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THE LIBRARY

THE UNIVERSITY
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Prof. G. C. Evans

ELEMENTS

OF THE

MATHEMATICAL THEORY

ELECTRICITY AND MAGNETISM

CAMBRIDGE UNIVERSITY PRESS

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ELEMENTS

OF THE

MATHEMATICAL THEOKY

ELECTRICITY AND MAGNETISM

SIR J. J. THOMSON, M.A., D.Sc., LL.D., Pn.D, F.R.S.,

FELLOW OF TRINITY COLLEGE, CAMBRIDGE;
CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS IN THE

UNIVERSITY OF CAMBRIDGE;

PROFESSOR OF NATURAL PHILOSOPHY IN THE
RQYAL INSTITUTION, LONDON

FOURTH EDITION

CAMBRIDGE :
AT THE UNIVERSITY PRESS

1909

First Edition 1895.
Second Edition 1897.
Third Edition 1904.
Fourth Edition 1909.

PREFACE TO FIRST EDITION

IN the following work I have endeavoured to give an
account of the fundamental principles of the Mathematical
theory of Electricity and Magnetism and their more
important applications, using only simple mathematics.
With the exception of a few paragraphs no more advanced
mathematical knowledge is required from the reader than
an acquaintance with the Elementary principles of the
Differential Calculus.

It is not at all necessary to make use of advanced
analysis to establish the existence of some of the most
important electromagnetic phenomena. There are always
some cases which will yield to very simple mathematical
treatment and yet which establish and illustrate the
physical phenomena as well as the solution by the most
elaborate analysis of the most general cases which could
be given.

The study of these simple cases would, I think, often
be of advantage even to students whose mathematical
attainments are sufficient to enable them to follow the
solution of the more general cases. For in these simple
cases the absence of analytical difficulties allows attention
to be more easily concentrated on the physical aspects
of the question, and thus gives the student a more vivid

236

VI PREFACE

idea and a more manageable grasp of the subject than he
would be likely to attain if he merely regarded electrical
phenomena through a cloud of analytical symbols.

I have received many valuable suggestions and much
help in the preparation of this book from my friends
Mr H. F. Newall of Trinity College and Mr G. F. C. Searle
of Peterhouse who have been kind enough to read the
proofs. I have also to thank Mr W. Hayles of the
Cavendish Laboratory who has prepared many of the
illustrations.

J. J. THOMSON.

CAVENDISH LABORATORY,
CAMBRIDGE.

September 3, 1895.

PREFACE TO THE SECOND EDITION

IN this Edition I have through the kindness of several
correspondents been able to correct a considerable number
of misprints. I have also made a few verbal alterations
in the hope of making the argument clearer in places
where experience has shown that students found unusual
difficulties.

J. J. THOMSON.

CAVENDISH LABORATORY,
CAMBRIDGE.

November, 1897.

PREFACE TO THE THIRD EDITION

THE most important of the alterations made in this
Edition is a new chapter on the properties of moving
electrified bodies ; many of these properties may be proved
in a simple way, and the important part played by moving
charges in Modern Physics seems to warrant a discussion
of their properties in even an Elementary Treatise.

I have much pleasure in thanking Mr G. F. C. Searle
of Peterhouse for many valuable suggestions, and for his
kindness in reading the proof sheets of the first five
chapters; to Mr P. V. Bevan of Trinity College I am
indebted for similar assistance with the subsequent
chapters.

J. J. THOMSON.

CAVENDISH LABORATORY,
CAMBRIDGE.

October 4, 1904.

PREFACE TO THE FOURTH EDITION

IN this Edition a few additions and corrections have
been made.

J. J. THOMSON.

CAVENDISH LABORATORY,
CAMBRIDGE.

April 26, 1909.

TABLE OF CONTENTS

CHAP. PAGES

I. General Principles of Electrostatics . . . 1 — 59

II. Lines of Force 60— 83

III. Capacity of Conductors. Condensers . . 84 — 119

IV. Specific Inductive Capacity .... 120 — 144

V. Electrical Images and Inversion . . . 145 — 190

VI. Magnetism 191—231

VII. Terrestrial Magnetism 232—245

VIII. Magnetic Induction 246—282

IX. Electric Currents 283-328

X. Magnetic Force due to Currents . . . 329—386
XL Electromagnetic Induction .... 387 — 456

XII. Electrical Units : Dimensions of Electrical

Quantities 457—479

XIII. Dielectric Currents and the Electromagnetic

Theory of Light 480—505

XIV. Thermoelectric Currents 506-518

XV. The Properties of Moving Electric Charges . 519—546

INDEX 547—550

ELEMENTS OF THE MATHEMATICAL

THEOEY OF
ELECTEICITY AND MAGNETISM

CHAPTER I

GENERAL PRINCIPLES OF ELECTROSTATICS

1. Example of Electric Phenomena. Electri
fication. Electric Field. A stick of sealing-wax after
being rubbed with a well dried piece of flannel attracts
light bodies such as small pieces of paper or pith balls
covered with gold leaf. If such a ball be suspended by
a silk thread, it will be attracted towards the sealing-wax,
and, if the silk thread is long enough, the ball will move
towards the wax until it strikes against it. When it has
done this, however, it immediately flies away from the
wax ; and the pith ball is now repelled from the wax
instead of being attracted towards it as it was before the
two had been in contact. The piece of flannel used to rub
the sealing-wax also exhibits similar attractions for the
pith balls, and these attractions are also changed into
repulsions after the balls have been in contact with the
flannel.

The effects we have described are called 'electric'
phenomena, a title which as we shall see includes an
T. E. 1

2 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

enormous number of effects of the most varied kinds. The
example we have selected, where electrical effects are pro
duced by rubbing two dissimilar bodies against each other,
is the oldest electrical experiment known to science.

The sealing-wax and the flannel are said to be electri
fied, or to be in a state of electrification, or to be charged
with electricity ; and the region in which the attractions
and repulsions are observed is called the electric field.

2. Positive and Negative Electrification. If we

take two pith balls A and B, coated with gold leaf and
suspended by silk threads, and let them strike against the
stick of sealing-wax which has been rubbed with a piece
of flannel, they will be found to be repelled, not merely
from the sealing-wax but also from each other. To
observe this most conveniently remove the pith balls to
such a distance from the sealing-wax and the flannel
that the effects due to these are inappreciable. Now
take another pair of similar balls, G and D, and let them
strike against the flannel; G and D will be found to
be repelled from each other when they are placed close
together. Now take the ball A and place it near C;
A and G will be found to be attracted towards each other.
Thus, a ball which has touched the sealing-wax is repelled
from another ball which has been similarly treated, but is
attracted towards a ball which has been in contact with
the flannel. The electricity on the balls A and E is thus
of a kind different from that on the balls G and D,
for while the ball A is repelled from B it is attracted
towards D, while the ball C is attracted towards B and
repelled from D ; thus when the ball A is attracted the
ball G is repelled and vice versd.

2] GENERAL PRINCIPLES OF ELECTROSTATICS 3

The state of the ball which has touched the flannel
is said to be one of positive electrification, or the ball is
said to be positively electrified ; the state of the ball which
has touched the sealing-wax is said to be one of negative
electrification, or the ball is said to be negatively electri
fied. '

We may for the present regard l positive ' and ' nega
tive ' as conventional terms, which when applied to electric
phenomena denote nothing more than the two states of
electrification described above. As we proceed in the
subject, however, we shall see that the choice of these
terms is justified, since the properties of positive and
negative electrification are, over a wide range of pheno
mena, contrasted like the properties of the signs plus and
minus in Algebra.

The two balls A and B must be in similar states of
electrification since they have been similarly treated;
the two balls C and D will also for the same reason be
in similar states of electrification. Now A and B are
repelled from each other, as are also C and D ; hence we
see that two bodies in similar states of electrification are
repelled from each other : while, since one of the pair A, B
is attracted towards either of the pair C, D, we see that
two bodies, one in a positive state of electrification, the other
in a negative state, are attracted towards each other.

In whatever way a state of electrification is produced
on a body, it is found to be one or other of the preceding
kinds ; i.e. the ball A is either repelled from the electrified
body or attracted towards it. In the former case the
electrification is negative, in the latter positive.

A method, which is sometimes convenient, of detecting
whether the electrification of a body is positive or negative

1—2

4 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

is to dust it with a mixture of powdered red lead and
yellow sulphur which has been well shaken ; the friction of
the one powder against the other electrifies both powders,
the sulphur becoming negatively, the red lead positively
electrified. If now we dust a negatively electrified surface
with this mixture, the positively electrified red lead will
stick to the surface, while the negatively electrified sulphur
will be easily detached, so that if we blow on the powdered
surface the sulphur will come off while the red lead will
remain, and thus the surface will be coloured red : if a posi
tively electrified surface is treated in this way it will be
come yellow in consequence of the sulphur sticking to it.

3. Electrification by Induction. If the negatively
electrified stick of sealing-wax used in the preceding ex
periments is held near to, but not touching, one end of an
elongated piece of metal supported entirely on glass or
ebonite stems, and if the metal is dusted over with the
mixture of red lead and sulphur, it will be found, after
blowing off the loose powder, that the end of the metal
nearest to the sealing-wax is covered with the yellow
sulphur, while the end furthest away is covered with red
lead, showing that the end of the metal nearest the
negatively electrified stick of sealing-wax is positively,
the end remote from it negatively, electrified. In this
experiment the metal, which has neither been rubbed
nor been in contact with an electrified body, is said to
be electrified by induction; the electrification on the
metal is said to be induced by the electrification on the
stick of sealing-wax. The electrification on the part of
the metal nearest the wax is of the kind opposite to that
on the wax, while the electrification on the more remote

GENERAL PRINCIPLES OF ELECTROSTATICS

parts of the metal is of the same kind as that on the
wax. The electrification on the metal disappears as soon
as the stick of sealing-wax is removed.

4. Electroscope. An instrument by which the
presence of electrification can be detected is called an
electroscope. All electroscopes give some indication of the
amount of the electrification, but if accurate measure
ments are required a special form of electroscope or a more
elaborate instrument, called an electrometer (Art. 60), is
generally used.

A simple form of electroscope, called the gold leaf
electroscope, is represented in Fig. 1. It consists of a

Fig. 1.

glass vessel fitting into a stand; a metal rod, with a
disc of metal at the top and terminating below in two
strips of gold leaf, passes through the neck of the vessel
the rod passing through a glass tube covered inside and
out with sealing-wax or shellac varnish and fitting tightly
into a plug in the mouth of the vessel.

6 GENEKAL PRINCIPLES OF ELECTROSTATICS [CH. I

When the gold leaves are electrified they are repelled
from each other and diverge, the amount of the divergence
giving some indication of the degree of electrification. It
is desirable to protect the gold leaves from the influence
of electrified bodies which may happen to be near the
electroscope, and from any electrification there may be on
the surface of the glass. To do this we take advantage of
the property of electrical action (proved in Art. 33), that a
closed metallic vessel completely protects bodies inside it
from the electrical action of bodies outside. Thus if the
gold leaves could be completely surrounded by a metal
vessel, they would be perfectly shielded from extraneous
electrical influence : this however is not practicable, as
the metal case would hide the gold leaves from obser
vation. In practice, sufficient protection is afforded by
a cylinder of metal gauze connected to earth, such as is
shown in Fig. 1, care being taken that the top of the
gauze cylinder reaches above the gold leaves.

If the disc of the electroscope is touched by an electri
fied body, part of the electrification will go to the gold
leaves; these will be electrified in the same way, and
therefore will be repelled from each other. In this case
the electrification on the gold leaves is of the same sign
as that on the electrified body. When the electrified
body does not touch the disc but is held near to it, the
metal parts of the electroscope will be electrified by induc
tion ; the disc, being the part nearest the electrified body,
will have electrification opposite to that of the body, while
the gold leaves, being the parts furthest from the elec
trified body, will have the same kind of electrification
as the body, and will repel each other. This repulsion
will cease as soon as the electrified body is removed.

4] GENERAL PRINCIPLES OF ELECTROSTATICS 7

If, when the electrified body is near the electroscope,
the disc is connected to the ground by a metal wire, then
the metal of the electroscope, the wire and the ground,
will correspond to the elongated piece of metal in the
experiment described in Art. 3. Thus, supposing the body
to be negatively electrified, the positive electrification will
be on the disc, while the negative will go to the most
remote part of the system consisting of the metal of the
electroscope, the wire and the ground, i.e. the negative
electrification will go to the ground and the gold leaves will
be free from electrification. They cease then to repel each
other and remain closed. If the wire is removed from
the disc while the electrified body remains in the neigh
bourhood, the gold leaves will remain closed as long as the
electrified body remains stationary, but if this is removed
far away from the electroscope the gold leaves diverge.
The positive electrification, which, when the electrified
body was close to the electroscope, concentrated itself on
the disc so as to be as near the electrified body as possible,
when this body is removed spreads to the gold leaves and
causes them to diverge.

If, when the electroscope is charged, we wish to deter
mine whether the charge is positive or negative, all we
have to do is to bring near to the disc of the electroscope
a stick of sealing-wax, which has been negatively electrified
by friction with flannel ; the proximity of the negatively
electrified wax, in consequence of the induction (Art. 3),
increases the negative electrification on the gold leaves.
Hence, if the presence of the sealing-wax increases the
divergence of the leaves, the original electrification was
negative, but if it diminishes the divergence the original
electrification was positive.

GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

5. Charge on an electrified body. Definition
of equal charges. Place on the disc of the electro
scope a metal vessel as nearly closed as possible, the
opening being only just wide enough to allow electrified

Fig. 2.

bodies to be placed inside. Then introduce into this vessel
a charged body suspended by a silk thread, and let it sink
well below the opening. The gold leaves of the electro
scope will diverge, since they will be electrified by in
duction (see Art. 3), but the divergence will remain the
same however the body is moved about in the vessel. If
two or more electrified bodies are placed in the vessel the
divergence of the gold leaves is the same however the
electrified bodies are moved about relatively to each other
or to the vessel. The divergence of the gold leaves thus
measures some property of the electrified body which re
mains constant however the body is moved about within
the vessel. This property is called the charge on the body,

6] GENERAL PRINCIPLES OF ELECTROSTATICS 9

and two bodies, A and B, have equal charges when the
divergence of the gold leaves is the same when A is inside
the vessel placed on the disc of the electroscope and B far
away, as when B is inside and A far away. A and B are
each supposed to be suspended by dry silk threads, for such
threads do not allow the electricity to escape along them ;
see Art. 6. Again, the charge on a body C is twice that
on A if, when C is introduced into the vessel, it produces
the same effect on the electroscope as that produced by
A and B when introduced together. B is a body whose
charge has been proved equal to that on A in the way
just described. Proceeding in this way we can test what
multiple the charge on any given electrified body is of
the charge on another body, so that if we take the latter
charge as the unit charge we can express any charge in
terms of this unit.

Two bodies have equal and opposite charges if when
introduced simultaneously into the metal vessel they pro
duce no effect on the divergence of the gold leaves.

6. Insulators and Conductors. Introduce into
the vessel described in the preceding experiment an elec
trified pith ball coated with gold leaf and suspended by a
dry silk thread : this will cause the gold leaves to diverge.
If now the electrified pith ball is touched with a stick of
sealing-wax, an ebonite rod or a dry piece of glass tube, no
effect is produced on the electroscope, the divergence of
the gold leaves is the same after the pith ball has been
touched as it was before. If, however, the pith ball is
touched with a metal wire held in the hand or by the
hand itself, the gold leaves of the electroscope immediately
fall together and remain closed after the wire has been

10 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

withdrawn from the ball. Thus the pith ball loses its
charge when touched with a metal wire, though not when
touched with a piece of sealing-wax. We may thus divide
bodies into two classes, (1) those which, when placed in
contact with a charged body, can discharge the electrifica
tion, these are called conductors ; (2) those which can not
discharge the electrification of a charged body with which
they are in contact, these are called insulators. The
metals, the human body, solutions of salts or acids are
examples of conductors, while the air, dry silk threads,
dry glass, ebonite, sulphur, paraffin wax, sealing-wax,
shellac are examples of insulators.

When a body is entirely surrounded by insulators it is
said to be insulated.

7. When electrification is excited by friction or by
any other process, equal charges of positive and negative
electricity are always produced. To show this, when the
electrification is excited by friction, take a piece of sealing-
wax and electrify it by friction with a piece of flannel ;
then, though both the wax and the flannel are charged
with electricity, they will, if introduced together into the
metal vessel on the disc of the electroscope (Art. 5), pro
duce no effect on the electroscope, thus showing that the
charge of negative electricity on the wax is equal to the
charge of positive electricity on the flannel. This can
be shown in a more striking way by working a frictional
electrical machine, insulated and placed inside a large
insulated metal vessel in metallic connexion with the
disc of an electroscope ; then, although the most vigorous
electrical effects can be observed near the machine inside
the vessel, the leaves of the electroscope remain unaffected.

7] GENERAL PRINCIPLES OF ELECTROSTATICS 11

showing that the total charge inside the vessel connected
with the disc has not been altered though the machine
has been in action.

To show that, when a body is electrified by induction,
equal charges of positive and negative electrification are
produced, take an electrified body suspended by a silk
thread, lower it into the metal vessel on the top of the
electroscope and observe the divergence of the gold leaves ;
then take a piece of metal suspended by a silk thread
and lower it into the vessel near to but not in con
tact with the electrified body ; no alteration in the diver
gence of the gold leaves will take place, showing that the
total charge on the piece of metal introduced into the
vessel is zero. This piece of metal is, however, electrified
by induction, so that its charge of positive electrification
excited by this process is equal to its charge of negative
electrification.

Again, when two charged bodies are connected by a
conductor, the sum of the charges on the bodies is unaltered,
i.e. the amount of positive electrification gained by one is
equal to the amount of positive electrification lost by the
other. To show this, take two electrified metallic bodies,
A and B, suspended from silk threads, and introduce A into
the metal vessel, noting the divergence of the gold leaves ;
then introduce B into the vessel and observe the diver
gence when the two bodies are in the vessel together : now
take a piece of wire wound round one end of a dry glass
rod and, holding the rod by the other end, place the wire
so that it is in contact with A and B simultaneously ; no
alteration in the divergence of the gold leaves will be pro
duced by this process, showing that the sum of the charges
on A and B is unaltered. Take away the wire and remove

12 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

B from the vessel, and now again observe the divergence
of the gold leaves ; it will not (except in very special cases)
be the same as it was before B was put into the vessel,
thus proving that, though a transference of electrification
between A and B has taken place, the sum of the charges
on A and B has not changed.

8. Force between bodies charged with elec
tricity. When two charged bodies are at a distance r
apart, r being very large compared with the greatest linear
dimension of either of the bodies, the repulsion between
them is proportional to the product of their charges and
inversely proportional to the square of the distance between
them.

This law was first proved by Coulomb by direct mea
surement of the force between electrified bodies; there
are, however, other methods by which the law can be
much more rigorously established ; as these can be most
conveniently considered when we have investigated the
properties of this law of force, we shall begin by assuming
the truth of this law and proceed to investigate some of
its consequences.

9. Unit charge. We have seen in Art. 5 how the
charges on electrified bodies can be compared with each
other; in order, however, to express the numerical value
of any charge it is necessary to have a definite unit of
charge with which the charge can be compared.

The unit charge of electricity is defined to be such
that when two bodies each have this charge, and are
separated by unit distance in air they are repelled from
each other with unit force. The dimensions of the charged

9] GENERAL PRINCIPLES OF ELECTROSTATICS 13

bodies are assumed to be very small compared with the
unit distance.

It follows from this definition and the law of force
previously enunciated that the repulsion between two
small bodies with charges e and e' placed in air at a
distance r apart is equal to

The expression ee/r* will express the force between
two charged bodies, whatever the signs of their electrifi
cations, if we agree that, when the expression is positive,
it indicates that the force between the bodies is a re
pulsion, and that, when this expression is negative, it
indicates that the force is an attraction. When the
charges on the bodies are of the same kind ee' is positive,
the force is then repulsive; when the charges are of
opposite sign ee' is negative, the force between the bodies
is then attractive.

Electric Intensity. The electric intensity at any
point is the force acting on a small body charged with
unit positive charge when placed at the point, the electri
fication of the rest of the system being supposed to be
undisturbed by the presence of this unit charge.

Total Normal Electric Induction over a Surface.

Imagine a surface drawn anywhere in the electric field,
and let this surface be completely divided up as in the
figure, into a network of meshes, each mesh being so small
that the electric intensity at any point in a mesh may be
regarded as constant over the mesh. Take a point in
each of these meshes and find the component of the
electric intensity at that point in the direction of the

14 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

normal drawn from the outside of the surface at that
point, and multiply this normal component by the area

Fig. 3.

of the mesh ; the sum of these products for all the meshes
on the surface is denned to be the total normal electric
induction over the surface. This is algebraically expressed
by the relation

where / is the total normal electric induction, N the com
ponent of the electric intensity resolved along the normal
drawn from the outside of the surface at a point in a
mesh, and w is the area of the mesh : the symbol S denotes
that the sum of the products Nco is to be taken for all the
meshes drawn on the surface.

With the notation of the integral calculus

I-JffdS,

where dS is an element of the surface, the integration
extending all over the surface.

10. Gauss's Theorem. We can prove all the pro
positions about the forces between electrified bodies, which
we shall require in the following discussion of Electro
statics, by the aid of a theorem due to Gauss. This
theorem may be stated thus: the total normal electric
induction over any closed surface drawn in the electric

10] GENERAL PRINCIPLES OF ELECTROSTATICS 15

field is equal to 4?r times the total charge of electricity
inside the closed surface.

We shall first prove this theorem when the electric
field is that due to a single charged body.

Let 0 (Fig. 4) be the charged body, whose dimensions
are supposed to be so small, compared with its distances

Fig. 4.

from the points at which the electric intensity is measured,
that it may be regarded as a point. Let e be the charge
on this body.

Let PQRS be one of the small meshes drawn on the
surface, the area being so small that PQRS may be regarded
as plane : join 0 to P, Q, R, S, and let a plane through
R at right angles to OR cut OS, OQ, OP respectively in
u,v,w: with centre 0 describe a sphere of unit radius, and
let the lines OP, OQ, OR, OS cut the surface of this sphere
in the points p, q, r, s respectively. The area PQRS is
assumed to be so small that the electric intensity may be

16 GENERAL PRINCIPLES OF ELECTROSTATICS fCH. I

i regarded as constant over it ; we may take as the value
of the electric intensity e/OR2, which is the value it
I has at R.

The contribution of this mesh to the total normal
induction is, by definition, equal to

area PQRS x JV,

where N is the normal component of the electric intensity
at.R

where 6 is the angle between the outward normal to the
surface at R, and OR the direction of the electric intensity.
The normal to the surface is at right angles to PQRS,
and OR is at right angles to the area Ruvw, and hence
the angle between the normal to the surface and OR is
equal to the angles between the planes PQRS and Ruvw.

Hence

area PQRS x cos 6 — the area of the projection of the
area PQRS on the plane Ruvw

= area Ruvw ..................... (1).

Consider the figures Ruwv and rspq. Ru is parallel
to rs since they are in the same plane and both at right
angles to OR, and for similar reasons Rv is parallel to rqt
vw to pq, uw to sp. The figure Ruwv is thus similar
to rspq : and the areas of similar figures are proportional
to the squares of their homologous sides. Hence

area Ruwv : area rspq = Ru2 : rs2

10] GENERAL PRINCIPLES OF ELECTROSTATICS 17

, , area Ruwv area pars

80 that -W -$-

= area pqrs ............... (2),

since Or is equal to unity by construction.

The contribution of the mesh PQRS to the total
normal induction is equal to

area PQRS x - x cos 9

area Ruvw , , . ,., .

bJ equation (1)

= e x area _pgrs by equation (2).

Thus the contribution of the mesh to the total normal
induction is equal to e times the area cut off a sphere of
unit radius with its centre at 0 by a cone having the
mesh for a base and its vertex at 0.

By dividing up any finite portion of the surface into
meshes and taking the sum of the contributions of each
mesh, we see that the total normal induction over the
surface is equal to e times the area cut off a sphere of
unit radius with its centre at 0 by a cone having the
boundary of the surface as base and its vertex at 0.

Let us now apply the results we have obtained to the
case of a closed surface.

First take the case where 0 is inside the surface.
The total normal induction over the surface is equal to
e times the sum of the areas cut off the unit sphere by
cones with their bases on the meshes and their vertices
at 0, and since the meshes completely fill up the closed
surface the sum of the areas cut off the unit sphere by
the cones will be the area of the sphere, which is equal

T. E. 2

18 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

to 4?r, since its radius is unity. Thus the total normal
induction over the closed surface is 4?r0.

Next consider the case when 0 is outside the closed
surface.

Draw a cone with its vertex at 0 cutting the closed
surface in the areas PQRS, P'QR'S'. Then the magni
tude of the total normal induction over the area PQRS

Fig. 5.

is equal to that over the area P'Q'R'S', since they are
each equal to e times the area cut off by this cone from a
sphere whose radius is unity and centre at 0. But over
the surface PQRS the electric intensity points along the
outward drawn normal so that the sign of the component
resolved along the outward drawn normal is positive ;
while over the surface P'Q'R'S' the electric intensity is in
the direction of the inward drawn normal so that the sign
of its component along the outward drawn normal is
negative. Thus the total normal induction over PQRS is
of opposite sign to that over P'Q'R'S', and since they are
equal in magnitude they will annul each other as far as
the total normal induction is concerned. Since the whole
of the closed surface can be divided up in this way by
cones with their vertices at 0, and since the two sections of
each of these cones neutralize each other, the total normal
induction over the closed surface will be zero.

10] GENERAL PRINCIPLES OF ELECTROSTATICS 19

We thus see that when the electric field is due to a
small body with a charge e, the total normal induction
over any closed surface enclosing the charge is 4t7re, while
it is equal to zero over any closed surface not enclosing
the charge. We have therefore proved Gauss's theorem
when the field is due to a single small electrified body.

We can easily extend it to the general case when the
field is due to any distribution of electrification. For we
may regard this as arising from a number of small bodies
having charges el, e^, es... &c. Let N be the component
along the outward drawn normal to the surface of the
resultant electric intensity, N! the component in the same
direction due to el} N2 that due to e2 and so on ; then

If o> is the area of the mesh at which the normal
electric intensity is N, the total normal induction over the
surface is

that is, the total normal electric induction over the surface
due to the electrical system is equal to the sum of the
normal inductions due to the small charged bodies of which
the system is supposed to be built up. But we have just
seen that the total normal induction over a closed surface
due to any one of these is equal to 4?r times its charge if
the body is inside the surface, and is zero if the body is
outside the surface. Hence the sum of the total normal
inductions due to the several charged bodies, i.e. that due
to the actual field, is 4?r times the charge of electricity
inside the closed surface over which the normal induction
is taken.

2—2

20 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

11. Electric intensity at a point outside a
uniformly charged sphere. Let us now apply the
theorem to find the electric intensity at any point in
the region outside a sphere uniformly charged with elec
tricity.

Let 0 be the centre of the sphere, P a point outside
the sphere at which the electric intensity is required.

Through P draw a spherical surface with its centre
at 0. Let R be the electric intensity at P. Since
the charged sphere is uniformly electrified, the direction
of the intensity will be OP, and it will have the same
value R at any point on the spherical surface through P.
Hence since at each point on this surface the normal
electric intensity is equal to R, the total normal induc
tion over the sphere through P is equal to R x (surface of
the sphere), i.e. R x 4?r . OP2. By Gauss's theorem this is
equal to 4?r times the charge enclosed by the spherical
surface, that is to 4?r times the charge on the inner
sphere. If e is this charge we have therefore

R x 4-TrOP2 - 47re,

7? e
= OP*'

Hence the intensity at a point outside a uniformly
electrified sphere is the same as if the charge on the
sphere were concentrated at the centre.

12. Electric intensity at a point inside a uni
formly electrified spherical shell. Let Q be a point
inside the shell, R the electric intensity at that point.
Through Q draw a spherical surface, centre 0; then as
before, the normal electric intensity will be constant all

13] GENERAL PRINCIPLES OF ELECTROSTATICS 21

over this surface. The total normal induction over this
sphere is therefore R x area of sphere, i.e.

R x 4vr . OQ2.

By Gauss's theorem this is equal to 4?r times the
charge of electricity inside the spherical surface passing
through Q ; hence as there is no charge inside this surface,

Hence the electric intensity vanishes at any point inside
a uniformly electrified spherical shell.

13. Infinite Cylinder uniformly electrified. We

shall next consider the case of an infinitely long circular
cylinder uniformly electrified. Let P be a point out
side the cylinder at which we wish to find the electric
intensity. Through P describe a circular cylinder coaxial
with the electrified one, draw two planes at right
angles to the axis of the cylinder at unit distance
apart, and consider the total normal induction over the
closed surface formed by the curved surface of the
cylinder through P and the two plane ends. Since the
electrified cylinder is infinitely long and is symmetrical
about its axis, the electric intensity at all points at the
same distance from the axis of the cylinder will be the
same, and the electric intensity at P will by symmetry
be along a radius drawn through P at right angles to the
axis of the cylinder.

Thus the electric intensity at any point on either plane
end of the cylinder will be in the plane of that end,
and will therefore have no component at right angles
to it; the plane ends will therefore contribute nothing

22 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

to the total normal induction over the surface. At each
point of the cylindrical surface the electric intensity is
at right angles to the curved surface and is equal to R.
The total normal induction over the surface is therefore

R x (area of the curved surface of the cylinder).

But since the length of the curved surface is unity
its area is equal to 2-Trr, where r is the distance of P
from the axis of the cylinder. If E is the charge per
unit length on the electrified cylinder, then by Gauss's
theorem the total normal induction over the surface is
equal to 4urE. The total normal induction is however
equal to R x 2?rr, hence

R X

*-§»

Thus, in the case of the cylinder, the electric intensity
varies inversely as the distance from the axis of the
cylinder.

We can prove in the same way as for the uniformly elec
trified spherical shell that the electric intensity vanishes
at any point inside a uniformly electrified cylindrical
shell.

14. Uniformly electrified infinite plane. In this
case we see by symmetry (1) that the electric intensity
will be normal to the plane, (2) that the electric intensity
will be constant at all points in a plane parallel to the
electrified one. Draw a cylinder PQRS, Fig. 6, the
axis of the cylinder being at right angles to the plane,
the ends of the cylinder being planes at right angles to

14] GENERAL PRINCIPLES OF ELECTROSTATICS 23

the axis. Since this cylinder encloses no electrification the
total normal induction over its surface is zero by Gauss's

Fig. 6.

theorem. But since the electric intensity is parallel to
the axis of the cylinder the normal intensity vanishes
over the curved surface of the cylinder. Let F be the
electric intensity at a point on the face PQ — this is
along the outward drawn normal if the electrification
on the plane is positive — F' the electric intensity at a
point on the face RS, o> the area of either of the faces
PQ or RS, then the total normal induction over the
surface PQRS is equal to

Fu-F'a)',

and since this vanishes by Gauss's theorem

F=F',

or the electric intensity at any point, due to the infinite
uniformly charged plane, is independent of the distance
of the point from the plane. It is, therefore, constant in
magnitude at all points in the field, acting upwards in the
region above the plane, downwards in the region below it.
To find the magnitude of the intensity at P. Draw
through P (Fig. 7) a line at right angles to the plane and
prolong it to Q, so that Q is as far below the plane as P
is above it. With PQ as axis describe a right circular
cylinder bounded by planes through P and Q parallel to
the electrified plane. Consider now the total normal
induction over the surface of this cylinder. The electric

24 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

intensity is everywhere parallel to the axis of the cylinder,
and has, therefore, no normal component over the curved

Fig. 7.

surface of the cylinder, the total normal induction over
the surface thus arises entirely from the flat ends. Let R
be the magnitude of the electric intensity at any point
in the field, CD the area of either of the flat ends of the
cylindrical surface. Then the part of the total normal
induction over the surface PQRS due to the flat end
through P is Rw. The part due to the flat end through
Q will also be equal to this and will be of the same sign,
since the intensity at Q is along the outward drawn
normal. Thus since the normal intensity vanishes over
the curved surface of PQRS the total normal induction
over the closed surface is 2Ra). If a is the quantity of
electricity per unit area of the plane the charge of elec
tricity inside the closed surface is aco ; hence by Gauss's
theorem

SRa) = 4:7T(7a),

or R = 27TO-.

By comparing this with the results given in Arts. 11 and
13 the student may easily prove that the intensity due
to the charged plane surface is half that just outside a
charged spherical or cylindrical surface having the same
charge of electricity per unit area.

16] GENERAL PRINCIPLES OF ELECTROSTATICS 25

15. Lines of Force. A line of force is a curve
drawn in the electric field, such that its tangent at any
point is parallel to the electric intensity at that point.

16. Electric Potential. This is defined as follows:
the electric potential at a point P exceeds that at Q by
the work done by the electric field on a body charged with
unit of electricity when the latter passes from P to Q. The
path by which the unit of electricity travels from P to Q
is immaterial, as the work done will be the same whatever
the nature of the path. To prove this suppose that the
work done on the unit charge when it travels along the
path PAQ is greater than when it travels along the path

Fig. 8.

PBQ. Since the work done by the field on the unit of
electricity when it goes from P to Q along the path PBQ
is equal to the work which must be done by applied
mechanical forces to bring the unit from QtoP along QBP,
we see that if we make the unit travel round the closed
curve PAQBP the work done by the field on the unit
when it travels along PAQ is greater than the work
spent by the applied forces in bringing it back from
Q to P along the path QBP. Thus though the unit of
electricity is back at the point from which it started,
and if the field is entirely due to charges of electricity,
everything is the same as when it started, we have, if our

26 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

hypothesis is correct, gained work. This is not in ac
cordance with the principle of the Conservation of Energy,
and we therefore conclude that the hypothesis on which
it is founded, i.e. that the work done on unit electric
charge when it travels from P to Q depends on the path
by which it travels, is incorrect.

Since electric phenomena only depend upon differences
of potential it is immaterial what point we take as the
one at which we call the potential zero. In mathematical
investigations it simplifies the expression for the potential
to assume as the point of zero potential one at an infinite
distance from all the electrified bodies.

If P and Q are two points so near together that the
electric intensity may be regarded as constant over the
distance PQ, then the work done by the field on unit
charge when it travels from P to Q is F x PQ, if F is the
electric intensity resolved in the direction PQ. If VP,
VQ denote the potentials at P and Q respectively, then
since by definition VP — VQ is the work done by the field
on unit charge when it goes from P to Q we have

VP-VQ=FxPQ,

hence

thus the electric intensity in any direction is equal to the
rate of diminution of the potential in that direction.

Hence if we draw a surface such that the potential is
constant over the surface (a surface of this kind is called
an equipotential surface) the electric intensity at any
point on the surface must be along the normal. For since
the potential does not vary as we move along the surface,

17] GENERAL PRINCIPLES OF ELECTROSTATICS 27

we see by equation (1) that the component of the electric
intensity tangential to the surface vanishes.

Conversely a surface over which the tangential com
ponent of the intensity is everywhere zero will be an
equipotential surface, for since there is no tangential in
tensity no work is done when the unit charge moves along
the surface from one point to another ; that is, there is no
difference of potential between points on the surface.

The surface of a conductor placed in an electric field
must be an equipotential surface when the field is in
equilibrium, for there can be no tangential electric in
tensity, otherwise the electricity on the surface would
move along the surface and there could not be equili
brium. It is this fact that makes the conception of the
potential so important in electrostatics, for the surfaces of
all bodies made of metal are equipotential surfaces.

17. Potential due to a uniformly charged sphere.

The potential at P is the work done by the electric field
when unit charge is taken from P to an infinite distance.
Let us suppose that the unit charge travels from P to an
infinite distance along a straight line passing through the
centre of the sphere. Let QRST be a series of points

ttt

Fig. 9.

very near together along this line. If e is the charge on
the sphere, 0 its centre, the electric intensity at Q is e/OQ2,
while that at R is e/OR2; as Q and R are very near together
these quantities are very nearly equal, and we may take

28 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

the average electric intensity between Q and R as equal
to e/OQ . OR, the geometric mean of the intensities at Q
and R. Hence the work done by the field as the unit
charge goes from Q to -R is equal to

~OQ OR'

Similarly the work done by the field as the charge goes
from R to 8 is

e _e_
OR OS'

as it goes from S to T

and so on. The work done by the field as the charge goes
from Q to T is the sum of these expressions, and this
sum is equal to

e _e_
OQ~W

and we see, by dividing up the distance between the points
into a number of small intervals and repeating the above
process that this expression will be true when Q and T are
a finite distance apart, and that it always represents the
work done by the field on the unit charge as long as
Q and T are two points on a radius of the sphere. The
potential at P is the work done by the field when the

18] GENERAL PRINCIPLES OF ELECTROSTATICS 29

unit charge goes from P to an infinite distance, and is
therefore by the preceding result equal to

This is also evidently the potential at P of a charge e
placed on a small body enclosing 0 if the dimensions of
the body over which the charge is spread are infinitesimal
in comparison with OP.

18. The electric intensity vanishes at any
point inside a closed equipotential surface which
does not enclose any electric charge. We shall first
prove that the potential is constant throughout the
volume enclosed by the surface; then it will follow by
equation (1), Art. 16, that the electric intensity vanishes
throughout this volume.

For if the potential is not constant it will be possible
to draw a series of equipotential surfaces inside the given
one; let us consider the equipotential surface for which
the potential is very nearly, but not quite, the same as for
the given surface. As the difference of potential between
this and the outer surface is very small the two surfaces
will be close together, and they cannot cut each other, for
if they did, any point in their intersection would have two
different potentials.

Suppose for a moment that the potential at the inner
surface is greater than that at the outer.

Let P be a point on the inner surface, Q the point
where the normal at P drawn outwards to the inner
surface cuts the outer surface. Then, since the electric
intensity from P to Q is equal to ( VP — VQ)/PQ and since
by hypothesis VP — VQ is positive, we see that the normal

30 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

electric intensity over the second surface is everywhere in
the direction of the outward drawn normal to the surface,
and therefore that the total normal electric induction over
the surface will be positive. Hence there must be a positive
charge inside the surface, as the total normal induction
over the surface is, by Gauss's theorem, proportional to the
charge enclosed by the surface. Hence, as by hypothesis
there is no charge inside the surface, we see that the
potential over the inner surface cannot be greater than
that at the outer surface. If the potential at the inner
surface were less than that at the outer, then the normal
electric intensity would be everywhere in the direction of
the inward normal, and, as before, we can show by Gauss's
theorem that this would require a negative charge inside
the surface. Hence, as there is no charge either positive
or negative the potential at the inner surface can neither
be greater nor less than at the outer surface, and must
therefore be equal to it. In this way we see that the
potential at all points inside the surface must have the
same value as at the surface, and since the potential is con
stant the electric intensity will vanish inside the surface.

19. It follows from this that if we have a closed
hollow conductor there will be no electrification on its
inner surface unless there are electrified bodies inside
the hollow. Let Fig. 10 represent the conductor with
a cavity inside it. To prove that there is no electrifica
tion at P a point on the inner surface, take any closed
surface enclosing a small portion a of the inner surface
near P; by Gauss's theorem the charge on a is pro
portional to the total normal electric induction over the
surface surrounding a. The electric intensity is however
zero everywhere over this surface. It is zero over the part

20]

GENERAL PRINCIPLES OF ELECTROSTATICS

of the closed surface which is in the material of the shell
because this part of the surface is in a conductor, and
when there is equilibrium the electric intensity is zero

Fig. 10.

at any point in a conductor. The electric intensity is zero
over the part of the closed surface which is inside the cavity
because the surface of the cavity being the surface of a
conductor is an equipotential surface, and as we have just
seen the electric intensity inside such a surface is zero
unless it encloses electric charges. Thus since the electric
intensity vanishes at each point on the closed surface
surrounding a, the charge at a must vanish ; in this way
we can see that there is no electrification at any point
on the inner cavity. The electrification is all on the
outer surface of the conductor.

20. Cavendish Experiment. The result proved in
Art. 18 that when the force between two charged bodies
varies inversely as the square of the distance between
them the electric intensity vanishes throughout the in
terior of an electrified conductor enclosing no charge,
leads to the most rigorous experimental proof of the
truth of this law.

32 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

Let us for simplicity confine our attention to the
case when the electrified conductor is a sphere posi
tively electrified.

Fig. 11.

Consider the state of things at a point P inside a
sphere whose centre is 0, Fig. 11 : through P draw a
plane at right angles to OP. The electrification on the
portion of the sphere above this plane produces an electric
intensity in the direction PO, while the electrification on
the portion of the sphere below the plane produces an
electric intensity in the direction OP. When the law of
force is that of the inverse square these two intensities
balance each other, the greater distance from P of the
electrification below the plane being compensated by the
larger electrified area.

Now suppose that intensity varies as r~p, then if p is
greater than 2 the intensity diminishes more quickly as
the distance increases than when the law of force is that of
the inverse square, so that if the larger area below the plane
was just sufficient to compensate for the greater distance
when the law of force was that of the inverse square it will
not be sufficient to do so when p is greater than 2 ; thus
the electrification on the portion of the sphere above the
plane will gain the upper hand and the resultant electric
intensity will be in the direction PO. Again, if p is less

20] GENERAL PRINCIPLES OF ELECTROSTATICS 33

than 2 the intensity will not diminish so rapidly when
the distance increases, as it does when p is equal to 2,
so that, if the greater area below the plane is sufficient
to compensate for the increased distance when the law of
force is that of the inverse square, it will be more than
sufficient to do so when p is less than 2 ; in this case the
electrification below the plane will gain the upper hand, and
the electric intensity at P will be in the direction OP.

Now suppose we have two concentric metal spheres
connected by a wire, and that we electrify the outer sphere
positively, then if p = 2 there will be no electric intensity
inside the outer sphere, and therefore no movement of
electricity to the inner sphere which will therefore remain
unelectrified. If p is greater than 2 we have seen that the
electric intensity due to the positive charge on the outer
sphere will be towards the centre of the sphere, i.e. the
force on a negative charge will be from the inner sphere
towards the outer. Negative electricity will therefore flow
from the inner sphere, which will be left with a positive
charge.

If however p is less than 2, the electric intensity due
to the charge on the outer sphere will be from the centre
of the sphere, and the direction of the force acting on a
positive charge will be from the inner sphere to the outer.
Positive electricity will therefore flow from the inner
sphere to the outer, so that the inner sphere will be left
with a negative charge.

Thus, according as p is greater than, equal to or less
than 2, the charge on the inner sphere will be positive,
zero or negative. By testing the state of electrification on
the inner sphere we can therefore test the law of force.
This is what was done by Cavendish in an experiment

T. E. 3

34 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

made by him, and which goes by his name*. The following
is a description of a slight modification, due to Maxwell,
of Cavendish's original experiment.

The apparatus for the experiment is represented in
Fig. 12.

Fig. 12.

The outer sphere A, made up of two tightly fitting
hemispheres, is fixed on an insulating stand, and the
inner sphere is fixed concentrically with the outer one by
means of an ebonite ring. Connection between the inner
and outer spheres is made by a wire fastened to a small
metal disc B which acts as a lid to a small hole in the
outer sphere. When the wire and the disc are lifted
up by a silk thread the electrical condition of the inner
sphere can be tested by pushing an insulated wire con
nected to an electroscope (or preferably to a quadrant
electrometer, see Art. 60) through the hole until it is in

* Mr Woodward (Nature, March 4, 1909) has pointed out that
Priestley (History of Electricity, 2nd Edition, 1769, p. 711) anticipated
Cavendish in this proof of the law of the inverse square.

20] GENERAL PRINCIPLES OF ELECTROSTATICS 35

contact with the inner sphere. The experiment is made
as follows : when the two spheres are in connection a
charge of electricity is communicated to the outer sphere,
fche connection between the spheres is then broken by
lifting the disc by means of the silk thread; the outer
sphere is then discharged and kept connected to earth ;
the testing wire is then introduced through the hole and
put into contact with the inner sphere. Not the slightest
effect on the electroscope can be detected, showing that
if there is any charge on the inner sphere it is too small
to affect the electroscope. To determine the sensitiveness
of the electroscope or electrometer, a small brass ball
suspended by a silk thread, is placed at a considerable
distance from the two spheres. After the outer sphere is
charged (suppose positively) the brass ball is touched and
then left insulated ; in this way the ball gets by induction
a negative charge amounting to a calculable fraction, say a,
of the original charge communicated to the outer sphere.
Now when the outer sphere is connected to earth this
negative charge on the ball will induce a positive charge
on the outer sphere which is a calculable fraction, say ft, of
the charge on the ball. If we disconnect the outer sphere
from the earth and discharge the ball this positive charge
on the outer sphere will be free to go to the electroscope
if this is connected to the sphere. When the ball is not
too far away from the sphere this charge is sufficient to
deflect the electroscope, i.e. a fraction aft of the original
charge on the sphere is sufficient to deflect the electro
scope, showing that the charge on the inner sphere in
the Cavendish experiment could not have amounted to
aft of the charge communicated to the outer sphere*. If

* Since the electroscope is connected with the inner sphere in the
first part of the Cavendish experiment and with the outer sphere in the

3—2

36 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

the force between two charges is assumed to vary as r~p, we
can calculate the charge on the inner sphere and express
it in terms of p, and then, knowing from the Cavendish
experiment that this charge is less than a/3 of the original
charge, we can calculate that p must differ from 2 by less
than a certain quantity. In this way it has been shown
that p differs from 2 by less than 1/20,000.

21. Definition of surface density. When the elec
trification is confined to the surface of a body, the charge
per unit area is called the surface density of the electricity.

22. Coulomb's Law. The electric intensity at a
point P close to the surface of a conductor surrounded
by air is at right angles to the surface and is equal to 4?ra-
where or is the surface density of the electrification.

The first part of this law follows from Art. 16, since
the surface of a conductor is an equipotential surface.

Fig. 13.

To prove the second part take on the surface a small area
around P (Fig. 13) and through the boundary of this

second part, the capacity of the electroscope and its connections will not
be the same in the two cases ; in estimating the sensitiveness of the
method a correction must be made on this account, this is easily
done by the method of Art. 61.

23] GENERAL PRINCIPLES OF ELECTROSTATICS 37

area draw the cylinder whose generating lines are parallel
to the normal at P. Let this cylinder be truncated at T
and S by planes parallel to the tangent plane at P.

The total normal electric induction over this cylinder is
Rw, where R is the normal electric intensity and co the area
of the cross section. For Ra> is the part of the total normal
induction due to the end T of the cylinder, and this is the
only part of the surface of the cylinder which contributes
anything to the total normal induction. For the intensity
along that part of the curved surface of the cylinder which
is in air is tangential to the surface and therefore has
no component along the normal, while since the electric
intensity vanishes inside the conductor the part of the
surface which is inside the conductor will not contribute
anything to the total induction. If cr is the surface density
of the electricity at P the charge inside the cylinder is
coo- ; hence by Gauss's theorem

Ra) = 4<7ra)(r
or R = 4-7TO-.

The result expressed by this equation is known as
Coulomb's Law. It requires modification when the con
ductor is not surrounded by air, but by some other in
sulator. See Art. 71.

23. Energy of an electrified system. If a number
of conductors are placed in an electric field, and if El is
the charge on the first conductor, Vi its potential, E2 the
charge on the second conductor, V2 its potential, and so
on, then we can show that the potential energy of this
system of conductors is equal to

To prove this we notice that the potentials of the

38 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

conductors will depend upon the charges of electricity on
the conductors, in such a way that if the charge on every
part of the system is increased m times, the potential at
every point in the system will also be increased m times.

To find the energy of the system of conductors we
shall suppose that each conductor is originally uncharged,
and at potential zero, and that we bring a charge E^n
from an infinite distance to the first conductor, a charge
Ez/n from an infinite distance to the second conductor,
a charge Es/n to the third conductor, and so on. After
this has been done, the potential of the first conductor
will be VJn, that of the second Vz/n, and so on. Let
us call this the first stage of the operation. Then
bring from an infinite distance charges E-Jn to the first
conductor, Ez/n to the second, and so on. When this
has been done the potentials of the conductors will be
2Fi/n, 2F2/n, .... Call this the second stage of the
operation. Repeat this process until the first conductor
has the charge El and the potential F1? the second con
ductor the charge E2 and the potential F2 , and so on.

Then in the first stage the potential of the first con
ductor is zero at the beginning, and V-i/n at the end ; the
work done in bringing up to it the charge EJn is therefore

77* V

greater than 0 but less than — . — ; similarly the work

spent in bringing up the charge Ez/n to the second con-

E V

ductor is greater than zero but less than — . — .

n n

If !$! be the work spent in the first stage of the
operations in charging the first conductor we have

23] GENERAL PRINCIPLES OF ELECTROSTATICS 39

In the second stage of the operations the potential of
the first conductor is V^n at the beginning, and 2 V^n at
the end, so that the work spent in bringing up the charge

Tf V~

Ei/n to the first conductor is greater than — . — but less
11 n n

P %V

than — . — * ; similarly the work spent in bringing up
n n

the charge E2/n to the second conductor is greater than

-2 — 2but less than ^.^Z?. Thus if Q is the work
n n n n

spent in this stage in charging the first conductor we have

Similarly if & is the work spent in the third stage in
charging the first conductor we have

,ft>!^,, &<^E,
and nQlt the work spent in the last stage, is

and <-,

Now Q! the total amount of work spent in charging the
first conductor is equal to & + 2Qi + . . . nQi , and is therefore

l + 2 + 8 + ...(«-l) EV l + Z + 9+...n

or >sll

40 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

If however we make n exceedingly great the two limits
coincide, and we see that Q, the total work spent in

charging the first conductor is equal to ^E^V^ and Q

the work done in charging the whole system is given by
the equation

The work done in charging the conductors is stored
up in the system as electrical energy, the potential
energy of the system being equal to the work done in
charging up the system ; the energy only depends on the
final state of the system and is independent of the way
that state is arrived at. Hence we see from the above
result that the energy of a system of conductors is one
half the sum of the products obtained by multiplying the
charge of each conductor by its potential.

24. Relation between the potentials and charges
on the conductors. Superposition of electrical
effects. Let V be the potential at any point P when
the first conductor has a charge E± and all the other
conductors are without charge, and V" the potential at P
when the second conductor has the charge E2 and all
the other conductors are without charge ; then when the
first conductor has the charge Elt the second the charge
E2, and all the other conductors are without charge, the
potential at P will be V + V".

The conditions to be satisfied in this case are that the
charges on the conductors should have the given values
and that the surfaces of the conductors should be equi-
potential surfaces.

25] GENERAL PRINCIPLES OF ELECTROSTATICS 41

Now consider the distribution of electrification when
the first conductor has the charge E1 and the rest are
without charge; this satisfies the conditions that the
conductors are equipotential surfaces, that the charge on
the first conductor is E^ and that the charges on the
other conductors are zero. The distribution of electri
fication when the second conductor is charged and the
rest uncharged satisfies the conditions that the conductors
are equipotential surfaces, that the charge on the first
conductor is zero, that the charge on the second con
ductor is E2, and that the charges on the other conductors
are zero. If we take a new distribution formed by super
posing the last two distributions, it will satisfy the con
ditions that the conductors are equipotential surfaces, that
the charge on each conductor is the sum of the charges
corresponding to the two solutions, i.e. that the charge on
the first conductor is El} that on the second conductor E2,
and that on each of the other conductors zero. In other
words, the new distribution will be that which occurs in
the case when the first conductor has the charge Elt the
second the charge Ez> while the rest of the conductors
are uncharged. But when two systems of electrification
are superposed, the potential at P is the sum of the
potentials due to the two systems separately, i.e. the
potential at P is V + V", and hence the theorem is true.

25. We can extend this reasoning to the general case
in which V is the potential at P when the first conductor
has the charge Elf the other conductors being uncharged,
V" the potential at P when the second conductor has the
charge EZt the other conductors being uncharged, V" the
potential at P when the charge on the third conductor is

42 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

Es, the other conductors being uncharged, and so on;
and we then see that when the first conductor has the
charge Ely the second the charge E2, the third the charge
E3, and so on, the potential at P is

26. When the first conductor has the charge Elt the
other conductors being uncharged and insulated, the
potentials of the conductors will be proportional to Elt
that is, the potentials of the first, second, third, &c. con
ductors will be respectively

where pn> pl2, pl3 are quantities which do not depend upon
the charges of the conductors or their potentials, but only
upon their shapes and sizes and their positions with
reference to each other. The quantities pn,pi2, PIS, &c.
are called coefficients of potential ; their properties are
further considered in Arts. 27 — 31. When the second
conductor has the charge E2) the other conductors being
uncharged and insulated, the potentials of the conductors
will be proportional to E2) and the potentials of the first,
second, third, &c. conductors will be

p2lE2, p22E2, p^E^, ....

When the third conductor has the charge E2, the other
conductors being uncharged and insulated, the potentials
of the first, second, third conductors will be

Hence by Art. 25, we see that when the first con
ductor has the charge Elt the second the charge E2, the

27] GENERAL PRINCIPLES OF ELECTROSTATICS 43

third the charge E3, and so on, Vl the potential of the
first conductor will be given by the equation
Vl=p11El+p2lE2+p3lEs+...,
Vz the potential of the second conductor by the equation

if F3 is the potential of the third conductor
If we solve these equations we get

where the q's are functions of the ps and only depend
upon the configuration of the system of conductors. The
qs are called coefficients of capacity when the two suffixes
are the same and coefficients of induction when the suffixes
are different.

27. We shall now show that the coefficients which
occur in these equations are not all independent, but that

To prove this let us suppose that only the first and
second conductors have any charges, the others being
without charge and insulated. Then we may imagine the
system charged, by first bringing up the charge El from
an infinite distance to the first conductor and leaving all
the other conductors uncharged, and then when this has
been done, bringing up the charge E2 from an infinite
distance to the second conductor. The work done in
bringing the charge El up to the first conductor will be

44 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

the energy of the system, when the first conductor has the
charge El and the other conductors are without charge ;
the potential of the first conductor is in this case pn E± , so
that by Art. 23 the work done is ^E1.pnEl or ^p^Ef.
To find the work done in bringing up the charge E2 to
the second conductor let us suppose that this charge is
brought up in instalments each equal to E2/n. Then
the potential of the second conductor before the first in

stalment is brought up is, by Art. 26, equal to pl2E1} and

-p
after the first instalment has arrived it is p^El +p^ — .

Hence the work done in bringing up the first instalment
will be between

Similarly the work done in bringing up the second
instalment E2/n will be between

E2\E2 ,
— — 2 and

and the work done in bringing up the last instalment of
the charge will be between

E2 . / nE,\E2

• -

2 .
-• and

Thus the total amount of work done in bringing up the
charge E2 will be between

, , 1 +2 + 3 ...+w-l

and P12E,E2 +

28J GENERAL PRINCIPLES OF ELECTROSTATICS 45

that is, between

puE.E, + 1 (l - i) p22E2* and &JB& + i (1 + 1) pJE*t
but if n is very great these two expressions become equal

to pnEJEi + ^p&E22, which is therefore the work done in
bringing up the charge E% to the second conductor when
the first conductor has already received the charge Elf
Hence the work done in bringing up first the charge E1
and then E2 is

It follows in the same way that the work done when
the charge Ez is first brought to the second conductor and
then the charge El to the first is

but since the final state is the same in the two cases, the
work required to charge the conductors must be the same ;
hence

i.e. P2i=pu-

It follows from the way in which the <?'s can be ex
pressed in terms of the ps, that g2i = ^2-

28. Now pl2 is the potential of the second conductor
when unit charge is given to the first, the other con
ductors being insulated and without charge, and p.2l is the
potential of the first conductor when unit charge is given
to the second. But we have just seen that p2l =pl2, hence
the potential of the second conductor when insulated and
without charge due to unit charge on the first is equal to
the potential of the first when insulated and without
charge due to unit charge on the second, the remaining
conductors being in each case insulated and without charge.

46 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

29. Let us consider some examples of this theorem.
Let us suppose that the first conductor is a sphere with
its centre at 0, and that the second conductor is very
small and placed at P, then if P is outside the sphere we
know by Art. 17 that if unit charge is given to the sphere
the potential at P is increased by I/OP. It follows from
the preceding article that if unit charge be placed at P
the potential of the sphere when insulated is increased
by I/OP.

If P is inside the sphere then when unit charge is
given to the sphere the potential at P is increased by 1 /a
where a is the radius of the sphere. Hence if the sphere
is insulated and a unit charge placed at P the potential of
the sphere is increased by I/a. Thus the increase in the
potential of the sphere is independent of the position of P
as long as it is inside the sphere.

Since the potential inside any closed conductor which
does not include any charged bodies is constant, by
Art. 18, we see by taking as our first conductor a closed
surface, and as our second conductor a small body placed
at a point P anywhere inside this surface, that since the
potential at P due to unit charge on the conductor is
independent of the position of P, the potential of the
conductor when insulated due to a charge at P is inde
pendent of the position of P. Thus however a charged
body is moved about inside a closed insulated conductor
the potential of the conductor will remain constant. An
example of this is afforded by the experiment described in
Art. 5 ; the deflection of the electroscope is independent
of the position of the charged bodies inside the insulated
closed conductor.

30] GENERAL PRINCIPLES OF ELECTROSTATICS 47

30. Again, take the case when the first conductor is
charged, the others insulated and uncharged ; then

so that j-f = — .

r 2 PIZ

Now suppose that the first conductor is connected to
earth while a charge E2 is given to the second conductor,
all the other conductors being uncharged; then since
V1 = 0 we have

*_ Pn_

E- p12~~F2

by the preceding equation.

Hence if a charge be given to the first conductor, all
the others being insulated, the ratio of the potential of
the second conductor to that of the first will be equal in
magnitude but opposite in sign to the charge induced on
the first conductor, when connected to earth, by unit
charge on the second conductor.

As an example of this result, suppose that the first
conductor is a sphere with its centre at 0, and that the
second conductor is a small body at a point P outside the
sphere ; then if unit charge be given to the sphere, the
potential of the body at P is a/ OP times the potential of
the sphere, where a is the radius of the sphere ; hence, by
the theorem of this article, when unit charge is placed at
P, and the sphere is connected to the earth, there will be
a negative charge on the sphere equal to a/ OP.

Another example of this result is when the first

48 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

conductor completely surrounds the second ; then since
the potential inside the first conductor is constant when all
the conductors inside are free from charge, the potential
of the second conductor when a charge is given to the first
conductor will be the same as that of the first. Hence
from the above result it follows that when the first con
ductor is connected to earth, and a charge given to the
second, the charge induced on the first conductor will be
equal and opposite to that given to the second.

Another consequence of this result is that if S be an
equipotential surface when the first conductor is charged,
all the others being insulated, then if the first conductor
be connected to earth the charge induced on it by a
charge on a small body P remains the same however P
may be moved about, provided that P always keeps on
the surface 8.

31. As an example in the calculation of coefficients
of capacity and induction, we shall take the case when the
conductors are two concentric spherical shells. Let a
be the radius of the inner shell, which we shall call the
first conductor, 6 the radius of the outer shell, which
we shall call the second conductor. Let E^t E2 be the
charges of electricity on the inner and outer shells re
spectively, V1} V2 the corresponding potentials of these
shells.

Then if there were no charge on the outer shell the
charge El on the inner would produce a potential EJa on
its own surface, and a potential EJb on the surface of the
outer shell ; hence, Art. 26,

1 1

Pn = ~; P» = I.

32] GENERAL PRINCIPLES OF ELECTROSTATICS 49

The charge E2 on the outer shell would, if there were
no charge on the inner shell, make the potential inside
the outer shell constant and equal to the potential at
the surface of the outer shell. This potential is equal
to Ez/b, so that the potential of the first conductor due
to the charge E2 on the second is E2/b, which is also
equal to the potential of the second conductor due to
the charge E2 ; hence, by Art. 26,

_ I _1

We have therefore

Solving these equations, we get

*- <-

Hence

ab ab

We notice that qlz is negative ; this, as we shall prove
later, is always true whatever the shape and position of
the two conductors.

32. Another case we shall consider is that of two
spheres the distance between whose centres is very large
compared with the radius of either. Let a be the radius
of the first sphere, b that of the second, R the distance

T. E. 4

50 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

between their centres, El, E2 the charges, Vlt F2 the
potentials of the two spheres. Then if there were no
charge on the second sphere, the potential at the surface
of the first sphere would, if the distance between the
spheres were very great, be approximately El/a, while the
potential of the second sphere would be approximately

E,/R; hence

_1 JL

P*-a> Pi*-£>

approximately.

Similarly, if there were no charge on the first sphere,
but a charge E% on the second, the potential of the first
sphere would be E2/R, that of the second Ez/b, approxi
mately ; hence we have approximately

So that approximately

*-§+*•

Solving these equations we get

abR

-ab l R*-ab 2'
abR bR*

hence when R is large compared with either a or b
aR* abR bR2

approximately.

33]

GENERAL PRINCIPLES OF ELECTROSTATICS

We see that as before ql2 is negative. We also notice
that qn and ql2 become larger the nearer the spheres are
together.

33. Electric Screens. As an example of the use
of coefficients of capacity we shall consider the case of
three conductors, A, B, 0, and shall suppose that the first
of these conductors A is, as in Fig. 14, inside the third

Fig. 14.

conductor 0, which is supposed to be a closed surface,
while the second conductor B is outside C. Then if
El} V1; E3, F2; E3> F3 denote the charges and potentials
of the conductors A, B, G respectively, qu, q^,... ql2... the
coefficients of capacity and induction, we have

E^quVi + qnVt + quV, (1).

Z^q^ + q^ + q^ (2).

E* = quVi + qnV* + q*V* (3).

Now let us suppose that the conductor C is connected
to earth so that F3 is zero ; then, since the potential
inside a closed conductor is constant if it contains no
charge, we see that if El is zero, Vl must vanish whatever

4—2

52 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

may be the value of V2. Hence it follows from equation
(1) that ql2 must vanish ; putting qlz and Vs both zero we
see from (1) that

Bi^toK,
and from (2) E2 = q^V2.

Thus, in this case, the charge on A if its potential is
given, or the potential if its charge is given, is entirely
independent of E2 and F2, that is a charge on B produces
no electrical effect on A, while a charge on A produces
no electrical effect at B. Thus the interaction between A
and B is entirely cut off by the interposition of the closed
conductor at potential zero.

C is called an electric screen since it screens off from
A all the effects that might be produced by B. This
property of a closed metallic surface at zero potential has
very important applications, as it enables us by sur
rounding our instruments by a metal covering connected
with earth to get rid entirely of any electrical effects arising
from charged bodies not under our control. Thus, in the
experiment described in Art. 4, the gold leaves of the
electroscope were protected from the action of external
electrified bodies by enclosing them in a surface made of
wire-gauze and connected with the earth.

34. Expression for the change in the energy of
the system. The energy of the system Q is, by Art. 23,
equal to y^EV\ hence we have, by Art. 27

Q = Ip^E* + fauEf + . . . p^E, +....

If the charges are increased to EI, Ez' &c. the energy Q'
corresponding to these charges is given by the equation

Qf = fa,E* + \p^ + . . . pvEiEJ + . . . .

35] GENERAL PRINCIPLES OF ELECTROSTATICS 53

The work done in increasing the charges is equal to
Q' — Q. By the preceding equations

+ (Ea' - E2) i {Pl2 (E, + #/) +pn (E, + E,')
4- ......

where F/ , F2' . . . are the potentials of the first, second, . . .
conductors when their charges are E-{, Ecf....

Thus the work required to increase the charges is
equal to the sum of the products of the increase in the
charge on each conductor into the mean of the potentials
of the conductor before and after the charges are in
creased.

If we express Q and Q' by Art. 26 in terms of the
potentials instead of the charge, we have

Q = !?!!?? + iftaF,' + q^V.V, + . .. ,
Q' = fen F/2+ i?22F2'2+ ?12F/ F2' + . . . ,
and we see that

So that the work required is equal to the sum of the pro
ducts of the increase of potential of each conductor into the
mean of the initial and final charges of that conductor.

35. Force tending to produce any displacement
of the system. When the conductors are not connected
with any external source of energy, i.e. when they are
insulated, then by the principle of the Conservation of
Energy, the work done by the system during any dis
placement will be equal to the electrical energy lost by
the system in consequence of the displacement; and in

54 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

this case the conductors will tend to move so as to make
the electric energy diminish.

When, however, the potentials of the conductors are
kept constant, as may be done by connecting them with
galvanic batteries, we shall show that the system moves
so that the electric energy increases. There is thus not
merely work done by the system when it is displaced,
but along with this expenditure of work there is an in
crease in the electric energy, and the batteries to which
the conductors are attached are drained of a quantity of
energy equal to the sum of the mechanical work done and
the increase in the electric energy.

36. We shall now prove that if any small displace
ment of the system takes place the diminution in the
electrical energy, when the charges are kept constant, is
equal to the increase in the potential energy when the
same displacement takes place and the potentials are
kept constant.

Let Elt Vi, E2, F2, ... be the charges and potentials
of the conductors before the displacement takes place,

El} F/, E2, F2', ... the charges and potentials of the
conductors after the displacement has taken place when
the charges are constant,

EI, Fj, E2', F2, ... the charges and potentials of the
conductors after the displacement when the potentials are
constant.

Then since the electric energy is one half the sum of
the product of the charges and the potentials, the loss in
electric energy by the displacement when the charges are
constant is

iUW - F/) + JS, (V, - TV) + ...}.

36] GENERAL PRINCIPLES OF ELECTROSTATICS 55

The gain in electric energy when the potentials are
constant is

The difference between the loss when the charges are
constant and the gain when the potentials are constant is
thus equal to
i K^-JZ/) (F-FO+...1 + iP^i-^i'Fi') + ...}.

Now for the displaced positions of the system Elt F/,
E2, V^ ... are one set of corresponding values of the
charges and the potentials, while EI , Vlt E2 , F2... are
another set of corresponding values. Hence if pnf, pl2' , . . .
denote the values of the coefficients of induction for the
displaced position of the system

and
Thus

and

+ E.'E,) + ...,
hence EM + E,V,+ ... -(ElfV1'+ ...) = 0.

Thus the difference between the loss in electric energy
when the charges are kept constant and the gain when
the potentials are kept constant is equal to

56 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

Now when the displacements are very small E — E' and
V — V will each be proportional to the first power of the
displacements, and hence the preceding expression is pro
portional to the square of the displacements, and may be
neglected when the displacements are very small. Hence
we see that the loss in electric energy for any small dis
placement when the charges are kept constant, is equal
to the gain in potential energy for the same displacement,
when the potentials are kept constant. When the poten
tials are kept constant, the batteries which maintain the
potentials of the conductors at their constant value, will
be called upon to furnish twice the amount of mechanical
work done by the electric forces. For they will have to
furnish energy equal to the sum of the mechanical work
done and the increase in the electric energy of the system ;
the latter is, as we have just seen, equal to the decrease
in the electric energy of the system while the charges are
kept constant, and this is equal by the principle of the
Conservation of Energy to the mechanical work done.

37. Mechanical Force on each unit of area of
a charged conductor. The electric intensity is at right
angles to the surface of the conductor, so that the force
on any small portion of the surface surrounding a point P
will be along the normal to the surface at P.

To find the magnitude of this force let us consider a
small electrified area round P. Then the electric intensity
in the neighbourhood of P may conveniently be regarded
as arising from two causes, (1) the electrification on the
small area round P, and (2) the electrification on the rest
of the surface of the conductor and on any other surfaces
there may be in the electric field. To find the force on

37] GENERAL PRINCIPLES OF ELECTROSTATICS 57

the small area we must find the value of the second part
of the electric intensity, for the electric intensity due to
the electrification on the small area will evidently not
have any tendency to move this area one way or another.

Let R be the total electric intensity along the out
ward drawn normal just outside the surface at P, RI that
part of it due to the electrification on the small area round
P, R2 the part due to the electrification of the rest of the
system. Then R^R^R^

Compare now the electric intensities at two points Q,
S (Fig. 15) close together and near to P, but so placed

Fig. 15.

that Q is just outside and 8 just inside the surface of
which the small area forms a part. Then the part of the
electric intensity at S in the direction of the outward
normal at P, which is due to the electrification on the
conductors other than the small area, will be equal to R2
its value at Q since these points are close together. The
part of the electric intensity due to the small area will
have at 8 the same magnitude as at Q, but will be in the
opposite direction, since Q is on one side of the small area,
while 8 is on the other. Thus the electric intensity at S

58 GENERAL PRINCIPLES OF ELECTROSTATICS [CH. I

due to this area in the direction of the outward drawn
normal will be — R1} that due to the rest of the electri
fication Rz. The total intensity at S will therefore be
— R!+ Rz. But this must be zero, since the intensity
inside a closed equipotential surface enclosing no charge
is zero. Thus R2 = R1} and therefore since

R — RI + R-2)
R, = ^R.

Now the force on the area o> in the direction of the
outward normal is R2o)cr if or is the surface density at P ;
thus if F is the mechanical force per unit area in the
direction of the outward normal

or F=\Ro- ........................ (1).

Since by Coulomb's Law, Art. 22,

R = 4-7TO-,

we have the following expressions for the force per unit
area

'-£ ........................ <*>•

..................... (3).

Since Coulomb's Law requires modification when the
medium surrounding the conductor is not air, the expres
sions (2) and (3) are only true for air: the equation (1) is
always true whatever be the insulator surrounding the
conductor.

When the electric intensity at the surface of a con
ductor exceeds a certain value the air ceases to insulate
and the electrification of the conductor is discharged.
The value of the electric intensity when the electrification

37] GENERAL PRINCIPLES OF ELECTROSTATICS 59

begins to escape from the conductor, depends upon a great
number of circumstances, such as the pressure of the air
and the proximity of other conductors. When the pres
sure of the air is about 760 mm. of mercury and the
temperature about 15° C., the greatest value of R is about
100, unless the conductor is within a fraction of a milli
metre of other conductors ; hence the greatest value of
F in dynes per square centimetre is

104/87T.

The pressure of the atmosphere is about 106 dynes
per square centimetre, hence the greatest tension along
the normal to an electrified surface in air is about
1/800-7T of the atmospheric pressure. That is, a pressure
due to about '3 of a millimetre of mercury would equal
in magnitude the greatest tension on a conductor placed
in air at ordinary pressure.

CHAPTER II

LINES OF FORCE

38. Expression of the properties of the Electric
Field in terms of Faraday Tubes. The results we
have hitherto obtained only depend upon the fact that
two charged bodies are attracted towards or repelled from
each other with a force varying inversely as the square
of the distance between them ; we have made no assump
tion as to how this force is produced, whether, for example,
it is due to the action at a distance of the charged bodies
upon each other or to some action taking place in the
medium between the bodies.

Great advances have been made in our knowledge of
electricity through the introduction by Faraday of the
view that electrical effects are due to the medium between
the charged bodies being in a special state, and do not
arise from any action at a distance exerted by one charged
body on another.

We shall now proceed to consider Faraday's method
of regarding the electric field — a method which enables
us to form a vivid mental picture of the processes going
on in such a field, and to connect together with great ease
many of the most important theorems in Electrostatics.

CH. ii. 38]

LINES OF FORCE

We have seen in Art. 15 that a line of force is a curve
such that its tangent at any point is in the direction of
the electric intensity at that point. As these lines of
force are fundamental in the method employed in this
and subsequent chapters for considering the properties
of the electric field, we give below some carefully drawn
diagrams of the lines of force in some typical cases.

Figure 16 represents the lines of force due to two
equal and opposite charges. In this case all the lines of
force start from the positive charge and end on the

Fig. 16.

negative. Fig. 17 represents the lines of force due to
two equal positive charges; in this case the lines of
force do not pass between the charged bodies, but lines
start from each of the bodies and travel off to an infinite
distance.

LINES OF FORCE

[CH. II

Figure 18 represents the lines of force due to a
positive charge equal to 4 at A, and a negative charge

Fig. 17.

Fig. 18.

38]

LINES OF FORCE

equal to — 1 at B. In this case all the lines of force
which fall on B start from A, but since the charge at
A is numerically greater than that at B, lines of force
will start from A which do not fall on B but travel off
to an infinite distance.

The lines of force which pass between A and B are
separated from those which proceed from A and go off
to an infinite distance by the line of force which passes
through C, the point of equilibrium, where

Fig. 19.

Figure 19 represents the lines of force due to a
charge 1 at A and 4 at B.

Figure 20 represents the lines of force due to a
charged conductor formed by two spheres intersecting at
right angles. The electric intensity vanishes along the
intersection of the spheres.

LINES OF FORCE

[CH. II

Figure 21 represents the lines of force between two
finite parallel places ; between the plates but away from

Fig. 20.

the edges of the plates the lines of force are straight
lines at right angles to the planes, but nearer the

Fig. 21.

39] LINES OF FORCE 65

edges of the plates they curve out ; some lines also pass
from the back of one plate to the back of the other.

39. Tubes of Force. If we take any small closed
curve in the electric field and draw the lines of force,
which pass through each point of the curve, these lines
will form a tubular surface which is called a tube of
force. These tubes possess the property that the electric
intensities at any two points on a tube are inversely
proportional to the cross sections of the tube, made by
planes cutting the tube at right angles at these points,
provided that the cross sections are so small that the
electric intensity may be regarded as constant over each
section. For let Fig. 22 represent a closed surface formed

Fig. 22.

by the tube and its normal sections. Let w^ be the area of
the cross section of the tube at P, &>2 its cross section at Q;
R!, R2 the electric intensities at P and Q respectively.
Now consider the total normal electric induction over
the surface. The only parts of the surface which con
tribute anything to this are the flat ends, as the sides
of the tube are by hypothesis parallel to the electric
intensity, so that this has no normal component over the
T. E. 5

66 LINES OF FORCE [CH. II

sides. Thus the total normal induction over the closed
surface PQ is equal to

the minus sign being given to the second term because, as
drawn in the figure, the electric intensity at P is in the
direction of the inward-drawn normal. Now, by Gauss's
theorem, the total normal electric induction over any
closed surface is equal to 4?r times the charge inside
the surface; hence if the surface does not include any

charge, we have

R2o).2 — Rlw1 = 0,

or the electric intensity at P is to that at Q inversely
as the cross section of the tube of force at P is to
that at Q.

The tubes of force will start from positive electrifica
tion and go on until they end on a negative electrified
body. If the points P and Q are on the surfaces of
positively and negatively electrified conductors, then if
crp is the surface density at P, O-Q that at Q,

thus the equation

is equivalent to

Now crpWi is the charge enclosed by the tube where
it leaves the positively electrified conductor, and - <7Q&>2
the charge enclosed by the tube where it arrives at the
negatively electrified conductor, hence we see that the
positive charge at the beginning of the tube is equal in
magnitude to the negative charge at the end. We may
draw these tubes so that they each enclose one unit of
electricity at their origin, each of these tubes will

40] LINES OF FORCE 67

therefore include unit negative charge at its end. Such
tubes are sometimes called unit tubes of force, we
shall for brevity, call them Faraday tubes. Each unit of
positive charge will be the origin, each unit of negative
charge the end of a Faraday tube. The total charge on
a conductor will be the excess of the number of tubes
which leave the conductor over the number which arrive
at the conductor.

Since the Faraday tubes run in the direction of the
electric intensity in air, they begin at places of high and
travel to places of low potential. No Faraday tube can
have its ends at the same potential, that is no Faraday
tube can pass from one surface to another if the two
surfaces are at the same potential.

40. The electric intensity at any point in the field
is proportional to the number of Faraday tubes which
pass through unit area of a plane drawn at right angles
to the direction of the electric intensity at the point
or, what is the same thing, through unit area of the
equipotential surface passing through the point.

For let A be a small area drawn at right angles to the
electric intensity, and let the tubes which pass through this
area be prolonged until they arrive at the positively elec
trified surface from which they start ; let B be the portion
of this surface over which these tubes are spread, R the
electric intensity at any point on B, co' the area at B.
Let F be the electric intensity, and co the area enclosed
by the tubes, at A. Then applying Gauss's theorem
(Art. 10) to the tubular surface formed by the prolonga
tions backwards of the tubes through A, we get

Fco - Rw = 0.

5—2

68 LINES OF FORCE [CH. II

But as a is the surface density of the electrification
at B, we have, by Coulomb's law (Art. 22), when the
medium surrounding B is air,

E = 47TO-,

and hence Fco = 47r<7o/.

But since era/ is the charge of electricity on B, it is
equal to N the number of Faraday tubes which start
from B, and which pass through A, hence

Fa = 4,7rN,
or if a) is unity F=4nrN.

Thus the electric intensity at any point in air is 4?r
times the number of Faraday tubes passing through unit
area of a plane drawn through the point at right angles
to the electric intensity.

41. The properties of the Faraday tubes enable us
to prove with ease many important theorems relating to
the electric field.

Thus, for example, we see that on the conductor
at the highest potential in the field the electrification
must be entirely positive. Any negative electrification
would imply that Faraday tubes arrived at the conductor ;
these tubes must however arrive at a place which is at
a lower potential than the place from which they start.
Thus, if the potential of the conductor we are considering
is the highest in the field it is impossible for a Faraday
tube to arrive at it, for this would imply that there was
some other conductor at a still higher potential from
which the tube could start.

Similar reasoning shews that the electrification on
the conductor or conductors at the lowest potential in the
field must be entirely negative.

42] LINES OF FORCE 69

When one conductor has a positive charge while all the
other conductors are connected to earth, we see from the
last result that the charges on the uninsulated conductors
must be all negative, and since the potentials of these
conductors are all equal and the same as that of the earth,
no Faraday tubes can pass from one of these conductors
to another, or from one of these to the earth. Hence
all the tubes which fall on these conductors must have
started from the conductor at highest potential. Thus
the sum of the number of tubes which fall on the un
insulated conductors cannot exceed the number which
leave the positively charged conductor, that is, the sum
of the negative charges induced on the conductors con
nected to earth cannot exceed the positive charge on
the insulated conductor.

42. These results give us important information as
to the coefficients of capacity and of induction defined
in Art. 26.

For let us take the first conductor as the insulated
one with the positive charge; then since F2, F3... are all
zero we have, using the notation of that Article,

Since El and Vl are positive, while E2) Es, &c. are all
negative, we see that qn is positive, while g12, q^, &c. are
all negative. Again, since the positive charge on the first
conductor is numerically not less than the sum of the
negative charges on the other conductors,

El is numerically not less than E2 + E3 + . . .,
i.e. qn is numerically not less than g12 + q™ + ^H + • • • •

70 LINES OF FORCE [CH. II

If one of the conductors, say the second, completely
surrounds the first, and if there is no conductor other
than the first inside the second, and if all the conductors
except the first are at zero potential, then all the tubes
which start from the first must fall on the second. Thus
the negative charge on the second must be numerically
equal to the positive charge on the first (see Art. 30).
There can be no charges on any of the other conductors,
for all the tubes which might fall on these conductors must
come from the first conductor, and all the tubes from this
conductor are completely intercepted by the second surface.
Thus if the second conductor encloses the first conductor,
and if there are no other conductors between the first
and the second, then qu = -qi-2> and qw, qu, ql5... are
all zero.

43. Expression for the Energy in the Field.

When we regard the Faraday tubes as the agents by which
the phenomena in the electric field are produced we are
naturally led to suppose that the energy in the electric
field is in that part of the field through which the tubes
pass, i.e. in the dielectric between the conductors. We
shall now proceed to find how much energy there must
be in each unit of volume if we regard the energy as
distributed throughout the electric field. We have seen in
Art. 23, that the electric energy is one half the sum of the
products got by multiplying the charge on each conductor
by the potential of that conductor. We may regard each
unit charge as having associated with it a Faraday tube,
which commences at the charge if that is positive and
ends there if the charge is negative. Let us now see how
the energy in the field can be expressed in terms of these

43] LINES OF FORCE 71

tubes. Each tube will contribute twice to the expression
for the electric energy ^EV, the first time correspond
ing to the positive charge at its origin, the second time
corresponding to the negative charge at its end. Thus,
since there is unit charge at each end of the tube, the
contribution of each tube to the expression for the
energy will be J- (the difference of potential between its
beginning and end). The difference of potential between
the beginning and end of the tube is equal to %R . PQ,
where PQ is a small portion of the length of the tube
so small that along it R, the electric intensity, may be
regarded as constant : the sign 2 denotes that the tube
between A and B, A being a unit of positive and B a unit
of negative charge, is to be divided up into small pieces
similar to PQ, and that the sum of the products of the
length of each piece into the electric intensity along it
is to be taken. Thus the whole tube AB contributes
^R . PQ to the electric energy, so that we may suppose
that each unit length of the tube contributes an amount
of energy equal to one half the electric intensity. Any
finite portion CD of the tube will therefore contribute
an amount of energy numerically equal to one half the
difference of potential between C and D. We may there
fore regard the electrical energy as distributed throughout
the field and that each of the Faraday tubes has associated
with it an amount of energy per unit length numerically
equal to one half the electric intensity.

Let us now consider the amount of energy per unit
volume. Take a small cylinder surrounding any point P
in the field with its axis parallel to the electric intensity
at P, its ends being at right angles to the axis. Then
if R is the electric intensity at P and I the length of the

72 LINES OF FORCE [CH. II

cylinder, the amount of energy due to each tube passing
through the cylinder is ^Rl. If w is the area of the
cross section of the cylinder, N the number of tubes passing
through unit area, the number of tubes passing through
the cylinder is Nw. Thus the energy in the cylinder is

but in air, by Art. 40,

so that the energy in the cylinder is

^-RHoy.

STT

But Ico is the volume of the cylinder, hence the energy
per unit volume is equal to

Thus we may regard the energy as distributed through
out the field in such a way that the energy per unit of
volume is equal to R^/STT.

44. If we divide the field up by a series of equi-

Fig. 23.

potential surfaces, the potentials of successive surfaces
decreasing in arithmetical progression, and if we then

45] LINES OF FORCE 73

draw a series of tubular surfaces cutting these equi-
potential surfaces at right angles, such that the number
of Faraday tubes passing through the cross section of
each of the tubular surfaces is the same for all the
tubes, the electric field will be divided up into a
number of cells which will all contain the same amount
of energy. For the potential difference between the
places where a Faraday tube enters and leaves a cell is
the same for all the cells, and thus the energy of the
portion of each Faraday tube passing through a cell will
be constant for all the cells, and since the same number
of Faraday tubes pass through each cell, the energy in
each cell will be constant.

45. Force on a conductor regarded as arising
from the Faraday Tubes being in a state of ten
sion. We have seen, Art. 37, that on each unit of area of
a charged conductor there is a pull equal to ^Rcr, where
a is the surface density of the electricity, and R the electric
intensity. Now a is equal to the number of Faraday
tubes which fall on unit area of the surface, and hence
the force on the surface is the same as if each of the
tubes exerted a pull equal to ^R. Thus the mechanical
forces on the conductors in the electric field are the same
as they would be if the Faraday tubes were in a state
of tension, the tension at any point being equal to one
half the electric intensity at that point. Thus the tension
at any point of a Faraday tube is numerically equal to
the energy per unit length of the tube at that point.

If we have a small area to, at right angles to the
electric intensity, the tension over this area is equal to

74 LINES OF FORCE [CH. II

where N is the number of Faraday tubes passing through
unit area, and R is the electric intensity. By Art 40

Hence the tension parallel to the electric intensity is

The tension across unit area is therefore equal to

46. This state of tension will not however leave the
dielectric in equilibrium unless the electric field is uni
form, that is unless the tubes are straight and parallel to
each other. If however there is in addition to this tension
along the lines of force a pressure acting at right angles to
them and equal to E^/STT per unit area the dielectric will
be in equilibrium, and since this pressure is at right angles
to the electric intensity it will not affect the normal force
acting on a conductor. To show that this pressure is in
equilibrium with the tensions along the Faraday tubes,
consider a small volume whose ends are portions of equi-
potential surfaces and whose sides are lines of force.

Fig. 24.

Let us now consider the forces acting on this small
volume parallel to the electric intensity at A. The forces
are the tensions in the Faraday tubes and the pressures at

46] LINES OF FORCE 75

right angles to the sides. Resolve these parallel to the
outward-drawn normal at A. The number ri of Faraday
tubes which pass through A is the same as the number
which pass through B. If R, R' are the electric intensities
at A and B respectively, then the force exerted on the
volume in the direction of the outward-drawn normal at
A by the Faraday tubes at A will be n'R/2, while the
force in the opposite direction exerted by the Faraday
tubes at B is n'R cose/2, where e is the small angle
between the directions of the Faraday tubes at A and B.
Since e is a very small angle we may replace cos e by
unity; thus the resultant force on the volume in the
direction of the outward-drawn normal at A due to the
tension in the Faraday tubes is

nf(R-K)/2.

Let N be the number of tubes passing through unit
area, &>, «' the areas of the ends A and B respectively ;
then, Art. 40,

/T\T R R f

n =iVft) = -7— co = - — co ,

47T 47T

so that the resultant in the direction of the outward-
drawn normal at A is

~<o(R

since R'a>' — Ru>,

we may write this as

RR' f i \

•g-V-o.),

or approximately, since R' is very nearly equal to R

76 LINES OF FORCE [CH. II

Let us now consider the effect of the pressure p
at right angles to the lines of force ; this has a com
ponent in the direction of the outward-drawn normal
at A as in consequence of the curvature of the tube
the normals to its surface are not everywhere at right
angles to the outward-drawn normal at A ; the angle
between the pressure and the normal at A will always

however be nearly a right angle. If this angle is ~ — 0

at a point where the pressure is p', the component of the
pressure along the normal at A will be proportional to
pf sin 9. But since p' only differs from p, the value of the
pressure at A, by a small quantity, and 6 is small, the
component of the pressure will be equal to p sin 0, if we
neglect the squares of small quantities ; that is, the effect
along the normal at A of the pressure over the surface
will be approximately the same as if that pressure were
uniform. To find the effect of the pressure over the sides
we remember that a uniform hydrostatic pressure over any
closed surface is in equilibrium; hence the force due to the
pressures over the sides C, D will be equal and opposite to
the force due to the pressures over the ends A and B. But
the force due to the pressure over these ends is pco' — pw ;
hence the resultant effect in the direction of the outward-
drawn normal at A of the pressure over the sides is
p(o) — ft>'). Combining this with the effect due to the
tension in the tubes we see that the total force on the
element parallel to the outward-drawn normal at A is

— (o)' - (*) +p (w - ft)') ;

E2 NR

this vanishes if p = ^— = -^- .

O7T ^

46] LINES OF FORCE 77

Thus the introduction of this pressure will maintain equi
librium as far as the forces parallel to the electric intensity
are concerned.

Now consider the force at right angles to the electric
intensity. Let PQRS, Fig. 25, be the section of the surface
in Fig. 24 by the plane of the paper, PS, QR being sections

Fig. 25.

of equipotential surfaces, and PQ, SR lines of force. Let
t be the depth of the volume at right angles to the plane of
the paper. We shall assume that the section of the figure
by the plane through PQ at right angles to the plane of
the paper is a rectangle. Let R be the electric intensity
along PQ, R' that along SR, s the length PQ, s' that of
SR. Since the difference of potential between P and Q
is the same as that between S and R,
Rs = R's'.

Consider the forces parallel to PS. First take the
tensions along the Faraday tubes ; the force due to those
at PS will have no component along PS : in each tube at
Q there is a tension R/'2, the component of which along
PS is (RsmO)l2, where 6 is the angle between PS and
QR. Since 9 is very small this component is equal to R0/2.
Let PS and QR meet in 0,

RS PQ PQ-SR s-s'
°~OR~OQ OQ-OR RQ'

78 LINES OF FORCE [CH. II

Thus the component along PS due to the tension at Q is

R 8-8'

2' RQ'

The number of tubes which pass through the end of
the figure through RQ at right angles to the plane of
the paper is N . QR . t, where N is the number of tubes
which pass through unit area.

The total component along PS due to the tensions
in these tubes is thus

R2 R'2

-•- 5*'

Now the component along PS due to the pressures at
right angles to the electric intensity is equal to

psi — p's't,
where p and p' are the pressures over PQ, RS respectively.

T, R2 , R2

P = 8^> P-tor'-

)

-t)t, (since Rs = R's'),

07T

or approximately, since R' is very nearly equal to R,

-£«-<><

Thus the component in the direction of PS due to the
tensions is equal and opposite to the component due to
the pressures ; thus the two are in equilibrium as far as
the component in the plane of the diagram at right angles

47] LINES OF FORCE 79

to the electric intensity is concerned ; we easily see that
the same is true for the component at right angles to the
plane of the paper. We have already proved that the
tensions and pressures balance as far as the component
along the direction of the electric intensity is concerned ;
thus the system of pressures and tensions constitutes
a system in equilibrium.

47. This system of tensions along the tubes of force
and pressures at right angles to them is thus in equilibrium
at any part of the dielectric where there is no charge, and
gives rise to the forces which act on electrified bodies
when placed in the electric field. Faraday introduced this
method of regarding the forces in the electric field; he
expressed the system of tensions and pressures which we
have just found, by saying that the tubes tended to con
tract and that they repelled each other. This conception
enabled him to follow the processes of the electric field
without the aid of mathematical analysis.

Since Rs = It's,

s OQ RQ

= =

R-K R

we have

Now OR is the radius of curvature of the line of force ;

denoting this by p we have

I__dv_
p~ ~ R

where dv is an element of length at right angles to the
electric force ; we see from this equation that the lines of
force are concave to the stronger parts of the field.

80 LINES OF FORCE [CH. II

The lines of force arrange themselves as a system of
elastic strings would do if acted on by forces whose
potential for unit length of string was jR/2.

48. The student will find much light thrown on the
effects produced in the electric field by the careful study
from this point of view of the diagrams of the lines of
force given in Art. 38. Thus, take as an example the
diagram given in Fig. 18, which represents the lines of
force due to two charges A and B of opposite signs, the
ratio of the charges being 4:1. We see from the diagram
that though more tubes of force start from the larger
charge A , and the tension in each of these is greater than
in a tube near the smaller charge 5, the tubes are much
more symmetrically distributed round A than round B.
The approximately symmetrical distribution of the tubes
round A makes the pulls exerted on A by the taut Faraday
tubes so nearly counterbalance each other that the resultant
pull of these tubes on A is only the same as that exerted
on B by the tubes starting from it; since these, though
few in number, are less symmetrically distributed, and
so do not tend to counterbalance each other to nearly
the same extent. The tubes of force in the neighbour
hood of the point of equilibrium are especially interesting.
Since the charge on A is four times that on B, only J of
the tubes which start from A can end on B, the remaining
| must go off to other bodies, which in the case given in
the diagram are supposed to be at an infinite distance.
The point of equilibrium corresponds as it were to the
'parting of the ways' between the tubes of force which
go from A to B and those which go off from A to an
infinite distance.

49] LINES OF FORCE 81

When the charges A and B are of the same sign, as
in Fig. 19, we see how the repulsion between similar tubes
causes the tubes to congregate on the side of A remote
from B, and on the side of B remote from A.

We see again how much more symmetrically the
tubes are distributed round A than round B] this more
symmetrical distribution of the tubes round A makes
the total pull on A the same as that on B.

We see too from this example that the repulsion
between the charges of the same sign and the attraction
between charges of opposite signs are both produced by
the same mechanism, i.e. a system of pulls ; the difference
between the cases being that the pulls are so distributed
that when the charges are of the same sign the pulls tend
to pull the bodies apart, while when the charges are of
opposite signs the pulls tend to pull the bodies together.

The diagram of the lines of force for the two finite
plates (Fig. 21) shows how the Faraday tubes near the
edges of the plates get pushed out from the strong parts of
the field and are bent in consequence of the repulsion
exerted on each other by the Faraday tubes.

49. As an additional example of the interpretation of
the processes in the electric field in terms of the Faraday
tubes, let us consider the effect of introducing an insulated
conductor into an electric field.

Let us take the field due to a single positively charged
body at A ; before the introduction of the conductor the
Faraday tubes were radial, but when the conductor is
introduced the tubes, which previously existed in the
region occupied by the conductor, are annulled ; thus the
repulsion previously exerted by these tubes on the sur-

T. E. 6

LINES OF FORCE

[CH. II

rounding ones ceases, and a tube such as AB, which was
previously straight, is now, since the pressure below it is
diminished, bent down towards the conductor ; the tubes
near the conductor are bent down so much that they strike
against it, they then divide and form two tubes, with
negative electrification at the end C, positive at the end D.

Fig. 26.

50. Force on an uncharged conductor placed in
an electric field. If a small conductor is placed in the
field at P, the Faraday tubes inside the conductor dis
appear, and, if the introduction of the conductor did not
alter the tubes outside it, the diminution of energy due
to the annihilation of the tubes in the conductor would
be proportional to R^/Str per unit volume, where R is the
electric intensity in the field at P before the conductor
was introduced. If the conductor is moved to a place
where the electric intensity is R', the diminution in the
electric energy in the field is R^j^ir per unit volume. Now
it is a general principle in mechanics that a system always

50] LINES OF FORCE 83

tends to move from rest in such a way as to diminish the
potential energy as much as possible, and the force tending
to assist a displacement in any direction is equal to the
rate of diminution of the potential energy in that direction.
The conductor will thus tend to move so as to produce the
greatest possible diminution in the electric energy, that is,
it will tend to get into the parts of the field where the
electric intensity is as large as possible ; it will thus move
from the weak to the strong parts of the field.

The presence of the conductor will however disturb the
electric field in its neighbourhood; thus R, the actual
electric intensity, will differ from R, the electric intensity
at the same point before the conductor was introduced.
By differentiating It2 /Sir we shall get an inferior limit to
the force acting on the conductor per unit volume. For
suppose we introduce a conductor into the electric field,
then R'^/STT would be the diminution in electric energy
per unit volume due to the disappearance of the Faraday
tubes from the inside of the conductor, the tubes outside
being supposed to retain their original position. In reality
however the tubes outside will have to adjust themselves
so as to be normal to the conductor, and this adjustment
will involve a further diminution in the energy, thus the
actual change in the energy is greater than that in R2/87r
and the force acting per unit volume will therefore be
greater than the rate of diminution of this quantity. If
we take the case when the force is due to a charge e at a
point, the rate of diminution of R2'/87r is e2/2-7rr5, and thus
the force on a small conducting sphere of radius a will be
greater than (47ra3/3) 02/27rr5), that is greater than 2e2a3/3r5.
The actual value (see Art. 87) is 2e2a*/r5.

6—2

CHAPTER III

CAPACITY OF CONDUCTORS. CONDENSERS

51. The capacity of a conductor is defined to be the
numerical value of the charge on the conductor when its
potential is unity, all the other conductors in the field
being at zero potential.

Two conductors insulated from each other and placed
near together form what is called a condenser; in this
case the charge on either conductor may be large, though
the difference between their potentials is small.

In many instruments the two conductors are so
arranged that their charges are equal in magnitude and
opposite in sign; in such cases the magnitude of the
charge on either conductor when the potential difference
between the conductors is unity is called the capacity of
the condenser.

If the difference of potential between two conductors,
produced by giving a charge + q to one conductor and - q
to the other, is V, then q/ V is defined to be the capacity
between the conductors.

52. Capacity of a Sphere placed at an infinite
distance from other conductors. Let a be the radius
of the sphere, V its potential, e its charge, the corre
sponding charge of opposite sign being at an infinite
distance. Then (Art. 17), the potential due to the charge
on the sphere at a distance r from the centre is e/r;
therefore the potential at the surface of the sphere is e/a.

CH. III. 53] CAPACITY OF CONDUCTORS. CONDENSERS 85

Hence we have

V=-

When V is unity, e is numerically equal to a : hence,
Art. 51, the capacity of the sphere is numerically equal to
its radius.

53. Capacity of two concentric spheres. Let

us first take the case when the outer sphere and any con
ductors which may be outside it are connected to earth,
while the inner sphere is maintained at potential V.
Then, since the outer sphere and all the conductors out
side are connected to earth, no Faraday tubes can start
from or arrive at the outer surface of the outer sphere,
for Faraday tabes only pass between places at different
potentials, and the potentials of all places outside the
sphere are the same, being all zero. Again, all tubes which
start from the inner sphere will arrive at the internal
surface of the outer shell, so that the charge on the inner
surface of this shell will be equal and opposite to the charge
on the inner sphere. Let a be the radius of the inner
sphere, b the radius of the internal surface of the outer
sphere, e the charge on the inner sphere, then — e will be
the charge on the interior of the outer sphere.

Consider the work done in moving a unit of electricity
from the surface 'of the inner sphere to the inner surface
of the outer sphere ; the charge on the outer sphere pro
duces no electric intensity at a point inside, so that the
electric intensity, which produces the work done on the
unit of electricity, arises entirely from the charge on the
inner sphere. The electric intensity due to the charge on
this sphere is, by Art. 11, the same as that which would be

86 CAPACITY OF CONDUCTORS [CH. Ill

due to the charge e collected at the centre 0. The work
done on unit of electricity when it moves from the inner
sphere to the outer one is thus the same as the work done
on a unit charge when it moves from a distance a to a
distance b from a small charged body placed at the centre
of the spheres; this, by Art. 17, is equal to

e _e

a b'

and is by definition equal to V, the potential difference
between the two spheres ; hence we have

V-6--6
~a b'

ab
or e =

T— - .
b — a

Thus, when b — a is very small, that is, when the radii of
the two spheres are very nearly equal, the charge is very
large. When 7=1, the charge is

6-a'

so that this is, by Art. 51, the capacity of the two spheres.
The value of this quantity when the radii of the two spheres
are very nearly equal is worthy of notice. In this case,
writing t for b — a, the distance between the spheres, the

capacity is equal to

ab a(a + t)

~i ~~ t

this, since t is very small compared with a, is approxi
mately

a2 4-Tra2

T =

surface of the sphere

53] CONDENSERS 87

Thus the capacity in this case is equal per unit area of
surface to l/4rr times the distance between the con
ductors. The case of two spheres whose distance apart is
very small compared with their radii is however approxi
mately the case of two parallel planes ; hence the capacity
of such planes per unit area of surface is equal to 1/4-7T
times the distance between the planes. This is proved
directly in Art. 56.

If, after the spheres are charged, the inner one is insu
lated, and the outer one removed to an infinite distance (to
enable this to be done we may suppose that the outer sphere
consists of two hemispheres fitted together, and that these
are separated and removed), the charge on the sphere will

remain equal to e, i.e. j— - V, but the potential of the
o — a

sphere will rise ; when it is alone in the field the potential
will be e/a, i.e.

Thus by removing the outer sphere the potential
difference between the sphere and the earth has been
increased in the proportion of b to b — a. By making 6 - a
very small compared with 6, we can in this way increase
the potential difference enormously and make it capable
of detection by means which would not have been suffi
ciently sensitive before the increase in the potential took
place.

It was by the use of this principle that Volta suc
ceeded in demonstrating by means of the gold-leaf electro
scope and two metal plates, the difference of potential
between the terminals of a galvanic cell ; this difference is

88 CAPACITY OF CONDUCTORS [CH. Ill

so small that the electroscope is not deflected when the
cell is directly connected to it; by connecting the ter
minals of the cell to two plates placed very close together,
and then removing one of the plates after severing the
connections between the plates and the cells, Volta was
able to increase the potential of the other plate to such
an extent that it produced an appreciable deflection of an
electroscope with which it was connected.

Work has to be done in separating the two con
ductors; this work appears as increased electric energy.
Thus, to take the case of the two spheres, when both
spheres were in position the electric energy, which, by
Art. 23 is equal to ^EV, is

1 ^L 72
26-a

When the outer sphere which is at zero potential is
removed the potential of the sphere is e/a, so that the
electric energy is

1 e2

2 a'

and has thus been increased in the proportion of b to
b — a.

54. Let us now take the case when the inner sphere
is connected to earth while the outer sphere is at the
potential V. In this case we can prove exactly as before
that the charge on the inner sphere is equal and opposite
to the charge on the internal surface of the outer sphere,
and that, if e is the charge on the inner sphere,

e—j^-V.

b-a

55] CONDENSERS 89

In this case, in addition to the positive charge on the
internal surface of the outer sphere, there will be a positive
charge on the external surface, since this surface is at a
higher potential than the surrounding conductors. If c
is the radius of the external surface of the outer sphere,
the sum of the charges on the two spheres must be Vc.
Since the charge on the inner surface of the outer sphere
is equal and opposite to the charge on the inner sphere,
the charge on the external surface of the outer sphere
must be equal to Vc. Thus the total charge on the outer
sphere is equal to

— a

55. The charge on the outside of the outer sphere
will be affected by the presence of other conductors. Let
us suppose that outside the external sphere there is a
small sphere connected to earth; let r be the radius of
this sphere, R the distance of its centre from 0 the centre
of the concentric spheres. Let e be the total charge on
the two concentric spheres, e" the charge on the small
sphere. The potential due to e at a great distance R
from 0 is e'/R, similarly the potential due to e" is at a
distance R equal to e"jR.

Since the surface of the outer sphere is at the po
tential F, we have

F-^'5,

c R

and, since the potential of the small sphere is zero, we

have

e' e"

90 CAPACITY OF CONDUCTORS [CH. Ill

hence

that is, the presence of the small sphere increases the
charge on the outer sphere in the proportion of

1 to l-rc/R2.

It is only the charge on the external surface of the
outer sphere which is affected. The charges on the inner
sphere and on the internal surface of the outer sphere are
not altered by the presence of conductors outside the
latter sphere.

56. Parallel Plate Condensers. Condensers are
frequently constructed of two parallel metallic plates;
the theory of the case, when the plates are so large in
comparison with their distance apart that they may be
regarded as infinite in area, is very simple.

In this case the Faraday tubes passing between the
plates will be straight and at right angles to the plates,
and the electric intensity between the plates is constant
since in passing from one plate to the other each Faraday
tube has a constant cross section ; let R be its value, then
if d is the distance between the plates, the work done
on unit charge of electricity as it passes from the plate
where the potential is high to the one where the potential
is low is ltd, and this by definition is equal to V, the
difference of potential between the plates. Hence

V=Rd.

57] CONDENSERS 91

If a is the surface-density of the charge on the plate
at high potential, that on the plate of low potential will
be — cr, and by Coulomb's law, Art. 22,

R = 47T<7.

Hence V = 4>Trcrd,

and if V is equal to unity, a is equal to

The charge on an area A of one of the plates when
the potential difference is unity is thus A/4<7rd, this by
definition is the capacity of the area A. We arrived at
the same result in Art. 53 from the consideration of
two concentric spheres. The electrical energy of the
condenser is, by Art. 23, equal to

which in this case is equal to

Swd'

or, if E is the charge on one of the plates, to

57. Guard Ring. In practice it is of course im
possible to have infinite plates, and when the plates are
finite, then, as the diagram, Fig. 21, Art. 38, shows, the
Faraday tubes near the edges of the plates are no longer
straight, and the electrification ceases to be uniform, and
is no longer given by the expression (1), Art. 56. Thus to
express the quantity of electricity on the finite plate, we

92 CAPACITY OF CONDUCTORS [CH. Ill

should have to add to the expression a correction for the
inequality of the distribution over the ends of the plates.
This correction can be calculated, but the necessity for it
may be avoided in practice by making use of a device due
to Lord Kelvin, and called a guard ring.

Fig. 27.

Suppose one of the plates, say the upper one, is divided
into three portions flush with each other and separated
by the narrow gaps E, F. Then if, in charging the
condenser the portions A, B, C are connected metallically
with each other, the places where the electrification
is not uniform will be on A and C, so that apart from the
effects of the narrow gaps E, F, the electrification on B
will, if we neglect the effect of the gaps, be uniform and
the total charge on B will be equal to SV/4>7rd, where S is
the area of the plate B. The capacity of B is thus equal
to Sj^ird.

If, as ought to be the case, the widths of the gaps
at E and F are very small compared with the distance
between the plates, we can easily calculate the effect
of the gaps. For if the gaps are very narrow the
electrification of the lower plate will be approximately
uniform. The Faraday tubes in the neighbourhood of
the gaps will be distributed as in Fig. 28. We see

58] CONDENSERS 93

from this, if we consider the gap E, that all the Faraday
tubes which would have fallen on a plate whose breadth

Fig. 28.

was E, if there had been no gap, will fall on one
or other of the plates A and B, Fig. 28, and from the
symmetry of the arrangement half of these tubes will
fall on B, the other half on A ; thus the actual amount
of electricity on B will be the same as if we supposed B
to extend halfway across the gap, and to be uniformly
charged with electricity whose surface density is V/4nrd.
We see then that, allowing for the effects of the gaps,
the capacity of B will be equal to Sf/4>7rd, where
S' = area of plate B

-f \ (the sum of the areas of the gaps E and F).
If the plate B is not at zero potential, there will be
some electrification on the back of the plate arising from
Faraday tubes which go from the back of B to other
conductors in its neighbourhood and to earth. The elec
trification of the back of B may be obviated by covering
this side of A, B, C with a metal cover connected with
A and G. It can also be obviated by making B the low
potential plate (i.e. the one connected to earth), care being
taken that the other conductors in the neighbourhood are
also connected to earth.

58. Capacity of two coaxial cylinders. Let us

take the case of two coaxial cylinders, the inner one being

94 CAPACITY OF CONDUCTORS [CH. Ill

at potential V, the outer one being at potential zero.
Then if E is the charge per unit length on the inner
cylinder, —E will be the charge per unit length on the
inner surface of the outer one, since all the Faraday tubes
which start from the inner cylinder end on the outer
one.

The electric intensity at a distance r from the axis of
the cylinders is, by Art. 13, equal to

Thus the work done on unit charge, when it goes from
the outer surface of the inner cylinder to the inner surface
of the outer cylinder, is equal to

2E ,
-dr,

where a is the radius of the inner cylinder, b the radius
of the inner surface of the outer cylinder.

This work is, however, by definition equal to V, the
difference of potential between the cylinders, and hence

= 1 —dr

a '

When V is unity, E, the charge per unit length, is
equal to

and this, by definition, is the capacity of the condenser
per unit length.

58] CONDENSERS 95

If the radii of the cylinders are nearly equal, and if
b — a = t,t will be small compared with a ; in this case the
capacity per unit length

a + t
2 log—

= - approximately

2-
a

la
2t

~ 4nrt'

Since 2?ra is the area of unit length of the inner
cylinder, the capacity per unit area is l/4<7r£; we might have
deduced this result from the case of two parallel planes.

When the two cylinders are coaxial, there is no
force tending to move the inner cylinder; thus since
the system is in equilibrium, the potential energy, if the
charges are given, must be either a maximum or a mini
mum. The equilibrium is, however, evidently unstable,
for, if the inner cylinder is displaced, the force due to
the electric field tends to make the cylinders come into
contact with each other and thus increase the displace
ment. Since the equilibrium is unstable the potential
energy is a maximum when the cylinders are coaxial.
The potential energy, however, is, by Art. 23, equal to

96 CAPACITY OF CONDUCTORS [CH. Ill

where C is the capacity of the condenser. Thus if the
potential energy is a maximum the capacity must be a
minimum. Thus any displacement of the inner cylinder
will produce an increase in the capacity, but since the
capacity is a minimum when the cylinders are coaxial,
the increase in the capacity will be proportional to square
and higher powers of the distance between the axes of the
cylinders.

59. Condensers whose capacities can be varied.

For some experimental purposes it is convenient to use a
condenser whose capacity can be altered continuously, and
in such a way that the alteration in the capacity can be
easily measured. For this purpose a condenser made of
two parallel plates, one of which is fixed, while the other
can be moved by means of a screw, through known dis
tances, always remaining parallel to the fixed plate, is
useful. In this case the capacity is inversely proportional
to the distance between the plates, provided that this
distance is never greater than a small fraction of the
radius of the plates.

Another arrangement which has been used for this
purpose is shown in Fig. 29. It consists of three

-D

B C
Fig. 29.

coaxial cylinders, two of which, AB, CD, are. of the same
radius and are insulated from each other, while the third,
EF, is of smaller radius and can slide parallel to its axis.
The cylinder EF is connected metallically with CD, so

60] CONDENSERS 97

that these two are always at the same potential, and the
cylinder AB is at a different potential, then when the
cylinder EF is moved about so as to expose different
amounts of surface to AB the capacity of the condenser
formed by AB and EF will alter, and the increase in the
capacity will be proportional to the increase in the area of
the surface of EF brought within AB.

60. Electrometers.

Consider the case of two parallel conducting plates;
let V be the potential difference between the plates, d
their distance apart. The force on a conductor per unit
area is, by Art. 37, equal to J Ra, where R is the electric
intensity at the conductor and a- the surface density ; but

V 1

R = -j , while <r = — - R by Coulomb's law ; we see there

of 4)7T

fore that the attraction of one plate on the other is per

1 F2

unit area equal to — -^ . Hence the force on an area A

of one of the plates is equal to

A F2

Thus, if we measure the mechanical force between the
plates, we can deduce the value of F, the potential differ
ence between them. This is the principle of Lord Kelvin's
attracted disc electrometer. This instrument measures
the force necessary to keep a moveable disc surrounded
by a fixed guard ring in a definite position; when this
force is known the value of the potential difference is
given by the expression (1).

Quadrant Electrometer. The effect measured by
the instrument just described varies as the square of the

T. E.

98 CAPACITY OF CONDUCTORS [CH. Ill

potential difference; thus when the potential difference
is diminished the attraction between the plates diminishes
with great rapidity. For this reason the instrument is
not suited for the measurement of very small potential
differences. To measure these another electrometer, also
due to Lord Kelvin, called the quadrant electrometer, is
frequently employed.

This instrument is represented in Fig. 30 : it consists
of a cage, made by the four quadrants A, B, C, D; each
quadrant is supported by an insulating stem, while the
opposite quadrants A and C are connected by a metal wire,
as are also B and D ; thus A and C are always at the same
potential and so also are B and D. Each pair of quadrants
is in connection with an electrode, E or F, by means of
which it can easily be put in metallic connection with any
body outside the case of the instrument. Inside the quad
rants and insulated from them is a flat piece of aluminium
shaped like a figure of eight. This is suspended by a
silk fibre and can rotate, with its plane horizontal, about
a vertical axis. A fine metal wire hangs from the lower
surface of this aluminium needle and dips into some
sulphuric acid contained in a glass vessel, the outside
of which is coated with tin-foil and connected with earth.
This vessel, with the conductors inside and outside, forms
a condenser of considerable capacity ; it requires therefore
a large charge to alter appreciably the potential of this jar,
and therefore of the needle. To use the instrument the jar
is charged to a high potential C; the needle will then also
be at the potential C. Now if the two pairs of quadrants
are at the same potential, the needle is inside a conductor
symmetrical about the axis of rotation of the needle, and
at one potential. There will evidently be no couple on

60] CONDENSERS 99

the needle arising from the electric field, and the needle
will take up a position in which the couple arising from
the torsion of the thread supporting the needle vanishes.
If, however, the two pairs of quadrants are not at the same
potential the needle will swing round until, if there is
nothing to stop it, the whole of its area will be inside the

Fig. 30.

pair of quadrants whose potential differs most widely
from its own. As it swings round, however, the torsion of
the thread produces a couple tending to bring the needle
back to the position from which it started. The needle
finally takes up a position in which the couple due to the
torsion in the thread balances that due to the electric

7—2

100 CAPACITY OF CONDUCTORS [CH. Ill

field. The angle through which the needle is deflected
gives us the means of estimating the potential difference
between the quadrants.

The way in which the couple acting on the needle
depends upon the potentials of the quadrants and the
needle can be illustrated by considering a case in which
the electric principles involved are the same as in the
quadrant electrometer, but where the geometry is simpler.

Let E, F (Fig. 31) be two large co-planar surfaces in
sulated from each other by a small air gap. Let G be
another plane surface, parallel to E and F, and free to
move in its own plane. Let t be the distance between G
and the planes E and F. Let A,B,C be the potentials of
the planes F, E, G respectively. Let I be the width of

Fig. 31.

the planes at right angles to the plane of the paper. If
XI is the force tending to move the plane G in the
direction of the arrow, then, if this plane be moved through
a short distance x in this direction, the work done by the
electric forces is Xlx. If the electric system is left to
itself, i.e. if it is not connected to any batteries, etc., so
that the charges remain constant, this work must have
been gained at the expense of the electric energy; we
have therefore, by the principle of the Conservation of
Energy,

Xlx = decrease in the electric energy of the system, the
charges remaining constant, when the plane G is
displaced through the distance x ;

60] CONDENSERS 101

or by Art. 36,

Xlx = increase in the electric energy of the system, the
potentials remaining constant, when the plane G is
displaced through the same distance x ......... (1).

Consider the change in the electric energy when the
plane G is moved through a distance an. The area of G
opposite to F will be increased by loc, and in consequence
the energy will be increased by the energy in a parallel
plate condenser, whose area is Ix, the potentials of whose
plates are A and C respectively, and the distance be
tween the plates is t ; this, by Art. 56, is equal to

At the same time as the area of G opposite to F is in
creased by lx, that opposite to E is decreased by the same
amount, so that the electric energy will be decreased by
the energy in a parallel plate condenser whose area is lxt
the potentials of the plates B and G and their distance
apart t ; this, by Art. 56, is equal to

Thus the total increase in the electric energy when G
is displaced through x, the potentials being constant,
is equal to

Thus, by equation (1),

102 CAPACITY OF CONDUCTORS [CH. Ill

If (C - A}2 is greater than (C - BY, X is positive, that
is, the plate G tends to bring as much of its surface as it
can over the plate from which it differs most in potential.

In the quadrant electrometer the electrical arrange
ments are similar to the simple case just discussed, and
hence the force will vary with the potential differences in
a similar way. Hence we conclude that if the needle in
the quadrant electrometer be at potential C, the couple
tending to twist it from the quadrant whose potential is B
to that whose potential is A, will be proportional to

we may put it equal to

where n is some constant.

When the needle is in equilibrium, this couple will
be balanced by the couple due to the torsion in the
suspension of the needle.

The torsional couple is proportional to the angle 6
through which the needle is deflected. Let the couple
equal md. Hence we have when the needle is in equi
librium

m0=n(B-A){O-z(A + B)

...(2).

If, as is generally the case when small differences of
potential are measured, the jar containing the sulphuric
acid is charged up so that its potential is very high com-

60] CONDENSERS 103

pared with that of either pair of quadrants, C will be very
large compared with A or B, and therefore with

\ (^ +B),
so that the expression (2) is very approximately

e = -(B-A)C.
m^

Hence, in this case, the difference of potential is pro
portional to the deflection of the needle. This furnishes
a very convenient method of comparing differences of
potential, and though it does not give at once the ab
solute measure of the potential, this may be deduced
by measuring the deflection produced by a standard po
tential difference of known absolute value such as that
between the electrodes of a Clark's cell.

The quadrant electrometer may also be used to
measure large differences of potential ; to do this, instead
of charging the jar independently, connect the jar and
therefore the needle to one pair of quadrants, say the pair
whose potential is A. Then, since C=A, the expression
(2) becomes

thus the needle is deflected towards the pair of quadrants
whose potential is B, and the deflection of the needle is, in
this case, proportional to the square of the potential differ
ence between the quadrants. Thus, if the quadrants are
connected respectively to the inside and outside coatings
of a condenser, the deflection of the electrometer will be
proportional to the energy in the condenser.

104 CAPACITY OF CONDUCTORS [CH. Ill

61. Use of the Electrometer to measure a charge
of electricity. Let a and ft denote the two pairs of
quadrants. If to begin with a and ft are both connected
with the earth, there will be a charge Q0 on the quadrants
a induced by the charge on the needle ; let a now be
disconnected from ft and from the earth, insulated, and
given a charge Qf of electricity, the needle will be deflected;
let 6 be the angle of deflection, A the potential of the
quadrants a, then if C is the potential of the needle, we
have, by Art. 26, since the charge on a. is QQ + Qf

Qo+Q' = qnA+qvC .................. (1),

where qu , qls are the coefficients of capacity and induction
for the displaced position of the needle. Since Q0 is the
charge on a when A is zero

where (g13)0 is the value of q13 when 6 = 0 ; hence by (1)

Q' = quA + (qls - (qu\) C.
Let ql3 - (q13\ = - p0,

6 being taken as positive when measured in the direction
of deflection due to a positive value of A, then if the
charge on the needle is negative Q0 the positive charge on
a induced by the needle will evidently increase with 6 so
that as C is negative p is a positive quantity; we have also
by equation (2), page 103, when C is large compared with A,

m
hence Q' = -e

61 a] CONDENSERS 105

It is interesting to notice that when the potential of
the needle is increased beyond a certain point the deflec
tion of the needle due to a given charge on the quadrants
diminishes as the potential of the needle increases, hence
to obtain the greatest sensitiveness when measuring elec
trical charges we must be careful not to charge the needle
too highly. We see from (2) that the greatest deflection
6' due to the charge Q' is given by the equation

when the deflection is greatest the potential of the needle

To get from the readings of the electrometer the value
of the charge in absolute measure, connect one plate of
a condenser whose capacity is F with the quadrants a, and
connect the other plate with the earth ; the coefficient qn
will now be increased by T and, if Oi is the deflection of
the electrometer for the same charge, then by (2)

Q'nC ,

"1-~(<?11 + r)m + ^G" '
Hence from (2) and (3)

If the deflection of the electrometer when the poten
tial of a is V is then

m
hence, from (4),

* /3 /3 *
C7 — C7-i

61 a. A gold leaf electroscope is for some purposes pre
ferable to an electrometer, on account of its much smaller

106 CAPACITY OF CONDUCTORS [CH. Ill

capacity, its portability and the ease with which it can be
shielded from external disturbances. With suitably designed
electroscopes it is possible to obtain with ease a deflection
of the gold leaf of 70 or 80 scale divisions for a change of
1 volt in the potential of the gold leaf, these divisions are
those of a micrometer eye-piece in a reading microscope
through which the gold leaf is observed. The behaviour
of these sensitive electroscopes may be illustrated by the
consideration of a very simple case. Suppose that we have
two parallel plates D and E maintained at potentials
A and - A respectively, let us represent the gold leaf by
another parallel plate 0 which can move backwards and
forwards and is pulled to a position midway between D and
E by a spring, which when C is displaced a distance x from
the mid-position pulls it back again with a force equal per
unit area of C to px.

If V is the potential of C, 2d the distance between D
and E, x the displacement of C towards the negative plate,
then for the equilibrium of the plate we must have
1 (A+ V)2 I (A- V)*_
STT (d - xf STT (d + x)* ~ ^'
or if V=yA, x = %d,

(1 + yy (l-

(I-?)2 (1 +

If y and f are small, this equation becomes

"'A

or x —

61 a] CONDENSERS 107

The equilibrium will be unstable unless
greater than 1, when this quantity exceeds unity by a
small fraction the denominator in the expression for x is
small so that x itself tends to become large, i.e. a small
potential difference Fwill produce a large displacement of
the plate. In addition to the value of x given above there
is a second value corresponding to another position of
equilibrium, the equilibrium in this case is unstable and if
C were in this position it would move up to D. When
V= 0 the two positions of equilibrium are given by x = 0
and

f-2=l-A

d* V/'

Thus when the instrument is very sensitive, i.e. when // is
nearly unity, the second value of x is very small, thus the
unstable position of equilibrium is close to the stable one,
so that a slight deflection from the latter will make the
gold leaf unstable, and it will fly up to one of the plates.
As V increases the value of x for the stable position
increases while that for the unstable one diminishes, so
that the two get nearer together, for a certain value of F
they coincide, while for greater values there is no position
of equilibrium.

In practice the office of the spring in the preceding
example is performed by the weight of the gold leaf; the
leaf is hung so as to be vertical when midway between the
plates, when it is disturbed from this position gravity tends
to bring it back. The successful use of instruments of this
type depends upon having means to keep the potential of
the fixed plates accurately constant. Except for very small
values of F, the deflection is not directly proportional to V,

108

CAPACITY OF CONDUCTORS

[CH. Ill

so that it is necessary to calibrate the instrument by
charging the gold leaf to known potentials and observing
the deflection.

The sensitiveness of the instrument can be adjusted by
altering V or d. In a type of instrument invented by
Mr C. T. R. Wilson and called the tilted electroscope
(Fig. 31 a), where the instrument can be tilted by means

Fig. 31 a.

of foot-screws, the adjustment is effected by altering the
tilt. The plate P is charged to a high potential, the case
of the instrument to earth, and initially the gold leaf is
to earth, it takes up a position of equilibrium from which
it is displaced as soon as its potential is altered.

62. Test for the equality of the capacities of
two condensers. The test can easily be made in the

63] CONDENSERS 109

following way. Suppose A and B, Fig. 32, are the plates
of one condenser, C and D those of the other. First
connect A to (7, and B to D, and charge the condensers by
connecting A and B with the terminals of a battery or
some other suitable means. Then disconnect A and B
from the battery. Disconnect A from C and B from D.
Then, if the capacities of the two condensers are equal,
their charges will be equal since they have been charged
to equal potentials. The charge on A will be equal and
opposite to that on D, while that on B will be equal and
opposite to that on C. Thus, if A be connected with D
and C with B, the positive charge on the one plate will
counterbalance the negative on the other, so that if after

Fig. 32.

this connection has been made A and B are connected
with the electrodes of an electrometer, no deflection will

occur.

63. Comparison of two condensers. If a con
denser whose capacity can be varied is available, the
capacity of a condenser can be compared with known
capacities by the following method.

Let A and B (Fig. 33) be the plates of the condenser
whose capacity is required, C and D, E and F, G and H,
the plates of three condensers whose capacities are known.
Connect the plates B and C together and to one electrode
of an electrometer, also connect F and G together and to
the other electrode of the electrometer. Connect D and

110

CAPACITY OF CONDUCTORS

[CH. Ill

E together and to one pole of a battery, induction coil or
other apparatus for producing a difference of potential,
and connect A and H together and to the other pole of
this battery. In general this will cause a deflection of
the electrometer; if there is a deflection, then we must
alter the capacity of the condenser whose capacity is
variable until the vanishing of this deflection shows that
the plates BO, FG are at the same potential. When
this is the case a simple relation exists between the
capacities.

Fig. 33.

Let Clf C2, Cs, C4 be the capacities of the condensers
AB, CD, EF, GH respectively, let VQ be the potential of
A and H, x the potential of B and G and y that of F and
G, V the potential of D and E. To fix our ideas, let us
suppose that V is greater than F0, then there will be a
negative charge on A, a positive one on B, a negative
charge on (7, and a positive one on D ; then since B and
C form an insulated system which was initially without
charge, the positive charge on B must be numerically
equal to the negative charge on C.

The positive charge on B

;-ft(ft-_FA

63] CONDENSERS 111

while the negative one on C is numerically equal to

which is a positive quantity ; hence, since these are equal,
we have

Cl(*-V.) = Ct(V-x) ............... (1).

Again, since F and G are insulated the positive charge
on G must be numerically equal to the negative charge
oiiF.

The positive charge on G is equal to

while the negative charge on F is numerically equal to

C,(V-y);

since these are equal

Ot(y-V.)=0,(V-y) ............... (2).

When there is no deflection of the electrometer the
potential of F and G is equal to that of B and G,
i.e. y — x. When this is the case we see by comparing
equations (1) and (2), that

C,_C2
Ct~G,'

n CzCj
or *— ZE"

Hence, if we know the capacities of the other condensers,
we know Q.

Thus, if we have standard condensers whose capacities
are known, we can measure the capacity of other con-
densers.

112

CAPACITY OF CONDUCTORS

[CH. Ill

There is a close analogy between the methods of
measuring capacity and those of measuring electrical
resistance. It is convenient to indicate that analogy
here, although the methods of measuring electrical re
sistance have not yet been discussed.

The arrangement of the condensers in the last method
can also be represented by the diagram (Fig. 34). In
this diagram C is the coil and G the electrometer. This
arrangement is analogous to that of resistances in a
Wheatstone's Bridge, see Art. 191, and the condition for

Fig. 34.

the balance of the condensers is the same as that of re
sistances in a Wheatstone bridge if each condenser were re
placed by a resistance inversely proportional to its capacity.
63 a. De Sauty's method. If two of the condensers
C3 and C4 in the last method are replaced by resist
ances R3 and R4, the electrometer by a galvanometer
and the induction coil by a battery with a key for making
and breaking the circuit, we get the arrangement known
as De Sauty's method, Fig. 35. In this method the re-

63 a]

CONDENSERS

113

sistances Rs and R4 are adjusted so that there is no kick
of the galvanometer on making the battery circuit. If i3
and {4, are the transient currents flowing through R3 and
RI at some short interval after making the circuit, then
neglecting self-induction, the potential difference at this
time between the terminals of the galvanometer will, by
Ohm's law, be R3i3 - R^, and this will be proportional to
the current through the galvanometer at this time. The
quantity of electricity flowing through the galvanometer
during charging will thus be proportional to f(R3i3—R^)dt.
when the integration extends over the time of charging.

Fig. 35.

If no current flows through the galvanometer, the current
i3 goes into the condenser (1) and i4 into condenser (2), so
that

where Ql and Q2 are the final charges in condensers (1)
and (2) respectively. Thus

i, - R4i4) dt =

T. E.

114

CAPACITY OF CONDUCTORS

[CH. Ill

since there is no kick of the galvanometer this vanishes,
so that

fl8Ql=^4&.

But when the condensers are charged there is the same
potential difference between the plates of (1) as between
those of (2), hence

Q:fe-4:4,

where Clt Ca are the capacities of the condensers, hence
when there is no kick of the galvanometer

thus the ratio G-JG^ is found as the ratio of two resistances.
We see that again the condition is the same as for the
balance in a Wheatstone bridge in which the condensers
have been replaced by resistances inversely proportional
to their capacity.

Other methods of determining capacity which require
for their explanation a knowledge of the principles of
electro-magnetism, will be described in the part of the
book dealing with that subject.

64. Ley den jar. A convenient form of condenser
called a Leyden jar is represented in Fig. 36. The

Fig. 36.

65] CONDENSERS 115

condenser consists of a vessel made of thin glass; the
inside and outside surfaces of this vessel are coated with
tin-foil. An electrode is connected to the inside of the
jar in order that electrical connection can easily be made
with it. If A is the area of each coat of tin-foil, t the
thickness of the glass, i.e. the distance between the surfaces
of tin-foil, then, if the interval between these surfaces was
filled with air, the capacity would be approximately

since this case is approximately that of two parallel
planes provided the thickness of the glass is very small
compared with the dimensions of the vessel. The effect of
having glass within the tin-foil surfaces will, as we shall
see in the next chapter, have the effect of increasing the
capacity so that the capacity of the Leyden jar will be

where K is a quantity which depends on the kind of
glass of which the vessel is made. K varies in value
from 4 to 10 for different specimens of glass.

SYSTEMS OF CONDENSERS.

65. If we have a number of condensers we can con
nect them up so as to make a condenser whose capacity
is either greater or less than that of the individual
condensers.

Thus suppose we have a number of condensers which
in the figures are represented as Leyden jars, and suppose
we connect them up as in Fig. 37, that is, connect all the

8—2

116 CAPACITY OF CONDUCTOKS [CH. Ill

insides of the jars together and likewise all the outsides ;
this is called connecting the condensers in parallel. We
thus get a new condenser, one plate of which consists of
all the insides, and the other plate of all the outsides of
the jars. If C is the capacity of the compound condenser,
Q the total charge in this condenser, V the difference of
potential between the plates, then by definition

Fig. 37.

If Qi> Qz> Qs> ••• are ^e charges in the first, second,
third, etc. condensers, Clt C2, C3, ... the capacities of these
condensers

but Q = Q, + Q, + Q, + ... = (C, + C, + C3+ ...) F,
hence C= C, + (72+ C3 + ... ,

or the capacity of a system of condensers connected in
this way, is the sum of the capacities of its components.
Thus the capacity of the compound system is greater
than that of any of its components.

Next, let the condensers be connected up as in Fig. 38,
where the condensers are insulated, and where the outside
of the first is connected to the inside of the second, the
outside of the second to the inside of the third, and so on.

65] CONDENSERS 117

This is called connecting the condensers up in cascade
or in series. One plate of the compound system thus
formed is the inside of the first condenser, the other plate
is the outside of the last.

Fig. 38.

Let C be the capacity of the system, Cl} Gz, C8) ... the
capacities of the individual condensers; then, since the
condensers are insulated, the charge on the outside of
the first is equal in magnitude and opposite in sign to
the charge on the inside of the second, the charge on the
outside of the second is equal in magnitude and opposite
in sign to the charge on the inside of the third, and so on.
Since the charge on the inside of any jar is equal and
opposite to the charge on the outside, we see that the
charges of the jars are all equal. Let Q be the charge
of any jar, Vlt V2... the differences of potential between
the inside and outside of the first, second, . . . jars. Then

F-« V = ® V-®

1 (V 2 cv 3~c3"

If V is the difference of potential between the outside of
the last jar and the inside of the first, then

F=F1+F2+F3...

118 CAPACITY OF CONDUCTORS [CH. Ill

so that Q = .

Cl C2 C3

But, since C is the capacity of the compound condenser of
which Q is the charge, and V the potential difference,

Q = GV,

hence = + ++....

w L^j 1^2 ^3

Thus the reciprocal of the capacity of the system made
by connecting up in cascade the series of condensers, is
equal to the sum of the reciprocals of the capacities of
the condensers so connected up.

We see that the capacity of the compound condenser
is less than that of any of its constituents.

66. If we connect a condenser of small capacity in
cascade with a condenser of large capacity, the capacity of
the compound condenser will be slightly less than that of
the small condenser ; while if we connect them in parallel,
the capacity of the compound condenser is slightly greater
than that of the large condenser.

67. As another example on the theory of condensers,
let us take the case when two condensers are connected in
parallel, the first having before connection the charge Qlt
the second the charge Q2. Let (7X and C2 be the capacities
of these condensers respectively. When they are put in
connection they form a condenser whose capacity is Gl + CZt
and whose charge is & + Q2.

Now the electric energy of a charged condenser is
one half the product of the charge into the potential
difference, while the potential difference is equal to

67] CONDENSERS

the charge divided by the capacity. Thus if Q is the
charge, C the capacity, the energy is

1Q2

20'

Thus the total electric energy of the two jars before
they are connected is

i&! i&2.

2 C, + 2 C, '
after they are connected it is

Now

1 /Q.' Q.'\_ 1 (ft + &
;+cU 2(4 + 0',

an essentially positive quantity which only vanishes if

ft/ft.- ft/ft,

that is, when the potentials of the jars before connection
are equal. In this case the energy after connection is
the same as before the connections are made. If the
potentials are equal before connection, connecting the
jars will evidently make no difference, as all that con
nection does is to make the potentials equal. In every
other case electric energy is lost when the connection
is made ; this energy is accounted for by the work done
by the spark which passes when the jars are connected.

CHAPTEE IV

SPECIFIC INDUCTIVE CAPACITY

68. Specific Inductive Capacity. Faraday found
that the charge in a condenser between whose surfaces a
constant difference of potential was maintained depended
upon the nature of the dielectric between the surfaces,
the charge being greater when the interval between the
surfaces was filled with glass or sulphur than when it was
filled with air.

Thus the ' capacity ' of a condenser (see Art. 51) de
pends upon the dielectric between the plates. Faraday's
original experiment by which this result was established
was as follows : he took two equal and similar condensers,
A and B, of the kind shown in Fig. 39, made of concentric
spheres ; in one of these, B, there was an opening by which
melted wax or sulphur could be run into the interval be
tween the spheres. The insides of these condensers were
connected together, as were also the outsides, so that the
potential difference between the plates of the condenser
was the same for A as for B. When air was the dielectric
between the spheres Faraday found, as might have been
expected from the equality of the condensers, that any
charge given to the condensers was equally distributed
between A and B. When however the interval in B was

CH. IV. 68] SPECIFIC INDUCTIVE CAPACITY

121

filled with sulphur and the condensers again charged he
found that the charge in B was three or four times that
in A, proving that the capacity of B had been in
creased three or four times by the substitution of sulphur
for air.

Fig. 39.

This property of the dielectric is called its specific
inductive capacity. The measure of the specific induc
tive capacity of a dielectric is defined as the ratio of the
capacity of a condenser when the region between its plates
is entirely filled by this dielectric, to the capacity of the
same condenser, when the region between its plates is
entirely filled with air. As far as we know at present,
the specific inductive capacity of a dielectric in a con
denser does not depend upon the difference of potential
established between the plates of that condenser, that is,
upon the electric intensity acting on the dielectric. We
may therefore conclude that, at any rate for a wide range

122 SPECIFIC INDUCTIVE CAPACITY [CH. IV

of electric intensities, the specific inductive capacity is
independent of the electric intensity.

The following table contains the values of the specific
inductive capacities of some substances which are fre
quently used in a physical laboratory:

Solid paraffin 2'29.

Paraffin oil T92.

Ebonite 315.

Sulphur 3'97.

Mica 6-64.

Dense flint-glass 7'37.

Light flint-glass 6'72.

Turpentine 2'23.

Distilled water 76.

Alcohol 26.

The specific inductive capacity of gases depends upon
the pressure, the difference between K, the specific in
ductive capacity, and unity being directly proportional to
the pressure.

The specific inductive capacity of some gases at
atmospheric pressure is given in the following table ; the
specific inductive capacity of air at atmospheric pressure
is taken as unity:

Hydrogen '999674.

Carbonic acid 1 '000356.

Carbonic oxide I'OOOl.

Olefiant gas 1-000722.

69. It was the discovery of this property of the di
electric which led Faraday to the view we have explained,
in Art. 38, that the effects observed in the electric field

69] SPECIFIC INDUCTIVE CAPACITY 123

are not due to the action at a distance of one electrified
body on another, but are due to effects in the dielectric
filling the space between the electrified bodies.

The results obtained in Chapters II and ill were
deduced on the supposition that there was only one
dielectric, air, in the field ; these require modification in
the general case when we have any number of dielectrics
in the field. We shall now go on to consider the theory
of this general case.

We assume that each unit of positive electricity, what
ever be the medium by which it is surrounded, is the
origin of a Faraday tube, each unit of negative electricity
the termination of one. Let us consider from this point
of view the case of two parallel plate condensers A and B,
the plates of A and of B being at the same distance apart,
but while the plates of A are separated by air, those
of B are separated by a medium whose specific inductive
capacity is K. Let us suppose that the charge per unit
area on the plates of the condensers A and B is the same.
Then, since the capacity of the condenser B is K times
that of A and since the charges are equal, the potential
difference between the plates of B is only I/ K of that
between the plates of A.

Now if VP is the potential at P, VQ that at Q, R the
electric intensity along PQ, then, whatever be the nature
of the dielectric, when PQ is small enough to allow of
the intensity along it being regarded as constant,

VP-VQ .................. (1),

for by definition R is the force on unit charge, hence the
left-hand side of this expression is the work done on unit

124 SPECIFIC INDUCTIVE CAPACITY [CH. IV

charge as it moves from P to Q, and is thus by definition,
(Art. 16), equal to the right-hand side of (1).

The electric intensity between the plates both of A
and of B is uniform, and is equal to the difference of
potential between the plates divided by the distance
between the plates; this distance is the same for the
plates A and B, so that the electric intensity between
the plates of A is to that between the plates of B as the
potential difference between the plates of A is to that
between the plates of B. That is, the electric intensity
in A is K times that in B.

Consider now these two condensers. Since the charges
on unit area of the plates are equal the number of
Faraday tubes passing through the dielectric between
the plates is the same, while the electric intensity in B
is only l/K that in air. Hence we conclude that when
the number of Faraday tubes which pass through unit
area of a dielectric whose specific inductive capacity is K
is the same as the number which pass through unit area
in air, the electric intensity in the dielectric is l/K of the
electric intensity in air.

By Art. 40, we see that if N is the number of Faraday
tubes passing through unit area in air, and R is the
electric intensity in air,

R = 4>7rN.

Hence, when N tubes pass through unit area in a medium
whose specific inductive capacity is K, the electric in
tensity, R, in this dielectric is given by the equation

71] SPECIFIC INDUCTIVE CAPACITY 125

70. Polarization in a dielectric. We define the
polarization in the direction PQ where P and Q are two
points close together as the excess of number of Faraday
tubes which pass from the side P to the side Q over the
number which pass from the side Q to the side P of a
plane of unit area drawn between P and Q at right
angles to PQ. We may express the result in Art. 68 in
the form

(electric intensity in any direction at P)

= -JJT (polarization in the dielectric in that direction at P).

The polarization in a dielectric is mathematically
identical with the quantity called by Maxwell the electric
displacement in the dielectric.

71. Thus the polarization along the outward-drawn
normal at P to a surface is the excess of the number
of Faraday tubes which leave the surface through unit
area at P over the number entering it. If we divide any
closed surface up as in Art. 9 into a number of small
meshes, each of these meshes being so small that the
polarization over the area of any mesh may be regarded
as constant, then if we multiply the area of each of the
meshes by the normal polarization at this mesh measured
outwards, the sum. of the products taken for all the
meshes which cover the surface is defined to be the total
normal polarization outwards from the surface. We see
that it is equal to the excess of the number of Faraday tubes
which leave the surface over the number which enter it.

Now consider any tube which does not begin or end
inside the closed surface, then if it meets the surface at
all it will do so at two places, P and Q ; at one of these

126 SPECIFIC INDUCTIVE CAPACITY [CH. IV

it will be going from the inside to the outside of the
surface, at the other from the outside to the inside. Such
a tube will not contribute anything to the total normal
polarization outwards from the surface, for at the place
where it leaves the surface it contributes + 1 to this
quantity, which is neutralized by the — 1 which it contri
butes at the place where it enters the surface.

Now consider a tube starting inside the surface ; this
tube will leave the surface but not enter it, or if the surface
is bent so that the tube cuts the surface more than once,
it will leave the surface once oftener than it enters it.
This tube will therefore contribute -f 1 to the total out
ward normal polarization: similarly we may show that each
tube which ends inside the surface contributes — 1 to the
total outward normal polarization. Thus if there are N
tubes which begin, and M tubes which end inside the
surface, the total normal polarization is equal to N — M.
But each tube which begins inside the surface corresponds
to a unit positive charge, each tube which ends in the
surface to a unit negative one, so that N — M is the differ
ence between the positive and negative charges inside the
surface, that is, it is the total charge inside the surface.

Thus we see that the total normal polarization over
a closed surface is equal to the charge inside the surface.
Since the normal polarization is equal to Kj^ir times
the normal intensity where K is the specific inductive
capacity, which is equal to unity for air, we see that when
the dielectric is air the preceding theorem is identical
with Gauss's theorem, Art. 10. In the form stated above
it is applicable whatever dielectrics may be in the field,
when in general Gauss's theorem as stated in Art. 10
ceases to be true.

73] SPECIFIC INDUCTIVE CAPACITY 127

72. Modification of Coulomb's equation. If a

is the surface density of the electricity on a conductor,
then cr Faraday tubes pass through unit area of a plane
drawn in the dielectric just above the conductor at right
angles to the normal. Hence a is the polarization in
the dielectric in the direction of the normal to the con
ductor. Hence, by Art. 69, if R is the normal electric
intensity

This is Coulomb's equation generalized so as to apply
to the case when the conductor is in contact with any
dielectric.

73. Expression for the Energy. The student
will see that the process of Art. 23 by which the expression
^(EV) was proved to represent the electric energy
of the system will apply whatever the nature of the
dielectric may be, as will also the immediate deduction
from it in Art. 43 that the energy is the same as it would
be if each Faraday tube possessed an amount of energy
equal per unit length to one-half the electric intensity.

The expression for the energy per unit volume how
ever requires modification. Consider, as in Art. 43, a
cylinder whose axis is parallel to the electric intensity
and whose flat ends are at right angles to it, let I be the
length of the cylinder, w the area of one of the ends,
P the polarization, R the electric intensity. Then the
portion of each Faraday tube inside the cylinder has an
amount of energy equal to

128 SPECIFIC INDUCTIVE CAPACITY [CH. IV

Now the number of such tubes inside the cylinder is equal
to Pa, hence the energy inside the cylinder is equal to

Since lw is the volume of the cylinder, the energy per
unit volume is equal to

but by Art. 69 P = -R,

4i7T

so that the energy per unit volume is equal to

Thus, for the same electric intensity the energy per
unit volume of the dielectric is K times as great as it
is in air. Another expression for the energy per unit
volume is

so that for same polarization the energy per unit volume
in the dielectric is only l/Kih part of what it is in air.

We see, as in Art. 45, that the pull along each Faraday
tube will still equal one-half the electric intensity R\
the tension across unit area in the dielectric will therefore

KR*

be -Q — , the lateral pressure will also be equal to KR2/87r.

74. Conditions to be satisfied at the boundary
between two media of different specific inductive
capacities. Suppose that the line AB represents the

74] SPECIFIC INDUCTIVE CAPACITY 129

section by the plane of the paper of the plane of separa
tion between two different dielectrics ; let the specific
inductive capacities of the upper and lower media re
spectively be Kl) K2.

Let us consider the conditions which must hold at
the surface. In the first place we see that the electric
intensities parallel to the surface must be equal in the
two media ; for if they were not equal, and that in the
medium Kt were the greater, we could get an infinite
amount of work by making unit charge travel round
the closed circuit PQRS, PQ being just above, and RS
just below the surface of separation. For, if PQ is the
direction of ^ the tangential component of the electric
intensity in the upper medium, the work done on unit

Fig. 40.

charge as its goes from P to Q is ^ . PQ ; as QR is ex
ceedingly small compared with PQ the work done on or
by the charge as it goes from Q to R may be neglected
if the normal intensity is not infinite ; the work required
to take the unit charge back from R to S is T2 . RS,
if T2 is the tangential component of the electric intensity
in the lower dielectric, and the work done or spent in
going from S to P will be equal to that spent or done in
going from Q to R and may be neglected. Thus since the
system is brought back to the state from which it started,
the work done must vanish, and hence 2\ . PQ — Tz . RS
must be zero. But since PQ = RS this requires that
Tl = T2 or the tangential components of the electric
intensity must be the same in the two media.

T. E. 9

ISO SPECIFIC INDUCTIVE CAPACITY [CH. IV

Next suppose that a is the surface density of the
free electricity on the surface separating the two media.
Draw a very flat circular cylinder shown in section at
PQRS, the axis of this cylinder being parallel to the
normal to the surface of separation, the top face of this
cylinder being just above, the lower face just below this
surface. As the length of this cylinder is very small com
pared with its breadth, the area of the curved surface
of the cylinder will be very small compared with the
area of its ends, and by making the cylinder sufficiently
short we can make the ratio of the area of the curved
surface to that of the ends as small as we please. Hence
in considering the total outward normal polarization over
the very short cylinder, we may leave out the effect of
the curved surface and consider only the flat ends of the
cylinder. But since the cylinder encloses the charge aw,
if o> is the area of one end of the cylinder, the total normal
polarization over its surface must be equal to crco. If Nl
is the normal polarization in the first medium measured
upwards the total normal polarization over the top of the
cylinder is N^; if N2 is the normal polarization measured
upwards in the second medium, the total normal polariza
tion over the lower face of the cylinder is — N2co ; hence
the total outward normal polarization over the cylinder is

Since, by Art. 71, this is equal to o-co, we have

N.-N^a.

When there is no charge on the surface separating
the two dielectrics, these conditions become (1) that the
tangential electric intensities, and (2) the normal polariza
tions, must be equal in the two media.

75] SPECIFIC INDUCTIVE CAPACITY 131

75. Refraction of the lines of force. Suppose
that R2 is the resultant electric intensity in the upper
medium, Rz that in the lower; and 01} 02 the angles these
make with the normal to the surface of separation. The
tangential intensity in the first medium is Rlsin01, that
in the second is R^ sin 02, and since these are equal

The normal intensity in the upper medium is R1 cos 0lt
hence the normal polarization in the upper medium is

cos

that in the second is K^RZ cos #2/4<7r, and since, if there
is no charge on the surface, these are equal, we have

(2);

dividing (1) by (2), we get

-JT tan &i = -fr tan #2-
Kl /i3

Hence, if Kl> K2, 01 is > #2, and thus when a Faraday
tube enters a medium of greater specific inductive capacity
from one of less, it is bent away from the normal.

This is shown in the diagram Fig. 41 (from Lord
Kelvin's Reprint of Papers on Electrostatics and Mag
netism), which represents the Faraday tubes when a sphere,
made of paraffin or some material whose specific inductive
capacity is greater than unity, is placed in a field of uni
form force such as that between two infinite parallel plates.

An inspection of the diagram shows the tendency of
the tubes to run as much as possible through the sphere ;
this is an example of the principle that when a system
is in stable equilibrium the potential energy is as small

9-2

132

SPECIFIC INDUCTIVE CAPACITY

[CH. IV

as possible. We saw, Art. 73, that when the polarization
is P the energy per unit volume is 27rP2/K, thus for the
same value of P, this quantity is less in paraffin than it
is in air. Hence when the same number of tubes pass
through the paraffin they have less energy in unit volume
than when they pass through air, and there is therefore
a tendency for the tubes to flock into the paraffin. The
reason why all the tubes do not run into the sphere is
that those which are some distance away from it would
have to bend considerably in order to reach the paraffin,

Fig. 41.

they would therefore have to greatly lengthen their path
in the air, and the increase in the energy consequent
upon this would not be compensated for in the case of the
tubes some distance originally from the sphere by the
diminution in the energy when they got in the sphere.

In Fig. 42 (from Lord Kelvin's Reprint of Papers on
Electrostatics and Magnetism) the effect produced on a
field of uniform force by a conducting sphere is given for
comparison with the effects produced by the paraffin

75]

SPECIFIC INDUCTIVE CAPACITY

133

sphere. It will be noticed that the paraffin sphere pro
duces effects similar in kind though not so great in
degree as those due to the conducting sphere. This obser
vation is true for all electrostatic phenomena, for we find
that bodies having a greater specific inductive capacity
than the surrounding dielectric behave in a similar way to
conductors. Thus, they deflect the Faraday tubes in the
same way though not to the same extent; again, as a con
ductor tends to move from the weak to the strong parts of

Fig. 42.

the field, so likewise does a dielectric surrounded by one
of smaller specific inductive capacity. Again, the electric
intensity inside a conductor vanishes, and just inside a
dielectric of greater specific inductive capacity than the
surrounding medium the electric intensity is less than that
just outside. As far as electrostatic phenomena are con
cerned an insulated conductor behaves like a dielectric
of infinitely great specific inductive capacity.

134 SPECIFIC INDUCTIVE CAPACITY [CH. IV

76. Force between two small charged bodies
immersed in any dielectric. If we have a small body
with a charge e immersed in a medium whose specific
inductive capacity is K, then the polarization at a dis
tance r from the body is e/4>7rr-. To prove this, describe
a sphere radius r, with its centre at the small body, then
the polarization P will be uniform over the surface of
the sphere and radial ; hence the total normal polarization
over the surface of the sphere will equal P x (surface
of the sphere), i.e. P x 4-Trr2; but this, by Art. 71, is equal

to e, hence

P x 4?rr2 = e,

But, if R is the electric intensity, then, by Art. 70

7? 4-7T p

R = g.P.

Hence, by (1), R = j^\

the repulsion on a charge e is Re', or ee'/Kr~ ; hence the
repulsion between the charges, when separated by a dis
tance r in a dielectric whose specific inductive capacity
is K, is only l/Kth part of the repulsion between the
charges when they are separated by the same distance in
air. Thus, when the charges are given, the mechanical
forces on the bodies in the field are diminished when the
charges are imbedded in a medium with a large specific
inductive capacity. We can easily show that the inter
position between the charges of a spherical shell of the
dielectric with its centre at either of the charges would
not affect the force between these charges.

77] SPECIFIC INDUCTIVE CAPACITY 135

77. Two parallel plates separated by a di
electric. Let us first take the case of two parallel
plates completely immersed in an insulating medium
whose specific inductive capacity is K. Let V be the
potential difference between the plates, cr the surface
density of the electrification on the positive plate, and
- a that on the negative. Let R be the electric intensity
between the plates, and d the distance by which they
are separated ; then, by Art. 72,

47TO- = KR

_KV

'' d

The force on one of the plates per unit area is, by Art. 37,

-27TQ-

Hence if the charges are given the force between the
plates is inversely proportional to the specific inductive
capacity of the medium in which they are immersed.
Again, since

we see that, if the potentials of the plates be given, the
attraction between them is directly proportional to the
specific inductive capacity. This result is an example of
the following more general one which we leave to the
reader to work out ; if in a system of conductors main
tained at given potentials and originally separated from
each other by air we replace the air by a dielectric whose
specific inductive capacity is K, keeping the positions of

136 SPECIFIC INDUCTIVE CAPACITY [CH. IV

the conductors and their potentials the same as before, the
forces between the conductors will be increased K times.

Thus, for example, if we fill the space between the
needles and the quadrants of an electrometer with a
fluid whose specific inductive capacity is K, keeping the
potentials of the needles and quadrants constant, the
couple on the needle will be increased K times by the
introduction of the fluid. Thus, if we measure the couples
before and after the introduction of the fluid, the ratio
of the two will give us the specific inductive capacity
of the fluid. This method has been applied to measure
the specific inductive capacity of those liquids, such as
water or alcohol, which are not sufficiently good insulators
to allow the method described in Art. 82 to be applied.

VllllllllllllllllllllllllllllllllllllllllllllllllJft
c

D
Fig. 43.

78. We shall next consider the case in which a
slab of dielectric is placed between two infinite parallel
conducting planes, the faces of the slab being parallel to
the planes.

Let d be the distance between the planes, t the
thickness of the slab, h the distance between the upper
face of the slab and the upper plane. The Faraday tubes
will go straight across from plane to plane, so that the
polarization will be everywhere normal to the conducting

78] SPECIFIC INDUCTIVE CAPACITY 137

planes and to the planes separating the slab of dielectric
from the air.

We saw in Art. 74 that the normal polarization does
not change as we pass from one medium to another, and
as the tubes are straight the polarization will not change
as long as we remain in one medium. Thus the polariza
tion which we shall denote by P is constant between the
planes. In air the electric intensity is 4-TrP; in the dielectric
of specific inductive capacity K, the electric intensity is
equal to 47rP/^T.

Thus between A and B the electric intensity is 4-TrP,

4<7rP

B and C — ~- ,

(7andD 4-rrP.

The difference of potential between the plates is the
work done on unit charge when it is taken from one plate
to the other. Now, when unit charge is taken across the
space AB, the work done on it is
4-TrP x h ;
when it is taken across the plate of dielectric the work

done is

4?rP

K Xt>

when it is taken across CD the work done is

4nrP{d-(h+t)}.

Hence V, the excess of the potential of the plate A
above that of D, is equal to

{d - (h + 1)}

138 SPECIFIC INDUCTIVE CAPACITY [CH. IV

If o- is the surface density of the electricity on the
positive plate, a = P, so that

(1).

Hence the capacity per unit area of the plate, i.e. the
value of o- when V = 1, is

4<7rld- t + ^

i.e. it is the same as if the plate of dielectric were re
placed by a plate of air whose thickness was t/K. The
presence of the dielectric increases the capacity of the
condenser. The alteration in the capacity does not depend
upon the position of the slab of dielectric between the
parallel plates.

Let us now consider the force between the plates;
the force per unit area

where R is the electric intensity at the surface of the
plate ; but, since the surface of the plate is in contact
with air, R = 47rcr, and thus the force per unit area on
either plate

= 27TO-2.

Hence if the charges on the plates are given, the attraction
between them is not affected by the interposition of the
plate of dielectric.

Next, let the potentials be given ; we see from equa
tion (1) that

O- = , _;

78] SPECIFIC INDUCTIVE CAPACITY 139

hence 27ro-'2, the force per unit area, is equal to

The force between the plates when there is nothing
but air between them, is

V_^

bird*'

Now since K is greater than 1, d—t + t/K is less
than d, so that l/(d - 1 + t/K)*- is greater than l/d2. Thus,
when the potentials are given, the force between the plates
is increased by the interposition of the dielectric.

If K be very great, tjK is very small, thus d — 1 4- t/K
is very nearly equal to d — t, and the effect of the inter
position of the slab of dielectric both on the capacity
and on the force between the plates is approximately
the same as if the plates had been pushed towards
each other through a distance equal to the thickness
of the slab, the dielectric between the plates being now
supposed to be air. This result, which is approximately
true whenever the specific inductive capacity of the slab
is very large, is rigorously true when the slab is made of
a conducting material.

Effect of the slab of dielectric on the potential
energy for given charges. The potential energy is, by
Art. 23, equal to

and thus the energy corresponding to the charge on each
unit of area of the plates is equal to

140 SPECIFIC INDUCTIVE CAPACITY [CH. IV

by equation (1) this is equal to

and it is thus when K > 1 less than 27r<r2rf, the value of
the energy for the same charges when no slab of dielectric
is interposed. The interposition of the slab thus lowers
the potential energy. We can easily see why this is the
case. When the charges are given the number of Faraday
tubes is given : and, when the plate of dielectric is in
terposed, the Faraday tubes in part of their journey
between the plates are in the dielectric instead of in air,
and we know from Art. 73 that when the Faraday tubes
are in the dielectric their energy is less than when they
are in air. Since the potential energy of a system always
tends to become as small as possible, there will be a
tendency to drag as much as possible of the slab of
dielectric between the plates of the condenser. Thus,
if the slab of dielectric projected on one side beyond the
plates it would be drawn between the plates until as
much of its area as possible was within the region between
the plates.

Effect of the slab on the potential energy for a
given difference of potential. The energy per unit
area of the plates is as we have seen equal to

this by equation (1) is equal to
1 F2

80] SPECIFIC INDUCTIVE CAPACITY 141

If the potential difference is given the energy when no
slab is interposed is

so that when the potential difference is kept constant the
electric energy is increased by the interposition of the slab.

79. Capacity of two concentric spheres with a
shell of dielectric interposed between them. If we

have two concentric conducting spheres with a concentric
shell of dielectric between them, and if e be the charge
on the inner sphere, a the radius of this sphere and b, c
the radii of the inner and outer surfaces of the dielectric
shell, and d the inner radius of the outer" conducting
sphere, then if V be the difference of potential between
the conducting spheres, and K the specific inductive
capacity of the shell, we may easily prove that

a bK\b c

Thus the capacity of the system is equal to
1

a d \ Kl\b

80. Two coaxial cylinders. As another example,
we shall take the case of two coaxial cylinders with a
coaxial cylindric shell of a dielectric, specific inductive
capacity K, placed between them. If V be the difference
of potential between the two conducting cylinders, E the
charge per unit length on the inner cylinder, a the radius
of this inner cylinder, b and c the radii of the inner and
outer surfaces of the dielectric shell and d the inner radius

SPECIFIC INDUCTIVE CAPACITY [CH. IV

of the outer cylinder, we easily find by the aid of Art. 58
that

1 C ,

so that the capacity per unit length of this system is

b 1 c .d}'

81. Force on a piece of dielectric placed in an
electric field. If a piece of a dielectric such as sulphur or
glass is placed in the electric field, then, when the Faraday
tubes traverse the dielectric there is, Art. 73, less energy
per unit volume than when the same number of Faraday
tubes pass through air. Thus, as we see in Fig. 39, the
Faraday tubes tend to run through the dielectric, because
by so doing the potential energy is decreased. If the
dielectric is free to move, it can still further decrease
the energy by moving from its original position to one
where the tubes are more thickly congregated, because the
more tubes which get through the dielectric the greater
the decrease in the potential energy. The body will tend
to move so as to make the decrease in the energy as great
as possible, thus it will tend to move so as to be traversed
by as great a number of Faraday tubes as possible. It
will therefore be urged towards the part of the field where
the Faraday tubes are densest, i.e. to the strongest parts
of the field. There will thus be a force on a piece of
dielectric tending to make it move from the weak to the
strong parts of the field. The dielectric will not move
except in a variable field where it can get more Faraday
tubes by its change of position. In a uniform field such

82] SPECIFIC INDUCTIVE CAPACITY 14-3

as that between two parallel infinite plates the dielectric
would have no tendency to move.

The force acting upon the dielectric differs in another
respect from that acting on a charged body, inasmuch
as it would not be altered if the direction of the electric
intensity at each point in the field were reversed without
altering its magnitude.

82. Measurement of specific inductive capacity.

The specific inductive capacity of a slab of dielectric can
be measured in the following way, provided we have a
parallel plate condenser one plate of which can be moved
by means of a screw through a distance which can be
accurately measured. To avoid the disturbance due to the
irregular distribution of the charge near the edges of the
plates (see Art. 57) care must be taken that the distance
between the plates never exceeds a small fraction of the
diameter of the plates. Let us call this parallel plate con
denser A ; to use the method described in Art. 63, first take
the condenser A and before inserting the slab of dielectric
adjust the other variable condenser used in that method
until there is no deflection of the electrometer. If the slab
of dielectric be now inserted between the plates of A the
capacity will be increased, A will no longer be balanced by
the other condensers and the electrometer will be deflected.
The capacity of A can be diminished by screwing the plates
further apart, and when the plates have been moved
through a certain distance, the diminution in the capacity
due to the increase in the distance between the plates
will balance the increase due to the insertion of the slab
of dielectric ; the stage when this occurs will be indicated
by there being again no deflection of the electrometer.

144 SPECIFIC INDUCTIVE CAPACITY [CU. IV

Suppose that when the deflection of the electrometer is
zero before the slab is inserted, the distance between the
plates of the condenser is d, while the distance after the
slab is inserted, when the electrometer is again in equili
brium, is d'. Then the capacity of A in these two cases
is the same. But if A is the area of the plate of A the
capacity before the slab is inserted is

If t is the thickness of the slab and K its specific inductive
capacity, the capacity after the insertion of the slab is (see
Art. 78) equal to

4?r (d! - t + g

but since the capacities are equal

so that d'-d =

But d' - d is the distance through which the plate has
been moved, so that if we know this distance and t we can
determine K the specific inductive capacity of the slab.
It should be noticed that this method does not require
a knowledge of the initial or final distances between
the plates, but only the difference of these quantities,
and this can be measured with great accuracy by the
screw attached to the moveable plate.

CHAPTER V.

ELECTRICAL IMAGES AND INVERSION.

83. We shall now proceed to discuss some geometrical
methods by which we can find the distribution of electricity
in several very important cases. We shall illustrate the
first method by considering a very simple example ; that
of a very small charged body placed in front of an infinite
conducting plane maintained at potential zero. Let P,
Fig. 44, be the charged body, AB the conducting plane.

Fig. 44.

Any solution of the problem must satisfy the following
conditions in the region to the right of the plane AB ;
(a) it must make the potential zero over the plane AB,
and (fi) it must make the total outward normal induction
taken over any closed surface enclosing P equal to 4>7re,
where e is the charge at P, while if the closed surface does
T. E. 10

146 ELECTRICAL IMAGES AND INVERSION [CH. V

not enclose P the total normal induction over it must
vanish. We shall now prove that there is only one solution
which satisfies these conditions. Suppose there were two
different solutions, which we shall call (1) and (2). Take
the solution corresponding to (2) and reverse the sign of
all the charges of electricity in the field, including that at
P ; this new solution, which we shall denote by (- 2), will
correspond to a field in which the electric intensity at any
point is equal and opposite to that due to the solution (2)
at the same point. The solution (— 2) corresponds to a
field in which the electric potential is zero over AB and
at any point at an infinite distance from P ; it also makes
the total normal induction over any closed surface enclos
ing P equal to — 4?re, that is equal and opposite to the
total induction over the same surface due to the solution
(1) ; and the total induction over any other closed surface
in the region to the right of AB zero. Now consider the
field got by superposing the solutions (1) and (— 2): it will
have the following properties ; the potential over AB will
be zero and the total normal induction over any closed
surface in the region to the right of AB will vanish.
Since the normal induction vanishes over all closed
surfaces in this region, there will in the field correspond
ing to this solution be no charge of electricity. We may
regard the region as the inside of a closed surface at zero
potential (bounded by the plane AB and an equipotential
surface at an infinite distance) : by Art. 18, however, the
electric intensity must vanish throughout this region as
there is no charge inside it. Thus, the electric intensity
in the field corresponding to the superposition of the
solutions (1) and (— 2) is zero : that is, the electric
intensity in the solution (1) is equal and opposite to that

84] ELECTRICAL IMAGES AND INVERSION 147

in (— 2). But the electric intensity in (— 2) is equal and
opposite to that in (2). Hence the electric intensity in
(1) is at all points the same as (2), in other words, the
solutions give identical electric fields. Hence, if we get in
any way a solution satisfying the conditions (a) and (/:?), it
must be the only solution of the problem.

84. Let P' be a point on the prolongation of the
perpendicular PN let fall from P on the plane, such that
P'N = PN, and let a charge equal to — e be placed at P'.
Consider the properties, in the region to the right of AB,
of the field due to the charge e at P and the charge — e
atP'.

The potential due to — e at P' and 4- e at P at a point
Q on the plane AB is equal to

But since AB bisects PP' at right angles PQ = P'Q, thus
the potential at Q vanishes. Again, any closed surface
drawn in the region to the right of the plane AB does not
enclose P', and thus the charge at P' is without effect
upon the total induction over any such surface. The total
induction over such a surface is zero or 4nre according as
the closed surface does not or does include P. In the
region to the right of AB the electric field due to e at P
and —e Sit P' thus satisfies the conditions (a) and (ft) and
therefore represents the state of the electric field. Thus
the electrical effect of the electricity induced on the
conducting plane AB will be the same as that of the
charge — e at P' at all points to the right of AB. This
charge at P' is called the electrical image of the charge P
in the plane.

10—2

148 ELECTRICAL IMAGES AND INVERSION [CH. V

The attraction on P towards the plane will be the
same as the attraction between the charges e at P, and
-e at P', that is

(2P)2 ~ 4

Thus the attraction on the charged body varies inversely
as the square of its distance from the plane.

To find the surface density of the electricity induced
on the plane AB we require the electric intensity at right
angles to the plane. The electric intensity at right angles
to the plane AB at a point Q on the plane due to the
charge e at P is equal to

_e_ PjV

P<22'P<2'

and acts from right to left. The electric intensity at Q
due to - e at P' in the same direction is
e P'N

P'Q*'P'Q'

Hence since PQ = P'Q and PN = P'N the resultant normal
electric intensity at Q is

PQ* '

This, by Coulomb's law, is equal to 4?ro-, if a is the
surface density of the electricity at Q, and hence

or the surface density varies inversely as the cube of the
distance from P.

The total charge of electricity on the plane is - e, as
all the tubes which start from P end on the plane.

85] ELECTRICAL IMAGES AND INVERSION 149

The electrical energy is equal to ^EV, so that if

the small body at P is a sphere of radius a, the energy
in the field is equal to

le* 1 e2

The dielectric in this case is supposed to be air. The
electric intensity vanishes in the region to the left of AB

85. Electrical images for spherical conductors

In applying the method of images to spherical conductors
we make great use of the following theorem due to Apol-
lonius. If S, Fig. 45, is a point on a sphere whose centre
is 0 and radius a, and P and Q are two fixed points on
a straight line passing through 0, such that OP . OQ = a2,
then QS/PS is constant wherever S may be on the sphere.

Fig. 45.
Consider the triangles QOS, POS. Since

OQ.OP-OS; <g-<g,

hence these triangles have the angle at 0 common and the
sides about this angle proportional. They are therefore
similar triangles, so that

QS _PS
OQ " 08 '

150 ELECTRICAL IMAGES AND INVERSION [CH. V

QS__OQ_OS
PS~OS OP'

Hence QS/PS is constant whatever may be the position
of S on the sphere.

86. Now suppose that we have a spherical shell (Fig.
45) at potential zero whose centre is at 0 and that a small
body with a charge e of electricity is placed at P and that
we wish to find the electric field outside the sphere.
There is no field inside the sphere, as the sphere is an
equipotential surface with no charge inside it.

Let OP =/, 08 = a. Consider the field due to a charge
e at P, and e' at Q where OQ.OP = a?. The potential at
a point 8 on the sphere due to the two charges is

But by Art. 85,

Thus the potential at $ = j e + e' -

Hence, if e' = - ea/f, the potential is zero over the
surface. Thus, under these circumstances the field satisfies
condition (a) of Art. 83, and it obviously satisfies the
condition that the total normal induction over any closed
surface not enclosing the sphere is zero or 4-Tre according
as the surface does not or does enclose P, so that, by
Art. 83 this is the actual field due to the sphere and the
charged body. Hence, at a point outside the sphere, the
effect of the electricity induced on the sphere by the

86] ELECTRICAL IMAGES AND INVERSION 151

charge at P is the same as that of a charge — ea/f at Q.
This charge at Q is called the electrical image of P in the
sphere. Since this charge produces the same effect as
the electrification on the sphere, the total charge on the
sphere must equal the charge at Q, i.e. it must be equal to
— ea/f (compare Art. 30). Thus of the Faraday tubes
which start from P the fraction a/f fall on the sphere.

The force on P is an attraction towards the sphere and
is equal to

a ez a e2 _ a e2 e^fa

fPQ* =f(OP-OQ)* = / (f o?

(f~f

We see from this result that, when the distance of P
from the centre of the sphere is large compared with the
radius, the force varies inversely as the cube of the
distance from the centre of the sphere : while when P
is close to the surface of the sphere the force varies
inversely as the square of the distance from the nearest
point on the surface of the sphere. When P is very
near to the surface of the sphere, the problem becomes
practically identical with that of a charge placed in front
of a plane at potential zero. We shall leave it as an
exercise for the student to deduce the solution for the
plane as the limit of that of the sphere.

If the body at P is a small sphere of radius b, then
since the electric energy is equal to y£EV, it is in this case

ea 1

1 <1_ a )

\ »/ /

152 ELECTRICAL IMAGES AND INVERSION [CH. V

87. To find the surface density at a point S on the
surface of the sphere, we must find the electric intensity
along the normal.

The electric intensity at 8 due to the charge e at P
can by the triangle of forces be resolved into the two com
ponents

/ x e OS i no
(«) -2 along OS,

parallel to PO,

while the electric intensity at S due to the charge — ea/f
at Q can be resolved into the components

, . ea I OS .

(7) ~ oiag08'

Hence the components of the resultant intensity are a 4- 7
along the normal OS, and ft + 8 parallel to PO.

Now the resultant intensity is along the normal, so
that the component /3 + S must vanish, and the resultant
intensity along the normal is equal to a + 7, i.e. to

r,of 1 a_I_)
e-08lFS>-fQ&\

Since PS/QS is constant, the quantity inside the brackets
is constant.

88] ELECTRICAL IMAGES AND INVERSION 153

If a- is the surface density of the electrification at 8,
then, by Coulomb's law,

eOS

f \Q8J j P&

so that the surface density of the electrification varies
inversely as the cube of the distance from P, and is, since
/ is greater than a, everywhere negative.

88. If the sphere is insulated instead of being at zero
potential, the conditions are that the potential over the
sphere should be constant and that the charge on the
sphere should be zero. The charge on the sphere in
the last case was — ea/f. Hence if we superpose on the
last solution the field due to a quantity of electricity
equal to ea/f placed at the centre of the sphere, which
will give rise to a uniform potential over the sphere, the
resulting field at points outside the sphere will have the
following properties; (1) the potential over the sphere is
constant, (2) the total charge on the sphere is zero,
(3) the total normal induction over any closed surface is
equal to 4?re if the surface encloses P and is zero if it
does not. Hence it is the solution in the region outside
the sphere when a charge e is placed at P in front of an
insulated conducting sphere. Thus, outside the insulated
sphere the electric field is the same as that due to the
three charges, e at P, — ea/f at Q, ea/f at 0. Let us
consider the potential of the sphere: the charges at P
and Q together produce zero potential over the sphere,
so that the potential will be that due to the charge ea/f,
at 0 ; this charge produces at any point on the sphere a
potential equal to e/f, so that by the presence of e at

154 ELECTRICAL IMAGES AND INVERSION [CH. V

P the potential of the sphere is raised by e/f. This
result was proved by a different method in Art. 29.

The force on P in this case is an attraction equal to
e2 a e2a

so that in this case, when/ is very large compared with a
the force varies inversely as the fifth power of the distance.
When the point is very close to the surface of the sphere
the force is the same as if the sphere were at zero
potential.

The potential energy, ^%EV is, if the body at P is a
small sphere of radius 6, equal to

ea ea

To find the surface density at $, we must superpose on
the value given in Art. 87, the uniform density

Thus

4"7T<7= —

PS3

At R the point on the sphere nearest to l\
PR=f-a,

88] ELECTRICAL IMAGES AND INVERSION 155

so that the surface density at R is equal to

JL *_ \ f+a II

47ra((/-a)2 f]

47T/(/-a)2'

At R' the point on the sphere most remote from P,

PR' = f+a,
and the surface density at R' is equal to

Since the total charge on the sphere is zero, the surface
density of the electricity must be negative on one part of
the sphere, positive on another part. The two parts will
be separated by a line on the sphere along which there is
no electrification. To find the position of this line put a
equal to zero in equation (1), we get if S is a point on
this line

= OP'-xPQ,

hence the points at which the electrification vanishes will
be at a distance (OP2 x PQ)4 from P.

The parts of the surface of the sphere whose distances
from P are less than this value are charged with electricity
of the opposite sign to that at P, the other parts of the
sphere are charged with electricity of the same sign as
that at P.

156 ELECTRICAL IMAGES AND INVERSION [CH. V

89. If the sphere instead of being insulated and with
out charge is insulated and has a charge E, we can deduce
the solution by superposing on the field discussed in Art.
88 that due to a charge E uniformly distributed over the
surface of the sphere ; this at a point outside the sphere
is the same as that due to a charge E at 0. Thus the
field outside the sphere is in this case the same as that
due to charges

„ ecu r ea n

E + y at 0, - -j at Q, e at P.

The repulsive force acting on P is equal to
ea\ e e2a

+7)f*~T^

When the point is very near the sphere we may put
/=« + #, where x is small, arid then the repulsion is
approximately equal to

Ee e2

a* ~4oT2'

and this is negative, i.e. the force is attractive unless

Thus, when the charges are given, and when P gets
within a certain distance of the sphere, P will be attracted
towards the sphere even though the sphere is charged with
electricity of the same sign as that on P. When we
recede from the sphere we reach a place where the attrac
tion changes to repulsion, and at this point there is no
force on P. Thus if P is placed at this point, it will be in

91] ELECTRICAL IMAGES AND INVERSION 157

equilibrium. The equilibrium will, however, be unstable,
for if we displace P towards the sphere the force on it
becomes attractive and so tends to bring P still nearer to
the sphere, that is to increase its displacement, while if we
displace P away from the sphere the force on it becomes
repulsive and tends to push P still further away from the
sphere, thus again increasing the displacement. This is an
example of a more general theorem due to Earnshaw that
no charged body (whether charged by induction or other
wise) can be in stable equilibrium in the electrostatic field
under the influence of electric forces alone.

90. If the potential of the sphere is given instead of
the charge, we can still use a similar method to find the
field round the sphere. Thus if the potential of the sphere
is F, then the field outside the sphere is the same as
that due to a charge Va at 0, — ea/f at Q, and e at P.

91. Sphere placed in a uniform field. As the

point P moves further and further away from 0 the
Faraday tubes due to the charge at P get to be in the
neighbourhood of the sphere more and more nearly parallel
to OP, thus when P is at a very great distance from the
sphere the problems we have just considered become in the
limit problems relating to the distribution of electricity on
a sphere placed in a uniform electric field.

Suppose that, as the charged body P travels away from
the sphere, the charge e increases in such a way that the
electric intensity at the centre of the sphere due to this
charge remains finite and equal to F, we have thus

158 ELECTRICAL IMAGES AND INVERSION [CH. V

Now consider the problem of an insulated sphere
without charge placed in this uniform field. We see by
Art. 88 that the electrification on the sphere produces the
same effect at points outside the sphere as would be pro
duced by two charges, one equal to ea/f placed at the
centre 0, the other equal to — ea/f at Q the image of P. If
we express these charges in terms of F we see that they are
equal respectively to + Fa/', when /is infinite they are also
infinite. Since OQ = az/fthe distance between these charges
diminishes indefinitely as/ increases, and we see that the
product of either of the charges into the distance between
them is equal to Fa3 and is finite. The electrification
over the surface of the sphere when placed in a uniform
field produces the same effect therefore as an electrical
system consisting of two oppositely charged bodies, placed
at a very short distance apart, the charges on the bodies
being equal in magnitude and so large that the product of
either of the charges into the distance between them is
finite. Such a system is called an electrical doublet and
the product of either of the charges into the distance
between them is called the moment of the doublet.

92. Electric field due to a doublet. Let A, B

be the two charged bodies, let e be the charge at A , — e

92] ELECTRICAL IMAGES AND INVERSION 159

that at B- let 0 be the middle point of AB, M the
moment of the doublet. Let C be a point at which the
electric intensity is required, and let the angle A 00 = 6.
The intensity at right angles to 00 is equal to

AC2 BO2

_ M sin 0
~00^'

approximately, since AO is very small compared with 00.
The intensity in the direction 00 is equal to

6 coaACO-~9coaBCO,

AC* BO2

but we have approximately

AG=OC-AOcos0,

BC =00+ BO cos 6.

Hence putting cos A CO = 1, cos BOO = 1 and using the
Binomial Theorem we find that the electric intensity
along 00 is approximately

2 A 0 ^\ _ _e_ ( _ 2BOcos

00* \ ' 00

_2eABcos0

00s

_ 2M cos 0

00s '

160 ELECTRICAL IMAGES AND INVERSION [CH. V

93. Let us now return to the case of the sphere
placed in the uniform field: the moment of the doublet
which represents the effect of the electrification over the
sphere is Fa?. Hence, when the sphere is placed in a
uniform field F parallel to PO, the intensity at a point C
is the resultant of electric intensities, F parallel to PO,
Fa*sin0/OC3 at right angles to OC, and 2 .Fa3 cos 0/0(7 3
along CO; 6 denotes the angle POO.

At the surface of the sphere where OG= a, the result
ant intensity along the outward drawn normal is

or -3Fcos0;

but by Coulomb's law, if <r is the surface density of the
electrification on the sphere,

or & = — -r-F cos 6.

4?r

Hence we see, that when an insulated conducting
sphere is placed in a uniform field, the surface density at
any point on the sphere is proportional to the distance of
that point from a plane through the centre of the sphere
at right angles to the electric intensity in the uniform
field.

On account of the concentration of the Faraday tubes
on the sphere the maximum intensity in the field is three
times the intensity in the uniform field.

94. We have hitherto supposed the electrified body
to be outside the sphere, but we can apply the same
method when it is inside. Thus, if we have a charge e

94] ELECTEICAL IMAGES AND INVERSION 161

at a point Q inside a spherical surface maintained at
zero potential, then the effect, inside the sphere, of the
electricity induced on the sphere will be the same as
that due to a charge — e . a/OQ at P where OP . OQ = a2.
The charge on the sphere is — e, since all the tubes which
start from Q end on the sphere.

Fig. 47.

If the sphere is insulated, then the charge on the
inside of the sphere and the force inside are the same
as when it is at potential zero; the only difference is
that on the outside of the sphere there is a charge equal
to e uniformly distributed over the sphere, and the field
outside is the same as that due to a charge e at the
centre.

Again, if there is a charge E on the sphere, the effect
inside is the same as in the two previous cases, only now
there is a charge E + e uniformly distributed over the
surface of the sphere raising its potential to (E + e)/a.

In all these cases the surface density of the electri
fication at any point on the inner surface of the sphere
varies inversely as the cube of the distance of that point
from P.

T. E. 11

162

ELECTRICAL IMAGES AND INVERSION

[CH. V

95. Case of two spheres intersecting at right
angles and maintained at unit potential. Let the

figure represent the section of the spheres, A and B being
their centres, and C a point on the circle in which they

Fig. 48.

intersect, CD a part of the chord common to the two
circles; then, since the spheres intersect at right angles
ACB is a right angle and CD is the perpendicular let fall
from C on AB.

Then we have by Geometry

Thus D and B are inverse points with regard to the
sphere with centre A, and A and D are inverse points
with regard to the sphere whose centre is B.

Let AC=a, BC = b, then CD.AB = AC.BC, so that

95] ELECTRICAL IMAGES AND INVERSION 163

Consider the effect of putting a positive charge at A
numerically equal to the radius AC, a positive charge
at B equal to BC, and a negative charge at D equal
to CD.

The charges at A and D will together, by Art. 86,
produce zero potential over the sphere with centre B.
For A and D are inverse points with respect to this
sphere, and the charge at D is to the charge at A as
- CD is to AC, i.e. as - BC is to AB, so that the ratio
of the charges is the same as that of those on a point
and its image, which together produce zero potential at
the sphere. Thus the value of the potential over the
surface of this sphere is that due to the charge at B, but
the charge is equal to the radius of the sphere, so that
the potential at the surface, being equal to the charge
divided by the radius, is equal to unity. Thus these
three charges produce unit potential over the sphere with
centre B; we can in a similar way show that they give
unit potential over the sphere with cerrtre A. The two
spheres then are an equipotential surface for the three
charges, and the electric effect of the conductor formed
by the two spheres, when maintained at unit potential, is
at a point outside the sphere the same as that due to the
three charges.

Capacity of the system. The charge on the system
is equal to the sum of the charges on the points inside it
which produce the same effect. Thus the capacity of the
system which, since the potential is unity, is equal to the
charge is equal to

, aJb

11—2

164

ELECTRICAL IMAGES AND INVERSION

[CH. V

96. If b is very small compared with a, the system
becomes a small hemispherical boss on a large sphere as
shown in Fig. 49. The capacity is equal to

ab
a + b —

or to

Fig. 49.

and, as in this case b/a is very small, the capacity is
approximately equal to

But

volume of boss

2 a3 volume of big sphere *

Thus we have, since a is the capacity of the large
sphere without the boss,

increase in capacity due to boss volume of boss
capacity of sphere volume of sphere '

97] ELECTRICAL IMAGES AND INVERSION 165

97. To compare the charges on the surface of
the two spheres. The charge on the spherical cap EFG
(Fig. 48) is, by Coulomb's law, equal to l/4nr of the total
normal induction over EFG. Now the total normal induc
tion is the sum of the total normal inductions due to the
charges at A, B, D. Since B is the centre of the cap
CFE the total normal induction due to B over CFE bears
the same ratio to 4?r6 (the total normal intensity over the
whole sphere) as the area of the cap CFE does to the
area of the sphere. But the area of the surface of a sphere
included between two parallel planes is proportional to
the distance between the planes, thus

area of EFC _b + BD
area of sphere 26

Hence the total normal induction over CFE due to the
charge at B

The total normal induction due to the charge A over
the closed surface CFEL is zero, therefore the total normal
induction due to A over CFE is equal in magnitude and
opposite in sign to the total normal induction over CLE,
that is, it is equal to the total normal induction over CLE
reckoned outwards from the side A. But CLE is a portion
of a sphere of which A is the centre, therefore the induction
over CLE is to 4?ra (the induction over the whole sphere
with centre A) as the area of CLE is to the area of the
sphere, that is as DL : 2a. Thus the induction due to A
over CFE is equal to

Next consider the total normal induction over CFE
due to the charge at D. Now of the tubes starting

166 ELECTRICAL IMAGES AND INVERSION [CH. V

from D as many would go to the right as to the left if it
were alone in the field, so that the induction over OFE
will be half that due to D over a closed surface entirely
surrounding it ; the latter induction is equal to 4?r times
the charge at D, i.e. to - 4?r . CD, hence the induction
due to D over the surface CFE is

-tor. CD.

Thus the total induction over CFE due to the three
charges is

DL-CD),

and the charge on CFE is therefore equal to
I/, fr2 a2 ab

4 '''-

The charge on CGE can be got by interchanging a and
b in this expression, and is thus equal to

98. In the case of a hemispherical boss on a large
sphere, b is very small compared with a ; in this case the
expression (1) becomes approximately

+--HS-'}

99] ELECTRICAL IMAGES AND INVERSION 167

This is equal to the charge on the boss. The mean
density on the boss is this expression divided by 2?r62, the
area of the surface of the boss, and is therefore

When b/a is very small the expression (2) is approxi
mately equal to a, thus the charge on the sphere is a and
the mean density is got by dividing a by 47ra2 the area of
the sphere. Thus the mean density on the sphere is

Hence the mean density on the boss is to the mean density
on the sphere as 3 : 2.

99. Since a plane may be regarded as a sphere of
infinite radius, this applies to a hemispherical boss of any
radius on a plane surface. It thus applies to the case

shown in Fig. 50. Since the mean density over the boss
is 3/2 of that over the plane, and since the area of the boss
is twice the area of its base ; there is three times as much
electricity on the surface occupied by the boss as there is,
on the average, on an area of the plane equal to the base
of the boss.

168 ELECTRICAL IMAGES AND INVERSION [CH. V

100. When b is very small compared with a, the points
B and D, Fig. 48, are close together, the distance between
them being approximately b*/a, which is small compared
with b ; the charge at B is 6, that at D is

and, when b is very small compared with a, this is
approximately equal to — b. Thus the charges at B and
D form a doublet whose moment is b3/a. The point A is
very far away and the force at B or D due to its charge
is I/a. Thus the moment of the doublet is b3 times this
force. This as far as the sphere is concerned is exactly
the case considered in Art. 93. Hence if F is the force at
the boss due to the charge A alone, the surface density at

o rr

a point P, Fig. 50, on the boss is — cos 6, where 9 is the

angle OP makes with the axis of the doublet. Now if <70
is the surface density on the plane at some distance from
the boss F ' = 47r<70. Hence, the surface density at P, a
point on the boss, is equal to

3o-0 cos 6,

where 6 is the angle OP makes with the normal to the plane.
The electric intensity due to the doublet at Q, a point
on the plane, is (Art. 92) equal to the moment of the
doublet divided by OQ3 and is at right angles to the
plane, thus the normal electric intensity at Q is

and cr, the surface density at Q, is given by the equation

103] ELECTRICAL IMAGES AND INVERSION 169

We have thus found the distribution of electricity on
a charged infinite plane with a hemispherical boss on it.

101. In the general case when the two spheres are of
any sizes the surface density on the conductor can be got
by calculating the normal electric intensity due to the
three charges. We shall leave this as an example for the
student, remarking that, since the potential of the con
ductor is the highest in the field, there can be no negative
electrification over the surface and that the electrification
vanishes along the intersection of the two spheres.

102. Effect of dielectrics. We have hitherto only
considered the case when the field due to the charge at
P was disturbed by the presence of conductors, but by
applying the principle that a solution which satisfies the
electric conditions is the only solution, we can find the
electric field in some simple cases when dielectrics are
present.

103. The first case we shall consider is that of a small
charged body placed in front of an infinite mass of uniform

dielectric bounded by a plane face. Let P be the charged
body, AB the plane separating the dielectric from air, the
medium to the right of AB being air, that to the left a

170 ELECTRICAL IMAGES AND INVERSION [CH. V

dielectric whose specific inductive capacity is K. From P
draw PN perpendicular to AB ; produce PN to P', so that
PN = P'N. Then we shall show that the field to the right
of AB can be regarded as due to e at P and a charge e'
at P', and that to the left of AB as due to e" at P; these
charges being supposed to produce the same field as if
there was nothing but air in the field.

In the first place this field satisfies the conditions that
the potential at an infinite distance is zero, also that the
induction over any closed surface surrounding P is 4-Tre,
while the induction over any closed surface not enclosing
P is zero. This is obvious if the surface is drawn entirely
to the left or entirely to the right of AB. If it crosses
this plane it can be regarded as two surfaces, one entirely
to the left bounded by the portion of the surface to the
left and the portion of the plane AB intersected by the
surface, the other entirely to the right bounded by the
same portion of the plane and the part of the surface to
the right.

The only other conditions we have to satisfy are that
along the plane AB the electric intensity parallel to the
surface is the same in the air as in the dielectric, and that
over this plane the normal polarization is the same in the
air as in the dielectric.

At a point Q in AB the electric intensity parallel to
AB is in the air

e' QN

This, since PQ = P'Q, is equal to

103] ELECTRICAL IMAGES AND INVERSION 171

The electric intensity at Q parallel to AB in the
dielectric is

this is equal to that in air if

e + e' = e" ..................... (1).

Again, the polarization at Q at right angles to AB
reckoned from right to left is in air

and that in the dielectric is

K. »PN

4>7re PQ3'
these are equal if

e-e' = Ke" ..................... (2).

Hence both the boundary conditions are satisfied if e'
and e" satisfy (1) and (2), i.e. if

l+K

K+L

The attraction of P towards the plane is equal to that
between e and e and is thus

ee K-l e*
(2PNY ~ K + 1
If K is infinite this equals

which is the same as when the dielectric to the left of AB
is replaced by a conductor.

172 ELECTRICAL IMAGES AND INVERSION [CH. V

Thus if K = 10, as is the case for some kinds of heavy
glass, the force on P when placed in front of the glass
would be about 9/11 of the attraction when P is placed
in front of a conducting plate. Inside the mass of
dielectric the tubes are straight and pass through P ; the
effect of the dielectric is, while not affecting the direction
of the electric intensity, to reduce its magnitude to 2/(l +K)
of its value in air when the dielectric is removed. The
lines of force when K=\'l are shown in Fig. 52.

Fig. 52.

104. Case of a dielectric sphere placed in a
uniform field. We have seen that, when a conducting
sphere is placed in a uniform field, the effect of the
electricity induced on the surface of the sphere can be
represented at points outside the sphere by a doublet
(see Art. 92) placed at the centre of the sphere. Since

104] ELECTRICAL IMAGES AND INVERSION 173

we have seen that the effects of a dielectric are similar
in kind though different in degree to those due to a
conductor, we are led to try if the disturbance produced
by the presence of the sphere cannot be represented at
a point outside the sphere by a doublet placed at its
centre. With regard to the field inside the sphere we
have as a guide the result obtained in the last article, that
in the case when the radius of the sphere is infinitely
large the field inside the dielectric is not altered in
direction but only in magnitude by the dielectric.

We therefore try if we can satisfy the conditions
which must hold when a sphere is placed in a uniform
electric field by supposing the field inside the sphere to
be uniform.

Let the uniform field before the insertion of the
sphere be one where the electric intensity is horizontal
and equal to H.

After the insertion of the sphere let the field outside
consist of this uniform field plus the field due to a
doublet whose moment is M placed at the centre of the
sphere, the dielectric being removed.

Inside the sphere let the intensity be horizontal and
equal to H'.

We shall see that it is possible to satisfy the con
ditions of the problem by a proper choice of M and H'.

The field at P due to the doublet is, by Art. 92, equiva-

2M
lent to an intensity jjp^ cos 6 along OP, and an intensity

M
-syp-z sin 0 at right angles to it, where 6 is the angle OP

174 ELECTRICAL IMAGES AND INVERSION [CH. V

makes with the direction of the uniform electric intensity.
Thus at a point Q just outside the sphere the intensity
tangential to the sphere is equal to

H sin 6 — —^ sin Q.
a?

where a is the radius of the sphere.

The intensity in the same direction at a point close
to Qlout just inside the sphere is

H'smQ.
The normal intensity at Q outside the sphere is

Hcos6 + — — cos 0,
a3

and at a point just inside the sphere it is H! cos 6.

The first boundary condition is that the tangential
intensity at the surface of the sphere must be the same
in the air as in the dielectric ; this will be true if

Hsm6 -- , sin 6 = H' sin 0,
a3

or H--3 = H' ..................... (1).

The second boundary condition is that the normal
polarization at the surface of the sphere must be the
same in the air as in the dielectric, thus

m-H' cos e,

4?r
H + =KH' ..................... (2).

105] ELECTRICAL IMAGES AND INVERSION 175

Equations (1) and (2) will be satisfied, if

and if

Thus, since, if H' and M have these values the con
ditions are satisfied, this will be the solution of the
problem. We see that the intensity inside the sphere
is %j(K + 2) of that in the original field, so that the in
tensity of the field is less inside the sphere than outside ;
on the other hand the number of Faraday tubes which
pass through unit area inside the sphere is 3K/(K +2)
times the number passing through unit area in the
original uniform field. When K is very great 3K/(K + 2)
is approximately equal to 3, so that the Faraday tubes
in this case will be 3 times as dense inside the sphere
as they are at a great distance away from it. This illus
trates the crowding of the Faraday tubes to the sphere.

The diagram of the lines of force for this case was
given in Fig. 41.

Method of Inversion.

105. This is a method by which, when we have
obtained the solution of any problem in electrostatics,
we can by a geometrical process obtain the solution of
another.

Definition of inverse points. If 0 is a fixed point,
P a variable one, and if we take P' on OP, so that

176

ELECTRICAL IMAGES AND INVERSION

[CH. V

where & is a constant, then P' is defined to be the inverse
point of P with regard to 0, while 0 is called the centre
of inversion, and k the radius of inversion.

If the point P moves about so as to trace out a surface,
then P' will trace out another surface which is called the
surface inverse to that traced out by P.

We shall now proceed to prove some geometrical pro
positions about inversion.

106. The inverse surface of a sphere is another
sphere. Let 0 be the centre of inversion, P a point
on the sphere to be inverted, C the centre of this sphere.

Fig. 53.

Let the chord OP cut the sphere again in P, let Q be
the point inverse to P, Q' the point inverse to P', R the
radius of the sphere to be inverted, then

OP.OQ = k\

But

and thus

£2
OG*-R

i OP' ;

similarly

106] ELECTRICAL IMAGES AND INVERSION 177

Thus OQ . OQ' - (QG^_Rif OP . OP'

Thus OQ bears a constant ratio to OP' ; hence the
locus of Q is similar to the locus of P', and is therefore a
sphere. Thus a sphere inverts into a sphere. If

k* = OC2 - .ft2
the sphere inverts into itself.

To find the centre of the inverse sphere, let the dia
meter OC cut the sphere to be inverted in A and B. Let
A', Bf be the points inverse to A and B respectively
and 0' the centre of the inverted sphere ; then

2 \OC -11 ' 00 +

-*• °°
'OC*-W

If D is the point where the chord of contact of tangents
from 0 to the sphere cuts OC, then

Hence D inverts into the centre of the sphere.
The radius of the inverse sphere

T. E. 12

178

ELECTRICAL IMAGES AND INVERSION

[CH. V

107. Since a plane is a particular case of a sphere
a plane will invert into a sphere ; this can be proved
independently in the following way :

Fig. 54.

Let AB be the plane to be inverted, P a point on that
plane, N the foot of the perpendicular let fall from 0 on
the plane and Q and N' the points inverse to P and N
respectively. Then since

OQ.OP^ON'.ON

OQ ON m

ON' ~ OP •

thus the two triangles QON, PON have the angle at 0
common and the sides about this angle proportional, they
are therefore similar, and the angle OQN' is equal to the
angle ON P. Hence OQN' is a right angle and therefore
the locus of Q is a sphere on ON' as diameter.

108] ELECTRICAL IMAGES AND INVERSION 179

108. Let 0 be the centre of inversion, PQ two points
and P'Q' the corresponding inverse points.

Then OP'_OQ.

thus the triangles POQ, Q'OP' are similar, so that

PQ_P'Q'
OP~ OQ"

Fig. 55.

If we have a charge e at Q, and a charge e' at Qf,
then if VP is the potential at P due to the charge at Q,
and F'P' the potential at P' due to the charge at Q',
P f>' P P'

V ' Vr* ----- - — _! _ •

V * PQ' PQ'~OP' OQ'
Take e:e'=OQ:k (1),

k
then V f — Vp j^p, .

If we have any number of charges at different points
and take the inverse of these points and place there
charges given by the expression (1), then, if VP be the
potential at a point P due to the original assemblage of
charges, VP> the potential at P' (the point inverse to P)
due to the charges on the inverted system,

V -V r A.

VP'~- VP Qp"

12—2

180 ELECTRICAL IMAGES AND INVERSION [CH. V

Thus, if the original assemblage of charges produces a
constant potential V over a surface S, the inverted system

Vk
will produce a potential -r-p? at a point P' on the inverse

of S. Hence, if we add to the inverted system a charge
— kV at the centre of inversion, the potential over the
inverse of S will be zero.

If the charges on the original system are distributed
over a surface instead of being concentrated at points the
charges on the inverted system will also be distributed over
a surface. Let cr be the surface density at Q, a place on
the original system, cr' the surface density at Q', the corre
sponding point on the inverted system, a. a small area at Q,
a the area into which it inverts ; then by (1)

ffa : a a! =OQ:k

and, since a and of are similar figures,

a : a' = OQ2 : OQ\

Hence a- : a = OQ'2 : WQ

kOQ k*
and thus a ~* 0^~*(vyi W-

This expression gives the surface density of the inverted
figure in terms of that at the corresponding point of the
original figure.

109. As an example of the use of the method of
inversion let us invert the system consisting of a sphere
with a uniform distribution of electricity over it, the
surface density being F/4?ra; where a is the radius of
the sphere. We know in this case that the potential is
constant over the sphere and equal to V. Take the
centre of inversion outside the sphere and choose the
radius of inversion so that the sphere inverts into itself.

110] ELECTRICAL IMAGES AND INVERSION 181

Then, if to the inverted system we add a charge - kV
at the centre of inversion the inverse sphere will be at
potential zero. By equation (2) a-' the surface density in
the inverted system at Q' is given by the equation

If we put e = — kV, this equals

where C is the centre of the sphere.

Thus a charge e at 0 induces on the sphere at zero
potential a distribution of electricity such that the surface
density varies inversely as the cube of the distance from
0. In this way we get by inversion the solution of the
problem which we solved in Art. 87 by the method of
images.

110. As an example illustrating the uses of the
method of inversion as well as that of images, let us
consider the solution, by the method of images, of a
charged body placed between two infinite conducting
planes maintained at potential zero.

Let P be the charged point, AB and CD the two planes
at potential zero, e the charge at P. Then if we place
a charge — e at P' where P' is the image of P in AB the
potential over AB will be zero, it will not however be
zero over CD', to make the potential over CD zero we
must place a charge — e at Q, the image of P in CD, and a
charge e at Qlf the image of P' in CD. These two charges
will however disturb the potential of AB; to restore zero
potential to AB we must introduce a charge +e at Plt
the image of Q in AB, and a charge - e at P", the image

182 ELECTRICAL IMAGES AND INVERSION [CH. V

of Qj in AB. The charges at Pl and P" will disturb the
potential over the plane CD ; to restore it to zero we must
place a charge — e at Qf, the image of Pl in CD, and a
charge +e at Q2> the image of P" in CD, and so on ; we get
in this way two infinite series of images to the right of
AB and to the left of CD.

The images to the right of AB are (1) charges — e, at
P', P", P". . . ; and (2) charges + e, at Plt P2, P3 . . . .

Now P" is the image of Ql in ^15, which is the image
of P' in CD and hence

PP" = FQ, = FE + EP' = 2FE + PP';

thus FP"-FP' = P'P" = 2FE = 2c, if c is the distance
between the plates.

Fig. 56.

Similarly P'P" = P"P'" = . . . = 2c and we can show in a
similar way that PP1=P1P2=P2P3= . . . = 2c. Thus on the
right of AB we have an infinite series of charges equal to
— e at the distance 2c apart, beginning at P' the image of

110] ELECTRICAL IMAGES AND INVERSION 183

P in AB, and a series of positive images at the same dis
tance 2c apart, beginning at Pl} a point distant 2c from P.

Similarly to the left of CD we have an infinite series
of images with the charge - e at the distance 2c apart,
beginning at Q, the image of P in CD, and an infinite series
of images each with the charge + e, at points at a distance
2c apart, beginning at Qlt a point distant 2c from P.

Now invert this system with respect to P. The two
planes invert into two spheres touching each other at P,
and maintained at a potential — e/k, the images to the
right of AB invert into a series of charged points inside
the sphere to the right of P and the images to the left of
CD invert into a system of charged points inside the
sphere to the left of P.

The system of charged points inside the spheres will
produce a constant potential - e/k over the surface of the
spheres, and therefore at a point outside the spheres the
electric field due to the two spheres in contact will be the
same as that due to the system of the electrified points.

If a, b are the radii of the spheres into which the
planes AB, CD invert, and if PF=d, then

26= - -,
c — d

Consider now the series of images to the right of AB.
The series of positive charges at the distance 2c apart
invert into a series of charges inside the sphere, whose
radius is a, of magnitudes

ek ek ek

2c' 4c' 60'""

184 ELECTKICAL IMAGES AND INVERSION [CH. V

since

charge at inverted point
charge at original point

= _ k _
distance of original point from centre of inversion '

The series of negative images at the distance 2c apart
invert into a series of negative charges

ek ek ek

Similarly, inside the sphere into which the plane CD
inverts, we have a series of positive charges

ek ek ek
2c' 4c' 6c''"'

and a series of negative ones

ek ek ek

" 2 (c - d) ' ~ 4c-2d' ~6c-2d'""

Thus El} the sum of the charges on the points inside
the first sphere, is given by the equation

.
4c 6c

fl. J_ J^ V m

\2d + 2c+2rf + 4c + 2rf^ ")}"

while EZ, the sum of the charges inside the second sphere,
is given by the equation

2c 4c 6c

110] ELECTRICAL IMAGES AND INVERSION 185

Rearranging the terms, we may write

d d d )

_ _

l~~ 2 d

+ ...

Expanding the expressions for E1 and E9 in powers
of d/c we get

1 , /I d „ d* d* \ .

-?*+?*-y*« •>•;-•

--i*!te+«4+*&+?ft+

where £fn = ^ 4- ^ + 3^ + ^ + • • • •

The values of 8n are given in De Morgan's Differential
and Integral Calculus, p. 554,

&=£ = 1-645, &
b

5, = 1-202, $

^4 = ^= 1*082, >Sf7 = 1-008.
Since El can be got from E2 by writing c-cZ for d, we get

(4).

Now, the total charge spread over the surface of the
first sphere is equal to the sum of the charges at the
points inside the sphere as these produce the same effect
at external points as the electrification over the surface
of the sphere : thus, El will be the charge on the first

186 ELECTRICAL IMAGES AND INVERSION [CH. V

sphere, Ez that on the second. If V is the potential of
the spheres

= Va(l — —

\ a + ba + 2b 2(a

E.-Vbll-

E,= Va\l-
and also

3(a + 6)3a + 46 '")
a a a a

\yj>

a + b 2a + b 2 (CL + 6) Set + 26
a a ^

(R\

3 (a + 6) 4a-f 36 " /

V°/'

(7)

Cf-ii-iQfi/iG'i /Q\

>S2 + ( :rTT ]89+( r-rgj 04+ — I- (8)-

The value of ^ can be got by interchanging a and 6
in the expressions (7) and (8).

Let us now consider some special cases. Take first
the case when a — b, then from equation (5) we have

_ 11_11_11_
~23~45~67 '

= Fa log 2,

the logarithm being the Napierian logarithm.
Since log 2 = '693

El = '693 Va.

110] ELECTRICAL IMAGES AND INVERSION 187

The charge on the second sphere is also E^\ thus the total
charge on the two spheres is

1-386 Fa.

When F*= 1 the charge on the two spheres is equal to
the capacity of the system; hence the capacity of two
equal spheres in contact is 2a log 2 or l'386a.

If the spheres had been an infinite distance apart, the
capacity of the two would have been 2a; if there had
only been one sphere the capacity would have been a.

We can find from this the work done on an uncharged
sphere when it moves under the attraction of a charged
sphere of equal radius from an infinite distance into con
tact with the charged sphere. Let a be the radius of each
sphere and e the charge on the charged sphere ; then,
when the spheres are at an infinite distance apart, the
potential energy is e*/2a and when the spheres are in
contact the potential energy is e2/2 x l"386a. Hence the
work done by the electric field while the uncharged
sphere falls from an infinite distance into contact with
the charged sphere is

le2 f 1 ] .#

2 a ( 1-386] a '

If one sphere has a charge E, the other the charge e,
then, when they are at an infinite distance apart, the

potential energy is — [E2+ e2}.

When the spheres are in contact the potential energy

188 ELECTRICAL IMAGES AND INVERSION [CH. V

Hence the potential energy is greater in the second
case than in the first by

If E = e, this is equal to

This is the work required to push the spheres together
against the repulsions exerted by their like charges.

The expression (9) vanishes when E/e is approximately
5 or 1/5 ; in this case the potential energy is the same
when the spheres are in contact as when they are an
infinite distance apart; thus no work is spent or gained
in bringing them together. The attraction due to the
induced electrification on the average balances the re
pulsion due to the like charges.

The next case we shall consider is where one sphere is
very large compared with the other. Let 6 be very large
compared with a. Now by (8) we have

or approximately, when b/a is large,

Va?

6 6

= 1-646

110] ELECTRICAL IMAGES AND INVERSION 189

Interchanging a and b in (7) we get

or approximately, when b/a is large,

The mean surface density over the small sphere is

= 1-645.
6

The mean surface density over the large sphere is
approximately

47T&2 47T&

and hence the mean surface density on the small sphere is
7r2/6 or 1'645 times that on the large sphere. We saw in
Art. 98 that, when a small hemisphere was placed on a
large sphere, the mean density on the hemisphere was
1-5 times that on the sphere.

Since a plane may be regarded as a sphere of infinite
radius, we see that if a sphere of any size is placed on a
conducting plane the mean surface density of the elec
tricity on the sphere is 7r2/6 of that on the plane.

We have

Vb jl + 2-404 5LJ approximately.

190

ELECTEICAL IMAGES AND INVERSION [CH. V

Thus, the capacity of the system of two spheres is
approximately

2-404-
We have thus

Increase of capacity due to small sphere
Capacity of large sphere

_ 9.404 v°lume °f small sphere
volume of large sphere'

Thus in this case, as in that discussed in Art. 96, the
increase of capacity due to the small body is proportional
to the volume of the small body.

From this result we can deduce the work done on a
small uncharged sphere of radius a when it moves from
an infinite distance up to a large sphere of radius b with
a charge E.

For, when they are at an infinite distance apart, the
potential energy is equal to

2 6 '

when the spheres are in contact the potential energy is

1 E*

2 ( n'3

b 1+2-404?-

The work done on the small sphere by the electrical
forces is the difference between these expressions, or ap
proximately,

1-202 J.

CHAPTER VI

MAGNETISM

111. A mineral called 'lodestone' or magnetic oxide
of iron, which is a compound of iron and oxygen, is often
found in a state in which it possesses the power of at
tracting small pieces of iron such as iron filings; if the
lodestone is dipped into a mass of iron filings and then
withdrawn, some of the iron filings will cling to the lode-
stone, collecting in tufts over its surface. The behaviour
of the lodestone is thus in some respects analogous to that
of the rubbed sealing-wax in the experiment described in
Art. 1. There are however many well-marked differences
between the two cases; thus the rubbed sealing-wax attracts
all light bodies indifferently, while the lodestone does not
show any appreciable attraction for anything except iron
and, to a much smaller extent, nickel and cobalt.

If a long steel needle is stroked with a piece of lode-
stone, it will acquire the power possessed by the lodestone
of attracting iron filings; in this case the iron filings
will congregate chiefly at two places, one at each end of
the needle, which are called the poles of the needle.

The piece of lodestone and the needle are said to be
magnetized ; the attraction of the iron filings is an example
of a large class of phenomena known as magnetic. Bodies
which exhibit the properties of the lodestone or the needle

192 MAGNETISM [CH. VI

are called magnets, and the region around them is called
the magnetic field.

The property of the lodestone was known to the
ancients, and is frequently referred to by Pliny and
Lucretius. The science of Magnetism is indeed one of the
oldest of the sciences and attained considerable develop
ment long before the closely allied science of Electricity ;
this was chiefly due to Gilbert of Colchester, who in his
work De Magnete published in 1600 laid down in an
admirable manner the cardinal principles of the science.

112. Forces between Magnets. If we take a
needle which has been stroked by a lodestone and suspend
it by a thread attached to its centre it will set itself so as
to point in a direction which is not very far from north
and south. Let us call the end of the needle which
points to the north, the north end, that which points to
the south, the south end, and let us when the needle is
suspended mark the end which is to the north; let us
take another needle, rub it with the lodestone, suspend it
by its centre and again mark the end which goes to the
north. Now bring the needles together; they will be
found to exert forces on each other, and the two ends
of a needle will be found to possess sharply contrasted
properties. Thus if we place the magnets so that the two
marked ends are close together while their unmarked ends
are at a much greater distance apart, the marked ends will
be repelled from each other ; again, if we place the magnets
so that the two unmarked ends are close together while
the marked ends are at a much greater distance apart,
the unmarked ends will be found to be repelled from
each other ; while if we place the two magnets so that the

113] MAGNETISM 193

marked end of one is close to the unmarked end of the
other, while the other ends are much further apart, the
two ends which are near each other will be found to be
attracted towards each other. We see then that poles of
the same kind are repelled from each other, while poles
of opposite kinds are attracted towards each other. Thus
the two ends of a magnet possess properties analogous to
those shown by the two kinds of electricity.

113. We shall find it conduces to brevity in the
statement of the laws of magnetism to introduce the term
charge of magnetism, and to express the property possessed
by the ends of the needles in the preceding experiment
by saying that they are charged with magnetism, one end
of the needle being charged with positive magnetism, the
other end with negative. We regard the end of the needle
which points to the north as having a charge of positive
magnetism, the end which points to the south as having
a charge of negative magnetism. It will be seen from the
preceding experiment that two charges of magnetism are
repelled from or attracted towards each other according as
the two charges are of the same or opposite signs. It must
be distinctly understood that this method of regarding
the magnets and the magnetic field is only introduced
as affording a convenient method of describing briefly
the phenomena in that field and not as having any
significance with respect to the constitution of magnets
or the mechanism by which the forces are produced : we
saw for example that the same terminology afforded a
convenient method of describing the electric field, though
we ascribe the action in that field to effects taking place
in the dielectric between the charged bodies rather than
in the charged bodies themselves.

T. E. 13

194 MAGNETISM [CH. VI

114. Unit Charge of Magnetism, often called pole
of unit strength. Take two very long, thin, uniformly
magnetized needles, equal to each other in every respect
(we can test the equality of their magnetic properties
by observing the forces they exert on a third magnet),
let A be one end of one of the magnets, B the like end
of the other magnet, place A and B at unit distance
apart in air, the other ends of the magnets being so far
away that they exert no appreciable effect in the region
about A and B : then each of the ends A and B is said
to have a unit charge of magnetism or to be a pole of
unit strength when A is repelled from B with the unit
force. If the units of length, mass and time are re
spectively the centimetre, gramme and second the force
between the unit poles is one dyne.

A charge of magnetism equal to 2, or a pole of
strength 2, is one which would be repelled with the force
of two dynes from unit charge placed at unit distance in
air.

If m and m! are the charges on two ends of two
magnets (or the strengths of the two poles), the distance
between the charges being the unit distance, the repulsion
between the charges is mm dynes. If the charges are of
opposite signs mm' is negative : we interpret a negative
repulsion to mean an attraction.

115. Coulomb by means of the torsion balance suc
ceeded in proving that the repulsion between like charges
of magnetism varies inversely as the square of the dis
tance between them. We shall discuss in Art. 132 a more
delicate and convenient method of proving this result.

Since the forces . between charges of magnetism obey

118] MAGNETISM 195

the same laws as those between electric charges we can
apply to the magnetic field the theorems which we proved
in Chap. n. for the electric field.

116. The Magnetic Force at any point is the
force which would act on unit charge if placed at this
point, the introduction of this charge being supposed not
to influence the magnets in the field.

117. Magnetic Potential. The magnetic potential
at a point P is the work which would be done on unit
charge by the magnetic forces if it were taken from P to
an infinite distance. We can prove as in Art. 17 that the
magnetic potential due to a charge m at a distance r from
the charge is equal to m/r.

118. The total charge of Magnetism on any
magnet is zero. This is proved by the fact that if a
magnet is placed in a uniform field the resultant force upon
it vanishes. The earth itself is a magnet and produces
a magnetic field which may be regarded as uniform over
a space enclosed by the room in which the experiments
are made. To show the absence of any horizontal resultant
force on a magnet, we may mount the magnet on a piece
of wood and let this float on a basin of water, then though
the magnet will set so as to point in a definite direction,
there will be no tendency for the magnet to move towards
one side of the basin. There is a couple acting on the
magnet tending to twist it so that the magnet sets in
the direction of the magnetic force in the field, but there
is no resultant horizontal force on the magnet. The
absence of any vertical force is shown by the fact that
the process of magnetization has no influence upon the

13-2

196 MAGNETISM [CH. VI

weight of a body. Either of these results shows that the
total charge on the body is zero. For let ml) m2) m3, &c.
be the magnetic charges on the body, F the external
magnetic force, then the total force acting on the body in
the direction of F is

This, since the field is uniform, is equal to

As this vanishes 2m = 0, i.e. the total charge on the
body is zero. Hence on any magnet the positive charge
is always equal to the negative one.

When considering electric phenomena we saw that it
was impossible to get a charge of positive electricity with
out at the same time getting an equal charge of negative
electricity. It is also impossible to get a charge of posi
tive magnetism without at the same time getting an
equal charge of negative magnetism ; but whereas in the
electrical case all the positive electricity might be on one
body and all the negative on another, in the magnetic
case if a charge of positive magnetism appears on a body
an equal charge of negative magnetism must appear on the
same body. This difference between the two cases would
disappear if we regarded the dielectric in the electrical
case as analogous to the magnets; the various charged
bodies in the electrical field being regarded as portions of
the surface of the dielectric.

119. Poles of a Magnet. In the case of very long
and thin uniformly magnetized pieces of iron and steel
we approximate to a state of things in which the magnetic
charges can be regarded as concentrated at the ends of
the magnet, which are then called its poles ; the positive

121] MAGNETISM 197

magnetism being concentrated at the end which points to
the north, which is called the positive pole, the negative
charge at the other end, called the negative pole.

In general however the magnetic charges are not
localized to such an extent as in the previous case, they
exist more or less over the whole surface of the magnet ;
to meet these cases we require a more extended definition
of ' the pole of a magnet.'

Suppose the magnet placed in a uniform field, then
the forces acting on the positive charges will be a series
of parallel forces all acting in the same direction, these
by statics may be replaced by a single force acting at a
point P called the centre of parallel forces for this system
of forces. This point P is called the positive pole of
the magnet. Similarly the forces acting on the negative
charges may be replaced by a single force acting at a
point Q. This point Q is then called the negative pole
of the magnet. The resultant force acting at P is by
statics the same as if the whole positive charge were
concentrated at P ; this resultant is equal and opposite to
that acting at Q.

120. Axis of a Magnet. The axis of a magnet is
the line joining its poles, the line being drawn from the
negative to the positive pole.

121. Magnetic Moment of a Magnet is the pro
duct of the charge of positive magnetism multiplied by
the distance . between the poles. It is thus equal to the
couple acting on the magnet when placed in a uniform
magnetic field where the intensity of the magnetic force
is unity, the axis of the magnet being at right angles to
the direction of the magnetic force in the uniform field.

198 MAGNETISM [CH. VI

122. The Intensity of Magnetization is the mag
netic moment of a magnet per unit volume. It is to be
regarded as having direction as well as magnitude, its
direction being that of the axis of the magnet.

123. Magnetic Potential due to a Small Mag
net. Let A and B, Fig. 57, represent the poles of a
small magnet, m the charge of magnetism at B, — m that

Fig. 57.

at A. Let 0 be the middle point of AB. Consider the
magnetic potential at P due to the magnet AB. The

(YV\

magnetic potential at P due to m at B is -^- , that due

to — m at A is — j-^ , hence the magnetic potential at
AJr

P due to the magnet is

m m
T$P~~AP'

From A and B let fall perpendiculars AM and BN
on OP : since the angles BPO, APO are very small and
the angles at M and N are right angles, the angles

124] MAGNETISM 199

PEN and PAM will be very nearly right angles, so that

approximately

= PO-ON,

m m
Then r=r=r

BP AP PO-ON PO + ON

2m. ON
"OP2 - ON* '
and this, since ON is very small compared with OP, is

approximately equal to

2m. ON

OP2

_ mAB cos 6
OP2 '
where 6 is the angle PO-B.

If M is the magnetic moment of the magnet

M=mAB,

hence the potential due to the magnet is equal to

M cos d
OP2 '

124. Resolution of Small Magnets.

We shall first prove that the moment of a small
magnet may be resolved like a force, i.e. if the moment
of the magnet is M, and if a force M acting along the
axis of the magnet be resolved into forces M^ M^, Ms, &c.
acting in directions OLlf OL2, OL3, &c., where 0 is the
point midway between the poles, then the magnetic
action of the original magnet at a distant point is the
same as the combined effects of the magnets whose
moments are Ml} Mz, Ms> &c., and whose axes are along
OLlt OL2, OL,, &c.

200 MAGNETISM [CH. VI

Now suppose a force M in the direction AB, Fig. 57, is
the resultant of the forces Mlf M9, M3 in the directions
OB1} OB2) OB3> &c., let OBlt OB2, OB3 make angles 0lt
09, 03 with OP, then

M cos 0 = M1 cos 0l + M2 cos 02 + . . . ,
M cos 6 _ M, cos 0l

cos

O OP2

Now Ml cos #!/OP2 is the magnetic potential at P
due to the magnet whose moment is Ml and whose axis
is along OBl} Jf2cos02/OP2 is the potential due to the
magnet whose moment is M2 and whose axis is OB2, and
so on;, hence we see that the original magnet may be
replaced by a series of magnets, the original moment being
the resultant of the moments of the magnets by which
the magnet is replaced. In other words, the moment
of a small magnet may be resolved like a force.

By the aid of this theorem the problem of finding
the force due to a small magnet at any point may be
reduced to that of finding the force due to a magnet at
a point on its axis produced, and at a point on a line
through its centre at right angles to its axis.

125. To find the magnetic force at a point on
the axis produced. Let AB, Fig. 58, be the magnet,
P the point at which the force is required. The magnetic
force at P due to the charge m at B is equal to

(OP -OB)*'
The magnetic force due to — m at A is equal to

m
~ (OP+OBf

126] MAGNETISM 201

The resultant magnetic force at P is equal to
m m lm.OB.OP

(OP - OB)* ~ (OP + OB)* ~ (OP* -

_ 4<mOB . OP
OP*

approximately, since OB is small compared with OP.
Q

Fig. 58.

If M is the moment of the magnet M = 2mOB, thus
the magnetic force at P is equal to

OP3'

The direction of this force is along OP.

126. To find the magnetic force at a point Q
on the line through O at right angles to AB. Since
Q is equidistant from A and B, Fig. 58, the forces due
to A and B are equal in magnitude; the one being

202 MAGNETISM [CH. VI

a repulsion, the other an attraction. The resultant of
these forces is equal to

2m OB M

_
BQ* ~BQ~

since BQ is approximately equal to OQ.

The direction of this force is parallel to BA and at
right angles to OQ.

If Q, a point on the line through 0 at right angles
to AB, is the same distance from 0 as P, a point on AB
produced, we see from these results that the force at P is
twice that at Q. This is the foundation of Gauss's method
(see Art. 132) of proving that the force between two poles
varies inversely as the square of the distance between them.

127. Magnetic force due to a small magnet at
any point. Let AB, Fig. 59, represent the small magnet,

Fig. 59.

let M be its moment, 0 its centre, P the point at which
the force is required, let OP make an angle 6 with AB,
the axis of the magnet. By Art. 124 the effect of M is

127] MAGNETISM 203

equivalent to that of two magnets, one having its axis
along OP and its moment equal to M cos 6, the other
having its axis at right angles to OP and its moment
equal to M sin 0. Let OP = r.

The force at P due to the first is, by Art. 125, along
OP and equal to 2M cos #/r3, the force at P due to the
second magnet is at right angles to OP and equal to
Msm0/rs, hence the force due to the magnet AB at
P is equivalent to the forces

2Jfcos<9 . nD
— along OP,

T M sin 6 . , , , ^ -r,

and — — — at right angles to OP.

Let the resultant magnetic force at P make an angle
<f> with OP, then

Msm0

tan 6 = 2777 ^ = i tan 6.

r 2M cos 6 2

Let the direction of the resultant force at P cut AB
produced in T, draw TN at right angles to OP, then

TN
PN'

and since tan <£ = \ tan 0, P^ = 20N. Thus ON= JOP.
Thus, to find the direction of the magnetic force at P,
trisect OP at N, draw NT at right angles to OP to cut
AB produced in T, then PT will be the direction of the
force at P.

204 MAGNETISM [CH. VI

The magnitude of the resulting force is

- V4cos20 + sin2<9 = ~ Vl + 3 cos20 ;

for a given value of r it is greatest when 6 — 0 or TT, i.e. at
a point along the axis, and least when 6 = Tr/2 or 3-7T/2,
i.e. at a point on the line at right angles to the axis.
The maximum value is twice the minimum one.

The curves of constant magnetic potential are repre
sented by equations of the form

cos 6 ~

— -°>

the lines of force which cut the equipotential curves at
right angles are given by the equations

where C is a variable parameter.

The radius of curvature of the line of force at a point
P can easily be proved to equal

3sin<£ (l + sin2</>)'

where <£ is the angle the line of force makes with OP.
Thus the radius of curvature at points on the line bisecting
the magnet at right angles is one-third of the distance of
the point from the magnet.

128. Couple on a Magnet in a Uniform Mag
netic Field. If a magnet is placed in a uniform field
the couple acting on the magnet, and tending to twist
it about a line at right angles both to the axis of the
magnet and the force in the external field, is

MHsiu0,
where M is the moment of the magnet, H the force in

129] MAGNETISM 205

the uniform field, and 0 the angle between the axis of
the magnet and the direction of the force.

Let AB be the magnet, the negative pole being at A,
the positive one at B. Then if ra is the strength of
the pole at B, the forces on the magnet are a force mH
at B in the direction of the external field and an equal
and opposite force at A. These two forces are equivalent
to a couple whose moment is HmNM, where NM is the
distance between the lines of action of the two forces.
But NM = AB sin 0, if 0 is the angle between A B and
H\ hence the couple on the magnet is
HmAB sin 0 = HM sin 0.

129. Couples between two Small Magnets.

Let AB, CD, Fig. 60, represent the two magnets; M,
Mf their moments ; r the distance between their centres
0, 0'. Let AB, CD make respectively the angles 0, 0'
with 00'.

Fig. 60.

Consider first the couple on the magnet OD.
The magnetic forces due to AB are

2Mcos0 ,

- along

at right angles to 00'.

206 MAGNETISM [CH. VI

These may be regarded as constant over the space
occupied by the small magnet CD.

The couple on CD tending to produce rotation in
the direction of the hands of a watch, due to the first
component, is

2Jfcos<9 .

M sin 9 ,

that due to the second is

M sin 0

- — M cos 9 \
r3

hence the total couple on CD is

MM'

— — (2 cos 9 sin & + sin 9 cos 9').

This vanishes if tan & — — \ tan 0, i.e. if CD is along
the line of force due to AB, see Art. 127.

We may show in a similar way that the couple on AB
due to CD tending to produce rotation in the direction of
the hands of a watch is

— — (2 cos ff sin 9 + sin & cos 9).

For both these couples to vanish, 0 = 0 or TT, 0' = 0

or TT, or 0 = + — 9' = ± ^ , so that the axes of the
- 2

magnets must be parallel to each other, and either
parallel or perpendicular to the line joining the centres
of the two magnets.

We shall find it convenient to consider four special
positions of the two magnets as standard cases.

129] MAGNETISM 207

CASE I.

Fig. 61.
Q = 0, & — 0, couples vanish, equilibrium stable.

CASE II.

Ji D

Fig. 62.

6 = — , 0' = — , couples vanish, equilibrium unstable.

CASE III.

A B

Fig. 63.

0 = 0, 6'= ^ , couple on (7D = — -r- , couple on ^1 5 = — ^- .

When the magnets are arranged as in this case, AB
is said to be ' end on ' to CD, while CD is ' broadside on '

CASE IV.

C D

Fig. 64.

0 = , 0'=0, couple on CD =~ , couple

208 MAGNETISM [CH. VI

In this case AB is broadside on to CD. We see that
the couple exerted on CD by AB is twice as great when
the latter is end on as when it is broadside on.

It will be noticed that the couples on AB and CD
are not in general equal and opposite; at first sight it
might appear that this result would lead to the absurd
conclusion that if two magnets were firmly fastened to
a board, and the board floated on a vessel of water, the
board would be set in rotation and would spin round
with gradually increasing velocity. The paradox will
however be explained if we consider the forces exerted
by one magnet on the other.

130. Forces between two Small Magnets. Let

AB, CD (Fig. 60) represent the two magnets, 0, 0' the
middle points of AB, CD respectively, 9, 6' the angles
which AB, CD respectively make with 00'. Let c£ be
the angle DOO', r= 00'] m, me the strengths of the poles
of AB and CD.

The force due to the magnet AB on the pole at D
consists of the component

cos (6 - (/>),

along OD, and

Mm . /zl ,N
sin (0 - </>),

at right angles to OD.

These are equivalent to a force equal to

2Mm cos (6 - <£) cos $ Mm' sin (6 - <ft) sin eft
OD* OD8

130] MAGNETISM 209

along 00' , and a force equal to

2Mm' cos (0 — <f>) sin <f> Mm' sin (6 — (/>) cos $
OD3 OD*

acting upwards at right angles to 00'.

Neglecting squares and higher powers of CD/ 00' we
have

cos (/>=!, sin cj) = — sin 0',

Substituting these values we see that the force exerted
by AB on D is approximately equivalent to a component

2Mm' cos 0 _ SMm' CD cos 6 cos 0' 3 Ifm' CD sin (9 sin 0'
v* r4 ^2" r4

along 00', and a component

7 sin d 3 Mm' CD sin 0 cos 0' 3 Mm'CDcos0sm0'

r3 2 r4 ~ 2~ r4

acting upwards at right angles to 00'.

We may show in a similar way that the force exerted
by A B on C is equivalent to a component

_ 23/w'cos 0 _ 3Mm'CDcos0cos0' 3 Mm' CD sin<9 sintf7
r3 "T4"" +2~ ^~

along 00', and a component

Mm' sin ^ 3 Mm' CD sin 0 cos 6>x 3 Mm7 CD cos (9 sin 6>7
r3 2 r4 "*" 2 " r4 '

acting upwards at right angles to 00'.

Hence the force on the magnet CD, which is the
T. E. 14

210 MAGNETISM [CH. VI

resultant of the forces acting on the poles C, D, is equi
valent to a component

^7— (2 cos 0 cos 6' - sin 0 sin 0'),

along 00', and a component

— - — (sin 6 cos 0' + cos 0 sin 0'),

acting upwards at right angles to 00'.

The force on the magnet AB is equal in magnitude
and opposite in direction to that on CD.

If we consider the two magnets as forming one system,
the two forces at right angles to 00' are equivalent to a
couple whose moment is

— - — (cos 0 sin 0' + sin 0 cos 0'),

this couple is equal in magnitude and opposite in direction
to the algebraical sum of the couples on the magnets AB,
CD found in Art. 129 : this result explains the paradox
alluded to at the end of that article.

131. Force between the Magnets in the four
standard positions. In the positions described in Art.
129, the forces between the magnets have the following
values.

CASE I. Fig. 61.

0=0, 6' = 0. Force between magnets is an attraction
along the line joining their centres equal to

132] MAGNETISM 211

CASE II. Fig. 62.

6 = ^ , 6' = ^ . Force is a repulsion along the line

joining the centres equal to

BMM'

r* '

CASE III. Fig. 63.

0 = 0, #' = J. Force is at right angles to the line

joining the centres and equal to

3MM'

CASE IV. Fig. 64.

0=—, & = 0. Force is at right angles to the line

joining the centres and equal to

ZMM'

r* '

The forces between the magnets vary inversely as
the fourth power of the distance between their centres,
while the couples vary inversely as only the cube of this
distance. The directive influence which the magnets
exert on each other thus diminishes less quickly with the
distance than the translatory forces, so that when the
magnets are far apart the directive influence is much the
more important of the two.

132. Gauss's proof that the force between two
magnetic poles varies inversely as the square of the
distance between them. We saw, Art. 129, that, the
distance between the magnets remaining the same, the

14—2

212 MAGNETISM [CH. VI

couple exerted by the first magnet on the second was
twice as great when the first magnet was ' end on ' to
the second as when it was ' broadside on.' This is equi
valent to the result proved in Art. 127, that when P and
Q are two points at the same distance from the centre of
the magnet, P being on the axis of the magnet and Q
on the line through the centre at right angles to the axis,
the magnetic force at P is twice that at Q. This result
only holds when the force varies inversely as the square
of the distance ; we shall proceed to show that if the force
varied inversely as the pih power of the distance the
magnetic force at P would be p times that at Q.

If the magnetic force varies inversely as the pih power
of the distance, then if ra is the strength of one of the
poles of the magnet, the magnetic force at P, Fig. 58, due
to the magnet AB is equal to

ra ra

"(OP +05)*

2mp . OB

approximately, if OB is very small compared with OP ; if
M is the moment of AB this is equal to

ra OB , ra OA
The force at Q = BQJ BQ+ AQ* AQ

M
88 OP*4*1

approximately.

132] MAGNETISM 213

Thus the magnetic force at P is p times that at Q.
We see from this that if we have two small magnets the
couple on the second when the first magnet is ' end on ' to
it is p times the couple when the first magnet is 'broadside
on.' Hence by comparing the value of the couples in
these positions we can determine the value of p.

This can be done by an arrangement of the following
kind. Suspend the small magnet which is to be deflected so
that it can turn freely about a vertical axis : a convenient
way of doing this and one which enables the angular motion
of the magnet to be accurately determined, is to place the
magnet at the back of a very light mirror and suspend the
mirror by a silk fibre. When the deflecting magnet is far
away the suspended magnet will under the influence of
the earth's magnetic field point magnetic north and south.
When this magnet is at rest bring the deflecting magnet
into the field and place it so that its centre is due east
or west of the centre of the deflected magnet, the axis of
the deflecting magnet passing through the centre of this
magnet. The couple due to the deflecting magnet will
make the suspended magnet swing from the north and
south position until the couple with which the earth's
magnetic force tends to bring the magnet back to its
original position just balances the deflecting couple.

Let H be the magnetic force in the horizontal plane
due to the earth's magnetic field. Then when the deflected
magnet has twisted through an angle 0 the couple due to
the earth's magnetic field is, see Art. 128, equal to

HM' sin 0,
where Mf is the moment of the deflected magnet.

214 MAGNETISM [CH. VI

The other magnet may be regarded as producing a
field such that the magnetic force at the centre of the
deflected magnet is east and west and equal to

where M is the moment of the deflecting magnet, r the
distance between the centres of the deflected and deflect
ing magnets. Thus the couple on the deflected magnet
due to this magnet is

MM' pcosO

rp+i •

The suspended magnet will take up the position in which
the two couples balance : when this is the case

Now place the deflecting magnet so that its centre is
north or south of that of the suspended magnet, and at the
same distance from it as in the last experiment, the axis
of the deflecting magnet being again east and west. Let
the suspended magnet be in equilibrium when it has
twisted through an angle 0'. The couple due to the earth's
magnetic field is

EM' sin!?'.

The couple due to the deflecting magnet is
MM' cos 0'

133] MAGNETISM 215

Since the suspended magnet is in equilibrium these
couples must be equal, hence

., MM' cos0'
HM sin 6 = ^ — ,

hence tan 0' = TT ^ (2).

tan 6
Thus

Hence if we measure 0 and 6' we can determine p.
By experiments of this kind Gauss showed that p - 2,
i.e. that the force between two poles varies inversely as
the square of the distance between them.

If we place the deflecting magnet at different dis
tances from the deflected we find that tan 6 and tan &
vary as 1/r3, and thus obtain another proof that p = 2.

133. Determination of the Moment of a Small
Magnet and of the horizontal component of the
Earth's Magnetic Force. Suspend a small auxiliary
magnet in the same way as the deflected magnet in the
experiment just described, and place the magnet A whose
moment is to be determined, so that its centre is due east
or west of the centre of the auxiliary magnet, and its axis
passes through the centre of the suspended magnet. Let
6 be the deflection of the suspended magnet, H the
horizontal component of the earth's magnetic force, M the
moment of A: we have, by equation (1), Art. 132, putting

hence if we measure r and 6 we can determine M/H.

216 MAGNETISM [CH. VI

To determine MH suspend the magnet A so that it
can rotate freely about a vertical axis, passing through its
centre, taking care that the magnetic axis of A is hori
zontal. When the magnet makes an angle 6 with the
direction in which H acts, i.e. with the north and south
line, the couple tending to bring it back to its position of
equilibrium is equal to

Hence if K is the moment of inertia of the magnet
about the vertical axis the equation of motion of the
magnet is

or if 6 is small

Hence T, the time of a small oscillation, is given by the
equation

,
MH

hence if we know K and T we can determine MH; and
knowing M/H from the preceding experiment we can
find both M and H. The value of H at Cambridge is
about '18 C.G.S. units.

134. Magnetic Shell of Uniform Strength. A

magnetic shell is a thin sheet of magnetizable substance
magnetized at each point in the direction of the normal
to the sheet at that point.

134]

MAGNETISM

217

The strength of the shell at any point is the product
of the intensity of magnetization into the thickness of
the shell measured along the normal at that point, it is
thus equal to the magnetic moment of unit area of the
shell at the point.

To find the potential of a shell of uniform strength.
Consider a small area a of the shell round the point Q,
Fig. 65, let / be the intensity of magnetization of the shell

Fig. 65.

at Q, t the thickness of the shell at the same point. The
moment of the small magnet whose area is a is lat, hence
if 6 is the angle which the direction of magnetization
makes with PQ, the potential of the small magnet at P
is by Art. 123 equal to

lat cos 0

PQ*
If (/> is the strength of the magnetic shell

hence the potential at P is

cos 6

PQ*

This, by Art. 10, is numerically equal to the normal
induction over a due to a charge of electricity equal to <f>

218 MAGNETISM [CH. VI

at P. Hence if c/> is constant over the shell the potential
of the whole shell at P is numerically equal to the total
normal electric induction over it due to a charge (/> at P.
This, by Art. 10, is equal to <£o>, where w is the area
cut off from the surface of a sphere of unit radius with
its centre at P by lines drawn from P to the boundary of
the shell; o> is called the solid angle subtended by the
shell at P ; it only depends on the shape of the boundary
of the shell.

If the shell is closed, then if P is outside the shell
the potential at P is zero, since the total normal electric
induction over a closed surface due to a charge at a point
outside the surface is zero ; if the point P is inside the
surface and the negative side of the shell is on the out
side, then since the total normal electric induction over
the shell due to a charge 0 at P is 47r</>, the magnetic
potential at P is 4?r(/>; as this is constant throughout
the shell, the magnetic force vanishes inside the space
bounded by the shell.

The signs to be ascribed to the solid angle bounded by
the shell at various points are determined in the following
way. Take a fixed point 0 and with it as centre describe
a sphere of unit radius. Let P be a point at which
the magnetic potential of the shell is required. The
contribution to the magnetic potential by any small area
round a point Q on the shell, is the area cut off from the
surface of the sphere of unit radius by the radii drawn
from 0 parallel to the radii drawn from P to the boundary
of the area round Q. The area enclosed by the lines from
0 is to be taken as positive or negative according as the
lines drawn from P to Q strike first against the positive or
negative side of the shell. By the positive side of the shell

134] MAGNETISM 219

we mean the side charged with positive magnetism, by the
negative side the side charged with negative magnetism.

With this convention with regard to the signs of the
solid angle, let us consider the relation between the
potentials due to a shell at two points P and P' ; P being
close to the shell on the positive side, P' close to P but

Fig. 66.

on the negative side of the shell. Consider the areas
traced out on the unit sphere by radii from 0 parallel to
those drawn from P and P' . The area corresponding to
those drawn from P will be the shaded part of the sphere,
let this area be w, the potential at P is tpco. The area
corresponding to the radii drawn from P' will be the
unshaded portion of the sphere whose area is 4-Tr — «,
but inasmuch as the radii from P' strike first against the
negative side of the shell the solid angle subtended at P'
will be minus this area, i.e. o> — 4?r ; hence the magnetic
potential due to the shell at P' is </> (w — 4?r). The
potential at P thus exceeds that at P' by 47r<£.

In spite of this finite increment in the potential in
passing from P' to the adjacent point P, there will be
continuity of potential in passing through the shell if we
regard the potential as given in the shell by the same
laws as outside.

Consider the potential at a point Q in the shell, and

220 MAGNETISM [CH. VI

divide the original shell into two, one on each side of Q.
Then as the whole shell is uniformly magnetized the

strength of the shells will be proportional to their thick
nesses. Thus if (/> is the strength of the original shell the

strength of the shell between P and Q will be </>

QP'
and that of the shell between Q and P' will be

The potential at Q due to the shell next to P' is

OP' OP

, that due to the shell next to P is (G> — 4?r) (/> -> >

the potential at Q is the sum of these, i.e.

this changes continuously as we pass through the shell from

0 (a) - 4-Tr) at P',
to 0ft> at P.

135. Mutual Potential Energy of the Shell and
an external Magnetic System. Let / be the intensity
of magnetization at a point Q on the shell ; consider a
small portion of the shell round Q, a being the area of
this portion. Let P, P' be two points on its axis of mag
netization, P being on the positive surface of the shell,
P' on the negative. Then we have a charge of positive
magnetism equal to /a at P, a negative charge - /a at

135] MAGNETISM 221

P'. If Vp , Vp> are the potentials at P and P' respectively
due to the external magnetic system, then the mutual
potential energy of the external system and the small
magnet at Q is equal to

Vpla-Vyla .................. (1).

If cf) is the strength of the shell

hence the expression (1) is equal to

PP'

But ( Vp — Vp>)/PP' is the magnetic force due to the
external system along PP', the normal to the shell. Let
this force be denoted by — Hn, the force being taken as
positive when it is in the direction of magnetization of
the shell, i.e. when the magnetic force passes from the
negative to the positive side through the shell, then
the mutual potential energy of the external system and
the small magnet at Q is equal to

Since the strength of the shell is uniform the mutual
potential energy of the external system and the whole
shell is equal to

na. being the sum of the products got by dividing the
surface of the shell up into small areas, arid multiplying
each area by the component along its normal of the
magnetic force due to the external system, this com
ponent being positive when it is in the direction of
magnetization of the shell. This quantity is often called
the number of lines of magnetic force due to the external
system which pass through the shell.

222

MAGNETISM

[CH. VI

It is analogous to the total normal electric induction
over a surface in Electrostatics, see Art. 9.

136. Force acting on the shell when placed in
a magnetic field. If X is the force acting on the shell
in the direction x, and if the shell is displaced in this
direction through a distance Sx, then XSx is the work
done on the shell by the magnetic forces during the
displacement ; hence by the principle of the Conservation
of Energy, XSx must equal the diminution in the energy
due to the displacement. Suppose that A, Fig. 68, re
presents the position of the edge of the shell before,

\ \

V \

I I

1 I

I J

A B

Fig. 68.

B its position after the displacement. The diminution
in the energy due to the displacement is, by the last
paragraph, equal to

$(Nf-N) (1),

where N and N' are the numbers of lines of magnetic
force which pass through A and B respectively. Consider

136] MAGNETISM 223

the closed surface having as ends the shell in its two
positions A and B, the sides of the surface being formed
by the lines PPf &c. which join the original position
of a point P to its displaced position. We see, as in
Art. 10, that unless the closed surface contains an excess
of magnetism of one sign ^Hna. taken over its surface
must vanish, Hn denoting the magnetic force along the
normal to the surface drawn outwards.

But ^Hna over the whole surface

= N' — N + ^LHna taken over the sides,
hence N'-N=-^Hn* .................. (2);

the summation on the right-hand side of this equation
being taken over the sides. Consider a portion of the
sides bounded by PQ, P'Q' ; P', Q' being the displaced
positions of P and Q respectively. Since

the area PQP'Q' is equal to

Sac x PQ x sin 0,

where 6 is the angle between PQ and PP'. If H is
the magnetic force at P due to the external system, the
value of Hna. for the element PQQ'P' is equal to

&c x PQ x sin 6 x H cos %,

where % is the angle which the outward-drawn normal
to PQQ'P' makes with H. Hence since Z&c = <j>(Nf- N)
we have by equation (2)

XSx = - 02 {&& x PQ x sin 0 x H cos%},
or since 8x is the same for all points on the shell
X = - </>2 {PQ x sin 6 x H cos %} .

224 MAGNETISM [CH. VI

Thus the force on the shell parallel to x is the same
as it would be if a force parallel to x acted on the
boundary of the shell, equal per unit length to

Since x is arbitrary this gives the force acting on
each element of the boundary in any direction ; to find
the resultant force on the element, we notice that the com
ponent along x vanishes if x is parallel to PQ, for in this
case 0 = 0, the resultant force is thus at right angles to
the element of the boundary. Again, if x is parallel to H,
% = ?r/2, and the force again vanishes, thus the resultant
force is at right angles to H. Hence the resultant force
on PQ is at right angles both to PQ and H. In order
to find the magnitude of this force we have only to
suppose that x is parallel to this normal, in this case

# = 7r/2 and %=~- — ^, where ty is the angle between

PQ and H\ the resultant force is therefore
— (f)H sin T/T.

Thus the force on the shell may be regarded as equiva
lent to a system of forces acting over the edge of the shell,
the force acting on each element of the edge being at
right angles to the element and to the external magnetic
force at the element, and equal per unit length to the
product of the strength of the shell into the component
of the magnetic force at right angles to the element of
the edge.

The preceding rule gives the line along which the
force acts ; the direction of the force is, in any particular
case, most easily got from the principle that since the
mutual potential energy of the shell and the external

136] MAGNETISM 225

magnetic system is equal to — <j)N, where N is the number
of lines of magnetic force due to the external system
which pass through the shell in the direction in which
it is magnetized, i.e. which enter the shell on the side
with the negative magnetic charge and leave it on the
side with the positive charge : the shell will tend to move
so as to make N as large as possible, for by so doing
it makes the potential energy as small as possible. The
force on each element of the boundary will therefore be
in such a direction as to tend to move the element of
the boundary so as to enclose a greater number of lines
of magnetic force passing through the shell in the positive
direction.

Thus if the direction of the magnetic force at the
element PQ is in the direction PT in Fig. 69, the force
on PQ will be outwards along PS as in the figure, for

Fig. 69.

if PQ were to move in this direction the shell would
catch more lines of force passing through it in the positive
direction.

Since XZx = <£ (N' - N)

Y A.dN

we get X = 4>-dx'

This expression is often very useful for finding the
total force on the shell in any direction.

T. E. 15

226 MAGNETISM [CH. VI

137. Magnetic force due to the shell. Suppose
that the external field is that due to a single unit pole
at a point A, the result of the preceding article will give
the force on the shell due to the pole, this must how
ever be equal and opposite to the force exerted by the
shell on the pole. If however the field is due to a unit
pole at A, H the magnetic force due to the external
system at an element PQ of the shell is equal to 1/J.P2
and acts along AP : hence by the last article the mag
netic force at A due to the shell is the same as if we
supposed each unit of length of the boundary of the shell
to exert a force equal to

where 6 is the angle between AP and the tangent to
the boundary at P, $ is the strength of the shell. This
force acts along the line which is at right angles both
to AP and the tangent to the boundary at P. The
direction in which the force acts along this line may be
found by the rule that it is opposite to the force acting
on the element of the boundary at P arising from unit
magnetic pole at A ; this latter force may be found by the
method given at the end of the preceding article.

138. If the external magnetic field in Art. 135 is
due to a second magnetic shell, then the mutual potential
energy of the two shells is equal to

where $ is the strength of the first shell, and N the
number of lines of force which pass through the first
shell, and are produced by the second. It is also equal to

139] MAGNETISM 227

where <£' is the strength of the second shell, and N' the
number of lines of force which pass through the second
shell and are produced by the first. Hence by making
<£ = <£' we see that, if we have two shells a and /3 of
equal strengths, the number of lines of force which pass
through a. and are due to /3 is equal to the number of
lines of force which pass through /3 and are due to a.

139. Magnetic Field due to a uniformly mag
netized sphere. Let the sphere be magnetized parallel
to as, and let / be the intensity of magnetization. We
may regard the sphere as made up, as in Fig. 70, of a
great number of uniformly magnetized bar magnets of
uniform cross section a, the axes of these magnets being
parallel to the axis of x. On the ends of each of these
magnets we have charges of magnetism equal to + la.
Now consider a sphere whose radius is equal to that of
the magnetized sphere and built up of bars in the same
way, each of these bars being however wholly filled with
positive magnetism whose volume density is p: consider

/\ r\

X—

Fig. 70.

also another equal sphere divided up into bars in the
same way, each of these bars being however filled with

15—2

228 MAGNETISM [CH. VI

negative magnetism whose volume density is — p ; suppose
that these spheres have their centres at 0' and 0, Fig. 71,
two points very close together, 00' being parallel to the
axis of x. Consider now the result of superposing these
two spheres : take two corresponding bars ; the parts of
the bars which coincide will neutralize each other's effects,
but the negative bar will project a distance 00' to the
left, and on this part of the bar there will be a charge of
negative magnetism equal to 00' x a. x p : the positive bar
will project a distance 00' to the right, and on this part
of the bar there will be a charge of positive magnetism
equal to 00' x a x p. If 00' is very small we may regard
these charges as concentrated at the ends of the bars, so
that if 00' x p = I the case will coincide with that of the
uniformly magnetized sphere.

We can easily find the effects of the positive and
negative spheres at any point either inside or outside.
Let us first consider the effect at an external point P.

The potential due to the positive sphere is equal to
4 7ra3p

zWF

if a is the radius of the sphere.

Fig. 71.

139] MAGNETISM 229

The potential due to the negative sphere is equal to
4

Hence the potential due to the combination of the
spheres is equal to

4 - 1-1- -±4

(O'P OP]

00' cose

approximately, if 00' is very small, and 6 is the angle
which OP makes with 00'.

Now we have seen that this case coincides with that
of the uniformly magnetized sphere if p x 00' = /, where
/ is the intensity of magnetization of the sphere ; hence
the potential due to the uniformly magnetized sphere
at an external point P is

4 , cos 6

§***•—-

where r = OP.

Comparing this result with that given in Art. 123 we
see that the uniformly magnetized sphere produces the
same effect outside the sphere as a very small magnet
placed at its centre, the axis of the small magnet being
parallel to the direction of magnetization of the sphere,
while the moment of the magnet is equal to the in
tensity of magnetization multiplied by the volume of
the sphere.

The magnetic force inside the sphere is indefinite
without further definition, since to measure the force on
the unit pole, we have to make a hole to receive the

230 MAGNETISM [CH. VI

pole and the force on the pole depends on the shape of
the hole so made : this point is discussed at length in
Chapter vm.

For the sake of completing the solution of this case,
we shall anticipate the results of that chapter and assume
that the quantity which is denned as ' the magnetic
force' inside the sphere is the force which would be
exerted on the unit pole if the sphere were regarded
as a spherical air cavity over the surface of which
there is spread the same distribution of magnetic charge
as actually exists over the surface of the magnetized
sphere. We may thus in calculating the effect of the
charges on the surface suppose that they exert the same
magnetic forces as they would in air.

To find the magnetic force at an internal point Q,
Fig. 71, we return to the case of the two uniformly charged
spheres.

The force due to the uniformly positively charged
sphere at Q is equal to

and acts along O'Q; the force due to the negatively
charged sphere is equal to

and acts along QO.

By the triangle of forces the resultant of the forces
exerted by the positive and negative spheres is equal to

f 7T/> . 00',

and is parallel to 00'. We have seen that the case of the
positive and negative spheres coincides with that of the

139]

MAGNETISM

231

uniformly magnetized sphere if O0'xp = l. Hence the
force inside the uniformly magnetized sphere is uniform
and parallel to the direction of magnetization of the sphere
and equal to

The lines of force inside and outside the sphere are
given in Fig. 72.

Fig. 72.

CHAPTER VII

TERRESTRIAL MAGNETISM

140. The pointing of the compass in a definite direc
tion was at first ascribed to the special attraction for iron
possessed by the pole star. Gilbert, however, in his work
De Magnete, published in 1600, pointed out that it showed
that the earth was itself a magnet. Since Gilbert's time
the study of Terrestrial Magnetism, i.e. the state of the
earth's magnetic field, has received a great deal of attention
and forms one of the most important, and undoubtedly
one of the most mysterious departments of Physical
Science.

141. To fix the state of the earth's magnetic field
we require to know the magnetic force over the whole
of the surface of the earth ; the observations made at a
number of magnetic observatories, scattered unfortunately
somewhat irregularly at very wide intervals over the earth,
give us an approximation to this.

To determine the magnitude and direction of the
earth's magnetic force we require to know three things :
the three usually taken are (1) the magnitude of the
horizontal component of the earth's magnetic force, usually
called the earth's horizontal force; (2) the angle which
the direction of the horizontal force makes with the
geographical meridian, this angle is called the declination ;

CH. VII. 142] TERRESTRIAL MAGNETISM 233

the vertical plane through the direction of the earth's
horizontal force is called the magnetic meridian ; (3) the
dip, that is the complement of the angle which the axis of
a magnet, suspended so as to be able to turn freely about
an axle through its centre of gravity at right angles to the
magnetic meridian, makes with the vertical. The fact that
a compass needle when free to turn about a horizontal
axis would not settle in a horizontal position, but ' dipped,'
so that the north end pointed downwards, was discovered
by Norman in 1576.

For a full description of the methods and precautions
which must be taken to determine accurately the values
of the magnetic elements the student is referred to the
article on Terrestrial Magnetism in the Encyclopaedia
Britannica : we shall in what follows merely give a general
account of these methods without entering into the details
which must be attended to if the most accurate results are
to be obtained.

The method of determining the horizontal force has
been described in Art. 133.

142. Declination. To determine the declination an
instrument called a declinometer may be employed ; this
instrument is represented in Fig. 73. The magnet —
which is a hollow tube with a piece of plane glass with a
scale engraved on it at the north end and a lens at the
south end — is suspended by a single long silk thread from
which the torsion has been removed by suspending from
it a plummet of the same weight as the magnet: the
suspension and the reading telescope can rotate about a
vertical axis and the azimuth of the system determined
by means of a scale engraved on the fixed horizontal base.

234 TERRESTRIAL MAGNETISM [CH. VII

The observer looks through the telescope and observes the
division on the scale at the end of the magnet with which
a cross wire in the telescope coincides; the magnet is
then turned upside down and resuspended and the division
of the scale with which the cross wire coincides again
noted ; this is done to correct for the error that would

Fig. 73.

otherwise ensue if the magnetic axis of the cylinder did
not coincide with the geometrical axis. The mean of the
readings gives the position of the magnetic axis. If now
we take the reading on the graduated circle and add to
this the known value in terms of the graduations on this
circle of the scale divisions seen through the telescope, we
shall find the circle reading which corresponds to the
magnetic meridian. Now remove the magnet and turn
the telescope round until some distant object, whose

143]

TERRESTRIAL MAGNETISM

235

azimuth is known, is in the field of view; take the reading
on the graduated circle, the difference between this and
the previous reading will give us the angular distance of
the magnetic meridian from a plane whose azimuth is
known : in other words, it gives us the magnetic declina
tion.

143. Dip. The dip is determined by means of an
instrument called the dip-circle, represented in Fig. 74. It

Fig. 74.

consists of a thin magnet with an axle of hard steel whose
axis is at right angles to the plane of the magnet, and
ought to pass through the centre of gravity of the needle ;
this axle rests in a horizontal position on two agate

236 TERRESTRIAL MAGNETISM [CM. VII

edges, and the angle the needle makes with the vertical
is read off by means of the vertical circle. The needle
and the vertical circle can turn about a vertical axis.
To set the plane of motion of the needle in the magnetic
meridian, the plane of the needle is turned about the
vertical axis until the magnet stands exactly vertical;
when in this position the plane of the needle must be
at right angles to the magnetic meridian. The instrument
is then twisted through 90° (measured on the horizontal
circle) and the magnet is then in the magnetic meridian ;
the angle it makes with the horizontal in this position is
the dip. To avoid the error arising from the axle of the
needle not being coincident with the centre of the vertical
circle, the positions of the two ends of the needle are read ;
to avoid the error due to the magnetic axis not being
coincident with the line joining the ends of the needle,
the needle is reversed so that the face which originally
was to the east is now to the west and a fresh set of
readings taken ; and to avoid the errors which would arise
if the centre of gravity were not on the axle, the needle
is remagnetized so that the end which was previously
north is now south and a fresh set of readings taken.
The mean of these readings gives the dip.

144. We can embody in the form of charts the deter
minations of these elements made at the various magnetic
observatories : thus, for example, we can draw a series of
lines over the map of the world such that all points on one
of these lines have the same declination, these are called
isogonic lines : we may also draw another set of lines so
that all the places on a line have the same dip, these are
called isoclinic lines. The lines however which give the

144]

TERRESTRIAL MAGNETISM

237

best general idea of the distribution of magnetic force over
the earth's surface are the lines of horizontal magnetic

Fig. 75.

•ld°20°East Variation

Fig. 76.

238 TERRESTRIAL MAGNETISM [CH. VII

force on the earth's surface, i.e. the lines which would be
traced out by a traveller starting from any point and
always travelling in the direction in which the compass
pointed; they were first used by Duperrey in 1836.

The isoclinic lines, the isogonic lines and Duperrey 's
lines for the Northern and Southern Hemispheres for 1876
are shown in Figs. 75, 76, 77, and 78 respectively.

145. The points to which Duperrey's lines of force
converge are called 'poles/ they are places where the
horizontal force vanishes, that is where the needle if freely
suspended would place itself in a vertical position.

Fig. 77.

Gauss by a very thorough and laborious reduction of
magnetic observations gave as the position in 1836, of
the pole in the Northern Hemisphere, latitude 70° 35',

146] TERRESTRIAL MAGNETISM 239

longitude 262° 1' E., and of the pole in the Southern
Hemisphere, latitude 78° 35', longitude 150° 10' E. The

Fig. 78.

poles are thus not nearly at opposite ends of a diameter
of the earth.

146. An approximation, though only a very rough one,
to the state of the earth's magnetic field, may be got by
regarding the earth as a uniformly magnetized sphere.

On this supposition, we have by Art. 139, if 6 is the
dip at any place, i.e. the complement of the angle between
the magnetic force and the line joining the place to the
centre of the earth, I the magnetic latitude, i.e. the com
plement of the angle this line makes with the direction of
magnetization of the sphere,

tan 6 = 2 tan I,

while the resultant magnetic force would vary as
[1 + 3 sin2 qi.

240 TERRESTRIAL MAGNETISM [CH. VII

These are only very rough approximations to the truth
but are sometimes useful when more accurate knowledge
of the magnetic elements is not available.

If M is the moment of the uniformly magnetized
sphere which most nearly represents the earth's magnetic
field, then in c.G.s. units

M ='323 (earth's radius)3.

147. Variations in the Magnetic Elements.

During the time within which observations of the mag
netic elements have been carried on the declination at
London has changed from being 11° 15' to the East of
North as in 1580 to 24° 38' 25" to the West of North as
in 1818. It is now going back again to the East, but
is still pointing between 16° and 17° to the West. The
variations in the declination and dip in London are
shown in the following table.

Date

Declination

Dip

1576

71° 50'

1580

11° 15' E.

1600

72° 0'

1622

6° O'E.

1634

4° 6'E.

1657

0° 0' E.

1665

1° 22' W.

1672

2° 30' W.

1676

73° 30'

1692

6° 30' W.

1723

14° 17' W.

74° 42'

1748

17° 40' W.

1773

21° 9'W.

72° 19'

1787

23° 19' W.

72° 8'

148] TERRESTRIAL MAGNETISM 241

Date

Declination

Dip

1795

23° 57' W.

1802

24° 6' W.

70C 36'

1820

24° 34' 30" W.

70° 3'

1830

24°

69° 38'

1838

69° 17'

1860

21° 39' 51"

68° 19'-29

1870

29° 18' 52"

67° 57''98

1880

18° 57' 59"

1893

17° 27'

67° 30'

1900

16° 52'-7

This slow change in the magnetic elements is often
called the secular variation. The points of zero declina
tion seem to travel westward.

148. Besides these slow changes in the earth's mag
netic force, there are other changes which take place with
much greater rapidity,

Diurnal Variation. A freely suspended magnetic
needle does not point continually in one direction during
the whole of the day. In England in the night from
about 7 p.m. to 10 a.m. it points to the East of magnetic
North and South (i.e. to the East of the mean position of
the needle), and during the day from 10 a.m. to 7 p.m. to
the West of magnetic North and South. It reaches the
westerly limit about 2 in the afternoon, its easterly one
about 8 in the morning, the arc travelled over by the
compass being about 10 minutes. This arc varies however
with the time of the year, being greatest at midsummer
and least at midwinter. There are two maxima in summer,
one minimum in winter.

The diurnal variation changes very much from one

T. E. 16

242

TERRESTRIAL MAGNETISM

[CH. VII

place to another, it is exceedingly small at Trevandrum,
a place near the equator.

In the Southern Hemisphere the diurnal variation is
of the opposite kind to that in the Northern, i.e. the
easterly limit in the Southern Hemisphere is reached in
the afternoon, the westerly in the morning.

In the following diagram, due to Prof. Lloyd, the
radius vector represents the disturbing force acting on
the magnet at different times of the day in Dublin, the

AM 10

Fig. 79.

forces at any hour are the average of those at that hour
for the year. The curve would be different for different
seasons of the year.

There is also a diurnal variation in the vertical

151] TERRESTRIAL MAGNETISM 243

component of the earth's magnetic force. In England the
vertical force is least between 10 and 11 a.m., greatest at
about 6 p.m.

The extent of the diurnal variation depends upon the
condition of the sun's surface, being greater when there
are many sun spots. As the state of the sun with regard
to sun spots is periodic, going through a cycle in about
eleven years, there is an eleven-yearly period in the
magnitude of the diurnal variation.

149. Effect of the Moon. The magnetic declina
tion shows a variation depending on the position of the
moon with respect to the meridian, the nature of this
variation varies very much in different localities.

150. Magnetic Disturbances. In addition to the
periodic and regular disturbances previously described,
rapid and irregular changes in the earth's magnetic field,
called magnetic storms, frequently take place ; these often
occur simultaneously over a large portion of the earth's
surface.

Aurorse are mostly accompanied by magnetic storms,
and there is very strong evidence that a magnetic storm
accompanies the sudden formation of a sun spot.

151. Very important evidence as to the locality of
the origin of the earth's magnetic field, or of its variations,
is afforded by a method due to Gauss which enables us to
determine whether the earth's magnetic field arises from
a magnetic system above or below the surface of the earth.
The complete discussion of this method requires the use of
Spherical Harmonic Analyses. The principle underlying

16—2

244 TERRESTRIAL MAGNETISM [CH. VII

it however can be illustrated by considering a simple case,
that of a uniformly magnetized sphere.

Let PQ be two points on a spherical surface concentric
with the sphere, then by observation of the horizontal
force at a series of stations between P and Q, we can
determine the difference between the magnetic potential
at P and Q. If £1P and HQ are the magnetic potentials
at P and Q respectively these observations will give us
HP - HQ. By Art. 139 if 0l9 02 are the angles OP and OQ
make with the direction of magnetization of the sphere

M
tip -nQ = - (cos 0! - cos <92) ............ (1),

where M is the magnetic moment of the sphere and

where 0 is the centre of the sphere.

If ZP) Zq are the vertical components of the earth's
magnetic force, i.e. the forces in the direction OP and
OQ respectively, then

<2 = 7T cos

Zp and Zq can of course be determined by observations
made at P and Q. By equations (1) and (2), we have

ClP-nQ = 1t(ZP-ZQ)r ............... (3),

hence if the field over the surface of the sphere through
P and Q were due to an internal uniformly magnetized
sphere, the relation (3) would exist between the horizontal
and vertical components of the earth's magnetic force.

151] TERRESTRIAL MAGNETISM 245

Now suppose that P and Q are points inside a uniformly
magnetized sphere, the force inside the sphere is uniform
and parallel to the direction of magnetization, let H be
the value of this force, then in this case
HP — HQ = Hr (cos 02 — cos #j),

Zq = H COS 62 ,

hence in this case

nP-nq = -r(ZP-ZQ) ............... (4).

Thus if the magnetic system were above the places at
which the elements of the magnetic field were determined,
the relation (4) would exist between the horizontal and
vertical components of the earth's magnetic force. Con
versely if we found that relation (3) existed between these
components we should conclude that the magnets pro
ducing the field were below the surface of the earth, while
if relation (4) existed we should conclude the magnets
were above the surface of the earth; if neither of these
relations was true we should conclude that the magnets
were partly above and partly below the surface of the
earth.

Gauss showed that no appreciable part of the mean
values of the magnetic elements was due to causes above
the surface of the earth. Schuster has however recently
shown by the application of the same method that the
diurnal variation must be largely due to such causes.
Balfour Stewart had previously suggested the magnetic
action of electric currents flowing through rarefied air in
the upper regions of the earth's atmosphere as the
probable cause of this variation.

CHAPTER VIII

MAGNETIC INDUCTION

152. When a piece of unmagnetized iron is placed
in a magnetic field it becomes a magnet, and is able to
attract iron filings; it is then said to be magnetized by
induction. Thus if a piece of soft iron (a common nail for
example) is placed against a magnet, it becomes mag
netized by induction, and is able to support another nail,
while this nail can support another one, and so on until a
long string of nails may be supported by the magnet.

If the positive pole of a bar magnet be brought near
to one end of a piece of soft iron, that end will become
charged with negative magnetism, while the remote end of
the piece of iron will be charged with positive magnetism.
Thus the opposite poles of these two magnets are nearest
each other, and there will therefore be an attraction be
tween them, so that the piece of iron, if free to move, will
move towards the inducing magnet, i.e. it will move from
the weak to the strong parts of the magnetic field due to
this magnet. If, instead of iron, pieces of nickel or cobalt
are used they will tend to move in the same way as the
iron, though not to so great an extent. If however we use
bismuth instead of iron, we shall find that the bismuth
is repelled from the magnet, instead of being attracted
towards it ; the bismuth tending to move from the strong

CH. VIII. 153] MAGNETIC INDUCTION 247

to the weak parts of the field; the effect is however
very small compared with that exhibited by iron ; and to
make the repulsion evident it is necessary to use a strong
electromagnet. When the positive pole of a magnet is
brought near a bar of bismuth the end of the bar next
the positive pole becomes itself a positive pole, while the
further end of the bar becomes a negative pole.

Substances which behave like iron, i.e. which move
from the weak to the strong parts of the magnetic field,
are called paramagnetic substances; while those which
behave like bismuth, and tend to move from the strong
to the weak parts of the field, are called diamagnetic
substances.

When tested in very strong fields all substances are
found to be para- or dia-magnetic to some degree, though
the extent to which iron transcends all other substances
is very remarkable.

153. Magnetic Force and Magnetic Induction.

The magnetic force at any point in air is defined to be
the force on unit pole placed at that point, or — what is
equivalent to this — the couple on a magnet of unit
moment placed with its axis at right angles to the
magnetic force. When however we wish to measure the
magnetic force inside a magnetizable substance, we have
to make a cavity in the substance in which to place the
magnet used in measuring the force. The walls of the
cavity will however become magnetized by induction, and
this magnetization will affect the force inside the cavity.
The magnetic force thus depends upon the shape of the
cavity, and this shape must be specified if the expression
magnetic force is to have a definite meaning.

248 MAGNETIC INDUCTION [CH. VIII

Let P be a point in a piece of iron or other mag
netizable substance, and let us form about P a cylindrical
cavity, the axis of the cylinder being parallel to the
direction of magnetization at P. Let us first take the case
when the cylinder is a very long and narrow one. Then
in consequence of the magnetization at P, there will be
a distribution of positive magnetism over one end of the
cylinder, and a distribution of negative magnetism over
the other. Let / be the intensity of the magnetization
at P, reckoned positive when the axis of the magnet is
drawn from left to right, then when the cylindrical cavity
has been formed round P there will be, if a. is the cross
section of the cavity, a charge la. of magnetism on the
end to the left, and a charge — la on the end to the right.
If 21, the length of the cylinder, is very great compared
with the diameter, then the force on unit pole at the
middle of the cylinder due to the magnetism at the ends
of the cylinder will be 2/a/Z2, and will be indefinitely
small if the breadth of the cylinder is indefinitely small
compared with its length. In this case the force on unit
pole in the cavity is independent of the intensity of
magnetization at P. The force in this cavity is defined
to be *' the magnetic force at P! Let us denote it by H.

Let us now take another co-axial cylindrical cavity,
but in this case make the length of the cylinder very
small compared with its diameter, so that the shape of
the cavity is that of a narrow crevasse. On the left end
of this crevasse there is a charge of magnetism of surface
density /, and on the right end of the crevasse a charge
of magnetism of surface density — /. If a unit pole be
placed inside the crevasse the force on it due to this
distribution of magnetism will be the same as the force

154] MAGNETIC INDUCTION 249

on unit charge of electricity placed between two infinite
plates charged with electricity of surface density + / and
— / respectively, i.e. by Art. 14, the force on the unit
pole in this case will be 4?r/. Thus in a crevasse the
total force on the unit pole at P will be the resultant of
the magnetic force at P and a magnetic force 47rJ in the
direction of the magnetization at P. The force on the
unit pole in the crevasse is defined to be the ' magnetic
induction' at P, we shall denote it by B. If we had taken
a cavity of any other shape the force due to the magnetiza
tion at 'P, would have been intermediate in value between
zero for the long cylinder and 4?r/ for the crevasse ; thus
if the cavity had been spherical the force due to the
magnetization would (Art. 139) have been 4?r//3.

The magnetic induction is not necessarily in the same
direction as the magnetic force, it will only be so when
the magnetization at P is parallel to the magnetic force.

154. Tubes of Magnetic Induction. A curve
drawn such that its tangent at any point is parallel to
the magnetic induction at that point is called a line
of magnetic induction : in non-magnetizable substances
the lines of magnetic induction coincide with the lines
of magnetic force. We can also draw tubes of magnetic
induction just as we draw tubes of magnetic force.

We shall choose the unit tube so that the magnetic
induction at any place whether in the air or iron is equal
to the number of tubes of induction which cross a unit
area at right angles to the induction.

Let us consider the case of a small bar magnet, the
magnetism being entirely at its ends. Suppose A and B
are the ends of the magnet, A being the negative, B the

250 MAGNETIC INDUCTION [CH. VIII

positive end, then in the air the lines of magnetic in
duction coincide with those of magnetic force and go
from B to A. To find the lines of magnetic induction
at a point P inside the magnet, imagine the magnet cut
by a plane at right angles to the axis and the two portions
separated by a short distance, the lines of magnetic force
in this short air space will be the lines of magnetic in
duction in the section through P. If the magnet is cut

as in the figure then the end G will be a positive pole of
the same strength as A, the end D a negative pole of the
same strength as B. Thus through the short air space
between C and D tubes of induction will pass running in
the direction AB. Draw a closed surface passing through
the gap between C and D and enclosing AC or DB. The
magnetic force at any point on this surface is equal to the
magnetic induction at the same point due to the undivided
magnet. Since this surface encloses as much positive as
negative magnetism, we see as in Art. 10 that the total
magnetic force over its surface vanishes. Hence we see
that the tubes of induction inside the magnet are equal
in number at each cross-section and this number is the

154] MAGNETIC INDUCTION 251

same as the number of those which leave the pole B and
enter A. In fact the lines of magnetic induction due
to the magnet form a series of closed curves all passing
through the magnet and then spreading out in the air,
the lines running from B to A in the air and from A to B
in the magnet.

Thus we may regard any small magnet, whose in
tensity is / and area of cross- section a, as the origin of
a bundle of closed tubes of induction, the number of
tubes being 4?r/a; every tube in this bundle passes
through the magnet, running through the magnet in
the direction of the magnetization.

It is instructive to compare the differences between
the properties of the tubes of electric polarization in
electrostatics and those of magnetic induction in mag
netism : the most striking difference is that whereas in
electrostatics the tubes are not closed but begin at posi
tive electrification and end on negative, in magnetism the
tubes of induction always form closed curves and have
neither beginning nor end.

A surface charged with electricity of surface density a
is the origin of a tubes of electric polarization per unit
area. A small magnet whose intensity of magnetization
is / is the origin of 4?rJ tubes of magnetic induction per
unit area of cross-section of the magnet, all these tubes
passing through the magnet which acts as a kind of girdle
to them.

The properties of these tubes are well summed up
by Faraday in the following passage (Experimental Re
searches, § 3117): "there exist lines of force within the
magnet, of the same nature as those without. What is
more, they are exactly equal in amount to those without.

252 MAGNETIC INDUCTION [CH. VIII

They have a relation in direction to those without and
in fact are continuations of them, absolutely unchanged
in their nature so far as the experimental test can be
applied to them. Every line of force, therefore, at what
ever distance it may be taken from the magnet, must be
considered as a closed circuit passing in some part of its
course through the magnet, and having an equal amount
of force in every part of its course." Faraday's lines of
force are what we have called tubes of induction.

155. We shall now proceed to consider the special
case, including that of iron and all non-crystalline sub
stances when magnetized entirely by induction, in which
the direction of the magnetization and consequently of
the magnetic induction is parallel to the magnetic force.
Let H be the magnetic force, B the magnetic induction,
and I the intensity of magnetization, then we have by
Art. 153,

The ratio of I to H when the magnetization is entirely
induced is called the magnetic susceptibility and is usually
denoted by the letter k. The ratio of B to H under the
same circumstances is called the magnetic permeability
and is denoted by the letter //,.

We thus have

and since B = H + 4?r/,

we have //, = / + 4>7rk.

The quantity //, which occurs in magnetism is analogous
to the specific inductive capacity in electrostatics; but

155]

MAGNETIC INDUCTION

253

while as far as our knowledge at present goes, the specific
inductive capacity at any time does not depend much, if at
all, upon the value of the electric intensity at that time,
nor on the electric intensity to which the dielectric has

12000
11000
10000
9OOO
8000
7OOO
dGOO
5000
4000
3000
2000
JOOO
O

I2345678G

Magnetic Force H.
Fig. SI.

previously been exposed; the permeability, on the other
hand, if the magnetic force exceeds a certain value (about
1/10 of the earth's horizontal force), depends very greatly
upon the magnitude of the magnetic force, and also upon
the magnetic forces which have previously been applied to

254 MAGNETIC INDUCTION [CH. VIII

the iron. The variations in the magnetic permeability
are most conveniently represented by curves in which the
ordinate represents the magnetic induction, the abscissa
the corresponding magnetic force. If P be a point on
such a curve, PN the ordinate, ON the abscissa, then
the magnetic permeability is PN/ON.

Such a curve is shown in Fig. 81, in which the
ordinates represent for a particular specimen of iron
the values of B, the magnetic induction, the abscissae
the values of H, the magnetic force. For small values
of H the curve is straight, indicating that the per
meability is independent of the magnetic force. When
however the magnetic force increases beyond about -fa
of the earth's horizontal force, or about '018 in C.G.S.
units, the curve begins to rise rapidly, and the value
of fjb is greater than it was for small magnetic forces.
The curve rises rapidly for some time, the maximum
value of p occurring when the magnetic force is about
5 C.G.S. units, then it begins to get flatter and there
are indications that for very great values of the mag
netic force the curve again becomes a straight line
making an angle of 45° with the axis along which the
magnetic force is measured. The relation between B and
H along this part of the curve is

B = H + constant :
comparing this with the relation

we see that it indicates that the intensity of magnetization
has become constant. In other words, the intensity of
magnetization does not increase as the magnetic force
increases. When this is the case the iron or other

155]

MAGNETIC INDUCTION

255

magnetizable substance is said to be 'saturated.' Thus
iron seems not to be able to be magnetized beyond a
certain intensity. In a specimen of soft iron examined by
Prof. Ewing, saturation was practically reached when the
magnetic force was about 2000 in c.G.s. units. For steel
the magnetic force required for saturation is very much
greater than for soft iron, and in some specimens of steel
examined by Prof. Ewing saturation was not attained
even when the magnetizing force was as great as 10000.

Induction B.
Fig. 82.

For a particular kind of steel called Hadfield's manganese
steel the value of /JL was practically constant even in
the strongest magnetic fields, this steel however is only
slightly magnetic, the value of /u, being about 1'4. The

256 MAGNETIC INDUCTION [CH. VIII

greatest value of p which has been observed is 20000 for
soft iron, in this case however the iron was tapped when
under the influence of the magnetic force. Fig. 82 re
presents the results of Ewing's experiments on the relation
between magnetic permeability and magnetic induction in
very intense magnetic fields.

156. Effect of Temperature on the Magnetic
Permeability. The permeability of iron depends very
much upon the temperature. Dr J. Hopkinson found that
as the temperature increases, starting from about 15° C.,
the magnetic permeability at first slowly increases; this
slow rate of increase is however exchanged for an exceed
ingly rapid one when the temperature approaches a 'critical
temperature ' which for different samples of iron and steel
ranges from 690° C. to 870° C., at this temperature the
value of the permeability is many times greater than
that at 15°C.: after passing this value the permeability
falls even more rapidly than it previously rose. Indeed
so fast is the fall that at a few degrees above the critical
temperature iron practically ceases to be magnetic. Just
below this temperature iron is an intensely magnetic sub
stance, while above that temperature it is not magnetic
at all. There are other indications that iron changes its
character in passing through' this temperature, for here
its thermo-electric properties as well as its electrical
resistance suffer abrupt changes. This temperature is
often called the temperature of recalescence from the fact
that a piece of iron wire heated above this temperature
to redness and then allowed to cool, will get dull before
reaching this temperature and will glow out brightly
again when it passes through it.

157] MAGNETIC INDUCTION 257

Though the value of ^ at higher temperatures (lower
however than that of recalescence) is for small magnetic
forces greater than at lower temperatures, still as it is
found that at the higher temperatures iron is much more
easily saturated than at lower ones, the value of ^ for
the hot iron might be smaller than for the cold if the
magnetic forces were large.

Hopkinson found that some alloys of nickel and iron
after being rendered non-magnetic by being raised above
the temperature of recalescence remained non-magnetic
when cooled below this temperature ; it was not until
the temperature had fallen far below the temperature of
recalescence that they regained their magnetic properties.
Thus these alloys can at one and the same temperature
exist in both the magnetic and non-magnetic states.

157. Magnet Retentiveness. Hysteresis. When
a piece of iron or steel is magnetized in a strong magnetic
field it will retain a considerable proportion of its mag
netization even after the applied field has been removed
and the iron is no longer under the influence of any ap
plied magnetic force. This power of remaining magne
tized after the magnetic force has been removed, is called
magnetic retentiveness; permanent magnets are a familiar
instance of this property. This effect of the previous
magnetic history of a substance on its behaviour when
exposed to given magnetic conditions has been studied in
great detail by Prof. Ewing, who has given to this property
the name of hysteresis. To illustrate this properly, let us
consider the curve (Fig. 83) which is taken from Prof.
Ewing's paper on the magnetic properties of iron (Phil.
Trans. Part II., 1885), and which represents the relation for

T. E. 17

258

MAGNETIC INDUCTION

[CH. VIII

a sample of soft iron between the intensity of magnetization
(the ordinate) and the magnetizing force (the abscissa),
when the magnetic force increases from zero up to ON,
then diminishes from ON through zero to — OM , and then
increases again up to its original value. When the force
is first applied we have the state represented by the por
tion OP of the curve, which begins by being straight, then
increases more rapidly, bends round and finally reaches P,
the point corresponding to the greatest magnetic force

Fig. 83.

applied to the iron. If now the force is diminished it
will be found that the magnetization for a given force is
greater than it was- when the magnet was initially under
the action of the same force, i.e. the magnet has retained
some of its previous magnetization, thus the curve PE>
when the force is diminishing, will not correspond to the
curve OP but will be above it. OE is the magnetization
retained by the magnet when free from magnetic force ;

157] MAGNETIC INDUCTION 259

in some cases it amounts to more than 90 per cent.
of the greatest magnetization attained by the magnet.
When the magnetizing force is reversed the magnet
rapidly loses its magnetization and the negative force
represented by OK is sufficient to deprive it of all
magnetization. When the negative magnetic force is in
creased beyond this value, the magnetization is negative.
After the magnetic force is again reversed it requires a
positive force equal to OL to deprive the iron of its
negative magnetization. When the force is again in
creased to its original value the relation between the
force and induction is represented by the portion LGP
of the curve. If after attaining this value the force is
again diminished to - ON and back again, the corre
sponding curve is the curve PEK.

From the fact that this curve encloses a finite area it
follows that a certain amount of energy must be dissipated
and converted into heat when the magnetic force goes
through a complete cycle. To show this let us suppose
that we have a small magnet whose intensity is /, cross-
section a, and length I, and that it is moved from a place
where the magnetic force is H to one where it is H + 8H.
We shall show that the work done on the magnet is

ISHal.

H is considered positive when it acts in the direction of
magnetization of the iron. For if Oj is the magnetic
potential at A, the negative pole; O2 that at B, the
positive pole, then the potential energy of the magnet
is equal to

17—2

260

MAGNETIC INDUCTION

[CH. VI1J

When the magnetic force is H + &H the potential energy
is equal to

Thus the diminution in the potential energy when
the magnet moves into the stronger field is lal&H, this is
equal to the work done by the magnet. If the intensity of
magnetization changes from / to /+£/ during the motion
of the magnet, the work done is intermediate between
lalSH and (/ + £/) alSH ; hence neglecting the small
terms depending upon SlSff, we may still take lal&H
as the expression for the work done. Since la is the
volume of the magnet the work done by the magnet
per unit volume is IBH.

If in Fig. 84 OS = H, OT=H + SH and SP = I, then
is represented on the diagram by the area SPQT.

Thus the total work done by the magnet when it
moves from a place where the force is OK to one where it
is OL is represented by the area CKLDE. Let the magnet
now be pulled back from the place where the force is OL
to the place from which it started where the force is OK,

157] MAGNETIC INDUCTION 261

work has to be done on the magnet and this work is re
presented by the area DFGKL. Thus the excess of the
work done on the magnet over that done by the magnet,
when the magnetic force goes through a complete cycle,
is represented by the area of the loop CEDFC. The
area of the loop thus represents the excess of the work
spent over that obtained : but since the magnetic force
and magnetization at the end of the cycle are the same
as at the beginning, this work must have been dissipated
and converted into heat. The mechanical equivalent of
the amount of heat produced in each unit volume of the
iron is represented by the area of the loop.

Another proof of this is given in Chapter xi.

If instead of a curve showing the relation between
/ and H we use one showing the relation between B and
H, there will be similar loops in this second curve and
the area of these loops will be 4?r times the area of the
corresponding loops on the / and H curve.

For the area of a loop on the first curve is

-fldH,

this is equal to

47J-J"

since jHdH = 0, as the initial and final values of H are
equal. The area of a loop on the B and H curve is
however equal to

-jBdH.

Hence we see that this area is 4?r times the area of the
corresponding loop on the / and H curve.

262 MAGNETIC INDUCTION [CH. VIII

158. Conditions which must hold at the bound
ary of two substances.

At the surface separating two media the magnetic
field must satisfy the following conditions.

1. The magnetic force parallel to the surface must be
the same in the two media.

2. The magnetic induction at right angles to the
surface must be the same in the two media.

To prove the first condition, let P and Q be two points
close to the surface of separation, Q being in the first,
P in the second medium. Now the magnetic force at a
point is by definition (see Art. 153) the force on a unit
pole placed in a cavity round the point, when the mag
netism on the walls of the cavity can be neglected : hence
since this magnetism is to be disregarded the difference
between the magnetic forces at P and Q must arise from
the magnetism on the surface between P and Q: but
though the forces at right angles to this portion of the
surface due to its magnetism are different at P and Q,
the forces parallel to the surface are the same. Hence
we see that the tangential magnetic forces will be the
same at P as at Q.

We shall now show that the normal magnetic induction
is continuous. All the tubes of magnetic induction form
closed curves. Hence any tube must cut a closed surface
an even number of times ; half these times it will be
entering the surface, half leaving it. The contributions of
each tube to the total normal magnetic induction will be
the same in amount but opposite in sign when it enters
and when it leaves the surface. Hence the total con
tribution of each tube is zero, and thus the total normal

158] MAGNETIC INDUCTION 263

magnetic induction over any closed surface vanishes.
Consider the surface of a very short cylinder whose sides
are parallel to the normal at P, one end being in the
medium (1), the other in (2). The total normal induction
over this surface is zero, but as the area of the sides is
negligible compared with that of the ends, this implies
that the total normal induction across the end in (1) is
equal to that across the end in (2), or that, since the
areas of these ends are equal, the induction parallel to the
normal in (1) is the same as that in the same direction
in (2). This is always true whether the magnet is per
manently magnetized or only magnetized by induction.

In Art. 74 we proved that the conditions satisfied at
the boundary of two dielectrics are

1. The tangential electric intensity must be the same
in both media.

2. When there is no free electricity on the surface
the normal electric polarization must be the same in
both. That is, if F, F' are the normal electric intensities
in the media whose specific inductive capacities are re
spectively K and K',

= K'F'.

If we compare these conditions with those satisfied at
the boundary of two media in the magnetic field and
remember that when the magnetization is induced, the
magnetic induction is equal to p times the magnetic
force, we see that we have complete analogy between the
disturbance of an electric field produced by the presence
of uncharged dielectrics and the disturbance in a magnetic
field produced by para- or dia-magnetic bodies in which
the magnetism is entirely induced.

264

MAGNETIC INDUCTION

[CH. VIII

Hence from the solution of any electrical problem
we can deduce that of the corresponding magnetic one
by writing magnetic force for electric intensity, and JJL
for K.

We can prove, as in Art. 75, that if 0, is the angle
which the direction of the magnetic force in air makes
with the normal at a point P on a surface, #2 the angle
which the magnetic force in the magnetizable substance
makes with the normal at the same point, then

//, tan 6l = tan 0.2.

Thus when the lines of force go from air to a para
magnetic substance they are bent away from the normal
in the substance, since in this case //- is greater than 1 ;
when they go from air to a diamagnetic substance they
are bent towards the normal, since in this case //, is less
than 1.

The effects produced when paramagnetic and diamag
netic spheres are placed in a uniform field of force are
shown in Figs. 39 and 85.

160] MAGNETIC INDUCTION 265

159. If fjb is infinite tan 01 vanishes, and then the lines
of force in air are at right angles to the surface, so that
the surface of a substance of infinite permeability is a
surface of equi-magnetic potential. The surface of such
a substance corresponds to the surface of an insulated
conductor without charge in electrostatics, and any pro
blem relating to such conductors can be at once applied
to the corresponding case in magnetism. In particular
we can apply the principle of images (Chap. V.) to find
the effect produced by any distribution of magnetic poles
in presence of a sphere of infinite magnetic permeability.

160. Sphere in uniform field. We showed in Art.
104 that if a sphere, whose radius is a, and whose specific
inductive capacity is K, is placed in a uniform electric
field, and if H is the electric intensity before the intro
duction of the sphere, then the field when the sphere is
present will at a point P outside the sphere, consist of H
and an electric intensity whose component along PO is
equal to

and whose component at right angles to PO in the direc
tion tending to increase 6 is

in these expressions OP = r, 0 is the angle OP makes
with the direction of H, 0 is the centre of the sphere.
Inside the sphere the electric intensity is constant,
parallel to H and equal to

3 H

266 MAGNETIC INDUCTION [CH. VIII

If we write //, for K the preceding expressions will give us
the magnetic force when a sphere of magnetic permea
bility yu, is placed in a uniform magnetic field where the
magnetic force is H.

A very important special case is when fj, is very large
compared with unity. In this case the magnetic forces
due to the sphere are approximately

ct
along PO, and H — sin 6

at right angles to it.

Inside the sphere the magnetic force is

and is very small compared with that outside. The mag
netic induction inside the sphere is 3H. Thus through
any area in the sphere at right angles to the magnetic
force, three times as many tubes of induction pass as
through an equal and parallel area at an infinite distance
from the sphere.

The resultant magnetic force in air vanishes round the
equator of the sphere.

161. Magnetic Shielding. Just as a conductor
is able to shield off the electric disturbance which one
electrical system would produce on another, so masses of
magnetizable material, for which JJL has a large value, will
shield off from one system magnetic forces due to another.

162] MAGNETIC INDUCTION 267

Inasmuch however as //, has a finite value for all sub
stances the magnetic shielding will not be so complete
as the electrical.

162. Iron Shell. We shall consider the protection
afforded by a spherical iron shell against a uniform mag
netic field. We saw in Art. 160 tha,t, when a solid iron
sphere is placed in a uniform magnetic field, the magnetism
induced on the sphere produces outside it a radial mag
netic force proportional to 2 cos 6/rs, and a tangential force
proportional to sin $/r3, and a constant force inside the
sphere. We shall now proceed to show that we can
satisfy the conditions of the problem of the spherical iron
shell by supposing each of the distributions of magnetism
induced on the two surfaces of the shell to give rise to
forces of this character.

Let a be the radius of the inner surface of the shell,
b that of the outer surface. Let H be the force in the
uniform field before the shell was introduced. Let the
magnetic forces due to the magnetism on the outer surface
of the shell consist, at a point P outside the sphere, of a

radial force

l cos 6

a tangential force

M-L sin 6

where r = OP and 6 is the angle OP makes with the
direction of H. The magnetic force due to this distribu
tion of magnetism will be uniform inside the sphere
whose radius is 6, it will act in the direction of H and
be equal to —

2G8 MAGNETIC INDUCTION [CH. VIII

Let the magnetization on the inner surface of the
shell give rise to magnetic forces given by similar ex
pressions with Mz written for M1 and a for b.

This system of forces, whatever be the values of
Mi and M^, satisfies the condition that as we cross the
surfaces of the shell the tangential components of the
magnetic force are continuous. We must now see if we
can choose Mlt M*. so as to make the normal magnetic
induction continuous.

The normal magnetic induction (reckoned positive
along the outward drawn normal) in the air just outside
the outer shell is equal to

# cos 0 + 2-~* cos 9 + -^cos 6,

O3 0*

the normal magnetic induction in the iron just inside the
outer surface of the shell is equal to

fj, (NCOS e -^ cos e + ^ cos e\ .

These are equal if

_ 2JT, 2Jf, /„ M, ZMt\
H+ +~= +~'

or, if

The normal magnetic induction in the iron just
outside the inner surface of the shell is

/ M ^M

a H cos 0 — -rf cos 6 + — - cos 0
V b3 a3

162] MAGNETIC INDUCTION 269

the normal magnetic induction in bhe air just inside the
shell is equal to

HCOS&- y-1 cos 0 -- -2 cos 6 ;
these are equal if

l=0*-l)tf ...... (2).

Equations (1) and (2) are satisfied if
M i ""

«i = 0* - U-ff -

The magnetic force in the hollow cavity is equal to

Substituting the values of M1 and M2 we see that this
is equal to

If fjb is very large compared with unity this is approxi
mately equal to

The denominator may be written in the form

2 volume of shell

7 ' volume of outer sphere '

270

MAGNETIC INDUCTION

[CH. VIII

Hence the force inside the shell will not be greatly
less than the force outside unless ^ is greater than the
ratio of the volume of the outer sphere to that of the
shell.

In the cases where /JL = 1000 and //,= 100, the ratio
of H', the force inside the sphere, to H for different values
of a/b is given in the following table.

a/6

H'/H
ti =1000

H'lH
At = 100

•99

3/23

9/15

•9

1/67

1/7

•8

1/109

1/12

•7

1/146

1/15

•6

1/175

1/18

•5

1/195

1/20

•4

1/209

1/22

•3

1/216

1/22

•2

1/221

1/23

•1

1/223

1/23

•o

1/223

1/23

Galvanometers which have to be used in places exposed
to the action of extraneous magnets are sometimes pro
tected by surrounding them with a thick-walled tube
made of soft iron.

We may regard the shielding effect of the shell as an
example of the tendency of the tubes of magnetic induc
tion to run as much as possible through iron ; to do this
they leave the hollow and crowd into the shell.

163. Expression for the energy in the magnetic
field. We shall suppose that the field contains per
manent magnets as well as pieces of magnetizable

163] MAGNETIC INDUCTION 271

substances magnetized by induction. If the distribution
of the permanent magnets is given, the magnetic field
will be quite determinate. The forces between magnetic
charges follow the same laws as those between electrical
ones. Hence the energy due to any system of magnetized
bodies will, if the magnetization due to induction is
proportional to the magnetic force, i.e. if JJL is constant,
be equal to the sum of one half the product of the
strength of each permanent pole into the magnetic
potential at that pole. Thus if Q is the potential energy
of the magnetic field,

where m is the strength of the permanent pole and II the
magnetic potential at that pole. Let us divide each of
the permanent magnets up into little magnets and con
sider the contribution of one of these to the energy. Let
/0 be the intensity of the permanent magnetization, and a
the area of the cross section : then the magnet has a pole
of permanent magnetism of strength 70a at A, another
pole of strength — 70a at B. If fl^, H# are the values of
the magnetic potentials at A and B, the contribution of
this magnet to the energy is therefore equal to

Now the magnet may be regarded as the origin of 47r/0a
tubes of magnetic induction forming closed curves running
through the magnet, leaving it at A and entering it at B ;
if ds is an element of one of these tubes, and R the
resultant magnetic force which acts along this element,
then

Rds,

272 MAGNETIC INDUCTION [CH. VIII

the integration being extended over the part of the tube
outside the magnet. Hence the contribution of this
magnet to the energy is the same as it would be if each
tube of which it is the origin had per unit length at P
an amount of energy equal to l/8?r of the resultant
magnetic force at P. The portion of the tube inside the
little magnet in which it has its origin, must not be taken
into account.

Now let us consider any small element of volume in
the magnetic field, let us take it as cylindrical in shape,
the axis of the cylinder being parallel to the resultant
magnetic force R at the element. Let / be the length of
this cylinder, co the area of its cross section. Now each
of the tubes of magnetic induction which pass through
the element and have not their origin within it, con
tributes R/STT units of energy for each unit of length of
the tube. Let J0 be the intensity of the permanent
magnetization of the element, / the induced magnetiza
tion, then the number of tubes of induction which pass
through unit area of the base of the cylinder is equal to
the value of the magnetic induction, i.e. it is equal to

but of these, 47r/0 have their origin in the element, and
hence the number of tubes per unit area which contribute
to the energy is equal to

and since / = kE and yu, = 1 + 4-7T&, this is equal to

fj,R,

therefore the number passing through the base of the
cylinder is equal to

164] MAGNETIC INDUCTION

The energy of the portion of each of the tubes within the
element is equal to 7?//87r, hence the energy contributed
by the element is

thus the energy per unit volume is equal to ^R^/S-rr. We
may then regard the energy of the magnetized system as
distributed throughout the magnetic field, there being
fjiR'2/87r units of energy in each unit volume of the field.

164. When a tube of induction enters a paramag
netic substance from air the resultant magnetic force is
— when the magnetization is entirely induced — less in
the paramagnetic substance than in air, the energy per
unit length will be less in the magnetic substance than
in the air since the energy per unit length of a tube of
induction is proportional to the resultant magnetic force
along it. Thus in accordance with the principle that
when a system is in equilibrium the potential energy is a
minimum, the tubes of induction will tend to leave the air
and crowd into the magnet, when this act does not involve
so great an increase in their length in the air as to
neutralize the diminution of the energy due to the parts
passing through the magnet.

Again, when a tube of induction enters a diamagnetic
substance the magnetic force inside this substance is
greater than it is in the air just outside, the tubes of
induction will therefore tend to avoid the diamagnetic
substance. Examples of this and the previous effect are
seen in Figs. 83 and 39.

A small piece of iron placed in a magnetic field where
the force is not uniform will tend to move from the weak

T. E. 18

2*74 MAGNETIC INDUCTION [CH. VIII

to the strong parts of the field, since by doing so it
encloses a greater number of tubes of induction and thus
produces a greater decrease in the energy. The direction
of the force tending to move the iron is in the direction
along which the rate of increase of R2 is greatest. This
is not in general the direction of the magnetic force.
Thus in the case of a bar magnet AB, the greatest
rate of increase in R* at C a point equidistant from A
and B is along the perpendicular let fall from C on A B,
and this is the direction in which a small sphere placed
at C will tend to move ; it is however at right angles to
the direction of the magnetic force at C.

There will be no force tending to move a piece of soft
iron placed in a uniform magnetic field.

A diamagnetic substance will tend to move from the
strong to the weak parts of the field, since by so doing
it will diminish the number of tubes of magnetic induc
tion enclosed by it and hence also the energy, for the tubes
of induction have more energy per unit length when they
are in the diamagnetic substance than when they are
in air.

165. Ellipsoids. We have hitherto only considered
the case of spheres placed in a uniform field. Bodies
which are much longer in one direction than another
have very interesting properties which are conveniently
studied by investigating the behaviour of ellipsoids placed
in a uniform magnetic field.

We saw in Art. 139 that the magnetic field, due to a
sphere uniformly magnetized in the direction of the axis
of a), might be regarded as due to two spheres, one of
uniform density p with its centre at 0', the other of

165] MAGNETIC INDUCTION 275

uniform density — p with its centre at 0, the points 0
and 0' being very close together and 00' parallel to the
axis of x\ the distance 00' is given by the condition
that pOO' is equal to the intensity of magnetization of
the sphere. An exactly similar proof will show that if
we have a body of any shape uniformly magnetized, the
magnetic potential due to it is the same as that due to
two bodies of the shape and size of the magnet, one
having the density p, the other the density — p, and so
placed that if the negative body is displaced through the
distance f in the direction of magnetization, it will coincide
with the positive body if pg = A, A being the intensity of
magnetization of the body.

Let us suppose that the body is uniformly magnetized
with intensity A in the direction of the axis of x, and let
pfl be the potential of the positive body at the point P,
then the potential of the negative body at P will be equal
to — p£l', where p£l' is the potential of the positive body
at P', if PP' is parallel to the axis of x and equal to f .

But since P'P is small,

The potential of the negative body is therefore

dfl

Thus the potential of the positive and negative bodies
together, and therefore of the magnetized body, will be

snce pg = A.

18—2

276 MAGNETIC INDUCTION [CH. VIII

If the body instead of being magnetized parallel to x
is uniformly magnetized so that the components of the
intensity parallel to x, y, z are respectively A, B, C, the
magnetic potential is

We shall now show that if an ellipsoid is placed in
a uniform magnetic field it will be uniformly magnetized
by induction. To prove this it will be sufficient to show
that if we superpose on to the uniform field, the field
due to a uniformly magnetized ellipsoid, it is possible to
choose the intensity of magnetization so as to satisfy
the two conditions, (1) that the tangential magnetic force
and (2) that the normal magnetic induction, are con
tinuous at the surface of the ellipsoid. The first of these
conditions is evidently satisfied whatever the intensity of
magnetization may be : we proceed to discuss the second
condition. The forces parallel to the axes of as, y, z (these
are taken along the axes of the ellipsoid) due to the
attraction of an ellipsoid of uniform unit density, are, see
Routh's Analytical Statics, vol. n. p. 112, equal to

Lx, My, Nz

respectively, where L, M, N are constant as long as the
point whose coordinates are x, y, z is inside the ellipsoid.

Hence by (1) since

cm T ,

-j— = - Lx, &c.,
dx

the magnetic potential inside the ellipsoid due to its
magnetization will be

BMy+CNz),

165] MAGNETIC INDUCTION 277

so that the magnetic forces parallel to the axes of #, y, z
due to the magnetization of the ellipsoid will be

- AL, - BM, - ON
respectively.

Hence if N-^ is the component of these forces along the
outward drawn normal to the surface of the ellipsoid,

N, = - (ALl + BMm + CNn\

where I, m, n are the direction cosines of the outward
drawn normal. If N2 is the force due to the magnetiza
tion on the ellipsoid in the same direction just outside the
ellipsoid, then

#„=#! + 4ir(lA + MB + nC)

= I A (47r - L) + mB (4nr-M) + nC(4iir - N).

Let X} Y, Z be the components of the force due to
the uniform field. Then NI, the total force inside the
ellipsoid along the outward drawn normal, will be given
by the equation

and if N? is the total force just outside the ellipsoid along
the outward drawn normal

If jju is the magnetic permeability of the ellipsoid, the
normal magnetic induction will be continuous if

that is if

- t*AL) + m (^Y-nBM) + n (^Z-

n{Z+C(4ar-N)

278 MAGNETIC INDUCTION [CH. VIII

But this condition will be satisfied if

XL — ~~

These equations give the intensity of magnetization of
an ellipsoid placed in a uniform magnetic field.

The force inside the ellipsoid due to its magnetization
has - AL, — BM, - CN for components parallel to the
axes of x, y, z respectively ; these components act in the
opposite direction to the external field and the force of
which these are the components is called the demagne
tizing force. We see from equations (2) that the
components of the demagnetizing force are

We shall now consider some special cases in detail.
Let us take the case of an infinitely long elliptic clyinder,
let the infinite axis be parallel to z, let 2«, 26 be the axes
in the direction of x and y ; then (Routh's Analytical
Statics, vol. n. p. 112)

L = 4,7r-^, M=4nr-^, N = 0.
a+b a+b

165] MAGNETIC INDUCTION 279

Thus A- . C*-1)* . - ** ,

where A; is the magnetic susceptibility.

We see from this equation that A/X is approximately
equal to k when (JJL — 1) 6/(a + 6) is very small, but only
then. A very common way of measuring k is to measure
A/X in the case of an elongated solid, magnetized along
the long axis; but we see that in the case of an elongated
cylinder this will to equal to k only when (//, - !)&/(« + b)
is very small. Now for some kinds of iron ^ is as great as
1000, hence if this method were to give in this case results
correct to one per cent., the long axis would have to be
100,000 times as long as the short one. This extreme
case will show the importance of using very elongated
figures when experimenting with substances of great
permeability. Unless this precaution is taken the ex
periments really determine the value of a/b and not any
magnetic property of the body.

When the body is an elongated ellipsoid of revolution
the ratio of the long to the short axis need not be so
enormous as in the case of the cylinder, but it must still
be very considerable. If the axis of x is the axis of revo
lution, then (Routh's Analytical Statics, vol. n. p. 112) we
have approximately

280 MAGNETIC INDUCTION [CH. VIII

Thus = - '• _ - _

Thus if fi were 1000, the ratio of a to b would have to
be about 900 to 1 in order that the assumption A/X = k
should be correct to one per cent.

166. Couple acting on the Ellipsoid. The mo

ment of the couple tending to twist the ellipsoid round
the axis of z, in the direction from x to y, is equal to
(volume of ellipsoid) ( YA - XB)

(47T + 0* - 1) L\ (47T + (fl - 1) M\ '

If the magnetic force in the external field is parallel
to the plane xy and is equal to H and makes an angle 6
with the axis of x,

and the couple is equal to

47rff 2 abc sin 0 cos 6 (//, - 1 )« (M - L)

3 (47T + (fj, - 1) L} (47T + (/A - 1) M\ '
If a > b, M is greater than L. Thus the couple tends to
make the long axis coincide in direction with the external
force, so that the ellipsoid, if free to turn, will set with its
long axis in the direction of the external force. This will
be the case whether /JL is greater or less than unity, i.e.
whether the substance is paramagnetic or diamagnetic,
so that in a uniform field both paramagnetic and dia
magnetic needles point along the lines of force. It
generally happens that a diamagnetic substance places
itself athwart the lines of magnetic force, this is due to
the want of uniformity in the field, in consequence of

167] MAGNETIC INDUCTION 281

which the diamagnetic substance tries to get as much of
itself as possible in the weakest part of the field. This
tendency varies as (/J, — 1); the couple we are investigating
in this article varies as (//, — I)'2, and as (//, — 1) is exceed
ingly small for bismuth, this couple will be overpowered
unless the field is exceptionally uniform.

167. Ellipsoid in Electric Field. The investiga
tion of Art. 165 enables us to find the distribution of
electrification induced on a conducting ellipsoid when
placed in a uniform electric field. To do this we must
make //- infinite in the expressions of Art. 165. The
quantity IA + mB + nC which occurs in the magnetic
problem corresponds to cr, the surface density of the elec
trification. Putting /u, =00 in equations (2) we find
(IX mY nZ}

If the force in the electric field is parallel to the axis of x

(7 = T'

Thus when the electric field is parallel to one of the axes
of the ellipsoid, the density of the electrification is, as in
the case of a sphere, proportional to the cosine of the
angle which the normal to the surface makes with the
direction of the electric intensity in the undisturbed field.
By Coulomb's law the normal electric intensity at the
surface of the ellipsoid is equal to 4?rcr, i.e. to

L '

Thus the electric intensity at the surface of the ellipsoid
is 4s7r/L times the electric intensity in the same direction
in the undistributed field.

282 MAGNETIC INDUCTION [CH. VIII

If the ellipsoid is a very elongated one with its longer
axis in the direction of the electric force, then by Art. 165

4-7T a2

Thus, when a/b is large, 4?r/Z is a large quantity, and
the electric intensity at the end of the ellipsoid is very
large compared with the intensity in the undisturbed
field. Thus if a/b = 100, the electric intensity at the
end is about 2500 times that in the undisturbed field.
This result explains the power of sharply pointed con
ductors in discharging an electric field, for when these are
placed in even a moderate field the electric intensity at
the surface of the conductor is great enough to overcome
the insulating power of the air, see Art. 37, and the
electrification escapes.

If an ellipsoidal conductor is placed in a uniform field
of force, at right angles to the axis c and making an angle
0 with the axis a, we see from § 166 that the couple round
the axis of c tending to make the axis of a move towards
the external force is equal to

when F is the external electric force.

When the ellipsoid is one of revolution round the axis
of a, and a is large compared with b, the couple is ap

proximately

1 a3^2 sin 20

CHAPTER IX.

ELECTRIC CURRENTS.

168. Let two conductors A and B be at different
potentials, A being at the higher potential and having
a charge of positive electricity, while .B is at a lower
potential and has a charge of negative electricity ; then
if A is connected to B by a metallic wire the potential
of A will begin to diminish and A will lose some of its
positive charge, the potential of B will increase and B will
lose some of its negative charge, so that in a short time
the potentials of A and B will be equalized. During the
time in which the potentials of A and B are changing the
following phenomena will occur : the wire connecting A
and B will be heated and a magnetic field will be pro
duced which is most intense near the wire. If A and
B are merely charged conductors, their potentials are
equalized so rapidly, and the thermal and magnetic effects
are in consequence so transient, that it is somewhat
difficult to observe them. If, however, we maintain A
and B at constant potentials by connecting them with the
terminals of a voltaic battery the thermal and magnetic
effects will persist as long as the connection with the
battery is maintained, and are then easily observed.

284

ELECTRIC CURRENTS

[CH. IX

The wire connecting the two bodies A and B at different
potentials is said to be conveying a current of electricity,
and when A is losing its positive charge and B its negative
charge the current is said to flow from A to B along the

wire.

Let us consider the behaviour of the Faraday tubes
during the discharge of the conductors A, B. Before the
conductors were connected by the wire these tubes may
be supposed to be distributed somewhat as in the figure.

Fig. 86.

When the conducting wire CD is inserted, the tubes which
were previously in the region occupied by the wire cannot
subsist in the conductor, they therefore shrink, their
ends travelling along the wire until the ends which were
previously on A and B come close together and the effect
of these tubes is annulled. The distribution of the tubes
in the field before the wire was inserted was one in which
there was equilibrium between the tensions along the
tubes and the lateral repulsion they exert on each other :

171] ELECTRIC CURRENTS 285

now after the tubes in the wire have shrunk the lateral
repulsion they exerted is annulled and there will therefore
be an unbalanced pressure tending to push the surround
ing tubes such as JEF, GH into the wire, where they will
shrink like those previously in the wire. This process
will go on until all the tubes which originally stretched
from A to B have been forced into the wire and their
effects annulled.

The discharge of the conductors is thus accompanied
by the movement of the tubes in towards the wire and
the sliding of the ends of these tubes along the wire.
The positive ends of the tubes move on the whole from
A towards B along the wire, the negative ends from
B towards A.

169. Strength of the current. If we consider
any cross-section of the wire at P, and if in the time &t
N units of positive electricity cross it from A towards B
and N' units of negative electricity from B towards A,
(N + N')/St is called the strength of the current at P.
When the wire is in a steady state the strength of the
current must be the same at all points along the wire,
for if it were not the same at P as at Q a positive or
negative charge would accumulate between P and Q and
the state of the wire would not be steady.

170. Electrodes. Anode. Cathode. If the ends
R, S of a body through which a current is flowing are
portions of equipotential surfaces, then R and S are called
the electrodes, and if the current is in the direction RS,
R is called the anode and S the cathode.

171. Electrolysis. In addition to the thermal arid
magnetic effects mentioned in Art. 168, there is another

286 ELECTRIC CURRENTS [CH. IX

effect characteristic of the passages of the current through
a large class of substances called electrolytes. Suppose
for example that a current passes between platinum plates
immersed in a dilute solution of sulphuric acid, then the
solution suffers chemical decomposition to some extent
and oxygen is liberated at the platinum anode, hydrogen
at the platinum cathode. There is no liberation of hydro
gen or oxygen in the portions of the liquid not in contact
with the platinum plates however far apart these plates
may be. Substances whose constituents are separated in
this way by the current are called electrolytes, and the act
of separation is called electrolysis. Electrolytes may be
solids, liquids, or, as recent experiments have shown, gases.
Iodide of silver is an example of a solid electrolyte, while
as examples of liquid electrolytes we have solutions of a
great number of mineral salts or acids as well as many
fused salts.

The constituents into which the electrolyte is separated
by the current are called the ions : the constituent which
is deposited at the anode is called the anion, that which
is deposited at the cathode the cation. With very few
exceptions, an element, or such a group of elements as
is called by chemists a ' radical/ is deposited at the same
electrode from whatever compound it is liberated; thus
for example hydrogen and the metals are cations from
whatever compounds they are liberated, while chlorine
is always an anion.

The amount of the ions deposited by the passage of a
current through an electrolyte was shown by Faraday to
be connected by a very simple relation with the quantity
of electricity which passes through the electrolyte.

173] ELECTRIC CURRENTS 287

172. Faraday's First Law of Electrolysis. The

quantity of an electrolyte decomposed by the passage of
a current of electricity is directly proportional to the
quantity of electricity which passes through it.

Thus as long as the quantity of electricity passing
through an electrolyte remains the same, it is immaterial
whether the electricity passes as a very intense current
for a short time or as a very weak current for a long time.

173. Faraday's Second Law of Electrolysis.

If the same quantity of electricity passes through different
electrolytes the weights of the different ions deposited will
be proportional to the chemical equivalents of the ions.

Thus, if the same current passes through a series
of electrolytes from which it deposits as ions, hydrogen,
oxygen, silver, and chlorine, then for every gramme of
hydrogen deposited, 8 grammes of oxygen, 108 grammes
of silver and 35'5 grammes of chlorine will be deposited.

If we define the electro-chemical equivalent of a sub
stance as the number of grammes of that substance depo
sited during the passage of the unit charge of electricity,
we see that Faraday's Laws may be comprised in the
statement that the number of grammes of an ion deposited
during the passage of a current through an electrolyte is
equal to the number of units of electricity which have
passed through the electrolyte multiplied by the electro
chemical equivalent of the ion.

Elements which form two series of salts, such as copper,
which forms cuprous and cupric salts, or iron, which forms
ferrous arid ferric salts, have different electro-chemical
equivalents according as they are deposited from solutions
of the cuprous or cupric, ferrous or ferric salts. The

288 ELECTRIC CURRENTS [CH. IX

electro-chemical equivalents of a few substances are given
in the following table ; the numbers represent the weight
in grammes of the substance deposited by the passage of
one electro-magnetic unit of electricity (see Chap. xn.).

Hydrogen '00010352.

Oxygen . , '000828.

Chlorine '003675.

Iron (from ferrous salts) '002898.

„ (from ferric salts) '001932.

Copper (from cuprous salts) '006522.

„ (from cupric salts) '003261.

Silver '01118.

The chemical composition of the portions of the elec
trolyte situated between the electrodes is unchanged by
the passage of the current. Imagine a plane drawn across
the electrolyte, there must pass in any time towards the
cathode across the plane an amount of the cation chemi
cally equivalent to that of the anion deposited in the
same time at the anode ; while a corresponding amount
of the anion must cross the plane towards the anode.
Thus in every part of the electrolyte the cation is moving
in the direction of the current, the anion in the opposite
direction.

Faraday's laws of electrolysis give a method of
measuring the quantity of electricity which has passed
through a conductor in any time and hence of measuring
the average current. For if we place an electrolyte in
circuit with the conductor in such a way that the current
through the electrolyte is always equal to that through the
conductor, then the amount of the electrolyte decomposed
will be proportional to the quantity of electricity which

174] ELECTRIC CURRENTS 289

has passed through the conductor; if we divide the
weight in grammes of the deposit of one of the ions by
the electro-chemical equivalent of that ion we get the
number of electro- magnetic units of electricity which has
passed through the conductor, dividing this by the time
we get the average current in electro-magnetic units.

An electrolytic cell used in this way is called a volta
meter ; the forms most frequently used are those in which
we weigh the amount of copper deposited from a solution
of copper sulphate, or of silver from a solution of silver
nitrate, or measure the amount of hydrogen liberated by
the passage of the current through acidulated water.

174. Relation between Electromotive Force
and Current. Ohm's Law. The work done by the
electric forces on unit charge of electricity in going
from a point A to another point B is called the electro
motive force from A to B. It is frequently written as
the E.M.F. from A to B.

Ohm's Law. The relation between the electromotive
force and the current was enunciated by Ohm in 1827,
and goes by the name of Ohm's Law.

This law states that if E is the electromotive force
between two points A and B of a wire, / the current
passing along the wire between these points, then

E=RI,

where R is a quantity called the resistance of the wire.
The point of Ohm's Law is that the quantity R defined
by this equation is independent of the strength of the
current flowing through the wire, and depends only upon
the shape and size of the wire, the material of which it is
made, and upon its temperature and state of strain.

T. E. 19

290 ELECTRIC CURRENTS [CH. IX

The most searching investigations have been made as
to the truth of this law when currents pass through
metals or electrolytes; these have all failed to discover
any exceptions to it, though from the accuracy with
which resistances can be measured (in several investiga
tions an accuracy of one part in 100,000 has been attained)
the tests to which it has been subjected are exceptionally
severe.

Ohm's Law does not however hold when the currents
pass through rarefied gases.

175. Resistance of a number of Conductors in
Series. Suppose we have a number of wires AB, CD,

A etc o«i FIG H

Fig. 87.

EF... (Fig. 87) connected together so that B is in contact
with (7, D with E, F with G and so on. This method of
connection is called putting the wires in series.

Let rlt TV, r3 ... be the resistances of the wires AB,
CD, EF... and let i be the current entering the circuit
AB, CD... at A, then the current i will flow through
each of the conductors. Let us consider the case when
the field is steady, then if VA) v£, VG, &c. denote the
potentials at A, B, C, &c. respectively, the .E.M.F. from
A to B is VA — VB ; thus we have by Ohm's Law,

But since B and C are in contact they will, if the wires

176] ELECTRIC CURRENTS 291

are made of the same substance, be at the same potential ;
hence VB = vC) VD = VE, and so on ; hence adding the pre
ceding equations we get

But if R is the resistance between A and b\ then by
Ohm's Law we have

VA — VF = Ri.

Comparing this expression with the preceding, we see
that

-8 = r1+r, + f,+ ....

Hence when a system of conductors are put in series, the
resistance of the series is equal to the sum of the resist
ances of the individual conductors.

176. Resistance of a number of Conductors
arranged in Parallel. If the wires instead of being
arranged so that the end of one coincides with the
beginning of the next, as in the last example, are arranged
as in Fig. 88, the beginnings of all the wires being in

Fig. 88.

contact, as are also their ends, the resistances are said to
be arranged in parallel, or in multiple arc.

We proceed now to find the resistance of a system of
wires so arranged. Let i be the current flowing up to A,
let this divide itself into currents il} ia, i3... flowing through

19—2

292 ELECTRIC CURRENTS [CH. IX

the circuits ACB, ADB, AEB... whose resistances are
n> ra, 7-3... respectively. Then if VA, VB are the potentials
of A and B respectively, we have by Ohm's Law

Now

But if R is the resistance of the system of conductors,
then by Ohm's Law,

Ri = (VA — VB) ;

hence comparing this expression with the preceding one
we see that

1 =1 I 1

R ~ i\ r2 rs

or the reciprocal of the resistance of a number of con
ductors in parallel is equal to the sum of the reciprocals
of the individual resistances. The reciprocal of the resist
ance of a conductor is called its conductivity, hence we
see that we may express the result of this investigation
by saying that the conductivity of a number of conductors
in parallel is equal to the sum of the conductivities of
the individual conductors.

In the special case when all the wires connected up in
multiple arc have the same resistance, and if there are n
wires, their resistance when in multiple arc is l/n of the
resistance of one of the individual wires.

178] ELECTRIC CURRENTS 293

177. Specific resistance of a substance. If we

have a wire whose length is I and whose cross section is
uniform and of area a, we may regard it as built up of
cubes whose edges are of unit length, in the following
way ; take a wire formed by placing I of these cubes in
series and then place a of these filaments in parallel ; the
resistance of this system is evidently the same as that
of the wire under consideration. If a is the resistance of
one of the cubes the resistance of the filament formed by
placing I such cubes in series is la, and when a of these
filaments are placed in parallel the resistance of the
system is IO-/OL ; hence the resistance of the wire is

la-
a

Since a only depends on the material of which the wire
is made we see that the resistance of a wire of uniform
cross section is proportional to the length and inversely
proportional to the area of the cross section.

The quantity denoted by a in the preceding expression
is called the specific resistance of the substance of which
the wire is made ; it is the resistance of a cube of the
substance of which the edge is equal to the unit of length,
the current passing through the cube parallel to one of its
edges.

178. Heat generated by the passage of a cur
rent through a conductor. Let A and B be two points
connected by a conductor, let E be the electromotive
force from A to B. By the definition of electromotive
force, work equal to E is done on unit positive charge
when it goes from A to B, and on unit negative charge

294 ELECTRIC CURRENTS [CH. IX

when it goes from B to A ; hence if in unit time N units
of positive charge go from A to B and N' units of nega
tive charge from B to A, the work done is E (N + Nf).
But N + N' is equal to C, the strength of the current
flowing from A to B, thus the work done is equal to EC.
If R is the resistance of the conductor between A and B,
E = RC ; thus the work done in unit time is equal to RC2.
We see that the same amount of work would be spent
in driving a current of the same intensity in the reverse
direction, viz. from B to A.

By the principle of the Conservation of Energy the
work spent by the electric forces in driving the current
cannot be lost, it must give rise to an equivalent
amount of energy of some kind or other. The passage
of the current heats the conductor, but if the heat is
caused to leave the conductor as soon as produced the
state of the conductor is not altered by the passage of
the current. The mechanical equivalent of the heat pro
duced in the conductor was shown by Joule to be equal
to the work spent in driving the current through the con
ductor, so that the work done in driving the current is
in this case entirely converted into heat. Thus if H is
the mechanical equivalent of the heat produced in time t,

The law expressed by this equation is called Joule's Law.
It states that the heat produced in a given time is pro
portional to the square of the strength of the current.

Since by Ohm's Law E = RC, the heat produced in
the time t is also equal to

179] ELECTRIC CURRENTS 295

179. Voltaic Cell. We have seen that in an electric
field due to any distribution of positive and negative
electricity, the work done when unit charge is taken
round a closed circuit vanishes; the electric intensity
due to such a field tending to stop the unit charge in
some parts of its course and to help it on in others.
Hence such a field cannot produce a steady current
round a closed circuit. To maintain such a current
other forces must come into play by which work can be
done ; this work may be supplied from chemical sources,
as in the voltaic battery ; from thermal sources, as in the
thermoelectric circuit ; or by mechanical means, as when
currents are produced by dynamos. We shall consider
here the case of the voltaic circuit. Let us consider
the simple form of battery consisting of two plates, one
of zinc, the other of copper, dipping into a vessel
containing dilute sulphuric acid. If the zinc and copper
plates are connected by a wire, a current will flow
round the circuit, flowing from the zinc to the copper
through the acid, and from the copper to the zinc
through the wire. When the current flows round the
circuit the zinc is attacked by the acid and zinc
sulphate is formed. For each unit of electricity that
flows round the circuit one electro-chemical equivalent of
zinc and sulphuric acid disappears and equivalent amounts
of zinc sulphate and hydrogen are formed. Now if a piece
of pure zinc is placed in dilute acid very little chemical
action goes on, but if a piece of copper is attached to
the zinc the latter is immediately attacked by the acid
and zinc sulphate and hydrogen are produced ; this action
is accompanied by a considerable heating effect, and we
find that for each gramme of zinc consumed a definite

ELECTRIC CURRENTS [CH. IX

amount of heat is produced. Now let us consider two
vessels (a) and (/8), such that in (a) the zinc and copper
form the plates of a battery, while in (0) the zinc has
merely got a bit of copper fastened to it : let a definite
amount of zinc be consumed in the latter and then let the
current run through the battery until the same amount
of zinc has been consumed in (a.) as in (13). The same
amount of chemical combination has gone on in the two
cells, hence the loss of chemical energy is the same in
(a) as in (/3). This energy has been converted into heat
in both cases, the difference being that in the cell (#)
the heat is produced close to the zinc plate, while in (a)
the places where heat is produced are distributed through
the whole of the circuit, and if the wire connecting the
plates has a much greater resistance than the liquid
between them, by far the greater portion of the heat
is produced in the wire, and not in the liquid in the
neighbourhood of the zinc. Though the distribution of
the places in which the heat is produced is different in the
two cases, yet, since the same changes have gone on in the
two cells, it follows from the principle of the Conservation
of Energy, that the total amount of heat produced in the
two cases must be the same. Thus the total amount of
heat produced by the battery cell (a) must be equivalent
to that developed by the combination of the amount of
zinc consumed in the cell while the current is passing
with the equivalent amount of sulphuric acid.

180. Electromotive Force of a Cell. If C is the

current, R the resistance of the wire between the plates,
r that of the liquid between the plates, t the time the
current has been flowing, then by Joule's law the mechanical

180] ELECTRIC CURRENTS 297

equivalent of the heat generated in the wire is RCH, that
of the heat generated in the liquid is rCH. We shall
see in Chapter xm, that when a current flows across
the junction of two different metals, heat is produced
or absorbed at the junction ; this effect is called the
Peltier effect. The laws governing the thermal effects
at the junction of two metals differ very materially from
Joule's law. The heat developed in accordance with
Joule's law in a conductor AB is, as long as the strength
of the current remains unaltered, the same whether the
current flows from A to B or from B to A. The thermal
effects at the junction of two metals G and D depend
upon the direction of the current ; thus if there is a
development of heat when the current flows across the
junction from C to D, there will be an absorption of heat
at the junction, if the current flows from D to C. These
heat effects which change sign with the current are called
reversible heat effects. The heat developed at the junction
of two substances in unit time is directly proportional to
the strength of the current and not to the square of the
strength.

In the case of the voltaic cell formed of dilute acid
and zinc and copper plates, the current passes across the
junctions of the zinc and acid and of the acid and copper
as well as across the metallic junctions which occur in the
wire used to connect up the two plates. Let P be the
total heat developed at these junctions when traversed
by unit current for unit time. Then the total amount
of heat developed in the voltaic cell is

RCH+rCH + PCt.

Since a current C has passed through the cell for a
time t, the number of units of electricity which have

298 ELECTRIC CURRENTS [CH. IX

passed through the cell is Ct, hence, if e is the electro
chemical equivalent of zinc: eCt grammes of zinc have been
converted into zinc sulphate. Let w be the mechanical
equivalent of the heat produced when one gramme of zinc
is turned into zinc sulphate, then the mechanical equi
valent of the heat which would be developed by the
chemical action which has taken place in the cell is eCtw ;
but this must be equal to the mechanical equivalent of
the heat developed in the cell, and hence we have

RCH + rCH + PCt = eCtw,
or (R + r)C=ew-P.

The quantity on the right-hand side is called the electro
motive force of the cell.

We see that it is equal to the sum of the products of
the current through the external circuit and the external
resistance and the current through the battery and the
battery resistance.

We shall now prove that if the zinc and copper plates
instead of being joined by a wire are connected to the
plates of a condenser, then if these plates are made of
the same material, they will be at different potentials,
and the difference between their potentials will equal the
electromotive force of the battery. For when the system
has got into a state of equilibrium, and any change
is made in the electrical conditions, the increase in the
electrical energy must equal the energy lost in making
the change. Suppose that the potential of the plate of
the condenser in connection with the copper plate in the
battery exceeds by E the potential of the other plate
of the condenser in connection with the zinc plate of
the battery ; and suppose now that the electrical state is

180] ELECTRIC CURRENTS 299

altered by a quantity of electricity equal to BQ passing
from the plate of the condenser at low potential to the
plate at high potential through the battery from the zinc
to the copper. The electrical energy of the condenser is
increased by EBQ, while the passage of this quantity of
electricity will develop at the junctions of the different
substances in the cell a quantity of heat whose mechani
cal equivalent is equal to PBQ. If t is the time this
charge takes to pass from the one plate to the other, the
average current will be equal to $Q/t, hence the heat
developed in accordance with Joule's law will be pro
portional to (&Q/t)2 x t or to (BQ)2/t ; by making &Q small
enough, we can make this exceedingly small compared with
either ESQ or P&Q which depend on the first powers of
SQ. The loss of chemical energy is eSQ x w, and this
must be equal to the heat produced plus the increase in
the electrical energy, hence we have

or E= ew - P,

that is, the difference of potential between the plates of
the condenser is equal to the electromotive force of the
battery. Hence we can determine this electromotive
force by measuring the difference of potential.

The simple form of voltaic cell just described does not
give a constant electromotive force, as the hydrogen pro
duced by the chemical action does not all escape from the
cell ; some of it adheres to the copper plate, forming a
gaseous film which increases the resistance and diminishes
the electromotive force of the cell.

The copper plate with the hydrogen adhering to it is
said to be polarized and to be the seat of a back electro-

300 ELECTRIC CURRENTS [CH. IX

motive force which makes the electromotive force of the
battery less than its maximum theoretical value. We
shall perhaps get a clearer view of the condition of the
copper plate with its film of hydrogen from the following
considerations. The hydrogen in an electrolyte follows
the current and thus behaves as if it had a positive
charge of electricity ; if now the hydrogen ions when
they come up to the copper plate, do not at once give up
their charges to the plate, but remain charged at a small
distance from it ; we shall have what is equivalent to
a charged parallel plate condenser at the copper plate,
the positively charged hydrogen atoms corresponding to
the positive plate of the condenser, and the copper to the
negative plate. If the positively charged hydrogen ions
charge up the positive plate of this condenser, driving off
by induction an equal positive charge from the copper
plate, instead of giving up their charge directly to this
plate ; the flow of electricity through the battery increases
the charge in the condenser. To increase this charge
work has to be done. If V is the potential difference
between the plates of the condenser, the rest of the nota
tion being the same as on p. 298, we have

(R + r) CH + VCt + PCt = ewCt
or (R + r)C=ew-P- V;

thus the electromotive force of the cell is diminished by
the potential difference between the plates of the condenser.
Another cause of inconstancy is that the zinc sulphate
formed acts as an electrolyte and carries some of the
current ; the zinc, travelling with the current, is deposited
against the copper plate and alters the electromotive force
of the cell.

181]

ELECTRIC CURRENTS

301

The deposition of hydrogen against the positive plate
of the battery, and its liberation as free hydrogen, can be
avoided in several ways ; in the Bichromate Battery the
copper plate is replaced by carbon, and potassium bichro
mate is added to the sulphuric acid ; as the bichromate is
an active oxidising agent it oxidises the hydrogen as soon
as it is formed, and thus prevents its accumulation on the
positive plate.

181. DanielPs Cell. In Daniell's cell, the zinc and
sulphuric acid are enclosed in a porous pot (Fig. 89) made

ZINC ROD
SULPHURIC ACID SOL

POROUS POT
CORPER SULPHATE SOL

COPPER CYLINDER

Fig. 89.

of unglazed earthenware ; the copper electrode usually
takes the shape of a cylindrical copper vessel, in which
the porous pot is placed. The space between the porous
pot and the copper is filled with a saturated solution of
copper sulphate, in which crystals of copper sulphate are
placed to replace the copper sulphate used up during the
working of the cell. When the sulphuric acid acts upon
the zinc, zinc sulphate is formed and hydrogen gas libe
rated ; the hydrogen following the current, travels through

302 ELECTRIC CURRENTS [CH. IX

the porous pot, where it meets with the copper sulphate,
chemical action takes place and sulphuric acid is formed
and copper set free. This copper travels to the copper
cylinder and is there deposited. Thus in this cell instead
of hydrogen being deposited on the copper, we have copper
deposited, so that no change takes place in the condition
of the positive pole and there is no polarization.

182. Calculation of E.M.F. of DanielPs Cell.

The chemical energy lost in the cell during the passage
of one unit of electricity may be calculated as follows :
in the porous pot we have one electro-chemical equivalent
of zinc sulphate formed while one equivalent of sulphuric
acid disappears ; in the fluid outside this pot one equiva
lent of sulphuric acid is formed and one equivalent of
copper sulphate disappears, thus the chemical energy lost
is that which is lost when the copper in one electro
chemical equivalent of copper sulphate is replaced by the
equivalent quantity of zinc.

Now the electro-chemical equivalent of copper is
•003261 grammes, and when 1 gramme of copper is
dissolved in sulphuric acid the heat given out is 909*5
thermal units, or 909*5 x 4*2 x 107 mechanical units, since
the mechanical equivalent of heat on the C. G. s. system
is 4*2 x 107. Thus the heat given out when one electro
chemical equivalent of copper is dissolved in sulphuric
acid is '003261 x 909*5 x 4*2 x 107 - 1*245 x 108 me
chanical units.

The electro-chemical equivalent of zinc is "003364
grammes, and the heat developed when 1 gramme of
zinc is dissolved in sulphuric acid is 1670 x 4*2 x 107
mechanical units. Hence the heat developed when one

184] ELECTRIC CURRENTS 303

'electro-chemical equivalent of zinc is dissolved in sulphuric
acid is '003364 x 1670 x 4'2 x 107 = 2'359 x 108 mechanical
units.

Thus the loss of chemical energy in the porous pot is
2'359 x 108 while the gain in the copper sulphate is
1-245 x 108, thus the total loss is 1114 x 108. Thus ew
in Art. 180 = 1114 x 108. The electromotive force of a
Daniell's cell is about T028 x 108. We see from the near
agreement of these values that the reversible thermal
effects (see Art. 180) are of relatively small importance,
though if we ascribe the difference between the two num
bers to this cause these effects would be much greater
than those observed when a current flows across the
junction of two metals.

183. In Grove's cell the hydrogen at the positive
pole is got rid of by oxidising it by strong nitric acid. The
zinc and sulphuric acid are placed in a porous pot, and this
is placed in a larger cell of glazed earthenware containing
nitric acid ; the positive pole is a strip of platinum foil
dipping into the nitric acid. This cell has a large electro
motive force, viz. T97 x 108.

Bunsen's cell is a modification of Grove's, in which
the platinum is replaced by hard gas carbon.

184. Clark's cell, which on account of its constancy
is very useful as a standard of electromotive force,
is made as follows. The outer vessel (Fig. 90) is a small
test-tube containing a glass tube down which a platinum
wire passes ; a quantity of pure redistilled mercury suffi
cient to cover the end of this wire is then poured into
the tube ; on the mercury rests a paste made by mixing

304 ELECTRIC CURRENTS [CH. IX

mercurous sulphate, saturated zinc sulphate and a little
zitic oxide to neutralize it ; a rod of pure zinc dips into
the paste and is held in position by passing through a

MARINE GLUE

ZINC SULPHATE SOLUTION
ZINC SULPHATE CRYSTALS
MERCUROUS SULPHATE

MERCURY
PLATINUM WIRE

Fig. 90.

cork in the mouth of the test-tube. The electromotive
force of this cell is T434 x 108 at 15° Centigrade.

Cadmium cell. In this cell the zinc of the Clark cell
is replaced by Cadmium, the negative electrode instead
of being zinc is an amalgam containing twelve parts by
weight of Cadmium in 100 of the amalgam; the zinc
sulphate solution is replaced by a saturated solution of
Cadmium sulphate; the rest of the cell is the same as
in the Clark cell. This all has a smaller temperature
coefficient than the Clark cell and is the one now most
frequently used as a standard ; its electromotive force at
t°C. is
T0185 - 0-000038 (t - 20) - 0*00000065 (t - 20)2 volts.

185. Polarization. When two platinum plates are
immersed in a cell containing acidulated water, and a
current from a battery is sent from one plate to the other

186] ELECTRIC CURRENTS 305

through the water, we find that the current for some time
after it begins to flow is not steady bat keeps diminishing.
If we observe the condition of the plates, we shall find
that oxygen adheres to the plate A, at which the current
enters the cell, while hydrogen adheres to the other plate
B, by which the current leaves the cell. If these plates
are now disconnected from the battery and connected by
a wire, a current will flow round the circuit so formed,
the current going from the plate B to the plate A
through the electrolyte and from A to B through the
wire. This current is thus in the opposite direction
to that which originally passed through the cell. The
plates are said to be polarized, and the E.M.F. round the
circuit, when they are first connected by the wire, is called
the electromotive force of polarization. When the plates
are disconnected from the battery and connected by the
wire the hydrogen and oxygen gradually disappear from
the plates as the current passes. In fact we may regard
the polarized plates as forming a voltaic battery, in which
the chemical action maintaining the current is the com
bination of hydrogen and oxygen to form water. Though
hydrogen and oxygen do not combine at ordinary tem
peratures if merely mixed together, yet the oxygen and
hydrogen condensed on the platinum plates combine
readily as soon as these plates are connected by a wire
so as to make the oxygen and hydrogen parts of a closed
electrical circuit. There are numerous other examples of
the way in which the formation of such a circuit facilitates
chemical combination.

186. A Finite Electromotive Force is required
to liberate the Ions from an Electrolyte. This follows

T. E. 20

306 ELECTRIC CURRENTS [CH. IX

at once by the principle of the Conservation of Energy
if we assume the truth of Faraday's Law of Electrolysis.
Thus suppose for example that we have a single Daniell's
cell placed in series with an electrolytic cell containing
acidulated water; then if this arrangement could produce
a current which would liberate hydrogen and oxygen from
the electrolytic cell, for each electro-chemical equivalent of
zinc consumed in the battery an electro-chemical equiva
lent of water would be decomposed in the electrolytic cell.
Now when one electro-chemical equivalent of hydrogen
combines with oxygen to form water, 1'47 x 108 mechanical
units of heat are produced, and the decomposition of one
electro-chemical equivalent of water into free hydrogen
and oxygen would therefore correspond to the gain of this
amount of energy. But for each electro-chemical equi
valent of zinc consumed in the battery the chemical energy
lost is (Art. 182) equal to I'll 4 x 108 mechanical units.
Hence we see that if the water in the electrolytic cell
were decomposed, 3'56 x 107 units of energy would be
gained for each unit of electricity that passed through
the cell : as this is not in accordance with the principle
of the Conservation of Energy the decomposition of the
water cannot go on. We see that electrolytic decom
position can only go on when the loss of energy in the
battery is greater than the gain of energy in the electro
lytic cell.

If we attempt to decompose an electrolyte, acidulated
water for example, by an insufficient electromotive force
the following phenomena occur. When the battery is
first connected to the cell a current of electricity runs
through the cell, hydrogen travelling with the current
to the plate where the current leaves the cell, oxygen

186] ELECTRIC CURRENTS 307

travelling up against the current to the other plate.
Neither the hydrogen nor the oxygen, however, is libe
rated at the plates, but adheres to the plates, polarizing
them and producing a back E.M.F. which tends to stop
the current ; as the current continues to flow the amount
of gas against the plates increases, and with it the polari
zation, until the E.M.F. of the polarization equals that of
the battery, when the current sinks to an excessively
small fraction of its original value. The current does
not stop entirely, a very small current continues to flow
through the cell. This current has however been shown
by v. Helmholtz to be due to hydrogen and oxygen
dissolved in the electrolytic cell and does not involve any
separation of water into free hydrogen and oxygen. The
way in which the residual current is carried is somewhat
as follows. Suppose that the battery with its small E.M.F.
has caused the current to flow through the cell until the
polarization of the plates is just sufficient to balance the
E.M.F. of the battery ; the oxygen dissolved in the water
near the hydrogen coated plate will attack the hydrogen
on this plate, combining with it to form water, and will,
by removing some of the hydrogen, reduce the polarization
of the plate; similarly the hydrogen dissolved in the water
or it may be absorbed in the plate, will attack the oxygen
on the oxygen coated plate and reduce its polarization.
The E.M.F. of the polarization being reduced in this way
no longer balances the E.M.F. of the battery; a current
therefore flows through the cell until the polarization
is again restored to its original value, to be again reduced
by the action of the dissolved gases. Thus in consequence
of the depolarizing action of the dissolved gases there
will be a continual current tending to keep the E.M.F.

20—2

308 ELECTRIC CURRENTS [CH. IX

of the polarization equal to that of the battery; the
current however is not accompanied by the liberation
of free hydrogen and oxygen and its production does not
violate the principle of the Conservation of Energy.

187. Cells in Series. When a series of voltaic cells,
Daniell's cells for example, are connected so that the zinc
pole of the first is joined up to the copper pole of the
second, the zinc pole of the second to the copper pole
of the third, and so on, the cells are said to be connected
up in series. In this case the total electromotive force of
the cells so connected up is equal to the sum of the
electromotive forces of the individual cells. We can see
this at once if we remember (see Art. 180) that the electo-
motive force of any system is equal to the difference
between the chemical energy lost, when unit of electricity
passes through the system, and the mechanical equivalent
of the reversible heat generated at junctions of different
substances : when the cells are connected in series the
same chemical changes and reversible heat effects go on
in each cell when unit of electricity passes through as
when the same quantity of electricity passes through the
cell by itself, hence the E.M.F. of the cells in series is the
sum of the E.M.F.'S of the individual cells.

The resistance of the cells when in series is the sum of
their resistances when separate. Thus if E is the E.M.F.
and r the resistance of a cell, the E.M.F. and resistance
of n such cells arranged in series are respectively nE
and nr.

188. Cells in parallel. If we have n similar cells
and connect all the copper terminals together for a new
terminal and all the zincs together for the other terminal

189J ELECTRIC CURRENTS 309

the cells are said to be arranged in parallel. In this case
we form what is equivalent to a large cell whose E.M.F.
is equal to E, that of any one of the cells, but whose
resistance is only r/n.

189. Suppose that we have N equal cells and wish to
arrange them so as to get the greatest current through a
given external resistance R. Let the cells be divided into
m sets, each of these sets consisting of n cells in series,
and let these m sets be connected up in parallel. The
E.M.F. of the battery thus formed will be nE, its resistance
nrjm, where E and r are respectively the E.M.F. and
resistance of one of the cells. The current through
the external resistance R will be equal to

nE E

n nr H r
R + — - + -

m n m

Now nm = N, hence the denominator of this expression
is the sum of two terms whose product is given, it will
therefore be least when the terms are equal, i.e. when

n
or

z? n
R = — r.

Since the denominator in this case is as small as possible
the current will have its maximum value. Since nr/m is
the resistance of the battery we see that we must arrange
the battery so as to make, if possible, the resistance of
the battery equal to the given external resistance. This
arrangement, though it gives the largest current, is not

310 ELECTRIC CURRENTS [CH. IX

economical, for as much heat is wasted in the battery as
is produced in the external circuit.

190. Distribution of a steady current in a
System of Conductors.

KirchhofPs Laws. The distribution of a steady
current in a network of linear conductors can be readily
determined by means of the following laws, which were
formulated by Kirchhoff.

1. The algebraical sum of the currents which meet at
any point is zero.

2. If we take any closed circuit the algebraical sum
of the products of the current and resistance in each of
the conductors in the circuit is equal to the electromotive
force in the circuit.

The first of these laws expresses that electricity is not
accumulating at any point in the system of conductors;
this must be true if the system is in a steady state.

The second follows at once from the relation (see
Art. 180)

where R is the external resistance, r the resistance of the
battery whose E.M.F. is E, and / the current through the
battery. For RI is the difference of potential between
the terminals of the battery, and by Ohm's law this is
equal to the sum of the products of the strength of
the current and the resistance for a series of conductors
forming a continuous link between the terminals of the
battery.

191. Wheatstone's Bridge. We shall illustrate
these laws by applying them to the system known as the

191] ELECTRIC CURRENTS 311

Wheatstone's Bridge. In this system a battery is placed
in a conductor AB, and five other conductors AC, BG, A D,
BD, CD are connected up in the way shown in Fig. 91.

Let E be the electromotive force of the battery, B the
resistance of the battery circuit AB, i.e. the resistance
of the battery itself plus the resistance of the wires con
necting its plates to A and B. Let G be the resistance
of CD, and b, a, OL, j3 the resistances of AC, BC, AD, BD
respectively. Let x be the current through the battery,
y the current through AC, z that through CD. By
Kirchhoff's first law the current through AD will be
x — yt that through CB y — z, and that through DB
x —y + z.

Since there is no electromotive force in the circuit
ACD we have by Kirchhoff's second law,
by 4- Gz — OL (x — y) — 0 ;

the negative sign is given to the last term because travelling
round the circuit in the direction ACD the current x-y
flows in the direction opposite to that in which we are
moving; rearranging the terms we get

(b + a)y + Gz-ouK = Q (1).

Since there is no electromotive force in the circuit
CDS, we have

Gz+ fi(x-y + z)-a (y - z) = 0,
or -(a + £).?/ +(£ + « + £)* + #z? = 0 (2).

312 ELECTRIC CURRENTS

From (1) and (2) we get

[CH. IX

«* + £) + (6 + «)(a

(3).

Since the electromotive force round the circuit ACB
is E, we have

hence by (3), we have

x = G (a + 6 +

(6 + a) (a

^ - y 4. 2- = ((7 (a + 6) + a (6 + a)} —
where

...(4),

+ G (a + b) (a + ft) + a (a + /3) (a + b) - a (aa - bj3)
= BG(a + b + a + /3) + B(b+a)(a + l3)

+ G(a + b)(ot+j3) + aboi + abfi + aa/3 + 6a&

A is the sum of the products of the six resistances
B, G, a, b, ct, /3, taken three at a time, omitting the product
of any three which meet in a point.

191] ELECTRIC CURRENTS 313

In the expressions given in equations (4) for the
currents through the various branches of the network
of resistances, we see that the multiplier of E/& in the
expression for the current through an arm (P) (other than
CD) is the sum of the products of the resistances other
than the battery resistance and the resistance of P taken
two and two, omitting the product of any two which meet
at either of the extremities of the battery arm or at either
of the extremities of the arm P.

From these expressions we see at once that if we keep
all the resistances the same, then the current in one arm
(A) due to an electromotive force E in another arm (B),
is equal to the current in (B) when the electromotive E
is placed in the arm A. This reciprocal relation is not
confined to the case of six conductors, but is true what
ever the number of conductors may be.

We may write the expression for x given by equation
(4) in the form

~B+R'
where

P _ G (a + b) (a + ff ) + gaff + aotb + a/3b + aftb

R is the resistance, between A and B, of the crossed
quadrilateral ACBD.

We see that R = (sum of products of the 5 resistances
of this quadrilateral taken 3 at a time, leaving out the
product of any three that meet in a point): divided by
the sum of the products of the same resistances taken two
at a time, leaving out the product of any pair that meet
in A or B.

314 ELECTRIC CURRENTS [CH. IX

192. Conjugate Conductors. The current through
CD will vanish if

in this case AB and CD are said to be conjugate to each
other, they are so related that an electromotive force in
AB does not produce any current in CD: it follows from
the reciprocal relation that when this is the case an
electromotive force in CD will not produce any current
in AB.

The condition that CD should be conjugate to AB
may be got very simply in the following way. If no current
flows down CD, C and D must be at the same potential;
hence since z = Q, we have by Ohm's law

by = a O - y),

since the difference of potential between A and C is
equal to that between A and D.

Since the difference of potential between C and B is
equal to that between D and B, we have

hence eliminating y and x — y, we get

- = -

or b& = aa.

When this relation holds we may easily prove that

which we may write as

193] ELECTRIC CURRENTS 315

where S is the resistance of ADB, ACB placed in series,
P the resistance of the same conductors when in parallel,
and P' the resistance of GA D, CBD in parallel.

When AB is conjugate to CD, then in whatever
part of the network an electromotive force is placed,
the current through one of these arms is independent
of the resistance in the other. We may deduce this
from the preceding expressions for the currents in various
arms of the circuit; it can also be proved in the following
way, which is applicable to any number of conductors.
Suppose that an electromotive force in some branch of
the system produces a current through AB, then we may
introduce any E.M.F. we please into AB without altering
the current through its conjugate CD. We may in par
ticular introduce such an electromotive force as would
make the current through AB vanish, without altering
the current in CD, but the effect of making the current
in AB vanish would be the same as supposing AB to
have an infinite resistance; hence we may make the
resistance of AB infinite without altering the current
through CD.

193. We may use Wheatstone's Bridge to get a differ
ence of potential which is a very small fraction of that
of the battery in the Bridge. The difference of potential
between C and D is equal to Gz, i.e. to
G (aa - bj3) E

~K~~

it thus bears to E the ratio of G (aa — b/3) to A. By
making aa — b/3 small we can without using either very
small or very large resistances make the ratio of the poten
tial difference between C and D to E exceedingly small ;

816 ELECTRIC CURRENTS [CH. IX

for example, let a =101, a = 99, 6 = 0=100, B=G = 1.
Thus we find that this ratio is nearly equal to 1/4 x 106,
or the potential difference between C and D is only about
one four-millionth part of the E.M.F. of the battery.

194. Heat produced in the System of Con
ductors. Assuming Joule's law (see Art. 178) we shall
show that for all possible distributions consistent with
Kirchhoff's first law, the one that gives the minimum rate
of heat production is that given by the second law.

For, consider any closed circuit in a network of con
ductors, let u, v, w... be the currents through the arms
of this circuit as determined by Kirchhoff's laws, and
7\, r2, ... the corresponding resistances. The rate of heat
production in this closed circuit is by Joule's law equal to
r1w2+r2?/2+ ..................... (1).

Now suppose that the currents in this circuit are
altered in the most general way possible consistent with
leaving the currents in the conductors not in the closed
circuit unaltered, and consistent also with the condition
that the algebraical sum of the currents flowing into
any point should vanish : we see that these conditions
require that all the currents in the closed circuit should
be increased or diminished by the same amount. Let
them all be increased by f ; the rate of heat production
in the circuit is now by Joule's law

Now since the currents u, v, w are supposed to be
determined by Kirchhoff's laws

r-flL + r2v + ••• =0,

195] ELECTRIC CURRENTS 317

if there is no electromotive force in the closed circuit
Hence the rate of heat production is equal to

r^ + r^ + ... + (r1 + ra + rg+ ...)f2 ...... (2).

Of the two expressions (1) and (2) for the rate of heat
production (2) is always the greater ; hence we see that
any deviation of the currents from the values determined
by KirchhofFs law would involve an increase in the rate
of heat production.

195. Use of the Dissipation Function. We may

often conveniently deduce the actual distribution of the
currents by writing down F the expression for the rate of
heat production and making it a minimum, subject to the
condition that the algebraical sum of the currents which
meet in a point is zero. Or we may by the aid of this
condition express, as in the example of the Wheatstone's
Bridge, the current through the various arms in terms of
a small number of currents xy y, z, then express the rate
of heat production in terms of x, y, z.

F is often called the Dissipation Function.

When there are electromotive forces Ep, Eq in the
arms through which currents up, uq are flowing respec
tively, then the actual distribution of current is that
which makes

a minimum. Thus in the case of the Wheatstone's
Bridge (Art. 191)

F =

ELECTRIC CURRENTS [CH. IX

and equations (4) of Art. 191 are equivalent to

which are the conditions that F - 2Ex should be a
minimum.

A very important example of the principle that
steady currents distribute themselves so as to make the
rate of heat production as small as possible, is that of
the flow of a steady current through a uniform wire ; in
this case the rate of heat production is a minimum when
the current is uniformly distributed over the cross section
of the wire.

196. It follows from Art. 194 that if two electrodes
are connected by any network of conductors, the equivalent
resistance is in general increased, and is never diminished,
by an increase in the resistance of any arm of this net
work.

If R is the resistance between the electrodes, i the
current flowing in at one electrode and out at the other,
then Ei2 is the rate of heat production. Let A and B
respectively denote the network before and after the
increase in resistance in one or more of its arms. With
out altering the resistance alter the currents until the
distribution of currents through A is the same as that
actually existing in B. The rate of heat production for

197] ELECTRIC CURRENTS 319

the new distribution is by Art. 194 greater than that
in A. Now take this constrained system and without
altering the currents suppose that the resistances are
increased until they are the same as in B. Since the
resistances are increased without altering the currents
the rate of heat production is increased, so that as this
rate was greater than in A before the resistances were
increased it will a fortiori be greater afterwards. But
after the resistances were increased the currents and
resistances are the same as B, hence the rate of heat
production in B and therefore its resistance is greater
than that of A.

197. The following proof of the reciprocal relations
between the currents and the electromotive forces in a
network of conductors is due to Professor Wilberforce.
Let A, B be two of the points in a network of con
ductors, let RAB denote the resistance of the wire joining
AB', VA the potential of A, VB that of B, GAB the
current flowing along the wire from A to B, EAS the
electromotive force of a battery in AB, tending to make
the current flow through the battery in the direction AB',
let currents from an external source be led into the net
work, the current entering at a point A being denoted by
I A- Then if 2,1 A denotes the sum of all these currents

si A =o.

We have by Ohm's law,

RABCAB=VA-VB + EAB (1).

Let us suppose that another distribution of currents,
potentials, and electromotive forces is denoted by dashed
letters. We have by (1),

RAB GAB @'AB — ( VA - VB) VAB + EAB V AB -

320 ELECTRIC CURRENTS [CH. IX

Taking the whole network of conductors we have
^nABCABG'AB = 2(VA-VS) G'AB + ^EABC'AB.
The coefficient of VA is the sum of all the currents
that flow outwards from A, this must be equal to I'A,
hence

2<RAB CAB C' AB = 2iV AI' A + 2*EABC' AB.

Since the left-hand side is symmetrical with respect
to the accented and unaccented letters we have

Now suppose that all the /'s and /"s are zero, and
that all the E's are zero except EAB, all the Ens except
E CD] (2) becomes

EAB VAB = E'CD CCD,

i.e. that when unit electromotive force acts in AB, the
current sent through another branch CD of the network
is the same as the current through A B when unit electro
motive force acts in CD. Again in equation (2) suppose
that all the E's and E"s are zero, that a current IA is led
in at A and out at B, all the other /'s vanishing, and that
in the distribution represented by the dashed letters a
current I' c is led in at C and out at D, all the other /"s
vanishing, then by (2)

Thus if unit current be led in at A and out at B the
potential difference between C and D is the same as the
potential difference between A and B when unit current
is led in at C and out at D.

198. Distribution of Current through an infinite
Conductor. We shall now consider the case when the
currents instead of being constrained to flow along wires

198] ELECTRIC CURRENTS 321

are free fco distribute themselves through an unlimited
conductor whose conductivity is constant throughout its
volume. We shall suppose that the current is introduced
into this conductor by means of perfectly conducting
electrodes, i.e. electrodes made of a material whose specific
resistance vanishes. The currents will enter and leave the
conductor at right angles to the electrodes, for a tangential
current in the conductor would correspond to a finite tan
gential electric intensity in the conductor and therefore in
the electrode, but in the perfectly conducting electrode a
finite electric intensity would correspond to an infinite
current. Let A and B be the electrodes, i the current which
enters at A and leaves at B', then we shall prove that the
current at any point P in the conductor is in the same direc
tion as, and numerically equal to, the electric intensity at the
same point, if we suppose the conducting material between
the electrodes to be replaced by air, and the electrodes A
and B to have charges of electricity equal to ij^Tr and
— i/4f7r respectively. For the current is determined by
the conditions (1) that it is at right angles to the surfaces
A and B, and (2) that since the current is steady, and there
is no accumulation of electricity at any part of the con
ductor, the quantity of electricity which flows into any
region equals the quantity which flows out. Hence we
see that the outward flow over any closed surface enclos
ing A and not B is equal to i, over any closed surface
enclosing B and not A is equal to — i, and over any closed
surface enclosing neither or both of these surfaces is zero.
But the electric intensity, when the conductor is replaced
by air and A has a charge i/4>7r of positive electricity,
while B has an equal charge of negative electricity,
satisfies exactly the same conditions, which are sufficient to

T. E. 21

322 ELECTRIC CURRENTS [CH. IX

determine it without ambiguity; hence the current in the
conductor is equal to the electric intensity in the air and
is in the same direction. A line such that the tangent to
it at any point is in the direction of the current at that
point is called a stream-line. The stream-lines coincide
with the lines of force in the electrostatic problem.

199. If q is the intensity of the current at any point
P (i.e. the current flowing through unit area at right
angles to the stream-line at P), a the specific resistance
of the conductor, ds an element of the stream-line, then
by Ohm's law the E.M.F. between the electrodes A and B
is equal to

faqds,

the integral being extended from the surface of A to that
of B. As <r is constant, this is equal to

a-fqds.

If F is the electric intensity at P in the electrostatic
problem, since F = q, the E.M.F. between A and B is

equal to

trfFda-,

but if V is the difference of potential between A and B
in the electrostatic problem,

V=JFds.

Hence the E.M.F. between A and B is equal to 0V.
But if C is the electrostatic capacity of the two conductors,
since these have the charges ^'/4<7^ and — i/4>7r respectively,

Hence the E.M.F. between A and B =

199] ELECTRIC CURRENTS 823

or the resistance between A and B is equal to

We see from this that the resistance of a shell bounded
by concentric spherical surfaces, whose radii are a and b,
is equal to

The resistance per unit length of a shell of conducting
material bounded by two coaxial cylindrical surfaces whose
radii are a and b is equal to

a- , b
^a'

The resistance between two spherical electrodes whose
radii are a and b and whose centres are separated by a
distance R, where R is very large compared with either
a or 6, is equal to

4-7T ja 6 R\
approximately.

The resistance per unit length between two straight
parallel cylindrical wires whose radii are a and b, and
whose axes are at a distance R apart, where R is very
large compared with a or 6, is approximately

a , R*

s^'a-

If we have two infinite cylinders, one with a charge
of electricity E per unit length, the other with the charge
— E ; then if A and B are the centres of the sections of
these cylinders by a plane perpendicular to the axis and

21—2

324 ELECTRIC CURRENTS [CH. IX

P a point in this plane, the electrostatic potential at P
will be equal to

if the cylinders are so far apart that the electricity
may be regarded as uniformly distributed over them.
Thus the lines along which the electrostatic potential is
constant are those for which

jD T)

= a constant quantity.

That is, they are the series of circles for which A and B
are inverse points. The lines of force are the lines which
cut these circles at right angles, i.e. they are the series of
circles passing through A and B. But the lines of force
in the electrostatic problem coincide with the lines along
which the currents flow between two parallel cylinders as
electrodes; hence these currents flow in planes at right
angles to the axes of the cylinders, along the circles
passing through the two points in which these planes
intersect the axes of the cylinders.

Since the resistance of unit length of the cylinders is

the resistance of a length t is

a . R2

This will be the resistance of a thin lamina whose thick
ness is t when the current is led in by circular electrodes
radii a and 6, if the thickness of the lamina is so small
that the currents are compelled to flow parallel to the

199]

ELECTRIC CURRENTS

325

lamina. The lines of flow in this case are circles
passing through A and B; they are represented in
Fig. 92.

Fig. 92.

Since the currents flow along these circles we shall
not alter the distribution of current if we imagine the
lamina cut along one or other of these circles; hence
if the lamina is bounded by two circular areas such as
APB, BQA the lines of flow will be circular arcs passing
through A and E.

To find the resistance of a lamina so bounded, con
sider for a moment the flow through the unlimited
lamina. The current will flow from out of each electrode
approximately uniformly in all directions; hence if we
draw a series of circles intersecting at the constant angle
a at A and B, we may regard the whole lamina as made
up of the conductors between the stream-lines placed in

2-77-

multiple arc ; the number of these conductors is —

and

326 ELECTRIC CURRENTS [CH. IX

since the same current flows through each, the resistance
of any one of them is 2?r/a of the whole resistance; thus
the resistance of one of these conductors is

a , R*
^loga&'

Thus, for example, if the electrodes are placed on
the circumference of a complete circle, a = TT and the
resistance of the lamina is

200. Conditions satisfied when a current flows
from one medium to another. Let AB be a portion
of the surface of separation of two media, al the specific
resistance of the upper medium, o-2 that of the lower, let
6 and (j) be the angles which the directions of the current
in the upper and lower media respectively make with the
normal to the surface. Let qlt q.2 be the intensities of
the currents in the two media, i.e. the amount of current
flowing across unit areas drawn at right angles to the
direction of flow. Then since, when things are in a steady
state, there is no increase or decrease in the electricity
at the junction of the two media, the currents along the
normal must be equal in the two media.

Thus qlcos 6 = q2cos (j> .................. (1).

Again, the electric intensity parallel to the surface
must be equal in the two media, and since the electric
intensity in any direction is equal to the specific resistance
of the medium multiplied by the intensity of the current
in that direction, we have

o-fli sin 6 = cy?2 sin (f> ............... (2),

200] ELECTRIC CURRENTS 327

hence from (1) and (2) we have

crl tan 6 = cr2 tan (j>.

This relation between the directions of the currents
in the two media is identical in form with that given in
Arts. 74 and 157, for the relation between the directions
of the lines of electric intensity and of magnetic force
when these lines pass from one medium to another.

We see that if o-l is greater than cr2, then c£ is greater
than 6 ; hence when the current flows from a poor con
ductor into a better one the current is bent away from
the normal.

The bending of the current as it flows from one
medium into another is illustrated in Fig. 93, which is
taken from a paper by Qudncke. The figure represents

the current lines in a circular lamina, one half of which is
lead, the other half copper, the electrodes E, E being placed
on the circumference. It shows how the currents in going
from the worse conductor (the lead) to the better one (the

328 ELECTRIC CURRENTS [CH. IX

copper) get bent away from the normal to the surface of
separation.

The electric intensity parallel to the normal in the
medium whose specific resistance is a^ is

flrtfj cos 6,

that in the medium whose specific resistance is a2 is
<T2q2 cos (/>. Since q± cos 6 is by equation (1), equal to
q2cos(f), we see that if ^ differs from cr2 the normal
electric intensity will be discontinuous at the surface of
separation.

If the normal electric intensity is discontinuous there
must be a distribution of electricity over the surface such
that 4?r times the surface density of this distribution is
equal to the discontinuity in the normal electric intensity;
hence if s is the surface intensity of the electricity on the
surface, and if the current is flowing from the first medium
to the second

4f7rs = a^q.2 cos <f) — o-^ cos 6

= (0-2-0-1)^1 cos 6.

CHAPTER X

MAGNETIC FORCE DUE TO CURRENTS

201. It was not known until 1820 that an electric
current exerted any mechanical effect on a magnet in its
vicinity. In that year however Oersted, a Professor at
Copenhagen, showed that a magnet was deflected when
placed near a wire conveying an electric current.

When a long straight wire with a current flowing
through it was held near the magnet, the magnet tended
to place itself at right angles both to the wire and the
perpendicular let fall from the centre of the magnet on
the wire.

The lines of magnetic force due to a long straight wire
may be readily shown by making the wire pass through
a hole in a card-board disc over which iron filings are
sprinkled. When the disc is at right angles to the wire,
the iron filings will arrange themselves in circles when
the current is flowing; these circles are concentric, having
as their centre the point where the wire crosses the plane
of the disc.

The connection between the direction of the current
and that of the magnetic force is such that if the axis

330 MAGNETIC FORCE DUE TO CURRENTS [CH. X

of a right-handed screw (i.e. an ordinary corkscrew) coin
cides with the direction of the current, then if the screw
is screwed forward into a fixed nut in the direction of the
current the magnetic force at a point P due to the current
is in the direction in which P would move if it were rigidly
attached to the screw.

Many students will find that they can remember the
connection between the direction of the current and
the magnetic force more easily by means of a figure
than by a verbal rule. The following figure exhibits
this relation.

Fig. 94.

202. Ampere's law for the magnetic field due
to any closed linear circuit. This may be stated as
follows : At any point P, not in the wire conveying the
current, the magnetic forces due to the current can be
derived from a potential H where O = Cico, i being the
current flowing round the circuit, w the solid angle sub
tended by the circuit at P, and C a constant which
depends on the unit in which the current is expressed.

When the unit of current is what is known as the
'electromagnetic unit,' see Chap. XIL, C is unity. We

202] MAGNETIC FORCE DUE TO CURRENTS 331

shall in the following investigations suppose that the
current is measured in terms of this unit.

We see from Art. 134 that this is equivalent to saying
that the magnetic field due to a current is the same
as that due to a magnetic shell whose strength is i,
the boundary of the shell coinciding with the circuit
conveying the current. The direction of magnetization
of the shell is related to the direction of the current in
such a way that if the observer stands on the side of the
shell which is charged with positive magnetism and looks
at the current, the current in front of him flows from
right to left.

The best proof of the truth of Ampere's law is that
though its consequences are being daily compared with
the results of experiments, no discrepancy has ever been
detected.

The potential due to the magnetic shell at a point
in the substance of the shell is not the same as that due
to the electric circuit, nor is the magnetic force at such a
point the same in the two cases. This however does not
cause any difficulty in determining the magnetic force due
to a circuit at any point P, for, since only the boundary
of the equivalent magnetic shell is fixed, we can always
arrange the shell in such a way that it does not pass
through P.

We can easily prove, however, that at any point,
whether in the substance of the shell or not, the mag
netic force due to the circuit is equal to the magnetic
induction due to the shell. For let P be a point in the
substance of the shell, then though the magnetic force
due to the shell will not be the same as at P', a point just

332 MAGNETIC FORCE DUE TO CURRENTS [CH. X

outside the shell, yet the force due to the current at P'
will differ from that at P by an amount which vanishes
when the distance PP' is indefinitely diminished. The
magnetic force at P' due to the current is the same as
the magnetic force at P' due to the shell. Since the shell
is magnetized along the normal, the tangential magnetic
force in the shell is equal to the tangential magnetic
induction. Now, by Art. 158, the tangential magnetic
force at P', a point just outside the shell, is equal to
the tangential magnetic force at P, a point just inside
the shell, and this, as we have just seen, is equal to the
tangential magnetic induction at P. Again, by Art. 158,
the normal magnetic force at P' is equal to the normal
magnetic induction at P. Thus since the normal force
at P' is equal to the normal induction at P, and the
tangential force at P' is equal to the tangential induction
at P, the magnetic force at P' is equal in magnitude
and direction to the magnetic induction at P. Since
the magnetic force at P due to the current is equal to
the magnetic force at P' due to the shell, we see that
the magnetic force due to the current at P is equal
to the magnetic induction due to the shell at P.

Thus since the lines of magnetic induction due to the
shell form a series of closed curves passing through the
shell, the lines of magnetic force due to the current flowing
round a closed linear circuit will be a series of closed
curves threading the circuit.

203. Work done in taking a magnetic pole
round a closed curve in a magnetic field due to
electric currents. Let EFGH be the closed curve
traversed by the magnetic pole; if this curve threads the

203] MAGNETIC FORCE DUE TO CURRENTS 333

circuit traversed by the current, then the magnetic shell
whose magnetic effect is equivalent to that of the current
must cut the curve, let it do so in PQ. Let a, 6, c be the
components of magnetic induction due to the shell at any
point, a, j3, 7 the components of the magnetic force at the
same point, and A, B, C the components of the intensity
of magnetization. Since the magnetic force due to the

Fig. 95.

circuit is the same as the magnetic induction due to
the shell, W, the work done on the unit pole when it
traverses the closed curve EFGH under the influence of
the electrical currents, is given by the equation

W = / (adx + bdy + cdx),

the integral being taken round the closed curve.
Hence we have by Art. 153

W = / {(a + 4mA )dx+($ + 4>7rB) dy + (7 + 4nrC) dz],
or since by Art. 134 the line integral of the magnetic
force due to the shell vanishes when taken round a closed
circuit, we have

f(adx + ftdy -f ydz) = 0;

hence W = 4?r/ ( Adx + Bdy + Cdz),

where the integral is now taken from P to Q, the points
where the shell cuts the curve EFGH, since it is only
between P and Q that A, B, C do not vanish.

334 MAGNETIC FORCE DUE TO CURRENTS [CH. X

If </> is the strength of the magnetic shell, and the
direction of integration is from the negative to the positive
side of the shell

J(Adx + Bdy + Cdz) = c/> ;
hence W=4<7r<j).

If i is the strength of the current which the shell
replaces

</> = !,

see Art. 202 ; hence

Thus the work done on unit pole when it travels
round a closed curve which threads the circuit once in
the positive direction, i.e. when the pole enters at the
negative side of the equivalent shell and leaves at the
positive, is constant whatever be the path, and is equal to

If the closed curve along which the unit pole travels
does not thread the circuit of the current, the work done
on the unit pole vanishes, for we can draw the equivalent
shell so as to be wholly outside the path of the pole, and
in this case At B, C vanish at all points of the path.

If the path along which the unit pole is taken threads
the circuit n times in the positive direction (the positive
direction being when the pole in its path enters the
equivalent magnetic shell at the negative side and leaves
it at the positive), and ra times in the negative direction,
the work done on the pole on its path is equal to
4>7ri (n — ra).

The value of f(adx + ftdy + <ydz) taken round a closed
circuit is independent of the nature of the material which
is traversed by the circuit ; it is the same, if the currents

204] MAGNETIC FORCE DUE TO CURRENTS 335

are unaltered, whether the circuit lies entirely in air,
entirely in iron or any other magnetizable medium, or
partly in air and partly in iron. For the field may be
regarded as made up of two parts, one, in which the
components of the magnetic force are alt &, 7! due to the
magnetic action of the currents when there is nothing
but air in the neighbourhood; the other, a field whose
components are «0, /30, 7o due to the magnetization in
duced or permanent of the iron.

Hence
j(cndx + ftdy + ^dz)

= /{(«! + «0) das + (ft + &) dy + (7x + 7o) dz}.

Since «0, /30) y0 are the forces due to a distribution
of magnets the work done by these forces on a unit pole
taken round a closed circuit must vanish, hence

f(a0dx + /30dy + yQdz) = 0,
when the integral is taken round any closed circuit.

Thus

j(a.dx
and

/(otjcb + Pidy +

= 4-7T (sum of currents embraced by the circuits).
Thus j(adx + fidy + <ydz) depends merely upon the
currents in the field and not upon the nature of the
material intersected by the circuit.

204. Magnetic force due to an infinitely long
straight current, in a field in which there are no
magnetizable substances. In this case the magnetic
force is numerically equal to the magnetic induction, and
hence the total normal magnetic force taken over any
closed surface vanishes. Take as the closed surface a

336 MAGNETIC FORCE DUE TO CURRENTS [CH. X

right circular cylinder with the current for axis, and let
R be the radial magnetic force at any point of the curved
surface of this cylinder ; by symmetry R is constant over
the curved surface. Since the current is infinitely long
the magnetic force will not vary as we move parallel to
the wire conveying the current ; hence the normal mag
netic force taken over one of the plane ends will cancel
that taken over the other. Thus, if S is the curved
surface of the cylinder, the total magnetic force taken
over the cylinder is RS, and since this vanishes, R must
vanish ; hence there is no radial magnetic force due to
the current.

To find T the tangential magnetic force, let P be
any point, and OP the perpendicular let fall from P
on the current ; T is the magnetic force at right angles
to OP and to the direction of the current. With 0 as
centre and radius OP describe in a plane at right angles
to the current a circle ; at each point on the circum
ference of this circle the tangential magnetic force will
by symmetry be constant, and equal to T. The work
done when unit pole is taken round this circle is 27rrT,
and since the path encircles the current once this must
by Art. 203 be equal to 4-Tn, if i is the strength of the
current; hence we have

or the tangential magnetic force varies inversely as the
distance from the current.

We shall now show that the magnetic force parallel to
the current vanishes.

We can do this by regarding the straight circuit as
the limit of a circular one with a very large radius.

204] MAGNETIC FORCE DUE TO CURRENTS 337

Consider the magnetic force at a point P due to the
circular current. Through P draw a circle in a plane
parallel to that of the current, so that the line joining 0
the centre of this circle, to the centre of the circle in
which the current is flowing, is perpendicular to the planes
of these circles. Then if T is the magnetic force along
the tangent to this circle at P, T will be, by symmetry,
the tangential force at each point of this circle. Hence
the work done in taking unit pole round the circumference
of this circle is 2-TrOP . T, this must however vanish as the
circle does not enclose any current, thus T must be zero.
Proceeding to the limit when the radius of the circle is
indefinitely increased we see that the magnetic force due
to a straight current has no component parallel to the
current.

Thus the lines of magnetic force due to the long
straight current are a series of circles whose centres are
on the axis of the current and their planes at right angles
to the current. The direction of the magnetic force is
related to that of the current in the way shown in the
diagram, Fig. 92 ; i.e. the directions of current and
magnetic force are related in the same way as the direc
tions of translation and rotation in a right-handed screw.

The magnetic force at a point P not in the current
itself is thus derivable from a potential H, where

11 = 2i0 + 4,7rni,

where 6 is the angle PO, the perpendicular let fall from
P on the axis of the current, makes with a fixed line in
the plane through 0 at right angles to the current : n is
an integer. The potential is a multiple-valued function
having at each point an infinite series of values differing
T. E. 22

338 MAGNETIC FORCE DUE TO CURRENTS [CH. X

from each other by multiples of 4?™, which is the work
done in taking unit magnetic pole round a closed circuit
embracing the current. This indeterminateness in the
potential arises from the fact that the work done on
unit pole as it goes from one point P to another point
Q, depends not merely on the relative positions of P and
Q but also on the number of times the pole in its path
from P to Q encircles the current.

205. Magnetic force inside the conductor con
veying the current. When the current is flowing
symmetrically through a circular cylinder, we can easily
find the magnetic force at a point inside the cylinder.
Let 0 be the centre of a cross section of the conductor,
and P a point at which the tangential force T is required ;
in the plane of the section draw a circle whose centre is
0 and radius OP. The work done in taking unit pole
round this circle is 2-TrOP. T, this by Art. 203 is equal to
4-7T times the current enclosed by the circle. Hence we
have
27rOP. T=4>TT (current enclosed by the circle with

centre 0 and radius OP).

If the current is all outside this circle, the right-hand
side of this equation vanishes : hence T vanishes and there
is no magnetic force. Thus there is no magnetic force in
the interior of a cylindrical tube conveying a current.

If the current is uniformly distributed over the cross
section, and i is the total current flowing through the
cylinder whose radius we shall denote by a, the current
through the circle whose radius is OP is equal to

. OP2

206] MAGNETIC FORCE DUE TO CURRENTS 339

OP2
Hence 2-TrOP. T= 4nri . ~-

Thus when the current is uniformly distributed, the
magnetic force inside the cylinder varies directly as the
distance from the axis ; outside the cylinder it varies
inversely as this distance.

206. The total normal magnetic induction through
any cylindric surface passing through two lines parallel
to the current is the same whatever be the shape of the

Fig. 96.

surface connecting these lines. This follows at once from
the principle that the total magnetic induction over any
closed surface is zero. To find an expression for the in
duction through the cylindric surface, let A and B be the
points where the two lines intersect a plane at right angles
to the current, 0 the point where the axis of the current
intersects this plane. Take the cylindrical surface such
that if B is the point nearest to 0, the normal section of
the surface is the circular arc BC and the radial portion
CA. Since the magnetic force is everywhere tangential to

22—2

340 MAGNETIC FORCE DUE TO CURRENTS [CH. X

BG no tube of force passes through the portion corre
sponding to BC ; if r is the distance of any point P on
CA from 0, the magnetic force at P is

r '

hence the number of tubes of magnetic force passing
through the portion corresponding to AC is

I2ij . OA
oc r °^0(7

= 2ilo —

and this represents the number passing through each
unit of length of any cylindric surface passing through
A and B.

207. Two infinitely long straight parallel cur
rents flowing in opposite directions. Let A and S,

Fig. 97, be the points where the axes of the currents
intersect a plane drawn at right angles to the direction
of the currents. Let the direction of the current at A

be downwards through the paper, that at B upwards ; if i
is the strength of either current, the magnetic potential
at a point P is, Art. 204, equal to

2i Z PAB ± 27rn - 2i TT - Z PBA

207] MAGNETIC FORCE DUE TO CURRENTS

This may be written

341

thus along an equipotential line the angle APB is
constant, hence the equipotential lines are the series of
circles passing through AB.

The lines of magnetic force are at right angles to the
equipotential lines, they are therefore the series of circles
having their centres along AB such that the tangents to
them from 0, the middle point of AB, are of the constant
length OA.

The lines of magnetic force and the equipotential
lines are represented in Fig. 98.

Fig. 98.

The direction of the magnetic force is easily found
as follows. If PT is the direction of the magnetic force
at P, then since PT is the normal to the circle round

342 MAGNETIC FORCE DUE TO CURRENTS [CH. X

APB, the angle BPT is equal to the complement of the
angle PAB.

The magnetic force R at P is the resultant of the
forces 2i/AP at right angles to AP and 2i/BP at right
angles to BP. Resolving these along PT, we have

os ABP + -2L cos BAP
nr

AP.BP'

Thus the intensity of the magnetic force at P varies
inversely as the product of the distances of P from A
and B.

At a point on the line bisecting AB at right angles
AP=BPy and along this line, which may be called the
axis of the current, the magnetic force is inversely pro
portional to the square of the distance from A or B',
the direction of the force is parallel to the axis.

At a point whose distances from A and B are large
compared with AB we may put AP = BP = OP, in this
case the magnetic force varies inversely as OP3, and the
direction of the force makes with OP the same angle as
OP makes with the line at right angles to A B.

208. Number of tubes of magnetic force due
to the two currents which pass through a circuit
consisting of two parallel wires. Let A, B be the

points where the two currents intersect a plane drawn
at right angles to them, C, D the points where the wires
of the circuit cut the same plane. Then, Art. 206,

208] MAGNETIC FORCE DUE TO CURRENTS 343

the number of tubes of magnetic force due to A which

pass through CD per unit length = 2i log -=-=: . Similarly

A.JJ

the number which pass through CD and are due to the
current B is

hence the number through CD per unit length due to the
current i at A and — i at B, is

AC BW

AC.BD

We see from the symmetry of the expression that this
is the number which would pass through the circuit AB
due to currents + i and — i at C and D respectively.

When the circuits AB, CD are so situated that the
total number of tubes passing through CD due to the
current in. A, B is zero, the circuits AB, CD are said to
be conjugate to each other. The condition for this is that

AC.BD ... . ,
log . ^ p ~ should vanish, or that

AC AD
~BC~ BD'

another way of stating this result is that C and D must
be two points on the same line of magnetic force due
to the currents at A and B; this is equivalent to the
condition that A and B should be points on a line of
magnetic force due to equal and opposite currents at
C and D. Since the lines of magnetic force due to the

344 MAGNETIC FORCE DUE TO CURRENTS [CH. X

currents A and B are a series of circles with their centres
on A B it follows that if CD is conjugate to AB it will
remain conjugate however CD is rotated round the point
0', 0' being the point where the line bisecting CD at right
angles intersects AB.

A case of considerable practical importance is when
we have two equal circuits AB and CD, the current
through A being in the same direction as that through
C and that through B in the same direction as that
through D.

Let us consider the case when AB and CD are equal
and parallel and so placed that the points A, B, D, C are
at the corners of a rectangle. Then if i is the current
flowing round each of the circuits, H the magnetic
potential at a point P will, by Art. 204, be given by
the equation

£1 = — 2i6 — 2i$ + constant,

where 6 and 4> are the angles subtended respectively
by AB and CD at P.

The lines of magnetic force are the curves which cut
these at right angles ; along such a line

is constant, where r1} rz, r3) r4 are the distances of a point
on the line from A, B, C, D respectively.

The lines of magnetic force are represented in Fig. 99.
There are two points E, F where the magnetic force
vanishes ; these points are on the line drawn through 0,
the centre of the rectangle, parallel to the sides A B and
CD ; we can easily prove that OE is equal to OA .

208]

MAGNETIC FORCE DUE TO CURRENTS

345

At a point P on the axis of the current, i.e. on the line
through 0 at right angles to AB, the magnetic force is
parallel to the axis and is by Art. 207 equal to

2t . AB 2i . CD
~~ ~CPr

if OP = x, AB = 2a, AC = 2d, the magnetic force at P is

equal to

4tia

o p

Fig. 99.

This is, neglecting the fourth and higher powers of x,
equal to

thus, if */3d = a, the term in a? disappears and the lowest
power of x which appears in the expression for the
magnetic force is the fourth. Thus with this relation
between the size of the coils and the distance between
them the force near 0 varies very slowly as we move
along the axis.

346

MAGNETIC FORCE DUE TO CURRENTS

[CH. X

The number of tubes of magnetic force which pass
through one circuit when a current % flows round the
other may, by using the result given on page 343, easily
be proved to be equal to

... BG

Fig. 100.

209. Direct and return currents flowing uni
formly through two parallel and infinite planes.

Let the two parallel planes be at right angles to
the plane of the paper and let this plane intersect them
in the lines AB, CD, Fig. 100. Let a current i flow upwards
at right angles to the plane of the paper through each
unit length of AB and downwards through each unit
length of CD. Let EF be the section of the plane
parallel to AB and CD and midway between them. We
shall prove that the magnetic force between the planes
is uniform and parallel to EF, being thus parallel to the
planes in which the currents are flowing and at right
angles to the currents.

We shall begin by proving that the magnetic force
has no component at right angles to the planes in which
the currents are flowing. This is evidently true by

209] MAGNETIC FORCE DUE TO CURRENTS 347

symmetry at all points in the plane midway between
AB and CD ; we can prove it is true at all points in
the following way. Take a rectangular parallelepiped one
of whose faces is in the plane whose section is EF, let
another pair of faces be parallel to the plane of the paper
and the third pair perpendicular to the line EF. The
total normal magnetic induction over this closed surface
vanishes. Since the currents are uniformly distributed
in the infinite planes, the magnetic induction will be the
same at all points in a plane parallel to those in which the
currents are flowing. Hence the total magnetic induction
over the pairs of faces of the parallelepiped which are at
right angles to the parallel planes will vanish : for the
induction at a point on one face will be equal to that at
a corresponding point on the opposite face, and in the
one case it will be along the inward normal, in the other
along the outward. Hence since the total induction over
the parallelepiped is zero the induction over one of the
faces parallel to the planes must be equal and opposite to
that over the opposite face. But one of these faces is
in the plane EF where the magnetic induction normal
to the face vanishes; hence the total normal induction
over the other face must vanish, and since the induction
is the same at each point on the face the induction can
have no component at right angles to this face, i.e. at
right angles to the planes in which the currents are
flowing. This proof applies to all parts of the field,
whether between the planes or outside them.

To prove that the force parallel to the currents
vanishes, we take a rectangle PQRS with two sides PQ,
RS parallel to the currents, the other sides PS, QR being
at right angles to the planes of the currents. No current

348 MAGNETIC FORCE DUE TO CURRENTS [CH. X

flows perpendicularly through this rectangle, hence (Art.
203) the work done when unit magnetic pole is taken
round its circumference is zero. But since the magnetic
force parallel to PS, RQ vanishes, the work done on unit
pole, if Fis the force along PQ, Ff that along RS, is equal to

(F-F')PQ.

Since this vanishes F=F', i.e. F is constant throughout
the field, and since by symmetry it vanishes along EF it
must vanish throughout the field.

We have now proved that throughout the field the
components of the magnetic force in two directions at
right angles to each other vanish, hence the magnetic
force, where it exists, must be parallel to EF, Fig. 100.

By drawing a rectangle in the space outside the planes
with one pair of its sides parallel to EF we can prove
that the force parallel to EF also vanishes outside the
planes, so that in this region there is no magnetic force.
To find the magnitude of the magnetic force H between
the planes, take a rectangle such as LMNK, Fig. 100,
cutting one of the planes, the sides of the rectangle being
respectively parallel and perpendicular to EF. The quan
tity of current flowing through this rectangle is i x LM,
since i flows through each unit of length of the plane;
hence 4-Tn x LM is equal to the work done in taking unit
magnetic pole round the rectangle. But this work is
H x LM, since no work is done when the pole is moving
along M N, NK and KL, hence we have

or H =

Thus the magnetic force is independent of the distance
between the planes.

210] MAGNETIC FORCE DUE TO CURRENTS 349

210. Solenoid. We can apply exactly the same
method to the very important case of an infinitely long
right circular solenoid, i.e. an infinitely long right circular
cylinder round which currents are flowing in planes
perpendicular to the axis. Such a solenoid may be con
structed by winding a right circular cylinder uniformly
with wire, the planes of the winding being at right angles
to the axis of the cylinder, so that between any two planes
at right angles to the axis and at unit distance apart there
are the same number of turns of wire. We can show by
the same method as in Art. 209, that inside the cylinder
the radial magnetic force vanishes, and that the force
parallel to the axis of the cylinder is uniform, that out
side the cylinder the magnetic force vanishes: and that
if H is the magnetic force inside the cylinder parallel to
the axis

H =4t7r (current flowing between two planes separated
by unit distance).

If there are n turns of wire wound round each unit
length of the cylinder and i is the current flowing through
the wire, this equation is equivalent to
H = 4>7rni.

The preceding result is true whatever be the shape
of the cross section of the cylinder on which the wire is
wound, provided the number of turns of wire between two
parallel planes at unit distance apart perpendicular to the
axis of the cylinder is uniform.

Endless Solenoids. Near the ends of a straight
solenoid the magnetic field is not uniform and ceases to be
parallel to the axis of the cylinder and equal to 4i7rni. We
can, however, avoid this irregularity if we wind the wire

350 MAGNETIC FORCE DUE TO CURRENTS [CH. X

on a ring instead of on a straight cylinder. Suppose the
ring is generated by the revolution of a plane area about
an axis in its own plane which does not cut it, and let the
ring be wound with wire so that the windings are in planes
through the axis of the ring and so that the number of
windings between two planes which make an angle 6 with
each other is equal to nOj^nr ; n is thus the whole number
of windings on the ring. Then we can prove as in Art.
209 that the magnetic force vanishes outside the solenoid,
and that inside the solenoid the lines of magnetic force
are circles having their centres on the axis of the solenoid
and their planes at right angles to the axis. Let H be
the magnetic force at a distance r from this axis; the
work done on unit pole when taken round a circle whose
radius is r and whose centre is on the axis and plane
perpendicular to it is %7rrH ; this by Art. 203 is equal to
4-Tr times the current flowing through this circle, and is
thus equal to 4>7rnit if i is the current flowing through one
of the turns of wire. Hence

2ni

or H = — .

Thus the force is inversely proportional to the distance
from the axis.

The preceding proof will apply if the solenoid is wound
round a closed iron ring; if however there is a gap in the
iron it requires modification.

Let Fig. 101 represent a section of the solenoid and
suppose that ABDC is a gap in the iron, the faces of
the iron being planes passing through the axis of the
solenoid. Let this axis cut the plane of the paper in 0.

210] MAGNETIC FORCE DUE TO CURRENTS 351

Let P be a point on the face of one of the gaps, B the
magnetic induction in the iron at right angles to OP,
then since the normal magnetic induction is continuous
B will also be the magnetic induction in the air. Hence
if fji is the magnetic permeability of the iron, the magnetic
force in the iron is BJJJ, while that in the air is B. If

Fig. 101.

OP = r, the work done in taking unit pole round a circle
whose radius is r is

— (27r

where 0 is the angle subtended by the air gap at the axis
of the solenoid. Hence by Art. 203 we have

or B =

This formula shows the great effect produced by even
a very small air gap in diminishing the magnetic induction.

352 MAGNETIC FORCE DUE TO CURRENTS [CH. X

Let us take the case of a sample of iron for which
At - 1 = 1000, then if 0/2*- = 1/100, i.e. if the air is only
one per cent, of the whole circuit, the value of B is only
one-eleventh of what it would be if the iron circuit were
complete, while even though 0/2-Tr were only equal to
1/1000 the magnetic induction would be reduced one-half
by the presence of the gap.

We can explain this by the tendency which the tubes
of magnetic induction have to leave air and run through
iron. If the magnetic force in the solenoid due to the
current circulating round it is in the direction of the
arrow, the face AB of the gap will be charged with
positive magnetism, the face CD with negative. If this
distribution of magnetism existed in air, tubes of mag
netic induction starting from AB and running through
the air to CD would be pretty uniformly distributed in
the field ; in this case they would only be in the solenoid
for a short part of their course. But as soon as the
solenoid is filled with soft iron these tubes forsake the air
and run through the iron, and as they are in the opposite
direction to the tubes due to the current they diminish
the magnetic induction in the iron.

Problems like the one just discussed can be easily
solved by making use of the analogy between the distribu
tion of magnetic induction in a field containing magnetic
and non-magnetic substances, and the distribution of
electric current in a field containing substances of different
electrical conductivity. This analogy is shown by the
following table, the properties stated on the left-hand side
relating to the magnetic field due to a magnetizing circuit
traversed by a current i, those on the right relating to the

210]

MAGNETIC FORCE DUE TO CURRENTS

353

distribution of current produced by a battery of electro
motive force E.

MAGNETIC SYSTEM.

1. The line integral of the
magnetic force round any closed
curve threading the magnetizing
circuit is 47ri, while round any
other closed curve it vanishes.

2. The lines of magnetic in
duction are closed curves thread
ing the magnetizing circuit.

3. The magnetic induction
is p. times the magnetic force,
where p. is the magnetic per
meability.

CURRENT SYSTEM.

1. The line integral of the
electric force round any closed
curve passing through the bat
tery is E, while round any other
closed curve it vanishes.

2. The lines of flow of the
current are closed curves passing
through the battery.

3. The intensity of the cur
rent is by Ohm's Law c times
the electric force, where c is the
specific conductivity of the sub
stance, i.e. the reciprocal of the
specific resistance.

4. At the junction of two
different media the normal elec
tric current and the tangential
electric force are continuous.

4. At the junction of two
different media the normal
magnetic induction and the
tangential magnetic force are
continuous.

From these results we see that the magnetic induction
due to a magnetizing circuit carrying a current i will be
numerically equal to the current produced by a battery
coinciding with the circuit, if the electromotive force of
the battery is 4?™', and if the specific conductivity of the
medium at any point in the surrounding field is numeri
cally equal to the magnetic permeability at that point.

Since the magnetic permeability of iron is so much
greater than that of air or other non-magnetic substances,
we may, when we use the analogy of the current, regard
the magnetic substances as good conductors, the non
magnetic substances as very bad ones.

T. E. 23

354 MAGNETIC FORCE DUE TO CURRENTS [CH. X

Thus in the case of a magnetizing coil round an iron
ring, the current analogue is a battery inserted in a ring
of high conductivity, the ring being surrounded by a very
bad conductor ; in this case practically all the current will
go round the ring, very little escaping into the surround
ing medium. If E is the E.M.F. of a battery, c the specific
conductivity of the ring, I its length, a the radius of its
cross section, the resistance of the ring is I /OTTO?, the
current through the ring is Ec7ra?/l, the average intensity
of the current is Ec/l: hence the magnetic induction in
an iron ring of length I due to a magnetizing circuit
traversed by a current is ^Trip/I. Suppose now that there
is a gap in this circuit, in the electric analogue this
would correspond to cutting the ring, inserting a disc of a
bad conductor in the opening, this would evidently greatly
reduce the current ; if c? is the width of the slit, Cj the
specific conductivity of the material with which it is filled,

then the resistance of the ring is - — + ^2 , the current

through the ring is equal to

Ewafo

l + d(--

the average intensity of current is equal to

The magnetic induction in the slit iron ring will therefore
since the magnetic permeability of air is unity, be

ad (/*-!)'

211] MAGNETIC FORCE DUE TO CURRENTS 355

Any problem in the distribution of currents has a
magnetic analogue. Thus take the problem of the
Wheatstone Bridge (Art. 191), in the magnetic analogue
we have six iron bars AB, BC, CA, AD, BD, CD (Fig. 89)
with a magnetizing circuit round AB; if ll} 12) 1B, 14 are
the lengths, a1} a2, a3) a4 the areas of the cross sections,
and yitj, /A2, yu3, yit4 the magnetic permeability of AC, CB,
AD, BD respectively, we see from the theory of the
Wheatstone Bridge that there will be no lines of magnetic
induction down CD if

a result which may be applied to the comparison of the
magnetic permeabilities of various samples of iron.

The student will find the use of this analogy between
magnetic and current problems of great assistance in
dealing with the former, and he will find it profitable to
take a number of simple cases of distribution of current
and find their magnetic analogues,. ._.

211. Ampere's Formula. We saw, Art. 137, that
the magnetic force exerted by a magnetic shell of uniform
strength c/>, is that which would be produced if each unit
of length at a point P on the boundary of the shell exerted
a magnetic force at Q equal to </> sin 0/PQ*, where 6 is the
angle between PQ and the tangent at P to the boundary
of the shell : the direction of the magnetic force at Q is
at right angles to both PQ and the tangent to the boundary
at P. Since the magnetic force due to the shell is by
Ampere's rule the same as that due to a current flowing
round the boundary of the shell, the intensity of the
current being equal to the strength of the shell, it follows

23—2

356 MAGNETIC FORCE DUE TO CURRENTS [CH. X

that the magnetic force due to a linear current may be
calculated by supposing an element of current of length ds
at P to exert at Q a magnetic force equal to ids sin 0/PQ2,
where i is the strength of the current, and 6 the angle
between PQ and the direction of the current at P : the
direction of the magnetic force being at right angles both
to PQ and to the direction of the current at P.

The direction of the magnetic force is related to the
direction of the current, like rotation to translation in
a right-handed screw working in a fixed nut.

212. Magnetic force due to a circular current.

The preceding rule will enable us to find the magnetic
force along the axis of a circular current.

Let the plane of the current be at right angles to the
plane of the paper. Let the current intersect this plane

Fig. 102.

in the points A, B, Fig. 102, flowing upwards at A and
downwards at B. Let 0 be the centre of the circle round
which the current is flowing, P a point on the axis of
the circle. The force at P will by symmetry be along OP.
If i is the intensity of the current, then the force at P
due to an element ds of the current at A will be at right
angles to the current at A, i.e. it will be in the plane
of the paper, it will also be at right angles to AP: the

212] MAGNETIC FORCE DUE TO CURRENTS 357

magnitude of this force is ids/AP2, hence the component
along OP is equal to

., OA

By symmetry each unit length of the current will furnish
the same contribution to the magnetic force along the
axis at P: hence the magnetic force due to the circuit

is equal to

D A 2

27H

Thus the force varies inversely as the cube of the
distance from the circumference of the circle. At the

Fig. 103.

centre of the circle AP= OA, hence the magnetic force
at the centre is equal to

2-Tn

OA'

and thus, if the current remains of the same intensity,
varies inversely as the radius of the circle.

MAGNETIC FORCE DUE TO CURRENTS [CH. X

The lines of magnetic force round a circular current
are shown in Fig. 103. The plane of the current is at
right angles to the plane of the paper and the current
passes through the points A and B.

213. A case of some practical importance is that of
two equal circular circuits conveying equal currents and
placed with their axes coincident. Let A, B; C, D be
the points in which the currents, which are supposed to
flow in planes at right angles to the plane of the paper,
cut this plane, the currents flowing upwards at A and (7,
downwards at B and D : let P be a point on the common
axis of the two circuits. The magnetic force at P is,
if i is the intensity of the current through either circuit,
equal to

AP*

where a is the radius of the circuits. If 2d is the
distance between the planes of the circuits, and x = OP,
where 0 is the point on the axis midway between the
planes of the currents, the magnetic force at P is

(a? + (d + x)*)* (a
1-£<V-

+ terms in x* and higher powers of x\ .

Thus if a = 2d, that is if the distance between the
currents is equal to the radius of either circuit, the
lowest power of x in the expression for the magnetic

214] MAGNETIC FORCE DUE TO CURRENTS 359

force will be the fourth. Thus near 0 where x is small
the magnetic force will be exceedingly uniform.

This disposition of the coils is adopted in Helmholtz's
Galvanometer.

214. Mechanical Force acting on an electric
current placed in a magnetic field.

The mechanical forces exerted by currents on a mag
netic system are equal and opposite to the forces exerted
by the magnetic system on the currents. Since the forces
exerted by the currents on the magnets are the same as
those exerted by Ampere's system of magnetic shells, it
follows that the mechanical forces on the currents must
be the same as those on the magnetic shells; hence the
determination of the mechanical forces on a system of
currents can be effected by the principles investigated
in Art. 136. Introducing the intensity of the current
instead of strength of the magnetic shell we see from
that article that the force in any direction acting on
a circuit conveying a current i is equal to i times the
rate of increase of the number of unit tubes of magnetic
induction passing through the circuit, when the circuit is
displaced in the direction of the force. In many cases the
deduction from this principle given on page 219 is useful,
as it shows that the forces on the current are equivalent
to a system of forces acting on each element of the circuit.
If i is the strength of the current, ds the length of an
element at P, B the magnetic induction at P, 0 the
angle between ds and B, then the force on the element
is equal in magnitude to idsB sin 0, and its direction is
at right angles both to ds and B. The relation between
the direction of the mechanical force and the directions
of the current and the magnetic induction is shown in

360

MAGNETIC FORCE DUE TO CURRENTS

[CH. X

the accompanying figure, where the magnetic induction is
supposed drawn upwards from the plane of the paper.

mechanical force

Fig. 104.

215. Couple acting on a plane circuit placed
in a uniform magnetic field. Let A be the area of
the circuit, i the intensity of the current, <£ the angle
between the normal to the plane of the circuit and the
direction of the magnetic induction. The number of unit
tubes of magnetic induction due to the uniform field
passing through the circuit is iAB cos <£, where B is the
strength of the magnetic induction in the uniform field,
and this does not change as the circuit is moved parallel
to itself; there are therefore no translatory forces acting
on the system. The number of tubes passing through
the circuit changes however as the circuit is rotated, and
there will therefore be a couple acting on the circuit ;
the moment of the couple tending to increase $ is by
the last article equal to the rate of increase with </> of
the number of unit tubes passing through the circuit,
that is to

-j-r (lAB COS <£)

= — iAB sin (>.

216] MAGNETIC FORCE DUE TO CURRENTS 361

The couple vanishes with (/>, and hence the circuit tends
to place itself with its normal along the direction of the
magnetic induction, and in such a way that the direction
of the magnetic induction through the circuit and the
direction in which the current flows round it are related
like translation and rotation in a right-handed screw
working in a fixed nut.

216. Force between two infinitely long straight
parallel currents. Let the currents be at right angles
to the plane of the paper, intersecting this plane in A
and B, let the intensity of the currents be i, i' respec
tively, and let the currents come from below upwards
through the paper. Then, by Art. 204, the magnetic
force at B due to the current through A is equal to

2i
AB>

and is at right angles to AB\ hence, by Art. 214, the
mechanical force per unit length on the current at B
is equal to

and since it acts at right angles both to the current and
to the magnetic force, it acts along AB. By the rule
given in Art. 214, we see that if the currents are in the
same direction the force between them is an attraction,
if the currents are in opposite directions the force between
them is a repulsion. Hence, we see that straight parallel
currents attract or repel each other, according as they are
flowing in the same or opposite directions, with a force
which varies inversely as the distance between them.

362

MAGNETIC FORCE DUE TO CURRENTS

[CH. X

217. Mechanical force between two circuits,
each circuit consisting of a pair of infinitely long
parallel straight conductors. Let the currents be
all perpendicular to the plane of the paper and let the
currents of the first and second pairs intersect the plane
of the paper in A, B and C, D respectively: we shall
consider the case when the circuits are placed symmetri
cally and so that the line EF bisects both AB and CD
at right angles. Let the current i flow upwards through

Fig. 105.

the paper at A, downwards at B, the current if upwards
through the paper at C, downwards at D. The force
between the circuits will by symmetry be parallel to EF.
Between the currents at A and C there is an attraction
along CA equal per unit length to

2n'

AC'

the component of this parallel to EF is

Btf

AC'

EF.

Between the currents B and C there is a repulsion along
BC equal per unit length to

BC'

218] MAGNETIC FORCE DUE TO CURRENTS 363

the component of this parallel to EF is

:- -$>•

Hence on each unit length of G there is a force parallel
to FE, and equal to

there is an equal force acting in this direction on each
unit length of D ; hence the total force per unit length on
the circuit CD is an attraction parallel to EF equal to

If EF = x, A E = a, CF= b, this is equal to
1 1

,..,

this vanishes when as = 0 and when x is infinite. Hence
there must be some intermediate value of x when the
attraction is a maximum. This value of x is easily found
to be given by the equation

X* = J {2 Va4 + 64-a262 - (a2 + 62)} :
when a — b is very small this gives

x — a — b,
when b/a is very small

% = —=. .
V3

218. Force between two coaxial circular cir
cuits.

The solution of the general case requires the use of
more analysis than is permissible in this work : there

364 MAGNETIC FORCE DUE TO CURRENTS [CH. X

are however two important cases which can be solved by
elementary considerations. The first of these is when the
radii of the circuits are nearly equal, and the circuits are
so close together that the distance between their planes
is a very small fraction of the radius of either circuit.
In this case the force per unit length of each circuit is
approximately the same as that between two infinitely
long straight parallel circuits, the distance between the
straight circuits being equal to the shortest distance
between the circular ones. Thus if i, i' are the currents
through the circular circuits, whose radii are respectively
a and 6, and x is the distance between the planes of
the circuits, the attraction between the parallel circuits
is at right angles to the planes of the circuits and is
approximately equal to

(a -

This is a maximum when x — a — b ; that is, when the
distance between the planes of the circuits is equal to
the difference of their radii.

Another case which is easily solved is that of two
coaxial circular circuits, the radius of one being small com
pared with that of the other, Let i be the intensity of the
current flowing round the large circuit whose radius is a,
i' the current round the small circuit whose radius is b ;
let x be the distance between the planes of the circuits.
Then since b is very small compared with a, the magnetic
force due to the large circuit will be approximately uniform

over the second circuit and equal to 27rm2/(a2 + #2)^, its
value at the centre of that circuit. Thus the number of

219] MAGNETIC FORCE DUE TO CURRENTS 365

unit tubes of magnetic induction due to the first circuit
which pass through the second circuit is equal to

Hence by Art. 214 the force on the second circuit
in the direction in which x increases, i.e. the repulsion
between the circuits, is equal to

Thus the attraction between the circuits is equal to

This is a maximum when x — a/2, so that the attraction
between the circuits is greatest when the distance between
their planes is half the radius of the larger circuit.

In the more general case when the radii have any
values, there is, unless the radii are equal, a position in
which the attraction is a maximum. When we use the
attraction between currents as a means of measuring
their intensities, the currents ought to be placed in this
position, for not only is the force to be measured greatest
in this case, but it is also practically independent of any
slight error in the proper adjustment of the distance
between the coils.

219. Coefficient of Self and Mutual Induction.

The coefficient of self-induction of a circuit is defined
to be the number of unit tubes of magnetic induction
which pass through the circuit when it is traversed by
unit current, there being no other current or permanent
magnet in its neighbourhood.

366 MAGNETIC FORCE DUE TO CURRENTS [CH. X

The coefficient of mutual induction of two circuits
A and B is defined to be the number of unit tubes of
magnetic induction which pass through B when unit
current flows round A, there being no current except
that through A, or permanent magnet in the neigh
bourhood of the circuits.

We see from Art. 138 that the coefficient of mutual
induction is also equal to the number of unit tubes of
induction which pass through A when unit current flows
round B.

If the circuit consist of several turns of wire, then in
the preceding definitions we must take as the number of
tubes of magnetic induction which pass through the circuit,
the sum of the number of tubes of magnetic induction
which pass through the different turns of the circuit.

We see from the preceding definitions that if we
have two circuits A and B, and if the currents it j flow
respectively through these circuits, then the numbers of
tubes of magnetic induction which pass through the
circuits A and B are respectively,

Li + Mjy and Mi 4- Njt

where L and N are the coefficients of self-induction of
the circuits A and B respectively, and M is the coefficient
of mutual induction between the circuits. The results
given in the preceding articles enable us to calculate the
coefficient of self-induction in some simple cases.

In the case of the long straight solenoid discussed in
Art. 210, when unit current flows through the wire the
magnetic force in the solenoid is kirn, where n is the
number of turns per unit length ; hence if A is the area
of the core of the solenoid, and if the core is filled with

219] MAGNETIC FORCE DUE TO CURRENTS 367

air, the number of unit tubes of magnetic induction
passing through each turn of wire is equal to ^irnA, and
since there are n turns per unit length, the coefficient of
self-induction of a length I of the solenoid is equal to

If the core were filled with soft iron of permeability //,,
then the number of unit tubes of magnetic induction
which pass through each turn of wire is 4*7rn/j,A, and the
coefficient of self-induction of a length I is ^irrPlpA.

If the iron instead of completely filling the core only
partially fills it, then if B is the area of the core occupied
by the iron, the coefficient of self-induction of a length
I is 4>7rri*l {>5 + A - B}.

Consider now the coefficient of mutual induction of
two solenoids a and (B with parallel axes. The coefficient
of mutual induction will vanish unless one of the solenoids
is inside the other, for the magnetic force due to a current
through a solenoid vanishes outside the solenoid. Hence
when a current flows through a no lines of induction will
pass through ft unless ft is either inside a or completely
surrounds it.

Let ft be inside a. Let B be the area of the solenoid ft,
and let m be the number of turns of wire per unit length.
Then if unit current flows through a, the magnetic force
inside is 4nrn, where n is the number of turns per unit
length. Hence if there is no iron inside the solenoids, the
number of tubes of magnetic induction passing through
each turn of ft is &7rnB, and since there are m turns
per unit length, the coefficient of mutual induction of
a length I of the two solenoids is 4vrnmlB.

We see, by Art. 218, that the coefficient of mutual

368 MAGNETIC FORCE DUE TO CURRENTS [CH. X

induction between a large circle of radius a and a small
one of radius 6, with their planes parallel and the line
joining their centres at right angles to their planes, is
equal to

where x is the distance between the planes.

If we have two circuits a, /3, each consisting of two
infinitely long, parallel straight conductors, the current
flowing up one of these and down the other, then by
Art. 208, the coefficient of mutual induction between a
and ft is, per unit length, equal to

AC. ED

where A, B, G, D are respectively the points where the
wires of the circuits a and /3 intersect a plane at right
angles to their common direction. The current through
the conductor intersecting this plane in A is in the same
direction as that through the conductor passing through G.

220. We can express the energy in the magnetic
field due to a system of currents very easily in terms of
the currents and the coefficients of self and mutual in
duction of the circuits. We proved, Art. 163, that the
energy per unit length in a unit tube of induction at P is
equal to R/STT, where R is the magnetic force at P. The
tube of induction is a closed curve, and the total amount
of energy in this tube is equal to

where ds is an element of length of the tube and
denotes the sum of all the products Rds for the tube.

220] MAGNETIC FORCE DUE TO CURRENTS 369

But ^Rds is the work done on unit pole when it is taken
round the closed curve formed by the tube of induction,
and this by Art. 203 is equal to 4?r times the sum of the
currents encircled by the curve. Hence the energy in
a tube of induction is equal to

J (the sum of the currents encircled by the tube).

Hence the whole energy in the magnetic field is equal to
half the sum of the products obtained by multiplying the
current in each circuit by the number of tubes of mag
netic induction passing through that circuit.

Thus if we have two circuits A and B, and if i, j are the
currents through A and B respectively, L, N the coefficients
of self-induction of A and B, M the coefficient of mutual
induction between these circuits, then the numbers of
tubes of magnetic induction passing through A and B
respectively are

Li + Mj,

and Mi + Nj.

Hence the energy in the magnetic field around this

circuit is

If we have only one circuit carrying a current i, then if
L is its coefficient of self-induction, the energy in the
magnetic field is

\Li\

Thus the coefficient of self-induction is equal to twice the
energy in the magnetic field due to unit current.

We may use this as the definition of coefficient of self-
induction, and this definition has a wider application than

T. E. 24

370 MAGNETIC FORCE DUE TO CURRENTS [CH. X

the previous one. The definition in Art. 219 is only
applicable when the currents flow through very fine wires,
the present one however is applicable when the current is
distributed over a conductor with a finite cross section.
Thus let us consider the case where we have a current
flowing through an infinitely long cylinder whose radius
is 0 A, the direction of flow being parallel to the axis of
the cylinder, and where the return current flows down
a thin tube, whose radius is OB, coaxial with this cylinder.

Fig. 106.

Let i be the current which flows up through the
cylinder and down through the tube, let us suppose that
the current through the cylinder is uniformly distributed
over its cross section. The magnetic force will vanish
outside the tube, for since as much current flows up
through the cylinder as down through the tube, the total
current flowing through any curve enclosing them both
vanishes, and therefore the work done in taking unit pole
round a circle with centre 0 and radius greater than
that of the tube will vanish. Since the magnetic force
due to the currents must by symmetry be tangential to
this circle and have the same value at each point on its

220] MAGNETIC FORCE DUE TO CURRENTS 371

circumference, it follows that the magnetic force vanishes
outside the tube. We can prove as in Art. 204 that at
a point P between the cylinder and the tube the magnetic
force is equal to

where r = OP.

At a point P inside the cylinder the magnetic force is

where a = OA, the radius of the cylinder.

By Art. 163 the energy per unit volume is equal to
/ULH^/STT, where H is the magnetic force ; hence if /x is the
magnetic permeability of the cylinder, the magnetic energy
between two planes at right angles to the axis of the
cylinder and at unit distance apart is equal to

4^2 ros 27rrdr 4i> f

o~~ — 5 -- f~ ~5~
8?r j OA r* 8?r J o

Hence, since the coefficient of self-induction per unit
length is twice the energy when the current is unity, it
is equal to

In this case the coefficient of self-induction will be very
much greater when the cylinder is made of iron than when
it is made of a non-magnetic metal like copper. For take
the case when OB = e.OA, where e = 2718, the base of the
Napierian logarithms ; then the self-induction for copper,
for which p is equal to unity, is equal to 2*5 per unit

24—2

372 MAGNETIC FORCE DUE TO CURRENTS [CH. X

length, but if the cylinder is made of a sample of iron
whose magnetic permeability is 1000, the coefficient of
self-induction per unit length is 502. Thus in this case
the material of the conductor through which the current
flows produces an enormous effect, much greater than it
does in the case of the solenoids.

The self-induction depends upon the way in which the
current is distributed in the cylinder ; thus if the current
instead of spreading uniformly across the section of the
cylinder were concentrated on the surface, the magnetic
force inside the cylinder would vanish, while that in the
space between the tube and the cylinder would be the
same as before, hence the energy would now be
•°* ZTrrdr OB

OA i VA.

so that the coefficient of self-induction would now be
2 log (OB/OA), thus it would be less than before and in
dependent of the material of which the cylinder is made.

221. Rational Current Element. In Ampere's
expression for the magnetic force due to a current, the
current is supposed to be divided up into elements, an ele
ment ds giving rise to a magnetic force equal to ids sin 0/r2.
Each of those elements when regarded as a separate
unit corresponds to an unclosed electric current, whereas
on the Modern Theory of Electricity such currents do not
exist. Thus the mathematical unit does not correspond
to a physical reality. To obviate this inconvenience
Mr Heaviside has proposed another interpretation of the
element of current ; he points out that the magnetic force
ids sin 0/r2 is that due to a system of closed currents
distributed through space like the lines of magnetic

221] MAGNETIC FORCE DUE TO CURRENTS 373

induction due to a small magnet, PQ, PQ being the ele
ment of current ds, and i representing the number of lines
of magnetic induction running through PQ, i.e. passing
through each cross section of the magnet ; the current at
any point in the field round the element of current is repre
sented in magnitude and direction by the magnetic induc
tion at that point due to the little magnet. The reader
will have no difficulty in proving this result, if he applies
the principle that the work done in taking the unit mag
netic pole round any closed circuit is equal to 4?r times the
current passing through the circuit. The element, PQ,
with its associated system of currents, Mr Heaviside calls
the rational current element, it has the advantage of corre
sponding to a possible physical system. It is important
to notice that this view of the element of current gives
us for closed circuits the same result as the old one, i.e.
the closed current is entirely confined to the closed
circuit and does not spread out at all into the surrounding
space ; for let PQ, RS be two elements, then if we place
these together so that the end Q of one coincides with the
beginning, R, of the other, then the analogy with the lines
of magnetic induction shows that the currents which
when PQ was alone in the field diverged from Q now run
through QS and diverge from S, hence if we put a number
of such elements together so as to form a closed circuit
the current will never leave the circuit.

We shall see that the magnetic force produced at P
when a charged particle 0 moves with a velocity v, is
ev sin 0/OP2, where e is the charge on the particle and
0 the angle between v and OP ; the direction of the force
is at right angles to the plane containing v and OP. Thus
another interpretation of the element ds of a circuit is,

374 MAGNETIC FORCE DUE TO CURRENTS [CH. X

that it is a place where n charged particles are moving in
the direction of the element with the velocity v ; n,e,v and
i being connected by the relation nev — ids.

MEASUREMENT OF CURRENT AND RESISTANCE.
Galvanometers.

222. The magnetic force produced by a current may
be used to measure the intensity of the current. This is
most frequently done by means of the tangent galvano
meter, which consists of a circular coil of wire placed with
its plane in the magnetic meridian. If the magnetic field
is not wholly due to the earth, the plane of the coil must
contain the resultant magnetic force. At the centre of
the coil there is a magnet which can turn freely about
a vertical axis. When the magnet is in equilibrium its
axis will lie along the horizontal component of the mag
netic force at the centre of the coil, thus when no current
is flowing through the coil the axis of the magnet will be
in the plane of the coil. A current flowing through the
coil will produce a magnetic force at right angles to the
plane of the coil, proportional to the intensity of the
current. Let this magnetic force be equal to Qi where
i is the intensity of the current flowing through the coil
and 0 a quantity depending upon the dimensions of the
coil. G is called the ' Galvanometer constant.' Let H
be the horizontal component of the magnetic force at the
centre of the coil. Then the resultant magnetic force at
the centre of the coil has a component H in the plane of
the coil and a component Qi at right angles to it, hence

222]

MAGNETIC FOKCE DUE TO CURRENTS

375

if 0 is the angle which the resultant magnetic force makes
with the plane of the coil,

tan 0 = ^f

.(l).

When the magnet is in equilibrium its axis will lie along
the direction of the resultant magnetic force, hence the
passage of the current will deflect the magnet through
an angle 6 given by equation (1). As the current is pro
portional to the tangent of the angle of deflection, this
instrument is called the tangent Galvanometer.

The smaller we can make H, the external magnetic
force at the centre of the coil, the larger will be the angle
through which a given current will deflect the magnet.
By placing permanent magnets in suitable positions in the
neighbourhood of the coil we can partly neutralize the
earth's magnetic field at the centre of the coil : in this way
we can reduce H and increase the sensitiveness of the
galvanometer. A magnet for this purpose is shown in
Fig. 107, which represents an ordinary type of galvano
meter.

Fig. 107.

376

MAGNETIC FORCE DUE TO CURRENTS

[CH. X

Another method of increasing the sensitiveness of the
instrument is employed in the ' astatic galvanometer.'
In this galvanometer (Fig. 108) we have two coils A and B
in series, so arranged that the current circulates round

Fig. 108.

them in opposite directions. Thus, if the magnetic force
at the centre of the upper coil is upwards from the plane
of the paper, that at the centre of the lower coil will be
downwards. Two magnets a, /?, mounted on a common
axis, are placed at the centres of the coils A and B re
spectively, the axes of magnetization of these magnets
point in opposite directions ; thus as the magnetic forces
at the centres of the two coils due to the currents are also
in opposite directions, the couples due to the currents
acting on the two magnets will be in the same direction.
The couples arising from the external magnetic field
will however be in opposite directions: if the external
magnetic field is uniform and the moments of the two
magnets very nearly equal, the couple tending to restore
the magnet to its position of equilibrium will be very
small, and the galvanometer will be very sensitive.

222] MAGNETIC FORCE DUE TO CURRENTS 377

The larger we make G the greater will be the sensi
tiveness of the galvanometer. If the galvanometer consists
of a single circle of radius a, then (see Art. 212) G = 2?r/a.
If there are n turns close together and arranged so that
the distance between any two turns is a very small fraction
of the radius of the turns, then G is approximately 2irn/a.
If the galvanometer consists of a circular coil of rectangular
cross section, the sides of the rectangle being in and at
right angles to the plane of the coil, and if 26 is the breadth
of this rectangle (measured at right angles to the plane of
the coil), 2a the depth in the plane of the coil, n the
number of turns of wire passing through unit area, then
taking as axis of x the line through the centre of the coil
at right angles to its plane, and as axis of y a line through
the centre at right angles to this, we have

G = 27m

•6 rc+a tfdxdy

b rc + a
_6 } c-a

where c is the mean radius of the coil.

If 26, 2</> are the angles subtended at the centre by
AB, CD, Fig. 109, this reduces to

cot-

G = 4f7rnb log — .

rn-^4- r

'*!

In sensitive galvanometers the hole in the centre for
the magnet is made as small as possible, so that the inner
windings have very small radii ; when this is the case, we

may put (j> = ^ , and then

n
G = ^Trnb log cot ^ .

378

MAGNETIC FORCE DUE TO CURRENTS

[CH. X

In this case when the area of the cross section of the
coil is given, i.e. when 262 cot 6 is given, we can prove that

G is a maximum when

f\
log cot - = 2 cos 6,

the solution of this equation is 0 = 16° 46': this makes the
breadth bear to the depth the ratio of 1 to 1*61.

H G

Fig. 109.

The sensitiveness of modern galvanometers is very
great, some of them will detect a current of 10~13 amperes.
It would take a current of this magnitude centuries to
liberate 1 c.c. of hydrogen by electrolysis.

Since

i = 77 tan 0,
Or

while

= sin 0 cos 6.

Thus for a given absolute increment of i, B0 will be
greatest when 6 is zero, and for a given relative increment,

223]

MAGNETIC FOKCE DUE TO CURRENTS

379

SO, or the change in deflection, will be greatest when
(9-45°.

In some cases it is important to have the magnetic
field near the magnet as uniform as possible. This can be
attained (see Art. 213) by using two equal coils placed
parallel to one another and at right angles to the line join
ing their centres, the distance between the coils being
equal to the radius of either. The magnet is then placed
on the common axis of the two coils and midway between
them.

223. Sine Galvanometer. In this galvanometer,
Fig. 110, the coil itself can move about a vertical axis, its

Fig. 110.

position being determined by means of a graduated hori
zontal circle. In using the instrument the coil is placed so
that when no current goes through it the magnetic axis
of the magnet at its centre is in the plain of the coil.
When a current passes through the coil, the magnet is
deflected out of this plane, and the coil is now moved

380 MAGNETIC FORCE DUE TO CURRENTS [CH. X

round until the axis of the magnet is again in the plane
of the coil. When this is the case the components of
the magnetic force at right angles to the plane of the
coil due respectively to the current and to the external
magnetic field must be equal and opposite. If H is
the external magnetic force, <j> the angle through which
the coil has been twisted when the axis of the magnet
is again in the plane of the coil, the external force at right
angles to the plane of the coil is H sin <£. If i is the
current through the coil, G the magnetic force at its
centre when the wires of the coil are traversed by unit
current, then the magnetic force at right angles to the
coil due to the current is Gi ; hence when this is in equi
librium with the component due to the external field,
H sin (j> = Gi,

. H .
or i = -~- sm (/>.

The advantage of this form of galvanometer is that the
magnet is always in the same position with respect to the
coil. For the same coils and magnetic field the deflection
is greater for the sine than for the tangent galvano
meter.

224. Desprez-d'Arsonval Galvanometer. In this
galvanometer the coil carrying the current moves while
the magnets are fixed. The galvanometer is represented
in Fig. 111. A rectangular coil is suspended by very fine
metal wires which also serve to convey the current to the
coil. The coil moves between the poles of a horse-shoe
magnet, and the magnetic field is concentrated on the coil
by a fixed soft iron cylinder placed inside the coil. When
a current flows round the coil, the coil tends to place itself

224]

MAGNETIC FORCE DUE TO CURRENTS

381

so as to include as many tubes of magnetic induction
as possible (Art. 215). It therefore tends to place itself so
that its plane is at right angles to the lines of magnetic
induction. The motion of the coil is resisted by the
torsion of the wire which suspends it, and the coil takes a
position in which the couple due to the torsion of the wire
just balances that due to the magnetic field. When the
magnetic field is uniform the relation between the de-

Fig, ill.

flection and the current is as follows. Let A be the area
of the coil, n the number of turns of wire, i the current
through the wire, B the magnetic induction at the coil.
When the plane of the coil makes an angle $ with the
direction of magnetic induction the number of tubes of
magnetic induction passing through it is

BAnsm<f>,

hence, by Art. 215, the couple tending to twist the coil is
iBAncoscf).

If the torsional couple vanishes when <£ is zero, the
couple when the coil is twisted through an angle <£

382 MAGNETIC FORCE DUE TO CURRENTS [CH. X

will be proportional to <£; let it equal T</>, then when
there is equilibrium, we have

iBAn cos <f> = r^>,

or j '=

J2JL71 cos (j> '
if <£ is small this equation becomes approximately

~BAn'

225. Ballistic Galvanometer. A galvanometer may
be used to measure the total quantity of electricity passing
through its coil, provided the electricity passes so quickly
that the magnet of the galvanometer has not time to
appreciably change its position while the electricity is
passing. Let us suppose that when no current is passing
the axis of the magnet is in the plane of the coil, then
if i is the current passing through the plane of the coil,
G the galvanometer constant, i.e. the magnetic force at the
centre of the coil when unit current passes through it,
m the moment of the magnet, the couple on the magnet
while the current is passing is

Qim.

If the current passes so quickly that the magnet has
not time sensibly to depart from the magnetic meridian
while the current is flowing, the earth's magnetic force
will exert no couple on the magnet. Thus if K is the
moment of inertia of the magnet, 9 the angle the axis
of the magnet makes with the magnetic meridian, the
equation of motion of the magnet during the flow of
the current is

225] MAGNETIC FORCE DUE TO CURRENTS 383

thus if the magnet starts from rest the angular velocity
after a time t is given by the equation

7 —>,.,„, icfa.
ac

If the total quantity of electricity which passes through
the galvanometer is Q and the angular velocity com
municated to the magnet w, we have therefore
Kco — GmQ.

This angular velocity makes the magnet swing out of
the plane of the coil : if H is the external magnetic force
at the centre of the coil, the equation of motion of the
magnet is, if there is no retarding force,

at*
Integrating this equation we get

1 -cos <9 = 0.

If S- is the angular swing of the magnet, the angular
velocity vanishes when 8 — ^, hence

= 2mH (1 - cos ^) = 4<mH sin2 - .
On substituting for co the value previously found we get

Q = 2 sin |S- 7^- VmH.R.
If T is the time of a small oscillation of the magnet,

hence

384 MAGNETIC FORCE DUE TO CURRENTS [cil. X

We have neglected any retarding force such as would
arise from the resistance of the air. Galvanometers which
are used for the purpose of measuring quantities of
electricity are called 'ballistic galvanometers,' and are
constructed so as to make the effects of the frictional forces
as small as possible. This is done eifcher by making the
moment of inertia of the magnet very large, or by making
the magnet so symmetrical about its axis of rotation that
the frictional forces are but small. The correction to be
applied when the frictional forces are not negligible is
investigated in Maxwell's Electricity and Magnetism,
Vol. II. p. 386.

226. Measurement of Resistance. The arrange
ment of conductors in the Wheatstone's Bridge (Art. 191)

Fig. 112.

enables us to determine the resistance of one arm of the
bridge, say BD, Fig. 89, in terms of the resistances of the
arms A C, GB and AD. For the measurement of resistances
by this method wires having a known resistance are used.
These are called resistance coils, and are made in the
following way. A piece of silk-covered German-silver
wire is taken and doubled back on itself (to avoid effects
due to electromagnetic induction, see Chap. XL) and then
wound in a coil. Its length is then carefully adjusted

226]

MAGNETIC FORCE DUE TO CURRENTS

385

until its resistance is some multiple of the standard
resistance, the ohm. Each end of this coil is soldered to
a stout piece of brass such as Ay B, or C, Fig. 112 ; these
pieces are attached to an ebonite board to insulate them
from each other. Two adjacent pieces of brass can be
put in electrical connection by inserting stout well-fitting
brass plugs between them. When the plug is out the
resistance between B and G is that of the wire, while
when the plug is in there is practically no resistance
between these places.

Fig. 113.

When there is no current through the arm CD of the
Wheats tone's Bridge there is, by Art. 191, a certain
relation between four resistances : hence to measure a
resistance by this method we require three known re
sistances. These resistances are conveniently arranged
in what is known as the Post-Office Resistance Box.
This is a box of coils arranged as in Fig. 113, and pro
vided with screws at A, B, C, D, to which wires can be
attached. To determine the resistance of a conductor
such as R connect one end to B and the other end to D ;
connect one terminal of a galvanometer to C and the other
to D, and one electrode of a battery to A, the other to B.
The arrangement of the conductors is the same as that in
the diagram in Art. 191, which is reproduced here by the

T. E. 25

386 MAGNETIC FORCE DUE TO CURRENTS [CH. X

side for convenience. To measure the resistance of R:
take one or more plugs out of CA and CB and then pro
ceed to take plugs out of AD until there is no deflection
of the galvanometer, when the battery circuit is completed.
As the current through CD vanishes, we must have by
Art. 191

resistance of BD x resistance of A G

= resistance of BG x resistance of AD.

As the resistances of AC, BC, AD are known, that of BD
is determined by this equation.

227. Resistance of a Galvanometer. A method
due to Lord Kelvin for measuring the resistance of a
galvanometer is an interesting example of the property
of conjugate conductors. We saw (Art. 192) that if CD
is conjugate to AB, then the current sent through any
arm of the bridge by a battery in AB is independent of
the resistance in CD, and the converse is also true. To
apply this to measure the resistance of a galvanometer,
place the galvanometer in the arm BD of the bridge and
replace the galvanometer in CD by a key by means of
which the circuit CD can be completed or broken at
pleasure. Then adjust the resistance of AD until the
deflection of the galvanometer is the same when the
circuit CD is completed as when it is broken. As in
this case the current through BD is independent of the
resistance of CD, CD must be conjugate to AB, and we
have therefore (Art. 191),

resistance of galvanometer x resistance of AC

= resistance of BC x resistance of A D.

CHAPTEE XI

ELECTROMAGNETIC INDUCTION

228. Electromagnetic Induction, of which the laws
were unravelled by Faraday, may be illustrated by the
following experiment. Two circuits A and B, Fig. 114, are
placed near together, but completely insulated from each

Fig. 114.

other; a galvanometer is in the circuit B, and a battery and
key in A. Suppose the circuit A at the beginning of the
experiment to be interrupted, press down the key and
close the circuit, the galvanometer in B will be deflected,
indicating the passage of a current through B, although B
is completely insulated from the battery. The deflection
of the galvanometer is not a permanent one, but is of the
same kind as that of a ballistic galvanometer when a finite

25—2

388 ELECTROMAGNETIC INDUCTION [CH. XI

quantity of electricity is quickly discharged through it,
that is, the magnet of the galvanometer is set swinging, but
is not permanently deflected, as it oscillates symmetrically
about its old position of equilibrium. This indicates that an
electromotive force, acting for a very short time, has acted
round B. The direction of the deflection of the magnet
of the galvanometer in B indicates that the direction of
the momentary current induced in B was opposite to that
started in A. After a time the motion of the magnet
subsides and the magnet remains at rest, although the
current continues to flow through A. If, after the magnet
has come to rest, we raise the key in A, so as to stop the
current flowing through the circuit, the galvanometer in
B is again affected, the direction of the first swing in this
case being opposite to that which occurred when the
current in A was started, indicating that when the current
in A is stopped, an electromotive force is produced round
B tending to start a current through B in the same
direction as that which previously existed in A. This
electromotive force, like the one produced when the circuit
A was completed, is but momentary.

These experiments show that the starting or the
stopping of a current in a circuit A is accompanied by
the production of another current in a neighbouring circuit
5, the current in B being in the opposite direction to that
in A when the current is started and in the same direction
when the current is stopped.

If instead of making or breaking the current in A, this
current is kept steadily flowing in the circuit, while the
circuit itself is moved about, then when A is moving away
from B an electromotive force is produced tending to send
round B a current in the same direction as that round A,

228] ELECTROMAGNETIC INDUCTION 389

while if A is moved towards B an electromotive force acts
round B tending to produce a current in the opposite
direction to that round A. These electromotive forces in
B only occur when A is moving, they stop as soon as it is
brought to rest. If we replace the circuit A, with the
current flowing through it, by its equivalent magnet, then
we shall find that the motion of the magnet will induce
the same currents in B as the motion of the circuit A. If
we keep the circuit A, or the magnet, fixed and move B,
we also get currents produced in B.

The currents started in B by the alteration in intensity
or position of the current in A, or by the alteration of the
position of B with respect to magnets in its neighbour
hood, are called induced currents ; and the phenomenon is
called electromagnetic induction.

A good deal of light is thrown on these phenomena
if we interpret them in terms of the tubes of magnetic
induction. Let us first take the case when the induction
is produced by starting a current in A. Then before the
current circulates through A no tubes of magnetic induc
tion pass through B ; when the current is started through
A this circuit is at once threaded by a number of tubes of
magnetic induction, some of which pass through B. The
induced current through B also causes B to be threaded
by tubes of magnetic induction, which since the induced
current is in the opposite direction to the primary one in
A, pass through the circuit in the opposite direction to
those sent through it by the current in A ; thus the effect
of the induced current in B is to tend to make the total
number of tubes of magnetic induction passing through B
zero; that is, to keep the total number of tubes of magnetic
induction through B the same as it was before the current

390 ELECTROMAGNETIC INDUCTION [CH. XI

was started in A. We shall find, when we investigate the
laws of induction more closely, that the tubes of magnetic
induction passing through B, due to the induced current,
are at the moment of making the primary circuit equal in
number and opposite in direction to those sent through B
by the current in A. The laws of the induction of currents
may thus be expressed by saying that the number of tubes
of magnetic induction passing through B does not change
abruptly.

Again, take the case when currents are induced in
B by stopping the current in A. Initially the current
flowing through A sends a number of tubes of magnetic
induction through B : when the current in A is stopped
these tubes cease, but the current induced in B in the
same direction as that in A causes a number of tubes of
magnetic induction to pass through B in the same direc
tion as those due to the original current in A. Thus the
action of the induced current is again to tend to keep the
number of tubes of magnetic induction passing through B
constant.

The same tendency to keep the number of tubes of
magnetic induction through B constant is shown by the
induction of a current in B when A is moved away from or
towards B. When A is moved away from B, the number
of tubes of magnetic induction due to A which pass
through B is diminished, but there is a current induced in
B in the same direction as that through A, which causes
additional tubes of magnetic induction to pass through B
in the same direction as those due to A : the production
of these tubes counterbalances the diminution due to the
recession of A, and thus the induced current again tends
to keep the number of tubes of magnetic induction passing

228] ELECTROMAGNETIC INDUCTION 391

through B constant. The same thing occurs when A is
moved towards B, or when currents are induced in B
by the motion of permanent magnets in its neighbour
hood.

Not only is there a tendency to keep the number of
tubes of magnetic induction passing through any circuit
in the neighbourhood of A constant, there is also the same
tendency with respect to the circuit A itself. Let us
suppose that A is alone in the field, then, when a current
is flowing round A, tubes of magnetic induction pass
through it. If the circuit is broken, and the current
stopped, the number of tubes would fall to zero; the
tendency, however, to preserve unaltered the number of
tubes passing through the circuit, will under suitable cir
cumstances, cause the current, in its effort to continue
flowing in the same direction, to spark across an air-gap
when the circuit is broken, even though the original
E.M.F., applied to send the current through A, was totally

Fig. 115.

inadequate to produce a spark. To show this effect
experimentally it is desirable to wind the coil A round a
core of soft iron, so as, with a given current, to increase the

392 ELECTROMAGNETIC INDUCTION [CH. XI

number of tubes of magnetic induction passing through
the circuit; the coil of an electro -magnet shows this effect
very well. The effect of this tendency is shown very
clearly in the following experiment. The coil of an electro
magnet E, Fig. 115, is placed in parallel with an electric
lamp L, the resistance of the lamp being very large com
pared with that of an electro-magnet; in consequence of
this, when the two are connected up to a battery, by far
the greater part of the current will flow through the coil,
comparatively little through the lamp, too little indeed to
raise the lamp to incandescence. If however the circuit is
broken at K, the tendency to keep the number of tubes of
magnetic induction passing through the circuit constant,
will send a current momentarily round the circuit HLGE,
which will be larger than that flowing through the lamp
when the battery is kept continuously connected up to
the circuit ; and thus though the lamp remains quite dark
when the current is steady, it can be raised to bright
incandescence by repeatedly making and breaking the
circuit.

229- The electromotive force round a circuit due
to induction does not depend upon the metal of which
the circuit is made. This may be proved by taking two
equal circuits of different metals, iron and copper, say,
placed close together and arranged so that the electro
motive forces due to induction in the two circuits tend
to oppose each other. When this circuit, connected up
to a galvanometer, is placed in a varying magnetic
field, no current passes, showing that the electromotive
forces in the two circuits are equal and opposite.

Faraday proved that in a magnetic field varying at

229] ELECTROMAGNETIC INDUCTION 393

an assigned rate, the electromotive force round a circuit
due to induction is proportional to the number of tubes of
magnetic induction passing through the circuit, by taking
a coil made of several turns of very fine wire, and in
serting in it a galvanometer whose resistance was small
compared with that of the coil : when this coil was placed
in a varying field the deflection of the galvanometer was
found to be independent of the number of turns in the
coil. As all the resistance in the circuit is practically in
the coil, the resistance of the circuit will be proportional to
the number of turns in the coil. Since the quantity of
electricity passing through the circuit is independent of
the number of turns, it follows that the E.M.F. round the
circuit must have been proportional to the resistance, i.e.
to the number of turns of the coil. Hence, since the turns
of the coils were so close together that each enclosed the
same number of tubes of magnetic induction, it follows
that when the rate of change is given the E.M.F. round
the circuit must be proportional to the number of tubes
of magnetic induction passing through it.

Faraday also showed by rotating the same circuit
at different speeds in the same magnetic field that the
E.M.F. round the circuit is proportional to the speed
of rotation, i.e. to the rate of change of the number of
tubes of magnetic induction passing through the circuit.

These investigations of Faraday's determined the
conditions under which induced currents are produced:
F. E. Neumann was however the first to give, in 1845, an
expression by which the magnitude of the electromotive
force could be determined. We may state the law of
induction of currents as follows — Whenever the number of
tubes of magnetic induction passing through a circuit is

394 ELECTROMAGNETIC INDUCTION [CH. XI

changing, there is an E.M.F. acting round 'the circuit
equal to the rate of diminution in the number of tubes of
magnetic induction which pass through the circuit. The
positive direction of the E.M.F. and the positive direction
in which a tube passes through the circuit are related to
each other like rotation and translation in a right-handed
screw.

We shall show later on (page 472) that this law can
be connected with Ampere's law (Art. 214) by dynamical
principles.

Let us apply this law of induction to the case of a
circuit exposed to a variable magnetic field. Let the
circuit contain a galvanic battery whose electromotive
force is E0, and let the resistance of the circuit, including
that of the battery, be R. If P is the number of tubes of
magnetic induction passing at any time t through the
circuit, there will be an E.M.F. equal to -dP/dt round
the circuit due to induction; hence by Ohm's law, we
have if i is the current round the circuit,

(1).

Suppose the magnetic field is due to two currents, one
circulating round this circuit and the other through a
second circuit in its neighbourhood; let j be the current
passing round the second circuit. Let L be the coefficient
of self-induction of the first circuit, N that of the second,
M the coefficient of mutual induction between the two
circuits. Then as the magnetic field is due to the two
circuits,

230] ELECTROMAGNETIC INDUCTION 395

and equation (1) becomes

If S is the resistance of the second circuit and EQ' the
electromotive force of any battery there may be in that
circuit, then we have similarly,

230. Let us compare these equations with the equa
tions of motion of a dynamical system having two degrees
of freedom, one degree being fixed by the coordinate x,
the other by the coordinate y ; these coordinates may be
regarded as fixing the positions of two moving pieces. Let
the first moving piece be acted upon by the external force
E0, the second by the force EJ. Let the motions of the
first and second moving pieces be resisted by resistances
proportional to their velocities, and let Rx, Sy be these
resistances respectively. The momenta corresponding to
the two moving pieces will be linear functions of the
velocities. Let the momentum of the first moving piece

Lx + My,
that of the second

MX + Ny.

Then, if L, M, N are independent of the coordinates x, y,
the equations of motion of the two systems will be

396 ELECTROMAGNETIC INDUCTION [CH. XI

Comparing these equations with those for the two currents
we see that they are identical if we make i, j the currents
round the two circuits coincide with x, y the velocities
of the two moving pieces. The electrical equations of a
system of circuits are thus identical with the dynamical
equations of a system of moving bodies, the current flowing
round a circuit corresponds to a velocity, the number of
tubes of magnetic induction passing through the circuit
to the momentum corresponding to that velocity, the
electrical resistance corresponds to a viscous resistance,
and the electromotive force to a mechanical force.

A further analogy is afforded by the comparison of
the Kinetic Energy of the Mechanical System with the
energy in the magnetic field due to the system of
currents. The Kinetic Energy of the Mechanical System
is equal to

The energy in the magnetic field is by Art. 220 equal
to

This expression becomes identical with the preceding
one if we write x for i and y for ;'.

Since the terms in the electrical equations which
express the induction of currents correspond to terms
in the dynamical equations which express the effects of
changes in the momentum, and as these latter effects arise
from the inertia of the system, we are thus led to regard
a system of electrical currents as also possessing inertia.
The inertia of the system will be increased by any circum
stance which, for given values of the currents, increases
the number of tubes of electromagnetic induction passing

231]

ELECTROMAGNETIC INDUCTION

397

through the circuits ; the inertia of the system may thus
be increased by the introduction of soft iron in the neigh
bourhood of the circuits.

231. We can illustrate by a mechanical model the
analogies between the behaviour of electrical circuits and
a suitable mechanical system. Models of this kind have
been designed by Maxwell and Lord Rayleigh; a simple
one which serves the same purpose is represented in
Fig. 116.

''"'2

Fig. 116.

It consists of three smooth parallel horizontal steel
bars on which masses m,, M, m2 slide, the masses being
separated from the bars by friction wheels: the three
masses are connected together by a light rigid bar, which
passes through holes in swivels fixed on to the upper part
of the masses ; the bar can slide backwards and forwards
through these holes, so that the only constraint imposed by
the bar is to keep the masses in a straight line.

This system will, if we regard the velocities of mlt m2
respectively along their bars as representing currents
flowing round two circuits, illustrate the induction of
currents. Let us start with the three masses at rest,
then suddenly move m± forward along its bar, m2 will then

398 ELECTROMAGNETTC INDUCTION [CH. XI

move backwards, an effect analogous to the production
of the inverse current in the secondary when the current
is started in the primary. If now mx is moved uniformly
forward the friction between mz and its bar will soon bring
it to rest and it will continue at rest as long as the motion
of mx remains uniform : this is analogous to the absence of
current in the secondary when the current in the primary
is uniform. If now we suddenly stop ml3 ra2 will start
off in the direction in which m1 was moving before being
brought to rest. This is analogous to the direct current in
the secondary produced by the stoppage of the current
in the primary. These effects are the more marked the
greater the mass M.

It is instructive to find the quantities in the dynamical
system which correspond to the coefficients of self and
mutual induction. Let us suppose that the bar on which
M slides is midway between the other two.

Then if xv is the velocity of ml along its bar, #2 that of
ra2, the velocity of M will be (a^ + #2)/2, and T the kinetic
energy of the system is given by the equation

The momentum along x^ is dTfdx^ and is therefore
equal to

M\. M .
l + -£,ri+ 4-**'
The momentum along #2 is dT/dx2 and is therefore
equal to

M_& f + M\

Thus m^ + if/4, m2 + if/4 correspond to the coefficients of

231] ELECTROMAGNETIC INDUCTION 399

self-induction of the two circuits, while M/4> corresponds
to the coefficient of mutual induction between the circuits.
The effect of increasing the coefficient of mutual induction
between the circuits, such an increase for example as may
be produced by winding the primary and secondary coils
round an iron core, may be illustrated by the effect pro
duced on the model by increasing the mass M relatively
to ml and w2.

The behaviour of the model will illustrate important
electrical phenomena. Thus suppose the mass m^ is struck
with a given impulse, it will evidently move forward with
greater velocity if m2 is free to move than if it is fixed,
for if m2 is free the large mass M will move very slowly
compared with mi} the connecting bar turning round the
swivel on M almost as if this were fixed: if however m2 is
fixed, then when ml moves forward it has to drag M along
with it, and will therefore move more slowly than in the
preceding case. When m^ is free to move it moves in
the opposite direction to ra^ Now consider the electrical
analogue, the case when m2 is free to move corresponds
to the case when there is in the neighbourhood of the
primary circuit a closed circuit round which a current can
circulate : the case when ra2 is fixed corresponds to the
case when this circuit is broken, when it can produce no
electrical effect as no current can circulate round it. The
greater velocity of rax when m2 was free than when it
was fixed shows that when an electrical impulse acts on
a circuit the current produced is greater when there
is another circuit in the neighbourhood than when the
primary circuit was alone in the field; in other words,
the presence of the secondary diminishes the effective
inertia or self-induction of the primary.

400 ELECTROMAGNETIC INDUCTION [CH. XI

232. Effect of a Secondary Circuit. As an

example in the use of the equations given in Art. 229 we
shall consider the behaviour of a primary and a secondary
coil when an electric impulse acts upon the primary.
Let us suppose that originally there were no currents in
the circuits. Let L, M, N be respectively the coefficients
of self-induction of the primary, of mutual induction
between the primary and the secondary, and the coefficient
of self-induction of the secondary : R, S the resistances
of the primary and secondary respectively, x and y the
currents through these coils. Then if P' is the external
electromotive force acting on the primary, we have by
the equations of Art. 229,

R0 = P' (1),

% = 0 (2).

The primary is acted on by an impulse, that is the force
P' only lasts for a short time, let us call this time r.
Then if a?0, y0 are the values of a?, y due to this impulse
we have by integrating equation (1) from £ = 0 to t = r

r r

Jo Jo

Since r is indefinitely small and x is finite
ft
\ xdt = 0;

let {TPfdt = P,

then we have Lx0 + My0 = P (3).

Similarly by integrating (2) we get

= 0 (4),

232] ELECTROMAGNETIC INDUCTION 401

hence

If the secondary circuit had not been present the
current in the primary due to the same impulse would
have been PjL: thus the effect of the secondary is to
increase the initial current in the primary : it diminishes
its effective self-induction from L to L - M2/N. This is
an illustration of the effect described in the last article.
Equation (4) expresses that the number of tubes of mag
netic induction passing through the second circuit is not
altered suddenly by the impulse acting on the first circuit.

When the impulse ceases, the circuits are free from
external forces, and the equations for x and y are

jt(Lx + My) + Rx = Q ............... (5),

Sy = Q...; ........... (6).

Let us now choose as the origin from which time is
measured the instant when the impulse ceases. Integrate
these equations from t = 0 to t = oo , then since x and y
will vanish when £ = oo we have

,00

R I xdt = Lx0 + MyQ

= P by equation (3),
but J^ xdt is the total quantity of electricity which passes

across any section of the primary circuit, if we denote
this by Q we have

•-$ .--:; ...... .-•:,.

hence Q is not affected by the presence of a secondary
circuit. Thus since the current is greater to begin with

T. E. ' 26

402 ELECTROMAGNETIC INDUCTION [CH. XI

when the secondary is present than when it is absent, it
must, since Q is the same in the two cases, die away
faster on the whole when the secondary is present.

The presence of the secondary increases the rate at
which the current dies away just after it is started, but
diminishes the rate at which the current ultimately dies
away.

Integrating (6) from t = 0 to t = oo we find

-00

J 0

= 0 by equation (4) ;

hence the total quantity of electricity passing across any
section of the secondary circuit is zero.
To solve equations (5) and (6) put
x = Ae~Kt,

eliminating A and B we find

(R-L\)(S-N\) = MW (7);

hence if Xj, X2 are the roots of this quadratic, we have

j 1 * 2

We notice that since \Lx> + Mxy + %Ny*t the expres
sion for the kinetic energy of the currents, must be positive
for all values of x and y, LN - Mz must be positive, and
therefore \± and X2 are positive quantities. If we deter
mine the values of the A's and .B's from the values of
x and y when t = 0, we find after some reductions
1 PN

-M-

232] ELECTROMAGNETIC INDUCTION 403

We see from the quadratic equation (7) that one of its
roots is greater than, the other root less than S/N, thus
\ — S/N, X2 — S/N are of opposite signs, and therefore
by (8), x the current through the primary never changes
sign; y the current through the secondary begins by
being of the opposite sign to x, it changes sign, and
finally x and y are of the same sign.

A very important special case of the preceding in
vestigation is when the two circuits are close together,
or when the circuits are wound round a core of soft iron
which completely fills their apertures ; in this case nearly
all the lines of magnetic force which pass through one
circuit pass through the other also ; this is often expressed
by saying that there is very little magnetic leakage between
the circuits. When this condition is fulfilled L — M2/N is
very small compared with L. In the limiting case when
this quantity vanishes we see by equation (7) that one of
the values of X, say X2> is infinite, while \ is equal to

LS + NR'

In this case we find from equations (8) and (9) that,
except at the very beginning of the motion,

The relation between the currents and the time, when

26—2

404

ELECTROMAGNETIC INDUCTION

[CH. XI

L — M2/N is small, is represented by the curves in
Fig. 117; the dotted curve represents the current in the
primary when the secondary is absent.

Fig. 117.

233. Currents induced in a mass of metal by
an impulse. Let us suppose that the impulse is due
to the sudden alteration of a magnetic system. Let N
be the number of tubes of magnetic induction due to
this system which pass through any circuit ; to fix our
ideas let us suppose this is the primary circuit in the
case considered in Art. 232. Then using the notation of
that article

" dt '
by Faraday's law.

Hence P = P P'dt = -(NT- N9),

J o

where NT and N0 represent respectively the number of
tubes of magnetic induction passing through the circuit

233] ELECTROMAGNETIC INDUCTION 405

at the times t = r and t = 0 respectively. We have,
however, by equation (3), Art. 232,

Lx, + %0 = P,

or Lx, + My, + Nr = N0.

Now the right-hand side is the number of tubes of
magnetic induction which pass through the circuit at the
time t = 0, i.e. the time when the impulse began to act ;
the left-hand side represents the number of tubes of
magnetic induction, some of them now being due to the
currents started in the circuit, which pass through the
circuit at the time t = T when the impulse ceases to
act. The equality of these two expressions shows that
the currents generated by the impulse are such as to
keep the number of tubes of magnetic induction which
pass through the circuit unaltered. The case we have
considered is one where there is only one secondary,
the reasoning is however quite general, and whenever an
impulse acts upon a system of conductors, the currents
started in these conductors are such that their electro
magnetic action causes the number of tubes of magnetic
induction passing through any of the conducting circuits
to be unaltered by the impulse.

Let us apply this result to the case of the currents
induced in a mass of metal by the alteration in an
external magnetic field.

The number of tubes passing through every circuit
that can be drawn in the metal is the same after the
impulse as before. Hence we see that the magnetic field
in the metal is the same after the impulse as before. This
will give an important result as to the distribution of
currents inside the metal. For we have seen (Art. 203)

406 ELECTROMAGNETIC INDUCTION [CH. XI

that the work done when unit pole is taken round a closed
circuit is equal to 4nr times the current flowing through
that circuit. Now as the magnetic field inside the metal,
and therefore the work done when unit pole passes round
a closed circuit, is unaltered by the impulse, the current
flowing through any such closed curve is also unaltered
by the impulse ; hence, as there were no currents through
it before the impulse acted, there will be none generated
by the impulse. In other words, the currents generated
in a mass of metal by an electric impulse are entirely on
the surface of the metal, and the inside of the conductor
is free from currents.

234. The currents will not remain on the surface,
they will rapidly diffuse through the metal and die away.
We can find the way the currents distribute themselves
after the impulse stops by the use of the two fundamental
principles of electro-dynamics, (1) that the work done
by the magnetic forces when unit pole travels round
a closed circuit is equal to 4-Tr times the quantity of
current flowing through the circuit, (2) that the total
electromotive force round any closed circuit is equal
to the rate of diminution of the number of tubes of
magnetic induction passing through the circuit.

Let u, v, w be the components of the electric current
parallel to the axes of x, y, z at any point ; a, /3, 7 the
components of the magnetic force at the same point. The
axes are chosen so that if x is drawn to the east, y to the
north, z is upwards. Consider a small rectangular circuit
ABCD, the sides AB, BC being parallel to the axes of z
and y respectively. Let AB=2h, BC='2k. Let a, /3, 7 be
the components of magnetic force at 0, the centre of the

234] ELECTROMAGNETIC INDUCTION 407

rectangle ; x,y,z the coordinates of 0 ; let the coordinates
of P, a point on AB, be x, y + k, z + f ; the z component
of the magnetic force at P will be approximately

dy ,, dry 1

v + -r-£+--,-k.

dz dy

Let now a unit magnetic pole be taken round the
rectangle A BCD, the direction of motion round A BCD
being related to the positive direction of x like rotation
and translation in a right-handed screw. The work done
on unit pole as it moves from A to B will be

which is equal to 2hy + 2hk -7- ;

the work done on the pole as it moves from G to D is

We may show similarly that the work done on unit
pole as it moves from B to G is equal to

and when it moves from D to A, to

Adding these expressions we see that the work done on unit
pole as it travels round the rectangle A BCD is equal to

dy dz

The quantity of current passing through this rectangle is
equal to 4tuhk,

408

ELECTKOMAGNETIC INDUCTION [CH. XI

hence since the work done on unit pole in going round
the rectangle is equal to 4?r times the current passing
through the rectangle, see Art. 203, we have

By taking rectangles whose sides are parallel to the axes
of x and z, and of as, y we get in a similar way

da. dy

-j --- ri .................. (2)

dz dx

If X, F, Z are the components of the electric intensity at
0, we can prove by a similar process that the work done
on unit charge of electricity in going round the rectangle
A B CD is equal to

If a, b, c are the components of magnetic induction
at 0, the number of tubes of magnetic induction passing
through the rectangle is ax^hk; hence the rate of
diminution of the number of unit tubes is equal to

But by Faraday's law of Electromagnetic Induction the
work done on unit charge in going round the circuit is
equal to the rate of diminution in the number of tubes
of magnetic induction passing through the circuit, hence

da ... fdZ dY

--rr4M= -= --- F-

at \dy dz

234]

ELECTKOMAGNETIC INDUCTION

409

similarly

da dZ
dt . dy
db_dX

dY \
dz '
dZ

dt dz
dc_dY
dt dx

dx '
dX
dy

(4).

Let us consider the case when the variable part of
magnetization is induced, so that

da_ da db _ d@ dc dy
dt dt ' dt dt dt dt '
where IJL is the magnetic permeability. If a is the specific
resistance of the metal in which the currents are flowing,
and if the currents are entirely conduction currents,

o-u = X, av = Y, aw = Z.
We have by equation (1)

du d dc d db

47TyLt -y- =

hence by putting Y= av, Z= aw in equation (4) we get
, du fd*u d2u d2u\ d /du dv dw^

We see from equations (1), (2), (3) that
du dv dw _

TH ~t~ ~J7. ~^~ ~j~ = V,

dx dy dz

hence

similarly

= „

dt
dw

d2u d*u
+ + -

df dz*J '
d2w d2w'

410 ELECTROMAGNETIC INDUCTION [CH. XI

WTe can also prove by a similar method that
da fd*a d2a dW

with similar equations for b and c.

These equations are identical in form with those which
hold for the conduction of heat, and we see that the
currents and magnetic force will diffuse inwards into
the metal in the same way as temperature would diffuse
if the surface of the metal were heated, and then the
heat allowed to diffuse.

235. We may apply the results obtained in the
conduction of heat to the analogous problem in the dis
tribution of currents. As a simple example let us take
a case in one dimension. Let us suppose that over the
infinite face of a plane slab we have initially a uniform
distribution of currents, and that these currents are left
to themselves. Then from the analogous problem in the
conduction of heat we know that after a time t has
elapsed the current, at a distance x from the face to which
the currents were originally confined, will be proportional
to

This expression satisfies the differential equation and
vanishes when t = 0 except at the face where x = 0.
The currents at a distance x will attain their maximum
value when

and the magnitude of the maximum current will be
inversely proportional to x.

236] ELECTROMAGNETIC INDUCTION 411

In the case of copper /JL= 1, a— 1600, hence the time
at which the current is a maximum at a place, one centi
metre from the surface, is 2-7T/1600 seconds, or about 1/250
of a second, a point '1 cm. from the surface would receive
the maximum current after about 1/25,000 of a second,
while at a point 10 cm. from the surface the current would
not reach its maximum for about 4/10 of a second.

Let us now consider the case of iron : for an average
specimen of soft iron we may put a = 104, //, = 103 ; hence
in this case, the time the current, 1 cm. from the surface,
will take to reach its maximum value is about 2?r/10
seconds, while a place 10 cm. from the face only attains
its maximum after 20?r seconds. Thus the currents
diffuse much more slowly through iron than they do
through copper. The diffusion of the currents is regu
lated by two circumstances, the inertia of the currents
which tends to confine them to the outside of the con
ductor, and the resistance of the metal which tends to
make the currents diffuse through the conductor; though
the resistance of iron is greater than that of copper,
this is far more than counterbalanced by the enormously
greater magnetic permeability of the iron which increases
the inertia of the currents, and thereby the tendency
of the currents to concentrate themselves on the outside
of the conductor.

When t is much greater than ^/(CT/TT/A), e~t^l^ differs
little from unity, in this case the currents are almost in
dependent of x and vary inversely as A, thus the currents
ultimately get nearly uniformly distributed, and gradually
fade away.

236. Periodic electromotive forces acting on

412 ELECTROMAGNETIC INDUCTION [CH. XI

a circuit possessing inertia. So far we have confined
our attention to the case of impulses; we now proceed
to consider the case when electromotive forces act on
a circuit for a finite time. If these forces are steady the
currents will speedily become steady also, and there will
be no effects due to induction; when, however, these forces
are periodic, induction will produce very important effects
which we shall now proceed to investigate. We shall
commence with the case of a single circuit whose co
efficient of self-induction is L and whose resistance is R ;
we shall suppose that this circuit is acted on by an
external electromotive force varying harmonically with
the time, the force at the time t being equal to E cos pt'}
this expression represents a force making p/27r complete
vibrations a second, it changes its direction p/Tr times per
second. If i is the current through the coil, we have

•j \ -M.VV JJ/ \J\JO JJU \ -*- / j

the solution of this equation is

,_#cos(^-a)

where tana = -^ ........................ (3).

The maximum value of the electromotive force is E,
while the maximum value of the current is

if a steady force E acted on the circuit the current
would be E/R. Thus the inertia of the circuit makes
the maximum current bear to the maximum electromotive
force a smaller ratio than a steady current through
the same circuit bears to the steady electromotive force

236] ELECTROMAGNETIC INDUCTION 413

producing it. The ratio of the maximum electromotive
force to the maximum current, when the force is periodic,

is equal to {R* 4- L2p2} 2 ; this quantity is called the
impedance of the circuit.

We see from equation (2) that the phase of the current
lags behind that of the electromotive force. When the
force oscillates so rapidly that Lp is large compared with
R, we see from equation (3) that a will be approximately
equal to Tr/2. In this case the current through the coil
will be greatest when the electromotive force acting on
the circuit is zero, and will vanish when the electromotive
force is greatest.

In this case, since Lp is large compared with R, we
have approximately

. E .
t-jjrinj*;

thus the current through the circuit is approximately
independent of the resistance and depends only upon
the coefficient of self-induction and on the frequency of
the electromotive force. Thus a very rapidly alternating
electromotive force will send far more current through
a short circuit with a small coefficient of self-induction,
even though it is made of a badly conducting material,
than through a long circuit with large self-induction,
even though this circuit is made of an excellent con
ductor. For steady electromotive forces on the other
hand, the current sent through the second circuit would
be enormously greater than that through the first.

The work done by the current per unit time, which
appears as heat, is equal to the mean value of either

414 ELECTROMAGNETIC INDUCTION [CH. XI

E cos pt . i or Ri2, and is equal to

Thus when the electromotive force changes so slowly that
Lp is small compared with R, the work done per unit
time varies inversely as R; while when the force varies
so rapidly that Lp is large compared with R, the work
done varies directly as R. If E, p and L are given the
work done is a maximum when

R = Lp.

237. Circuit rotating in the Earth's field. An

external electromotive force of the type considered in the
last article is produced when a conducting circuit rotates
with uniform velocity o> in the earth's magnetic field about
a vertical axis. If 6 is the angle the plane of the circuit
makes with the magnetic meridian, H the horizontal com
ponent of the earth's magnetic force, A the area of the
circuit, then the number of tubes of magnetic induction
passing through the circuit is

H A sin 6 :
the rate of diminution of this is

.
dt

If the circuit revolves wifch uniform angular velocity &>,
0 = wt, and the rate of diminution in the number of tubes
of magnetic induction passing through the circuit is

— H A a) cos at,

as this, by Faraday's law, is the electromotive force
acting on the circuit. The case is identical with that just
considered if we write co for p and — HA a) for E '; thus

237] ELECTRO MAGNETIC INDUCTION 415

if L is the coefficient of self-induction of the circuit,
R the resistance, i the current through the circuit,

— a)

The motion of the circuit is resisted by a couple whose
moment is, by Art. 214, equal to the current multiplied
by the differential coefficient with respect to 6 of the
number of tubes of magnetic induction due to the earth's
field passing through the circuit ; thus the moment of
the couple is

iHA cos 6,

H*A2o) cos cot cos (cot — a)

[LW + R^
Thus the couple always tends to oppose the rotation

of the coil unless 6 is between ^ and ^ + a or between
STT , STT

To maintain the motion of the circuit work must be
spent ; the amount of work spent in any time is equal
to the mechanical equivalent of the heat developed in
the circuit.

The mean value of the retarding couple is
*c0 cos PL H*AzRa>

it vanishes when co is zero or infinite and is greatest
when co = RjL.

If the circuit rotates so rapidly that LCD is large
compared with R, a. is approximately equal to Tr/2, and
we see that

HA sin cot

416 ELECTROMAGNETIC INDUCTION [CH. XI

Now by definition Li is the number of tubes of
magnetic induction due to the currents which pass
through the circuit, while HA sin wt is the number pass
ing through the same circuit due to the earth's magnetic
field ; we see from the preceding expression for i that the
sum of these two quantities, which is the total number of
tubes of magnetic induction passing through the circuit,
remains zero throughout the whole of the time. This is
an illustration of the general principle that when the
inertia effects are paramount the number of tubes passing
through any conducting circuit remains constant.

238. Circuits in parallel. Suppose that two points
A and B are connected by two circuits in parallel. Let
R be the resistance of the first circuit, S that of the
second ; let the first circuit contain a coil whose coefficient
of self-induction is L, the second one whose coefficient of
self-induction is N. Let the coils be so far apart that
their coefficient of mutual induction is zero. Then if
a difference of potential Ecospt be maintained between
the points A and B we see by the preceding investigations
that i and j, the currents in the two circuits, will be given
by the equations

. _ E cos (pt — a)

~~~

._Ecos(pt-@)
J~ {N*p + 8*}*''

where tan a = -~ , tan ft = —£- .

Jl> b

If the external electromotive force varies so rapidly that

239] ELECTROMAGNETIC INDUCTION 417

Lp and Np are large compared with R and 8 respectively >
then

. _ E sin pt

~ Np '

or the currents flowing through the two circuits are
inversely proportional to their coefficients of self-induc
tion. Thus with very rapidly alternating currents the
distribution of the currents is almost independent of
their resistances and depends almost entirely on their
self-inductions. Thus if one of the coils had a moveable
iron core, the current through the coil would be very
much increased by removing the iron, as this would
greatly diminish the self-induction of the circuit.

239. Transformers. We have hitherto confined our
attention to the case when the only circuit present was
the one acted upon by the periodic electromotive force.
We shall now consider the case when in addition to the
circuit acted upon by the external electromotive force,
which we shall call the primary circuit, another circuit is
present in which currents are induced by the alternating
currents in the primary: we shall call this circuit the
secondary circuit, and suppose that it is not acted upon
by any external electromotive force beyond that due to
the alternating current in the primary. A very important
example of this is afforded by the ' transformer.' In this
instrument a periodic electromotive force acts on the
primary, which consists of a large number of turns of
wire ; in the ordinary use of the transformer for electric
lighting this electromotive force is so large that it would

T. E. 27

418 ELECTROMAGNETIC INDUCTION [CH. XI

be dangerous to lead the primary circuit about a building ;
the current for lighting is derived from a secondary circuit
consisting of a smaller number of turns of wire. The
primary and secondary circuits are wound round an iron
core as in Fig. 118.

Fig. 118.

The tubes of magnetic induction concentrate in this
core, so that most of the tubes which pass through the
primary pass also through the secondary.

The current in this secondary is larger than that in
the primary, but the electromotive force acting round it
is smaller. The current in the secondary bears to that in
the primary approximately the same ratio as the electro
motive force round the primary bears to that round the
secondary.

Let L, M, N be respectively the coefficients of self-
induction of the primary, of mutual induction between the
primary and the secondary and of self-induction of the
secondary, let R and S be the resistances of the primary
and secondary respectively, x and y the currents through
these coils. Let Ecospt be the electromotive force acting

239] ELECTROMAGNETIC INDUCTION 419

on the primary. To find x and y we have the following
equations :

(1),

The values of x and y are

x = A cos (pt - a) (3),

By substituting these values in equations (1) and (2),
we find

,_ T

L'p
tan a = -=£-

Y + s^

an/-« = -.
Np

From the expressions for A and a in terms of E we see
that the effect of the secondary circuit is to make the
primary circuit behave like a single circuit whose co
efficient of self-induction is L' and whose resistance is R '.
We see from the expressions for L' and R', that L' is less
than L, while R' is greater than R. Thus the presence of
the secondary circuit diminishes the apparent self-induc
tion of the primary circuit, while it increases its resistance.

27—2

420 ELECTROMAGNETIC INDUCTION [CH. XI

When the electromotive force changes so rapidly that Np
is large compared with 8, we have approximately

ft - a = TT.

This value of the apparent self-induction is the same
as that under an electrical impulse, see Art. 231. In a
well-designed transformer L — M*/N is exceedingly small
compared with L. When the secondary circuit is not
completed S is infinite ; in this case L' = L. When the
secondary circuit is completed through electric lamps
&c., 8 is in practice small compared with Np, so that
L' — L — M*/N. Thus the completion of the circuit
causes a great diminution in the value of the apparent
self-induction of the primary circuit. The work done per
unit time in the transformer is equal to the mean value
of Ecospt.x, it is thus equal to
1 Ez cos a

E*R

When the secondary circuit is broken S is infinite and
therefore L' = L, R' = R, and the work done on the trans
former per unit time or the power spent on it is equal to
1 E*R

When the circuit is completed, and 8 is small compared

239] ELECTROMAGNETIC INDUCTION 421

with Np, L' = L- M*IN, R' = R + M2S/N*, and then the
power spent is equal to

£rV

This is very much greater than the power spent
when the secondary circuit is not completed; this must
evidently be the case, as when the secondary circuit is
completed lamps are raised to incandescence, the energy
required for this must be supplied to the transformer. The
power spent when the secondary circuit is not completed
is wasted as far as useful effect is concerned, and is spent
in heating the transformer. The greater the coefficient of
self-induction of the primary, the smaller is the current
sent through the primary by a given electromotive force,
and the smaller the amount of power wasted when the
secondary circuit is broken. When the secondary circuit
is closed the self-induction of the primary is diminished
from L to Lr ; since there is less effective self-induction
in the primary, the current through it, and consequently
the power given to it, is greatly increased.

We see from the expression just given that the power
absorbed by the transformer is greatest when

that is, when

When there is no magnetic leakage, i.e. when

422 ELECTROMAGNETIC INDUCTION [CH. XI

the power absorbed continually increases as the resist
ance in the secondary diminishes ; when however LN is
not equal to Mz the power absorbed does not necessarily
increase as S diminishes, it may on the contrary reach
a maximum value for a particular value of S, and any
diminution of 8 before this value will be accompanied
by a decrease in the energy absorbed by the transformer.
The greater the frequency of the electromotive force, the
larger will be the resistance of the secondary when the
absorption of power by the transformer is greatest. When
the frequency is very great, such as, for instance, when
a Leyden jar is discharged (see page 436), the critical value
of the resistance in the secondary may be exceedingly
large. In this case the difference between the maximum
absorption of power and that corresponding to S = 0 may
be very great. Thus when S = 0, the power absorbed

is equal to

1 E*R

or approximately for very high frequencies

while the maximum power absorbed is

which exceeds that when S = 0 in the proportion of L'p
to 2R.

The currents x, y in the primary and secondary are
represented by the equations

x — A cos (pt — a),

239] ELECTROMAGNETIC INDUCTION 423

Thus the ratio of the maximum value of the current
in the primary to that in the secondary is B/A : by
equation (5), we have

or, when Np is large compared with S,
AM
B~ N'
13 — a = TT.

If the primary and secondary coils cover the same
length of the core, and are wound on a core of great
permeability, then MjN is equal to m/nt where m is the
number of turns in the primary, and n the number in
the secondary.

If we have a lamp whose resistance is s in the secondary
the potential difference between its electrodes is sy, i.e.

sB cos (pt - /9).

The maximum value of this expression is sB; substi
tuting the value of B, we find that when Np is large com
pared with 8 this value is equal to

M ET
SNE

This is greatest when L' = 0, in which case it is equal to

and this, as S is small compared with Np, is equal to

424 ELECTROMAGNETIC INDUCTION [CH. XI

If R is small compared with SM*/N2 this is ap
proximately

Thus if for example M/N=<20, the maximum current
through the secondary is 20 times that through the
primary ; while . the electromotive force between the
terminals of the lamp is approximately

s F
20S

Now s is always smaller than S, as S is the resistance
of the whole secondary circuit, while s is the resistance
of only a part of it : the electromotive force between the
terminals of any lamp is thus in this case always less than
1/20 of the electromotive force between the terminals of
the secondary. In getting this value we have assumed the
conditions to be those most favourable to the production
of a high electromotive force in the secondary; if there
is any magnetic leakage, i.e. if L' is not zero, then at
high frequencies the electromotive force in the secondary
would be very much less than the value just found, in
fact where there is any magnetic leakage, the ratio of the
electromotive force in the secondary to that in the primary
is indefinitely small when the frequency is infinite.

240. Distribution of rapidly alternating currents.

When the frequency of the electromotive force is so great
that in the equations of the type

L -^ + M -~ + ... Rx = external electromotive force,
dt dt

the term Rx depending on the resistance is small com
pared with the terms Ldxjdt, Mdyjdt depending on

240] ELECTROMAGNETIC INDUCTION 425

induction, which, if the electromotive force is supposed to
vary as cospt, will be the case when Lp, Mp are large
compared with R ; the equations determining the currents
take the form

-7- (Loc + My + ...) = external electromotive force,
cLt

dN
" dt '

where N is the number of tubes of induction due to the
external system passing through the circuit whose co
efficient of self-induction is L.

We see from this that

Lx -f My + . . . + N = constant,

and since a?, y . . . N all vary harmonically, the constant
must be zero. Now Lx + My + ... is the number of tubes
of magnetic induction which pass through the circuit
we are considering due to the currents flowing in this
and the neighbouring circuits, while N is the number of
tubes passing through the same circuit due to the ex
ternal system. Hence the preceding equation expresses
that the total number of tubes passing through the circuit
is zero. The same result is true for any circuit.

Now consider the case of the currents induced in a
mass of metal by a rapidly alternating electromotive force.
The number of tubes of magnetic induction which pass
through any circuit which can be drawn in the metal is zero,
and hence the magnetic induction must vanish through
out the mass of the metal. The magnetic force will con
sequently also vanish throughout the same region. But
since the magnetic force vanishes, the work done when unit
pole is taken round any closed curve in the region must
also vanish, and therefore by Art. 203 the current flowing

426 ELECTROMAGNETIC INDUCTION [CH. XI

through any closed curve in the region must also vanish ;
this implies that the current vanishes throughout the mass
of metal, or in other words, that the currents generated
by infinitely rapidly alternating forces are confined to the
surface of the metal, and do not penetrate into its interior.

We showed in Art. 235 that the currents generated
by an electrical impulse started from the surface of the
conductor and then gradually diffused inwards. We may
approximate to the condition of a rapidly alternating force
by supposing a series of positive and negative impulses
to follow one another in rapid succession. The currents
started by a positive impulse have thus only time to
diffuse a very short distance from the surface before the
subsequent negative impulse starts opposite currents from
the surface ; the effect of these currents at some distance
from the surface is to tend to counteract the original
currents, and thus the intensity of the current falls off
rapidly as the distance from the surface of the conductor
increases.

The amount of concentration of the current depends
on the frequency of the electromotive force and of the
conductivity of the conductor. If the frequency is infinite
and the conductivity finite, or the frequency finite and
the conductivity infinite, then the current is confined to
an indefinitely thin skin near the surface of the conductor.
If, however, both the frequency and the conductivity are
finite, then the thickness of the skin occupied by the
current is finite also, while the magnitude of the current
diminishes rapidly as we recede from the surface. Any
increase in the frequency or in the conductivity increases
the concentration of the current.

240] ELECTROMAGNETIC INDUCTION 427

The case is analogous to that of a conductor of heat,
the temperature of whose surface is made to vary har
monically, the fluctuations of temperature corresponding
to the alterations in the surface temperature diminish in
intensity as we recede from the surface, and finally cease
to be appreciable. The fluctuations, however, with a long
period are appreciable at a greater depth than those with
a short one. We may for example suppose the temperature
of the surface of the earth to be subject to two variations,
one following the seasons and having a yearly period, the
other depending on the time of day and having a daily
period. These fluctuations become less and less apparent
as the depth of the place of observation below the surface
of the earth increases, and finally they become too small
to be measured. The annual variations can, however, be
detected at depths at which the diurnal variations are
quite inappreciable.

This concentration of the current near the surface of
the conductor, which is sometimes called 'the throttling of
the current,' increases the resistance of the conductor to
the passage of the current. When, for example, a rapidly
alternating current is flowing along a wire, the current
will flow near to the outside of the wire, and if the
frequency is very great the inner part of the wire will
be free from current ; thus since the centre of the wire is
free from current, the current is practically flowing through
a tube instead of a solid wire. The area of the cross
section of the wire, which is effective in carrying this
rapidly alternating current, is thus smaller than the
effective area when the current is continuous, as in this
case the current distributes itself uniformly over the whole
of the cross section of the wire. As the effective area for

428 ELECTROMAGNETIC INDUCTION [CH. XI

the rapidly alternating currents is less than that for con
tinuous currents, the resistance, measured by the heat pro
duced in unit time when the total current is unity, is greater
for the alternating currents than for continuous currents.

241. Distribution of an alternating current in
a Conductor. The equations given in Art. 234 enable
us to find how an alternating current distributes itself
in a conductor. We shall consider a case in which the
analysis is simple, but which will serve to illustrate the
laws of the phenomenon we are discussing. This case
is that of an infinite mass of a conductor bounded by a
plane face. Take the axis of x at right angles to this
face, and the origin of coordinates in the face; let the
currents be everywhere parallel to the axis of z, and the
same at all points in any plane parallel to the face of the
conductor. Then if yu, is the magnetic permeability and
cr the specific resistance of the conductor, w the current at
the point as, y, z at the time t parallel to the axis of z,
we have by the equations of Art. 234,

dw (d?w d2w d2w

or, since w is independent of y and

We shall suppose that the currents are periodic, making
r complete alternations per second. We may put,

writing i for V — 1,

w =€ipt0)^

where co is a function of xt but not of t. Substituting this
value of w in equation (1) we get

241] ELECTROMAGNETIC INDUCTION 429

or if n2

2 _^2&)

The solution of this is

a, = Ae-nx + Benx,
where A and B are constants.

Now ?i = j

V41

We shall suppose that the conductor stretches from
x = 0 to x = oo and that the cause which induces the
currents lies on the side of the conductor for which x is
negative. It is evident that in this case the magni
tude of the current cannot increase indefinitely as we
recede from the face nearest the inducing system ; in
other words, w cannot be infinite when x is infinite : this
condition requires that B should vanish ; in this case we
have

and therefore

w = e a e
Thus if w = A cos pt when x — 0,

. ,

Ae \ " J cos-f — ^-\ x

at a distance x from the surface.

430 ELECTROMAGNETIC INDUCTION [CH. XI

This result shows that the maximum value of the
current at a distance x from the face is proportional to

_nnw\*x

€ \ <r J t Thus the magnitude of the current diminishes
in geometrical progression as the distance from the face
increases in arithmetical progression.

In the case of a copper conductor exposed to an electro
motive force making 100 alternations per second, /A = 1,

<7 = 1600, ^ = 27rxlOO; hence {2*v&p/<r}* = ir/2, so that

_TT«

the maximum current is proportional to e 2 . Thus at
1 cm. from the surface the maximum current would only
be '208 times that at the surface, at a distance of 2 cms.
only "043, and at a distance of 4 centimetres less than
1/500 part of the value at the surface.

If the electromotive force makes a million alternations

per second [Zirftp/o]* = 50?r; the maximum current is thus
proportional to e~50wx, and at the depth of one millimetre
is less than one six-millionth part of its surface value.

The concentration of the current in the case of iron
is even more remarkable. Consider a sample of iron
for which JJL = 1000, a = 10000, exposed to an electro
motive force making 100 alternations per second, so that

p = 2-7T x 100. In this case {27r/i/)/<rp = 20 approximately,
and thus the maximum current at a depth of one milli
metre is only *13 times the surface value, while at
5 millimetres it is less than one twenty-thousandth part
of its surface value.

If the electromotive force makes a million alternations
per second, then for this specimen of iron {29r/*p/<r}

242] ELECTROMAGNETIC INDUCTION 431

is approximately 2000, and the maximum current at the
distance of one-tenth of a millimetre from the surface
is about one five-hundred-millionth part of its surface
value.

We see from the preceding expressions for the current
that the distance required to diminish the maximum cur
rent to a given fraction of its surface value is directly
proportional to the square root of the specific resistance,
and inversely proportional to the square root of the number
of alternations per second.

242. Magnetic Force in the Conductor. The

currents in the conductor are all parallel to the axis of z,
and are independent of the coordinates y, z.

Now the equations of Art. 234 may be written in the

form

da _ fdw dv\ db fdu
~'dt = (7 \dy~dz)' ~dt

dc fdv du
dt \dx dy

where a, 6, c are the components of the magnetic induc
tion, u, v, w those of the current. In the case we are
considering u = v — 0, and w is independent of y and z ;
hence a = c = 0, and the magnetic induction is parallel
to the axis of y. Thus the currents in the plate are
accompanied by a magnetic force parallel to the surface
of the plate and at right angles to the direction of the
current.

From the above equations we have

db _ dw
dt~ dx '

432 ELECTROMAGNETIC INDUCTION [CH. XI

and by Art. 241 w = Ae~mx cos (pt — mx),

where m =

*J2crm .

Hence b = —

TT\

— mx—-r}
4/

cos [pt - mx — -

m \f 4

Thus the magnetic force in the conductor diminishes
as we recede from the surface according to the same law
as the current.

243. Mechanical Force acting on the Con
ductor. When a current flows in a magnetic field a
mechanical force acts on the conductor carrying the
current (see Art. 214). The direction of the force is
at right angles to the current and also to the magnetic
induction, and the magnitude of the force per unit volume
of the conductor is equal to the product of the current
and the magnetic induction at right angles to it.

In the case we are considering the magnetic induction
and the current are at right angles. If w is the intensity
of the current, the current flowing through the area
dxdy is wdxdy, hence the force on the volume dxdydz
parallel to x, and in the positive direction of x, is equal

— wbdxdydz.

The total force parallel to x acting on the conductor is
— II I wbdxdydz,

but since b and w are both independent of y and £, the
force acting on the conductor per unit area of its face is

— I wbdx,
J o

243] ELECTROMAGNETIC INDUCTION 433

Now if a, ft, 7 are the components of the magnetic
force _ dft da.

dx dy '

hence, since b = fift, we see that the force on the con
ductor parallel to x is

where ft0 is the value of ft when as = 0, i.e. at the surface
of the conductor, and ftx is the value of ft when x — oo .
But it follows from the expression for b given in the
last article that ftM = 0 ; hence the force on the conductor
parallel to x per unit area of its face due to the action of
the magnetic field on the currents is equal to

STT '

The magnetic force is not uniform in the conductor
but diminishes as we recede from the surface ; hence, if the
conductor is a magnetic substance, there will, in addition
to the mechanical force due to the action of the magnetic
field on the currents, be a force due to the effort of
the magnetic substance to move towards the stronger
parts of the field. The magnitude of the force parallel to

x per unit volume is by Art. 164 equal to ^=- — --^— :

OTT ax

thus the force acting per unit area of the face of the
slab due to ^his cause is

-00

J 0

8?r dx

T. B. 28

434 ELECTROMAGNETIC INDUCTION [CH. XI

LL/3 2

Adding this to the force ^— due to the action of the

07T

magnetic field on the currents we find that the total
force parallel to x is per unit area of surface of the slab
/302/87r, which for equal values of /30 is the same for
magnetic as for non-magnetic substances.

This force is always positive, and hence the conductor
tends to move along the positive direction of x\ in other
words, the conductor is repelled from the system which in
duces the currents in the conductor. These repulsions have
been shown in a very striking way in experiments made
by Professor Elihu Thomson and also by Dr Fleming.
In these experiments a plate placed above an electro
magnet round which a rapidly alternating current was
circulating, was thrown up into the air, the repulsion
between the plate and the magnet arising from the cause
we have just investigated.

6 2
The expression ^~ is the repulsion at any instant,

but since /30 is proportional to cos (pt + e) the mean value
of /302 is H2/2 if H is the maximum value of /30. Hence
the mean value of the repulsion is equal to

16-7T'

244. The screening off of Electromagnetic In
duction. We have seen in Art. 242 that the magnetic
force diminishes rapidly as we recede from the surface
of the conductor, and becomes inappreciable at a finite
distance, say d, from the surface. At a point P whose
distance from the surface is greater than d we may neglect
both the current and the magnetic force. Thus the electro-

244] ELECTROMAGNETIC INDUCTION 435

magnetic action of the currents in the sheet of the con
ductor whose thickness is d just counterbalances at P
the electromagnetic action of the original inducing system
situated on the other side of the face of the conductor.

Hence the slab of thickness d may be regarded as
screening off from P the electromagnetic effect of the
original system. In the investigation in Art. 242 we sup
posed that the conductor was infinitely thick, but since
the currents are practically confined to the slab whose
thickness is d, it is evident that the screening is done
by this layer and that no appreciable advantage is gained
by increasing the thickness of the slab beyond d. The
thickness d of the slab required to screen off the magnetic
force depends upon the frequency of the alternations and
on the magnetic permeability and specific resistance of the
conductor. By Arts. 241 and 242 the current and magnetic
force at a distance x from the surface are proportional to
e~mx) where m = (2v/ip/0>} , hence for a thickness d to
reduce the magnetic force to an inappreciable fraction of
its surface value md must be considerable. If we regard
the system as screened off when the magnetic effect is
reduced to a definite fraction of its undisturbed value,
then d the thickness of the screen is inversely propor
tional to m. The greater the frequency the thinner the
screen. Thus from the examples given in Art. 241 we
see that if the system makes a million oscillations a
second, a screen of copper less than a millimetre thick
will be perfectly efficient, while a screen of iron a very
small fraction of a millimetre in thickness will stop prac
tically all induction. If the system only makes 100 alter
nations a second, the screen if of copper must be several
centimetres and if of iron several millimetres thick.

28—2

436 ELECTROMAGNETIC INDUCTION [CH. XI

245. Discharge of a Leyden Jar. One of the

most interesting applications of the laws of induction of
currents is to the case of a Leyden jar, the two coatings of
which are connected by a conducting circuit possessing
self-induction. Let us consider a jar whose inside A is
connected to the outside B by a circuit whose resistance
is R and whose coefficient of self-induction is L. Let i
be the current flowing through the circuit from A to B\
VA and VB the potentials of A and B respectively. Then
by the laws of the induction of currents

L -r + Ri = electromotive force tending to increase i
dt

= VA-VB .................................... (1).

If Q is the charge on the inside of the jar, and C the
capacity of the jar, then

or (VA-Ve)=%.

The alteration in the charge is due to the current
flowing through the conductor, and i is the rate at which
the charge is diminishing, so that

di*

Substituting this value of i in equation (1), we get

The form of the solution of this equation will depend upon
whether the roots of the quadratic equation

are real or imaginary.

245] ELECTROMAGNETIC INDUCTION 437

Let us first take the case when they are imaginary,
i.e. when

In this case the solution of (2) takes the form

where A and a are arbitrary constants.

We see from this expression that Q is alternately
positive and negative and vanishes at times following
one another at the interval

The charge Q is thus represented by a harmonic function
whose amplitude decreases in geometrical progression as
the time increases in arithmetical progression.

The discharge of the jar is oscillatory, so that if, for
example, to begin with, the inside of the jar is charged
positively, the outside negatively; then on connecting by
the circuit the inside and the outside of the jar, the posi
tive charge on the inside diminishes; when however it has
all disappeared there is a current in the circuit, and the
inertia of this current keeps it going, so that positive
electricity still continues to flow from the inside of the jar;
this loss of positive electricity causes the inside to become
charged with negative electricity, while the outside gets
positively charged. Thus the jar which had originally
positive on the inside, negative on the outside, has now
negative on the inside, positive on the outside. The poten
tial difference developed in the jar by these charges tends

438 ELECTROMAGNETIC INDUCTION [CH. XI

to stop the current and finally succeeds in doing so. When
this happens the charges on the inside and outside would
be equal and opposite to the original charges if the re
sistance of the circuit were negligible ; if the resistance
is finite the new charges will be of opposite sign to the
old ones, but smaller. The current now begins to flow
in the opposite direction, and goes on flowing until the
inside is again charged positively, the outside negatively ;
if there were no resistance the charges on the inside and
outside would regain their original values, so that the
state of the system would be the same as when the dis
charge began; if the resistance is finite the charges are
smaller than the original ones. The system goes on then
as before until the charges become too small to be ap
preciable. The charges in the jar and the currents in the
wire are thus periodic, the charges surging backwards and
forwards between the coatings of the jar.

The oscillatory character of the discharge was sus
pected by Henry from observations on the magnetization
of needles placed inside a coil in the discharging circuit.
The preceding theory was given by Lord Kelvin in 1853.
The oscillations were detected by Feddersen in 1857.
The method he used consisted of putting an air break
in the wire circuit joining the inside to the outside of
the jar. This air break is luminous when a current passes
through it, shining out brightly when the current passing
through it is great, while it is dark when the current
vanishes. Hence if we observe the image of this air space
formed by reflection at a rotating mirror, it will, if the
discharge is oscillatory, be drawn out into a band with
dark and bright spaces, the interval between two dark
spaces depending on the speed of the mirror and the

245] ELECTROMAGNETIC INDUCTION 439

frequency of the electrical vibrations. Feddersen observed
that the appearance of the image of the air break formed
by a rotating mirror was of this character. He showed
moreover that the oscillatory character of the discharge
was destroyed by putting a large resistance in the circuit,
for he found that in this case the image of the air space
was a broad band of light gradually fading away in
intensity instead of a series of bright and dark bands.

When the discharge is oscillatory the frequency of the
discharges is often exceedingly large, a frequency of a
million complete oscillations a second being by no means
a high value for such cases. We see by the expression (3)
that when R = 0, the time of vibration is 2ir*/LC ; thus
this time is increased when the self-induction or the
capacity is increased. By inserting coils with very great
self-induction in the circuit, Sir Oliver Lodge has produced
such slow electrical vibrations that the sounds generated
by the successive discharges form a musical note.

In the preceding investigation we have supposed that
R2 was less than 4L/C; if however R is greater than this
value, the solution of equation (2) changes its character,
and we have now

where — Xi, —^ are the roots of the quadratic equation

i=0.

Xl =

440 ELECTROMAGNETIC INDUCTION [CH. XI

If we take t = 0 when the circuit is closed, then dQ/dt
vanishes when t = 0 and we get, if Q0 is the value of Q
when t = 0,

Hence dQ/dt never vanishes except when t = 0 and when
t = oo . Thus Q which is zero when £ = oo never changes
sign. The charge in this case instead of becoming positive
and negative never changes sign but continually diminishes,
and ultimately becomes too small to be observed. This
result is confirmed by Feddersen's observations with the
rotating mirror.

The behaviour of the Leyden jar is analogous to that
of a mass attached to a spring whose motion is resisted
by a force proportional to the velocity. If M is the mass
attached to the spring, x the extension of the spring, nx
the pull of the spring when the extension is x, rdxjdt the
frictional resistance, then the equation of motion of the
spring is

*d*x dx

Comparing this with the equation for Q we see that if
we compare the extension of the spring to the charge
on the jar, then the coefficient of self-induction of the
circuit will correspond to the mass attached to the spring,
the electrical resistance of the circuit to the frictional
resistance of the mechanical system, and the reciprocal of
the capacity of the condenser to n, the stiffness of the
spring.

246] ELECTROMAGNETIC INDUCTION 441

The pulling out of the spring corresponds to the charg
ing of the jar, the release of the spring to the completion
of the circuit between the inside and the outside of
the jar ; when the spring is released it will if the friction
is small oscillate about its position of equilibrium, the
spring being alternately extended and compressed, and
the oscillations will gradually die away in consequence
of the resistance ; this corresponds to the oscillatory dis
charge of the jar. If however the resistance to the motion
of the spring is very great, if for example it is placed in a
very viscous liquid like treacle, then when it is released it
will move slowly towards its position of equilibrium but
will never go through it. This case corresponds to the
non-oscillatory discharge of the jar when there is great
resistance in the circuit.

We have seen that the resistance of a conductor to a
variable current is not the same as to a steady one, and
thus since the currents which are produced by the dis
charge of a condenser are not steady, R, which appears in
the expression (2), is not the resistance of the circuit to
steady currents. Now R the resistance depends upon the
frequency of the currents, while as the expression (3)
shows, the frequency of the electrical vibrations depends
to some extent on the resistance ; hence the preceding
solution is not quite definite, it represents however the
main features of the case. For a complete solution we
may refer the reader to Recent Researches in Electricity
and Magnetism, J. J. Thomson, Art. 294.

246. Periodic Electromotive Force acting on
a circuit containing a condenser. Let an external
electromotive force equal to E cos pt act on the circuit

442 ELECTROMAGNETIC INDUCTION [CH. XI

which connects the coatings of the jar, let C be the capacity
of the jar, L the coefficient of self-induction, and R the
resistance of the circuit connecting its coatings. Then if
x is the charge on one of the coatings of the jar (which
of the coatings is to be taken is determined by the con
dition that an increase in x corresponds to a current in
the direction of the external electromotive force), we can
prove in the same way as we proved equation (2) Art. 245,
that

The solution of this equation is

- . *si"^-g) ., ...(2),

, ,1 dx Ecos(pt-a)

and thus -=- = -^- -

where tan a =

Comparing these equations with those of Art. 234 we
see that the circuit behaves as if the jar were done
away with and the self-induction changed from L to
L — l/(7p2. We also see from (3) that if Cp2 is greater
than 1/2Z, the current produced by the electromotive force
in the circuit broken by the jar (whose resistance is
infinite) is actually greater than the current which would
flow if the jar were replaced by a conductor of infinite con
ductivity. If Cp'2 = \JL the apparent self-induction of the

247] ELECTROMAGNETIC INDUCTION 443

circuit is zero, and the circuit behaves like an induction-
less closed circuit of resistance R. Thus by cutting the
circuit and connecting the ends to a condenser of suitable
capacity we can increase enormously the current passing
through the circuit. We can perhaps see the reason for
this more clearly if we consider the behaviour of the
mechanical system, which we have used to illustrate the
oscillatory discharge of a Leyden jar, viz. the rectilinear
motion of a mass attached to a spring and resisted by a
frictional force proportional to the velocity. Suppose that
X, an external force, acts on this system; then at any
instant X must be in equilibrium with (1) the resultant
of the rate of diminution of the momentum of the mass,
(2) the force due to the compression or extension of the
spring, (3) the resistance. If the frequency of X is very
great, then for a given momentum (1) will be very large,
so that unless (1) is counterbalanced by (2) a finite force
of very great frequency will produce an exceedingly small
momentum. Suppose however the frequency of the ex
ternal force is the same as that of the free vibrations of
the system when the friction is zero, then when the mass
vibrates with this frequency, (1) and (2) will balance each
other, so that all the external force has to do is to balance
the resistance ; the system will therefore behave like one
without either mass or stiffness resisted by a frictional
force.

247. A circuit containing a condenser is parallel
with one possessing self-induction.

Let ABC, AEG, Fig. 119, be two circuits. Let L be
the coefficient of self-induction of ABC, R the resistance
of this circuit, C the capacity of the condenser in AEG, r

444 ELECTROMAGNETIC INDUCTION [CH. XI

the resistance of wires leading from A and 0 to the plates.
Then if i is the current through ABC, a; the charge on the

Fig. 119.

plate nearest to A, we have, neglecting the self-induction
of the circuit AEG,

T di -p . doc x
Ldt + R* = rdt + C'

since each of these quantities is equal to the electromotive
force between A and C.

If i = cos pt,

(LY + Rrf .
then x = — *• T sin (pt + a),

Lp 1

where a = tan"1 -^5- + tan"1 — ~ .

R rpC

dx /Zy + R2
Hence - = /— •£• - cos (pt + a).

Thus the maximum current along A EC is to that
along ABC as \/Zy + .R2 is to A/7^-2+r2, or, if we can

neglect the resistances of the wires to the condenser, as
R* : l/Cp. We see that for very high frequencies

248] ELECTROMAGNETIC INDUCTION 445

practically all the current will go along the condenser
circuit.

Thus when the frequency is very high a piece of a
circuit with a little electrostatic capacity will be as
efficacious in robbing neighbouring circuits of current
as if the places where the electricity accumulates were
short-circuited by a conductor.

248. Lenz's Law. When a circuit is moved in a
magnetic field in such a way that a change takes place
in the number of tubes of magnetic induction passing
through the circuit, a current is induced in the circuit ;
the circuit conveying this current being in a magnetic
field will be acted upon by a mechanical force. Lenz's
Law states that the direction of this mechanical force is
such that the force tends to stop the motion which gave
rise to the current. The result follows at once from the
laws of the induction of currents. For suppose Fig. 120

Fig. 120.

represents a circuit which, as it moves from right to left,
encloses a larger number of tubes of induction passing
through it from left to right. The current induced will
tend to keep the number of tubes of induction unaltered,
so that since the number of tubes of magnetic induction
due to the external magnetic field which pass through
the circuit from left to right increases as the circuit
moves towards the left, the tubes due to the induced

446 ELECTROMAGNETIC INDUCTION [CH. XI

current will pass through the circuit from right to left.
Thus the magnetic shell equivalent to the induced current
has the positive side on the left, the negative on the
right. Since the number of tubes of induction due to
the external field which pass through this shell in the
negative direction, i.e. which enter at the positive and
leave at the negative side, increases as the shell is moved
to the left, the force acting on the shell is, by Art. 214,
from left to right, which is opposite to the direction of
motion of the circuit.

There is a simple relation between the mechanical
and electromotive forces acting on the circuit. Let P be
the electromotive force, X the mechanical force parallel
to the axis of x, i the current flowing round the circuit,
u the velocity with which the circuit is moving parallel
to x, N the number of unit tubes of magnetic induction
passing through the circuit. Then

~ dt '
and if the induced current is due to the motion of the

circuit dN dN

-jr =-T- • u;
dt ax

hence P = -.

Again, by Art. 214, we have

v .dN
X =i -j— ,
dx

so that Xu = -Pi.

If we wish merely to find the direction of the current
induced in a circuit moving in a magnetic field, Lenz's law
is in many cases the most convenient method to use.

248] ELECTROMAGNETIC INDUCTION 447

An example of this law is afforded by the coil revolving
in a magnetic field (Art. 237) ; the action of the magnetic
field on the currents induced in the coil produces a couple
which tends to stop the rotation of the coil. The magnets
of galvanometers are sometimes surrounded by a copper
box, the motion of the magnet induces currents in the
copper, and the action of these currents on the magnets
by Lenz's law tends to stop the magnet, and thus brings
it to rest more quickly than if the copper box were
absent. The quickness with which the oscillations of
the moving coil in the Desprez-D'Arsoiival Galvanometer
(Art. 224) subside is another example of the same effect ;
when the coil moves in the magnetic field currents are
induced in it, and the action of the magnetic field on these
currents stops the coil. Again, if a magnet is suspended
over a copper disc, and the disc is rotated, the movement
of the disc in the magnetic field induces currents in the
disc; the action of the magnet on these currents tends
to stop the disc, and there is thus a couple acting on the
disc in the direction opposite to its rotation. There must,
however, be an equal and opposite couple acting on the
magnet, i.e. there must be a couple on the magnet in
the direction of rotation of the disc; this couple, if the
magnet is free to move, will set it rotating in the
direction of rotation of the disc, so that the magnet and
the disc will rotate in the same direction. This is a
well-known experiment ; the disc with the magnet freely
suspended above it is known as Arago's disc. Another
striking experiment illustrating Lenz's law is to rotate
a metal disc between the poles of an electro-magnet, the
plane of the disc being at right angles to the lines of
magnetic force ; it is found that the work required to turn

448 ELECTROMAGNETIC INDUCTION [CH. XI

the disc when the magnet is ' on ' is much greater than
when it is ' off.' The extra work is accounted for by the
heat produced by the currents induced in the disc.

249. Methods of determining the coefficients of
self and mutual induction of coils. When the coils
are circles, or solenoids, the coefficients of induction can
be calculated. When, however, the coils are not of these
simple shapes the calculation of the coefficients would be
difficult or impossible ; they may, however, be determined
by experiment by means of the following methods.

250. Determination of the coefficient of self-
induction of a coil. Place the coil in BD, one of the

Fig. 121.

arms of a Wheatstone's Bridge, and balance the bridge
for steady currents, insert in CD a ballistic galvanometer,
and place a key in the battery circuit. When this key
is pressed down so as to complete the circuit, although
there will be no current through the galvanometer when
the currents get steady, yet a transient current will flow
through the galvanometer, in consequence of the electro
motive forces which exist in BD arising from the self-
induction of bhe coil. This current though only transient

250] ELECTROMAGNETIC INDUCTION 449

is very intense while it lasts and causes a finite quantity
of electricity to pass through the galvanometer, producing
a finite kick. We can calculate this quantity as follows :
an electromotive force E in BD will produce a current
through the galvanometer proportional to E, let this cur
rent be kE. In consequence of the self-induction of the
coil there will be an electromotive force in BD equal to

where L is the coefficient of self-induction of the coil and
i the current passing through the coil. This electromotive
force will produce a current q through the galvanometer
where q is given by the equation

If Q is the total quantity of electricity which passes
through the galvanometer

= -kffi(Li)dt,

the integration extending from before the circuit is com
pleted until after the currents have become steady. The
right-hand side of this equation is equal to

where i0 is the value of i when the currents are steady.
By the theory of the ballistic galvanometer, given in
Art. 225, we see that if 9 is the kick of the galvanometer

T. E. 29

450 ELECTROMAGNETIC INDUCTION [CH. XI

where T is the time of swing of the galvanometer needle,
G the galvanometer constant, and H the horizontal com
ponent of the earth's magnetic force.

Hence we have

TTrr

(1).

TTUT

Let us now destroy the balance of the Wheatstone's
Bridge by inserting a small additional resistance r in
BD, this will send a current p through the galvanometer.
To calculate p we notice that the new resistance has
approximately the current i0 running through it, and the
effect of its introduction is the same as if an electromotive
force ri0 were introduced into DB, this as we have seen
produces a current kri0 through the galvanometer ; hence

p = kri0.

This current will produce a permanent deflection </> of
the galvanometer, and by Art. 222

±H

p = tan <p -Q ,

or &n'0 = tan<£ -~ ..................... (2).

Hence from equations (1) and (2), we get

sin \Q T
L = r --\- .
tanc£ TT

251. Determination of the coefficient of mutual
induction of a pair of coils. Let A and B, Fig. 122,
represent the pair of coils of which A is placed in series
with a galvanometer, and B in series with a battery ; this

251] ELECTROMAGNETIC INDUCTION . 451

second circuit being provided with a key for breaking or
closing the circuit.

Let R be the resistance of the circuit containing A.
Suppose that originally the circuit containing B is broken
and that the key is then pressed down, and that after
the current becomes steady the current i flows through
this circuit. Then before the key is pressed down no

Fig. 122.

tubes of magnetic induction pass through the coil A,
while when the current i flows through B the number
of such unit tubes is Mi, where M is the coefficient of
mutual induction between A and B. Thus the circuit
containing A has received an electrical impulse equal to
Mit so that Q, the quantity of electricity flowing through
the galvanometer, will be Mi/R, and if 6 is the kick of
the galvanometer, we have

using the same notation as before. We can eliminate
a good many of the quantities by a method somewhat
similar to that used in the last case. Cut the circuit con
taining the coil A and connect its ends to two points on the
circuit B separated by a small resistance $ ; then if R is

29—2

452

ELECTROMAGNETIC INDUCTION

[CH. XI

very large compared with S this will not alter appreciably
the current flowing round J5; on this supposition the
current flowing round the galvanometer circuit will be

8 .

and if $ is the corresponding deflection of the galvano
meter

8 . H

.(2).

Hence from equations (1) and (2), we get

RS sin i<9 T

R + S tan </> TT '

252. Comparison of the coefficients of mutual
induction of two pairs of coils. Let A, a be one pair
of coils, B, b the other. Connect a and b in one circuit
with the battery, and connect the points P and Q (Fig. 123)

Fig. 123.

to the two electrodes of a ballistic galvanometer. Insert
resistances in PAQ and PBQ until there is no kick of
the galvanometer when the circuit through a and b is
made or broken. Let R be the resistance then in PAQ,

252] ELECTROMAGNETIC INDUCTION 453

8 that in PBQ, and let Mlt M2 be the coefficients of mutual
induction between the coils Aa, Bb respectively, then

ML M,
R~ S *

To prove this we notice that, by Art. 190, if we have
any closed circuit consisting of various parts, the sum of
the products obtained by multiplying the resistance of
each part by the current passing through it is equal to the
electromotive force acting round the circuit. In the case
when the electromotive forces are transient, we get by
integrating this result, that the sum of the products got
by multiplying the resistance of each part of the circuit by
the quantity of electricity which has passed through it is
equal to the electromotive impulse acting round the circuit.
Let us apply this to our case : if i is the steady current
flowing through the coils a and b, the electromotive impulse
acting on A due to the closing of the circuit is M^i, while
that on B is M2i. If x is the quantity of electricity which
passes through A when the circuit through a, b is closed,
y that through B, x — y will be the quantity which passes
through the galvanometer ; hence applying the above rule
to the circuit APQ, we have if K is the resistance of the
galvanometer circuit

Rx + K(x-y) = Mli.
Applying the same rule to the circuit BPQ, we get

But if the total quantity which passes through the
galvanometer is zero, we have x — yt and therefore

Ml_M1

R~ S '

454 ELECTROMAGNETIC INDUCTION [CH. XI

253. Comparison of the coefficients of self-
induction of two coils. Place the two coils whose
coefficients of self-induction are L and N respectively in
the arms AB, BD of a Wheatstone's Bridge, Fig. 121,
balanced for steady currents, then adjust the resistances
in AD, BD so that no kick of the galvanometer occurs
when the battery circuit is made ; these alterations in
the resistances of AD and BD will entail proportional
alterations in those of AC and BG in order to keep the
bridge balanced for steady currents. Then when there is
no kick of the galvanometer when the circuit is made, and
no steady deflection when it is kept flowing, we have

L_P_R

N~ Q~ S'

where P, Q, R, S are the resistances of the arms AD, BD,
AC, BC respectively.

We can see this as follows : suppose we have a
balanced Wheatstone's Bridge with the resistances in as
above, then for steady currents the balance will be un
disturbed if P and Q are altered in such a way that their
ratio remains unchanged ; but the alteration of P and Q
in this way is equivalent to the introduction into AD
and BD of electromotive forces proportional to P and Q.
For since no current flows through the galvanometer
the same current flows through AD as through BD} and
the preceding statement follows by Ohm's Law. Hence
we see that the introduction into the arms AD and BD
of electromotive forces proportional to P and Q, will not
alter the balance of the bridge, and, conversely, that if
this balance is not altered by the introduction of an

254] ELECTROMAGNETIC INDUCTION 455

electromotive force A into the arm AD, and another, B,
into the arm BD, then A/B must be equal to P/Q.

Now if we have coils in AD and BD whose coefficients
of self-induction are L, N, then since after the current
gets steady, the same current, i say, flows through each of
these coils, there must be, whilst the current is getting
steady, an impulse Li in AD, and another equal to Ni
in BD. Since these impulses do not send any electricity
through the galvanometer they must, by the preceding
reasoning, be proportional to P and Q, hence

£=P

N Q'

254. Heat developed by the hysteresis of iron.

We can, as Dr John Hopkirison showed, deduce from the
law of Electromagnetic Induction the expression given on
p. 261 for the heat produced in iron per unit volume when
the magnetic force undergoes a cyclical change. Take
the case of a solenoid filled with iron and carrying a
current whose value i is changing cyclically ; let I be the
length of the solenoid, n the number of turns of wire per
unit length, a the area of cross section of the core and
B the magnetic induction. The electromotive force in

the solenoid due to induction is —nla-j-, hence the work

Cit

spent by the current in time T in consequence of the
presence of the iron is

[T- 7 dB j+

\ inla —r- . at.

Jo dt
But if H is the magnetic force

H = 4}7TW,

456 ELECTROMAGNETIC INDUCTION [CH. XI

so that the work spent by the current, appearing as heat
in the iron, is equal to

— | laH-j-dt.

4-7T J o dt

Since the volume of the iron is la, the heat produced
per unit volume is

±!Hd4dt

This is the value already obtained on p. 261.

CHAPTER XII

ELECTRICAL UNITS:
DIMENSIONS OF ELECTRICAL QUANTITIES

255. In Art. 9 we denned the unit charge of elec
tricity, as the charge which repelled an equal charge with
unit mechanical force when the two charges were at unit
distance apart and surrounded by air at standard tem
perature and pressure. When we know the unit charge
the various other electrical units easily follow. Thus the
unit current is the one that conveys unit charge in unit
time; unit electric intensity is that which acts on unit
charge with unit mechanical force ; unit difference of
potential is the potential between two points when unit
work is done by the passage of unit charge from one point
to the other. Unit resistance is the resistance between
two points of a conductor between which the potential
difference is unity when the conductor is traversed by
unit current.

The step from the electrical to the magnetic quanti
ties is made by means of the law that the work done
when unit magnetic pole is taken round a closed circuit is
equal to 4?r times the current flowing through the circuit.
This law is to some extent a matter of definition. All
that is shown by experiment is that the work done when

458 ELECTRICAL UNITS [CH. XII

unit pole is taken round the circuit is proportional to the
current flowing through the circuit, and, as long as the
current remains the same, is independent of the nature
of the substances passed through by the pole in its tour
round the circuit. If we said that p times the work done
was equal to 4?r times the current, these conditions would
still be fulfilled provided p was independent of the current,
the magnetic force and the nature of the substances in
the field. Though, as we shall see later, it would be
possible to get a somewhat more symmetrical system of
units by a proper choice of p, yet in practice, to avoid
the introduction of an unnecessary constant, p is always
taken as unity. When p — 1, it follows from Art. 210 that
the magnetic force at the centre of a circle of radius a
traversed by a current i is 27ri'/a; thus unit magnetic
force will be the force at the centre of a circle of radius
2-7T traversed by unit current. Thus knowing the unit
current we can at once determine the unit magnetic force.
Having got the unit magnetic force, the unit magnetic
pole follows at once, since it is the pole which is
acted on by unit magnetic force with the unit mechani
cal force. From these units we can go on and deduce
without ambiguity the units of the other magnetic quan
tities. The System of units arrived at in this way is
called the Electrostatic System of Units.

Starting from the unit charge as defined in Art. 9,
we thus arrive at a unit magnetic pole. In Art. 114,
however, we gave another definition of unit magnetic pole
deduced from the repulsion between two similar poles.
The unit magnetic pole as defined in Art. 114 does not
coincide with the unit pole at which we arrive, starting,
as we have just done, from the unit charge of electricity.
The numerical relation between the two units depends

255] ELECTRICAL UNITS 459

upon what units of length and time we employ ; if these
are the centimetre and second, then the unit magnetic
pole on the electrostatic system of units is about 3 x 1010
times as great as the unit pole defined in Art. 114.

Instead of starting with unit charge of electricity we
may start with unit magnetic pole as defined in Art. 114.
The units of the other magnetic quantities would at once
follow from considerations similar to those by which we
deduced the unit electrical quantities from the unit
electrical charge. The electrical units would follow from
the magnetic ones, by the principle that the magnetic
force at the centre of a circular current of radius a is
2iri/a, where i is the strength of the current; thus the
unit current is that which produces unit magnetic force
at the centre of a circle whose radius is 2-jr. In this
way we can get the unit current, and from this the units
of the other electrical quantities follow without difficulty.
The System of units got in this way is called the Electro
magnetic System of Units.

The electromagnetic system of units does not coincide
with the electrostatic system. The electromagnetic unit
charge of electricity bears to the electrostatic unit charge
a ratio which depends on the units of length and time ; if
these are the centimetre and second the electromagnetic
unit of electricity is found to be about 3 x 1010 times the
electrostatic unit. The ratio of the electromagnetic unit
of charge to the electrostatic unit is equal to the ratio of
the electrostatic unit pole to the electromagnetic unit.

In the following table the relations between the
electrostatic and electromagnetic units of various electric
and magnetic quantities are given. Here v is the ratio
of the electromagnetic unit charge of electricity to the
electrostatic unit.

460 ELECTRICAL UNITS [CH. XII

Electrostatic unit

Quantity Symbol in terms of

Electromagnetic

Quantity of Electricity e l/v

Electric intensity F v

Potential difference V v

Current i l/u

Resistance of a conductor R v2

Electric Polarization D l/v

Capacity of a condenser C 1/V2

Strength of Magnetic Pole m v

Magnetic force H l/v

Magnetic induction B v

Magnetic permeability /A v2

Coefficient of Self-Induction L v2

Certain combinations of these quantities are equal
to purely geometrical or dynamical quantities, such as
length, force, energy. The numerical expression of such
combinations must evidently be the same whatever system
of units we employ; thus, for example, the mechanical
force on a charge e placed in a field of electric intensity
is Fe, but this force is a definite number of dynes, quite
independent of any arbitrary system of measuring electric
quantities, thus F x e must be the same whatever system
of electrical units we employ.

The following are examples of such combinations.

Time =?.

Length = -^

Force = Fe ; mH.
Energy = %Ve ; j£; Rift; \Li\
Energy per unit volume = FD/Str • pH*/8ir.

256] DIMENSIONS OF ELECTRICAL QUANTITIES 461

Thus since Fe is independent of the electrical units
chosen, if we adopt a new system in which the unit
of e is v times the old unit, the new unit of F must be
l/v times the old unit. Again, Ri2 is another quantity
unaltered by the change of units, so that if the new
unit of i is v times the old, the new unit of R must
be 1/fl2 times the old unit.

Dimensions of Electrical Quantities

256. For the general theory of Dimensions we shall
refer the reader to Maxwell's Theory of Heat, Chap. IV. ;
we shall in this chapter confine our attention to the
dimensions of electrical quantities.

It may be well to state at the outset that the
'dimensions' of electrical quantities are a matter of
definition and depend entirely upon the system of units
we adopt. Thus we shall find that on the electromagnetic
system of units a resistance has the same dimensions as
a velocity, while on the electrostatic system of units it
has the same dimensions as the reciprocal of a velocity.
In fact we might choose a system of units so as to make
any one electrical quantity of any assigned dimensions ;
when the dimensions of this are fixed that of the others
becomes quite determinate.

A symbol representing an electrical quantity merely
tells us how much of the quantity there is, and does not
tell us anything about the nature of the quantity; this
would require a dynamical theory of electricity. A theory
of dimensions cannot tell us what electricity is; its
object is merely to enable us to find the change in the
numerical measure of a given charge of electricity or any

462 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

other electrical quantity when the units of*length, mass
and time are changed in any determinate way.

We have to fix the electrical quantities by one or
other of their properties. Thus, to take an example, we
may fix a charge of electricity by the repulsion it exerts
on an equal charge, as is done in the electrostatic system
of units, or by the force experienced by a magnetic pole
when the charge is being transferred from one place to
another by a current, as is done in the electromagnetic
system ; these two measures are of different dimensions.
To take a simpler case we might fix a quantity of water
by the number of hydrogen atoms it contains, by its
mass, or by its volume at a definite temperature; all
these measures would be of different dimensions.

On the electrostatic system of units the force between
two equal charges e, separated by a distance L in a
medium whose specific inductive capacity is K, is e2/KL2,
and since this is of the dimensions of a force we have the
dimensional equation

m
KL? T*

M, L, T representing mass, length and time.

This result, with the meaning assigned to K in Art.
68, is only true on the electrostatic system of units. We
may, however, generalize the meaning of K and say that
whatever be the system of units, the repulsion between
the charges is e^lKL?, where K is defined as the ' specific
inductive capacity of the medium on the new system of
units.' We may regard this as the definition of K on this
system. The ratio of the K's for two substances on this
system is of course the same as the ratio of the K's on the

256] DIMENSIONS OF ELECTRICAL QUANTITIES 463

electrostatic system. We shall regard the dimensions of
K as indeterminate and keep them in the expression for
the dimensions of the electrical quantities1. From equa
tion (1) we have the dimensional equation

Similarly on the electromagnetic system of units the
repulsion between two poles of strength m separated by a
distance L in a medium whose magnetic permeability is /JL
is m^f/jiL2, fjb for this system of units being a quantity of
no dimensions. We shall suppose that whatever be the
system of units the force between the poles is equal to
m^lfjuD: where /JL thus determined is defined as the
magnetic permeability of the medium on this system of
units. Thus, for example, if m is the measure, on the
electrostatic system of units, of the strength of a pole,
the force between two equal poles separated by unit
distance in air is not m2 but 9 x 1020m2. Hence we say
the magnetic permeability of air on the electrostatic
system of units is 1/9 x 1020. We shall regard the di
mensions of fji as being left undetermined and retain JJL
in the expressions for the dimensions of the electric
quantities. Since w2//zZ2 is of the dimensions of a force
we have the dimensional equation

We shall find it instructive to suppose that the electric
and magnetic units are connected together by the rela
tion that p times the work done by unit pole in traversing
a closed circuit is equal to 4?r times the current flowing
through the circuit: the convention made on both the
electrostatic and magnetic systems is that p is a quantity

1 Kiicker, Phil. Mag. vol. 27, p. 104.

464 DIMENSIONS OF ELECTKICAL QUANTITIES [CH. XII

of no dimensions and always equal to unity. We shall for
the present leave the dimensions of p undecided.

The dimensional equation connecting the electric and
magnetic quantities is therefore

p x H x L = i,
where H is magnetic force, L a length and i a current.

Taking this relation and starting with the electric
charge, we can get by the equations given in Art. 255 the
dimensions of all the electrical and magnetic quantities in
terms of M, L, T, p,K : or starting with the magnetic pole
we can get them in terms of M, L, T, p, /JL. The results
for some of the more important electrical quantities are
given in the following table.

Quantity Symbol Dimensions in Dimensions in terms

terms of K and p of /u, and p

Charge e

Electric intensity F

Potential difference V

Current i

Resistance R

Electric polarization D

Capacity C KL

Specific inductive

capacity K K

Strength of Mag
netic pole m pK'

Magnetic force H

Magnetic induction B

Magnetic per
meability p

256] DIMENSIONS OF ELECTRICAL QUANTITIES 465

We see from this table that the dimensions of K, /UL, and
p must on all systems of measurement be connected by
the relation

On Maxwell's theory of the electric field p/^^K is equal
to the velocity with which electric disturbances travel
through a medium whose magnetic permeability is //, and
specific inductive capacity K.

On the electrostatic system of units K is of no dimen
sions, as the specific inductive capacity of air is taken as
unity whatever may be the units of mass, length and time.
Also on this system p is by hypothesis of no dimensions,
being always equal to unity. Hence the dimensions of
the electrical quantities on this system of units are got
by omitting p and K in the third column of the table.

On the electromagnetic system of units ^ is of no
dimensions, the magnetic permeability of air being taken
as unity whatever the units of mass, length and time ; p is
also of no dimensions on this system. Hence the dimen
sions of the electrical quantities on this system of units
are got by omitting //, and p from the fourth column in
the table.

Another system of units could be got by taking //, and
K as of no dimensions and p a velocity. If this velocity
were taken equal to the ratio of the electromagnetic unit
charge to the electrostatic unit, then the unit of electric
charge on this system would be the ordinary electrostatic
unit of that quantity, while the unit magnetic pole would be
the unit as defined on the electromagnetic system. This
system would thus have the advantage that the electric

T. E. 30

466 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

quantities would be as defined in the electrostatic system,
while the magnetic quantities would be as defined in the
magnetic system, and we should not have to introduce
any new definitions : whereas if we use the electrostatic
system we have to define all the magnetic quantities
afresh, and if we use the electromagnetic system we have
to re-define all the electrical ones1.

This system is however never used in practice; the
electromagnetic system or one founded upon it is uni
versally used in Electrical Engineering, and the electro
static system is used for special classes of investigations.

257. The units of resistance, of electromotive force,
of capacity on the electromagnetic system are either too
large or too small to be practically convenient : hence new-
units which are definite multiples or submultiples of the
electromagnetic units are employed. These units and their
relation to the electromagnetic system of units (when
the units of length, mass and time are the centimetre,
gramme and second) are given in the following table.
The unit of resistance is called the Ohm and is equal
to 10° electromagnetic units.

1 It should be noticed that it is only when the electromagnetic system
of units is used that ' magnetic induction ' has the meaning assigned
to it in Art. 153. If we use any system of units in which we start
from electrical quantities, the ' magnetic induction through unit area '
appears as the quantity whose rate of variation is equal to p times the
electromotive force round the boundary of the area. The magnetic
induction defined in this way is always proportional to the magnetic
induction as defined in Art. 153. The two are however only identical
on the electromagnetic system of units. With the definition of Art. 153
the magnetic induction is of the same dimensions as magnetic force,
since they are both the mechanical force on a unit pole when placed in
cavities of different shapes.

258] DIMENSIONS OF ELECTRICAL QUANTITIES 467

The unit of electromotive force is called the Volt and is

equal to 108 electromagnetic units.
The unit of current is called the Ampere and is equal

to 10"1 electromagnetic units.
The unit of charge is called the Coulomb and is equal

to 10"1 electromagnetic units.
The unit of capacity is called the Farad and is equal to

10~9 electromagnetic units.

The Microfarad is equal to 10~15 electromagnetic units.
The Ampere is the current produced by a Volt through

an Ohm.

We shall now proceed to explain the methods by
which the various electrical quantities can be measured in
terms of these units : when the quantity is so measured it
is said to be determined in absolute measure.

258. Determination of a Resistance in Absolute
Measure. The method given in Art. 226 enables us
to compare two resistances, and thus to find the ratio
of any resistance to that of an arbitrary standard such as
the resistance of a column of mercury of given length and
cross section when at a given temperature. In order to
make use of the electromagnetic system of units we must
find the number of electromagnetic units in our standard
resistance, or what amounts to the same thing we must
be able to specify a conductor whose resistance is the
electromagnetic unit of resistance.

The first method we shall describe, that of the re
volving coil, was suggested by Lord Kelvin, and carried
out by a committee of the British Association, who were
the first to measure a resistance in absolute measure. The

30—2

468 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

method was also one of those used by Lord Rayleigh
and Mrs Sidgwick in their determination of the Ohm.

When a coil of wire spins about a vertical axis in the
earth's magnetic field, currents are generated in the coil ;
these currents produce a magnetic force at the centre
of the coil. If a magnet is placed at the centre of the
coil, this magnetic force gives rise to a couple on the
magnet tending to twist the magnet in the direction in
which the coil is rotating. The resistance of the coil may
be deduced from the deflection of the magnet as follows.

Let H be the horizontal component of the earth's
magnetic force, A the area enclosed by one turn of the
coil, n the number of turns, 9 the angle the plane of
the coil makes with the magnetic meridian ; let the coil
revolve with uniform velocity &>, so that we may put

0 = at.

The number of tubes of magnetic induction passing
through the coil is equal to

nAH sin 6,

and the rate of diminution of this is
— nAHco cos cot.

Hence, if L is the coefficient of self-induction of the
coil, R its resistance, and i the current flowing through the
coil, the current being taken as positive when the lines of
magnetic force due to the current and those due to the
earth pass through the circuit in the same direction, we have

L -y- + Ri = — nAHco cos cot.
dt

Hence, as in Art. 237, we have

nAHco fr) ,

i = — — — [R cos cot + Leo sin a>t\.

258] DIMENSIONS OF ELECTRICAL QUANTITIES 469

Now if unit current through the coil produces a mag
netic force G at the centre, the current i through the coil
will produce a magnetic force Gi cos cot at right angles to
the magnetic meridian, and a force Gi sin cot along the
magnetic meridian, since 6 = cot. Hence the magnetic
force due to the currents in the coil has a component
nAHGcoR nAHGco , D r-

{R cos 2arf + Lo> sm

at right angles to the magnetic meridian ; and a component
nAHGLco2 nAHGco

2 (JP + «•£») 2

along the magnetic meridian.

Now suppose we have a magnet at the centre of the coil,
and let the moment of inertia of this magnet be so great
that the time of swing is very large compared with the
time of revolution of the coil. The magnetic force acting
on the magnet due to the current induced in the coil
consists, as we see, of two parts, one constant, the other
periodic, the frequency being twice that of the revolution
of the coil. By making the moment of inertia of the
magnet great enough we may make the effect of the
periodic terms as small as we please ; we shall suppose
that the magnet is heavy enough to allow us to neglect
the effect of the periodic terms; when this is done the
magnetic force at the centre has a component equal to

nAHGcoR
2 (tt* + ft>2Z2)

at right angles to the magnetic meridian, and one equal to

nAHGLco*

along it.

470 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

Hence if $ is the angle the axis of the magnet at the
centre of the coil makes with the magnetic meridian,

1 nAHGwR

- 2~

~^|

1 nAGcoR
or tancf>= -

1 nAGLfaP
I ^^^_______

This equation enables us to find R} as A, G, L can be
calculated from the dimensions of the rotating coil. When
Leo is small compared with R the equation reduces to
the simple form

1 nAGa)

When the coil consists of a single ring of wire of
radius a, n = 1, A = Tra2, G = 2?r/a ; hence

Thus by this method we compare R, which, by Art. 256,
is of the dimensions of a velocity, with the velocity of a
point on the spinning coil.

The preceding investigation is only approximate as
we have neglected the magnetic field due to the magnet
placed at the centre of the ring.

259. Lorenz's Method. This was also one of the
methods used by Lord Rayleigh and Mrs Sidgwick in
their determination of the Ohm. It depends upon the
principle that if a conducting disc spins in a magnetic
field which is symmetrical about the axis of rotation, and
if a circuit is formed by a wire, one end of which is

259] DIMENSIONS OF ELECTRICAL QUANTITIES 471

connected to the axis of rotation while the other end presses
against the rim of the disc, an electromotive force propor
tional to the angular velocity will act round the circuit.

We can determine this electromotive force by finding
the couple acting on the disc when a current flows round
this circuit.

Let / be the current flowing through the wire. When
this current enters the disc at its centre it will spread
out ; let q be the radial current crossing unit length of the
circumference of a circle of radius r at the point defined by
0. Let rdr dO be an element of the area of the disc. The
radial current flowing through this area is equal to qrdd.
Hence by Art. 214, if H is the magnetic force normal to
the disc at this area, the tangential mechanical force
acting on the area is equal to Hqrdr dQ. The moment of
this force about the axis of the disc is equal to

hence the couple acting on the disc is equal to
rHqr*drd0,

the integration being extended over the area of the disc.

Since the current flowing across a circle drawn on the
disc, with its centre at the centre of the disc, must equal
the current I flowing into the disc, we have

Since the magnetic field is symmetrical about the axis
of rotation, H is independent of 6, hence the couple acting
on the disc is equal to

IJHrdr.

472 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

If N be the number of tubes of magnetic induction
passing through the disc

and thus the couple acting on the disc is equal to

Now suppose there is a battery whose electromotive
force is E in the circuit, then in the time St the work
done by the battery is EI§t\ this work is spent in heating
the circuit and in driving the disc. The angle turned
through by the disc in this time is wbt, if co is the angular
velocity of the disc ; hence the mechanical work done is
equal to

2^ INo)St.

By Joule's law the mechanical equivalent of the heat
produced in the circuit is equal to

where R is the resistance of the circuit. Hence we have
by the Conservation of Energy

ERt = Rl^t + ~ INn&t,

ZTT

hence there is a counter-electromotive force in the circuit
equal to

259] DIMENSIONS OF ELECTRICAL QUANTITIES 473

This case illustrates the remark made on page 394,
since from Ampere's law of the mechanical force acting on
currents on a magnetic field we have deduced, by the aid
of the principle of the Conservation of Energy, the expres
sion for the electromotive force due to induction, and have
thus proved by dynamical principles that the induction of
currents is a consequence of the mechanical force exerted
by a magnet on a circuit conveying a current.

In Lord Rayleigh's experiments, the disc was placed
between two coils through which a current passed, and
the axis of the disc and of the two coils were coincident.
The magnetic field acting on the disc may be considered
as approximately that due to the current through the coils,
as this field is very much more intense than that due to
the earth. Hence if i is the current through the coils,
M the coefficient of mutual induction between the coils
and a circuit coinciding with the rim of the disc,

N=Mi.

So that the electromotive force due to the rotation of the
disc is

Mica

~2^'

The experiment was arranged as in the diagram, Fig.
1 24 ; a galvanometer was placed in the circuit connecting
the centre of the disc and the rim, and this circuit was
connected to two points P, Q in the circuit in series with
the coils, and the resistance between P and Q was adjusted
until no current passed through the galvanometer. If R
is the resistance between P and Q, and if a current i flows
through PQ the E.M.F. between P and Q will be Ei, but,
since there is no current through the galvanometer, this

474 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

balances the electromotive force due to the rotation of the
disc; hence

P . _ Miw

= ^T'

T> MM

or R = — — .

Fig. 124.

Since M can be calculated from the dimensions of the
coil and the disc, this formula gives us R in absolute
measure.

260. The method given in Art. 251 for determining a
coefficient of mutual induction in terms of a resistance may
be used to determine a resistance in absolute measure. If
we use a pair of coils whose coefficient of mutual induction
can be determined by calculation, then equation (2) of
Art. 251 will give the absolute measure of a resistance.
This method has been employed by Mr Glazebrook.

The result of a large number of experiments made by
the preceding methods is that the Ohm is the resistance
at 0° C. of a column of mercury 106'3 cm. long and 1 sq.
millimetre in cross section.

For a comparison of the relative advantages of the
preceding methods the student is referred to a paper by

263] DIMENSIONS OF ELECTRICAL QUANTITIES 475

Lord Rayleigh in the Philosophical Magazine for November,

1882.

261. Absolute Measurement of a Current. A

current may be determined by measuring the attraction
between two coils placed in series with each other and
with their planes parallel and at right angles to the line
joining their centres. If i is the current through the
coils, M the coefficient of mutual induction between the
coils, x the distance between their centres, the attraction
between the coils is equal to

dM .,

--T-**.

By attaching one of the coils to the scale-pan of a
balance and keeping the other fixed we can measure this
force, and hence if we calculate dMjdx from the dimensions
of the coils we can determine i in absolute measure.

The unit current is very conveniently specified by the
amount of silver deposited from a solution of silver nitrate
through which this current has been flowing for a given
time.

Lord Rayleigh found that the Ampere is the current
which flowing uniformly for one second would cause the
deposition of '001118 gramme of silver.

262. The unit electromotive force is that acting on a
conductor of unit resistance when conveying unit current.
A practical standard of electromotive force is the Clark
cell (Art. 183), whose electromotive force at t° Centigrade
is equal to

1-434 {i _ -00077 (* - 15)} volts.

263. Ratio of Electrostatic and Electromag
netic Units. The table given on page 460 shows that

476 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

the ratio of the measure of any electrical quantity on the
electrostatic system of measurement to the measure of the
same quantity on the electromagnetic system, is always
some power of a certain quantity which we denoted by
"v" and which is the ratio of the electromagnetic unit
of electric charge to the electrostatic unit.

The measurement of the same electrical quantity on
the two systems of units will enable us to find u/y." The
quantity which has most frequently been measured with
this object is the capacity of a condenser. The electro
static measure of the capacity can be calculated from
the dimensions of the condenser; thus the electrostatic
measure of the capacity of a sphere is equal to its radius ;
the capacity of two concentric spheres of radii a and b is
ab/(b — a); the capacity of two coaxial cylinders of length I,
radii a and b, is %1/log b/a. Thus if we choose a condenser
of suitable shape the electrostatic measure can be calculated
from its dimensions.

The electromagnetic measure can be determined by the
following method due to Maxwell. One of the arms AC of
a Wheatstone's Bridge is cut at P and Q (Fig. 125), one
plate of the condenser is connected to P, the other to a
vibrating piece R which oscillates backwards and forwards
between P and Q ; when R comes into contact with Q the
condenser gets charged, when into contact with P it gets
discharged. The current through the galvanometer may
be divided into two parts. There is first a steady current
which flows through A D when no electricity is flowing into
the condenser, this we shall denote by y. Besides this there
is at times a transient current which flows while the con
denser is being charged. We shall suppose that each time
the condenser is being charged a quantity of electricity

263] DIMENSIONS OF ELECTRICAL QUANTITIES 477

equal to Y flows through D A in the opposite direction to y.
Then if the condenser is charged n times a second the
amount which flows through the galvanometer owing to
the charging of the condenser is nY. If the time of swing

Fig. 125.

of the galvanometer needle is very long compared with
l/n of a second this will produce the same effect on the
galvanometer as a steady current whose intensity is nY
flowing from D to A. Thus if nY=y, the current due to
the repeated charging of the condenser will just balance
the steady current and there will be no deflection of the
galvanometer.

We now proceed to find Y. This is evidently equal to
the quantity of electricity which would flow from A to D
if there were no electromotive force in the wire BG and
the plates of the condenser with the greatest charge they
acquire in the experiment were connected to P and Q
respectively.

Let Z be the current from the condenser along PA
during the discharge, Y the current along AD, W the
current along BD. Let the resistances of AB, BG, GD,
DB, DA be c, a, 7, /3, a respectively. Let the coefficients

478 DIMENSIONS OF ELECTRICAL QUANTITIES [CH. XII

of self-induction of these circuits be Z1} L.2, L3, L^, L6
respectively. Then from the circuit ABD, we have

tiff dZ] dW_

" \dt~Tt] P~dt~

Integrating from just before discharging until after the
condenser is completely discharged, and remembering that
both initially arid finally Y, Z, W vanish, we have

aY+c(Y-Z)-j3W = 0 ............ (1),

where Y, Z, W are the quantities of electricity which have
passed during the discharge through AD, PA, and BD
respectively.

Similarly from the circuit DBG, we have

(/3 + 7-fa)F+(7 + a)F-a£ = 0 ...... (2).

We find from equations (1) and (2)

Now Z is the maximum charge in the condenser;
hence if C is capacity of the condenser, and A and C
the potentials of A and C respectively when the charge
is a maximum, i.e. when no current is flowing into the
condenser,

Z=C{A-C}.

If y is the current flowing through AD when no
current is flowing in the condenser, and D denotes the
potential of D,

A - D = a,

.-. A-C=

263] DIMENSIONS OF ELECTRICAL QUANTITIES 479

Hence by equation (3)

J+

But when there is no deflection of the galvanometer

nF = y;
hence

If we know the resistances and n, we can deduce from
this equation the value of C in electromagnetic measure.
In practice the resistance of the battery a is very small
compared with the other resistances, hence putting a = 0,
we find that approximately

1 +

cy \ 7 (a

By this method we find the electromagnetic measure
of the capacity of a condenser; the electrostatic measure
can be found from its dimensions.

Now by Art. 255

electrostatic measure of a condenser
electromagnetic measure of the same condenser '

Experiments made by this method show that
v — 3 x 1010 cm./sec. very nearly.

CHAPTER XIII

DIELECTRIC CURRENTS AND THE ELECTROMAGNETIC
THEORY OF LIGHT

264. The Motion of Faraday Tubes. Dielectric
Currents. In Chapter XL we considered the relation
between the currents in the primary and secondary circuits
when an alternating current passes through the primary
circuit, we did not however discuss the phenomena occurring
in the dielectric between the circuits. As we regard the
dielectric as the seat of the energy due to the distribution
of the currents, the study of the effects in the dielectric
is of primary importance. We owe to Maxwell a theory,
now in its main features universally accepted, by which
we are able to completely determine the electrical con
ditions, not merely in the conductors but also in every
part of the field. We shall also see that Maxwell's
views lead to a comprehensive theory of optical as well
as of electrical phenomena, and enable us by means of
electrical principles to explain the fundamental laws of
Optics.

Before specifying in detail the principles of Maxwell's
theory, we shall endeavour to show by the consideration of
some simple cases that in considering the relation between
the work done in taking unit magnetic pole round a closed

264] DIELECTKIC CURRENTS 481

circuit and the current flowing through that circuit
(see Art. 203), we must include under the term current,
effects other than the passage of electricity through con
ducting media, if we are to retain the conception that
the dielectric is the seat of the energy in electric and
magnetic phenomena.

Let us consider the case of a long, straight, cylindrical
conductor carrying an alternating electric current. In
the dielectric around this wire there is a magnetic field,
and, according to the views enunciated in Art. 163, there
is in a unit volume of the dielectric at a place where the
magnetic force is H an amount of energy equal to /j,H2/87r.
As the alternating current changes in intensity, the energy
in the surrounding field changes, and this change in the
energy must be due to the motion of energy from one part
of the field to another, the energy moving radially towards
or away from the wire conveying the current. If the
dielectric medium possesses inertia, and if its properties
in any way resemble those of any kind of matter with
which we are acquainted, the energy cannot travel from
one place to another with an infinite velocity.

As the alternating current changes, the energy in the
field will change also ; when the current is passing through
its zero value, it is evident that the magnetic energy
cannot now vanish throughout the field, for we assume
that the energy travels at a finite rate, and it is only a
finite time since the current was finite. If the magnetic
energy did vanish it would imply that the energy could
travel over a distance, however great, in a finite time.
If, however, the magnetic energy does not vanish simul
taneously all over the field, there must be places where

T. E, 31

482 DIELECTRIC CURRENTS [CH. XIII

the magnetic force does not vanish. But the current
through the conductor vanishes and there are no magnetic
substances in the field. Hence we conclude that unless
we assume that the energy in the magnetic field can
travel from one place to another with an infinite velocity,
we must admit that in a variable field magnetic forces
can arise apart from magnets or electric currents through
conductors.

265. Let us now see if we can find any clue as to what
produces the magnetic field under these circumstances.
Let us consider the following simple case. Let A, B
(Fig. 126) be two vertical metal plates forming a parallel

Fig. 126.

plate condenser, and let the upper ends of these plates be
connected by a wire of high resistance. Suppose that
initially the plate A is charged with a uniform distribution
of positive electricity while B is charged with an equal
distribution of negative electricity. If the plates are dis
connected, horizontal Faraday tubes at rest will stretch
from one plate to the other. When the plates are
connected by the wire the horizontal Faraday tubes will
move vertically upwards towards the wire. Let v be the
velocity of these tubes, and a the surface density of the

265] DIELECTRIC CURRENTS 483

electricity on the plates, then the upward current passing
across unit length in the plate A and the downward
current in B are equal to vcr. By Art. 209 these currents
will produce a uniform magnetic field between the plates,
the magnetic force being at right angles to the plane
of the paper and its magnitude equal to ^TTVCT. If N is
the number of Faraday tubes passing through unit area
of a plane in the dielectric parallel to the plates of the
condenser N = <r. Thus the magnetic force between the
planes is equal to 4<7rNv. The condition of things between
the plates is such that we have the Faraday tubes moving
at right angles to themselves, and that we have also a
magnetic force at right angles both to the Faraday tubes
and to the direction in which they are moving ; while the
intensity of this force is equal to 4?r times the product
of the number of tubes passing through unit area and the
velocity of these tubes.

Let us now see what are the consequences of gene
ralizing this result, and of supposing that the relation
between the magnetic force and the Faraday tubes which
exists in this simple case is generally applicable to all
magnetic fields. Suppose then that whenever we have
movements of the Faraday tubes we have magnetic force
and conversely, and that the relation between the magnetic
force and the Faraday tubes is that the magnetic force
is equal to 4?r times the product of the ' polarization '
(Art. 70) and the velocity of the Faraday tubes at right
angles to the direction of polarization ; and that the direc
tion of the magnetic force is at right angles to both the
direction of polarization and the direction in which the
Faraday tubes are moving.

31—2

484

DIELECTRIC CURRENTS

[CH. XIII

We shall begin by considering what on this view is
the physical meaning of H' x 00', where 00' is a line so
short that the magnetic force may be regarded as constant
along its length, and H1 is the component of the magnetic
force along 00'.

Let OA (Fig. 127) represent in magnitude and direction
the velocity of the Faraday tubes, and OP the polarization ;

then if OB represents the magnetic force, OB will be at
right angles to OA and OP and equal to
4<7r.OA. OP sin 0,

where <£ is the angle POA. The component H1 of the
magnetic force along 00' will be

4<7r.OA. OP sin </> cos 0,
where 6 is the angle BOO'. Thus we have

H' x 00' = ^7r.OA.OP. 00' sin (/> cos 0

= 247rA (1),

where A is the volume of the tetrahedron three of whose
sides are OA, OP, 00'.

265] DIELECTRIC CURRENTS 485

Let us now find the number of Faraday tubes which
cross 00' in unit time. To do this, draw 00 and O'D
equal and parallel to AO, OA being the velocity of the
Faraday tubes. Then the number of tubes which cross
00' in unit time is the number of tubes passing through
the area OCDO'.

The area of the parallelogram 00 DO' is equal to
OA x 00' smAOO'.

The number of tubes passing through it is therefore
OPxsmO'xOA xOO'smAOO' (2),

where 6' is the angle between OP and the plane of the
parallelogram OCDO'', this is the same as the angle
between OP and the plane AOO'. But

6A = OP x sin & x 0 A x 00' sin AOO',

where A as before is the volume of the tetrahedron POO' A.
Hence from (1) and (2) we see that

H' x 00' = 4-7T (number of Faraday tubes crossing 00' in
unit time).

Thus IH'ds where the integral is taken round a closed

curve is equal to 4?r times the number of tubes which pass
inwards across the curve in unit time.

In Art. 203 iH'ds was taken as equal to 4?r times the

currents flowing through the space enclosed by the curve,
and the only currents discussed in that article were
currents flowing through conductors : we shall now con
sider what interpretation we must attach to the new

expression we have just found for IH'ds.

486

DIELECTRIC CURRENTS

[CH. XIII

In the first place, any tube which in unit time passes
inwards across one part of the curve and outwards across
another part, will not contribute anything to the total
number of tubes passing across the closed curve, for its
contribution when it passes inwards is equal and opposite
to its contribution when it passes outwards. Hence all
the tubes we need consider are those which only cross
the curve once, which pass inwards across the curve and
do not leave it within unit time. These tubes may be

tube

Fig. 128.

divided into two classes, (1) those which remain within
the curve, (2) those which manage to disappear without
again crossing the boundary. The first set will increase the
total polarization over any closed surface bounded by the
curve, and the number of those which cross the boundary
in unit time is equal to the rate of increase in this total
polarization. The existence of the second class of tubes
depends upon the passage of conductors, or of moving
charged bodies, through the area bounded by the curve.

265] DIELECTRIC CURRENTS 487

Thus suppose we have a metal wire passing through the
circuit, then the tubes which cross the boundary may run
into this wire and be annulled, the disappearance of each
unit tube corresponding to the passage of unit electricity
along the wire ; or a tube might have one end on the wire
and cross the circuit, its end running along the wire ; the
passage of such a tube across the boundary means the pas
sage of a unit of electricity along the wire ; again, one end
of a tube might be on a charged body which moves through
the circuit. Thus the number of tubes of class (2) which
cross the circuit in unit time is equal to the number of
units of electricity which pass in that time along conductors
or on charged bodies passing through the circuit, i.e. it is
equal to the sum of the conduction and convection currents
flowing through the circuit.

Hence the work done when unit pole is taken round a
closed circuit is equal to 4?r times the sum of the conduction
and convection currents flowing through that circuit plus
the rate of increase of the total polarization through the
circuit. From this we see that a change in the polariza
tion through the circuit produces the same magnetic effect
as a conduction current whose intensity is equal to the
rate of increase of the polarization. We shall call the
rate of increase in the polarization the dielectric current.
The recognition of the magnetic effects due to these
dielectric currents is the fundamental feature of Maxwell's
Theory of the Electric Field. We have given a method
of regarding the magnetic field which leads us to expect
the magnetic effects of dielectric currents. It must be
remembered, however, Maxwell's Theory consists in the
expression of this result and is not limited to any particular
method of explaining it.

488 DIELECTRIC CURRENTS [CH. XIII

266. Propagation of Electromagnetic Disturb
ances. We shall now proceed to show that Maxwell's
Theory leads to the conclusion that an electric disturbance
is propagated through air with the velocity of light.

We can employ the equations we deduced in Art. 234,
if we regard u, v, w the components of the current, as the
components of the sum of the dielectric, convection, and
conduction currents. If X, F, Z are the components of the
electric intensity, and K its specific inductive capacity, then
the x, y, z components of the polarization are respectively

the components of the dielectric currents are therefore

K^dX^ I^dY K^dZ
4<7r dt ' 4?r dt ' 4?r ~dt '

If cr is the specific resistance of the medium, the
components of the conduction current are

X Y Z

a' a1 a'

Hence u, v, w the components of the total effective
current are given by the equations

_%idXL X

U~^TT dt + a- '

47T dt + O-

K dZ Z

W= T- -j7 + -

4-7T dt cr

Hence substituting these values of u, v, w in the equa
tions of Art. 234 we get, using the notation of that Article,

DIELECTRIC CURRENTS

489

4w 4^-=£

r dt
i j.L dZ

4?H-; 7T

266]

the following equations as the expression of Maxwell's
Theory,

(K dX X\_dy d^

4?T -\ -j — — 77 — | r — -^ 7- ,

(4?r dt a) dy dz

Y\ da. __ dy

a]~dz dx'

Z\_dj3_da.
a) ~ dx dy'

_da,_dZ_dY

dt dy dz '
_db_dX _dZ^

dt dz dx '
_dc=dY_dX

dt ~ dx dy'

Let us now consider the case of a dielectric for which
o- is infinite, so that all the currents are dielectric currents ;
putting a infinite in the preceding equations, and a = pa,
b = /A/3, c = py, we get

(1),

, T dX dy
K — =- = -=
dt dy

rrdY da.

dp'
dz
dry

dt dz
KdZ _d/3
dt dx

da_dZ
dt dy
dp dX

dx
da.

dz
dZ

dt dz
dy_dY
dt dx

dx
dX
dy

(2).

490 DIELECTRIC CURRENTS [CH. XIII

Differentiating the first equation in (1) with respect
to t, we get

d*X _ d dy d d/3
~di? ~ dy~di ~dz dt '

Substituting the values of dy/dt, d/3/dt, and noticing
that by (1)

dx dy dz
is independent of the time, we get

We may by a similar process get equations of the same
form for Y, Z, a, b, c.

To interpret these equations let us take the simple
case when the quantities are independent of the coordi
nates xy y. Equation (3) then takes the form

If we put

and change the variables from z and t to f and rj, we get

The solution of which is

where F and / denote any arbitrary functions.

266] DIELECTRIC CURRENTS 491

Since F (z — t/^/pK) remains constant as long as
z — £/V pK is constant, we see that if a point travels along
the axis of z in the positive direction with the velocity
l/VyLL&r, the value of F(z — t/^^K) will be constant at this
point. Hence the first term in equation (5) represents a
value of X travelling in the positive direction of the axis
of z with the velocity 1/V ^K. Similarly the second term
in (5) represents a value of X travelling in the negative
direction along the axis of z with the velocity l/VyuJfT.
For example, suppose that when t — 0, X is zero except
between z=+e, z — — e where it is equal to unity, and
suppose further that dX/dt is everywhere zero when t = 0.
Then equation (5) shows that after a time t

X = between z = -=== - e, and z = -== + e,

and between z — -- 7= — e, and z = -- := +

and is zero everywhere else. Thus the quantity repre
sented by X travels through the dielectric with the
velocity

It is shown in treatises on Differential Equations that
equation (3), the general form of the equation (4), represents
a disturbance travelling with the velocity l/V/iJT.

Thus Maxwell's Theory leads to the result that electric
and magnetic effects are propagated through the dielectric
with the velocity l/V/Aj&T.

Let us see what this velocity is when the dielectric is
air. Using the electromagnetic system of units we have

492 DIELECTRIC CURRENTS [CH. XIII

for air /*= 1, K— — , where v is the ratio of the electro
magnetic unit of electricity to the electrostatic unit
(Art. 255). Hence on Maxwell's Theory electric and
magnetic effects are propagated through air with the
velocity "v" Now experiments made by the method
described in Art. 263 lead to the result that, within the
errors of experiment, v is equal to the velocity of light
through air. Hence we conclude that electromagnetic
effects are propagated through air with the velocity of light.
This result led Maxwell to the view that since light travels
with the same velocity as an electromagnetic disturbance,
it is itself an electromagnetic phenomenon ; a wave of light
being a wave of electric and magnetic disturbances.

267. Plane Electromagnetic Waves. Let us con
sider more in detail the theory of a plane electric wave.
If/, g, h are the components of the electric polarization in
such a wave, I, m, n the direction cosines of the normal to
the wave front, and X the wave length, then we may put

/= /0 cos -£- (Ix + my+nz- Vt),

2-7T

g = g0 cos — (Ix + my + nz- Vt),

h = hQ cos — (Ix + my + nz — Vt)',
A,

where V is the velocity of propagation of the wave, and
fo> 9o> h0 quantities independent of x, y, z or t. Since
df dq dh

-J- _| 2. _| — 0

dx dy dz

we have If0 + mg0 + nh0 = 0,

and therefore lf+mg + nh = 0.

267] DIELECTRIC CURRENTS 493

Thus the electric polarization is perpendicular to the
direction of propagation of the wave.

By equation (2), Art. 266, we have

= dZdY
dt

and

= _

dt dy dz '

Hence

da 4-7T 2?r , , , . 2-Tr /7 Tr .

= -— {mh0 - ng0] sin — (Ix +my + nz- Vt),

g0 - mh0) cos — (las + my + nz- Vt) ;

or snce

a = 4nrV(ng — mh) ;
similarly ^ = ^irV(lh — nf),

7=4>7rV(mf-lg).
Hence la + m/3 + ny = 0,

so that the magnetic force is at right angles to the
direction of propagation of the wave, and since

the magnetic force is perpendicular also to the electric
polarization.

Since {a2 + /32 + 72}* = 4-TrF f/2 + #2 + A2),
the resultant magnetic force is 4vrF times the resultant
electric polarization.

Hence in a plane electric wave, and therefore on
Maxwell's Theory in a plane wave of light, there is in

494

DIELECTKIC CURRENTS

[CH. XIII

the front of the wave an electric polarization, and at
right angles to this, and also in the wave front, there is a
magnetic force bearing a constant ratio to the polarization.
We shall see in Art. 270 that in a plane polarized light
wave the electric polarization is at right angles to, and
the magnetic force in, the plane of polarization.

In strong sunlight the maximum electric intensity is
about 10 volts per centimetre, and the maximum magnetic
force about one-fifth of the horizontal magnetic force due
to the earth in England.

268. Propagation by the Motion of Faraday
Tubes. The results obtained by the preceding analysis
follow very simply from the view that the magnetic force

A B

Fig. 129.

is due to the motion of the Faraday tubes. The electro
motive force round a circuit moving in a magnetic field
is eo^ual to the rate of diminution of the number of tubes

268] DIELECTRIC CURRENTS 495

of magnetic induction passing through the circuit. Thus
let P, Q (Fig. 129) be two adjacent points on a circuit, P', Q'
the positions of these points after the lapse of a time St.
Then the diminution in the time Bt of the number of
tubes of magnetic induction passing through the circuit
of which PQ forms a part may, as in Art. 136, be shown
to be equal to the number of tubes which pass through
the sum of the areas PP'Q'Q. The number passing
through PP'Q'Q is equal to

PQxPP'x 5 sin £ sin 0,

where B is the magnetic induction, $ the angle it makes
with the plane PP'Q'Q, and 6 the angle between PP' and
PQ. If V is the velocity with which the circuit is moving
PP' = V8t. Thus the rate of diminution in the number
of tubes passing through the circuit is

2PQ . VB sin $ sin 0.

Hence we may regard the electromotive force round
the circuit as equivalent to an electric intensity at each
point P of the circuit whose component along PQ is equal
to VB sin <£ sin 9. As the component of this intensity
parallel to B and V vanishes, the resultant intensity is
at right angles to B and V and equal to

where ^r is the angle between B and V. In this case
the circuit was supposed to move, the tubes of induction
being at rest ; we shall assume that the same expression
holds when the circuit is at rest and the tubes of mag
netic induction move with the velocity V across an element
of the circuit at rest.

Let us now introduce the view that the magnetic force
is due to the motion of the Faraday tubes. Let 0 A (Fig.

496

DIELECTRIC CURRENTS

[CH. XIII

130) represent the velocity of the Faraday tubes, OP the
electric polarization, and OB the magnetic induction, which

Fig. 130.

in a non-crystalline medium is parallel to the magnetic
force and therefore (see page 483) at right angles to OP
and OA. By what we have just proved the electric in
tensity is at right angles to OB and OA, and therefore
along OC. Now in a non-crystalline medium the electric
intensity is parallel to the electric polarization; hence
OP and 00 must coincide in direction ; thus the Faraday
tubes move at right angles to their length.

Again, if E is the electric intensity, by what we have
just proved

E = BV. ....................... (1).

But if H is the magnetic force, //, the magnetic permea
bility,

and by Art. 265

where P is the electric polarization.

(2),

269] DIELECTRIC CURRENTS 497

Hence by (1) and (2)

If K is the specific inductive capacity of the dielectric

hence we have F2 = 1/yuJJT. The tubes therefore move with
the velocity 1/A///JT at right angles to their length.

269. Evidence for Maxwell's Theory. We shall
now consider the evidence furnished by experiment as to
the truth of Maxwell's theory.

We have already seen that Maxwell's theory agrees
with facts as far as the velocity of propagation through
air is concerned. We now consider the case of other
dielectrics.

The velocity of light through a non-magnetic dielectric
whose specific inductive capacity is K is on Maxwell's
theory equal to I/VX

Hence

velocity of light in this dielectric
velocity of light in air

specific inductive capacity of air
specific inductive capacity of dielectric '

But by the theory of light this is also equal to

1
n'

T. E. 32

498 DIELECTRIC CURRENTS [CH. XIII

where n is the refractive index of the dielectric. Hence

on Maxwell's theory

7i2 = electrostatic measure of the specific inductive capacity.

In comparing the values of n2 and K we have to re
member that the electrical conditions under which these
quantities are on Maxwell's theory equal to one another,
are those which hold in a wave of light where the electric
intensity is reversed millions of millions of times per
second. We have at present no means of directly measur
ing K under these conditions.

To make a fair comparison between n2 and K we ought
to take the value of K determined for electrical oscilla
tions of the same frequency as those of the vibrations of
the light for which n is measured. As we cannot find K
for vibrations as rapid as those of the visible rays, the
other alternative is to use the value of n for waves of very
great wave length ; we shall call this value n^.

The process by which n^ is obtained is not however
very satisfactory. Cauchy has given the formula

connecting n with the wave length X, which holds accurately
within the limits of the visible spectrum, unless the refract
ing substance is one which shows the phenomenon known
as ' anomalous dispersion.' To find nm we apply this em
pirical formula to determine the refractive index for waves
millions of times the length of those used to determine
the constants A, B, C which occur in the formula. For
these reasons we should expect to find cases in which K
is not equal to ri^, but though these cases are numerous
there are many others in which K is approximately equal
to r£. A list of these is given in the following table :

270] DIELECTRIC CURRENTS 499

Name of Substance K ri^

Paraffin 2'29 2'022

Petroleum spirit T92 T922

Petroleum oil 2'07 2'075

Ozokerite 213 2'086

Benzene 2'38 2'2614*

Carbon bisulphide 2'67 2'678*

As examples where the relation does not hold, we
have

Glass (extra dense flint) 101 2'924*
Calcite (along axis) 7'5 2'197*

Quartz (along optic axis) 4'55 2*41*
Distilled water 76 1-779*

Sir James Dewar and Professor Fleming have shown
that the abnormally high specific inductive capacities of
liquids such as water, disappear at very low temperatures,
the specific inductive capacities at such temperatures
becoming comparable with the square of the refractive
index.

Maxwell's Theory of Light has been developed to a
considerable extent and the consequences are found to
agree well with experiment. In fact the electromagnetic
is the only theory of light yet advanced in which the
difficulties of reconciling theory with experiment do not
seem insuperable.

270. Hertz's Experiments. The experiments made
by Hertz on the properties of electric waves, on their

* These are the values of w02 where n0 is the refractive index for
sodium light.

32—2

500 DIELECTRIC CURRENTS [CH. XIII

reflection, refraction, and polarization, furnish perhaps the
most striking evidence in support of Maxwell's theory,
as it follows from these experiments that the properties
of these electric waves are entirely analogous to those
of light waves. We regret that we have only space
for an exceedingly brief account of a few of Hertz's
beautiful experiments; for a fuller description of these
and other experiments on electric waves with their
bearings on Maxwell's theory, we refer the reader
to Hertz's own account in Electrical Waves and to
Recent Researches in Electricity and Magnetism by
J. J. Thomson.

We saw in Art. 245 that when a condenser is dis
charged by connecting its coatings by a conductor, elec
trical oscillations are produced, the period of which is
approximately 2?r \/LC where C is the capacity of the
condenser, and L the coefficient of self-induction of the
circuit connecting its plates. This vibrating electrical
system will, on Maxwell's theory, be the origin of elec
trical waves, which travel through the dielectric with the
velocity Vand whose wave length is 2?rF V LC. By using
condensers of small capacity whose plates were connected
by very short conductors Hertz was able to get electrical
waves less than a metre long. This vibrating electrical
system is called a vibrator.

Hertz used several forms of vibrators ; the one used
in the experiment we are about to describe consists of two
equal brass cylinders placed so that their axes are coinci
dent. The two cylinders are connected to the two terminals
of an induction coil. When this is in action sparks
pass between the cylinders. The cylinders correspond to

270] DIELECTRIC CURRENTS 501

the plates of the condenser, and the air between the
cylinders (whose electric strength breaks down when
the spark passes) to the conductor connecting the plates.
The length of each of these cylinders is about 12 cm.,
and their diameters about 3 cm. ; their sparking ends are
well polished.

To detect the presence of the electrical waves, Hertz
used a very nearly closed metallic circuit, such as a piece
of wire, bent into a circle, the ends of the wire being ex
ceedingly close together. When the electric waves strike
against this detector very minute sparks pass between
the terminals ; these sparks serve to detect the presence
of the waves. Recently Sir Oliver Lodge has introduced
a still more sensitive detector. It is founded on the fact
discovered by Branly that the electrical resistance of a
number of metal turnings, placed so as to be loosely in
contact with each other, is greatly affected by the impact
of electric waves, and that all that is necessary to detect
these waves is to take a glass tube, fill it loosely with iron
turnings, and place the tube in series with a battery and
a galvanometer. When the waves fall on the tube its
resistance, and therefore the deflection of the galvano
meter, is altered.

The analogy between the electrical waves and light
waves is very strikingly shown by Hertz's experiments
with parabolic mirrors.

If the vibrator is placed in the focal line of a parabolic
cylinder, and if the Faraday tubes emitted by it are
parallel to this focal line ; then if the laws of reflection
of these electric waves are the same as for light waves,
the waves emitted by the vibrator will, after reflection

502

DIELECTRIC CURRENTS

[CH. XIII

from the cylinder, emerge as a parallel beam, and will
therefore not diminish in intensity as they recede from
the mirror. When such a beam falls on another parabolic
cylinder, the axis of whose cross section coincides with
the axis of the beam, it will be brought to a focus on the
focal line of the second mirror.

The parabolic mirrors used by Hertz were made of
sheet zinc, and their focal length was about 12*5 cm.
The vibrator was placed so that the axes of the cylin
ders coincided with the focal line of one of the mirrors.
The detector, which was placed in the focal line of an
equal parabolic mirror, consisted of two pieces of wire ;
each of these wires had a straight piece about 50 cm.
long, and was then bent at right angles so as to pass
through the back of the mirror, the length of the bent
piece being about 15 cm. The ends of the two pieces
coming through the mirror were bent so as to be exceed
ingly near to each other. The sparks passing between
these ends were observed from behind the mirror. The
mirrors are represented in Fig. 131.

J
1

Fig. 131.

270] DIELECTRIC CURRENTS 503

Reflection of Electric Waves.

To show the reflection of these waves the mirrors were
placed side by side so that their openings looked in the same
direction and their axes converged at a point distant about
3 metres from the mirrors. No sparks passed between the
points of the detector when the vibrator was in action. If
however a metal plate about 2 metres square was placed
at the intersection of the axes of the mirrors, and at right
angles to the line which bisects the angle between the axes,
sparks appeared at the detector. These sparks however
disappeared if the metal plate was turned through a small
angle. This experiment shows that the electric waves are
reflected and that, approximately at any rate, the angle of
incidence is equal to the angle of reflection.

Refraction of Electric Waves.

To show the refraction of these waves Hertz used
a large prism made of pitch. This was about T5 metres
high, and it had a refracting angle of 30° and a slant side
of 1'2 metres. When the electric waves from the mirror
containing the vibrator passed through this prism, the
sparks in the detector were not excited when the axes of
the two mirrors were parallel, but sparks were produced
when the axis of the mirror containing the detector made
a suitable angle with that containing the vibrator. When
the system was adjusted for minimum deviation, the sparks
were most vigorous in the detector when the angle between
the axes of the mirrors was equal to 22°. This would
make the refractive index of pitch for these electrical
waves equal to 1*69.

504 DIELECTRIC CURRENTS [CH. XIII

Electric Analogy to a plate of Tourmalins.

If a properly cut tourmaline plate is placed in the
path of a plane polarized beam of light incident at right
angles on the plate, the amount of light transmitted
through the tourmaline plate depends upon its azimuth.
For one particular azimuth all the light will be stopped,
while for an azimuth at right angles to this the maximum
amount of light will be transmitted.

If a screen be made by winding metal wire round a
large rectangular framework so that the turns of the wire
are parallel to one pair of sides of the frame, and if this
screen be interposed between the mirrors when they are
facing each other with their axes coincident, then it will
stop the sparks in the detector when the turns of the wire
are parallel to the focal lines of the mirrors, and thus to the
Faraday tubes proceeding from the vibrator : the sparks
will however recommence if the framework is turned
through a right angle so that the wires are perpendicular
to the focal lines of the mirror.

If this framework is substituted for the metal plate
in the experiment on the reflection of waves, the sparks
will appear in the detector when the wires are parallel
to the focal lines of the cylinders and will disappear when
they are at right angles to them. Thus this framework
reflects but does not transmit Faraday tubes parallel to
the wires, while it transmits but does not reflect Faraday
tubes at right angles to them. It thus behaves towards
the transmitted electrical waves as a plate of tourmaline
does towards light waves. By using a framework wound
with exceedingly fine wires placed very close together
Du Bois and Rubens have recently succeeded in polarizing

270] DIELECTRIC CURRENTS 505

in this way radiant heat, whose wave length, though greater
than that of the rays of the visible spectrum, is exceedingly
small compared with that of electric waves.

Angle of Polarization.

When light polarized in a plane at right angles to the
plane of incidence falls upon a plate of refracting substance,
and the normal to the wave front makes with the normal
to the refracting surface an angle tan~1/ct> where //. is the
refractive index, all the light is refracted and none re
flected. When light is polarized in the plane of incidence
some of the light is always reflected.

Trouton has obtained a similar effect with electric
waves. From a wall 3 feet thick reflection was ob
tained when the Faraday tubes proceeding from the
vibrator were perpendicular to the plane of incidence,
while there was no reflection when the vibrator was
turned through a right angle so that the Faraday tubes
were in the plane of incidence. This proves that on
the electromagnetic theory of light we must suppose
that the Faraday tubes are at right angles to the plane
of polarization.

A very convenient arrangement for studying the
properties of electric waves is described in a paper by
Professor Bose in the Philosophical Magazine for January
1897.

CHAPTER XIV

THERMOELECTRIC CURRENTS

271. Seebeck discovered in 1821 that if in a closed
circuit of two metals the two junctions of the metals are
at different temperatures, an electric current will flow
round the circuit. If, for example, the ends of an iron
and of a copper wire are soldered together and one of the
junctions is heated, a current of electricity will flow round
the circuit ; the direction of the current is such that the
current flows from the copper to the iron across the hot
junction, provided the mean temperature of the junctions
is not greater than about 600° Centigrade.

The current flowing through the thermoelectric circuit
represents a certain amount of energy, it heats the circuit
and may be made to do mechanical work. The question
at once arises, what is the source of this energy ? A dis
covery made by Peltier in 1834 gives a clue to the answer
to this question. Peltier found that when a current
flows across the junction of two metals it gives rise to
an absorption or liberation of heat. If it flows across
the junction in one direction heat is absorbed, while if it
flows in the opposite direction heat is liberated. If the
current flows in the same direction as the current at the

272] THERMOELECTRIC CURRENTS 507

hot junction in a thermoelectric circuit of the two metals
heat is absorbed; if it flows in the same direction as
the current at the cold junction of the circuit heat is
liberated.

Thus, for example, heat is absorbed when a current
flows across an iron-copper junction from the copper to
the iron.

The heat liberated or absorbed is proportional to the
quantity of electricity which crosses the junction. The
amount of heat liberated or absorbed when unit charge
of electricity crosses the junction is called the Peltier
Effect at the temperature of the junction.

Now suppose we place an iron-copper circuit with one
junction in a hot chamber and the other junction in a
cold chamber, a thermoelectric current will be produced
flowing from the copper to the iron in the hot chamber,
and from the iron to the copper in the cold chamber.

Now by Peltier's discovery this current will give rise
to an absorption of heat in the hot chamber and a libera
tion of heat in the cold one. Heat will be thus taken
from the hot chamber and given out in the cold. In this
respect the thermoelectric couple behaves like an ordinary
heat-engine.

272. The experiments made on thermoelectric currents
are all consistent with the view that the energy of these
currents is entirely derived from thermal energy, the
current through the circuit causing the absorption of heat
at places of high temperature and its liberation at places
of lower temperature. We have no evidence that any
energy is derived from any change in the molecular state

508 THERMOELECTRIC CURRENTS [CH. XIV

of the metals caused by the passage of the current or
from anything of the nature of chemical combination
going on at the junction of the two metals.

Many most important results have been arrived at
by treating the thermoelectric circuit as a perfectly re
versible thermal engine, and applying to it the theorems
which are proved in the Theory of Thermodynamics to
apply to all such engines. The validity of this application
may be considered as established by the agreement be
tween the facts and the result of this theory. There are
however thermal processes occurring in the thermoelectric
circuit which are not reversible, i.e. which are not reversed
when the direction of the current flowing through the
circuit is reversed. There is the conduction of heat along
the metals due to the difference of temperatures of the
junctions, and there is the heating effect of the current
flowing through the metal which, by Joule's law, is pro
portional to the square of the current and is not reversed
with the current. Inasmuch as the ordinary conduction
of heat is independent of the quantity of electricity passing
round the circuit, and the heat produced in accordance
with Joule's law is not directly proportional to this
quantity, it is probable that in estimating the connection
between the electromotive force of the circuit, which is
the work done when unit of electricity passes round the
circuit, and the thermal effects which occur in it, we
may leave out of account the conduction effect and the
Joule effect and treat the circuit as a reversible engine.
If this is the case, then, as Lord Kelvin has shown, the
Peltier effect cannot be the only reversible thermal effect
in the circuit. For let us assume for a moment that the
Peltier effect is the only reversible thermal effect in the

272] THERMOELECTRIC CURRENTS 509

circuit. Let Pl be the Peltier effect at the cold junction
whose absolute temperature is Tl} so that Pl is the
mechanical equivalent of the heat liberated when unit of
electricity crosses the cold junction ; let P2 be the Peltier
effect at the hot junction whose absolute temperature is
Tz, so that P2 is the mechanical equivalent of the heat
absorbed when unit of electricity crosses the hot junction.
Then since the circuit is a reversible heat-engine, we have
(see Maxwell's Theory of Heat)

work done when unit of electricity goes round the circuit
= jp ^ — .

•*2~ -M

But the work done when unit of electricity goes round
the circuit is equal to E, the electromotive force in the
circuit, and hence

E-M-TJ.Z.

Thus on the supposition that the only reversible
thermal effects are the Peltier effects at the junctions,
the electromotive force round a circuit whose cold junction
is kept at a constant temperature should be proportional
to the difference between the temperatures of the hot
and cold junctions. Gumming, however, showed that
there were circuits where, when the temperature of the
hot junction is raised, the electromotive force diminishes
instead of increasing, until, when the hot junction is
hot enough, the electromotive force is reversed and the
current flows round the circuit in the reverse direc
tion. This reasoning led Lord Kelvin to suspect that
besides the Peltier effects at the junction there were

510 THERMOELECTRIC CURRENTS [CH. XIV

reversible thermal effects produced when a current flows
along an unequally heated conductor, and by a laborious
series of experiments he succeeded in establishing the
existence of these effects. He found that when a current
of electricity flows along a copper wire whose tempera
ture varies from point to point, heat is liberated at any
point P when the current at P flows in the direction of
the flow of heat at P, i.e. when the current is flowing
from hot places to cold, while heat is absorbed at P
when the current flows through it in the opposite direc
tion. In iron, on the other hand, heat is absorbed at
P when the current flows in the direction of the flow
of heat at P, while heat is liberated when the current
flows in the opposite direction. Thus when a current
flows along an unequally heated copper wire it tends to
diminish the differences of temperature, while when it
flows along an iron wire it tends to increase those differ
ences. This effect produced by a current flowing along
an unequally heated conductor is called the Thomson
effect.

Specific Heat of Electricity.

273. The laws of the Thomson effect can be con
veniently expressed in terms of a quantity introduced by
Lord Kelvin and called by him the ' specific heat of the
electricity in the metal.' If a- is this 'specific heat of
electricity,' A and B two points in a wire, the temperatures
of A and B being respectively ^ and t2> and the difference
between ^ and t2 being supposed small, then a is defined
by the relation,

<r(A - £2) = heat developed in AB when unit of electricity
passes through AB from A to B.

273] THERMOELECTRIC CURRENTS 511

The study of the thermoelectric properties of con
ductors is very much facilitated by the use of the thermo
electric diagrams introduced by Professor Tait. Before
proceeding to describe them we shall enunciate two
results of experiments made on thermoelectric circuits
which are the foundation of the theory of these circuits.

The first of these is, that if El is the electromotive
force round a circuit when the temperature of the cold
junction is t0 and that of the hot junction tl} E2 the electro
motive force round the same circuit when the temperature
of the cold junction is t1} and that of the hot junction t2,
then El + E2 will be the electromotive force round the
circuit when the temperature of the cold junction is t0,
and that of the hot junction tz. It follows from this
result that E, the electromotive force round a circuit
whose junctions are at the temperatures t0 and ^, is

equal to

t
Qdt,

r
J

where Qdt is the electromotive force round the circuit
when the temperature of the cold junction is t — ^dt,
and the temperature of the hot junction is t + ^dt. The
quantity Q is called the thermoelectric power of the
circuit at the temperature t.

The second result relates to the electromotive force
round circuits made of different pairs of metals whose
junctions are kept at assigned temperatures. It may
be stated as follows : If EAc is the electromotive force
round a circuit formed of the metals A, C, Esc that round
a circuit formed of the metals B, C, then EAC — EBC is the
electromotive force acting round the circuit formed of the

512

THERMOELECTRIC CURRENTS [CH. XIV

metals A and J5; all these circuits being supposed to work
between the same limits of temperature.

274. Thermoelectric Diagrams. The thermo
electric line for any metal (.A) is a curve such that the
ordinate represents the thermoelectric power of a circuit
of that metal and some standard metal (usually lead) at a
temperature represented by the abscissa. The ordinate is
taken positive when for a small difference of temperature
the current flows from lead to the metal A across the
hot junction.

It follows from Art. 273, that if the curves a and /3
represent the thermoelectric lines for two metals A and B,
then the thermoelectric power of a circuit made of the
metals A and B at an absolute temperature represented
by ON will be represented by RS, and the electromotive
force round a circuit formed of the two metals A and B

when the temperature of the cold junction is represented
by OL, that of the hot junction by OM, will be repre
sented by the area EFGH.

Let us now consider a circuit of the two metals A and
B with the junctions at the absolute temperatures OLl}
OL2) Fig. 133, where OL^ and OL2 are nearly equal. Then

274]

THERMOELECTRIC CURRENTS

513

the electromotive force round the circuit (i.e. the work
done when unit of electrical charge passes round the
circuit) is represented by the area EHGF. Consider now
the thermal effects in the circuit. We have Peltier effects

Fig. 133.

at the junctions ; suppose that the mechanical equivalent
of the heat absorbed at the hot junction when unit of
electricity crosses from B to A it is represented by the area
Pl, let the mechanical equivalent of the heat liberated at
the cold junction be represented by the area P2. There
are also the Thomson effects in the unequally heated
metals ; suppose that the mechanical equivalent of the
heat liberated when unit of electricity flows through the
metal A from the hot to the cold junction is represented
by the area Kl} and that the mechanical equivalent of
the heat liberated when unit of electricity flows through
B from the hot to the cold junction is represented by

T. E.

514 THERMOELECTRIC CURRENTS [CH. XIV

the area Kz. Then by the First Law of Thermodynamics,
we have

&TeB,EFGH = P1-Pi + Ka-K1 ......... (1).

The Second Law of Thermodynamics may be expressed
in the form that if H be the amount of heat absorbed
in any reversible engine at the absolute temperature t,
then

In our circuit the two junctions are at nearly the same
temperature, and we may suppose that the temperature
at which the absorption of heat corresponding to the
Thomson effect takes place is the mean of the tempera
tures of the junctions, i.e. \ (OL^ + OZ2).

Hence by the Second Law of Thermodynamics, we
have

Hence from (1) and (2) we get

area EFGH = 1 j A + A J (0L, - OLJ,

or since OL^ is very nearly equal to OLZ and therefore
is very nearly equal to P2) this gives approximately

area EFGH = (01* - OL2).

UJui

But when OL^ is very nearly equal to OL2 , the area

so that

274] THERMOELECTRIC CURRENTS 515

thus P1 is represented by the area GHVU. Now Pl is
the Peltier effect at the temperature represented by OL1}
hence we see that at any temperature

Peltier effect = (thermoelectric power) (absolute

temperature),
or P=Qt,

where t is the absolute temperature.

By the definition of Art. 273 we see that if ^ is the
specific heat of electricity for the metal A, cr2 that for B,
then

K! - KZ = (o-i - o-2) LJj^.

But by (1)

area EFGH = P, - P2 +- K, - Klt
and P1 = area GHVU,

P2 = area FEST.

Hence K^ - K2 = area SEE V - area TFG U

— (tan 01 — tan 02) OL^ x L2L1}

where 0lf 02 are the angles which the tangents at E and F
to the thermoelectric lines for A and B make with the axis
along which temperature is measured. Hence

o-i - <ra = (tan 0! - tan 02) 0A ............ (3).

When the temperature interval L^L^ is finite the areas
UGHV and FEST will still represent the Peltier effects at
the junctions, and the area TFGU the heat absorbed when
unit of electricity flows along the metal B from a place
where the temperature is OL2 to one where it is OL^

33—2

516 THERMOELECTRIC CURRENTS [CH. XIV

The preceding results are independent of any assump
tion as to the shape of the thermoelectric lines. The
results of the experiments made by Professor Tait and
others show, that over a considerable range of tempera
tures, these lines are straight for most metals and alloys,
while Le Roux has shown that the 'specific heat of
electricity ' for lead is excessively small. Let us assume
that it is zero and suppose that the diagram represents
the thermoelectric lines of metals with respect to lead:
then since these lines are straight, 0 is constant for any
metal and cr2 vanishes when it refers to lead, the value of
o- the ' specific heat of electricity ' in the metal is by (3)
given by the equation

<r = tan 0 . t,

where t denotes the absolute temperature.

The thermoelectric power Q of the metal with respect
to lead at any temperature t is given by the equation

Q = tan<9(£-£0),

where t0 is the absolute temperature where the line of
the metal cuts the lead-line ; t0 is defined as the neutral
point of the metal and lead.

Let us consider two metals; let 0lt 02 be the angles
their lines make with the lead- line, and ^ and t2 their
neutral temperatures, then Ql and Q2 their thermoelectric
powers with respect to lead are given by the equations

Q, = tan 0! (t - t,\
Q, = tan <92 (I - t.2) ;

hence Q, the thermoelectric power of a circuit consisting
of the two metals, is given by the equation

- tan

250 200 150 100 50

0 50

Fig. 134.

150 200 250

518 THERMOELECTRIC CURRENTS [CH. XIV

where T0 is the neutral temperature for the two metals
and is given by the equation

y _ £1 tan ft - t2 tan ft
tan ft - tan ft '

The electromotive force round a circuit formed of
these metals, the temperatures of the hot and cold junc
tions being Tl} T2) respectively, is equal to

l\dt = (tan ft - tan ft) (T, - T,) (£ (T, + T2) - T0).

This vanishes when the mean of the temperatures
of the junctions is equal to the neutral temperature.
If the temperature of one junction is kept constant the
electromotive force has a maximum or minimum value
when the other junction is at the neutral temperature.

In Fig. 134 the thermoelectric lines for a number of
metals are given. The figure is taken from a paper by
Noll, Wiedemanris Annalen, vol. 53, p. 874. The abscissae
represent temperatures, each division being 50° C., the
ordinates represent the E.M.F. for a temperature difference
of 1° C., each division representing 2'5 microvolts. To
find the E.M.F. round a circuit whose junctions are at
<! and t2 degrees we multiply the ordinate for ^ (^ + £2)
degrees by (t2 — ^).

CHAPTER XV

THE PROPERTIES OF MOVING ELECTRIC CHARGES

275. As the properties of moving electric charges are
of great importance in the explanation of many physical
phenomena, we shall consider briefly some of the simpler
properties of a moving charge and other closely allied
questions.

Magnetic Force due to a Moving Charged Sphere.

The first problem we shall discuss is that of a uniformly
charged sphere moving with uniform velocity along a
straight line. Let e be the charge on the sphere, a its
radius, and v its velocity ; let us suppose that it is moving
along the axis of z> then when things have settled down
into a steady state the sphere will carry its Faraday tubes
along with it. If we neglect the forces due to electro
magnetic induction, the Faraday tubes will be uniformly
distributed round the sphere and the number passing
normally through unit area at a point P will be e/4?rOP2,
0 being the centre of the charged sphere. These tubes
are radial and are moving with a velocity v parallel to the
axis of z, hence the component of the velocity at right
angles to their direction is v sin 0, where 0 is the angle OP

520 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

makes with the axis of z ; by Art. 265 these moving
tubes will produce a magnetic force at P equal to

4-7T 0/4-7T . OP2) vsinO = ev sin 0/OP*.

The direction of this force is at right angles to the tubes,
i.e. at right angles to OP; at right angles also to their direc
tion of motion, i.e. at right angles to the axis of z ; thus
the lines of magnetic force will be circles whose planes are
at right angles to the axis of z and whose centres lie along
this axis. Thus we see that the magnetic field outside
the charged sphere is the same as that given by Ampere's
rule for an element of current ids, parallel to the axis of z,
placed at the centre of the sphere, provided ev = ids.

276. As the sphere moves, the magnetic force at P
changes, so that in addition to the electrostatic forces there
will be forces due to electromagnetic induction, these will
be proportional to the intensity of the magnetic induction
multiplied by the velocity of the lines of magnetic induc
tion, i.e. the force due to electromagnetic induction at a
point P will be proportional to p (ev sin 0/OP2) x v, where
ft is the magnetic permeability of the medium ; while the
electrostatic force will be e/K.OP*, where K is the specific
inductive capacity of the medium. The ratio of the force
due to electromagnetic induction to the electrostatic force
is pKv*am 0 or sin 0t;2/F2, where Fis the velocity of light
through the medium surrounding the sphere; hence in
neglecting the electromagnetic induction we are neglecting
quantities of the order v2/ F2. The direction of the force due
to electromagnetic induction at P is along NP, ifPNis the
normal drawn from P to the axis of z ; this force tends to
make the Faraday tubes congregate in the plane through
the centre of the sphere at right angles to its direction

277] PROPERTIES OF MOVING ELECTRIC CHARGES 521

of motion ; when the sphere is moving with the velocity
of light it can be shown that all the Faraday tubes are
driven into this plane.

Increase of Mass due to the Charge on the Sphere.

277. Returning to the case when the sphere is moving
so slowly that we may neglect v*/V2; we see that since H,
the magnetic force at P, is ev sin 0/OP2, and at P there
is kinetic energy equal to fj,H2/87r per unit volume (see
Art. 163), the kinetic energy per unit volume at P is

yLteV sin2 O/STT . OP4.

Integrating this for the volume outside the sphere, we find

u62/y2

that the kinetic energy outside the sphere is £§ — , where a

oft

is the radius of the sphere. Thus if m be the mass of the
uncharged sphere the kinetic energy when it has a charge e
is equal to

Thus the effect of the charge is to increase the mass of
the sphere by 2/ie2/3a. It is instructive to compare this
case with another, in which there is a similar increase in
the effective mass of a body; the case we refer to is that
of a body moving through a liquid. Thus when a sphere
moves through a liquid it behaves as if its mass were
m + £m', where m is the mass of the sphere, and m' the
mass of liquid displaced by it. Again when a cylinder
moves at right angles to its axis through a liquid its
apparent mass is m + m', where m' is the mass of the liquid
displaced by the cylinder. In the case of an elongated

522 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

body like a cylinder, the increase in mass is much greater
when it moves sideways than when it moves point fore
most, indeed in the case of an infinite cylinder the increase
in the latter case vanishes in comparison with that in the
former; the increase in mass being m sin2 0, where 6 is the
angle the direction of motion of the cylinder makes with
its axis. In the case of bodies moving through liquids the
increase in mass is due to the motion of the body setting
in motion the liquid around it, the site of the increased
mass is not the body itself but the space around it where
the liquid is moving. In the electrical problem we may
regard the increased mass as due to the Faraday tubes
setting in motion the ether as they move through it.
From the expression for the energy per unit volume we
see that the increase in mass is the same as if a mass
^Tr/jiN2 were bound by the tubes, and had a velocity given
to it equal to the velocity of the tubes at right angles
to themselves, the motion of the tubes along their length
not setting this mass in motion. Thus on this view the
increased mass due to the charge is the mass of ether set
in motion by the tubes. If we regard atoms as made
up of exceedingly small particles charged with negative
electricity, embedded in a much larger sphere of positive
electricity, the positive charge on this sphere being equal
to the sum of the negative charges embedded in it, it is
possible to regard all mass as electrical in its origin, and
as arising from the ether set in motion by the Faraday
tubes connecting the electrical charges of which the atoms
are supposed to be made up. For a development of this
view the reader is referred to the author's Conduction of
Electricity through Gases and Electricity and Matter.

278] PROPERTIES OF MOVING ELECTRIC CHARGES 523

Momentum in the Electric Field.

278. The view indicated above, that the Faraday
tubes set the ether moving at right angles to the direction
of these tubes, suggests that at each point in the field
there is momentum whose direction is at right angles to
the tubes, and by symmetry in the plane through the
tube and the line along which the centre of the charged
sphere moves. As the mass of the ether moved per unit
volume at P is 4?r pN2 where N is the density of the
Faraday tubes at P, the momentum per unit volume
would, on this view, be 4<7r//JVr2 v sin 6. This is equal to
BN where B is the magnetic induction and N the density
of the Faraday tubes at P, the direction of the momentum
being at right angles to B and N. We shall now prove
that this expression for the momentum is general and is
not limited to the case when the field is produced by a
moving charged sphere.

279. Since the magnetic force due to moving Faraday
tubes is (Art. 265) equal to 4?r times the density of the
tubes multiplied by the components of the velocity of the
tubes at right angles to their direction, and is at right
angles both to the direction of the tubes and to their
velocity; we see if a, fi, 7 are the components of the mag
netic force parallel to axes of xy yy z at a place where the
densities of the Faraday tubes parallel to x, y, z are/", g, h,
and where u, v, w are the components of the velocity of
the tubes, a, /3, 7 are given by the equations

a = 4?r (hv — gw), {3 = 4?r ( fw — hu), 7 = 4?r (gu —fv).

524 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

If all the tubes are not moving with the same velocity
we shall have

a =

with similar expressions for fi, 7. Here u^v^ wl are the
components of the velocity of the tubes/!, gl} h±\ -u.2, v>2, w2
those of the tubes /2, g2, h., and so on.

Now T the kinetic energy per unit volume at P is
equal to

~ («2 + P + 72) = £: x 167T2 . ((S (hv -

= ZTTJJ, . {(2 (hv - gw))* + (2 (fw - hu)? + (2 (gu -fv))2} ;
the momentum per unit volume parallel to x due to the

dT
tubes /,#!, /*! is equal to ^— , i.e. to

£ (fw - hu) - g£ (gu -fv)\

= P (ffiV ~ ^i/3).
Similarly that due to the tubes /2, g2, h2 is equal to

and so on, thus P the total momentum parallel to x per
unit volume is given by the equation

where / g, h are the densities parallel to x, y, z of the whole
assemblage of Faraday tubes. Similarly Q, R, the com
ponents of the momentum parallel to y and z, are given
respectively by the equations

280] PROPERTIES OF MOVING ELECTRIC CHARGES 525

Thus we see that the vector P, Q, R is perpendicular
to the vectors a, /3, y, f, g, h, and its magnitude is BN sin 6
where B is the magnetic induction at the point, N the
density of the Faraday tubes and 0 the angle between B
and N', hence we see that each portion of the field possesses
an amount of momentum equal to the vector product of
the magnetic induction and the dielectric polarization.

280. Before considering the consequences of this
result, it will be of interest to consider the connection
between the momentum and the stresses which we have
supposed to exist in the field. We have seen (Arts. 45, 46)
that the electric and magnetic forces in the field could be
explained by the existence of the following stresses :

(1) a tension — — along the lines of electric force ;

O7T

a 1 KR*

(2) a pressure -r— at right angles to these lines;

here K is the specific inductive capacity, and R the
electric force ;

(1) a tension ^ — along the lines of magnetic force;

(2) a pressure ^ — at right angles to these lines;

here p is the magnetic permeability of the medium and H
the magnetic force.

Let us consider the effect of these tensions on an
element of volume bounded by plane faces perpendicular
to the axes of x, y, z. The stresses a are equivalent to a
hydrostatic pressure KR*/87r and a tension KR2/4<7r along

526 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

the lines of force. The effect of the hydrostatic pressure
on the element of volume is equivalent to forces
d

parallel to the axes of x, y, z respectively, A#, Ay, A^ being
the sides of the element of volume.

Let us now consider the tension KR^j^-jr. We know
that a stress N in a direction whose direction cosines are
lt m, n is equivalent to the following stresses :

(Nl2 acting on the face AyA,? parallel to x,

Nmn

Nln
Nmn

Thus the effect of these stresses on the element of
volume is equivalent to a force parallel to x equal to

the forces parallel to y and z are given by symmetrical
expressions.

In our case the tension is along the lines of force,

V T7" p

hence l==>m = > n=> where X> Y> % are tne

280] PROPERTIES OF MOVING ELECTRIC CHARGES 527

components of the electric force, hence substituting these
values for I, m, n and putting N = — — , we see that the

~r7T

tension produces a force parallel to x equal to

fd XX* d KXY d KXZ\ ,

-, — -A 1- ~i -A h i * A# AT/ A£.

\dx 4-7T dy 4?r dz 4vr /

The force parallel to x due to the hydrostatic pressure
and this tension is equal to

f--

V dx

d K (Z2 + F2 + Z*) d_ KX*

~\ 7

dx 8?r dx 4?r

d KXY d KXZ\
dy 4>7r dz 4>7r J

when the medium is uniform, this may be written

K_\Y (dX_dY\ _ (dZ_dX\
4-7T | \dy dx J . \dx dz J

v (dX dY dZ\\ .
+ X -j— + -r- + -j- }> A# Ay A^.
\dx ay dz J)

Now KX)KY)KZ=^f,
and by equation (4) Art. 234,

dX dY dc dZ dX db

dy dx dt ' dx dz ~ dt' dz dy dt '

(7 T7" ^71^ J <7\

(LA. Cil Ci/i\ .
1 H -T- + -T- = 4<7rP ;
dx dy dz]

thus the force parallel to x due to the electric stresses may
be written

/ dc ,db TrA

(*ar*«+z'v

528 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

In the same way the magnetic stresses may be shown
to give a force parallel to x equal to

4-7T | \dy dx) '\dx dz)

fda. d/3 dy

\dx dy dz
since by Art. 234

dy dz~ ^ dt ' dz dx~ ^ ~di ' ~dx~ ~dy~ 1T~dt)

and „>,__,.__ _p __

ix dy dz

where a is the density of the magnetism, the magnetic
stresses give rise to a force parallel to x equal to
dg , dh

hence the system of electric and magnetic stresses together
gives rise to a force parallel to x equal to

\dt ^ ~ ^ + Xp +

The terms Xp and acr represent the forces acting on the
charged bodies and the magnets in the element of volume,
and are equal to the rate of increase of momentum parallel
to x of these bodies, the remaining term

-r (eg — bh) A# kykz

equals the rate of increase of the x momentum in the
ether in the element of volume. This agrees with our
previous investigation ; for we have seen (p. 524) that
the momentum parallel to x per unit volume is equal to
gc — hb,

282] PROPERTIES OF MOVING ELECTRIC CHARGES 529

281. A system of charged bodies, magnets, circuits
carrying electric currents &c. and the ether forms a self-
contained system subject to the laws of dynamics; in
such a system, since action arid reaction are equal and
opposite, the whole momentum of the system must be
constant in magnitude and direction, if any one part of
the system gains momentum some other part or parts
must lose an equal amount. If we take the incomplete
system got by leaving out the ether, this is not true.
Thus take the case of a charged body struck by an electric
wave, the electric force in the wave acts on the body and
imparts momentum to it, no other material body loses
momentum, so that if we leave out of account the ether
we have something in contradiction to the third law of
motion. If we take into account the momentum in the
ether there is no such contradiction, as the momentum
in the electric waves after passing the charged body is
diminished as much as the momentum of that body is
increased.

282. Another interesting example of the transference
of momentum from the ether to ordinary matter is afforded
by the pressure exerted by electric waves, including light
waves, when they fall on a slab of a substance by which
they are absorbed. Take the case when the waves are
advancing normally to the slab. In each unit of volume
of the waves there is a momentum equal to the product
of the magnetic induction B and the dielectric polariza
tion N ; B and N are at right angles to each other, and
are both in the wave front; the momentum which is at
right angles to both B and N is therefore in the direction
of propagation of the wave. In the wave B = ^ir^NY, so

T. E. 34

530 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV
1 B2

that BN = -- TT, V being the velocity of light; B is a

periodic function, and may be represented by an expression
of the form B0 cos (pt — nan), x being the direction of propa
gation of the wave ; the mean value of B2 is therefore
-|502. Thus the average value of the momentum per unit

1 B2

volume of the wave is ^ -- ^ , the amount of momentum
STT /juy

that crosses unit area of the face of the absorbing

1 B 2
substance per unit time is therefore 5 --- ~ x V, or

OTTyLt V

-- B<?. As the wave is supposed to be absorbed by the
STT/LI

slab no momentum leaves the slab through the ether, so

T> 2

that in each unit of time ^— units of momentum are

communicated to the slab for each unit area of its face
exposed to the light : the effect on the slab is the same
therefore as if the face were acted upon by a pressure
Bfl&TTii. It should be noticed that //, is the magnetic
permeability of the dielectric through which the waves
are advancing, and not of the absorbing medium.

If the slab instead of absorbing the light were to
reflect it, then if the reflection were perfect each unit
area of the face would in unit time be receiving Bfl&irp
units of momentum in one direction, and giving out an
equal amount of momentum in the opposite direction ;
the effect then on the reflecting surface would be as if
a pressure 2 x Bf/STr/j, or B02/4>7r/j, were to act on the
surface. This pressure of radiation as it is called was
predicted on other grounds by Maxwell ; it has recently
been detected and measured by Lebedew and by Nichols
and Hull by some very beautiful experiments.

283] PROPERTIES OF MOVING ELECTRIC CHARGES 531

283. If the incidence is oblique and not direct, then
if the reflection is not perfect there will be a tangential
force as well as a normal pressure acting on the surface.
For suppose i is the angle of incidence, B0 the maximum
magnetic induction in the incident light, B0' that in the
reflected light, then across each unit 'of wave front in the
incident light JB02/87r/j, units of momentum in the direction
of the incident light pass per unit time, therefore each
unit of surface receives per unit time cosiBi/Sjr^ units
of momentum in the direction of the incident light, or
cos i sin i'502/87r/u. units of momentum parallel to the re
flecting surface. In consequence of reflection

cos i sin iB0'2/87Tfj,

units of momentum in this direction leave unit area of
the surface in unit time, thus in unit time

cos i sin i (£02 - B^ftTriJb

units of momentum parallel to the surface are communi
cated to the reflecting slab per unit time, so that the slab
will be acted on by a tangential force of this amount.
Professor Poynting has recently succeeded in detecting
this tangential force.

Since the direction of the stream of momentum is
changed when light is refracted, there will be forces
acting on a refracting surface, also when in consequence
of varying refractivity the path of a ray of light is not
straight the^refracting medium will be acted upon by
forces at right angles to the paths of the ray; the de
termination of these forces, which can easily be accom
plished by the principle of the Conservation of Momentum,
we shall leave as an exercise for the student.

34—2

532 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

284. We shall now proceed to illustrate the distribu
tion of momentum in some simple cases.

Case of a Single Magnetic Pole and an Electrified Point.

Let A be the magnetic pole, B the charged point, m
the strength of the pole, e the charge on the point, then
at a point P the magnetic induction is m/AP2 and is
directed along AP, the dielectric polarization is
and is along BP, hence the momentum at P is
me sin APB

and its direction is the line through P at right angles
to the plane APB. The lines of momentum are therefore
circles with their centres along AB and their planes at
right angles to it, the resultant momentum in any direction
evidently vanishes. There will however be a finite moment
of momentum about AB : this we can easily show by
integration to be equal to em. Thus in this case the
distribution of momentum is equivalent to a moment of
momentum em about AB. The distribution of momentum
is similar in some respects to that in a top spinning about
AB as axis. Since the moment of momentum of the
ether does not depend upon the distance between A and
B it will not change either in magnitude or direction
when A or B moves in the direction of the line joining
them. If however the motion of A or B is not along
this line, the direction of the line AB and therefore the
direction of the axis of the moment of momentum of the
ether, changes. But the moment of momentum of the
system consisting of the ether, the charge point, and the
pole must remain constant ; hence when the momentum in

284] PROPERTIES OF MOVING ELECTRIC CHARGES 533

the ether changes, the momentum of the system consisting
of the pole and the charge must change so as to com
pensate for the change in the momentum of the ether.
Thus suppose the charged point moves from B to Bf in
the time 8t, then in that time the moment of momentum

Fig. 135.

in the ether changes from em along AB to em along AB' ';
this change in the moment of momentum of the ether is
equivalent to a moment of momentum whose magnitude
is em&6, where $0 = /.BABf, and whose axis is at right
angles to AB in the plane BAB'. The change in the
moment of momentum of the pole and point must be equal
and opposite to this. Since the resultant momentum of
the ether vanishes in any direction, the change in the
momentum of the pole must be equal and opposite to the
change in momentum of the point, and these two changes
must have a moment of momentum equal to emW : we see
that this will be the case if S/ the change in momentum
of the point is at right angles to the plane BAB' and

fs s\

equal to p , while the change in momentum in the

pole is equal and opposite to this. This change in
momentum — r^- occurring in the time 8t may be re-

534 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

garded as produced by a force F acting on the point at
right angles to the plane BAB' and given by the equation

„_ em $0
* AB'Bt'

,. BB'smABB'

]\OW 00= — -r-TT— - ,

An
or if v be the velocity of the point,

^ vSts'mABB'

Sd v sin ABB'
Tt = -AB~

thus F-

where H is the magnetic force at the point and </> the
angle between H and the direction in which the point
is moving ; from this we see that a moving charged point
in a magnetic field is acted on by a force at right angles
to the" velocity of the point, at right angles also to the
magnetic force at the point, and equal to the product of
the charge, the magnetic force and the velocity of the
point at right angles to the magnetic force. Thus we
see that we can deduce the expression for the force acting
on a charged point moving across the lines of magnetic
force directly from the principle of the Conservation of
Momentum. We should have got an exactly similar
expression if we had supposed the charge at rest and
the pole in motion ; in this case we must take v to be the
velocity of the pole and <p the angle between v and AB.

285] PROPERTIES OF MOVING ELECTRIC CHARGES 535

285. From the expression given on page 524 for the
momentum in the field we can prove that the momentum
in the ether due to a charged point at P and the magnetic
force produced by a current flowing round a small closed
circuit, is equivalent to a momentum passing through P
whose components F, G, H parallel to the axes of x, y, z
respectively are given by the equations

TJ, • ( d 1 dl\

$ = Liia. m -: n-^-~

\ dzr dyr)

n - f d 1 7 d 1

Cr = LLIOL in -j I -= —

\ ax r dzr

TT • /7 d 1 d 1\

H = /MO. (l-j m -y- - 1 ,

\ dy r dxr)

where i is the current flowing round the circuit, a the
area of the circuit and I, m, n the direction cosines of the
normal to its plane, x, y, z are the coordinates of P and r
the distance of P from the centre of the circuit, the charge
at P is supposed to be the unit charge. We see that

dF dG . ( d2 1 / d* 1 d2 1 \ d'2

= „££ J JYI — — n I _ 4. _]_i_/

dy dx r { -7~-J- " 7 "

orsnce

dx* r + dy*

dz* r

• li ^

d* 1

\ dxdz

dydz

+ n

dz*r.

. d /L d

+ d I

dz \ dx

m dyr

n-j-
dz

c being the z component of the magnetic induction at P

536 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

due to the small circuit. We have similarly if a and b
are the x and y components respectively of this induction

dH dF _
dx dz

dG dH

~? r~ = a-

dz dy

The usual expression for the electromotive force due
to induction follows at once from the principle of the
Conservation of Momentum. For the momentum in the
ether is equivalent to a momentum through P whose
components are F, G, H. Suppose that in consequence
of the motion of the circuit or the alteration of the current
through it, F, G, H become F+SF, G + SG, H + SH,
then the momentum in the ether still passes through P
but has now components F-\-SF, G + SG, H+SH instead
of F,G,H; but the momentum of the whole system, point
circuit and ether must remain constant ; thus to counter
balance the changes in momentum SF, SG, SH at P due
to the ether, we must have changes in momentum of the
unit charge at P equal to — SF, -SG, -SH. Suppose
that the time taken by the changes SF, SG, SH is St, then
in the time St the x momentum of the unit charge at P
must change by - SF, i.e. the unit charge must be acted on

by force — — . Thus there is at P an electric force whose

? ET

component parallel to x is - -^- , similarly the components

parallel to y and z are - — ^ , — The electric force

at at

whose components we have just found is the force due to
electromagnetic induction, and its magnitude is that given

286] PROPERTIES OF MOVING ELECTRIC CHARGES 537

by Faraday's law. To prove this we notice that the line
integral of the electric force round a fixed circuit of which
ds is an element is equal to

_dFdx dG_dy_ dH dz\
dt ds dt ds dt ds)

dtj \ as ds ds

d(dG dH\ dH dF dF dG

J T~j --- j

dtj { \dz ay) \dx dz) \dy dx

by Stokes' theorem ; here I, m, n are the direction cosines
of the normal to a surface filling up the closed curve, dS
is an element of this surface. Substituting the values

already given for -= -- -=— , &c. the preceding expression
becomes

— -7- 1 (la + mb 4- nc) d8 ;

dt J

the integral in this expression is the number of lines of
magnetic induction passing through the closed circuit,
hence we see that the line integral of the electric force
due to induction round a closed circuit equals the rate of
diminution in the number of lines of magnetic induction
passing through the circuit ; this however is exactly
Faraday's law of induction (see Art. 229).

286. When a charged particle is moving so rapidly
that v*/V'2 cannot be neglected, the distribution of the
Faraday tubes round the particle is no longer uniform
and the expression 2pe*v/a given in Art. 277 for the
momentum of the charged sphere has to be modified.
For an investigation of this case we refer the reader to

538 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

Recent Researches in Electricity and Magnetism, where,
page 21, it is shown that when the velocity of the charged
sphere is w, R the momentum parallel to z is in the
general case given by the equation

R=l^

2 a w(p_«*)irV 4™2'

+ lsin2*(l+- —
2 V 4 w2

• ^ w
where sm;j=^.

From this value of R we see that when w approaches
V, the value of Rjwt the apparent mass, increases rapidly
with w ; thus if an appreciable amount of the mass of a
body is due to electric charge, the mass of the body will
increase with the velocity, it is only however when the
velocity of the body approaches that of light that this
increase becomes appreciable, in the limiting case where
the velocity is that of light the apparent mass would be
infinite. The influence of velocity on the apparent mass
of particles travelling with great velocities has been
detected by Kaufmann by some very interesting experi
ments, a short account of which will be found in the
author's Conduction of Electricity through Gases, page 533.
Kaufmann found that a particle moving with a velocity
about five per cent, less than the velocity of light, had a
mass about three times that with small velocities.

The increase in the mass of a slowly moving charged
sphere is 2/me2/3a, i.e. 4 (potential energy of the sphere )/3 F2,
thus if this mass were to move with the velocity of light
its kinetic energy would be two-thirds of the electrical
potential energy. The same proportion between the in-

287] PROPERTIES OF MOVING ELECTRIC CHARGES 539

crease in the mass due to electrification and the electrical
potential energy can be shown to hold for any system of
electrified bodies as well as for the simple case of the
charged sphere.

287. Effects due to changes in the velocity of
the moving charged body. We shall take first the
case of a charged sphere moving so slowly that the lines of
force are symmetrically distributed around it, and consider

Fig. 136.

what will happen when the sphere is suddenly stopped.
The Faraday tubes associated with the sphere have inertia
and are in a state of tension, thus any disturbance com
municated to one end of a tube will travel along the tube
with a finite and constant velocity — the velocity of light.
Let us suppose that the stoppage of the particle takes a
finite small time r. We can find the configuration of the
tubes, after a time t has elapsed since the sphere began

540 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

to be stopped, in the following way. Describe with the
centre of the charged sphere as centre two spheres, one
having the radius Vt, the other the radius V(t—r).
Then since no disturbance can have reached the portions
of the Faraday tubes situated outside the surface of the
outer sphere these tubes will be in the positions they
would have occupied if the sphere had not been stopped,
while since the disturbance has passed over the tubes
within the inner sphere, these tubes will be in their
final position. Thus consider a tube which when the
particle was stopped was along the line OPQ, 0 being
the centre of the charged sphere, this will be the final
position of the tube ; hence at the time t the portion of
this tube inside the inner sphere will be in the position
OP, the portion P'Q' outside the outer sphere will be in
the position it would have occupied if the sphere had not
been stopped, i.e. if 0' is the position to which 0 would
have come if the sphere had not been stopped, P'Q' will
be a straight line passing through 0'. Thus to preserve
its continuity the tube must bend round in the shell
between the surfaces of the two spheres, and take the
position OPP'Q'. Thus the tube which before the sphere
was stopped was radial, has now, in the shell, a tangential
component, and this implies a tangential electric force;
this tangential force is, as the following calculation shows,
much greater than the radial force at P before the sphere
was brought to rest.

Let us suppose that 8, the thickness of the shell, is so
small that the portion of the Faraday tube inside it may
be regarded as straight, then, if T is the tangential force
inside the pulse, R the radial force, we have
T P'N' 00' sin d wtsm6

R PN'

.(1),

288] PROPERTIES OF MOVING ELECTRIC CHARGES 541

where w is the velocity with which the sphere was moving
before it was stopped, and 9 the angle OP makes with the
direction of motion of the sphere ; t is the time since the
sphere was stopped. Since OP = Vt and R = e/K . OP2,
K being the specific inductive capacity of the medium, we
have, writing r for OP,

„ __ ew sin 6
~ KV.rS'

Thus the tangential force varies inversely as the distance
and not as the square of the distance.

The tangential Faraday tubes move radially outwards
with the velocity V, they will therefore produce a mag
netic force at right angles to the plane of the pulse and in
the opposite direction to the magnetic force at P before
the sphere was stopped ; this force is equal to

Jr KT ew sin 0

Vx 47r.-: — = s — ;

4-7T r8

the magnetic force before the sphere was stopped was
ew sin 6/r2, thus the magnetic force in the pulse, which
however only lasts for a very short time, exceeds that in
the steady field in the proportion of r to 8.

Thus the pulse produced by the stoppage of the sphere
is the seat of very intense electric and magnetic forces ;
the pulses formed by the stoppage of the regularly elec
trified particles of the cathode rays form, in my opinion,
the well-known Rontgen rays.

288. Energy in the Pulse. The energy due to the
magnetic force in the field is per unit volume

542 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

integrating this through the pulse we find that the energy
due to the magnetic force in the pulse is

The energy due to the tangential electric force in the
pulse is per unit volume

KT2 eVsiQ2l9

integrating this through the pulse we find that this energy

, . . Tr

is equal to ^ , since /juK = -^ .

Thus the total energy in the pulse is ^ ^-g— ; and this

energy radiates away into space. The energy in the field
before the sphere was stopped was l/juehv^/a, where a is
the radius of the sphere (see Art. 277). Thus if 3 is not
much greater than the diameter of the sphere a very con
siderable fraction of the kinetic energy is radiated away
when the particle is stopped. *

289. Distribution of Momentum in the Field.

There is no momentum inside the surface of the sphere
whose radius is b (t — T), there is a certain amount of
momentum in the pulse, and momentum in the opposite
direction in the region outside the pulse ; we shall leave
it as an exercise for the student to show that the mo
mentum in the pulse is equal and opposite to that outside
it, so that as soon as the sphere is reduced to rest the
whole momentum in the field is zero.

290. Case of an Accelerated Charged Body.

The preceding method can be applied to the case when
the charged body has its velocity altered in any way, not

290] PROPERTIES OF MOVING ELECTRIC CHARGES 543

necessarily reduced to zero. Thus if the velocity instead
of being reduced to zero is diminished by $w, we can show
in just the same way as before that the magnetic force H
in the pulse is given by the equation
TT e&w . sin 6

£J_ = — ,

and the tangential electric force T by
m eAw sin 6
~ KVrS '

Now 8 = V&t if Bt is the time required to change the
velocity by Aw, hence we have

„ _ e Aw sin 6 „ e

but Aw/&£ = -f, where/ is the acceleration of the particle,
hence

efsmO T_ e /sin 6
V r KV2 r

It must be remembered that / is not the acceleration
of the sphere at the time when H and T are estimated
but at the time r/V before this. We see that when the
velocity of the sphere is not uniform, part of the magnetic
and electric force will vary inversely as the distance from
the centre of the sphere, while the other part will vary
inversely as the square of this distance ; at great distances
from the sphere the former part will be the most im
portant.

The energy in the pulse emitted whilst the velocity is
changing is equal to

3 V2 '
where d is the thickness of the pulse ; since d = VSt, where

544 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

Bt is the time the acceleration lasts, the energy emitted in
the time St is

thus the rate of emission of energy is 2e*f*/3 V.

291. Magnetic and Electric Forces due to a
charged particle vibrating harmonically through
a small distance. The magnetic force proportional to
the acceleration which we have just investigated arises
from the motion of the tangential part of the Faraday
tubes — the portion P'N' of Fig. 134 ; the radial tubes are
however also in motion, their velocity at right angles to
their length being w sin 0, where w is the velocity of the
particle when its acceleration is /, i.e. at a time r/V
before the force is estimated. This motion of the radial
tubes produces a magnetic force ew sin 0/r2 in the same
direction as that due to the acceleration. Thus H the
magnetic force at P is equal to

ewsiuO efsind

-pr- HPF"'

and is at right angles to OP and to the axis of z along
which the particle is supposed to be moving. Let the
velocity of the particle along this line be w sin.pt and its
acceleration therefore wpcospt. The magnetic force at P
at the time t will depend upon the velocity and accelera

tion of the particle at the time t — ^, these are respectively

w siup (t - y J and wp cosp (t - y j , thus H the magnetic
force at P is given by the equation

/ T\ (

eo> sin 0 ship it — ^J ew sin 6p cos p It — -^

TT _ \ "/ , __ ^

Hm - H - + - Pr

291] PROPERTIES OF MOVING ELECTRIC CHARGES 545

If a, /3, 7 are the components of this force parallel to
the axes of x, y, z, then

= 0.

»• — • s\ •*--*- j A-/ — • /\ •*••*- *

r sm 6 r sin 6

Hence

/ , r \
^ eo) sinjp 1 1 — -^1 ,eo)S.^^p ,,

a = T-

If JT, F, ^T are the components of the electric force, we
have by equation (1), page 489,

/ r

jv j JQ j> ecosmp U — !
^ V

dz dxdz

eo) sn » —

^ V.

___ ^
dt dz dx dydz

j-dZ _d(3 _da __f d- d*\

dt ~ dx dy \da? dy'2) r

Hence the periodic parts of X, F, Z are given by the
equations

p dxdz

,,
1 ft"

T j^> j> eeocos p t — -TT

i(d* d2 \ ^

11 Z = - --. -- H -j —

-. .

p \dx2 df) r

T. E. 35

546 PROPERTIES OF MOVING ELECTRIC CHARGES [CH. XV

In addition to these there are the components

e d 1 e d 1 e d I

~' ~Kdzr'

of the electrostatic force due to the charge at 0. In this
investigation co is supposed to be so small compared with
V that ca2/F2 may be neglected.

INDEX

(The numbers refer to the pages)

Absolute measurement

of a resistance 467, 470, 473

of a current 475
Alternating currents, distribution

of 424, 428

Ampere's law 330, 355
Angle, Solid 218
Anode 285
Axis of a magnet 197

Ballistic galvanometer 382
Boundary conditions

for two dielectrics 128
for two magnetizable sub
stances 262
for two conductors carrying

currents 326
Bunsen's Cell 303

Cadmium Cell 304
Capacity

of a condenser 84

of a sphere 84

of two concentric spheres 85,
141

of two parallel plates 90, 135

of two coaxial cylinders 93,
141

specific inductive 120
Capacities, comparison of two 108

determination of, in electro
magnetic measure 476
Cathode 285

Cavendish experiment 31
Charge of electricity 8
Charge, unit 12

Circuit, magnetic 352
Circular currents

magnetic force due to 356

force between two 358
Clark's Cell 303, 475
Coefficients

of capacity 43

of induction 43

of potential 42

of self-induction 365, 448

of mutual induction 365, 450
Condensers 84

comparison of two 109

in parallel 116

in cascade 117

parallel plate 90
Condenser in an alternating current

circuit 443
Conductors 9
Conjugate conductors 314
Coulomb's Law 36, 127
Couples

between two magnets 205

on a current in a magnetic

field 360
Currents

electric 283

strength of 285

magnetic force due to 329, 356

distribution of steady 310, 320

distribution of alternating 424,
428

dielectric 480
Cylinder

electric intensity due to 21

capacity of 93, 141

548

INDEX

Darnell's Cell 301

Declination, magnetic 233

Dielectric

currents 480

plane and an electrified point

169
sphere in an electric field 172

Dimensions of electrical quantities
461

Dip 235

Discharge of Leyden jar 436

Dissipation function 317

Distribution of

steady currents 310, 320
alternating currents 424, 429
currents due to an impulse 404

Diurnal variation 241

Doublet, electric field due to 158

Duperrey's lines 238

Dynamical system illustrating in
duction 397

Electric

intensity 13

potential 25

screens 51

images 145

currents 283
Electrification

by friction 1

positive and negative 2

by induction 4
Electrolysis 285, 287
Electrolyte, E.M.F. required to

liberate ions of 305
Electromagnetic

induction 387

Faraday's law of 393
Neumann's law of 393

screening 434

wave, plane 492
Electrometers 97

quadrant 98

Electromotive force of a cell 296
Electroscope 5

Element, Rational current 372
Ellipsoids in magnetic field 274
Energy

in the electric field 37, 70, 127

of a shell in a magnetic field
222

Energy

in the magnetic field 270

due to a system of currents 369

in a pulse 541

due to a moving charged sphere

521
Equipotential surface 29

Faraday's

laws of Electrolysis 287
laws of electromagnetic induc
tion 393

tubes 67, 480, 494
tubes, tension in 73
tubes, pressure perpendicular

to 74
Force

lines of 60

tubes of 65

on an uncharged conductor 82

on an electrified system 53

between electrified bodies 12

on charged conductor 56

between bodies in a dielectric

134

on a dielectric 142
between magnets 192
due to a magnet 202
between two small magnets 208
on a shell in a magnetic field

222

on a current in a magnetic
field 359

Galvanometer

tangent 375

sine 379

ballistic 382

Desprez-D'Arsonval 380

resistance of 386

Gauss's proof of law of force be
tween poles 211
Gauss's theorem 14
Grove's Cell 303

Heat produced by a current 293,

316

Hertz's experiments 499
Hysteresis 257

Impedance 413

INDEX

549

Impulse, distribution of currents

induced by 404
Induction

magnetic 247

electromagnetic 387

total normal electric 13
Insulators 9
Intensity

electric IB

of magnetization 198
Inversion 175
Ions 286

Isoclinic lines 236
Isogonic lines 236

Jar, Ley den 114

discharge of, 436
Joule's Law 293

Kirchhoff s Laws 310

Law of force between electrified

bodies 12, 31
Lenz's Law 445
Ley den jar 114

in parallel 116

in cascade 117

discharge of 436
Light, Maxwell's Theory of 480 et

seq.
Lines of force 25, 60

refraction of 131
Lorenz's Method 470

Magnet

pole of 196

axis of 197

moment of 197, 215

potential due to 198

resolution of 199

force due to 200, 202

couple on 204
Magnetic

force 195

disturbances 243

potential 195

shell 216

shell, force due to 226

shell, force acting on 222

shielding 266

induction 246

Magnetic

induction, tubes of 249

permeability 252

retentiveness 257

susceptibility 251

declination 233

dip 235
Magnetic field

energy in 270

due to current 329

due to two straight currents
342

due to circular current 356
Magnetization, intensity of 198
Magnetized sphere, field due to

227
Magnets 191

action between two small 205
Mass due to electric charge 516
Maxwell's Theory 480 et seq.
Model illustrating magnetic in
duction 397
Moment of magnet, determination

of 215

Momentum in electric field 523
Moving electric charges 519 et seq.
Mutual induction, coefficient of
365

determination of 450

comparison of 452

Neumann's Law of electromagnetic

induction 393
Neutral temperature 518

Ohm, determination of 470, 474

Ohm's Law 289

Oscillating electric charge 543

Parallel plate condenser 90
Peltier Effect 507
Periodic electromotive force 411
Permeability, magnetic 252

affected by temperature 256
Plane uniformly electrified 22
Plane and electrified point 145
Planes

parallel, separated by dielectric
135

two parallel and electrified
point 181

550

INDEX

Planes

magnetic force due to currents

in parallel 346
Polarization

in a dielectric 125

of a battery 304
Pole, unit 194
Poles of a magnet 196
Potential

electric 25

of charged sphere 27

of a magnet 198

Propagation of electromagnetic dis
turbance 488 et seq.
Pulse due to stopping or starting
charge 539

Ratio of units 475

Refraction of lines of force 131

Resistance

electric 289

of conductors in series 290

of conductors in parallel 291

specific 293

measurement of 374

absolute 467, 473
Resolution of a magnet 199
Retentiveness, magnetic 257
Rotating circuit 414

Saturation, magnetic 255
Screening

electric 51

electromagnetic 434

magnetic 266

Secondary circuit, effect of on
apparent self-induction and re
sistance 419
Self-induction

coefficient of 365

coefficient of, of a solenoid 367

coefficient of, of two parallel
circuits 368

determination of 448

comparison of 454
Shell, magnetic 216
Sine galvanometer 379
Solenoid 349
Solid angle 218

Specific inductive capacity 120

determination of 143
Specific resistance 293
Specific Heat of Electricity 510
Sphere

electric intensity due to 20

potential due to 27

capacity of 84, 85

and an electrified point 149

in a uniform electric field 157

inversion of 176

magnetic field due to 227

in a uniform magnetic field 265
Spheres

intersecting at right angles 162

in contact 183
Surface density 36
Susceptibility 252

Tangent galvanometer 375

Temperature, effect of, on magnetic
permeability 256

Terrestrial magnetism 232

Thermoelectric
currents 506
diagrams 512

Thomson Effect 510

Transformers 417

Tubes

of electric force 65
Faraday 67, 480, 494
Faraday, tension in 73
Faraday, pressure perpendicu
lar to 74
of magnetic induction 249

Units

electrostatic system 465
electromagnetic system 465

Variation

in magnetic elements 240

diurnal 241
Voltaic cell 295

Wave, Electromagnetic 492 et seq.
Wheatstone's Bridge 310, 384
Work done when unit pole is taken
round a circuit 332

CAMBRIDGE : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.

7 DAY USE

RETURN TO

ASTRON-MATH-STAT. LIBRARY

Tel. No. 642-3381

This publication is due before Library closes on
the LAST DATE and HOUR stamped below.

^^141975

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Roc'd UC3 A/M/S

1986

RB17-5m-2'75
(S4013slO)4187— A-32

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