A CURRENT— ELECTRIC OSCILLA-
TIONS—RADIATION—WIRELESS TELEGRAPHY AND
TELEPHONY.

WE have already shown (see p. 368) that, if Z lines of magnetic force
cut N turns in t seconds, the average value of the induced E.M.F. is
given by

/ x = — . absolute units.

(average) f

If the rate of cutting is not uniform, the induced E.M.F. at any
instant depends upon the rate of cutting at that instant, and we may
write

6 =— -N m

(instantaneous) dt

This expression holds good for any induced E.M.F. We have now
to apply it more particularly to the phenomena attending the rise and
fall of a current in an inductive circuit.

Let a steady current i flow in a coil of N turns, producing
a flux of Z lines. Then equation (1) gives the value of the E.M.F.
of self-induction at any instant whilst the current is starting or
stopping (or more generally varying in strength). In this case, it is
customary to write the equation with a negative sign in order to
indicate that, when Z is increasing, e is a back E.M.F. Now, by
definition (see p. 376)

ZN-Li

and e=-L~
dt

We have frequently pointed out, that this relation holds good only

593 2 P

594 VOLTAIC ELECTRICITY

when the conditions are such that L may be regarded as constant,
whereas equation (1) is always true.

Hence, in order that a current i may flow in a circuit (or part of
a circuit) of resistance II and inductance L, we must have, at any
instant, an impressed E.M.F., E, which, while the current is increas-
ing, is numerically equal to e'R + L— , and while the current is decreas-
ed

ing, is equal to iE, — L — ; or algebraically

Ct v

= eR + L— (3)

(instantaneous)

where the sign of the second term depends upon the sign of — ; i.e.

Ctlr

positive for an increasing, and negative for a decreasing, current.

From this equation, we may deduce the law of rise of current in
a circuit, when a steady E.M.F. is suddenly applied to it, i.e. the law
for direct-current circuits.

Helmholtz's Equation. — Writing expression (3) in the form

E-»R=L*

by transposition we obtain

di R,.

and integrating both sides of this equation,

lo&( i ~ *} = ~ ? +
\R J L

•n*

Now, ^=0 when t = 0, and, therefore, the constant is loge —

R

E

R

/
or

E,/ L

in _K*

(where e is the base of natural logs.)

STARTING AND STOPPING A CURRENT 595

From this equation we can calculate the value of i at any time t after the
start. We see that the second term gradually decreases as t increases, and
becomes zero when t is infinite, which means that the current takes an infinite
time to rise to its steady value. When L = 0, the expression reduces to the
ordinary form of Ohm's law, but as no circuit is absolutely non-inductive,
equation (4) may be taken as holding good generally.

Time Constant of an Inductive Circuit— Putting t=^ in the

a

above equation, it becomes

hence, — is the time required by the current to reach a certain fraction
R

(l - -, or 1 - — L_, or -632 V of its final value.
\ c 2-7182 /'

As the dimensions of self- inductance are [length x /JL] and of resistance

[LT"V]» it follows that the ratio —has the dimensions of time, and is known

R
as the time constant of the circuit.

This ratio has been already met with, e.g. on p. 454 it was shown that, if
a steady current i be suddenly interrupted, a quantity, </, rushes round the

circuit, such that q= — xi. Hence, an inductive circuit behaves as if a

R

quantity were stored up in it equal to the quantity conveyed by the full
current flowing for a time equal to the time constant of that circuit.

In a non-inductive circuit, the current would rise instantly to its full
strength, and, during t seconds after the start, a quantity, i.t, would have
passed round the circuit. In an inductive circuit, a smaller quantity would
have passed during this time, and it can be shown that the difference is

equal to — , or the circuit behaves as if the missing quantity were stored up

R

in it. Of course, it is not " quantity," but " energy " that is really stored up,
in the form of a magnetic field. An expression for this energy has already
been found by simple reasoning (see p. 377) ; it may also be evaluated as
follows : —

The current rises from 0 to i in time t against a back E.M.F., whose

value at any instant is L ^.
dt
If the current and the back E M.F. were steady, the work done against

the latter would be ix L— x t, and this would be the energy stored up in the

field. But, as both current and back E.M.F. vary, we must evaluate the
above expression over the whole time, by writing,

Work-l ixL^xdt

:— \ i',
Jo

Dying Away of a Current in an Inductive Circuit. —

Equation (4) suggests the converse problem — given a current i0 flowing
steadily in the circuit, what occurs when the applied E.M.F. is suddenly

596 VOLTAIC ELECTRICITY

removed without opening the circuit ? To answer this question, we must put
e=0 in equation (3), then we have

dt

from which, by rearrangement,

T=~Lrf*

and integrating we obtain

T>

loge i =-—.£ + a constant.
L

Now, when <=0, the current is i0t therefore the constant is log v

e i0 L

or i=i0xe~iT (5)

From which, we may calculate the current strength at any instant as it dies
away.

Rise of Current in a Circuit containing Capacity and

Resistance. — Let a steady E.M.F., E0, be suddenly applied to a circuit
containing a condenser of capacity K in series with a resistance R, and let
E be the P.D. between the condenser terminals at any instant. At first
E will be zero and will gradually rise to E0 in a direction opposed to E0,
while the current will gradually decrease and become zeio at the instant
E = E0.

Let Q be the quantity in the condenser at any instant, then

Q=EK
dQ -u-dE

• . — r— = -IA. --

dt dt

but i = S, where i is the current at that instant,
at

'dt~ R
rfE _ dt
E0-E~KR

Integrating, loge (E0 - E)= - ~ + a constant.

KK

Now, when t=0, E=0, .'. constant is log E0

.. ioge?»d?=— L

E0 KR

-p j_
or 1-J^e-IS

Bg

f

or E=El-g"KR

STARTING AND STOPPING A CURRENT 597

but as i= <J°~
R

.'. f~**xe-K (6)

The product KR has the dimensions of time. It is called the time con-
stant l of the circuit. Evidently it is the time the P.D. across the condenser

lakes to reach 1 — or '632 of its steady value.

Dying Away of a Current in a Circuit containing Capa-
city and Resistance. — Let the condenser be charged to a P.D. equal
to E0, and then be suddenly short-circuited through a resistance R. Then,
if Q and E be the respective values at any instant during discharge, Q = EK.

Also i= - -~, the negative sign being due to the fact that, in this case,
ctt

Q is decreasing as t is increasing.

As before, — - = K— - , and as both Q and E are decreasing with the time,
dt clt

both sides of this equation are negative— and that sign cancels out.

dQ vd~E E

_

~dt dtR

KR

T71 j.

Integrating, log* — = - —

from which, by the methods already given, we obtain

and t=4 = *2xriE (7)

R R

This case is important, as it may be applied to the measurement of very

high resistances.

•pi j

For this purpose, the expression log =- = - ==— may be written

R- -U _JL_

S"Klo Eo (8)

Hence, if a well-insulated condenser of known capacity is charged to
any convenient potential, and then short-circuited by a resistance R so
great that the charge leaks through it comparatively slowly, the value of
R can be calculated by observing the P.D. at the beginning and end of any

measured time, t seconds. As ~ is a ratio, their values can be expressed

1 This implies that L = 0, just as a previous statement (that time con-
stant = -) implies that K=0.
R

598 VOLTAIC ELECTRICITY

in any arbitrary units. If K is expressed in farads, R will be in ohms ;
if K is expressed in static units of capacity, R will be in static units of
resistance. Now, a reference to p. 580 will show that, in static units, the
dimensions of resistance are those of the reciprocals of a velocity (putting
spec. ind. cap.= l), and the above result illustrates the meaning of this fact,
for we see that the greater the value of R, the slower will the charge leak
away through it.

As an interesting example of the use of this formula, the following well-
known method may be mentioned. It was required to find the " insulation
resistance" of a certain specimen of electric light cable, 110 yards in length,
wound in the form of a coil. For such purposes the coil is immersed in a
tank of water, with both ends protruding, and the resistance to be measured
is that between the copper wire and a plate placed in the water. The
best method for the purpose would be that given on p. 275, but, in this case,
the resistance was much too great to be measured in that way with the
apparatus available. The leakage method was, therefore, employed, and as
such a coil has a capacity and acts like a Ley den jar (the wire forming one
coating and the water the other coating), it was unnecessary to use a separate
condenser.

The first step was to measure the capacity of the coil by comparing it
with a condenser of known capacity, using the method described on p. 313.
This was found to be ^ microfarad. (In order to obtain a small known
capacity for comparison, several condensers were connected in series, and
use was made of the formula given on p. 65.)

Then the throw was taken, (1) when the coil, after being charged by a
suitable battery, was immediately discharged through a galvanometer ; (2)
when three minutes were allowed to elapse between charge and discharge,
during which period the charge was leaking slowly through the insulation.
These throws were 54 and 42 scale divisions respectively.

Therefore, we have — = —
E 42

where R= *

3x60
Hence, R = ^ ohms,

1 loSio 43-

32xlO« -4343
which, in round numbers, is 23,000 x 106 ohms, or 23,000 megohms. It may

be mentioned that the value for 1 mile of such cable would be ' =1440.

lo

This result would be expressed by saying that the insulation resistance was
about 1400 megohms per mile.

In such experiments, all the apparatus must be carefully insulated in
order to avoid large errors due to leakage other than that through the
insulation, and in any case no great accuracy is likely to be obtained. It
should be regarded as a method of obtaining approximate values for enor-
mously great resistances, useful when no other plan is available.

Equation (8) may also be applied to compare the capacities of two con-
densers by means of an arrangement similar to Wheatstone's bridge. For
this purpose, the condensers whose capacities are to be compared (C and C1,
Fig. 405 *) are placed in the two arms of a bridge, the other two arms being

1 From Carey Foster and Porter's Electricity and Magnetism.

STARTING AND STOPPING A CURRENT 599

occupied by resistance boxes a and a'. The circuit may be earthed at points
shown in the figure, or these points may be connected by a wire. When the
key is depressed, the condensers are charged by the battery, and when it
is allowed to fly back they are discharged again. The resistances are
adjusted until no throw is produced on working the key. The resistances
should be fairly large, and it is preferable to rely upon balance during
" charge," for, if the condensers have different dielectrics, absorptive effects
may interfere with balance during discharge.

Evidently the condition for balance is that the potential at B should, at
any instant, be equal to that at B1. This implies that the P.D. across each

< >— earth

FIG. 405.
condenser rises or falls at exactly the same rate. When this is the case, the

"171

ratio -~ in equation (8) is the same for each condenser. Hence, if R and R:

are the resistances unplugged in the boxes a and a1, and K and K^ are the
capacities of C and C1 respectively, we have at any very short time t, after
the commencement of charge or discharge,

_ * _ * 2. = ?!

Klog

E

i.e. KR=K1R1, which means that the "time constants" of the two paths are
adjusted to equality.

Discharge of a Condenser when the Circuit contains

Self-inductance but not Resistance. — In this case, there are two
E.M.F.'s in existence during discharge: (1) the condenser E.M.F.; (2) the
E.M.F. of self-induction, which we know is in opposition to the rising cur-
rent, and, therefore, also in opposition to the condenser E.M.F., which
is the cause of that current. Also, as there is no resistance, there is no
resultant voltage, and, therefore, at any instant, the sum of the two E.M.F.'s
must be zero.

Let i be the current at a given instant ; then the E.M.F. of self-induction

-r di

Also, if Q be the quantity in the condenser at that instant, the condenser

E.M.F. island i=™, or ™=?£
K' dt' dt dt*

di

^ = 0 (algebraically)
K

(9)

600 VOLTAIC ELECTRICITY

This is a familiar type of differential equation, which is known to represent
an undamped simple harmonic motion, whose periodic time, T, is given by

T = 27rx/LK (10)

Hence, we find that such a discharge is oscillatory. It is not difficult to
understand the physical meaning of this result.

The charged condenser contains a certain amount of energy stored up
in the form of an electrostatic field in the dielectric. During discharge,
the static field disappears, but, in its place, a magnetic field comes into
existence, and as, owing to the assumed absence of resistance, no energy
is lost, the whole of the energy, at the instant the condenser is discharged,
is stored up in this magnetic field. Then, the magnetic field begins to
disappear, and in so doing sets up an E.M.F. in the same direction as the
discharge current which is dying away, but which is thereby prolonged, and
evidently this means that the condenser begins to charge up again, but in
the opposite direction to its previous charge. Hence, the energy oscillates
between the two forms of field, much as a spring oscillates when suddenly
released. In actual practice, however, there must be some resistance, and,
therefore, some energy (due to C2r heat in the conductor) will be lost in
each transformation, so that the oscillations will gradually decrease in
amplitude until they disappear.

It is instructive to derive the same result from the fundamental principle
for all vibrations, that, assuming no loss, the sum of the potential and kinetic
energies is constant at any given instant. Now the potential energy, at any

instant, is |-i-, and the kinetic energy is |Li2
K.

Q2

. •. J ~ + %Li2 = a constant.
.K.

Differentiating with respect to time, we obtain

.-. L + = 0, as before.
dt K.

Discharge of a Condenser through a Circuit contain-
ing Self-induction and Resistance. — in this case, there is, at

any instant, a resultant E.M.F. equal to in, and the expression becomes

This is the characteristic equation of a damped vibration. Its solution
is somewhat difficult, and need not be given here, but it leads to the result
that the periodic time is given by

T — v

~W (12)

rs?

ELECTRIC OSCILLATIONS 601

It will be seen that when K = 0, this equation reduces to the same value
as before (equation 10).

Again, when -i_=2L, the discharge is just non-oscillatory.
LK 4L

It will be noticed that the effect of damping is to alter the period of
vibration. This is a characteristic property of all simple harmonic vibration,
and it is possible to express it in another way, by writing

when I is the logarithmic decrement.

The alteration in period due to damping is of very great importance in
connection with wireless telegraphy.

Electric Oscillations. — The student should clearly realise that
loth inductance and capacity must be present in a circuit, in order
that current oscillations may occur in it. As we have seen, when
only one of these factors is present, the current rises and falls
according to some exponential function of the time, but without
oscillations. Similarly, in order that a material system may vibrate,
it must possess both elasticity and inertia, and wTe have already seen
(p. 584) that capacity (or rather its reciprocal) and self-induction
represent analogous quantities in electric circuits.

Lord Kelvin, in 1853, first obtained mathematically the result
given in equation (12), and from it, he inferred that under certain
conditions electric discharges were oscillatory. Five years later, the
fact was confirmed experimentally by Fedderson, who photographed
the image of a Leyden jar spark in a rotating mirror, and found that
it was drawn out into a series of images due to sparks following each
other in rapid succession.

High-frequency Currents.— Alternating currents of very
high frequency are most readily obtained by means of condenser
discharges. A very convenient form of apparatus l for experimental
purposes on a small scale is shown diagrammatically in Fig. 406.

FIG. 406.

A is an induction coil — the more powerful the better — worked by
a battery in the usual way. B is a condenser, consisting of one or
more Leyden jars, in parallel with the coil ; and a discharge circuit
is provided through a spark-gap and an inductance, C. (The coil

1 Although this apparatus is described here for the sake of convenience,
such arrangements were not devised until after the discoveries of Hertz,
mentioned subsequently.

602 VOLTAIC ELECTRICITY

shown on the right of C may be ignored for the moment.) The
coil charges the condenser until a spark occurs across the gap —
each discharge being oscillatory and of a frequency depending only
upon the capacity, inductance, and resistance in the discharge circuit.
The spark is short, but very bright and noisy, and the current
strength (as read on a hot-wire ammeter) may be several amperes,
very much greater than the current in the secondary of the induction
coil. The maximum value of the current is of course not given by
the ammeter, but it is remarkably great and may amount to 50-100
amperes with jars of quite moderate capacity. This is because the
actual time of discharge is very small, and hence a comparatively
small quantity of charge passing round the circuit may be equivalent
to a large current during that time.

Owing to the high frequency, inductive effects can be produced
with this apparatus in a very striking manner. For instance, let,
the inductance (C in Fig. 406) consist of 4 or 5 turns of well-
insulated copper wire in the form of a square, with sides about
2 feet long, supported on a rough wooden frame; and let another
similar coil be made with its circuit closed through a 10 or 20 volt
glow-lamp. It will be found that the lamp lights up, when its coil
is held parallel to the first coil, and anywhere reasonably near it.
When they are brought quite close together, the lamp may be
burnt out.

In a certain case, the joint capacity of two Leyden jars, which
were used, was -0038 microfarads. The coil consisted of 5 turns
wound on a square frame, each side being 65 centimetres long, and
its inductance was found to be 130,000 absolute units (often written
130,000 centimetres, as the dimensions of L are those of a length).
Neglecting the effect of resistance, which would be relatively small,
we have

also frequency ^^

In substituting numerical values in this equation, L and K
must be expressed in absolute units. It was shown on p. 583 that
1 microfarad = 10~15 absolute units of capacity, and hence we have

n = — = 230,000 alternations per second.

2^130^00 x-g*

With this arrangement, the oscillations are superposed upon a
certain amount of non-oscillatory discharge derived from the coil, and
for high-frequency currents, it is convenient to use a simple trans-
former of the kind introduced by Tesla. The primary may consist

ELECTRIC OSCILLATIONS 603

of a very few turns (perhaps less than a dozen) of thick copper wire,
without an iron core, and the secondary is a single layer of fine
insulated copper wire ending in well-insulated discharge terminals,
the windings being immersed in oil to obtain sufficient insulation.
The iron core is omitted because experience shows that its presence
does not in any way increase the power of the coiL In fact, at these
high frequencies, the effect of eddy currents in the iron is great
enough to prevent the magnetic induction from following the rise and
fall of the current with sufficient rapidity, and thus it becomes practi-
cally inert.

The two windings of this coil are shown on the right of Fig. 406,
but the secondary is there shown short-circuited, whereas it would
have been better to have introduced discharge terminals.

This transformer raises the voltage without altering the frequency,
and, when in action, a highly oscillatory discharge, several inches
long, passes across the secondary terminals. If a piece of glass be
held between them, it does not interrupt the discharge, which behaves
very much as if the glass were not there. The glass, however, soon
becomes very hot, and has, in fact, been acting as a condenser of
small capacity. It is merely an instance of the result obtained on.
p. 448, where it is shown that

In this case, although K is small, n is so great that the impedance
of the glass is scarcely appreciable. The heating of the glass shows
that some dielectric loss occurs in it at each charge and discharge (as
pointed out on p. 447).

If the hand be brought near one of the terminals, the sparks are
distinctly more painful than ordinary sparks of the same length, but
if a piece of metal on which the spark is received be held in the
hand, nothing whatever is felt. Short sparks can then be drawn from
any part of the body of the operator ; he can light a gas jet with
his finger, without being insulated from the ground; and a vacuum
tube merely held in the hand lights up more or less brilliantly. It
is a remarkable fact, first demonstrated by Tesla, that at these high
frequencies, very powerful discharges may be passed through the
body without any sensation whatever being felt. Using the apparatus
just described, with a 10-inch spark coil worked by six chromic acid
cells, one of the writers has frequently performed the experiment
of short-circuiting the secondary terminals of the Tesla coil through
his body. If the terminals are touched whilst the discharge is
passing, a shock will be felt at that instant. To avoid this, it is
better to first bring the terminals into contact with each other
(this does not injure the coil), and then they may be grasped in

604 VOLTAIC ELECTRICITY

the hands and separated without anything being felt. Before releas-
ing the terminals, they should be again brought into contact. To
indicate that a discharge is really passing through the body, a small
incandescent lamp (about 10 volts) may easily be introduced between
one hand and a terminal; its filament then glows more or less
brilliantly.

Impedance at High Frequencies.— In consequence of the
high frequency, the " choking " effect of a few turns of wire is very
great, and even straight conductors possess considerable impedance
(for reasons explained on p. 434). For instance, when the secondary
terminals were joined by a coil of well-insulated copper wire, having
only 1-5 ohms resistance, the spark discharge, although shorter than
before, still persisted, thus indicating that a very considerable P.D.
still existed across the gap.

Surface Distribution of Current at High Frequencies.—

At very high frequencies, however, another important effect comes into
operation, which makes the resistance of a conductor greater than its actual
value for steady currents. A current starts at the surface of a conductor, and
takes an appreciable, though very short, time to penetrate to the interior.
When the alternations are very rapid, it may never reach the centre at all,
and the effective cross section of the conductor is thus decreased. In fact, at
very high frequencies, a wooden rod covered with tinfoil conducts as well as
copper.

The theory of this effect has been worked out by Lord Rayleigh, to whom
the following formula is due : —

Let K-the resistance to steady currents in ohms,
l=the length of the conductors in centimetres,
w = frequency,
R! = the effective resistance,

then' E1=
This expression applies only to straight conductors.

Considering a round wire, let A = area of section = -r-, and let s = specific
resistance.

Now B--'=x'

or =

4s

Substituting this value of I, we obtain

which shows that the effect increases with the diameter of the conductor, a
fact that we might expect. If d=l centimetre, and n = 106, Rx is about 40
times R ; but if d = TV centimetre, and n=106, R^ is practically the same as
R. Hence, all conductors for use with such currents should be laminated,
i.e. made up of separately insulated small wires or strips. As permeability
influences the result, this "skin effect " is much greater for iron than for non-

RADIATION 605

magnetic metals. For instance, at a frequency of 100, a current will penetrate
to a depth of about 26 millimetres in copper, but only about 2 millimetres in
iron ; and at a frequency of 106, these values become about ^ millimetre for
copper, and -^ millimetre for iron.

Electro-magnetic Radiation.— Fitzgerald appears to have
first pointed out that the oscillatory discharge of a condenser must
set up electro-magnetic waves in the surrounding space, and on the
basis of Clerk-Maxwell's theory, he inferred that such radiations must
be identical with ordinary light, in all essential respects except
frequency. Little progress was made, however, until, in 1888,
Hertz brought the subject within the range of experiment by devising
a convenient method of producing oscillations of extremely high
frequency, and (what was even more important) a method of detecting
the waves thereby radiated into space.

His " oscillator " took the form shown in Fig. 407. Two large
metal plates (or two conducting spheres) are fitted with long metal rods

or wires, between the ends of
which is taken the spark from
an ordinary induction coil. The
capacity is mainly provided by
the plates, and the inductances
by the rods — both quantities
being very small. The plates
may, in fact, be regarded as the
two coatings of a condenser
FIG. 407 separated by a very long air-

space. The induction coil charges

up the plates until the P.D. across the gap rises to the sparking
value, and then the spark practically amounts to connecting the rods,
very suddenly, by a path of low resistance, across which the discharge
oscillates at a rate which (as already stated) does not in any way
depend upon the induction coil.

The magnitude of the various quantities involved will be realised
more clearly if we make a rough estimate of their value in a particular
case. Suppose (merely for convenience of calculation) that the plates
are circular and 50 centimetres in diameter, and that the rods are
each 50 centimetres in length and '5 centimetre in diameter. Now,

'2?*

we know that the capacity of a circular disc in static units is — , where

7T

2 x 25
r is the radius. Hence, the capacity of each plate is — — = 16 static

7T

units. The two plates are really in series, so that the total capacity

1 s*

is — = 8 static units (assuming, as a first approximation, that all the

2

capacity is in the plates)

It is somewhat difficult to calculate the self-inductance of a

606 VOLTAIC ELECTRICITY

straight wire, and here it will be sufficient to give (without proof)
the following expression for it —

where I is the length and d the diameter of the wire.

We, therefore, have for a total length of 100 centimetres of wire —

L = 2 x 100 (loge i^ - 1\ = 1 140 absolute units.
As before, the frequency, n, is given by

—I-

27TVLK

In this equation, L and K must be expressed in the same system of
units (although it is immaterial which system we use). On p. 582,
we have shown that the absolute unit of capacity in the magnetic
system is v2 times greater than the static unit, and hence K is

(3 x i0io)a magnetic units.
* = 500 millions per second.

This is the rate at which electric charges surge backwards and for-
wards in the plates and rods. To understand how such oscillations
may give rise to wave motion, we must remember that during the
very brief time required to charge the plates, an electrostatic field has
been forming in the surrounding space, which must be regarded as
extending without limit, although it is very weak except in the
immediate neighbourhood of the plates. At the instant they are fully
charged, we may think of these lines as extending in sweeping curves
from one plate to the other, and, for the moment, as being stationary.
Then, as the plates begin to discharge, the ends of these lines move
towards each other along the rods (which is the same thing as saying
that a current flows), and a magnetic field begins to form and to
extend throughout all space, which, as already explained (see p. 600),
must in its turn disappear, re-forming another electrostatic field
reversed in sign of charge.

Under ordinary circumstances, i.e. at moderate frequencies, the
energy represented by the two forms of field merely surges backwards
and forwards without sensible loss ; but at very high frequencies, a
certain portion may never return to the circuit, but may be thrown
off into space. If, for example, the two ends of an electrostatic line
move together with sufficient rapidity along the rods and across the
spark-gap, we may regard them as meeting and crossing before the
line itself has had time to contract and disappear. The result will be

RADIATION 607

the formation of a closed loop, which is detached and thrown off into
space. Such a loop represents a certain amount of energy, although
it is essentially unstable ; it must immediately begin to contract, and
the contraction must continue until finally it disappears. We have
seen that the contraction and disappearance of an electrostatic field
always produce a magnetic field linked with it. Hence, a magnetic
field forms, which reaches its maximum strength at the instant the
static loop vanishes, but this, in its turn, is unstable and must
immediately begin to contract, with the formation of another static
field. As no energy has been dissipated by resistance, the cycle of
changes persists indefinitely, the result being an electro-magnetic
wave.

It is now certain that ordinary light- waves are of the same nature,
but of very much greater frequency. For instance, the smallest
frequency of a visible light-wave is about 10 million times greater
than that of the waves radiated from the Hertzian oscillator described
above. Again, as the wave-length is obtained by dividing the velocity
of propagation by the frequency, in the case worked out it will be

3 x 1010

= 60 centimetres, whereas the wave-length of the longest

5 x 10b

visible light- wave is about T4Q00 centimetre,

Damping" Due to Radiation. — It is now evident that, even
if oscillations occur in a path of no resistance, they will not persist
indefinitely if radiation occurs, and the more powerfully the system
radiates, the greater will be the damping,, The rate of radiation of
an oscillatory circuit does not depend merely upon the frequency, but
also upon its form. Just as ordinary substances at a given temperature
differ in their radiating powers for heat, so do different types of
circuit differ as regards their readiness to emit electro-magnetic waves.
Those circuits in which the plates of the condenser (or its equivalent)
are widely separated, so that the lines of electrostatic force extend far
into space, are the best radiators ; and those in which the condenser
plates are close together (as in the arrangement shown in Fig. 406,
using ordinary jars) are poor radiators. In the first type, energy is
rapidly lost and the oscillations die out quickly ; in the second, the
amplitude decreases much more slowly and the oscillations tend to
persist. For this reason, in a Hertzian oscillator — which is a good
radiator — only a few strongly damped vibrations occur at each spark,
and there is a distinct interval between the sparks during \vhich no
oscillations are taking place. In fact, the actual time, during which
oscillations are occurring and giving rise to radiation, is an extremely
small fraction of the total time. Hence, although the amount of
energy radiated per second may be quite small, the rate at which it is
radiated is usually very great. In the particular case taken, it would
probably be more than 50 horse-power.

Detectors of Radiation. — Hertz, by devising a form of de-

608 VOLTAIC ELECTRICITY

tector for electro-magnetic waves, was the first to show that such
radiation actually existed. His receiver took various forms, the
simplest being a circle of wire, two or three feet in diameter, ter-
minated in knobs which were separated by a small air-gap. Such

a circuit has inductance and ca-

^M^MBM , , ••••• pa™'*-y — both, of course, small —

and when held -in a suitable posi-
tion near the oscillator, as shown
in Fig. 408, the radiation falling
upon it sets up a rapidly alter-
nating P.D. between the terminals,
and the surging thus produced
becomes great enough to spark
across the gap. The best effect
FIG. 408. ^s obtained when the dimensions

of the receiver are such that its

natural period is the same as that of the oscillator, but, as a matter
of fact, the period of the oscillator becomes somewhat indefinite on
account of the excessive damping, and anything like exact " tuning "
is impossible in practice, so that a very fair approximation to the
exact dimensions, found by trial, gives good results, when the distance
is not greater than a few yards. In the position shown in the figure,
it will be evident that the inductive effect on the ring is entirely due
to the magnetic component, for the direction of the static field is such
that it cannot produce a P.D. between the knobs. If the ring were
rotated through 90° in an anti-clockwise direction, the static compo-
nent would also become effective, for then static lines would pass
from knob to knob. But if it were to be tilted through 90° into a
vertical plane, both components become inoperative, and the sparking
disappears.

Such a receiver is very inconvenient and troublesome in practice,
but with'its aid Hertz was enabled to demonstrate that the radiations
from his oscillator travelled at sensibly the same speed as light, and
also to measure their wave-length and frequency. This was ac-
complished as follows : The vibrator was placed before a metallic
reflector, so that the waves returned back upon themselves. In this
case the interference between direct and reflected waves formed
stationary nodes and vibrating segments, whose positions could be
determined by recurrent maxima and minima of sparking in the
resonator. The distance between two positions of maximum sparking
being half a wave-length, the velocity of propagation could be easily
found, when the number of vibrations per second was known. This
gave a velocity nearly equal to that of light, although the conditions
of experimenting were such as to permit of approximate measurements
only.

Hertz also showed that the radiation was almost perfectly reflected

RADIATION 609

by metallic surfaces, whereas it readily passed through non-conductors,
such as paraffin-wax, pitch, or even a brick wall. He also showed
that the ordinary laws of reflection and refraction were obeyed ; for
the latter purpose he used a large prism of pitch.

These researches, here very briefly described, supplied the experi-
mental proof which was required to establish the theories of Maxwell.

Although Hertz's form of oscillator is not used for commercial pur-
poses, little has been changed since his time in the method of producing
electro- magnetic radiation, except in matters of detail. Perhaps the
greatest advance has been made in the direction of obtaining un-
damped and persistent trains of waves by making use of certain
properties of the arc,, to be subsequently described. On the other
hand, enormous progress has been made in devising delicate and
convenient receivers or detectors of radiation, and it is chiefly owing
to these improvements that wireless telegraphy on a commercial scale
has become possible.

The " Coherer." — The second important step was the invention
of the "coherer," which is based upon a peculiar microphonic be-
haviour of loose metal contacts to electric waves, detected some twenty
years ago by Hughes, but first brought into prominent notice by
Branly, and afterwards more thoroughly investigated by Sir Oliver
Lodge, to whom the term "coherer" is due. Its action may be
easily demonstrated by fitting, say, two pins into a narrow glass
tube about an inch long, which is partially filled with iron or nickel
filings.1 If this tube be arranged in circuit with a single cell and a
reflecting galvanometer — in which it is best to include, for convenience
of adjustment, a steadying resistance of, say, 500 or 1000 ohms —
there will possibly be no deflection, or, at any rate, a very slight one,
owing to the almost infinite resistance offered by the filings. If,
however, an electric spark be produced in the neighbourhood, the
impact of its waves upon the filings enormously reduces in some way
their resistance, and a sudden deflection of the galvanometer is pro-
duced, the coherer retaining its low resistance until its particles are
disturbed by tapping or shaking. This simple form is not very
reliable, and it was much improved by Marconi. His pattern consists
of a narrow glass tube, about _^

an inch and a half long, con- - JJJ jj| ^)O

taining accurately-fitting silver u^ m *™

terminals (A, B, Fig. 409) FIG. 409.

about half a millimetre apart,

between which are the metal filings. The tube is partially exhausted

before sealing — it is, in fact, a small vacuum tube.

Suppose now that an ordinary sensitive relay is inserted in the
coherer circuit instead of the galvanometer. Then evidently the

1 The best results are obtained with iron or nickel mixed with a small
proportion of silver or gold filings.

2Q

610 VOLTAIC ELECTRICITY

impact of an ether-wave will close the relay, and thus actuate an
independent battery working an ordinary telegraphic receiver. If to
this an automatic tapper be added to " decohere " after each signal,
we have a workable method of signalling.

The great drawback of this form of coherer lies in the fact that it
does not automatically return to 'its original state. Modifications
possessing this property have been devised, among which may be
mentioned Castelli's form, consisting of a globule of mercury between
two iron surfaces. Sir Oliver Lodge uses a steel disc, revolved by
clockwork, which just dips into a vessel containing mercury covered
with a thin layer of paraffin-oil. A cell is connected through some
form of receiving instrument, such as a syphon recorder, to the disc
and mercury respectively, but the layer of oil forms a bad contact and
interrupts the circuit. When, however, electric waves fall on it, the
oil film is pierced and a current flows of sufficient strength to give
signals, the arrangement returning at once to its former state. With
such detectors, it is now a common practice to dispense with ordinary
telegraphic apparatus, and to receive the signals as clicks in a tele-
phone suitably connected to it.

Wireless Telegraphy. — The general arrangements adopted for
"sending" purposes in Marconi's system are indicated in Fig. 410.

A is a long vertical wire, or

i group of wires, insulated at the

£ i upper end and earth-connected

i at the lower end. It is known

as the "antenna" or "aerial."

1^ In it is inserted (1) an induc-

^ tance, L, so arranged that the

^ number of turns may be readily

adjusted ; (2) the secondary of
a special kind of transformer, T,
in which alternating currents of
high frequency are produced.
The primary of this transformer
has only one turn. The aerial pos-
£> I £? ~ B sesses a certain small capacity,

~~^~ and, therefore, also a natural
period of oscillation, which may
be adjusted within certain limits
by varying the inductance, L.
Its function is to radiate into

FIG 410 space, in the form of waves, the

energy supplied to it by means
of the transformer, T. The earth connection is very important,
because it ensures that the lines of electrostatic force are thrown off
as loops closed through the earth, and hence the waves creep along

~R r i

tit!

WIRELESS TELEGRAPHY AND TELEPHONY 611

the earth's surface, instead of being propagated in straight lines.
Were it not for this fact, the rotundity of the earth would be a
serious obstacle to long-distance telegraphy. It follows, however,
that the conductivity of the earth's surface is an important factor.
When it is a good conductor, as at sea, the best results are obtained,
but over long stretches of dry or rocky ground signalling is difficult
or impossible.

The aerial is usually made of copper or of aluminium ; iron is not
suitable because its magnetic properties damp down the oscillations
too rapidly, although, as high-frequency currents are practically con-
fined to the surface, it will serve if thinly plated with copper.

It will be seen that the remainder of the apparatus is essentially
the same in principle as that already described (in Fig. 406).

B (Fig. 410) is a battery working an induction coil, I, the signals
being sent by means of the key, Jc. The secondary is connected to a
circuit containing a condenser, K, a spark-gap, S, and the primary of
the transformer, T, which feeds j
the aerial. The capacity of the
condenser is adjustable, and the
frequency is thereby adjusted to
bring it in "tune" with that of
the aerial.

At the more powerful stations
used for long-distance work, the
most recent development is that
of charging the condenser directly *-
from a battery of 6000 accumu-
lators in series, giving 11,000 to
12,000 volts, and the spark-gap
is replaced by the "disc dis-
charger" shown diagrammatically
in Fig. 411.

B is the battery which feeds
(1) the condenser through chok-
ing coils, ccy (to prevent back-
rush), whilst the discharge circuit
contains the primary of the aerial
transformer, T, and the disc dis-
charger. This last consists of a
metal disc, D, having a number of
copper studs, SS, fixed at regular
intervals around its periphery,

and rotating at a very high FlG 411

speed between two other discs,

dd, which rotate slowly in a plane at right angles to that of D.
The studs just touch the side discs in passing and momentarily bridge

612 VOLTAIC ELECTRICITY

the gap. The sparks occur between the studs and side discs before
contact, and the immediately ensuing contact enables oscillations to
take place without loss of energy due to the resistance of the spark-
gap. The latest form of condenser consists of insulated metal plates
suspended in air, thus dispensing with the use of glass. This
eliminates a serious loss of energy due to "dielectric hysteresis" pro-
duced in solid dielectrics ; and is also economical from another point
of view, for all condensers of the ordinary type soon deteriorate
with use.

The arrangements at the receiving station differ considerably in
detail, according to the form of detector adopted. In all cases, how-
ever, the essential parts are (1) a second aerial, which receives the
incident radiation and converts it into a rapidly oscillating current ;
(2) a transformer, having its primary in series with the aerial, and its
secondary in connection with the detector and auxiliary apparatus,
the best results being obtained when the receiving aerial is tuned in

unison with that of the emitting station.
Fig. 412 shows one of the simpler ar-
rangements adopted by Marconi. The
aerial, A, is connected to earth through
the adjustable inductance, L, "and the

L primary of the transformer, T. The

^ secondary of this transformer is in

circuit with a condenser, K, and a
coherer, C, from which wires, WW, are
led off to some form of receiver. For
instance, they may be connected to a
cell and a sensitive relay governing
ordinary telegraphic apparatus, or to a
~v7 cell and a telephone, in which case a
click is heard when the incidence of a
W wave reduces the resistance of the
'"" ~ coherer and thus permits the cell to
send a current through the telephone.
With the telephone, however, as already
stated, it is necessary to use some form
of detector which automatically and
instantly regains its original state.

For very long-distance work, the aerials are now usually pro-
longed horizontally, as it has been found that this arrangement,
instead of radiating in all directions alike as does the vertical form,
tends to radiate most strongly in a direction opposite to that in
which the aerial points, thus effecting a partial, but very useful, con-
centration in any desired direction.

Perhaps the most important applications of wireless telegraphy
is in connection with signalling between vessels at sea, or between

WIRELESS TELEGRAPHY AND TELEPHONY 613

such vessels and shore stations. In ship installations, the details are
simplified as much as possible to lessen the risk of a breakdown, and,
by international agreement, the normal wave-length for that purpose
is 300 metres (corresponding to a frequency of one million per
second), although 600 metres is also permitted. Every coast station
employs one or other of these wave-lengths. For transatlantic pur-
poses a longer wave-length is usually employed.

Marconi has recorded certain important phenomena, which as yet
cannot be fully explained. For instance, it is much more difficult to
transmit signals over long distances by day than it is by night.
Another very curious fact is that short wave-lengths are more affected
than longer ones. The difficulty itself may be due to a partial
conductivity of the air owing to ionisation produced by sunlight,
but no satisfactory explanation has been suggested regarding the
latter fact.

We have briefly outlined the methods adopted in one system of
wireless telegraphy, but many other systems, based upon the same
fundamental principles, are in use. Again, the number of detectors
which have been or are actually employed in the various systems is
so great, that we can only enumerate the more important, without
attempting any explanation of their action. They may be regarded
as devices for detecting very feeble alternating currents of high
frequency.

Detectors of Electro-magnetic Waves. —

(1) Those depending upon the production of a visible spark.
(The original Hertz receivers are instances. These are not very
sensitive.)

(2) Those depending upon the coherer principle, i.e. upon a bad
contact in some form.

(3) Magnetic detectors, depending upon the demagnetising effect
of a feeble oscillatory current upon the magnetism of a steel bar.
(Introduced by Rutherford, materially improved by Marconi— very
reliable and very sensitive.)

(4) Electrolytic detectors. (A thick platinum wire dips into a
vessel containing dilute sulphuric acid, and another extremely fine
wire dips only a fraction of a millimetre into it. Owing to the back
E.M.F. of polarisation, a single cell can send practically no current
through this arrangement, but the polarisation is temporarily reduced
by the passage of a feeble oscillatory discharge. This is very
sensitive and convenient in use.)

(5) Thermal detectors, in which the oscillatory current produced
in the receiving aerial is passed through an extremely fine wire,
and is detected by its heating effect. In some forms, use is made
of the change of resistance due to the heating; in others, a small
thermopile is in contact with the wire. Duddell's thermo-galvano-
meter (see p. 298) can be used in this way, the current from the

614 VOLTAIC ELECTRICITY

aerial passing through the fine- wire heater. Such detectors are not
so sensitive as either of the types (3) or (4).

(6) Rectifying detectors, which depend upon a difference in
conductivity in different directions, especially marked in certain
crystals. As a result, an oscillatory current can be partially trans-
formed into a uni-directional current by stopping the flow in one
direction. A cell in circuit with a telephone may be arranged to
maintain a P.D. across the arrangement, the current then being
increased by superposing an oscillatory discharge. Among these,
we may mention (1) a carborundum crystal between two brass plates,
(2) a corner of a fragment of zincite (native oxide of zinc) pressing
against a piece of chalcopyrite (iron-copper sulphide), (3) a pointed
piece of graphite touching the face of a crystal of galena. These
receivers are very sensitive, and are largely used in actual practice.

(7) Those depending upon certain properties of an electric
discharge in gases at low pressures. The most important of these
is Professor Fleming's oscillation valve, which depends upon an
effect discovered many years ago by Edison. It is really a carbon
filament glow-lamp, containing also a metal plate. When the carbon
filament is heated to incandescence by a local battery, a single cell
will send a current in one direction between the cold plate and
the hot filament, but not in the other direction. The cell must
be joined up so that its negative terminal is connected with the
negative end of the filament. When the aerial replaces the cell, the
oscillatory currents induced in it are converted into uni-directional
currents, and may be used to deflect a galvanometer or to work a
telephone. This is an excellent detector and is much used.

Wireless Telephony. — This is a much more difficult problem,
which up to the present has been only partially solved, the greatest
distance as yet covered being about 200 miles. Without going
into details, it may be pointed out that in order to transmit speech,
and not merely audible signals, it is essential (1) that the sending
apparatus should emit a continuous and uniform series of undamped
waves of high frequency, and (2) that these waves must be controlled
in duration and amplitude by some microphonic arrangement set in
action by the voice. (3) The effect on the receiver must be propor-
tional to the strength of the current induced in the aerial, i.e. it must
respond to rapid and slight changes in the waves.

Undamped Waves. — Undamped wave-trains are also ex-
tremely useful in wireless telegraphy, on account of the readiness
with which exact tuning can be obtained, the consequent sharpness
of resonance between the receiving and the emitting circuits greatly
reducing thereby the power required for signalling over a great
distance. It is merely an instance of the well-known fact in ordinary
mechanics that very feeble impulses, if only persistent and regularly
timed, will set up vibrations of great amplitude in a body capable

WIRELESS TELEGRAPHY AND TELEPHONY 615

of vibrating with the same frequency, whereas very much more
powerful impulses, if irregular, will produce only a comparatively
small effect. At the same time, although several methods of pro-
ducing undamped trains of waves are now known, they have not yet
been very extensively employed in wireless telegraphy.

The most direct method of producing such waves would be to
excite the aerial by means of an alternating dynamo designed to
give the necessary high-frequency. Machines of this type have been
made, but the practical difficulties are very great, and at present
there is little prospect of overcoming them. The earliest successful
method, due to Poulsen, is based upon the properties of Duddell's
musical arc.

Duddell's Musical Arc. — Duddell in 1900 discovered that, if
a continuous current arc, fed by some steady source of E.M.F. such as
a battery of storage cells, was established between solid carbons

~B

SAMA/WVVW"

FIG. 413.

(ordinary carbons have a core of special composition running through
them centrally), and shunted with a capacity and a self-inductance,
persistent oscillations were set up in the shunt circuit, whose
frequency depended entirely upon the constants of that circuit.
When the frequency was suitable, the arc emitted a loud musical
note, and by arranging a variable disposition of capacity and in-
ductance controlled by a simple keyboard, it was possible, for instance,
to make the arc play a simple tune. His arrangement is shown in
Fig. 413 (taken from J. A. Fleming's Electric Wave Telegraphy],
in which B is a battery supplying the arc A through a resistance R
of about 40 ohms. C is a condenser, which may be from 1 to 5
microfarads capacity, and L a coil having an inductance of about
'005 henry and a small resistance of about *5 ohm. According to
Duddell's theory, such oscillations depend primarily upon the peculiar
relation between the current through the arc and the P.D. across

616 VOLTAIC ELECTRICITY

it (already alluded to on p. 553). Imagine a condenser to be
suddenly connected across the arc. It at once begins to charge up,
and in doing so slightly reduces the current through the arc. This
reduction, however, implies a rise in the P.D. between the carbons,
and therefore tends to charge the condenser still further. When
the charging is completed, the current rises to its old value, and,
in consequence, the P.D. between the carbons falls and the condenser
begins to discharge. The result is an increase of current, which
lowers the P.D., and thus facilitates the discharge.

Thus, the condenser alternately charges and discharges, and as it
is part of a circuit possessing a natural period of vibration, these
actions tend to set up oscillations of the corresponding frequency,
which differ from those produced by a spark in being regular and
persistent, instead of being merely a series of separate and damped
wave-trains. The frequencies readily obtained in this way, vary
from about 500 to 10,000, and are not great enough for the purposes
of wireless telephony. It might be thought that any desired fre-
quency could be obtained by giving suitable values to the capacity
and inductance of the shunt circuit, but, as a matter of fact, the
very small capacity required for high frequencies does not produce a
sufficient change in the strength of the arc current to maintain the
action properly.

In 1903, Poulsen of Copenhagen pointed out the great advantage
of surrounding the arc with a hydrogen atmosphere, when very high
frequencies are desired. In his form of apparatus, the arc is pro-
duced between a negative electrode of carbon and a hollow water-
cooled positive electrode of copper. It is placed in a chamber filled
with coal gas (or the vapour of any hydrocarbon), and a strong
transverse magnetic field is brought to bear upon it, which tends
to blow it out and thereby increases the P.D. required to maintain
the arc between the electrodes. A small arc is used, taking 1J to
2 amperes at about 220 volts. As a result, Poulsen obtained much
higher frequencies — of from 100,000 to 1,000,000 per second.

Apparently these modifications facilitate the very rapid variations
in the intensity of the arc current which are required — the hydrogen
atmosphere by its cooling effect, and the magnetic field by its tendency
to quicken the dying away of the arc by blowing it out.

As regards the working details of telephonic systems, the oscil-
latory circuit is linked with an aerial, in essentially the same way as
that shown in Fig. 410, and a carbon microphone is often used as
a transmitter. This may be connected up in several ways ; for
instance, it may be placed in series with the aerial between the
latter and the earth, thus controlling the amplitude of the current
induced in the aerial itself.

The arrangement of the receiving circuit is also essentially the
same as that already described. It is, however, necessary that the

WIRELESS TELEGRAPHY AND TELEPHONY 617

detector should be of a kind which will respond quantitatively to
the small changes in intensity of the current induced in the receiving
aerial. When only the Morse code of dots and dashes has to be
signalled, it does not signify whether the effect produced on the
detector varies exactly with the current strength or not, as long as
it responds at all ; but for reproducing articulate speech, it is evident
that such a condition is of the utmost importance. Fleming's
oscillation valve or the electrolytic detectors are very suitable for
the purpose.

Hitherto, the greatest obstacle to the practical development of
wireless telephony has been the relatively great cost of the apparatus
required as compared with that with ordinary telephony. It seems
probable that this difficulty will be eventually overcome ; in fact,
at the time of writing, several new methods, of which particulars are
not yet available, are being brought forward which appear to have
been fairly successful over short distances.

The "Telefunken" or "Quenched Spark" System.—

This is a " spark " method of producing wave-trains which are only moderately
damped. To explain the principle involved, it is necessary to mention a well-
known effect which occurs in all cases when two vibrating systems —
mechanical or electrical — are tuned to resonance. For instance, let the
aerial in Fig. 410 be tuned to the same frequency as the primary cir-
cuit containing the spark-gap. Then the oscillations induced in it react
inductively again on the primary circuit itself, and the result is an inter-
change of energy between them, which leads to phenomena essentially
similar to "beats" between musical notes, the amplitude of the vibrations
waxing and waning alternately. It can be shown that this is equivalent
to the production in the aerial of two independent vibrations of nearly
equal frequency, and, as the receiving systems can only be tuned into
unison with one of these frequencies, there is a consequent waste of
energy.

Any two vibrating bodies, which can thus react upon each other, form a
"coupled system." When the coupling is " loose " — i.e. when the arrange-
ment is such that the motions of one only slightly affect the other — the
above action is negligible, and thus only one frequency is induced when
resonance, occurs. Two electric circuits are " loosely coupled " when their
mutual inductance is small, and " tightly coupled" when it is large. In the
limiting case of exceedingly tight coupling — equivalent to a rigid connec-
tion—there is again only one induced frequency at resonance; but this is
not, as a rule, a possibility.

This action and reaction between the aerial and the primary may be
avoided by abruptly stopping the primary oscillations as soon as the maxi-
mum amount of energy has been transferred to the aerial ; and in 1906, Wien
showed that this could be done in a simple manner by using exceedingly
short sparks between large masses of metal. With this arrangement, the
resistance of the spark-gap is very quickly restored by cooling, and the
circuit is thereby broken at the first or second oscillation. The result is
equivalent to a single impulse given to the aerial, which then vibrates without
further disturbance at its own natural frequency, and with very much less
damping.

Fleming's Cymometer. — The actual frequency of the oscillations
in an aerial or in any circuit, can be readily measured by the aid of this

618 VOLTAIC ELECTRICITY

instrument. It consists of an inductance and a condenser in series with each
other, the circuit being completed by a straight metal bar.

The inductance is a wire solenoid with a sliding contact, and the con-
denser is formed by one cylinder sliding inside another. The arrangements
are such that the inductance and capacity can be simultaneously altered by
the same motion of a handle, to which is attached a pointer moving over a
fixed scale.

When the straight bar is placed near and parallel to any conductor in
which oscillations are occurring, induced oscillations of identical frequency
are produced in it (the coupling being designedly very loose), and the in-
ductance and capacity can be adjusted until the maximum effect is obtained.
When resonance occurs, the maximum P.D. exists between the coatings of
the condenser ; and a vacuum tube connected across them (preferably con-
taining neon on account of its brightness), then lights up most brilliantly.
Evidently the natural period of the cymometer circuit is then equal to that of
the aerial, and the frequency is obtained at once from the scale reading and
the constant of the instrument.

APPENDIX

HINTS ON WIRELESS TELEGRAPHY FOR TEACHERS
AND PRIVATE STUDENTS

SINCE this book was written rapid advances have been made in wire-
less telegraphy. Until the interruption occasioned by the war, daily
time signals and weather reports w^ere sent out by various stations at
home and abroad.

It is a comparatively easy matter (after obtaining the needful
license from the Postmaster-General) to arrange apparatus for receiv-
ing these signals, and such work interests many persons who have no
special desire to transmit messages themselves, besides being of the
greatest value in illustrating and elucidating many of the fundamental
principles of electricity.

We, therefore, append a brief outline of the simpler methods in
use. It must, however, be understood that we are writing not for
the professional operator, to whom complexity of detail is immaterial
and absolute certainty all-important, but for teachers and private
students.

The Aerial or Antenna. — Most of the stations we desire to
pick up use long wave lengths (of the order of 2000 metres and
upwards), because long waves carry farther than short ones, i.e. are
less scattered by obstacles, such as houses, &c. Hence the natural
wave length of the aerial must also be long. Now the wave length
of a single vertical wire, in the limiting case when it is infinitely thin,
and, therefore, of negligible capacity, is four times its actual length, and
in the case of real wires this number lies somewhere between 4 and 5.
Taking 4' 5 as an approximate value, it follows that we require an
aerial about 450 metres in height — obviously an impossible value.
By connecting an inductance, or "tuning coil," between its lower end
and the earth, the wave length of a shorter aerial can be raised to
any required value ; but this method must not be pushed too far, for
the tuning coil itself plays no part in picking up radiation. It is,
therefore, desirable that the aerial should have a reasonably long
wave length to begin with, a result most easily obtained by using
several wires in parallel, thereby increasing the capacity.1 Again,

1 It may be noticed that adding wires in parallel decreases the inductance,
but this is readily made up by the tuning coil.

619

620 APPENDIX

it is seldom convenient to use the vertical position ; most likely
the aerial will be slung between two chimney stacks, or poles
attached thereto, or even between two trees. In any case, the
height should be as great as possible, for on it the power of picking
up a distant station mainly depends ; and if one end has to be higher
than the other, it will be better to lead into the instruments from
the higher end. We are thus led to some such design as that shown
in Fig. 414, it being understood that, although three wires in parallel
are shown, the precise number is to some extent immaterial, and is
determined by convenience. The individual wires should be spaced
at least 3 feet apart, for proximity still further reduces self-inductance ;
and although a single leading-in wire may be used, it is preferable to
continue the three wires downwards, and to join them to a single lead-
ing-in wire just before they enter the building containing the instru-
ments. The span is usually dictated by circumstances, and may be
from 50 to 100 feet or more.

In Fig. 414 BB are two light bamboo rods, 8 or 9 feet long and

B

FIG. 414.

about IJinch in diameter. (These can be obtained from makers of
bamboo furniture at about 6d. each.) To these are attached three
18-gauge wires, preferably of phosphor-bronze on account of its great
tensile strength. About 4 Ibs. of wire may be required, costing per-
haps 6s. The three wires should be of exactly the same length,
counting to the place where they join the leading-in wire. They are
sometimes joined together at the other end also, but it is rather better
to keep them separate. The two rods can be slung in the required
positions by means of galvanised iron wire or rope (wire rope is the
better, but it is somewhat expensive), merely intercalating insulators
EE. These consist preferably of ebonite rod, 8 or 10 inches long and
1 J inch in diameter, with a hole or groove at each end to affix the
suspensions. The leadiug-in wire should be well insulated, and close
proximity to the wall of a house should be avoided. A convenient
method of bringing the wire into an ordinary room is to fit a board
under the raised window sash, carrying an ebonite tube (or an
ebonite plate with central hole) for entrance, and having another hole,
which need not be insulated, for the outgoing earth connection. Inside

APPENDIX 621

the room the leading-in wire terminates in some form of switch, which
enables it to be joined either directly to earth or through the receiving
apparatus and then to earth. For safety during thunderstorms it is
desirable to arrange some device which allows the aerial to be con-
nected directly to earth outside the building, in which state it should
be left when not in use. For the " earth " itself, in the case of
private stations, there is probably nothing better than a stout copper
wire well soldered to the water pipe system ; but it must be remem-
bered that a really satisfactory earth is all-important, and it is well
worth while taking pains to obtain it. It may be pointed out that
the wave length depends upon the total length of aerial, which
includes the leading-in wires and the earth connection.

Receiving Apparatus. — This may be either of two distinct
types — (1) a tuning coil alone maybe used to provide both tuning
inductance and the inductance for the receiving circuit; (2) an
"oscillation transformer," commonly called a "jigger," may be
used.

The first method is the simpler; the second admits of more
selective tuning, but it requires more skill to operate, and is not so
well adapted to the needs of a beginner.

Tuning Coil Method. — The tuning coil itself may be made
by winding a single layer of (say) 22-gauge enamelled copper wire
on a cardboard tube about 15 inches long and 5 inches in diameter,
which will give a total inductance of about 9000 micro-henries. In
two places the enamel is carefully removed with emery-paper along
the whole length of the coil, and two sliding contacts are provided.
In addition, a variable condenser, some form of crystal detector, and
a pair of high resistance telephones with headgear are required.

Condenser. — The pattern is immaterial, but a cylindrical form
is very convenient, in which the coatings are brass tubes, one sliding
within the other. The outer tube may be 5 or 6 inches long, and
about f inch in diameter ; the inner tube, of the same length and of
such diameter as to slide easily in or out when covered with paraffined
paper for insulation. The dielectric is thus partly paraffined paper
and partly air, and the insulation is ample for the voltage it will have
to stand. It is convenient to have two condensers of this size, which
may be joined in parallel if required. In case of emergency, a useful
variable condenser may be made from two glass test tubes, one sliding
within the other, and each provided with a suitable tinfoil coating.

Telephones. — These are specially wound for the purpose, and
should be purchased from a reliable firm. The resistance should not
be less than 1500 to 2000 ohms. It may be pointed out that the
unscientific custom of rating such telephones in terms of their resist-
ance leads to curious misconceptions. Ordinary telephones have a
resistance of 100 to 200 ohms, and the only reason why they are
unsatisfactory for the present purpose is because they have not a

622 APPENDIX

sufficient number of turns. In fact, with a crystal detector, the
current available is so extremely small that an enormous number of
turns is required to give the needful ampere-turns. Now, in order
to get these turns into the available space extremely fine wire must
be used, and this inevitably means a high resistance. The high re-
sistance itself is merely a necessary evil — not an advantage — but
instances have occurred in which people who should know better have
wound their telephones with high resistance wires, such as eureka or
platinoid, in -order to obtain more readily what they deem to be the
needful resistance.

Crystal Detectors. — These have been briefly referred to already
on p. 614. As they are especially suitable for use by beginners, it
is desirable to explain their action more fully. In the first place,
whatever form of receiving apparatus may be adopted, the final result
will be a series of feeble alternating currents. If these currents pass
through the telephones no sound will be heard, not because they are
alternating currents, but because their frequency is so great (e.g.
150,000 cycles per second for a 2000 metre wave) that the movement
of the diaphragm cannot possibly follow it.

Spark and Wave Frequency. — Further, a clear distinction
must be made between "spark frequency" and "wave frequency."
When a transmitting circuit is closed, sparks succeed each other at
a rate which, for an ordinary coil, is determined by the type of
contact breaker used, or with the arrangement shown in Fig. 411,
p. 611, by the speed of the disc discharger. In any case, it is never
more than a few hundred per second. This is " spark frequency."
Each spark represents a series of electrical oscillations, or a " wave
train " of enormously greater " wave frequency." If now we place in
series with the telephones some rectifying device which transmits
current in one direction only, or which transmits it in one direction
much better than in the other, then each wave train becomes resolved
into a series of currents in the same direction through the telephone,
succeeding each other so rapidly that the effect on the more slowly
vibrating diaphragm is equivalent to a single impulse. These impulses
are repeated at the rate of the spark frequency, and this is the rate
at which the diaphragm vibrates. Now, the human ear is most
sensitive to sounds having a frequency of about 500 per second,
and hence it is very usual to adjust the spark frequency to something
like this value. The diaphragm then emits a musical note as long
as the transmitting spark persists. In fact it soon becomes possible
to recognise several of the more important transmitting stations by
the pitch of the note alone. Thus the Eiffel Tower works with a
relatively low spark frequency giving a low grunting note, whilst
Cleethorpes gives a clear musical note of much higher pitch.

Arrangement and Working of Crystals. — Not only certain
crystals but also various other substances possess more or less rectify-

APPENDIX

623

ing power. One of the most sensitive combinations is a lump of
fused silicon, on which rests lightly a gold point. Another form in
common use is the " perikon " detector, which is a crystal of zincite
touching another of bornite. These substances are conveniently
mounted in little brass cups, imbedded in Wood's fusible alloy (an
alloy which melts below the boiling point of water, and is, therefore,
easily managed), and are held in some form of spring holder, which
permits ready adjustment of pressure and of the point of contact.

The above detectors work well alone, but most crystals give
better results when a steady E.M.F. is applied to them by a local
battery. As it is most important that this auxiliary E.M.F. should
have exactly the right value — which varies from J to 2 or 3 volts
according to the nature of the crystal — it is usual to regulate its
application by means of a potentiometer.

Personally, we have found that the most generally reliable and
convenient — although not the most sensitive — detector consists of a
crystal of carborundum between a steel and a brass surface, in con-
junction with a potentiometer.

An auxiliary E.M.F. enables us to turn to useful account not
only the rectifying power of the crystal, but also the fact that the
current through it does not follow Ohm's
law, e.g. its resistance is not a constant, -y ,
but decreases as the current increases.
If we plot the relation between applied
E.M.F. and current for carborundum,
for instance, the result is somewhat as
shown in Fig. 415, and to obtain the
best signals we must give the auxiliary
E.M.F. the value exactly corresponding
to the bend of the curve at P.

During the receipt of signals, the
effect is to superpose upon this steady
E.M.F. a small alternating E.M.F., and
we see from the figure that if the voltage is thereby increased by an
amount ab, the current increases by an amount cd. On the other
hand, an equal decrease in E.M.F., ablt only decreases the current by
the amount cd^ ; that is, by working at the bend of the curve we
get the maximum change of current for a given change in voltage.
In practice, there is no difficulty in finding the correct voltage to
apply ; it is simply necessary to adjust it by means of the potentio-
meter until either an increase or a decrease of E.M.F. weakens the
received signals.

Receiving Circuit Connections. — One of the most con-
venient arrangements (but by no means the only one) is shown in
Fig. 416a. The aerial is connected at A to one end of the tuning
coil I, and one of the sliders Sx completes its path to earth. The

Current

FIG. 415.

624

APPENDIX

condenser K is joined up between A and the other slider S2, so that
the turns between A and S2 not only serve as aerial inductance, but
also constitute, in conjunction with the condenser, an independent
oscillatory circuit. (As the period of this circuit can be altered by
moving S2, it is not essential that the condenser should be of variable
capacity, although this is a convenience.) D is the crystal detector,
and T represents diagrammatically the pair of telephones. (Amateurs
often place the crystal between A and K, forgetting that they are

FIG. 41 6a.

FIG. 4166.

HUB

introducing a large resistance in the oscillatory circuit, and thereby
more or less damping out the oscillations.) When a potentiometer is
used some care is necessary to avoid connecting it up incorrectly.
One would naturally apply it to the crystal in the manner already
• familiar, leading to the arrangement shown in

Fig. 416c, in which P is the potentiometer wire
and B the battery ; but a little thought will show
that a by-path is thus provided for the alternating
current, enabling it to avoid the crystal. The
correct method is shown in Fig. 416ft. It is
convenient to insert a current reverser between
B and P, in order that the best direction of the
added E.M.F. may be readily found by trial. A
very compact form of potentiometer is obtained by using an ordinary
single layer adjustable resistance, such as is fitted with a terminal at
each end of the winding, and a sliding contact rubbing on the bare
wire. A total resistance of from 100 to 200 ohms will be suitable,
and if carborundum is used, B may consist of three or four dry or
Leclanche cells.

Adjustment of Crystal. — It is evidently desirable to make
sure that the crystal is acting properly before attempting to pick up
signals, and the 'easiest method of procedure is to use a "buzzer."

FIG. 416c.

APPENDIX

625

Connect up an ordinary electric bell with a cell and a tapper key,
remove the gong, and if necessary limit the swing of the armature by
a stop to prevent too much rattling. Let this device be placed any-
where near the crystal, which is supposed to be connected up in
accordance with previous diagrams. On depressing the key feeble
waves will be produced, which at such short range will produce a
sound in the telephones with almost any setting of the sliders, and
it is merely necessary to listen and adjust the pressure, or point of
contact, until the best result is obtained.1 Similarly, adjust the
potentiometer, if one is used. When this is done, attempts may be
made at a suitable time to pick up some distant station by moving
the earth slider. If anything is heard, adjust the other slider until
the sounds are most distinct. It will be found that the latter slider
will require but little adjustment for different stations, whereas the
earth slider will require a good deal. At certain times a number of
stations may be transmitting simultaneously, and each is picked up
without confusion (unless two frequencies are nearly equal) at its
particular point on the tuning coil.

Oscillation Transformer or "Jigger" Method of
Receiving". — For our purpose, the primary of this transformer may
be a single-layer coil of 22-gauge enamelled copper* wire wound on a
cardboard tube, 5 inches in
diameter and 5 or 6 inches
long. It should be provided
with one sliding contact.
The secondary is also a
single-layer coil wound on
another tube small enough
to slide freely, without actual
contact, in or out of the
primary. It may be of 30-
gauge enamelled wire, 10 or
12 inches long, and should
be tapped with soldered
connections in four or five

places, in order that various fractions of its length may be employed
as desired. The connections are shown in Fig. 417, in which I is
the tuning coil already described, and P and S the primary and
secondary of the jigger respectively. In this case the condenser must
be variable, and it is customary to indicate this by a diagonal line
across it in the diagram.

It may be thought that the tuning coil is unnecessary, and that
its functions might be fulfilled by the primary of the jigger, but for
various reasons the arrangement shown is preferable. It is best to

1 A still better method of testing or adjusting crystals involves the use of a
wavemeter, and will be described subsequently.

2 R

FIG. 417.

626 APPENDIX

join the crystal to that end of the secondary which is farthest away
from the primary (i.e. from earth), for this will be the point of highest
potential.

As regards the capacity required, it might be thought that any
combination of L and K. might be equally satisfactory, provided the
all-important product LK was unaltered, but it should be noticed
that a crystal detector acts in virtue of the P. D. set up across the
condenser coatings, and the higher this P. D., the greater will be the
telephone current. Now, for a given amount of energy induced in
the secondary, the smaller the capacity of the condenser the greater
will be the P. D. to which it is charged, and vice versa. Hence we
infer that the smaller the capacity and the greater the inductance the
better will be the result. In fact, as the secondary coil itself must
have some capacity, and has, therefore, a natural period of its own,
ideal sensitiveness would be obtained in the limiting case when the
condenser is zero ; but some adjustable capacity must be retained for
the purpose of tuning, and a tubular condenser of the type already
described answers very well. These considerations will show why
the connections shown in the figure, although the simplest to make,
and very commonly used, are not the best possible, for although the
inductance can be altered by using various portions of the secondary,
this does not get rid of the capacity of the idle part. It is, therefore,
better in practice, although a little more troublesome, to arrange
matters so that the idle part can be actually cut off from the remainder.
The great advantage of the jigger lies in the sharper tuning obtain-
able, and thus it becomes possible to avoid confusion, or, as it is
commonly called, "jamming," by tuning in or out stations which are
transmitting simultaneously at nearly the same wave length. This is
partly because the " coupling " can be so readily altered by sliding the
secondary in or out of the primary. The latter process may weaken
the induced E.M.F., but at the same time gives sharper signals, for
reasons indicated on p. 617. In commercial practice the jigger method
is generally used, but it is not so well adapted to the purposes of the
private student as the simple arrangement first described.

Working Range of Wave Length. — In addition to picking
up messages transmitted by stations using long wave lengths, the
student may desire to receive signals from neighbouring private
stations, and these are usually limited by the terms of their licence
to wave lengths not exceeding 200 metres. An aerial suitable for
the former purpose may be quite incapable of picking up such short
waves, even when the distance is very small. Now we know that the
capacity of a condenser is reduced by connecting another in series
with it, and also that the joint capacity is less than that of the
smaller one. Hence we see that the effective capacity, and therefore
the wave length, of the aerial may be reduced and adjusted by insert-
ing a variable condenser between its lower end and the earth. The

APPENDIX

627

method is useful, but it has its disadvantages and limitations ; for
instance, it reduces the energy which can be picked up by the aerial,
and also, by insulating the latter, prevents the free escape of static
charge derived from the atmosphere ; and even then it is often impos-
sible to reduce the wave length as much as is required. Perhaps the
easiest and most satisfactory plan is to erect a shorter aerial for private
transmission and receiving — preferably at right angles to the other.

Wavemeter or Cymometer.— The original form, due to
Fleming, has already been mentioned (see p. 617). A more recent

FIG. 418.

type, supplied by the Marconi Company, is shown in Fig. 418, and the
scheme of connections in Fig. 419, in which D
is a carborundum crystal, and T represents a
pair of telephones. The inductance L is a small
square coil, just fitting into the larger compart-
ment of the lid of the box, and, in order to
increase the working range, it is divided into two
parts (connected in the two-way plug key, shown
in the smaller compartment of the lid), so that
a portion of the whole may be used as required.
The condenser alone is continuously variable.
This is contained in the circular ebonite case
(seen on the right), and is adjusted by rotating the upper disc, which

628 APPENDIX

is provided with a scale. Such condensers usually consist of two sets
of semicircular plates, one set fixed and the other attached to a rotat-
ing spindle so as to move between or away from the fixed set — an
arrangement similar in principle to that illustrated in Fig. 394,
p. 559. A great increase in convenience and sensitiveness is ob-
tained by using a carborundum crystal and telephones to detect
resonance in place of a vacuum tube. The crystal is placed between
the two upright brass strips just seen in the figure, and is sensitive
enough without using a potentiometer.

The open box is placed a short distance from the aerial, or circuit
in which oscillations are occurring, and the graduated disc turned
until a portion is found which gives the loudest sound in the tele-
phones. The corresponding wave length is then read off directly
from a calibration card supplied with the instrument. Using the first
portion of the inductance, the range is from 110 to 763 metres; and
with the whole inductance, from 695 to 2699 metres.

Uses of Wavemeter. — This instrument is by no means re-
stricted to the measurement of wave lengths emitted by an 'aerial or
other oscillating circuit. It can be used for a variety of purposes,
and is rapidly becoming as indispensable as a Wheatstone bridge.
For example, it may be employed to emit waves of known length,
very feeble waves it is true, but sufficient for the purpose (say) of
calibrating the secondary of a jigger.

For suppose we make and fit up a receiving set as shown in Fig.
417. It is evident that we shall be quite ignorant of the working
range of its secondary circuit, and we may waste time in trying to
pick up waves outside that range. We may estimate the values of
L and K by approximate calculation, or we may measure them by
suitable methods and then calculate the wave length from the for-

mulae N = - - -=andV = NW, where Y = 3xl010, and W is the

wave length in centimetres (see pp. 606 and 607) ; but these processes
are either troublesome or uncertain, whereas the wavemeter gives the
required information at once.

To understand this application of the instrument, let us suppose
that in Fig. 419 we replace D and T by a tapper key and cell. When
the key is depressed a current flows through L, and when the circuit
is broken oscillations are set up in the path LK of the particular
frequency determined by L and K, but of course only persisting for
a fraction of a second. To obtain a series of such oscillations we
evidently need a rapid make-and-break, and this is readily supplied
by an ordinary bell with the gong removed. Then a difficulty arises,
for the coils of the bell electro-magnet give rise to an inductive rush
when contact is broken, and this muddles up the effect we require.
We can get over the difficulty by shunting the bell winding with a
large non-inductive resistance (easily made by winding some fine gauge

APPENDIX 629

high resistance wire on a flat piece of cardboard). Care must be
taken to avoid the mistake of connecting this shunt across the spark
gap. See Fig. 191, p. 252. One end should be secured under the
screw F, and the other to the left-hand terminal. The action of this
shunt will be understood by referring to Fig. 275, p. 360, which may
be interpreted as representing the coils of the bell, the cell, and the
contact-breaker, the non-inductive shunt replacing the metal handles.
It will be seen that the shunt provides a by-path for the inductive
rush, which in the experiment referred to produces a shock.

The method of procedure is, therefore, to connect up the bell
thus shunted with two Leclanche or dry cells in place of the wave-
meter telephones, and to short-circuit the crystal gap by means of a
bit of stiff copper wire. A plug key may also be introduced (see Fig.
420). The open lid of the wavemeter is then placed against an end
of the secondary of the jigger (which is con-
nected up exactly as for receiving signals), and
the primary is left disconnected and inactive.
On inserting the plug key some faint sound will
probably be heard in the telephones of the jigger,
and the wavemeter condenser is adjusted until
a well-defined maximum is obtained. We then
know that the natural wave length of the
secondary circuit is equal to the wave length
emitted by the wavemeter, found at once from ^ FIG 420
the scale reading. In this way values can be
rapidly obtained for every possible setting of the secondary and its
condenser. (The necessity for the non-inductive shunt will be
realised if we repeat the experiment with it disconnected. The result
is a noise in the telephones for all wavemeter readings, but without
any distinct maxima.)

The following results were obtained in an actual experiment on
a five-part secondary and two tubular condensers — in parallel — of
the dimensions already given : —

Portion of Secondary
ill Use.

The whole
4 parts
3 „

2 „

Range of "Wave Length obtained by
adjusting Variable Condenser.

1300 to 1000 metres

1140 to 870 „

900 to &50 ,,

700 to 490 ,,

420 to 370 „

Calibration of Jigger Primary.— It is evident that with any
of the above settings, the aerial and the primary must be adjusted to

630 APPENDIX

the same frequency. The approximate positions of the sliders can be
found as follows. The wavemeter is placed with the lid close to a
convenient portion of the aerial, which is now connected up to the
jigger primary, and the latter to earth. (Care must be taken that
the wavemeter does not also act directly on the secondary.) Let us
suppose the secondary to be set in the way already found to possess
a wave length of (say) 1 300 metres. The scale of the wavemeter is
then adjusted to the mark corresponding to this wave length, and the
plug key is inserted. The wavemeter is now acting as a primary on
the aerial, inducing in the latter oscillations of its own period, and
this is again inducing similar oscillations in the jigger secondary.
Whilst listening to the telephones of the latter, the slider of the
tuning inductance is moved until the sound is a maximum. Then
we know that the natural period of the aerial circuit also corresponds
to a wave length of 1300 metres. A final adjustment can be made
by means of the slider on the jigger primary. The secondary is then
set to another wave length, say 1000 metres, the wavemeter adjusted
to give this length, and the operations repeated.

Testing and Setting Crystals. — A "buzzer" method has
already been mentioned, but crystals can be tested and adjusted with
much greater facility by means of a wavemeter. For this purpose it
is convenient to place the open lid against one end of the tuning
coil, the connections being as shown in Fig. 420. Then, with either
receiving arrangement, sounds will be heard in the telephone, and the
best setting of crystals and potentiometer can be found with the
utmost ease.

Finally, it may be pointed out that it is quite easy to make a
simple form of wavemeter, although it is difficult to calibrate it in
actual wave lengths. Such an instrument is of very great usefulness
quite apart from its calibration.

Again, it should be noted that experimental work can be carried
out with almost equal value to students between transmitting and
receiving stations in adjacent rooms, or even in the same room, thus
avoiding the deadlock due to the war without infringing the temporary
regulations issued by the Postmaster- General.

Meaning of the Word " Tuning."— On p. 448 it is shown
that the impedance of a circuit containing self-induction and capacity

reduces to its ohmic resistance when 2?™L = - =^- where n is the

frequency. When we "tune" a circuit to any given wave length,
we are really satisfying this condition. Another way of regarding
the matter is to say that we are making its natural period of oscilla-
tion equal to the impressed period. It is easy to show that the two
statements are identical, for the condition may be written

APPENDIX 631

T = 2?r

which we know is the natural period when the resistance is small.

Calculation of Inductance.4— It is often required to calculate
the approximate value of L for a single layer coil, when it is
difficult to count the actual number of turns, but easy to find the
number of turns per centimetre with fair accuracy. For this pur-
pose it is convenient to transform the equation given in p. 376,

T 47rN2A
L= -J—K

into the form

L = Z(7r2N2D2),

where D is the diameter of the coil, and N is the number of turns
per centimetre (as no iron is present /x is unity). It will be a useful
exercise for the student to verify this transformation.

ANSWERS

EXERCISE III (p. 29)

1. No effect. 3. (a) No effect ; (6) partial collapse.

4. Parallel to the line joining the centres of the spheres. 5. + 24, - 12 units.
6. 5 dynes, TV dyne. 7. 29'3 and 207. 8. '1414 dyne. 140

10 '5 mg. 11. 121-3. 12. 4'9. 13. "151. 9- "T^

14. 37 : 13 with similar charges ; 7 : 1 with opposite charges.

EXERCISE IV (p. 47)

1. 3 ergs. 2. '436. 3. 23-093. 4. V1) = 5'414. Y0 = 8'484. Work = 3'07 ergs.

5. 45 units. 6. 5'2. 7. 7'13. 8, 2'94 dynes. 9. 36 and 12 units.
10. 18 and 2 units. 11. 26 units. 12. 176. 13. 1 : 14. 14. 500 ergs.
16. 33, 5. 18. 5 units. 21. 6'7. 22. 250 dynes.

25. (a) The leaves of the electroscope diverge with electricity similar in kind to

that on the knob of the Leyden jar ; (6) on the potential of the knob and
its distance from the end of the tube.

26. Quantities are as 1 : 5 ; densities are as 5 : 1.

27. When the two balls are placed near together, the potential of the charged

ball depends on the quantity of its charge and on its capacity ; the
potential of the other depends on the distance of its centre from the first.
When touched with the finger, it acquires zero potential. The difference
of potential is thus increased, and a spark passes.

28. The leaves of an electroscope diverge when we have a difference of potential

between the cap and the base. When the electrified rod is held under the
• can, there is a uniform potential over the can and throughout its interior,
and the cap and base are at the same potential. When the cap is touched
with the finger, it is brought to zero potential, thus producing the neces-
sary difference of potential to cause the leaves to diverge.

29. (1) No electricity will pass, as the potential is the same as that of the con-

ductor. (2) Electricity will pass when they are removed to a distance
from it ; they remove unequal charges, and as their capacities are equal,
their potentials are different.
2 .__ 31. Second ball, 14 ; third ball, 24. 32. 1 : 1'8.

30. -TfcP)i.e. P V22. 33. - 20 units. 34.4cm. 35. 144 ergs. 36. TV

38. (1) A deflection; (2) leaves collapse; (3) a deflection — larger than at first ;
(4) deflection unaltered ; (5) deflection unaltered ; (6) might be first a
collapse and then a re-divergence more or less marked with a charge of
opposite sign.

EXERCISE V (p. 70)

1.3. 2. From the larger jar. 3. 6 '6 ergs. 4.26. 5. fV.

6. 636T4r static units or centimetres. 7. 4:9. 8. 25,000 ergs. 9. 2:1.
10. Charge = •£• ; potential = \ ; energy = £. 11. 1 ; 2. 12. 50. 13. 84: 325. 14. 7'2

633

634 ANSWERS

15. The divergence of the leaves in connection with the charged plate diminishes

very slightly, and that of the other increases, when the sulphur is intro-
duced.

16. (a) The divergence will increase ; (b) the leaves will collapse.

17. (a) The amount of induced electricity depends upon the amount of charge on

the inducing body, the distance between the two bodies, and the specific
inductive capacity of the dielectric. When the glass plate is introduced,
the specific inductive capacity increases. (b) Move the plate away.

18. HeatcciQV. When the plates are brought in contact, their capacity is

diminished, and the potential is therefore greater, which makes the heat

greater. 19. 1 : 9.

20. The heat produced by the discharge varies as the potential. When the ball

is brought near the wall its capacity increases, and therefore its potential

is less than it was in the first discharge. 21. 5 turns. 22. 625 ergs.
23. 144 units. 24. 69'6. 25. 233-3. 26. 4000 ergs.

27. Energy of discharge of A = 16'53 ; of B = 19'83.

28. Energy of A = 3 x 107 ; of B = T5 x 107. The energy lost is represented by the

spark which passes at contact.

29. 40. 30. 566'6 x 103 ergs. 31. 16?r lines per sq. cm.

EXERCISE VI (p. 101)

1. Depends on the capacity of the instrument. A gold-leaf electroscope can be

made of smaller capacity than an electrometer, and therefore a given

charge will raise its potential to a higher value.

2.4-2. 6.35. 7. Field = 80?r lines per sq. cm. V=1607r.

8. In second case the attraction is (-55)2 = -3 of its first value.

EXERCISE VIII (p. 170)

1 mA = 4
m-z 1

2. Three times greater with the straight spring than with the bent one.

3. 2 sin a = sin b, where a is the angle between the long magnet and the

meridian, and b the angle between the short magnet and the meridian.

4. '707 : 1. 5. 1 : T732. 6. 1:2. 7. 2'52 dynes. 8. '414. 9. 205'4.
10. 84-5. 11. 14-4 and 21'6 inches 12. 230° 13. 750°.

14. ^A = ^ is. 293-3°.
MB 11

17. No difference. Times depend on (a) moment of inertia of the magnet,

(6) strength of earth's field, (c) magnetic moment of system. The first
two are the same ; the resultant couples in the third are also equal.

18. 1 : 1-21. 19. 1 : T56 (nearly). 20. 64 : 81 : 36 21. 1'76 : 1.
22. 1-75 : 1. 23. 20 : 9. 24. 3'26 : 1 25. 13'57 min.
26. 17 : 24. 27. 64 : 125. 29. 30°. 30, '01.

„ Ho 1-331 32. -143. mi44vl1*H- ^99xl°3xH

31- H; =~T~ 34. 370-3. 33' (1) l 49

35. Field at A = earth's field x 6'25. 36. 125 : 216. 37. 24.

38. 20 x/2 cm. 39. 150. 40. 27 : 64.

EXERCISE IX (p. 193)

3. Equilibrium in any position. 5. No difference.

8. (a) Needles must be parallel ; similar poles in the same direction ; if
moments are unequal, similar poles need not be in the same direction.
(b) Needles must be parallel, of equal moment, and similar poles in oppo-
site directions. 11. '256 dyne. 16. 205,000 nearly.

ANSWERS 635

EXERCISE X (p. 215)

(i. ) "No effect ; (ii.) leaves of electroscope connected with platinum end
collapse, and those connected with zinc end diverge further ; (iii.) leaves
connected with zinc collapse, those with platinum diverge further.

(i.) No resultant difference of potential ; (ii.) difference of potential which is
maintained by the energy of the chemical action between zinc and acid.

EXERCISE XI (p. 227)

1. £ ampere. 2. P.D. at terminals is reduced from 4 volts to 3 volts.

3. J ohm. 4. 4, I. 5. 6'6 ohms. 6. '4, -36. 7. '088, -142, '1 (ampere).

8. '1 ohm. 9. '055 ampere (nearly). 10. In series. 11. 2 : 1. 12. £ volt.

13. 1-95 volt. 14. 1-2 volt. 15. 80, 50, 25.

17. (a) '35 ampere, (b) 4 volts, (c) 1'43 ohm.

18. (1) deflection, (2) no deflection. 19. '183 ampere.

20. C = -V<55- ampere, C^f-g- ampere flowing backwards through weaker battery.

21. 4 : 3. 22. f of 35 feet from copper terminal. 23. 1 '3 volt.
24. (1) '0478, (2) -0239 (ampere). 25. C = '3 ampere. P.D. = 1'3 volt.
26. Four rows of six cells in series. 27. g ampere.

28. With P1 the deflection is d and negative compared with the middle point of

the wire ; with P2, no deflection ; with P3 the deflection is d in positive
direction ; with P4 deflection 2d in positive direction.

29. 1 wrongly connected. 31. 49 lamps.

32. 275 ohms ; 60 watts in battery, 40 watts in leads, 1100 watts in lamps.

EXERCISE XII (p. 238)

1. Rise in temperature in cell with plates wide apart will be twice that in other

cell.

2. The number of units of heat is the same in both cases, but they are distri-

buted over a much greater weight of metal in the thick wire than in the
thin one.

3. The relative amounts of heat per unit time are inversely as their lengths.

4. 15:24. 5. 2'4 units. 6. 108 units. m Ha_2. ,„* H«_3
8. 2:1. 9. 2143 units. 10. If ampere. ' ( ' H^~3 ' H6~2

11. (a) 1 : 3, (b) 1 : 2. 12. 10 : 1. 14. 2x 109 ergs per second. 15. 1'9 ohm.

16. 4'15 x 103 grams. 17. '77 ampere. 18. '9 volt ; y^j- ohm. 19. 4^d. (nearly).

20. '37 ampere, 2 cells, '917 ohm.

EXERCISE XIII (p. 254)

6. (a) Becomes magnetised, (i) no effect.

7. The equilibrium will probably be disturbed, owing to attraction or repulsion

of parallel currents.

8. Time required for 20 oscillations depends on the strength of the field, being

smaller when field is stronger. If the current is flowing downwards, the
field due to current and that of the earth strengthen each other on the
west and weaken each other on the east, . *. time of vibration is shorter
on the west than on the east.

636 ANSWERS

EXERCISE XIV (p. 280)

1. 3'13 ohms. 2. 22 '5 metres. 3. 52*008 ohms. 4. 1'08 mm.

5. 14;34 ohms (nearly). 6. 1 : 100 7. 2 ohms. 8. 5 '955. 2 -134.

9. 6'6 ohms. 10. 9'1 ohms. 11. 60 ohms. 12. '999 ohm. 13. 52 '5 metres.

14. The branches BA, AD, and BC, CD each give 2 ohms resistance. Their joint

resistance is 1 ohm. There is no modification when A and C are connected
by a wire.

15. '62, '31 (ampere). 5l = ?

H2 1

16. C = J ampere. Potential at middle point of FA = 0. Let A be the copper.

Potential at A= + J volt (above earth) ; at F = - \ volt (below earth) ; at
C= - 1 9 volt ; at B= - f| volt. 20. ip- ohms, 1 ampere.

21. 70 divisions. 22. f ohm. 23. ^ ampere open or closed ; '141 ampere
open ; 1--337 ampere closed. 24. 45 '4° 0. 26. '0058 sq. inch.

EXERCISE XV (p. 301)

1. l : V3.

2. Strength of current through large galvanometer is twice that of the small

one. A_J_ 5- '716 ampere. 6. '034 and '34. 7. £ ohm.

8. '083 metre. B~78 9- 30°. 10. '022 in direction of axis.

11. -18 (nearly). 12. 2V3 : 1. ,

13. Increased or decreased according to direction of current. 14. 1 : */2.

EXERCISE XVI (p. 316)

3. 1-88 (nearly) A_3 5. "24 ohm. 6. 1 '03 ohms. 7.30°.

B~5

8. 3*9 ampere. q A — 12.

' B~7

EXERCISE XVII (p. 341)

1. 6'78 amperes. 2. 14*15 grams. 3. 6'16 amperes. 4. 1*4 ampere.

5. 2-3. 7. -OOl. 8. -000828. 9. '0_003.

10. Weight of ion in first case : weight of ion in second case : : 1 : ^2.

11. '0104 grs. 13. (a) 195 grains, (I) 97 -5 grains. 14. 260 grains.
15. 1: v/SL 16. 1-49 volt. 17. 2'39 volts. 18. 1 '53 ohm. 19. 84,175 watts.

21. Weaker in cell with platinum plates, because back E.M.F. is set up in that

cell and not in the other.

22. Current is diminished in strength, as electrolysis is 'set up, giving an

opposing E.M.F.

23. With one cell C = - ~ e, where e is the back E.M.F., which is smaller than

E.

E (the E.M.F. of a Grove's cell) ; with two cells Cx - J ~ e, where r is

' the internal resistance of the added cell. Consider a numerical example :
if E = 2, e=l-5, R = l ohm, and r = '2 ohm, in case (i.) C = '5, and in case
(ii.) C1 = 2-08.

24. 66 volts. 25. 1 "03 ampere in A, '365 ampere in B.

ANSWERS 637

EXERCISE XVIII (p. 366)

5. At middle part of both magnets, practically no current ; when crossing the

the poles, at a maximum.

6. The effect of the iron is practically to make the magnet longer, (i.) The

magnet pole must be thrust further into the ring ; (ii.) the ring must be .
moved towards the neutral line of the magnet.

7. From S to N in the upper moving side.

8. Greatest when axis of rotation is at right angles to the line of dip ; least

when axis is parallel to that direction.

EXERCISE XIX (p. 380)

1. 12-57 x 10-5 volts.

2. 250 dynes acting at. right angles to the field and to the length of conductor.

3. '0007 volt (nearly).

4. Force on coil produces a couple whose moment is i.n.ir.rz.H, where i~ current

in absolute units ; n = number of turns ; r= radius of coil ; H = horizontal
component.

5. (i.) With steady current, currents are as 4 : 3 ; (ii.) self-induction, acting as

a momentary resistance, is set up, which will be greater in the wire coiled
round the iron than in the zigzag wire. The data are not sufficient to
give any ratio between the currents.

6. H = 5. Diameter has no effect.

7. e = Hlv. If conductor moves through distance s in t seconds, v=-;
V e= — — , but H.Z.s is number of lines cut in t seconds.

t

8. L — -05 henry (nearly), M = '018 henry (nearly). Reduced.

9. liri ergs, where *= current in absolute units.

EXERCISE XX (p. 423)
5. 3130 ampere-turns. 6. 2511 ampere-turns. 7. 100 Ibs. (nearly).

EXERCISE XXI (p. 451)

1. 140-5 volts. 2. '028 henry, 3. 3 ohms. 4. 16jr volts.

10. 0 = 1 ampere. D. P. across condenser =800 volts (nearly). D. P. across resist-

ance =600 volts (nearly).

EXERCISE XXII (p. 479)

4. At the junction of A with antimony, heat is absorbed, and therefore junction

is cooled ; junction of B with antimony is warmed. The effect is propor-
tional to the strength of the current (not to its square).

5. Not the same strength, as difference of potential depends not only on the

difference of temperatures, but also upon the mean temperature.

11. E= -00147 volt (nearly) ; neutral point = 223° C.

638 ANSWERS

EXERCISE XXIII (p. 500)
6. 2?rr x 10~10 dynes, where r— radius of orbit ; r= s, =. cm.

^/ 2tlT X 10

EXERCISE XXV (p. 545)

5. The rate of consumption of zinc decreases. Kate oc current, but C= ~e

R

where E = E.M.F. of battery, e = back E.M.F., and R = resistance of the
circuit. Now, E and R are practically constant, while e is directly pro-
portional to speed of engine ; . • . if e is increased, C is diminished.
10. 82'8 amperes. 13. 5 x 106 lines.

EXERCISE XXVI (p. 556)
3. 288 amperes. 4. 2s. 4d. (nearly). 5. 55 watts ; 25 watts.

EXERCISE XXYII (p. 572)

2. "We notice that a current of ^ or r^ of an ampere deflects pointer to its
full extent, and we must arrange matters so that a difference of potential
of 50 volts sends r^T ampere through the instrument ; hence, its resist-
ance must be increased by r ohms, where — = — : i.e. r = 4500

100 500 fr
ohms. 3. 14'7 divisions ; 19*09 divisions.

EXERCISE XXVIII (p. 591)

6. '003 volt.

7. Electrical energy costs 226 times as much as heat energy.

INDEX

ABSOLUTE electrometer, 94

— measurements of M and H, 159

— potential, 35

— units, 206, 207, 230, 377, 582
Absorption in dielectrics, 68
Acceleration, 573
Accumulators, 336-341
Aclinic line, 187

Action at starting and stopping a
current, 35 /

— between magnetic fields and

currents, 343

— of magnetic and electric fields

on cathode rays, 485, 486

— of lightning conductors, 83

— magnetic poles, 114

— of points, 22, 81
Agonic line, 185
Air condenser, 51
Alloys, 256, 423
Alternating currents, 425

— current generators, 540

— E.M.F.,425

— motors, 543
Aluminium rectifier, 323
Amalgamation, 196
Ammeters, 558

" Amount of cutting," 368
Ampere, the, 207, 590

— turns, 248
Ampere's rule, 242

— theory of magnetism, 118, 248
Anderson's method of measuring

self -inductance, 455
Anion, 319
Anode, 317
Arago's rotations, 356
Arc, the electric, 552

— lamps, 553

— theory of, 553
Arcs, flame, 553

— open and closed, 553

— magnetite and titanium, 554
Armature, drum, 529

— Gramme ring, 527

Armature, Siemens', 525

— reaction, 533
Astatic needle, 164

— galvanometer, 291
Attenuation and distortion of speech,

522
Atmospheric electricity, 102

sources of, 105

Atomic magnets, 117
Attracted disc electrometer, 94
Attraction, electrical, 1

— magnetic, 112

— between conductors carrying

currents, 243

— between magnet pole and arma-

ture, 417

- between parallel plates, 92, 96
Aurora, the, 108
Ayrton-Mather galvanometer, 295

BACK E.M.F., 371

- in electrolyte, 330

of motor, 537

Ballistic, meaning of, 296

— galvanometers, 382-391

Barlow's wheel, 345

Becqueral rays, -494

Bell, electric, 252

Bell's telephone, 515

Best grouping of cells, 218, 234

Bifilar suspension, 86

Biot's experiment, 18

Board of Trade unit, 234, 591

Boys' radio-micrometer, 474

Broca galvanometer, 292

Bunsen's cell, 210

CALLENDAR'S radio-balance, 474
Calorie, the, 574
Canal rays, 489

Candle-power, meaning of, 556
Capacity, electrical, 42, 50

— of cables, 522

— of circular disc, 61

640

INDEX

Capacity of condenser, measurement
of, 390

— of condensers in parallel and in

series, 63

— of parallel plate condenser, 61

— of spherical condenser, 60

— of telegraphic lines, 507

— of two Leyden jars, comparison

of, 85

Carey Foster's method of comparing

resistances, 267

of measuring mutual in-
ductance, 457

Cathode, the, 317

— glow, 483

— rays, 484

velocity of, 486

Cation, 319

Cavendish's experiment, 18

Cell, Benko, 212

— bichromate or chromic acid,

210

— Bunsen, 210

— Callaud, 205

— Clark's standard, 213

— Daniell, 202

— dry, 209

— Edison-Lalande, 212

— Grove, 210

— Leclanche, 208

— Minotto, 205

- simple, 195

- Weston cadmium standard, 214
Central battery system in telegraphy,

512

Chemical equivalent, 325
Choking coil, 362

— effect, 434, 604
Clerk-Maxwell's relation between

dielectric constant and refrac-
tive index, 586

relation between conductivity

and opacity, 586

hypothesis about residual

charges, 55

Coercive force, 121, 414
Coherer, 609
Condenser, 50

— effect of dielectric on, 62

— energy of, 66

— in induction coil, 364

— limit to charge on, 52

— key, 314

— measurement of internal re-

sistance by, 313

— method of comparing E.M.F.'s,

313

Condenser method of comparing
capacities, 313

— Moscicki, 57

- spherical, 60
Condensing electroscope, 57, 197

...Conductivity of air, 105, 481
Conductors, 2

— lightning, 107
Consequent poles, 119

Constant of tangent galvanometer,
288, 335

— of ballistic galvanometer, 386
Construction for equipotential sur-
faces, 38-41

— • for lines of force (electric), 41

— of magnetic curves, 161
Contact breaker, 362
Commutation, 533
Commutator, 526, 528
Compound winding in dynamos, 532
Cooper-Hewitt mercury lamp, 555
Coulomb, the, 207, 591
Coulomb's law and torsion balance,

22, 135

Couple acting on coil carrying cur-
rent in magnetic field, 379

Critical temperature of iron, 422

Crookes' dark space, 483

Cup radio-balance, 474

Current balances, 567-570

- measurement of, 300, 305, 335

— reverser, 253

Cycles of magnetisation, 412
Cylinder machine,- 73

DANIELL'S cell, 202
D'Arsonval galvanometer, 293
Damping, 147, 297, 354

— correction, 392

— due to radiation, 607
Dead-beat, meaning of, 296
Deckert type of transmitter, 519
Declination, 175, 177

Deflection of compass-needle by mag-
netic field, 142

Demagnetising effect of poles, 118
Density, surface, 20

— on differently shaped conductors,

21

— on spheres, 44

Detectors of electromagnetic waves,

607, 613
Deviation, 192
Diamagnetism, 419-423
Dielectric, 3

— constant, 25

— • — measurement of, 91

INDEX

641

Dielectric constant of liquids, 92

and refractive index, 586

— constants, table of, 69

Differential duplex working in tele-
graphy, 511

Dimensional equation, 574

Dimensions of derived units, 575

— of electromagnetic units, 578"

— of static units, 577
Dip, 174, 177

— circle, 178, 181

— determination of, 178, 183
Discharge of a condenser through

circuit containing self-induc-
tion and resistance, 600

— of a condenser through circuit

containing self-induction but
not resistance, 589

— through gases, 483
Discharging an electrified body, 7,

22

— tongs, 53
Distribution, 18
Direct-current dynamos, 525

motors, 537

Duddell's thermo-galvanometer, 298

— musical arc, 615
Duplex working in telegraphy, 511
Dying away of current in an induc-
tive circuit, 595

in circuit containing

capacity and resistance, 597
Dynamometer, Siemens', 562

— application to wattmeters, 564
of measurement of power in

alternating circuits, 565
Dynamos, 525
Dyne, the, 573

EAETH inductor coil, 369, 389
Edison's accumulator, 340

— incandescent lamp, 548
Eddy currents, 356, 527

Effect of change of temperature on
resistance, 257, 265

— of changing the dielectric, 62

— of condensers on alternating

currents, 446

— of iron core in solenoid, 247

— of iron on alternating currents,

437

— of iron ships on compass, 192

— of shunting a galvanometer, 299
Efficiency of arc lamps, 554

— of a cell or battery, 232

— of generators, 536

— of incandescent lamps, 547, 548

Einthoven's string galvanometer, 297
Electric bell, 252

— density, 20

— field, 11,25

— force, 26

— oscillations, 601

— whirl, 82

Electrification by pressure and cleav-
age, 7

Electro-chemical equivalents, 326
Electrodes, 317
Electrolysis, 317
— • of dilute sulphuric acid, 318

— outline of theory of, 318

— of copper sulphate, 320

— of sodium sulphate, 321

— of lead acetate, 321

— practical applications of, 322

— laws of, 325
Electrolyte, 317

Electromagnetic theory of light, 583
— waves, 583, 605, 607
Electromagnets, 248

— winding of, 250
Electrometer, quadrant, 86

— attracted disc, 94

— Dolezalek's, 87
Electromotive force, 199, 205

absolute unit of, 350

direction of induced, 354

— induced in armatures, 530

magnitude of induced, 350

measurement of, by chemical

action, 333

- by condenser, 313

— by electrometer, 309

— by Lumsden's method,
306

— by potentiometer, 303
by reflecting galvano-
meter, 310

-by Wheatstone's
method, 307

by sum and difference,

308
by volmeter, 309

— produced by motion of con-
ductor in magnetic field, 348

thermo-electric, 462

Electromotive series, 199
Electrons, 10, 488

— and metallic conduction, 493
Electroplating, 325

Elect rophorus, 14
Electroscope, 4

— charging by induction, 14

— condensing, 57, 197

2s

642

INDEX

Electroscope measures difference of

potential, 33
Electrotyping, 324
Energy, electrical, 234

— expended in circuit, 230

— influence of dielectric on, 62

— of charged body, 58

— of charged .condensers, 66

— of field per unit volume, 98

— • of two conductors sharing a
charge, 59

— stored up in circuit, 377

— unit of, 574
Equipotential surfaces, 38
Ether, the, 586

E wing's hysteresis tester, 415

— theory of magnetisation, 416

FARAD, the, 583

Faraday's butterfly-net experiment, 18

— dark space, 483

— effect, 497

— experiments on specific inductive

capacity, 67

— ice-pail experiment, 12

— laws of electrolysis. 325

— room, 19

Faure's formation of secondary cells,

338

Ferromagnetic bodies, 422
Field, electric, 11, 25

— magnetic, 125

— due to current, 240

— due to current in circular wire,

244

— due to solenoid, 245

— strengths by oscillations, 156

— strength inside solenoid, 374

— strength near a straight wire

carrying a current, 373
- — strength-near a bar magnet, 139,
161

— strength near charged conductor,

45

— strength of earth's, 133

— strength of magnetic, 133, 136
Fleming's oscillation valve, 614

— cymometer, 617
Floating battery, 245
Flux, 28, 406
Force, electric, 26

— lines of magnetic, 125

— between conductors carrying cur-

rents, 374

— on conductor, 46

— on conductor carrying a current

in magnetic field, 372

Force, unit of, 5, 73

— very close to charged conductor,

45

Foucault's currents, 356

Franklin's experiments on atmos-
pheric electricity, 102

— plate or fulminating pane,

53

Free and bound electricity, 12
Frequency defined, 353

GALVANOMETER, astatic, 291

— Ayrton-Mather, 295

— ballistic, 296, 382

— Broca, 292

— D' Arson val, 293

— dead-beat, 296

— Duddell's thermo-, 298.

— Einthoven's string, 297

— Helmholtz, 290

— moving coil, 293

— simple, 198, 282

- sine, 289

— tangent, 284

— Thomson, 292

— vibration, 459
Gauss, the, 137

— A and B positions of, 143
Gold-leaf electroscope, 4
Gramme ring armatures, 527
Grouping of. cells, 216-218
Grove's cell, 210

- gas battery, 337
Gyro-compass, 193

HALL effect, 498
Heat, unit of, 574
Heating effect of current, 234
Helmholtz equation, 594

— galvanometer, 290
Henry, the, 377, 591
Hertzian oscillator, 605

— receiver, 608
High-frequency currents, 601
Horizontal component of earth's field.

133, 137, 159, 174
Horse-power, 231
Hughes' microphone, 517
Hysteresis, 414

— Ewing's tester, 415

— loss, 415

IMPEDANCE, 434

— at high frequencies, 604
Incandescent lamps, 547-550
Inclination, 174, 177
Induced polarity, 120, 399

INDEX

643

Inductance, 375

— measurement of, 453-459

— variable standards of, 459
Induction coils, 362

— coils in cables, 523

— coils in telephony, 519

— electric, 9

— magnetic, 120

— in telephony, 521
Inductive circuits, 361

— process of charging an electro-

scope, 14
Influence of dielectric, 62

— of medium on magnetic induc-

tion, 122, 149
Insulators, 2

— in telegraphic lines, 501
Intensity of magnetisation, 168, 408
Intermediate poles, 119
Inverse-cube law, 159
Inverse-square law for point charges,

28

— poles, 137, 150, 158
Ionic velocities, 328
Ions, 82, 203, 318, 327, 552
lonisation, sources of, 494
Iron alloys, 423
Isoclinic lines, 187
Isodynamic lines, 187
Isogonic lines, 185

JOULE, the, 234, 591
Joule's law, 236

KEEPERS, 118
Kelvin, Lord:

— attracted disc electrometer, 94

— balance, 568

— experiment on Volta effect, 58

— galvanometer, 292

— mariner's compass, 191

— method of measuring resistance

of galvanometer, 279

— multicellular voltmeters, 559

— portable electrometer, 96

— quadrant electrometer, 86

— water-dropping apparatus, 103
Kerr effect, 498

Kirchhoff's laws, 276

LAG, angle of, 434
Lamp and scale, 147
Law, Coulomb's, 22, 135

— Joule's, 236

— Lenz's, 355

— of successive temperatures, 462

— of intermediate contacts, 462

Leakance, 508
Lenz's law, 355
Leyden battery, 55
Leyden jar, 52

seat of charge, 54

residual charges, 54

Lichtenberg's figures, 53
Life of incandescent lamp, 548
Lightning, 107

— conductors, 83, 107
Lines of force, electric, 11, 15

construction of, 41

refraction of, 99

magnetic, 125, 397

properties of, 129

through magnetic sub-
stances, 128
Lines of magnetic induction and of

magnetisation, 397
Load, explanation of, 443

— automatic adjustment of current

to, 445

— calculation of no-load current,

443

— secondary, 444
Local action, 200
Lodestones, 111

Lodge, Sir Oliver, on lightning con-
ductors, 83, 108

Logarithmic decrement, 386, 393
Loss of magnetisation, 123
Lullin's experiment, 56

MACHINE, cylinder, 73

— dynamo, 525

— magneto, 520, 525

— plate, 75

— Voss, 75

— Wimshurst, 78
Magnetic alloys, 423

- balance, 137

— battery, 119

- chain, 121

— circuit, 405

— effect of moving charge, 485

— elements at various places, 187

— field and force, 125

— field due to current, 240-250

— meridian, 116, 176

— needle, to make, 113

— shell, 379
— storms, 185

— substances, 119, 419
Magnetisation :

- by electric current, 122, 240-252
— • by single and separate touch, .113

— curves, 403, 410

644

INDEX

Magnetisation, cycles of, 412

— effects of, 124

— Ewing's theory of, 416

— loss of, 123
Magneto-bell. 521

— geneiators, 520, 525
Magnetometer, 143, 146, 153
Magnetomotive force, 406

Mance's method of measuring internal
resistance of a cell, 279

Mansbridge's condenser, 57

Marconi's arrangement for wireless
telegraphy, 610

— coherer, 609
Mariner's compass, 190
Maximum rate of working, 233
Measurement of current, 300, 305,

335

— of extremely small currents, 491

— of declination, 177

— of dip, JJ8

— of E.M.F. by chemical action, 333

— • by condenser, 313

by electrometer, 309

— • — — • by Lumsden's method, 306

— by potentiometer, 303

by reflecting galvanometer,

310

-by Wheatstone's method,

307 '

-by sum and difference, 308

— by voltmeter, 309

— of insulation resistance, 273, 598

— of internal resistance of cells, 311,

313, 315

— of magnetising force (H), 401

— of magnetic induction (B), 401

— of B in iron core, 443

— of mutual inductance, 456

— of resistance, in electrolytes, 332

— by substitution, 259

— by voltmeter and ammeter,

259

-by Wheatstone's bridge,

261 '

— of very low resistances, 270

— of very high resistances, 273

— of self-inductance, 453

— of specific inductive capacity, 91

— of specific resistance, 264

— of temperature coefficient of a

magnet, 151

— - of temperature by thermo-elec-
tricity, 473

— of thermo-electromotive force,

462, 475
Measuring instruments, 558-572

Medium, influence of on magnetic
force, 122, 149

Mercury lamps, 555

Metallic filaments in lamps, 549

Metre bridge, 262

Microphone, 517

Molecular conductivity, 330

Moment of coil carrying a current, 378
— • of couple on deflected compass-
needle, 152

— of inertia, 155

— of magnet, 140, 144, 149, 157
Moore light, 555

Moscicki condenser, 57
Motors, alternating, 543

— direct current, 537
Moving coil galvanometers, 293
Multipolar generators, 534

Mutual action between magnetic

fields and currents, 343
— • inductance, coefficient of, 378
measurement of, 456

NATUEAL magnets, 111
Nernst lamp, 550

Neutral points in magnetic field,
128

in thermo-electricity, 468

Non-conductors, 3
Non-inductive circuits, 361
Null method, 272, 305

OERSTED'S experiment, 241
Ohm, the, 206

— in terms of column of mercury,

588

— evaluation of, 587
Ohm's law, 207, 216
Open circuit, 219
Oscillation apparatus, 153, 156
Oscillations, electric, 601

— of magnet, 153
Oscillator,"Hertz's, 605
Oscillatory discharge, 600
Oscillographs, 571

PARALLEL circuits, 271
Paramagnetic bodies, 419
Peltier effect, 464

— — coefficient of, 467
Period of galvanometer, 297

— of radio-active bodies, 496
Permeability, 136, 403

— curves, 403
Pith-ball pendulum, 1
Plante's secondary cell, 337
Plate machine, 75

INDEX

645

Points, action of, 22, 81

Polarisation, 201

Poles, consequent or consecutive, 119

— of a magnet, 112

— position of, 115

— no-isolated, 116
Polyphase windings, 542
Potential (electrostatic), 31

— at a point due to a number of

charges, 37

— at a point near a charged body. 35

— between terminals of a cell, 219

— definition of, 34, 35

— energy of field, 418
- gradient, 104

— of isolated sphere, 37

— (magnetic), 168, 375
Potentiometer, 303

— Lord Rayleigh's form of, 306
Poundal, the", 577

Power, 230

— in alternating current, 450
Practical standards of current and

E.M.F., 589

of resistance, 587

Proof-plane, 9
Pyro-electricity, 8

QUADRANT electrometer, 86
Quadrantal deviation, 192

RADIATION, electro-magnetic, 605

detectors of, 607, 613

Radio-active bodies, 494
Radio-balance, Callendar's, 474
Radio-micrometer, Boys', 474
Ratio of charge to mass, 487
Reactance, 434
Recording instruments (magnetic),

187

Reduction factor of tangent galvano-
meter, 288

Reflecting magnetometer, 146
Refraction of lines of force, 99, 419
Relay, 506, 508

Relation between density and poten-
tial on spheres, 45

— — dielectric constant and re-

fractive index, 586

heat, current, and resistance,

236
— horse-power and watts, 231

— — mutual and self-induction, 458

opacity and conductivity, 586

resistance, reactance, and im-
pedance, 434

Reluctance, 406

Repulsion (electric), lv 6

— (magnetic), 114

- between currents, 243
Residual charges, 54
Resistance, 206, 255

— boxes, 260, 267

- — change of temperature on, 257

— of lamp filament, 549

— measurement of, 259, 261, 270,

273 .

— specific, 255
Retentivity, 121, 414
Return wires in telephony, 521

Rise of current in circuit containing
capacity and resistance, 596

Robison's magnets, 139

Roget's vibrating spiral, 247

Rontgen rays, 489

Rotation of liquid conductor in mag-
netic field, 347

— of magnet in its q^n field, 347
— • of wires in magnetic field, 345

SCALAE quantity, 26 .
Seat of condenser charge, 54
Secondary battery, 336-341
Seebeck effect, 461
Self-exciting principle, 531
Self-induction, 360
coefficient of, 375

— — measurement of, 453
Self-induced E.M.F. in terms of cur-
rent, 433

Semi-circular deviation, 192
Series-winding in dynamos, 531
Short-circuiting, 216, 222
Shunt-winding in dynamos, 532
Shunts, 272, 299
Simple cell, 195
— • circuit in telegraphy, 504

— galvanometer, 198, 282
Simultaneous development of both

kinds of electrification, 6
Sine curve, 425

— galvanometer, 289
Single and separate touch, 113
"Skin effect," 604
Slide-wire bridge, 262
Soldering, 4

Solenoid, standard, 395'
Sounder, 503

— polarised, 512

Sources of atmospheric electrifica-
tion, 105

— of ionisation, 494

Spark discharge at ordinary pressure,
482

646

INDEX

!Specific inductive capacity, 66-69
measurement of, 91

— resistance, 255

measurement of, 264

Square root of mean squares, 429
Standards of inductance, 459

— of current and E.M.F., 589
Standards of resistance, 587
Strength of electric field, 25

— of magnetic field, 133, 136
Stress in dielectric, 98
Subdivision of charges on spheres,

43
Surface density, 20

— on spheres, 44

— of earth, 104 -

Surface distribution of current at

high frequencies, 604
Susceptibility, 409
Swan's incandescent lamp, 548

TANGENT galvanometer, 284

— — Helmholtz, 290
Tantalum lamp, 550

" Telefunken " system of producing

wave-trains, 617
Telegraphic lines, 501, 506

— batteries, 502

— repeaters or translators, 510

— sending and receiving instru-

ments, 503

Telegraphy, wireless, 610
Telephony, 514
— • receivers and transmitters, 517-

520

— wireless, 614

Temperature coefficient of magnet,

151
of metal, 264

— effect of, on resistance, 257

— measurement of, by thermo-

couple, 473

Terrestrial magnetism, 174-193
Theory of alternating currents, 425—

451

— of electrolysis, 318

— of electrostatics, 10

- of induction, 368-380

— of magnetisation, 397-423

— of quadrant electrometer, 89

— of transformer, 442

— of voltaic cells, 202
Thermo-electric diagram, 469

— generators, 475

— heights, 468
Thermo-dynamics, 476

— -electricity, 461-479

Thermo-electromotive force, measure-
ment of, 462, 475

galvanometer, 298

pile, 473

Thomson effect, 466

— coefficient of, 468

— galvanometer, 292
Thunder, 107

Time constant, 595, 597, 599
Torsion balance, 22, 135
Total intensity (T), 176, 184
Transformers, 365, 439, 550

— efficiency of, 445
Transmitters in telegraphy, 503

— in telephony, 517, 518
Tubes of force, 28

Two states of electrification, 2

UNDAMPED waves, 614
Unit of acceleration, 573

- Board of Trade, 234, 591

— of capacity, 43, 582

— of charge (electrostatic), 24 ;

(practical), 207, 230

— of current, 207, 591

— dimensions of, 574-591

— of electromotive force, 206, 351,

591

— of energy, 234

— of force, 573

— of heat, 574

— magnetic pole, 135, 136

— of inductance, 377, 591

— of power, 231, 591

— of quantity (electrostatic), 24 ;

(practical), 207, 230

— of resistance, 206, 587

— of velocity, 573

— of weight, 574

- of work, 574
Universal discharger, 55

" v " (ratio of electrical units), 581

Valency, 325

Variations in decimation, 184, 185

dip, 185

Vector diagrams, 431

— quantity, 26
Verdet's constant, 497

Vertical component of earth's field,

133, 174

— determination of,

389
Vibration galvanometer, 459

— of compound pendulum, 154
Virtual volts and amperes, 429
Volt, the, 206

INDEX

647

Volta effect, 58, 478

Voltaic cells, 195, 202, 205, 208-215

Voltameter, 318

— copper, 335

— silver, 335
Voltmeters, 558
Voss machine, 75

WATCH receiver in telephony, 517
Watt, the, 231
Wattmeters, 564, 570
Wave-motion, 514, 583
Welsbach lamp, 549
Weston cadmium cell, 214
Wheatstone's bridge, 261
Post Office pattern of, 267

Wimshurst machine, 78

Winding of alternator armatures, 542

— of electromagnets, 250

— of field magnets in dynamos, 531,

532

Winding, polyphase, 542
Wireless telegraphy, 610, 619-631

— telephony, 614
Work, unit of, 574

X-KAYS, 489

— ionising power of, 490

ZEEMAN effect, 499
Zero-keeping property of galvano-
meters, 297

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at Paul's Work, Edinburgh

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