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r Physics

PART II.

K.S.CARHART

PHYSICS

FOR

University Students

HENRY S. CARHART, LL.D.

PROFESSOR OF PHYSICS IN THE UNIVERSITY OF MICHIGAN

PART II.

HEAT, ELECTRICITY, AND MAGNETISM

ALLYN AND BACON

Boston ano Chicago

PREFACE. V

The present volume has been written in pursuance of
the plan outlined in Part I. ; it is , from the same point
of view, and the same method has been followed as far as
practicable.

The favor with which the first part of the book has been
received by teachers of experience leads the author to hope
that this second part may be found to furnish a satisfactory
completion of the elementary course in Physics which the
two were designed to furnish.

The author would take this occasion to acknowledge the
courtesy of The Cambridge Press in granting permission
to reproduce a few illustrations from Glazebrook's Heat;
he would also make acknowledgment of similar indebted-
ness to Professor Barker for illustrations from his Physics ;
and to Assistant Professor G. W. Patterson, Jr., for cordial
assistance in proof-reading.

University of Michigan, February, 1896.

(1H)

CONTENTS.

HEAT.

CHAPTER PAGE

I. Nature of Heat 1

II. Temperature and its Measurement 9

III. Expansion 23

IV. Measurement of the Quantity of Heat .... 42
V. Fusion 54

VI. Vaporization 65

VII. Transmission of Heat 90

VIII. Radiation and Absorption 104

IX. Thermodynamics 126

X. Kinetic Theory of Gases i . 142

ELECTRICITY AND MAGNETISM.

XI. Electric Charges 150

XII. Electrification by Influence 171

XIII. Electrical Potential . . . . . . .188

XIV. Capacity and Condensers 201

XV. Atmospheric Electricity 224

XVI. Primary Cells 233

XVII. Electrolysis 255

XVIII. Ohm's Law and its Applications . . . . . 273

XIX. Thermal Relations 286

XX. Properties of Magnets 305

XXI. Magnetic Effects of a Current 329

XXII. Electrodynamics 343

XXIII. Electromagnetism 352

XXIV. Electromagnetic Induction 372

XXV. Dynamos and Motors 397

XXVI. Electric Oscillations and Waves 422

Appendix 433

Index ... 437

REFERENCES.

The letters, enclosed in parentheses accompanying the headings
of articles, refer to the following books, numerals denoting pages :

B., Barker's Physics.

G., Glazebrook's Heat.

J. J. T., J. J. Thomson's Elements of Electricity and Magnetism.

M., Maxwell's Theory of Heat (Tenth Edition).

Max., Maxwell's Treatise on Electricity and Magnetism.

P., -Preston's Theory of Heat.

S., Stewart's Elementary Treatise on Heat (Sixth Edition).

T., Tait's Heat.

Th., Thompson's Elementary Lessons in Electricity and Magnetism.

Tyn., Tyndall's Heat as a Mode of Motion.

The numerals enclosed in parentheses in the body of the text
refer to articles. When the reference is to Part I., it is indicated
bv the letter I. before the number denoting the article.

(*)

HEAT.

CHAPTER I.

NATURE OF HEAT.

1. Heat a Form of Energy. — The conclusion to which
many remarkable investigations of the present century
lead is that heat is a form of energy, and that it can be
transformed into mechanical work. We are not at liberty
to regard it as a substance, because it can be produced
from something which is not a substance, and it is inex-
haustible in amount. Heat is not motion, but the energy
of motion. It depends on the confused and incessant
activity of the molecules of matter.

Heat is, moreover, the lowest form of energy, or the
form which all other kinds of energy tend to assume
whenever any transformation occurs. It is the form taken
by unavailable energy when work is spent in friction, and
by the unconverted residue when available energy is em-
ployed to do work, as in the heat-engine. When energy
is transformed in any operation in such a way that it is
not directed by the mechanism of the transformation into
some specialized form, it always manifests itself as heat.
Thus, when a piece of zinc is acted on by sulphuric acid,
the energy of the chemical union appears as heat, unless
the conditions are such as to constitute a voltaic cell, when
most of it appears first as the energy of an electric current.

2 HEAT.

The kinetic energy of a bullet is converted into heat when
it strikes the target; the energy of meteors becomes heat
by friction with the air ; the energy of combustion is heat,
and only a small portion of it can be reconverted into
useful forms. The energy of sound, of winds, and of
waves, of lightning and of falling water, ultimately fritters
down into diffused heat.

2. Heat in Material Bodies (P., 34). — In primitive
times heat was supposed to be a subtle fluid. The excess
of it in a body caused it to be hot ; its deficiency left it
cold. After many controversies it was demonstrated to
be without weight, and was therefore included among the
imponderables. It was assumed that this heat-fluid, or
caloric, was indestructible. The quantity of heat in the
universe was, therefore, considered to be constant.

To explain the physical changes produced by heat, it
was imagined that caloric entered into combination with
material bodies. Thus, water was conceived to be a com-
pound of ice and caloric, and steam was ice with a larger
proportion of caloric. The heat generated by friction,
grinding, or compression was said to be forced out of
bodies or to be due to their lessened capacity for heat.
Such explanations, which now seem to partake of the
grotesque, were regarded by many philosophers as plau-
sible and satisfactory, and the theory persisted down into
the present century. But the burden of proof which it
had to sustain became at length too great, and it was aban-
doned, as the material theory of light had been before it.

The real nature of heat had been foreshadowed in early
times. Bacon expressed the opinion that heat consists in
a "brisk agitation" of the parts of a body, and Robert
Boyle concurred in this opinion. The non-materiality of

N A TUBE OF HEAT. 3

heat was demonstrated by Count Rumford (Benjamin
Thompson) and by Sir Humphrey Davy at the beginning
of the present century, but their demonstration was not
accepted till Joule had determined the " mechanical equiv-
alent " of heat, or the work that the quantity of heat con-
stituting the unit of measurement is capable of doing
(86). This equivalence between heat and work is inde-
pendent of any theory of molecular motion. It is a de-
monstration that heat is a form of energy, because heat
and energy in other forms are reciprocally convertible.

The precise theory of the molecular motions concerned
in heat has not yet been made out. We know that the
ultimate particles or molecules of a body are in a state
of perpetual agitation. In gases this motion is in part
vibrational, in part probably rotational, and in part motion
of translation, the molecules colliding and rebounding, but
having at a given temperature a mean velocity of which
we have a fair knowledge (98). In liquids their move-
ments are much more restricted, but diffusion shows that
they enjoy a good degree of freedom of motion. In solids
the molecules are still more limited in their movements.
Each molecule is restricted to a very small space which it
never leaves, and is within the limits of the action taking
place among contiguous molecules. In solids and in
liquids a part of the heat-energy is potential, since each
molecule is acted on by its neighbors. The remainder is
the kinetic energy of molecular motion. The heat-energy
of gases is all kinetic, but it is not known how the total
kinetic energy of a molecule is divided between the three
components of motion — vibration, rotation, and translation.

3. Bumford's Experiment. — During the operation of
boring brass cannon at the military arsenal in Munich in

4 HEAT.

1799, Rumford was impressed with the large amount of
heat generated by the abrasion of material with the boring
tool. The calorists ascribed the heat to the diminished
capacity of the abraded metal for heat. Rumford sought
to test this explanation by comparing the amount of heat
contained in equal masses of the solid and the abraded
metal by raising them to the same temperature of boiling
water and observing the rise of temperature of the equal
masses of water in which they were cooled. No difference
could be detected.

In this famous experiment, which disproved the material
theory of heat, a blunt steel borer 3£ inches wide was
turned by horse power 32 times a minute inside a brass
cylinder weighing 113 pounds. In two and a half hours
the water surrounding the cylinder and weighing 18|
pounds was heated from 60° F. to the boiling point. Only
4,145 grains of the metal were abraded. Rumford cor-
rectly concluded that this large amount of heat, which
appeared to be inexhaustible, could not have been derived
from the abraded metal, which at the same time had not
lost any of its capacity for heat. After showing that all
other conceivable explanations were excluded by the con-
ditions, he concludes as follows : " It is hardly necessary to
add that anything which any insulated body, or system of
bodies, can continue to furnish without limitation cannot
possibly be a material substance ; and it appears to me
extremely difficult, if not quite impossible, to form any
distinct idea of anything capable of being excited and com-
municated in the manner heat was excited and communi-
cated in these experiments, except it be MOTION."

Rumford's experiment was complete except that he did
not proceed to determine the numerical relation between
the work done and the heat generated; but it must be

NATURE OF HEAT. 5

remembered that the law of Conservation of Energy was
not then known.

4. Davy's Experiment. — About the time of Rum-
ford's experiment, Sir Humphrey Davy devised another
to test the current explanation of the heat generated by
friction. By means of clock-work he arranged to produce
friction between two blocks of ice, in such a manner that
no heat could be received from external objects ; and he
thus demonstrated that the ice could be melted by the
friction of one block against the other. The fusion took
place only at the surface of contact between the two, and
they were almost completely converted into water, whose ca-
pacity for heat, according to the supposition of the calorists,
was diminished. Davy reasoned that heat must have
been generated, because water by universal concession con-
tains more heat than ice. But this heat could not have
come from the diminution of thermal capacity, because the
thermal capacity of water is much greater than that of ice.
This was Davy's first contribution to science ; and he con-
cluded, though with some apparent lack of confidence, that
friction produces heat, and that there is no such thing as
caloric, or the matter of heat* It was not till 1812 that he
asserted with firm conviction that

"The fundamental cause of the phenomena of heat is
motion, and the laws of its communication are precisely
the same as the laws of the communication of motion."

The first of these propositions should now be amended
in view of the doctrine of the Conservation of Energy.
Heat is not motion, or a mode of motion, but the Energy
of Motion. The second of Davy's statements remains
entirely correct.

These experiments of a public officer in the prosecution

6 HEAT.

of his official duties, and of a young scientific man, des-
tined later to become famous in both physics and chem-
istry, laid the foundation of the modern dynamical theory
of heat, which was developed later by Joule, Hirn, Clausius,
and Maxwell.

5. Radiant Heat. — One of the ways in which a hot
body loses heat is by radiation. We may feel the warmth
of the sun's rays when the temperature of the air is below
freezing. Approach to a hot stove is readily perceived in
the dark without contact. The air is not the medium by
which the heat is conveyed, for radiant energy is trans-
mitted more readily through the most perfect vacuum than
through air. The process by which heat is transferred from
one body to another without heating the medium through
which it passes is called radiation. It is customary to
speak of the radiant energy, which becomes sensible heat
when absorbed by material bodies, as Radiant Heat. But
while heat is certainly communicated from one body to
another by the vibrational process of radiation, we are not
at liberty to speak of what passes between them as heat,
since it does not warm the air through which it passes ;
for the passage of heat through any medium as heat always
warms the medium. When heat leaves a radiating body,
it is wholly transformed into radiant energy. Energy in
the radiant state or form of transmission is not the heat
which gives rise to our sensation of warmth. It is recon-
verted into heat only when it reaches our bodies or other
absorbing substances. We have no evidence that the
radiant energy from the sun is heat during its passage
through interplanetary space. Heat is converted into
radiant energy at the sun, and it is transmitted as radiant
energy through the intangible ether as a medium. It

NATURE OF HEAT. 7

becomes heat again only when it is absorbed by material
bodies and becomes the energy of irregular molecular
vibrations. But the term Radiant Heat has come into
scientific use on account of the intimate connection between
heat and radiation; its use does not imply the existence
of a new kind of heat, but refers to the thermal aspect of
radiation.

6. Radiant Heat and Light Identical. — It was for-
merly supposed that the radiations from the sun, or any other
self-luminous body, consisted of three distinct kinds, having
different distributions in the spectrum ; viz., the luminous,
the heat, and the actinic rays. The last were supposed to
be the only ones concerned in the process of photography ;
but the progress of physical science has shown that the
differences ascribed to radiations are rather differences in
the receptive apparatus (I., 217). Radiant heat and light
are identical, but are perceived by us through different
avenues of sensation. Radiations of identically the same
wave-length produce the impression of light when received
through the eye, and of radiant heat when detected by the
sense of heat or by a thermometer. All radiations when
stopped by the appropriate absorbing body are transformed
into sensible heat ; that is, heat which affects the sense of
heat or a thermometer. The limit in their effect upon the
eye is imposed by the receptive mechanism ; and their pos-
sibilities in initiating chemical changes are determined by
the sensitizing substances used. The most widely appli-
cable method of exploring the spectrum is by heat effects.
Direct optical methods can be applied to only about three
or four per cent, of the wave-lengths actually measured ;
and as yet photography is much more limited than the
method of exploration developed by Langley, which depends

8 HEAT.

on changes produced by small variations in temperature due
to absorbed heat. The only difference between heat and
light objectively is not a fundamental one, but at most
only a difference of wave-length.

It is known that there are waves too short to produce
vision, and these have heat energy, though it is small in
amount and difficult to measure. There are other waves
too long to excite the eye, and they represent more energy
than those lying within the range of vision. But all waves
of ethereal commotion are propagated by the same physical
process. Waves which are too long to excite vision may
yet warm our bodies, or give rise to electromagnetic or
electrical phenomena.

The distinguishing characteristic of radiant heat is that
it travels through any uniform medium in straight lines or
rays like light, for it is intercepted by a screen in the same
manner as light. It is also reflected in accordance with the
same laws as light, for the focus of a mirror for radiant
heat is the same as its focus for light. It is not propagated
instantaneously, but its speed in a vacuum is identical with
that of light. This is demonstrated by the simultaneous
disappearance and reappearance of the light and heat at
the time of a solar eclipse. In fact, since radiant heat and
light are absolutely identical throughout the visible spec-
trum, it follows that all the physical laws which have been
demonstrated to hold for light must also apply to radiant
heat, for none of these laws depend on wave-length.

TEMPERATURE AND ITS MEASUREMENT.

CHAPTER II.

TEMPERATURE AND ITS MEASUREMENT.

7. Definition of Temperature. — The words hot and
cold are primitive ones, and refer to our impressions
received through the sense of heat. It is now generally
conceded that this sense is independent of the sense of
touch, with which it has often been confused. When we
are warmed by radiation from a fire, or by the rays of the
sun, this change of physical condition is made known to us
as a sensation received through a specialized sense-organ,
which is distinct from the visual sense, but more closely
related to it than to that of touch or smell, since the
impressions of warmth and of light are both excited by the
same radiations from a hot body.

When the surface of our bodies is brought into contact
with other bodies, they may give to us the feeling of either
coldness or hotness, and we may be able to assert that one
body is hotter than another. By means of these sensations
we might arrange a collection of bodies of the same kind in
a series of relative hotness, and should be able to assert
that any one of them is hotter than all others which we
place below it in the series. This order of hotness is scien-
tifically expressed by means of the word temperature. The
body which gives to us the sensation of superior hotness is
said to be of a higher temperature than another of the same
kind which feels cooler.

If now one of two bodies of equal hotness be heated by

10 BEAT.

a flame, we ascribe its rise of temperature to the possessio
of a larger amount of what we call heat. This simple ii
ference is entirely justifiable, and is independent of an
theory of heat. Imagine, now, this heated body placed i
contact with a cooler one. We can readily determine tha
the hotter one becomes cooler and the cooler one hotter
and if sufficient time be allowed, the process continues ti
both are of the same temperature. Hence we say that i
the attainment of this equilibrium heat has passed from tli
hotter body to the cooler one. This inference is Justine
by the fact that the application of heat to a body withou.
change of state makes it feel hotter. Hence, temperatui
may be defined by reference to this phenomenon of tli
transfer of heat as follows : Temperature is the ihermt
condition of a hody which determines the transfer of hec
between it and other bodies. If two bodies, A and J5, ai
placed in thermal communication with each other, one (
three results will follow: A will become cooler and .
warmer ; B will become cooler and A warmer ; or neith(
will change in relative hotness to the other. In the fir!
case the temperature of A is said to be higher than tiu
of B; in the second the temperature of B is high*
than that of A; and in the third their temperatures ai
equal, and the two bodies are said to be in thermal equ
librium.

Temperature is analogous to pressure of gases. If t\\
vessels in which air has been unequally compressed ai
made to communicate with each other, air is forced fro:
the vessel of higher pressure to the one of lower pressm
till an equilibrium of pressures has been established. Tl
direction of the flow is determined entirely by pressure
and not in the least by the relative volumes of the tw
vessels.

TEMPERATURE AND ITS MEASUREMENT. 11

Temperature may be compared to potential in elec-
tricity, where the flow is from places of higher to places
of lower potential. Similarly heat flows from bodies of
higher to bodies of lower temperature.

The sense of heat is, however, a very unreliable means
of determining relative temperatures. It may be totally
misleading when the comparison is made between bodies
of different kinds, having different capacities for heat and
different conductivities. Aside from this source of unrelia-
bility, the sensations cannot be made an accurate measure
of physical properties. In the most favorable cases the
judgment can only be trained by frequent comparison
with the data furnished by the use of a unit of measure.
It is necessary, therefore, to have recourse to some physical
change produced by heat for the construction of an in-
strument to serve as an accurate measure of temperature.

8. Expansion. — One of the most familiar changes due
to the increase of the temperature of a body is its increase
of volume, or expansion by heat. This physical change is
the one commonly employed to measure temperature.

With but few exceptions, an increase in the temperature
of a body is attended by an increase in volume. Thus the
rails of a railway are not laid in contact end to end in cold
weather, but a small space is left for expansion by heat.
The tire of a wagon wheel is put on hot, and it shrinks
and compresses the wheel on cooling. Gravesand's appa-
ratus consists of a metallic ball which closely fits a ring of
the same material when both are at the same temperature.
If now the ball be slightly heated, it will expand to such
an extent that it will no longer pass through the ring.
Clocks and watches, unless carefully compensated, have a
slower rate when warm than when cold. This change in

12 HEAT.

rate is because the pendulum increases in length with
temperature, or the vibrating parts have their moments of
inertia increased by linear expansion.

A very interesting illustration of expansion is furnished
by the creeping downward of heavy metal roofs. If they
are free when they expand by heat they expand downward
because gravity aids this movement ; but when they con-
tract again in cooling, the upper edge is pulled in. The
result is that the metal sheet has the motion of a common
earthworm, and creeps down the incline by alternately
pushing forward its lower edge and drawing its upper one
after it. In this way the sheet lead covering the choir of
Bristol cathedral is reported by Tyndall to have crept
downward at the rate of nine inches a year.

Liquids and gases expand in volume only. Their appar-
ent dilatation is the difference between that of the gas or
liquid and the containing vessel. If a liquid and its en-
velope expanded at the same rate, the liquid would show
no relative dilatation and could not be employed in the
construction of an instrument for measuring temperature.
Let a large glass bulb, or a small Florence flask with a
long narrow stem, be completely filled with water up to a
convenient point on the stem. On suddenly plunging the
flask into hot water the liquid in the tube will at first de-
scend, but as soon as the heat penetrates into the liquid
the index in the stem will stop moving downward and will
then begin to ascend. The envelope first expands by heat,
its increase of volume being indicated by the apparent
shrinkage of the water; but finally the dilatation of the
water exceeds that of the glass and the index rises. The
movement of the index in the stem indicates then only
the apparent expansion of the liquid, or the excess of its
expansion over that of the glass envelope. This relative

TEMPERATURE AND ITS MEASUREMENT.

expansion of a liquid contained in a glass envelope is the
phenomenon most commonly employed in the thermometer^
the instrument to measure temperature.

9. The Mercurial Thermometer. — The mercurial
thermometer consists of a closed capillary glass tube ter-
minating in a bulb or reservoir of a cylin-
drical, spherical, or other form (Fig. 1).
The bulb and a part of the stem are filled
with mercury ; the remainder of the stem
contains only the vapor of mercury. A
cylindrical bulb is preferable to a spherical
one because the mercury then exposes a
larger surface relative to its mass, and so
acquires more promptly the temperature of
surrounding bodies. A small change in
the volume of the mercury in the bulb is
readily indicated by the motion of the end
of the column in the narrow stem.

All such an instrument can do is to in-
dicate its own temperature ; but if it is in
sufficiently intimate contact with another
body, as when it is immersed in a liquid,
it may indicate also the temperature of this
other body with which it is in equilibrium.

Mercury is a very suitable thermometric
substance, for it fulfils most of the necessary requirements.
It can be readily procured in a state of purity. Its co-
efficient of expansion is large and nearly uniform between
the limits within which it remains liquid, and those limits
represent a wide range of temperature. It readily transmits
heat through itself, so that all the mercuiy in the ther-
mometer rapidly comes to the same temperature. It requires

Fig. I.

14 HEAT.

less heat to raise the temperature of any mass of mercury
through any range than is required for equal masses of
most other liquids — that is, its thermal capacity is small.
When therefore it is brought into contact with a warmer
body, at whose expense its temperature rises, this body in
general loses but little heat and its temperature is not
changed by the application to it of the instrument intended
to measure its temperature. Moreover, it does not stick to
the tube so much as other liquids, and it is opaque and can
easily be seen as a fine thread in the bore. On the other
hand, mercury is very heavy, and its weight brings great
stress on the bulb. Also its meniscus is not the same
when the column rises as when it falls, and on a falling
temperature the column is known to descend with an
irregular jerky movement.

10. The Two Fixed Points on a Thermometer. — For
the purpose of making different thermometers comparable,
it is necessary to have fixed points of temperature which
are invariable and easy of reproduction. The two points
universally employed are the temperature of melting ice
and the temperature of steam from water boiling under the
pressure of a standard atmosphere (I., 101). The former
is called for brevity the freezing point, and the latter the
boiling point. The employment of these two points as
standards of reference was first suggested by Hooke ; they
were adopted by Newton in 1701.

The first point is obtained by placing the thermometer
in a vessel filled with pounded ice at the melting tempera-
ture. It is desirable that the interstices between the lumps
of ice should be filled with water while all excess drains
off. The thermometer must be completely immersed in
the ice and water, and must remain there till the mercury

TEMPERATURE AND ITS MEASUREMENT. 15

becomes stationary in the tube. The top of the column is
then marked by a fine scratch on the glass.

To determine the boiling point, the thermometer is
passed through a hole in the top of a tall vessel, the bottom
of which contains boiling water. The thermometer must
be completely enveloped in steam, no part of it touching
the water. When it has acquired the temperature of the
steam it is drawn up till the top of the mercury thread is
visible and the point is marked by a scratch. The upper
portion of the tall vessel is made double so that the steam
may circulate round the inner tube containing the thermom-
eter as a steam jacket to keep the steam up to the boiling
point at every part of the thermometer. The bulb of the
thermometer is not allowed to touch the water, because
the temperature at which water boils varies somewhat
with the material of the containing vessel, while the steam
escaping from boiling water is always at the same tem-
perature for the same pressure. If the atmospheric pressure
is not 760 mms. a correction must be applied, the boiling
point rising 1° C. for every 26.8 mms. increase of pressure.

11. Thermometer Scales (T., 103). — The distance
between the two fixed points on a thermometer must be
subdivided into some convenient number of divisions, each
of which represents one degree of temperature. The
volume of the capillary bore of the tube between the
freezing point and the boiling point represents the total
expansion of the mercury from the one temperature to the
other. A degree of temperature is then that rise of
temperature which causes the mercury to expand some
definite fraction of its entire expansion between the
freezing and boiling points.

Three scales of uniform graduation are in common use :

HEAT.

Fahrenheit's Scale. Fahrenheit about 1714 constructed
the first thermometers with a uniform graduation of the
scale, and this scale is still the one most commonly used in
English-speaking countries. The distance between the
two fixed points is divided into 180 parts of equal volume.
The freezing point is marked 32°, and the boiling point is
therefore 212°. The graduation is- usually continued
below 32°. One degree F. is that rise of temperature
which causes xsiyth of the expansion in volume between
the freezing and boiling points.

The Centigrade Scale. Celsius of Upsala divided the
scale between the fixed points into 100 equal parts. The
freezing point he marked 0° and the boiling point 100°.
This scale is obviously simpler than that of Fahrenheit,
and is in general use among scientific men in connection
with the metric system of measurement. One degree C.
is longer than 1° F. in the ratio of 9 to 5.

R4aumur>8 Scale. In this scale the freezing point is
marked 0°, and the boiling point 80°. It is in use for
domestic purposes on the continent of Europe, but has
little to commend it, except that it avoids Fahrenheit's
fault of a misplaced zero.

In all three scales the graduation is often extended below
zero and above the boiling point.

'^ —

= »

Fahrenheit g

F 2

Centigrade

C l

Reaumur

Fig. 2.

12. Comparison of Thermometer Scales (G., 12 ;
T., 105). — do compare corresponding readings on the
three scales, let us suppose the three attached to the same

TEMPERATURE AND ITS MEASUREMENT. 17

thermometer (Fig. 2). Let A be the freezing point,
B the boiling point, and P the head of the mercury column ;
also let F, C, and M be the readings on the three scales
respectively corresponding to the point P.

Then, since AP is the same fraction of AB measured by
either scale,

.y-32 0 B
180 ~100~80*

The readings on either scale below zero must be treated
as negative. It must be noted also that the zero of Fahren-
heit's scale is displaced 32° in comparison with the zero of
the other two. For Fahrenheit readings therefore 32° must
be subtracted algebraically to find the number of degrees
between the freezing point and the reading. Thus, 50° F.
is 50° -32° =18° above freezing; and -10° F. is -10°
— 32°= -42°, or 42° below freezing.

13. Change of Zero (P., 115). — A thermometer should
not be graduated for several months after filling with mer-
cury. It has been found that the volume of the bulb
slowly decreases for a long period after being strongly
heated. Glass is in some degree plastic, and a gradual
molecular readjustment goes on after it has been strained
or heated. This decrease of the capacity of the bulb raises
the zero point on the stem. The correction at the zero
point even on standard thermometers may often amount to
as much as 0°.7 C, though it rarely equals 1° C.

Besides this progressive and permanent change, there is
another temporary one which may be observed after a
thermometer has been heated in boiling water. It is there-
fore customary to determine first the freezing point and
then the boiling point. , If the freezing point is determined
immediately after immersion in boiling water, it will be

18 HEAT.

found that it may have been depressed as much as 0°.3 C,
and it will not recover its former value until ten days or
more have elapsed.

If the fixed points have been found with the thermometer
in a horizontal position, it should be used horizontally ; or
if they have been found in a vertical position, the ther-
mometer should be used vertically. The reason is that the
hydrostatic pressure of the mercury column compresses
the mercury and enlarges the bulb in the vertical position,
and so lowers all the readings. For a similar reason the
readings of unprotected deep-sea thermometers are too
high, because the bulb is compressed by the pressure of
the water.

14. The Alcohol Thermometer. — Since mercury freezes
at — 38°.8 C. and boils at about 350° C, the mercury ther-
mometer cannot be employed for temperatures beyond
these limits. For temperatures lower than — 38° C. absolute
alcohol has often been used because it freezes only at about
— 130° C. and its dilatation is even greater than that of mer-
cury. But since the dilatation of alcohol is not uniform
at different temperatures, the alcohol thermometer must be
graduated by comparison with a standard mercurial ther-
mometer. It can be used only in a vertical position, bulb
downward, because the alcohol wets the tube, and time
must be allowed after a fall of temperature to permit the
liquid to run down.

15. The Air Thermometer (M., 46; S., 70). — For
high temperatures and for accurate scientific purposes some
form of air thermometer is often used. If a volume of
gas Vo be heated from 0° to 1° under a constant pressure
and its increase of volume be v, then its dilatation will be

TEMPERATURE AND ITS MEASUREMENT. 19

the same volume v for an equal rise of temperature at any-
other part of the scale. This law, called the law of
Charles, is not rigorously exact, but gases approach it more
and more closely at low pressures and high temperatures,
or, in other words, in a highly rarefied state. Within cer-
tain limits, however, all gases, sufficiently removed from
their condensing points, may be regarded as
expanding equally. The ratio v / V0 for one
degree Centigrade was found by Regnault to
be 0.003665 for air.

This property of uniform expansion may be
employed in the construction of a thermometer.
The first air thermometer was made by Galileo
before 1597. The air was contained in a bulb
from which a tube descended to a bottle filled
with a colored liquid (Fig. 3), or was bent
twice at right angles and terminated in an open
bulb. This thermometer is filled by heating the
bulb before the stem is inserted in the liquid.
On cooling, the air contracts and the liquid
rises in the stem. Then if the temperature
changes, the liquid column moves. But un-
fortunately the instrument is also affected by
any change in atmospheric pressure, and can
therefore be used only as a thermoscope unless it be greatly
modified and made more complicated.

The first use to which the air thermometer was applied
was by physicians to obtain the temperature of the human
body. The patient took the air bulb in his mouth, and the
extent to which the liquid column descended indicated to
the observer whether the patient had a fever.

The simplest form of air thermometer is the one employed
by Boyle in 1665. It was composed of a glass bulb from

20 HEAT.

which rose a long stem containing a drop of mercury or
sulphuric acid to separate the air within from the external
atmosphere (Fig. 4). As the temperature rises, the air
within expands and drives the liquid index before it.

The dilatation of air is about twenty times as great as
that of mercury for the same range of temperature.
Hence a thermometer filled with air is much more sensitive
than one filled with mercury. For any given
range of temperature it has been found that air
and mercury thermometers agree closely, though
not exactly. •

It is worth while to point out that the only
reason we have for asserting that the thermal value
of the successive degrees of a well-calibrated mer-
cury thermometer are the same is that they cor-
respond closely with those of the air thermometer.
But, strictly speaking, it is impossible to prove
the law of Charles with precision, for its experi-
mental demonstration implies the possession of an
accurate instrument for measuring temperature.
There are, however, theoretical reasons for believ-
ing this law to be exact when the gas is in a state
of extreme tenuity and the molecules are so far
apart as to exert no influence upon one another. It is
then called a perfect gas.

The practical methods of using air as a thermometric
substance are described in memoirs and large treatises. A
description of one form will be found in a later chapter
(26).

16. The Absolute Zero (M., 48, 213). — The air
thermometer in the form of a straight tube of uniform
bore may be employed to illustrate the meaning of the

TEMPERATURE AND ITS MEASUREMENT. 21

"absolute zero of temperature," or, better, the "zero of
absolute temperature."

Let a long narrow tube be closed at one end, and let air
be confined and separated from the external air by a short
cylinder of oil, mercury, or sulphuric acid. We shall
assume that the pressure on this enclosed air is maintained
constant.

Let the point marked F (Fig. 5) be the position
of the surface of the enclosed air or index cylinder
when the tube is in melting ice, and let B mark
the position of the index for the temperature of
boiling water. The question then arises, What
temperature will be indicated at the bottom of the
tube, if the uniform graduation is carried down
there, and what is its meaning ?

The first question is easily answered. Let x
equal the ^ngth AF on the same scale as FB is
100 divisions. Then we know that the volume of
the portion of the tube between A and F is to the
volume of AB as 1 is to 1.3665, since 0.3665 is the
dilatation of air for 100° C. Then

z:z-f 100 ::1 : 1.3665.

Whence x = 272.85, or in whole numbers 273. The
bottom of the tube will then be marked — 273°.
This point is called " absolute zero." The meaning
of it is that if the law of Charles should continue
to hold down to the temperature — 273° C, the volume of
the gas would become zero, or the air would be entirely
devoid of heat. Now, while it is not supposed that the
contraction of a gas would continue at the same rate down
to any such temperature, still this is a convenient point
from which to reckon temperatures, because the volume
of a perfect gas is simply proportional to its temperature

22 HEAT.

measured on this scale. Temperatures on the Centigrade
scale are converted into corresponding readings on the
absolute scale by adding 273°.

It is important to know that the scale of the air ther-
mometer agrees almost exactly with that derived from
thermodynamical considerations. The agreement has been
experimentally verified between the limits 0°C. and 100° C.

PROBLEMS.

1 . Convert the following readings on the Fahrenheit scale into
the corresponding degrees Centigrade : 60°, 28°, — 20°.

2. Convert the following readings on a Centigrade thermometer
into degrees of the Fahrenheit scale: 15°, — 10°, — 20°.

3. At what temperature will the Fahrenheit and Centigrade
scales read the same?

4. At what temperature will the reading of the Fahrenheit scale
be double that of the Centigrade ?

5. At what temperature will the reading of the Centigrade scale
be double that of the Fahrenheit ?

6. If a thermometer scale were marked 10° at the freezing point
and 60° at the boiling point, what would 40° on this scale mean in
Centigrade degrees ?

7. A thermometer tube with uniform bore has 5 C. divisions to a
cm. ; how many F. divisions to the cm. would there be?

8. The testing of a Centigrade thermometer shows that the
freezing point reads -f- 0°-6 and the boiling point 101°. What is the
meaning of 50° on this scale if the tube is uniform ?

9. The latent heat of fusion of ice on the Centigrade scale is 80;
find it on the Fahrenheit scale.

EXPANSION. 23

CHAPTER III.

EXPANSION.

17. The Cubical Dilatation of Solids (S., 27 ; P., 157).
— The expansion of solids and liquids has already been
alluded to in the last chapter. The property of a thermo-
metric substance which is utilized to indicate temperature
is its increase in volume with heat.

Let V0 be the volume of a body at zero and F"its volume
at t°. Then if the increase of volume v for an increase of
one degree in temperature is constant at different parts of
the scale, we have

vt=V-V0,

or V=V0+vt=V0(l + ^.€)=,V0(l + lct).

* 0

The constant k is called the coefficient of cubical ex-
pansion. It is equal to v / V0 , or the expansion per unit of
volume when the temperature rises from 0° to 1° C. This
is sometimes called the zero coefficient. If, for example,
1 c.c. of iron at 0° becomes 1.003546 c.c. at 100° C, then
0.00003546 denotes the mean coefficient of cubical dilata-
tion of iron between these two temperatures.

While the equation V— V0 (1 + k€) is a very near ap-
proximation, it is not rigorously exact. Each substance
has its own constant k.

24 HEAT.

Since the volume of any mass of a substance is inversely
as its density, we may write

d0 = d(l + kt).

Whence k = — ?— — -.

This formula is the basis of a method of measurement
which depends on the determination of the density of a
solid at different temperatures.

The general law of the dilatation of solids assumes that
they expand when heated and recover their initial volume
when restored to their initial temperature ; that is, that
under a constant pressure the volume is a function of the
temperature.

Neither of these assumptions is rigorously correct. It
has been found that Rose's fusible* metal expands to a
maximum, after which, if the temperature be increased, it
contracts. So also Fizeau found that iodide of silver con-
tracts regularly when heated between 10° and 70° C. It
has since been determined that it reaches a point of maxi-
mum density at 116° C, at which point on cooling it passes
from the amorphous into the crystalline state.

Neither is it true that the restoration of an antecedent
temperature always restores a body to the corresponding
volume. If some bodies, like glass, are cooled suddenly,
the molecules have insufficient time to arrange themselves
in accordance with their mutual attractions. Hence certain
stresses are set up which may produce a slow change in
volume as they adjust themselves to zero.

The purpose of annealing glass and metals by slow cool-
ing is to give time for the forces of cohesion to adjust
themselves without constraint. The annealed body is then
much tougher. It is not much in error to say that when

EXPANSION. 25

bodies are heated and then very slowly cooled, they return
to the same volumes at the same temperatures.

18. Linear Expansion. — If the distance between two
transverse parallel lines on a metallic bar is l0 at a tempera-
ture of 0° and I at £°, the increase in length is I —lQ. This
linear expansion is found to be nearly proportional to the
length and to the rise of temperature; and the constant
which defines this proportionality, and which depends upon
the nature of the body, is called the coefficient of linear
expansion. If this coefficient is denoted by a, then

I — ?0 = al0t, or a =

l-lo

Whence I = l0 (1 + at).

It is obvious from the equation for a that the coefficient
of linear expansion is tne increase which occurs in unit
length of a solid when the temperature rises from 0° to 1°
C. This is very nearly the same as the mean coefficient
between 0° and 100° C. It is the ratio of the increase in
length for one degree to the total length at 0°. In the
metric system it is the increase in the length of one cm.
due to a rise of temperature of one degree C.

Fig. 6.

The expression 1 + at is called the expansion-factor. It
is the ratio of the final to the initial length.

A simple method of showing the expansion of a wire in

26 HEAT.

length is illustrated in Fig. 6. The wire, which should be
about one metre long, is rigidly attached at one end A to
the stand, and at the other is fastened to a small screw-eye
in the long, light wooden pointer BO. The pointer is free
to turn around a smooth pin at B, a point very near the
screw eye. Heat the wire by passing through it a current of
electricity from some appropriate source. The expansion
will be indicated by a wide sweep of the pointer. The
wire will cool quickly when the current is off, and the
pointer will return to its initial position.

The expansion of a bar may be conveniently illustrated
by supporting one end A rigidly, as by a weight (Fig. 7),
while the other end rests on a thin, straight sewing-needle,

which in turn lies on
a sheet of plate glass.
A slender pointer of
5~7/l straw or foil may be
attached to the eye
of the needle by a bit

/31

Fig. 7. of sealing wax, and it

should be counterbalanced.

When the bar is heated by a lamp or a Bunsen burner
it lengthens, and the free end advancing rolls the needle.
The movement of the pointer indicates the expansion. It
should return to its original position when the bar cools.

19. Relation between the Coefficients of Length and
of Volume (P., 199). — The coefficient of volume-expan-
sion is three times that of linear expansion ; for the volume
of a cube, whose side is l0 at zero, is 11 (1 + at)* at t°.
This volume is also V0 (1 + kt~). But V0 = 11. Hence

1 + kt = (1 + at)3 = 1 + Zat + BaH* + aH\

EXPANSION. 27

But since a is a very small quantity, its higher powers
may be neglected in comparison with the first, or

1 + Jet = 1 + Sat,
and Jc = Sa nearly.

This relation assumes that the body is isotropic, or has
the same physical properties and expands equally in all
directions. In the case of crystals this is true only for
those of the regular cubic system, which do not cause
double refraction of light (I., 226). These dilate uniformly
in all directions in the same manner as amorphous bodies.

In general crystals have three rectangular axes of dilata-
tion, and the linear coefficients in these three directions
are not identical ; the voluminal coefficient is then equal to
the sum of the three linear coefficients. It follows that a
crystalline sphere at one temperature ceases to be spherical
at any other temperature, and a cubical portion of a crys-
talline body at one temperature will not remain cubical
when the temperature changes, unless the crystal belongs
to the cubic system.

Crystals belonging to the rhombic system have an axis
of crystalline symmetry, and the two coefficients of expan-
sion perpendicular to this axis are equal, or the crystal has
the same properties in all directions perpendicular to" the
axis of symmetry. In this case

k = ax + 2a2 •

Here ax is the coefficient of expansion parallel to the axis,
and a2 is the coefficient perpendicular to it.

Optically biaxial crystals dilate unequally in the direc-
tion of the three principal axes. Iceland spar and beryl
expand in the direction of their principal axis, but contract
transversely with rise of temperature.

Mitscherlich concluded that the effect of heat on crystals

HEAT.

is a tendency to separate the molecules in the direction in
which their distance is the least, so as to equalize their
distances, and to give to the crystal identical properties in
all directions.

If such crystals as quartz are strongly heated, their
unequal expansion in different directions causes them to
buret into small pieces.

rV>

20. Measurement of Linear Expansion. — All meth-
ods of determining
coefficients of linear
expansion involve the
- exact measurement of
the change in length
of a body, or some
definite portion of it,
produced by a known
change of tempera-
ture. The variations
among them consist in
the methods adopted
to measure this change
of length.

By the "interfer-
ential method " of Professors Morley and Rogers the small
difference in the length of the two bars compared is
measured by counting the corresponding number of wave-
lengths of monochromatic light of known refrangibility.
The instrument by which such measurements are made
was invented by Professor Michelson, and is called the
" interferential comparator." The elements of it are shown
in plan in Fig. 8, where 6 is a plate of plane parallel glass
so silvered in front that half of an incident ray of light

rig. 8.

EXPANSION. 29

from S is transmitted and half is reflected. To the near
ends of the bars are attached plane mirrors 5 and 8, silvered
in front, and to the remote ends, 4 and 9, with the silvered
portion extending out at one side of the bar. At 7 is a
plate of plane parallel glass of the same thickness as 6,
but unsilvered.

In using the apparatus a ray of monochromatic light from
S is incident at 6. Half of it is reflected and goes to the
mirror 5, from which it is reflected back to 6, where half of
this reflected portion is transmitted and passes to the eye
of the observer at /. The transmitted half of the incident
ray at 6 is reflected from 8 back to 6, where half of it is
reflected and enters the eye along with the other com-
ponent from 5. Since the mirror 6 is silvered on the side
facing S, the portion of the light which returns from 8
traverses the glass 6 three times, while the first portion
reflected from 5 traverses it but once. Hence the plane
plate 7 is introduced to equalize the thickness of glass
traversed by the two components which enter the eye of
the observer.

If now the two rays have travelled exactly equal dis-
tances from the first incidence at 6 to the eye, they will
interfere, because a difference of phase of half a wave-
length has been impressed on them by the fact that one has
suffered internal and the other external reflection at the
mirror 6 (I., 214). Any difference of path of the two
portions of the incident ray reflected from 5 and 8 will
produce a difference of phase at I; and when this differ-
ence of path amounts to an even number of half wave-
lengths for the particular color employed interference will
result.

The form of interferential comparator shown in Fig. 9
was devised by Professor Morley for the determination of

30 HEAT.

the absolute coefficient of expansion of metals between the
freezing and the boiling points of water. The bars to be
compared are mounted as shown. Plate 14 is moved by a
weight which keeps it in contact with a cross-plate actuated
by a precision screw. By means of the interference phe-
nomena described, 5 and 8 and then 4 and 9 are made
equidistant from 6. The motion of the bar 8 and 9 in
passing from the first position to the second can be

Fig. 9.

measured by counting the interference bands during the
motion. A microscope and graduated scale shown at 10
are used to measure the length corresponding to any
observed number of wave-lengths of the monochromatic
light.

If now one of these bars be kept at a constant tempera-
ture and the other one be compared with it in the way de-
scribed, first at the freezing point and then at the boiling
point, the expansion for 100 degrees C. will be measured in
terms of a particular wave-length of light as a unit of
length.

21. Dilatation of Liquids. — Liquids in general are more
expansible than solids. In the case of liquids and gases

EXPANSION.

the only expansion to deal with is volume expansion. The
approximate formula V= V0 (1 + kf) holds as in the case
of solids.

An instrument like a thermometer is well suited to
measure the apparent expansion of a liquid, or the excess
of its expansion over the volume expansion of the glass
envelope. If the absolute coefficient of expansion of the
glass is known, the absolute expansion of the liquid can be
deduced from the apparent expansion.

The absolute coefficient of ex-
pansion of mercury has been de-
termined by Regnault with great
accuracy by means of the principle
that the heights of two liquids in
communicating tubes .above their
common surface of separation are
inversely as their densities (I.,
80). The actual investigation
involved some modifications and
many minute details.

Two vertical iron tubes, ab and
a'b', about 150 cms. long, were con-
nected near their upper ends by a horizontal cross-tube aa'
(Fig. 10). The cross-tube joining the lower ends b and b'
was interrupted at its middle, and two vertical glass
tubes were inserted and connected with each other and
with a reservoir filled with air, the pressure of which
could be varied at pleasure, ^yhen the two columns of
mercury filling the apparatus are at different temperatures,
the mercury will stand at different heights d and d' in the
glass tubes ; while their upper surfaces near a and a' will
be at levels to produce equilibrium at the upper horizontal
cross-tube by hydrostatic pressure. The tubes were all

Fig. 10.

32 . HEAT.

enclosed in water jackets, and the two glass tubes were at
the same temperature.

The pressures at d and d' are the same, because the two
surfaces of mercury are in contact with air under pressure.
We may therefore place the pressures on the two sides at
d and d' equal to each other. Since the pressure of the
short column above a is equal to the one above a', because
they are in equilibrium through the tube aa', we need to
consider only the long columns from a and a' respectively
down to the horizontal plane through bb'. Let H be this
common height, and let h and h' be the heights of d and d'
respectively above the same level through bb'. Also let t
be the temperature of the mercury in ab, and t' the temper-
ature of the mercury in all the other tubes. Then

H , h_ H h'

1 + kt 1 + kt' ~ 1 + kt' 1 + kf*

The division of H by the expansion-factor reduces it to the
height at zero, and this multiplied by the density at zero
gives pressure. The same is true of the other terms. But
the density is a common factor and disappears.
From this equation

E _H-(h'-K)

Hence k=

1 + kt 1 + kt'

h-h'

m'-t(H+h-h')

It is not necessary to see the tops of the long columns,
since the parts above aa' are in equilibrium. H is deter-
mined from the apparatus itself, though a correction is
needed for a change in temperature. In addition the two
temperatures and the difference of level between d and d<
must be observed.

EXPANSION.

By means of this apparatus Regnault made measure-
ments which enabled him to draw up a table of the dilata-
tion of mercury for every 10° from 0° to 350° C. (Appendix,
Table I.).

22. Dilatation of Water (B., 293 ; S.f 52 ; P., 176).
— Water shows the anomalous property of contracting
when heated at the freezing point. This contraction con-
tinues up to 4° C. ; at this point expansion sets in, so that
the greatest density of water is at a temperature of 4°, and
its density at 8° is nearly the same as at 0° C.

This peculiar behavior of water is illus-
trated by Hope's apparatus (Fig. 11). It
consists of a glass jar with a tubulure near
the top and the bottom to admit thermom-
eters. About its middle is placed an annular
reservoir. If the vessel is filled with water
at about 10° C, the upper thermometer will
show at first a slightly higher temperature
than the lower one. If now the trough at
the middle be filled with a freezing mixture,
the first effect will be the gradual fall of the
lower thermometer to 4° C. without much
change of the upper one. After the lower thermometer
becomes stationary, the upper one falls rapidly till its
temperature is reduced to zero and ice forms at the surface.
The water at 4° C. sinks to the bottom, while that below
4° is lighter and rises to the top, where the freezing first
takes place. For this reason ice forms at the surface of a
body of cold water which freezes from the surface down-
ward, instead of from the bottom upward.

The relation between the volume and the temperature of
water near the freezing point may be determined by means

Rg. II.

HEAT.

of a large thermometer filled with distilled water. If the
apparent volumes of the water in glass are plotted as ordi-
nates and the corresponding temperatures as abscissas, the
curve is approximately a parabola (abc, Fig. 12). The
vertex is somewhat above 4° C. This is then the tempera-
ture of the least apparent volume. But the observations
for this curve include the dilatation
of both the glass and the water.
The real volume-temperature curve
of water may be found by adding to
the ordinates of this one the expan-
sion of the glass. For this purpose,
if the glass is assumed to expand
uniformly for the small range of tem-
perature included within the obser-
vations, it is only necessary to draw
a line OD, making with the axis of
temperatures an angle whose tan-
gent, expressed in terms of the two
scales, is the dilatation of the glass
Jx for one degree. If the vertical ordi-
nates between OX and OD are
added to the corresponding ones of
D abc, the result is the curve adf.
The point of least volume, or great-
est density, will correspond to the shortest ordinate between
OD and the curve abc. This may be found by drawing a
tangent to the curve parallel to OD. This tangent touches
the curve at b, and this is the point of least volume. It
corresponds very closely to 4° C.

When the pressure is increased above one atmosphere,
the temperature of maximum density of water recedes
toward zero. Amagat found the mean rate of recession to

110

120

SO
CO

1 c

>bs

Fig. 12.

EXPANSION. 35

be about 0.025 degree C. per atmosphere. At 144.8 atmos-
pheres the temperature of greatest density was 0.6° C.

Table II. in the Appendix contains the volumes and
densities of water from 0° to 100° C. deduced from Rosetti's
experiments.

23. Dilatation of Gases — Law of Charles (P., 186;
S., 61; G., 100). — The law first enunciated by Charles in
1787 and confirmed later by Rudberg and Regnault is the
following : The volume of a given mass of any gas, under
constant pressure, increases from the freezing to the boiling
point by a constant fraction of its volume at zero. This is
therefore known as the law of Charles. For the Centi-
grade scale the constant fraction is 0.3665 for dry air.
This is equivalent to 0.003665 for one degree C. A near
approximation is ^1-^. Hence 30 c.c. at 0° become about
41 c.c. at 100° C.

It follows from this law that the formula of dilatation,
which has already served for solids and liquids, may be
applied to a gas under constant pressure, or

v = v0 (1 + kt).

The investigations of Regnault and others have shown that
this law, like that of Boyle, is not absolutely exact, but is
a close approximation to the truth.

For a perfect gas obeying Boyle's law (I., 103), the
product pv of the pressure and volume, for a constant
temperature, is a constant. This product is then some
function of the temperature, or

It is obvious from this expression that the changes pro-
duced by the application of heat to a gas may be investi-
gated by observing the changes of volume under constant

UFA T.

pressure, or the changes of pressure at constant volume.
These two methods have been found to give nearly, though
not absolutely, identical results.

The method of a constant volume is more readily applied
than the other to determine the laws relating to gases.
Regnault's apparatus consisted essentially of a large glass
bulb of some 600 to 800 c.c. capacity, connected with an
open mercury manometer (Fig. 13). At the point h was

a mark, and the mer-
cury was kept at this
height by enlarging
or contracting the size
of the reservoir at the
bottom by means of
the screw S, which
moved a piston out
or in, or by some
equivalent method.
The bulb b was first
placed in melting ice,
the mercury in T was
brought to the point
h, and the difference
between the levels of
the mercury in T and T was measured. By adding the
height of the barometer, the pressure on the gas in the bulb
was determined.

The bulb was then enveloped in steam and the operations
were repeated to determine the total pressure at 100° C.
Then, knowing the several temperatures and the volume of
the bulb at the different temperatures employed, as well
as that of the stem, it was possible to calculate the coeffi-
cient of increase of pressure. Regnault found that for dry

Fig. 13.

EXPANSION. 37

air an initial pressure of one atmosphere at 0° C. became
1.3665 atmospheres at 100° C.

With slight modifications in the operations, Regnault
found the dilatation in volume under constant pressure.
Between 0° and 100° C. the increase in volume was 0.3670.

The table exhibits the results with several gases.

Coefficients of Dilatation and Pressure between

0° AND 100° C.
Gas. Constant pressure. Constant volume.

Hydrogen 0.003661 0.003667

Air 0.003670 0.003665

Nitrogen 0.003668

Carbon monoxide 0.003669 0.003667

Carbon dioxide 0.003710 0.003668

Nitrous oxide . 0.003719 0.003676

Sulphur dioxide 0.003903 0.003845

Cyanogen 0.003877 0.003829

The easily liquefiable gases at the bottom of the list
have a somewhat larger coefficient of dilatation than those
which are liquefied with great difficulty. Regnault con-
cluded from his elaborate investigations —

(1) That all gases have not the same coefficient of
expansion, and that for the same gas there is a slight
difference between the coefficient under constant pressure
and that at constant volume.

(2) That the coefficient of all gases, except hydrogen,
increases with the initial pressure of the gas.

(3) That the coefficients of the gases investigated
approach equality as the pressure decreases.

These conclusions correspond with the fact that all gases
depart more or less from Boyle's law ; but as they are more
highly rarefied by reduced pressure, they approximate more
nearly to the ideal limit of exact obedience to this law.

38 BEAT.

24. Volume of a Mass of Gas proportional to Abso»
lute Temperature. — Under the condition of a constant
pressure, the law of expansion of a perfect gas is such that
increments of volume are proportional to increments of
temperature, or

t—t0 = A (y — v0),

where A is a constant. If now the temperature of the
least volume of the gas be taken as the zero of the scale
(16), and the temperature on this scale be denoted by T,
then t0 is zero, and

T=A(v-v0-).

For an ideal gas following Boyle's law rigorously, the
volume would become zero at the zero of this absolute
scale, or

T=Av.

Hence, under a constant pressure, the volume of a given
mass of such a gas is proportional to the temperature on
the absolute scale.

The zero of this scale can be calculated from the
formula of Art. 23,

v = v0 (1 + kf) = v0 (1 + 0.0036650-
To find the value of t on the Centigrade scale at which the
volume v becomes zero, we have

0 = 1 + 0.003665^,
or £ = -273°.

25. The Laws of Boyle and Charles combined. —
The application of the law of Charles enables us to com-
bine both it and the law of Boyle into one expression, viz.,
that the product of the volume and pressure of any mass
of a gas is proportional to its absolute temperature. This
result may be reached in the following manner :

Let v0, jt?0, T0, be the volume, pressure, and absolute

EXPANSION. 39

temperature of the gas under standard conditions, as, for
example, 0° C. and 760 rams, pressure.

Also let v, jo, and T be the corresponding quantities at
temperature T.

Then, applying Boyle's law to increase the pressure to
the value p, the temperature remaining constant, we have

v0 : v' : : p : p0 .
By changing the pressure from p0 to p the volume has
changed to v'.

Next apply the law of Charles, keeping the pressure con-
stant at the value p, and starting with volume v'. Then

v' : v : : T0 : T.
It must be observed that these changes have taken place
by two independent, successive steps.

From the first proportion — = £. ; and from the second

v' p0

v' T

— =-£. Multiplying the two equations together member

v T

by member, we have

« = £T. jMi £, a constant

v p«T % T

or the product pv is proportional to T, the temperature on
the absolute scale. We may therefore write

pv = RT,
where B. is a constant. We see from this expression that
in a perfect gas, following these two laws, both the press-
ure at constant volume and the volume under constant
pressure vary directly as the absolute temperature.

26. The Constant- Volume Air Thermometer. — Pro-
fessor Jolly has devised a constant-volume air thermometer,
which is similar in principle to Regnault's apparatus for

HEAT.

the determination of the coefficient of dilatation of gases.
It is shown in Fig. 14. The capillary tube is bent twice
at right angles, and at B is joined to another tube of larger
diameter, on which a mark is made near the junction with
the capillary. CE is a glass tube of the same diameter as
BD, and the two are connected by a
piece of strong, flexible rubber tubing,
which permits CE to be raised or
lowered so as to keep the level of the
mercury at B. CE may be clamped
in any position by the screw clamp S.
The difference in level of the mercury
at, B and E, added to the height of the
barometer, both corrected for temper-
ature, gives the pressure of the aii
in the thermometer. The air in the
bulb must be very dry and free from
carbonic acid.

For ordinary measurements the dif-
ference of level of B and E may be
obtained with sufficient accuracy by
means of a scale engraved on a strip
of glass before it is silvered. This
scale is mounted on the frame sup-
porting the thermometer and tubes.
In reading, the observer avoids parallax by reading the
point on the scale touched by the line joining the top of
the mercury column and its image in the mirror.
From the relation

pv = RT,

it is obvious that the pressures of a fixed volume of gas
are proportional to the corresponding absolute temperatures,
since R is a constant.

Fig. 14.

EXPANSION. 41

If, therefore, p0 be the pressure at 0° C. and p the press-
ure at some higher temperature t° C, then since the abso-
lute zero is 273 degrees below zero C, we may write

273 : 273 + t : : p0 : p.

Whence t = 273

(f.-1)

If we employ Regnault's coefficient 0.003665, the abso-
lute zero is -272°.85 C. instead of —273°.

The pressure at zero must be determined by surrounding
the bulb of the thermometer with ice and taking readings.
Any other temperature is then measured by observing the
pressure necessary to keep the mercury at the fiducial
point near B.

PROBLEMS.

1. A glass flask holds 200 c.c. of water at 0° C. How much will
it hold at 100° C. ? The coefficient of linear expansion for glass is
0.0000083.

2. The density of a piece of silver at 0° is 10.5. Find its den-
sity at 100° C. if its coefficient of cubical expansion is 0.0000583.

3. The volume of a mass of copper at 50° C. is 500 c.c. ; find its
volume at 300° C. Coefficient of cubical expansion, 0.0000565.

4. A brass pendulum keeps correct time at 15° C, but at 35° C.
it loses 16 seconds a day. Find the linear coefficient of expansion
of brass.

5. A solid displaces 500 c.c. when immersed in water at 0° C. ;
but in water at 30° C. it displaces 503 c.c. ; find its mean coefficient
of cubical expansion.

42 HEAT.

CHAPTER IV.

MEASUREMENT OF THE QUANTITY OF HEAT.

27. Unit Quantity of Heat. — Heat as a physical
quantity is subject to measurement. For this purpose no
knowledge of the ultimate nature of heat is required, but
the methods of measurement are based on some established
property or effect attributed to heat. Twice as much heat
is required to raise the temperature of two grammes of
water one degree as of one gramme one degree. The
thermal element of such a comparison is limited to an
observation of temperatures. The measurement of heat is
called Calorimetry.

Heat, like other physical quantities, must be expressed
in terms of some unit. The unit quantity of heat is the
heat required to raise the temperature of unit mass of
water one degree. If the unit of mass is the gramme and
the unit of temperature the degree Centigrade, the unit of
heat is called the calorie.

The number of units required to raise the temperature
of m gms. of water 1° C. is then m calories ; and since the
heat necessary to effect the same increase of temperature
of 1 gm. of water at any part of the scale is nearly the
same,' the heat which will warm 50 gms., for example, one
degree is almost the same as the heat required to raise
1 gm. 50 degrees. This is demonstrated by mixing equal
masses of water of different temperatures and observing

MEASUREMENT OF THE QUANTITY OF HEAT. 43

whether the temperature of the mixture is the mean of
the two contributing temperatures. It is found that the
quantity of heat given out by the warmer mass in cooling
through any range raises the cooler mass through the same
range.

Since the heat which will warm one gramme of water one
degree at different temperatures is not rigorously the same,
the definition of unit quantity is often as follows : The unit
quantity of heat is the heat required to raise the tempera-
ture of 1 gm. of water from 4° C. to 5° C. The same
quantity of heat is given out by 1 gm. of water in cooling
from 5° C. to 4° C.

28. Thermal Capacity. — The thermal capacity of a
body is the number of heat units required to raise its tem-
perature one degree. The thermal capacity of any body of
water is numerically equal to its mass in grammes, since
the thermal capacity of unit mass of water is the heat unit.
But the case is very different with other substances. If
equal masses of mercury at 80° C. and water at 20° C. be
mixed, the temperature of the whole will be only about
22° C. The "heat which the mercury gives up in cooling 58
degrees will heat the water only about 2 degrees, or the
thermal capacity of water is about thirty times that of
mercury.

This difference in thermal capacities may be further shown
as follows : Take a number of metal balls of equal mass,
such as lead, tin, zinc, copper, and iron, and place them in
boiling water. By means of fine attached wires place them
all simultaneously on a flat cake of paraffin supported at
the edges, and observe the extent to which the paraffin is
melted by each ball. If the plate is not too thick the iron,
copper, and zinc balls may melt through, but they will not

44 HEAT.

go through in exactly the same time. The tin ball will
not sink into the wax so deeply, while the lead will melt
less than any of the others. The thermal capacity of the
lead ball is the smallest, while that of the iron one is the
greatest of the series.

The thermal capacity of a substance is the heat required
to raise the temperature of unit mass of it one degree.
When the unit of heat is denned as above, the thermal
capacity of unit mass is numerically equal to the specific
heat of a substance.

The specific heat of a substance is generally denned as
the ratio between the thermal capacities of equal masses of
the substance and of water. Since specific heat is a ratio,
it is independent of the unit of measurement employed.
The thermal capacity of a body is the product of its specific
heat and its mass.

Liquids exhibit differences of specific heat similar to those
of solids. If one kilo, of bisulphide of carbon at 0° C. be
mixed with one kilo, of water at 60° C, the temperature
of the mixture will be about 48°. 25 C. The number of calo-
ries lost by the water in cooling 11.75 degrees is 1,000 x
11.75 or 11,750 ; hence the thermal capacity of the kilo, of

carbon bisulphde is ' or 240, and the thermal capacity

of 1 gm. of it is 0.240. This is therefore its specific
heat. If heat be applied at the same rate to equal masses
of water and carbon bisulphide, the temperature of the
latter will rise about four times as rapidly as that of the
former.

29. Specific Heat by the Method of Mixtures (P.,
221; G., 34). — The last example illustrates roughly
the method of determining specific heats by the method

MEASUREMENT OF THE QUANTITY OF HEAT. 45

of mixtures. It is desirable to describe the method some*
what more fully for the purpose of illustrating the thermal
principles involved.

Let Ax and A., be two bodies of masses ml and m2, tem-
peratures tr and £>, and specific heats sx and s2. If they
are placed in contact they will arrive at some intermediate
temperature t. The quantity of heat lost by A2 will be
m^ (t2 — £), and the quantity gained by Ai will be
to,»i (t — £1). If we assume that the only interchange of
heat going on is between Ai and A2 , the heat lost by A2 will
be equal to that gained by Ax , and consequently

W2S2 (£2 — 0 — misi (P ~ t0-

If J.! be a mass of water, its specific heat by definition is
unity, and therefore

Si _ml(t- t{)
m2 (t2 — f)

This equation gives the mean specific heat between the
temperatures tx and,£2 obtained by means of the water
calorimeter.

It has been assumed that the thermal equilibrium be-
tween Ax and A2 is reached without loss of heat to other
bodies during the period of equalization of temperatures.
In practice there will be interchange of heat with other
bodies. There will be some loss by radiation, and the
heat given to the calorimeter and its fittings must be taken
into account. The thermal capacity of the calorimeter is
usually expressed in terms of the quantity of water which
the number of heat units expressing that capacity would
heat one degree. This is called its " water equivalent."
The gain of heat by the calorimeter and its fittings must
be added to that gained by the water.

46 HEAT.

Let the water equivalent be m. Then the heat acquired
by the calorimeter and its contents will be

m(t — 1{) + Wj (t — Q ,

and we have m2s2 (t2 — tj = (m + m{) (t — 1{) ,

or ^(m + CT,)(i-t,)t

rn2(t2 — 0

The correction appears in the formula as an addition to
the water in the calorimeter.

To correct for radiation, Rumford arranged the experi-
ment so that the initial temperature of the water in the
calorimeter shall be as much below that of the surround-
ing air as the final temperature is above it. Then the heat
gained by absorption during the first part of the experiment
will be nearly equal to that lost by radiation during the
latter part.

The specific heat of liquids, of powders, and of sub-
stances soluble in water may be determined by sealing
them in thin glass or metal tubes and proceeding as before.
The slowness with which they may then acquire the tem-
perature of the water increases the correction for radiation
and reduces the accuracy.

In the case of solids of poor conductivity and soluble in
water, another liquid of known specific heat in which they
are insoluble may be used in the calorimeter.

There are three other methods of measuring specific heat.
The first is founded on the mass of ice which a known
mass of the substance will melt. The second depends on
the relative rate of cooling of equal masses of water and of
the substance. The third is based on determining the
amount of steam condensed in raising the temperature of
the body through any observed range of temperature. The
last method has lately been developed into one of great

MEASUREMENT OF THE QUANTITY OF HEAT. 47

scientific value and accuracy. The details will be found
in Preston's Theory of Heat, p. 236.

30. Variation of Specific Heat with Temperature
(S., 307; P., 258). — The specific heat of a substance in
general increases with the temperature. This increase
becomes quite large in solids near the temperature of
fusion. The law governing the variation of specific heat
with temperature has not yet been discovered; but the
specific heat of any substance may be expressed by the
empirical formula

8 = a -f bt + ct*

in which a, 6, c, etc., are constants determined by experi-
ment. Such a formula is used only to express the results
of a series of experiments, and cannot be regarded as con-
taining any law which holds beyond the range of the
experimental series.

The following table embodies the results of Dulong and
Petit's experiments :

Substance. Mean Specific Heat.

Between 0* and 100* C Between 0* and 300* C

Iron 0.1098 0.1218

Glass 0.1770 0.1990

Copper 0.0949 0.1018

Zinc 0.0927 0.1015

Silver 0.0557 0.0611

Antimony 0.0507 0.0549

Platinum 0.0355 0.0355

Bismuth 0.0308 ....

For higher temperatures platinum has since been found
to exhibit a variation, but it is less marked than with other
metals. For this reason, a piece of platinum may be used
to determine the temperature of a furnace. When it has

48 HEAT.

acquired the temperature of the furnace, it is quickly re-
moved and plunged into a known mass of ice-cold water.
By noting the rise of temperature of the water, it is easy to
calculate the approximate temperature of the platinum and
hence of the furnace. Such an instrument for measuring
high temperatures is called a pyrometer.

According to Hirn the thermal capacity of alcohol attains
the value 1.11389 at 160° C, a value superior even to that
of water at 100° 0.

31. Specific Heat of Carbon (P., 260). — A few sub-
stances, notably carbon, exhibit large variations of specific
heat with temperature. Weber conducted a series of careful
experiments on the specific heat of diamond, and found the
following formula for the mean specific heat between 0°
and 200° C. :

s = 0.0947 + 0.000497* - 0.00000012*2 ... (a)

The total quantity of heat required to raise one gramme
of diamond from 0° to f C. is then

^ = 0.0947^+0.000497^-0.00000012^ ... (6)

The mean specific heat between 0° and t° C. is obtained by
dividing q by t. If the specific heat at any definite tem-
perature is required, it is necessary to find the limiting
value of the mean specific heat as the range of temperature
is indefinitely diminished ; or

8 = -i)

dt
where dq is the indefinitely small quantity of heat required
to raise the temperature of unit mass through the indefinitely
small range of temperature dt. Therefore L

1 The formula is obtained by finding from (b) the differential coefficient _? •

MEASUREMENT OF THE QUANTITY OF HEAT. 49

s =0.0947 + 0.000994* -0.00000036*2 ...(c)

At 200° C. the thermal capacity of diamond is therefore
nearly three times as great as at 0° C.

Weber showed further that the specific heat of carbon at
600° Is about seven times as great as at — 50° C. As the
temperature rises it approaches a maximum value of about
0.46.

The following are the specific heats of carbon in its dif-
ferent states of aggregation :

Animal charcoal 0.2608

Wood charcoal 0.2415

Coke 0.2008

Graphite 0.2018

Diamond 0.H68

32. Specific Heat of Wacer (P. 262). — Water has
the highest thermal capacity of any known substance except
hydrogen, unless it be a mixture of water and twenty per
cent of alcohol, which Dupre" and Page found to have a
thermal capacity five per cent higher than water.

The thermal capacity of water is nearly twice as great
as that of ice (0.504), and more than twice as great as that
of steam under constant pressure (0.477). Generally
speaking, the specific heat of a substance when liquid is
higher than when solid.

The heat which will warm a gm. of water one degree
will warm 9 gms. of iron, or 18 gms. of silver, or 28 gms.
of platinum or gold, or 31 gms. of lead one degree.

The distribution of large quantities of heat in buildings
by means of hot water is made possible because of the
large thermal capacity of this agent. " The vast influence
which the ocean must exert as a moderator of climate here
suggests itself. The heat of summer is stored up in the

50 HEAT.

ocean, and slowly given out during winter. This is one
cause of the absence of extremes in an island climate."

Water exhibits a- marked peculiarity in the variation of
its specific heat with temperature. The formula of Reg-
nault, which is still often quoted, indicates a gradual in-
crease of specific heat as the temperature rises from the
freezing to the boiling point. But the experiments of
Rowland, in his exhaustive investigation of the dynamical
equivalent of heat, were the first ones of sufficient accuracy
to show that the specific heat of water first decreases from
0° to about 30° C, and then a gradual increase begins.
Rowland's conclusion has been confirmed by Griffiths and
by Bartoli and Stracciati, who found a minimum value
for the specific heat of water at 20° C. The precise posi-
tion of this minimum is difficult of determination, since
the change in the specific heat near this point is very
minute.

33. Atomic Heat of Simple Bodies (P., 256 ; S., 313).
— In 1819 Dulong and Petit made experiments on simple
substances to determine whether their specific heats could
be connected by any simple law. From an examination of
the specific heats of such substances as iron, lead, gold,
silver, etc., these physicists concluded that the atoms of all
simple substances have the same thermal capacity. The
number of atoms of simple substances in the same mass
is inversely as the atomic weight. If therefore the thermal
capacity of the atom is the same, the specific heat must be
inversely proportional to the atomic weights, " or the prod-
uct of the specific heat by the atomic weight is the same
for all the elementary substances."

This law has been found to hold approximately true for
most of the elements which occur in the solid state at

MEASUREMENT OF THE QUANTITY OF HEAT. 51

ordinary temperatures, if the specific heats be taken at
temperatures sufficiently below the point of fusion. For
thirty-two of these substances the mean product is 6.38 and
the extremes are 6.76 and 5.7. The atomic weight of
hydrogen is the unit.

Since the specific heats of solids are not constant, but
vary with the temperature and the physical state, it is to
be expected that the product of the atomic weights and
the specific heats will exhibit a similar variation from
constancy. .

34. Specific Heat of Gases. — The specific heat of
a gas may be measured in two different ways. It may
be measured under the condition of a constant pressure or
of a constant volume. The former is called the specific
heat under constant pressure and the latter the specific heat
at constant volume. The two are by no means the same.
In the latter all the heat applied goes to increase the mo-
lecular kinetic energy, while in the former the gas does
work in expanding by heat under a constant pressure ; and
heat must be supplied not only to increase the kinetic
energy of the molecules to the same extent as when the
volume is kept constant, but in addition enough to do the
external work. The specific heat under constant pressure
is therefore greater than the specific heat at constant
volume. The ratio of the one to the other for air is about
1.41. The importance of this ratio, to which reference has
already been made in Sound (I., 120), will be discussed in
a later chapter.

It has been found very difficult to measure the specific
heat of gases at a constant volume, and till quite recently
the difficulties have not been surmounted.
> The specific heat of a gas under a constant pressure has

52 HEAT.

been determined by conducting the dry gas at a uniform
flow and constant pressure through two spirals. In the
first it is heated to a known temperature, and in the latter
it is cooled to the temperature of the bath. The heat given
up in the second spiral, or series of chambers, is determined
by measuring the rise of temperature of a known mass of
water, or by passing the gas through till the temperature
becomes stationary, when the heat gained from the gas
equals the heat lost by radiation. The mass of gas flowing
through is determined by measuring the change of pressure
taking place in the known constant volume of the gas-
holder. The experimental difficulties are largely due to
the small density of gases, so that a large volume must be
passed through the calorimeter to produce a measurable
change of temperature. This requires time, and the errors
due to conduction and radiation are greatly augmented.

The following are Regnault's conclusions respecting the
specific heat of gases under constant pressure :

1. The specific heat of all approximately perfect gases,
like air, does not vary with the temperature.

2. The thermal capacity of a given mass of such a gas
does not vary with its pressure ; and therefore the thermal
capacity of a given volume of such a gas is proportional to
its density.

3. The thermal capacity of equal volumes of the simple
gases which are not easily condensible are equal. This
equality does not hold for easily condensible gases.

4. The specific heat of easily condensible gases increases
with the temperature, like that of solids and liquids.

The specific heat of air is sensibly constant for all tem-
peratures between — 30° and 225° C, and under pressures
from 1 to 10 atmospheres. The specific heat of carbon
dioxide is about doubled at 2,000° C.

MEASUREMENT OF THE QUANTITY OF HEAT. 53

The table is from Regnault's results.

Specific Heat op Simple Gases.

Hydrogen 3.4090 Oxygen 0.2175

Nitrogen 0.2438 Chlorine 0.1210

Air 0.2374 Bromine 0.0555

Specific Heat of Compound Gases.

Ammonia 0.5084 Carbon dioxide . . . 0.2169

Carbon monoxide . . 0.2450 Hydrochloric acid . . 0.1852

Hydrogen sulphide . . 0.2432 Sulphur dioxide . . . 0.1544

PROBLEMS.

1. If 3 kilos, of iron (specific heat, 0.11) at 95° C. are put into
3 litres of water at 10° C, what will' be the rise of temperature of
the water ?

2. The specific heat of mercury h ^5th. If 10 kilos, of mercury
be cooled from 100° to 25° C. in 1 kilo, of water, at what tempera-
ture was the water before the addition of the mercury ?

3. A mass of 500 gins, of copper at 98° C. put into 500 gms. of
water at 0° C, contained in a copper vessel weighing 150 gms.,
raises the temperature of the water to 8°. 3 C. Find the specific heat
of copper.

4. If 20 gms. of iron at 98° C. (specific heat, 0.11) are immersed
in 80 gms. of water at 10° C, contained in a copper vessel whose
mass is 15 gms., find the resulting temperature, the specific heat of
copper being 0.095.

5. 250 gms. of turpentine, enclosed in a copper vessel whose
mass is 25 gms., are heated to 100° C. and immersed in 589 gms. of
water at 13° C. in a copper calorimeter weighing 110 gms. The tem-
perature rises to 27.5° C. Assuming the specific heat of copper to be
0.1, find that of turpentine (Glazebrook's Heat). .

64 11EAT.

CHAPTER V.

FUSION.

35. The Fusing Point. — When heat is applied to a
crystalline solid, its temperature rises till it reaches the
point where it begins to pass into the liquid form. The
temperature then remains sensibly constant till the entire
mass has fused or melted, when with continued application
of heat it rises again. Conversely when the temperature
falls, a stationary point is again reached where the crystal-
lization or solidification sets in, and the body continues to
give up heat while the temperature remains fixed. Under
the same conditions of pressure the two stationary tem-
peratures coincide, and this point is called the normal
fusing point of the substance under the given conditions.
Above this temperature the substance will be in the liquid
state, and below it in the solid state.

This temperature is called the normal fusing point be-
cause under different conditions the fusing point may be
different. Thus ice melts normally at 0° C, but under
pressure it melts at a lower temperature, and water may be
cooled several degrees below zero before it freezes. Other
substances present similar abnormal features, and the liquid
state may persist at a temperature considerably below the
normal point of solidification.

The melting point of ice is sharply marked, and there
is no appreciable difference of temperature between the

FUSION. 55

melting ice and the water into which it passes. This is gen-
erally true of crystalline substances, but the case is very
different with amorphous solids, like wax, glass, and iron,
which cannot be said to have a definite melting point.
Such substances soften and become plastic before reaching
a more or less viscous liquid state. It is because of this
property that glass can be bent, moulded, drawn out into
rods and tubes, or blown into various forms. Similarly the
softening of wrought iron at a temperature far below the
liquefying point permits the metal to be rolled, forged,
and welded. In the fusion of wax the outer portions are
softer than the interior and presumably at a higher tem-
perature. The experiments of Person go to show that ice
begins to soften and to increase in specific heat between
— 2° and 0° C, and that there is a certain very small range
of temperature within which ice softens and melts. The
difference between it and wax is then one of degree. Ice
represents one extreme of this transition state, and wrought
iron perhaps the other.

In general, however, crystalline bodies have a definite
fusing point, or a temperature at which they may exist
either as a solid or a liquid ; while amorphous bodies pass
gradually from the solid to the liquid state.

36. Condition of Instability (P., 270). — A liquid
which has a definite point of solidification, or whose pas-
sage from the solid to the liquid state is abrupt, may be
slowly and carefully cooled several degrees below the
normal freezing point without solidifying. This condition
is an unstable one, and if the under-cooled liquid be jarred,
or if a solid fragment of the same substance be dropped
into it, solidification will at once set in, with the disengage-
ment of heat. The temperature then rises to the normal
freezing point.

56 HE A T.

Fahrenheit observed that water sealed in a glass bulb
remained liquid at a temperature below freezing, but on
breaking off the stem rapid solidification followed. Gay-
Lussac found that water placed in a small vessel and
covered with oil remained liquid down to —12° C. Depretz
cooled water down to — 20° C. in fine capillary tubes, and
Dufour obtained a like result by suspending small drops of
water in a liquid of the same density, with which it would
not mix.

On the other hand, the surface of very still water freezes
sooner than one which is disturbed by the wind. A
running stream freezes less readily than a placid one.
There is no evidence, however, that the temperature of
running water is ever below 0° C. The surface layers of
still water cool down to the freezing point by rapid radi-
ation, while the poor conductivity of water (64) prevents
the replenishing of the heat from below.

This property of under-cooling is not peculiar to water.
It has been observed also in the case of phosphorus. If an
over-saturated solution of sodium sulphate, prepared by
dissolving the salt in hot water, be placed in a clean flask,
it will remain liquid on cooling if undisturbed. But a
slight jar, or the introduction of a small crystal of the salt,
will start the solidification. When the unstable equilibrium
is disturbed crystallization proceeds rapidly with a rise of
temperature. The potential energy of the unstable liquid
mass is converted into heat.

37. Change of Volume during Fusion. — In passing
from the liquid to the solid state bodies undergo a change of
volume. In most cases the volume diminishes. Ice, bismuth,
type metal, and cast iron are among the exceptions. Cast
iron and type metal expand on solidifying, and this expansion

FUSION.

causes them to fill every little line and crevice of the mould.
The powerful expansion of ice is attested by the bursting
of water-pipes and the rending of rocks by frost. If a
short piece of gas-pipe, with a screw cap fitted to each end?
be completely filled with water and placed in a freezing
mixture, it will burst with a loud report when the water
congeals. '

Major Williams at Quebec filled a 12-inch shell with
water and closed it with a wooden plug driven in with a
mallet. When the shell was exposed in the air at —28° C.
the stopper was
projected to a
distance of 300
feet, and a cylin-
der of ice about
8 inches in
length pro-
truded from the
hole. Probably
some of the
water remained
liquid till actu-
ally relieved of pressure by the giving way of the wooden
plug. The time required for the water to follow the
plug a distance of 8 inches was the interval from the
liquid to the solid state.

The change of volume of ice has been followed by Erman
from the solid to the liquid state by enclosing it in a large
bulb like a thermometer and taking readings on the long
stem. The continuous change in volume is represented in
Fig. 15, where AB represents the expansion of ice. At
0° C. there is a rapid diminution in rolume, which con-
tinues after the whole mass is liquefied, but at a reduced

Fig. 15.

58 HEAT.

rate, up to 4° C, the temperature of the maximum density
of water. Beyond this point the liquid at first dilates
rapidly along BE, and then the uniform expansion of the
liquid sets in along the line EF. The slope of the line AB
is greater than that of EF, or ice expands by heat more
rapidly than water. It is probable from the later experi-
ments of Kopp that the change in Volume at zero is much
more abrupt than that found by Erman.

In the case of phosphorus the dilatation in the solid state
is less rapid than in the liquid; while for a fusible alloy,
consisting of one part of tin, one of lead, and two of bis-
muth, the coefficient of expansion of the solid is the same
as that of the liquid.

Numerous examples of substances investigated go to
show that there is generally an anomalous dilatation at
the fusing point, but that the curve connecting volume
and temperature is probably in all cases continuous.

38. Influence of Pressure on the Melting Point (P.,
275; T., 119). — That the melting point is affected by
pressure was deduced from theory by James Thomson in
1849. His conclusions were verified by his brother, Lord
Kelvin, in the same year. The theoretical conclusion was
as follows : Bodies which contract on melting have their
melting points lowered by pressure, while those which
expand have their melting points raised by the same means.
Such a result might have been anticipated from the simple
consideration that if a substance like water expands on
freezing, any pressure which prevents this expansion at the
same time prevents congelation. But if the substance con-
tracts on solidifying, then increase of pressure is favorable
to this change of state.

Thomson calculated that the freezing point of water

FUSION. 59

should be lowered by 0.0075 of a degree Centigrade for an
increase of pressure of one atmosphere ; and the experi-
ments of Dewar, later than those of Lord Kelvin, show a
mean reduction of 0.0072 degree Centigrade per atmos. up
to 700 atmos. Mousson by enormous pressure lowered the
freezing point of water to — 20° C.

A rough numerical statement of the result is that under
a pressure of one ton per square inch, or of 144,600 gms.
per square cm., ice melts one degree Centigrade below its
normal melting-point.

The converse conclusion has also been verified by Bun-
sen, who found that paraffin wax, which melted at 463.3 C.
under atmospheric pressure, melted at 49°.9 C. when the
pressure was raised to 100 atmos.

Lord Kelvin has shown that the rigidity of the earth is
greater than if it were composed of glass. This conclusion
is derived from the phenomena of the tides, which show
that the earth does not yield appreciably to the forces
which raise them. The great rigidity, and therefore the
solidity, of the earth can be accounted for if it be assumed
that the materials composing the earth have their fusing
point raised by pressure. It has been ascertained that this
is true of ordinary lava.

39. Regelation. — The phenomenon of the re-freezing
of water from ice melted by pressure, when the pressure is
relieved, is called regelation. It was first noticed by Fara-
day. Familiar illustrations of it are the hardening of a
snowball under the pressure of the hands, and the passage
of snow into compact ice in a roadway, where it is com-
pressed by vehicles and the hoofs of horses. Frozen foot-
forms may often be seen to persist in compact ice after the
loose snow has melted, and the bottom of a snowbank is
not infrequently compressed into clear ice.

60 HEAT.

Unless the pressure is very great, this solidification occurs
only when the snow is soft or near the melting point. The
pressure applied then reduces the freezing point and melts
those portions of the snow that are subjected to stress,
while the water again freezes when the pressure is removed.
If two pieces of ice at 0° C. be firmly pressed together, they
will adhere by freezing after the pressure is relieved.
This may be done even under the surface of warm water.
If there is a small range of temperature within which lique-
faction takes place, as Person supposed, then the interior of
a lump of ice is at a slightly lower temperature than the
surface ; and when two such surfaces are pressed together,
even lightly, they are brought sufficiently near together to
give them the temperature of the interior of the block, and
as soon as the stress is removed the intervening film of
water freezes.

Bottomley's experiment on regelation is instructive.
Support a stout bar of ice horizontally by wooden supports
at the two ends, and hang on it a weight by means of a
copper wire passed over the ice at the middle. The press-
ure will melt the ice under the wire, and the water passing
around it and relieved of the stress will freeze. In this
way the wire will cut its way through the ice and the
weight will fall, but the bar of ice will remain intact,
though the track of the wire through it remains visible.
It is well to put a non-conducting link between the weight
and the wire to prevent the flow of heat upward from the
weight.

When a large body of ice melts in the spring, it will
sometimes be found to have a columnar structure consisting
of long slender prisms standing vertically. These can be
readily detached a foot in length without making more
than a small hole through the weak ice. It seems not

FUSION. 61

improbable that this peculiar structure has been caused by
lateral pressure and incipient melting.

Regelation has been invoked to explain the motion of a
glacier down its uneven, tortuous channel. A glacier
makes its way down its course by very irregular move-
ments. Ice is undoubtedly to some extent plastic, but it
is quite probable that regelation plays an important r61e in
glacial motion. The ice melts where it is subjected to the
pressure of enormous masses above it. This relief by press-
ure at many points permits the ice to accommodate itself
to changes in the channel, and a slow ice-flow is permitted.
As soon as the pressure is relieved at any surface the water
again freezes. The motion thus takes place by alternate
melting and freezing. The middle of the ice stream moves
faster than the sides because the weight there is greater
and the consequent melting more extensive.

40. The Latent Heat of Fusion. — The nearly sta-
tionary temperature maintained by a solid during its
passage into the liquid state has already been described.
The heat that fuses a crystalline solid does not sensibly
raise its temperature. In the language of the caloric theory
it becomes latent or concealed. The term latent heat has
been retained in the modern theory of heat, but we now
know that the heat which disappears during fusion ceases
to be heat, and is the energy expended or converted into
the potential form in the work of giving mobility to the
molecules. .

The manner of measuring the heat of fusion may be
illustrated by the method of mixtures. If two kilogrammes
of water, one at 0° C. and the other at 100° C, be mixed,
the result will be two kilogrammes at a mean temperature
of 50° C. But if a kilogramme of water at 100° C. be

62 BEAT.

mixed with one of ice at the freezing point, the ice will
melt and there will be two kilogrammes of water at about
10°.4 C. The heat lost by the hot water is 1,000 x 89.6
calories. A portion of this, viz., 1,000 x 10.4 calories, has
been employed to raise the ice-cold water from 0° to 10°.4
C, but the remainder has disappeared in the melting of the
ice. Therefore to melt 1,000 gms. of ice, 1,000 (89.6 - 10.4)
equals 1,000 x 79.2 calories of heat are required. This
is equivalent to 79.2 calories per gm. of ice. In an actual
experiment of this kind, the water equivalent of the calo-
rimeter must be taken into account. It is here supposed
to be included in the kilogramme of hot water. Experi-
ments of this kind have demonstrated that for every unit
of ice melted about 79.2 units of heat disappear.

The latent heat of fusion is denned as the number of
calories required to convert one gramme of a substance
from the solid to the liquid state without change of
temperature.

Let mx be the mass and tx the temperature of the water
and the calorimeter ; also let m be the water equivalent of
the calorimeter, and let m., be the mass of the ice whose
heat of fusion I is to be found. If the resulting tempera-
ture of the mixture is t° C, then the heat lost by the cal-
orimeter and its contents may be equated to the heat of
fusion of the ice and its gain of heat in rising from zero
to t° C, or

(m + Mi) (£i — f) = lm-2 + m2t.

Whence I = O + O Q. ~ 0_ u

The correction for radiation may be avoided by Rumford's
method.

The most probable value of the heat of fusion is 79.25,
though Person found 80.02 and Bunsen 80.03, the mean

FUSION. 63

specific heat of water between 0° and 100° C. being taken
by Bunsen as the unit.

41. Heat absorbed in Solution (S., 94). — We have
seen that when a solid is changed to the liquid form, heat
is absorbed. If the liquefaction is accomplished by solu-
tion in a proper solvent without chemical action, heat is
still required to give mobility to the molecules, and the
temperature of the solution falls. This effect is often
masked by the generation of heat by chemical action
between the solid and the solvent. If a delicate thermo-
scope be used, such as a thermopile and a galvanometer
(see " Thermal Electricity "), the heat absorbed by the solu-
tion of sugar in water may be readily detected. A still
larger effect is produced by dissolving common salt, while
quite a notable reduction of temperature is produced by
the dissolving of nitrate of sodium. When glacial acetic
acid is dissolved in water, the absorption of heat necessary
to increase the fluidity exceeds the evolution of heat by
chemical action.

Freezing mixtures are based on the absorption of heat
necessary to give fluidity. Salt water freezes at a lower
temperature than pure water. Hence, when salt and snow
or pounded ice are mixed, both of them become fluid and
absorb heat in the transition from one state to the other.
By this means a temperature of — 22J C. may be obtained.

Many other chlorides, as well as some nitrates, form
freezing mixtures with snow or ice. Among them, in the
order of effectiveness, are the chlorides of calcium, copper,
strontium, ammonium, potassium, and barium.

PROBLEMS.
1. Into a mass of water at 0° C. are introduced 100 gms. of
ice at — 12° C. ; 7.5 gms. of ice are frozen and the temperature of

64 HEAT.

all the ice is raised to 0° C. If the latent heat of fusion is 80, find
the specific heat of ice.

2. How much ice at 0° C. will be melted by 30 gms. of copper
(specific heat, 0.095) at 200° C. ?

3. A mass of 100 gms. of platinum (specific heat, 0.0355) is
heated in a furnace and is then dropped into 200 gms. of water at
0° C. ; the temperature of the water rises to 26° C. What was the
temperature of the furnace ?

4. If a kilo, of copper at 100° C. be placed in a cavity in a
block of ice at 0° C, and if 119 gms. of ice are melted, find the heat
of fusion of ice.

5. 100 gms. of ice at — 20° C. were thrown into 1 kilo, of water
at 20° C. contained in a copper vessel weighing 100 gms. When the
ice was melted the temperature of the water was 10°. 15 C. Find
the latent heat of fusion of ice.

VAPORIZATION. 65

CHAPTER VI.

VAPORIZATION.

42. Four Varieties of Vaporization. — The passage
of a substance into the state of a gas or a vapor is called
vaporization. There are four distinct types depending
upon the conditions under which the process goes on:

1. Evaporation, where a liquid is converted into a gas
quietly at a relatively low temperature and without the
formation of bubbles.

2. Ebullition, or boiling, a rapid evaporation at a
higher thermal equilibrium, when bubbles of gas form in
the mass of the liquid.

3. The Spheroidal State, where quiet vaporization, at a
rate between evaporation and ebullition, goes on when the
liquid is in apparent contact with a body of relatively high
temperature.

4. Sublimation, in which a solid passes directly into
the gaseous form without going through the intermediate
state.

Whether the gaseous condition is reached by one of
these processes or another, heat is always absorbed in
considerable quantity, although the vapor is at the same
temperature as the solid or liquid from which it comes ;
we have therefore the expression, "latent heat of vapori-
zation." The latent heat of gases is greater than that of
liquids ; this fact prevents a disastrously sudden conversion

66 HEAT.

either from the liquid to the gaseous state without chemical
change, or the reverse condensation of vapors to liquids,
since the heat involved in either operation must be supplied
for evaporation, or must be disposed of when generated by
condensation (44).

43. Evaporation in a Closed Space. — In a solid the
molecules are free to vibrate about fixed positions of equilib-
rium, but have no motion of translation. In a liquid the
conditions of molecular freedom are much more extended.
A molecule is so far released from rigid cohesion that it
may make its way throughout the entire mass; but its
progress is slow because most of its time is spent in en-
counters with other molecules, of which it is never inde-
pendent. There is practically no free path to molecular
motion in liquids, and the migratory track of any mole-
cule depends upon its innumerable chance encounters with
other molecules.

In the interior of a liquid mass a molecule is equally
obstructed in its movements in all directions ; but at the
surface the resultant molecular attraction is normal, and
there results the phenomenon of a surface film, called sur-
face tension (I., 93).

Whenever a molecule at the surface of the liquid has a
normal component of motion sufficient to carry it through
the surface film, it may escape from bondage and wander
about in free space. It is then independent of its fellows,
except for numerous collisions with them, which determine
all the properties of the gaseous state, without, however,
absorbing a large portion of time as compared with that of
the free motion of the molecule. Such molecules consti-
tute the vapor or the gaseous form of the substance, and
the process of entering this state is called evaporation.

VA PO RIZA TION. 67

If evaporation takes place in a closed space, then the free
molecules may again come within the range of molecular
action at the liquid surface, and may be again entangled
and return into the liquid. This process is called con-
densation. When the number of molecules making their
escape equals the number returning through the surface
film, there is an equilibrium between the loss and the gain,
and at this stage the evaporation is said to cease. This
vapor in contact with its liquid is then said to be saturated.
Its density will remain unchanged unless there is a change
of temperature. An elevation- of the temperature causes
more of the liquid to assume the form of a vapor. If the
volume of the saturated vapor is diminished without change
of temperature, some of the vapor will condense to a
liquid ; and if the volume is increased, more of the liquid
will evaporate so as to maintain the same vapor density.

Dalton concluded that the presence of inert gases, like
air, has no influence on the final density of the vapor ; that
its only effect is to increase the time required to reach the
equilibrium between evaporation and condensation. But
Regnault has shown that the maximum pressure of the
saturated vapor of water, ether, and some other substances,
is slightly diminished when air is present.

The maximum pressures of aqueous vapor in millime-
tres of mercury are given in the Appendix, Table III.

44. Ebullition. — Each molecule carries away heat in
evaporation represented by the additional potential energy
which it gains in entering the gaseous state. If only a
moderate amount of heat is applied, the evaporation is con-
fined to the surface, and it increases till the rate at which
heat is supplied equals the rate of loss by evaporation.
When the evaporation takes place into open space, the

68 HEAT.

molecules escaping from the surface may never return to
the liquid. There is then no saturated vapor, and the
evaporation continues so long as the heat is supplied and
any liquid remains. But if the surface is limited in area
and the heat supply is in excess, this equilibrium of quiet
evaporation cannot be established. The temperature rises
till bubbles of vapor begin to form in the interior of the
liquid, or at points on the inner surface of the containing
vessel. • If the vapor pressure is not sufficient to support
them as they rise, they collapse and produce the familiar
sound of " simmering." With a slightly higher temperature
they rise to the surface, expanding in the ascent under re-
duced hydrostatic pressure ; and if the evaporation into
them from their enlarged surface is sufficiently rapid, they
burst through the surface film and escape. This process of
rapid evaporation from the interior, as well as at the
surface, is called ebullition or boiling. An equilibrium is
thus established at a higher temperature than the preced-
ing, and this temperature is called the boiling point of
the liquid. It is constant for the same pressure.

If the heat be supplied at a still more rapid rate the tem-
perature of the liquid does not rise higher, but the boiling
is more violent. So long as the pressure remains the same,
ebullition goes on at any rate of heat supply in excess of
the rate at which silent evaporation at the surface can dis-
pose of it. The vaporization is then no longer confined to
the free surface, but takes place in the interior into small
bubbles initiated by expanding air, disengaged from the
liquid by heat, or by other bodies which are very active in
separating vapor from the heated liquid.

When the air has been all boiled out and the containing
vessel is clean, the temperature of water may rise several
degrees above the normal boiling point. Ebullition then

VAPORIZATION.

sets in with almost explosive violence, and proceeds till the
excess of heat, due to the elevation of temperature above
the normal boiling point, is disposed of. This abnormally
high boiling point of air-free water probably accounts for
many explosions of stationary boilers at the moment when
steam is first drawn from them after fresh firing. As a
measure of precaution, a fresh supply of water containing
air should be pumped in before the temperature rises to the
boiling point.

The boiling point is the temperature at which the liquid
boils or gives off bubbles of its own vapor. It has been
found by experiment that at the boiling point the saturated
vapor is given off at a pressure equal to that sustained by the
surface of the liquid.

45. Effect of Pressure on the Boiling Point (T., 135).
— The work done by
heat in ebullition is
partly internal and
partly external. The
internal work consists
in separating the mole-
cules beyond the range
of molecular attraction.
The external work de-
pends upon the fact that
the liberated vapor is
formed under pressure.
The work done is meas-
ured by the volume of
vapor formed multi- I

plied by the pressure on ^^w^— e

it per unit area. Since Fig. to.

HEAT.

this external work is diminished by diminishing the pres-
sure, the lower the pressure the lower the boiling point.

Under diminished pressure water boils at a reduced
temperature. A familiar form of experiment to demon-
strate this fact consists in boiling water in an open flask
till the air is nearly all expelled by the steam. The flask
is then tightly corked and inverted (Fig. 16). The boiling

ceases, but is renewed by
applying cold water to
the flask. The cold water
condenses the vapor and
reduces the pressure with-
in the flask so that the
boiling begins again. If
the air has been thorough-
ly expelled, the water may
be kept boiling till the
temperature has fallen to
that of the air of the
room.

A convenient modifica-
tion of this experiment
consists in fitting into the

Fig. 17. o

flask a rubber stopper
traversed by a small glass tube, so that the flask is air-tight
except through the tube. The tube should be bent twice
at right angles in order that the outer end may dip down
below the surface of cold water in a beaker (Fig. 17).
Boil the water in the flask till the air is expelled, and then
dip the open end of the tube under water, at the same time
removing the lamp. If the apparatus is air-tight the cold
water will rise in the tube as the flask cools, and will at
length pass the bend and pour into the flask in a stream.

VAPORIZATION.

The cold water condenses the vapor and causes violent
ebullition, which will continue, though the water all the
time becomes cooler. The only precaution to be observed
is to make sure that the air shall enter the tube before the

SSao

£
J
^

150° 300°

Temperatures C.

Fig. 18.

flask is filled; otherwise the shock due to the sudden
stopping of the stream when the flask is full will break it.
Beyond the normal boiling point the pressure of saturated
vapor rises rapidly with the temperature. The rise of
temperature from 100° to 180° C. increases the pressure of
water vapor from one to nearly ten atmospheres, and an

72 HEAT.

additional rise of 40° C. raises the pressure to about 23
atmospheres.

The relation between temperature and the vapor press-
ure of water is represented by the curve of Fig. 18.
Above 150° C. this curve rises very rapidly as the tem-
perature increases.

46. The Spheroidal State. — When a drop of water is
placed on a clean hot stove it will often take a flattened
globular form and roll around with rapid but silent
evaporation. This phenomenon is known as the spheroidal
state. It may be beautifully exhibited by heating a small
flat platinum dish red hot over a Bunsen burner, and care-
fully placing in it a large globule of water by means of a
pipette. It will not boil, but will assume the spheroidal
state. The globule is not in contact with the hot metal,
but rests on a cushion of its own vapor, which escapes
rhythmically from its edge and often throws it into beauti-
ful undulations. If the lamp be removed the temperature
will fall till a point is reached at which the drop comes into
contact with the hot metal, when violent ebullition will
take place.

If the drop is not too large, light may be projected
through between it and the hot metallic surface, thus
demonstrating that the drop is not in contact with the
metal.

Boutigny, by placing a small thermometer in the drop
of water, found that its temperature remained below the
boiling point ; and Berger afterwards found that in a
large globule the temperature varied from 96° or 98° C.
near the bottom to about 90° C. at the upper surface.
Budde has shown that under the exhausted receiver of
an air-pump the spheroidal state of water may persist at

VAPORIZATION. i 73

temperatures as low as 80° or 90° C. The vapor pressure
under the drop is then only what is required to support the
drop itself, the air pressure having been removed.

Water is not the only substance that assumes the
spheroidal form. The temperature of spheroidal sulphur
dioxide is low enough to freeze a drop of water placed in
it. This may happen in a red-hot crucible because the
sulphur dioxide in the spheroidal state is below its boiling
point, and this is below the freezing point of water.

Solid carbon dioxide may be touched with the hand or
even the tongue without danger if no pressure is applied,
because it is kept out of contact by an intervening film of
the substance in the gaseous form. Faraday succeeded in
freezing mercury in a mixture of ether and solid carbon
dioxide contained in a red-hot crucible. The contents of
the crucible were cushioned on their vapor in a state
analogous to the spheroidal form.

Quite a remarkable example of the spheroidal state,
because it takes place at a low temperature, is exhibited by
liquid oxygen on water. The liquid oxygen boils gently
at about —180° C, and when placed on water it imme-
diately exhibits all the aspects of a globule of water on
a hot plate. The water is at a high temperature relative
to the oxygen. So much heat, however, is abstracted from
the water by the evaporation of the liquid oxygen that the
spheroidal globule soon encases itself in an envelope of ice,
with only a small blowhole for the escape of the gas.

47. Sublimation. — The usual course from the solid to
the gaseous state is through that of a liquid. But a num-
ber of solids slowly waste away by evaporation without
liquefying. Ice and snow at temperatures below freezing
gradually lose in volume by evaporation. So carbon dioxide

74 HEAT.

snow when exposed in the air wastes away by evaporation,
and can be liquefied only with difficulty in an open tube.
A solid brick of it will remain unmelted for many hours
even in warm weather. It evaporates only so fast as it
gets heat to do the work of evaporation.

Other substances, such as camphor and ammonium car-
bonate, sublime at ordinary temperatures. Iodine, ammo-
nium chloride, and arsenic sublime when heated under
atmospheric pressure. But if the pressure is increased
arsenic may be fused ; and below a certain critical pressure
for each, ice, mercuric chloride, and camphor do not melt,
but pass directly into the gaseous state.

If by reduction of pressure the boiling point of a liquid
is reduced to the fusing point of its solid, then the solid
may pass directly into the gaseous state. A solid will,
therefore, sublime when the pressure upon it is less than
the vapor pressure of its saturated vapor at the tempera-
ture of fusion. The vapor pressure of carbon dioxide at
its fusing point of — 65° C. is three atmospheres ; under a
lower pressure than three atmospheres it therefore sub-
limes. If the pressure of mercuric chloride is below 420
mms. it must evaporate without liquefaction. The same
is true of iodine at pressures under 90 mms., and of ice
under 4.6 mms. of mercury.

48. Latent Heat of Vaporization (P., 304). — The
latent heat of vaporization is the quantity of heat required
to convert one gramme of the liquid into vapor without
change of temperature. The temperature at which the
vaporization takes place is often understood to be the
boiling point of the liquid under a pressure of one stand-
ard atmosphere. The investigations of Regnault on the
latent heat of steam enable us to express the latent heat of

VAPORIZATION. 75

vaporization, for water vapor at least, by one formula appli-
cable through a considerable range of temperature. By
the total heat of steam at any temperature Regnault meant
the amount of heat necessary, first, to raise one gramme of
water to that temperature without evaporation, and then
to convert it wholly into saturated vapor at the same tem-
perature. If L is the latent heat of vaporization, tx and t
the initial and final temperatures, and s the mean specific
heat between these temperatures, then the total number of
heat units required to convert a gramme of water at t°
into saturated vapor at t° is

H=L + s(t — t^.

Regnault's experiments were conducted under pressures
ranging from 0.22 to 13.625 atmospheres, and from 0° to
230° C. ; and between these limits he found that the total
heat was represented by the equation

ff= 606.5 + 0.305*.

Taking the mean specific heat of water to be unity and the
initial temperature zero, the formula for the latent heat at
any temperature t° C. becomes

L= H-t = 606.5 - 0.695*.

If t is 100° C. the latent heat of steam is therefore 537 ; that
is, 537 calories are required to convert one gramme of water
at 100° C into steam at the same temperature.

Taking into account the very small variation in the
specific heat of water, the latent heat of steam falls from
606.5 at 0° C. to 536.5 at 100°, and to 464.5 at 200° C.

Prior to Regnault's investigations it was generally ad-
mitted on insufficient evidence that the heat required to
change a gramme of water at 0° C. into steam was inde-
pendent of the pressure. If that were true, the sums of the

76 HEAT.

three pairs of numbers above, representing latent heats
and temperatures, should be approximately the same. On
the contrary they increase by nearly five per cent for every
100 degrees rise of temperature.

Andrews found the latent heat of evaporation of a few
common liquids boiling under atmospheric pressure to be
as follows :

Water 536.0

Alcohol . 202.4

Ether 90.5

Bromine . . . . . . . 45.6

49. Cold due to Evaporation. — If the heat required
is not supplied from some external source, the evaporation
of a liquid will be accompanied by a
lowering of its temperature. This fact
accounts for the coolness felt when ether,
alcohol, or benzine evaporates from the
hand. While their latent heat of evapo-
ration is smaller than that of steam, their
boiling points are lower and the rapid
evaporation absorbs much heat.

In Leslie's experiment a thin flat dish,
containing about 10 c.c. of water, is supported by a tripod
over a large shallow glass vessel containing strong sulphuric
acid, and the whole is placed under the receiver of an air-
pump (Fig. 19). The dish should be held in such a manner
that it cannot receive heat from below by conduction.

On exhausting the air rapidly the pressure is reduced till
the boiling point falls to the temperature of the water.
The water then begins to boil briskly ; and if the vapor is
removed rapidly, both by working the pump and by ab-
sorption by the acid, the pressure may be kept low enough
to keep the water boiling till it freezes. The conditions are

VAPORIZATION.

such as to produce rapid evaporation, while the heat re-
quired to do the internal work is drawn entirely from the
water itself and the thin dish. Not infrequently the
bubbles may be frozen before they burst.

Wollaston's cryophorus (Fig. 20) is also designed to
show the freezing of water by evaporation. It consists of
a bent tube with a bulb at each end. Before it is sealed
some water is introduced and boiled to expel all the air
from the tube, which is then sealed by fusing the glass.

In performing the experiment the water is all collected
in the upper bulb A, and the lower one
is imbedded in a freezing mixture. The
vapor condenses rapidly in B and forms at
the same rate in A. Heat is thus carried
by the vapor from A to B ; and as A parts
with its heat, if none is supplied to it, the
temperature of the water in it will fall to
the freezing point.

Much lower temperatures may be se-
cured by the rapid evaporation of liquids
which boil at a temperature below the
boiling point of water. Thus with liquid sulphur diox-
ide, which boils under atmospheric pressure at- —10° C,
mercury may be frozen. By the rapid evaporation of
liquid carbon dioxide Pictet obtained a temperature of
— 140° C. Liquid oxygen boiling in air reduces the tem-
perature to — 182° C. ; and by increasing the rate of evap-
oration by reducing the pressure, Dewar has reached a
temperature of — 200° C, or even lower.

This property of heat absorption by liquids that evap-
orate at a low temperature has been applied to the con-
struction of ice machines. Ammonia is first condensed by
pressure and cooling to a liquid with about one-tenth of

Fig. 20.

78 HEAT.

its weight of water. It is then evaporated under reduced
pressure secured by powerful pumps, and its temperature
falls low enough to freeze water in vessels about it or
within it. The process is made continuous by returning
the ammonia to a condensing chamber cooled with water.
It thus passes repeatedly through the same cycle of changes.

50. Relative Humidity. — The atmosphere always con-
tains aqueous vapor, whose pressure is the same as if it
alone were present. When its pressure at any temperature
of the air equals the saturation pressure for that tempera-
ture, it will condense on the surface of bodies, or fall as
rain or snow.

The humidity or dampness of the air does not depend
alone on the quantity of aqueous vapor present, but on the
nearness of the vapor pressure to the saturation point.
The saturation pressure at any temperature is the same as
that under which water boils at that temperature.

The saturation pressure rises rapidly with the tempera-
ture (Appendix, Table III.). Thus the maximum pressure
of aqueous vapor at 10° C. is 9.17 mms., while at 21° C. it
is 18.5 mms., or a little more than twice as great. There-
fore the quantity of aqueous vapor that would saturate the
air at the lower temperature would only half saturate it at
the higher. The air is said to be damp when it is nearly
saturated with vapor. Hence the heating of the atmos-
phere, while the quantity of aqueous vapor remains un-
altered, removes it further from the saturation point and
diminishes its dampness. When damp air from outdoors
passes through a hot-air furnace it becomes dry air, not
because it has lost any aqueous vapor, but because its
capacity to take up vapor of water has been increased by
the rise of temperature. The requisite saturation pressure

VAPORIZATION. 79

of the aqueous vapor is then much higher ; and hence the
necessity of adding more vapor of water to bring the air
of the rooms nearer to the saturation point. In winter the
humidity is usually greater than in summer, not because
the quantity of vapor present is greater, but because the
temperature is lower and the amount of vapor required to
produce saturation is less.

Humidity must therefore be expressed relatively as the
ratio of the pressure of the aqueous vapor present at a
given temperature to the saturation pressure at the same
temperature. This ratio can be measured by determining
the actual pressure of the aqueous vapor in the air and
comparing it with the maximum pressure at the same
temperature obtained from the tables. This is the method
applied by means of all dew-point instruments, called
hygrometers.

51. The Dew-Point. — If a mass of air containing
aqueous vapor be gradually cooled, a temperature will at
length be reached at which the vapor will begin to con-
dense. This temperature is called the dew-point. Con-
densation of aqueous vapor may be beautifully illustrated
by passing a beam of strong light through a large glass
receiver on an air-pump in a darkened room. If the air
be only moderately moist, a single stroke of the pump will
produce a thick cloud of precipitated vapor with splendid
iridescent diffraction effects. The expansion of the air
under pressure cools it below the dew-point, and the vapor
at once condenses as a visible cloud, consisting of water in
a state of fine division. Each minute mote of dust floating
in the air serves as a nucleus of condensation and acquires
a coating of liquid.

Aitken has shown that the presence of such particles of

80 HEAT.

dust is necessary to produce condensation of moisture,
and that a dustless atmosphere may be supersaturated
without the formation of a cloud.

52. Dew. — Any cool body lowers the temperature of
the air in contact with it ; and if the temperature is by this
means reduced to the dew-point, the cool body will become
covered with a film of water. Hoar frost is formed when
the temperature of deposition is below freezing. If the
reduction of temperature to the dew-point occurs in the
interior of a mass of air, the condensation results in rain
or snow ; but if it be in contact with bodies on the earth's
surface, the condensation takes the form of dew or frost,
according as the temperature of deposition is above or below
the freezing point.

The first correct explanation of the conditions attend-
ing the formation of dew was given by Wells. He ex-
plained the free deposition of dew on cloudless nights by
the uncompensated radiation of heat from the earth toward
a clear sky. Hence objects which readily lose heat by
radiation, particularly if their specific heat be low, receive
the largest deposit of dew. On cloudy nights the clouds
absorb heat and radiate it back to the earth, or return it
by reflection, so that the ground does not cool to the same
extent as when the sky is clear. ,

Another condition favoring a heavy dew is a quiet atmos-
phere. When the wind blows, the air in contact with any
body is replenished so rapidly that it has not time to be
chilled to the dew-point.

53. Regnault's Hygrometer. — The hygrometer is an
instrument for determining the relative humidity of the
atmosphere. The form devised by Regnault is considered
superior to all others.

VAPORIZATION.

It consists of two thin polished silver thimbles into
which are fitted glass tubes open at both ends (Fig. 21).
The tube A is half filled with sulphuric ether, and is closed
with a stopper through which pass a thermometer V and a
bent tube C extending down nearly to the bottom of the
silver thimble. The other tube contains only a ther-
mometer t. The two are connected
by means of the cross tube sup-
ported by the exhaust tube DJE,
which is connected to an aspirator.

To make an observation, the
air is drawn in through 0 by the
aspirator and bubbles up through
the ether, causing it to evaporate
rapidly. The temperature of A
is thus lowered; and when the
dew-point is reached, it is indicated
by a dimming of the silver tube A
in comparison with B, which re-
mains at the temperature of the
atmosphere, as indicated by the
thermometer t. The agitation of
the ether makes it certain that
the thermometer t' indicates the
correct temperature of A. The thermometer t' is read as
soon as the dimming is apparent. The aspiration is
stopped at the same time, and the temperature is again
read at the instant when the dew disappears. The obser-
vations are made with a telescope at a distance.

The temperature given by V is then the dew-point.
The corresponding vapor pressure for both temperatures
read on tf and t may then be taken from the table, and
their ratio is the relative humidity. For example, if the

Fig., 21.

HEAT.

dew-point were 7° and the temperature of the air 20° C,
the corresponding saturation pressures are 7.49 and 17.39
mms. respectively. The pressure of the aqueous vapor
present would be therefore 7.49, and the maximum possible
pressure at 20° C. is 17.39. Hence the relative humidity
7.49

would be

17.39

or 0.431.

Regnault's hygrometer may be
roughly imitated by using two test-
tubes (Fig. 22) and forcing air
through by means of a foot bellows.
The escaping vapor may be condensed
in a cooled flask further removed
from the apparatus than the figure
shows.1

54. Liquefaction of Gases. — Un-
der atmospheric pressure a number of
substances are known to us in both
the liquid and the gaseous states.
Water is liquid below 100° and a
vapor at higher temperatures. Alco-
hol is a liquid below 78° C. and a vapor above. Sulphuric
ether is a liquid below 35° C. and a vapor above. If we had
no means of obtaining temperatures below freezing, sulphur
dioxide would be known to us only as a gas at atmospheric
pressure, since it boils at — 10° C. In the cold of Arctic
regions it would always remain liquid, since under a press-
ure of one atmosphere it is always liquid below —10° C.

The two facts that some vapors condense to liquids by
lowering their temperatures, and that the boiling point of
a liquid is raised by pressure, suggest the combined appli-

i Wright's Heat, p. 201.

Fig. 22.

VAPORIZATION. 83

cation of cold and pressure to effect the liquefaction of
substances which are ordinarily known only in the gaseous
form. When the temperature of a substance in the form
of a gas is lowered by artificial means, and its boiling
point is raised by pressure, the two temperatures approach
each other; and if the two simultaneous processes are
carried far enough to make the two temperatures coincide,
liquefaction ensues.

Faraday was the first to liquefy chlorine, carbon dioxide,
cyanogen, and ammonia. His apparatus was of the simplest
character, consisting merely of a bent tube (Fig. 23) into
which the materials to produce the
gas could be placed and hermeti-
cally sealed. The pressure employed
was the pressure of the gas itself.
The shorter limb of the tube was
surrounded with a freezing mixture
for lowering the temperature.

When crystals of hydrate of chlo-
rine, made by passing chlorine gas

into water just above the freezing point, were heated in
the longer limb a, they decomposed and formed a greenish
liquid floating on a clear one. The lighter liquid distilled
over and condensed in the shorter arm b. When the
tube was opened, this condensed liquid was found to
be liquid chlorine.

Carbon dioxide was condensed to a liquid in a similar
way by heating sodium carbonate in the limb a. When
cyanide of mercury was placed in a and heated, cyanogen
was liberated and was liquefied in b. To liquefy ammonia
advantage was taken of the fact that chloride of silver
absorbs about 200 times its volume of this gas. Before
sealing either end of the tube, the longer limb was nearly

84 HEAT.

filled with dry precipitated silver chloride. Dry ammonia
was then passed through the tube, and when the air had
been expelled and the chloride was fully charged, both
ends were sealed. The end b was then placed in a freez-
ing mixture, and a Bunsen flame was carefully applied to
a. The silver chloride melts at 38°, and begins to part
with its ammonia at about 115° C. As the pressure of the
liberated ammonia rose, the gas was condensed to a clear,
highly refrangible liquid in b.

The pressure at which condensation took place was
determined by introducing into the experimental tube a
smaller one, open at one end, and containing air confined
by a small piston of mercury. The pressure was indicated
by the extent to which the air in the small tube was com-
pressed. In every case the pressure was observed to
increase up to the point where condensation began, and
after that it remained constant so long as the condensed
liquid was kept at the same temperature. This pressure
was that of the saturated vapor at the given temperature.

55. Continuity of the Liquid and Gaseous States
(P., 368; M., 119; S., 130). — If water or other liquids
be heated in a closed vessel, it is well known that the
pressure of the vapor rises very rapidly with the tempera-
ture (45). Steam formed at 100° C. has a density of only
ttW» while steam formed at 231° C. has a density of ^,
the maximum density of water being the unit. Hence not
very far above this latter temperature there will be no
difference in density between the steam and the water.
At such a temperature liquefaction will not be accom-
panied by condensation, and the usual distinctions between
water and steam vanish.

In 1822 Cagniard de la Tour heated water and other

VAPORIZATION. 85

liquids in closed tubes and observed that they appeared to
be converted into a gas occupying only from two to four
times the volume of the liquid. When a tube, about one-
fourth full of water, was slowly heated to 360° C, the
curvature of the surface gradually diminished and finally
all demarkation between the liquid and the vapor dis-
appeared. When the gas had cooled a little a thick cloud
suddenly made its appearance, and soon the surface of
separation between the liquid and the vapor was again
visible. De la Tour found the same phenomenon with
ether, alcohol, and bisulphide of carbon, but the tempera-
ture at which the liquid disappeared was different in each
case. This temperature has since been called the critical
temperature, and the corresponding pressure is the criti-
cal pressure. The inference is easy that above its critical
temperature a gas cannot be liquefied by any pressure,
however great.

This conclusion was fully justified by the extended
investigations of Dr. Andrews ' on the conditions of the
liquefaction of a gas, and especially of carbon dioxide,
for which he found a critical temperature of 30°.92 C.
If the pressure on this gas, when above this temperature,
be increased to 150 atmospheres, a steady decrease of
volume will be observed, but there will be no sudden
change of volume at any point. The temperature may
then be gradually lowered until the carbon dioxide has
reached the temperature of the air. It will then be found
to be a liquid. The substance has passed from the gaseous
state to the liquid state by imperceptible gradations and
without the sudden evolution of heat. Andrews concluded
that a gas and a liquid are only widely separated forms of
the same condition of matter, and that the passage from

» Phil. Trans., 1869, Part 2, p. 575.

86 HEAT.

one to the other may be made without breach of
continuity.

The following are the critical temperatures for several
substances :

Ether

196°.2 C

Acetone

246.1

Alcohol

258.6

Carbon bisulphide ....

276.1

Water

365

56. Distinction between a Gas and a Vapor. — The
discovery of Andrews permits us to distinguish between a
gas and a vapor. By a vapor is meant a substance in the
gaseous state at any temperature below the critical point.
A vapor can be reduced to a liquid by pressure alone, and
can therefore exist in contact with its own liquid. A gas,
on the other hand, cannot be liquefied by pressure alone,
but only by combined pressure and cooling. A gas is
the form which any liquid assumes above its critical tem-
perature. A substance can exist partly in the liquid and
partly in the vaporous state in contact only at tempera-
tures below the critical point. Thus, below 30°.92 C.
carbon dioxide may exist as a vapor, but above that tem-
perature it cannot be reduced to the liquid state and is a
gas.

Below the critical temperature the liquid and the vapor

of any substance may be readily distinguished ; above

that temperature they have not as yet been differentiated

by any decisive characteristics. They have apparently

v the same density and refrangibility, and their molecular

-> attractions are equalized to the extent that there is no

;V /surface tension. At the critical temperature the latent

heat of vaporization is reduced to zero.

VAPORIZATION.

57. Liquefaction of Oxygen and Nitrogen. — On
Dec. 24, 1877, two announcements were made to the Paris
Academy of Sciences by Cailletet and Pictet that they
had liquefied oxygen. It had previously resisted low tem-
peratures and enormous pressures because its low critical
temperature had not been reached.

Their plan of operations was to reduce the temperature
of carbon dioxide by the rapid evaporation of liquid
sulphur dioxide under reduced pressure secured by a
vacuum pump ; then to
carry the lowering of the
temperature one step fur-
ther by the similar rapid
evaporation of the cooled
liquid carbon dioxide. The
gas was carried back in
each case and again con-
densed by a compression
pump. The cycle of oper-
ations was thus complete.

Fig. 24 illustrates Pic-
tet's apparatus. The oxy-
gen was produced in the heavy iron retort L, and toward
the close of the decomposition of the potassium chlorate the
manometer indicated a pressure of 500 atmospheres in the
copper tube MN. IT and K are filled with carbon dioxide
and C and D with sulphur dioxide. The two double-acting
pumps, A and B, are coupled together so that A exhausts
the sulphur dioxide vapor from the cylinder C, and B
compresses it under a pressure of 3 atmospheres in the
receiver 2), where it is cooled by a stream of cold water.
From D it is returned by the small pipe d to C as a liquid.
Its rapid evaporation in C lowers the temperature of the

88 HEAT.

liquid to —65 or —70° C. The purpose of this operation
is to produce and maintain a sufficient quantity of liquid
carbon dioxide in H and K. The two pumps, E and F,
perform the same offices as A and B. As fast as E ex-
hausts the vapor from IT, F compresses it in K, where it
condenses under a pressure of from 4 to 7 atmospheres
on account of the low temperature produced by the evap-
oration of the liquid sulphur dioxide. The evaporation
of the carbon dioxide reduces the temperature to — 130° C.
At this stage the pressure of the manometer R sinks to
320 atmospheres, indicating that the oxygen begins to
liquefy. On opening the stop-cock N the liquid issues
with great violence as a white jet, and is further cooled
by the evaporation and expansion to such an extent that
some of it may be obtained in the liquid state.

Professor Dewar has more recently improved on the
older process by the employment of nitrous oxide and
ethelene in the two successive cycles. The chamber con-
taining the oxygen is protected by a heavy felt covering
and is surrounded by two tubular circuits, one traversed
by nitrous oxide and the other by ethelene. After the
two successive reductions of temperature by the evapora-
tion of first the one liquid and then the other, the cold
oxygen under pressure is allowed to rush out through a
stop-cock at the bottom of the chamber. It is received in
a flask and becomes in part liquid by the further cooling
due to the work done in pushing back the atmosphere to
make way for itself. It is mixed with some solid carbon
dioxide from which it is freed by filtering through an ordi-
nary filter paper. It has a delicate sky-blue color, and its
temperature when evaporating under atmospheric pressure
is —182° C. Nitrogen is liquefied by the same apparatus.
The advantage over the older method is in point of the
quantity of gas condensed.

VAPORIZATION. 89

The critical temperature of oxygen is about — 112° C,
and its critical pressure 50 atmospheres. The critical tem-
perature of nitrogen is — 145° C. ; that of hydrogen is still
lower. Thus gases which are condensed only with great
difficulty have very low critical temperatures, while sub-
stances ordinarily liquid have very high ones.

PROBLEMS.

1. If a mass of aqueous vapor occupies a volume of 500 c.c.
under a pressure of 5.9 mms. at 25° C, find the pressure when the
volume has been reduced to 200 c.c. ; also the volume at which
the vapor becomes saturated at the same temperature (Appendix,
Table III.).

2. A vessel is filled with a gas at 15° C. and a pressure of 100
mms. of mercury ; find the pressure at 100° C.

3. If 25 gms. of steam at the boiling point be passed into 500
gins, of ice-cold water, to what tempevature will the water be raised?
The latent heat of steam is 536.5.

4. A block of ice weighing 100 gms. is enveloped in steam at
100° C, and when the ice is all melted the water has a temperature
of 50° C. Assuming no loss of heat, how many gms. of steam have
been condensed ? The latent heat of water is 79.25.

5. How manj- calories are required to evaporate 100 gms. of ice-
cold water if the evaporation takes place under a pressure of 91.98
mms. ? (Appendix, Table III.).

6. How many calories are required to change 100 gms. of ice at
— 15° C. into steam at 150° C. ? (Art. 48).

90 HEAT.

CHAPTER VH.

TRANSMISSION OF HEAT.

58. Three Modes of Transmission. — The distribu-
tion of heat takes place by three distinct modes, which
are called conduction, convection, and radiation. By the
first method heat is transmitted from particle to particle
of a body, or from one body to another in contact with it,
by a slow process, which depends upon difference of tem-
perature between contiguous parts, and upon the nature
of the conducting substance.

In convection heat is taken up by matter, and is carried
with it in its motion. Convection is the- transfer of heat
from place to place by sensible masses of matter. In this
way buildings are heated by the circulation of hot water,
and heat is conveyed by hot air. It is chiefly in this way
that a uniform temperature in large masses of fluid is
established.

Heat is also distributed as radiant energy, which is
propagated by a wave-motion in the ether, and by the same
physical process as the one involved in the transmission of
light. It is by this method that heat and light are con-
veyed to us from the sun, or from a lamp or a fire. During
the transit the heat and light are both radiant energy, or
simply radiation. By the first two modes heat is dis-
tributed through the agency of matter ; while in the third
method the ether is the medium of propagation.

TBANSMISSION OF HEAT. 91

59. Conduction by Solids (T., 178; S., 268; P., 505;
G., 160). — Heat should not be confused with its effects.
The melting of iron, the boiling of water, the energetic
outrush of steam under pressure, and the leaping aloft of
flames are not heat, but the results of converting the
motion of heat into mechanical motion. Heat is the
energy of molecular motion ; and when the molecules of
a solid or a liquid are agitated by the motion of heat, they
are not free to oscillate without imparting motion to other
molecules. The slow transmission of the motion of heat
from molecule to molecule of ordinary matter is con-
duction.

If one end of an iron rod
be placed in a fire, the other
end will in course of time
become hot ; the heat travels
slowly along the rod from
particle to particle, and finally
appears at the distant end.
This mode of conveyance, by
which heat is transmitted

Fig. 25.

from the hotter to the colder

parts of a body, or from one body to another of lower
temperature, is called conduction. It tends to establish
equilibrium of temperatures.

Different substances possess this power of transmitting
heat in very different degrees. As a general rule metals
are the best conductors, while glass, wood, chalk, fire-clay,
gypsum, water, wool, and feathers are very poor conduc-
tors. If a cylinder be made one-half of wood and the
other half of brass, joined end to end, and if a piece of
thin writing-paper be wrapped tightly round it and a
flame be applied to the junction (Fig. 25), the paper round

92 HEAT.

the wood will soon be scorched, while round the brass it
will not be injured. The metal conducts away the heat so
rapidly that the paper remains below the temperature
of ignition ; but the sluggishness of the wood in the same
process of passing on the heat permits it to accumulate in
the paper.

A Norwegian cooking-stove is a box heavily lined with
felt, into which fits a metallic dish with a cover. The dish
is covered with a felt cushion. The materials to be cooked
are placed in the dish with water, which is first boiled for
a short time. The dish is then transferred to the box, and
is enclosed in it. The conductivity of the felt and the
imprisoned air is so poor that in three hours the tempera-
ture does not fall more than 10 or 15 degrees C, and the
cooking is completed without further application of heat.

Differences in the apparent temperature of bodies are
due to their different conductivities. If pieces of metal,
marble, wood, and woollen cloth in the same room be
touched with the hand, the metal will feel cold, the marble
less so, and the woollen cloth least so of all. The sensation
of coldness is due to the rapid withdrawal of heat from
the hand by the good conducting power of the metal and
the marble. In a similar way, if the temperature of these
objects were higher than that of the hand, the metal
would feel the warmest of the series, because the rate at
which heat would flow from it to the hand would be
greatest. For this reason we handle hot objects by inter-
posing a poor conductor, like flannel, between them and
the hand ; and ice is kept from melting by wrapping in
woollen cloth or embedding in sawdust.

60. The Experiment of Ingenhausz. — One of the
earliest methods of comparing the thermal conductivities of

TRANSMISSION OF HEAT. 93

metals was suggested by Franklin and executed by Ingen-
hausz over 100 years ago. A number of rods of the same
length and diameter were fitted into the side of a long
trough (Fig. 26). The external portions were thinly coated
with wax. Hot water or hot oil was then poured into the
trough, and the distances to which the wax was melted on
the several bars waSJtfflteasured after their temperatures had
attained a permanent state. The relative rates at which
the wax is melted at first on the several rods are not the
same as their relative conductivities for heat. The rods on
which the wax melts most rapidly are not necessarily the
ones on which the melting finally
proceeds the farthest. If all the
rods had the same conductivity, or
transmitted the same quantities of
heat in unit time, the temperatures
of the rods would then be inversely
as their densities and specific heats.
On prolonged immersion the rods Fi 26

reach a permanent state, and all

the heat entering them by conduction leaves them by
convection and radiation. The rate of flow must be dis-
tinguished from the rate of rise of temperature. The
wave of temperature travels faster in bismuth than in
iron, but the thermal conductivity of iron is much greater
than that of bismuth. While the density of bismuth is
somewhat greater than that of iron, its specific heat is
only about one-fourth as great ; and though the heat reach-
ing it is smaller, its temperature rises more rapidly.

The thermal conductivities of the several rods will not be
directly proportional to the lengths on which the wax has
been melted after prolonged immersion, but to the squares
of. those lengths, if the rods have the same rates of radiation.

HEAT.

61. Coefficient of Thermal Conductivity (M., 253;
P., 509; B., 334; G., 168; S., 271). — The precise
meaning of the expression coefficient of thermal conductiv-
ity, or specific thermal conductivity, may be best obtained
by considering the transmission of heat through a homo-
geneous wall with plane parallel faces, one of which is
maintained permanently at a temperature t and the other
at t'. Let AB and CD (Fig. 27) be the two parallel faces
of the wall, and let the line AB represent the temperature
of one side and CD that of the other.
Since the temperatures are maintained,
there will be a permanent state and a uni-
form flow of heat across the wall in the
direction AC. Then if the conducting
power of the wall is independent of its
temperature between t and tf, the flow of
heat will be uniform ; and it may be
taken as established by experiment that
the quantity of heat traversing any im-
aginary plane EF in the interior of the
wall is proportional to the temperature
difference t — if. The rate of flow across
any section of unit thickness perpendicular to the faces of the
wall will therefore be inversely as the thickness of the wall.
The rate at which the temperature falls from one side of
the wall to the other, or the temperature gradient, will
then be uniform, and will be represented in the figure by
the slope of the line BD. The total flow of heat through
any area S of the wall of unit thickness in time T will be
proportional to S and to T. Consequently we have for the
quantity H which flows through area 8 and thickness e in

time T

t-t'

Fie. 27.

H=KS

TRANSMISSION OF HEAT. 95

The coefficient K is the specific thermal conductivity
and depends on the nature of the substance. It may be
defined as numerically equal to the quantity of heat which
flows in a unit of time through unit area of a plate of unit
thickness when unit difference of temperature is main-
tained between its faces. If the temperature is measured
in Centigrade degrees, the dimensions in centimetres,
and the time in seconds, the quantity of heat will be in
calories.

The practical methods of measuring thermal conductivi-
ties are not applied to such a wall, but to the flow of heat
along a bar, one end of which is maintained at a constant
temperature and the other is at the temperature of the
room. The temperature gradient will then be represented
by the tangents to a curve obtained by measuring the tem-
peratures at equal distances along the bar. The heat
flowing past any cross-section of the bar is all dissipated
from the surface beyond the section. The relative con-
ductivities of two bars can be determined by obtaining
their temperature gradients ; but to measure the absolute
conductivity another experiment is necessary for the
purpose of finding the rate of cooling, so as to be able to
calculate the total quantity of heat traversing any section
of the bar.

62. Comparison of Thermal and Electrical Con-
ductivities. — The order of conductivities for the pure
metals is the same for heat as for electricity, though the
relative values of these conductivities are not the same in
the two cases. Both these facts are clearly displayed
by the following table, in which the electrical conductivi-
ties are those of Lenz and the thermal conductivities those
of Wiedemann and Franz :

96 HEAT.

Names of metals. Electrical conductivity. Thermal conductivity.

Silver 100.0 100.0

Copper 73.3 73.6

Gold 58.5 53.2

Brass ........ 21.5 23.6

Tin 22.6 14.5

Iron 13.0 11.9

Lead 10.7 8.5

Platinum 10.3 6.4

Bismuth 1.9 1.8

A further question of much interest is the change of
thermal conductivity with increase of temperature. The
electrical conductivity of all the metals diminishes with
increase of temperature. The same law applies to the
thermal conductivity of iron. Matthiessen found that
the electrical conductivity of iron decreased 38.26 per cent
between 0° and 100° C. Forbes found a thermal decre-
ment for iron between the same limits of temperature of
24.5 per cent in one case and 15.9 in another. But Tait
has shown that Forbes overlooked the large change in the
specific heat of iron with change of temperature ; and when
allowance is made for this, the variation in the thermal
conductivity obtained by Forbes is reduced to about £ of
the original value.

Professor Tait has shown that the thermal conductivity
of iron reaches a minimum somewhere about red-heat;
also that copper and lead show a much smaller change
with change of temperature than iron does, and that this
change is an increment rather than a decrement. Mitchell
has recently repeated the measurements with the same
bars nickel-plated, so as to preserve the surfaces from
oxidation at high temperatures, and he found the tempera-
ture coefficient for iron to be positive, as it is for the other
metals examined.

TRANSMISSION OF HEAT. 97

63. Conduction in Wood and Crystals (Tyn., 189).
— Tyndall has shown that in thirty-two kinds of wood
investigated heat is conducted much better along the
fibres than across them. Further, the conductivity per-
pendicular to the fibres and to the ligneous layers or rings
is greater in every case than in a direction tangential to
them. The conductivity in the first of these three rectan-
gular directions is from two to four times as great as in
the last.

A similar difference of conductivity
has been found in the case of lami-
nated rocks, conduction being better
along the planes of cleavage than
across them. The same statement
may be made with regard to bismuth.

If two plates be cut from quartz
crystals, one perpendicular to the „ 28

crystallographic axis or axis of sym-
metry, and the other parallel to it, and if a minute hole be
made through each plate for the admission of a fine wire
which can be heated by an electric current, then a film of
wax on the plate of crystal will be melted in the form of a
circle when the section is at right angles to the axis, but as
an ellipse when the section is parallel to the axis (Fig. 28).
Quartz and caic spar conduct heat best along the axis and
equally in all directions perpendicular to it, while tourma-
line conducts best at right angles to the axis.

64. Conduction by Liquids (P., 557). — If a liquid
be heated at the bottom, the expansion by heat diminishes
the density and convection currents are set up. The heat
distributed by convection masks any distribution by con-
duction. This difficulty has been overcome in part by

HE A T.

heating at the top. Even then the results are complicated

with diffusion and with conduction by the containing

vessel.

All liquids except molten metals are poor conductors.

The upper strata of water in a test-tube may be boiled for

some time without melting a lump
of ice confined at the bottom of
the tube. If a simple air-thermom-
eter have its bulb surrounded by
water in a funnel (Fig. 29), and
if alcohol be burned in the small
porcelain or platinum crucible at
the top, it will be found that the
thermometer is scarcely affected,
even though its bulb be near the
surface of the water. So feeble
is the flow of heat through liquids
that the result is always open to
the suspicion that the transport is
accomplished by diffusion and con-
yHfr J ~^=afj|i|i vection.
// <C_2 JH^* ^^H No very concordant results have
«4-agsi7T- — : — ■ — ■-/ ever iJeen obtained. The best

agreement is perhaps the follow-
ing, where the conductivity is calculated in C.G.S.
units :

Conductivity.

Substance.

Temperature.

Lundquist. Weber.

Water

. 40.8 C.

0.00156 0.00159

Salt solution, density 1.178 .

. 43.9

0.00149 0.00150

Zinc sulphate, " 1.382 .

. 45.3

0.00144 0.00145

65. Conduction by Gases. — The difficulties encoun-
tered in measuring the conductivity of liquids are exagger-

TRANSMISSION OF HEAT. 99

ated in the case of gases, so that they become almost
insuperable. Many familiar facts, however, go to show
that heat is conveyed very imperfectly by gases, except
under conditions favorable to convection. The interstices
filled with air in bodies made of wool, hair, feathers, fur,
and some vegetable fibres, render them poorer conductors
than when they have been compressed so as to diminish
the air spaces. So some solids which conduct fairly well
are very poor conductors when reduced to a powder,
because of the interstices containing air.

The dynamical theory of heat leads to the conclusion
that the coefficient of conductivity for air is 0.000055, or
about Ttihrs that of copper; also that the coefficient for
hydrogen is 7.1 times as great as for air. These conclu-
sions have been approximately verified. But this theory
does not demonstrate that gases conduct heat in the same
sense as do solids, for it is based on molecular convection,
or the energy transferred by the exchange of motion
among molecules when they collide.

66. Convection in Liquids. — The distribution of heat
by currents of warm water may be illustrated by heating
a large beaker filled with water and containing some bits
of cochineal. A stream of warm water will be observed
ascending along the axis above the burner, and currents
of cooler water descending along the sides. Faraday's
apparatus to illustrate convection currents is shown in
Fig. 30. If the flask and connecting tubes are completely
filled with water above the open end of the tube AB, the
water will begin to circulate up AB and down CD as soon
as heat is applied to the flask by means of a Bunsen
burner. To make the circulation visible, the liquid in the
flask may be colored red with some aniline dye, and that

100

HEAT.

in the large open tube at the top may be colored blue.
The red liquid will ascend and the blue one descend.

This experiment illustrates the method of heating by
hot water. A pipe rises from the top of the boiler to an
expansion tank in the upper part of the
building. From this tank the water is
distributed through the several radiators,
and finally again enters the boiler at the
bottom. The water loses heat in the
radiators and so becomes denser. The
heat of the boiler and the loss by radiation
and convection at the radiators produce
unequal hydrostatic pressures, which give
rise to continuous currents so long as the
heat is applied.

The Gulf Stream, even though it may
be largely produced by wind, is a convec-
tion current on a gigantic scale, and it
transports enormous quantities of heat
from the equatorial regions and distributes it over the
British Islands and the western part of the continent of
Europe. A stream of cold water flows south from Green-
land and washes the Atlantic coast of America. Hence
the contrast between the climate along the Hudson and
the Tiber.

Fig. 30.

67. Convection in Gases. — Since the mobility of
gases is greater than that of liquids, convection currents
are all the more easily set up in them. The heated air
over a gas flame or a fire rises rapidly, and its place is
supplied by the inrush of cold air from the sides. The
same action goes on near the sea-coast on a large scale.
The ground is heated by the sun, and it in turn heats the

TRANSMISSION OF HEAT.

101

air in contact with it. The heated air rises and the cooler
air from the sea flows landward to take its place, giving
rise to the sea-breeze. As soon as the sun sets the ground
cools rapidly by uncompensated radiation and the air
above it becomes cooler than that over the sea. Hence
the pressure is outward from the land, and the land-breeze
sets in.

Under the vertical rays of a tropical sun the earth and
the atmosphere are highly heated ; the latter expands,
rises, and overflows toward either
hemisphere. The denser air flows
in from north and south to replace
the ascending mass. This inflow
has the velocity toward the east of
those parts of the earth's surface
from which it comes. This is less
than the velocity at the equator.
Hence, the currents of air approach-
ing the equatorial belt lag behind the
rotating sphere and arrive as north-
east and southeast Trade Winds.

A belt of calms advances a few
degrees toward the north in summer,
while in winter it recedes somewhat toward the south, fol-
lowing the declination of the sun. The overflow north and
south veers toward the east by reason of its greater eastern
velocity than the successive points at which it arrives. It
therefore constitutes the southwest and the northwest
upper trades, which gradually sink toward the earth.

The principle of convection explains ventilation. The
heated air in a chimney rises because it is warmer than
the air without. The external pressure is therefore only
partly counterbalanced by that of the air in the flue. If

Fig. 31.

102

HEAT.

the chimney happens to be colder than the external air,
there is a downdraft, or the chimney smokes.

Place a lighted candle at the bottom of a lamp chimney.
Ingress of air at the bottom may be prevented by pouring
a little water in the outer dish (Fig. 31). The flame soon
goes out for lack of air. If the T-shaped partition be now
inserted in the chimney and the jcandle be relighted, it
will continue to burn ; and if a piece of smouldering brown
paper be held over the tube, the smoke will descend on
one side of the partition and ascend on the
other, a true convection current supplying
oxygen to the candle and carrying off the
products of combustion.

68. Convection by Hydrogen. — The rapid
distribution of heat by hydrogen was the sub-
ject of a celebrated experiment by Dr.
Andrews. A thin platinum wire, which
could be heated by an electric current, was
stretched along the axis of a tube. The corks
at the ends were provided with an inlet and
an outlet tube (Fig. 32.) When the tube
was exhausted of air, the current was ad-
justed so as to heat the wire to vivid bright-
ness without fusing it. The introduction of
air diminished the brightness of the wire
somewhat ; but when the tube was filled with
hydrogen the wire was scarcely red hot. In
a vacuum the wire loses heat almost entirely
by radiation, but in an atmosphere of hydro-
gen, even though it be very attenuated, the
light and rapidly moving molecules carry

frequent cargoes of heat from the wire to the cooler walls

of the tube.

TRANSMISSION OF HEAT. 103

The slow rate of cooling of a heated platinum wire in
an exhausted globe, as compared with its rate in the open
air, illustrates the loss of heat by convection currents.
The wire remains visible for a sensibly longer time in a
vacuum than in the air after the heating current of elec-
tricity is cut off.

The incandescent lamp is ordinarily made with a high
vacuum to avoid the loss of heat by the convective pro-
cess. For this reason an inert gas like nitrogen cannot be
used In the bulb, because the energy is then rapidly con-
veyed from the filament to the envelope, and heats it at
the expense of the brightness of the filament. The glass
globe heats to a still higher temperature when the carbon
filament is enclosed in an atmosphere of hydrogen. Hydro-
carbon gases have sometimes been used in glow lamps
because of their reparative func'cion when decomposed by
heat, since they deposit carbon on the filament as an offset
to the waste going on in the normal operation of the
lamp.

PROBLEMS.

1. How many calories of heat will be conducted in one hour
through an iron plate one metre square and 0.3 cm. thick if the two
sides are kept at the temperatures 0° and 60° C, the coefficient of
conductivity of iron being 0.175?

2. One side of a brass plate 1 cm. thick and 100 sq. cms. in
area is kept in contact with boiling water on one side and with melt-
ing ice on the other; it was found that 22.9 kilos, of ice were melted
in 10 minutes. Find the coefficient of conductivity of brass in C.G.S.
units.

3. How much water will be evaporated per hour at 100° C. from
a boiler 0.5 cms. thick and with a heating surface of 1,000 sq. cms.,
its outer surface being kept at 150° C. ?

4. A plate of glass 2 cms. thick and 3X4 metres in area sepa-
rates two rooms which are kept at 15° and 50° C. respectively. If
the coefficient of conductivity of the glass is 0.015, find the quantity
of heat given off per minute by the glass.

104 HEAT.

CHAPTER VIII.

RADIATION AND ABSORPTION.

69. Appliances for the Study of Radiation. — The
physical identity of radiant heat and light has already been
dwelt upon in an earlier chapter (6). The transmission of
heat-energy through a medium without affecting it is an
operation identical with that of the transmission of light ;
but since radiations of longer wave-length than about 7,600
tenth-metres (I., 217) do not excite vision, the study of
radiations of long wave-length falls within the domain of
Heat; for to produce any effect these radiations must first
be absorbed, and by this process energy is imparted to the
substance on which they fall. The most general effect of
this absorption is heat.

In light the effects are directly visible, but we need some
means of recognizing the presence of those radiations which
do not excite vision.

Since most substances exhibit the same selective prefer-
ence in the absorption of radiation generally, as they do for
those wave-lengths which lie within the visible spectrum
and give rise to color, it becomes of prime importance in
studying heat effects to find some substance which will
absorb all radiations alike. Such a substance is lampblack.

A thermometer with a blackened bulb is sufficient for
many purposes in the study of radiant heat. A still more
sensitive receiving apparatus is the thermopile, which will

RADIATION AND ABSORPTION. 105

be fully described later. It will be sufficient to explain
here that if a junction of two dissimilar metals, such as
antimony and bismuth, be heated, an electromotive force
will be generated, which will give rise to a current through
a closed circuit. If a number of thin bars of the two
metals, alternating with each other, be joined together and
arranged so that the alternate junctions all fall on one side
of a cube, as in Fig. 33, then when this
face is heated a current will flow through
the circuit and it will be indicated by an
appropriate galvanometer. If the same
face be cooled, a current will flow in the
reverse direction.

Boys' radiomicrometer consists of a single
thermal junction and a galvanometer com-
bined in one instrument.' It hat been made so sensitive
as to indicate readily the heat radiated from a candle on
the opposite side of a large hall. In both instruments the
receptive portion must be covered with lampblack.

70. Invisible Radiation reflected like Light. — The
essential identity of radiant heat and light becomes evident
when it is demonstrated that the various phenomena of
optics may be reproduced by those radiations which do not
directly affect the eye. Aside from the simplest observa-
tion that radiant heat like light travels in straight lines
through a uniform medium, the most obvious analogy
between the two is found in their common obedience to
the law of reflection.

Let two large concave mirrors, usually of brass or
copper, be placed several metres apart and facing each

1 Preston's Theory of Heat, p. 497.

106

HEAT.

other, as in Fig. 34. If a candle be placed at the principal
focus of one mirror, the two may be adjusted in position,
and the image of the candle may be found at the focus of
the second one by means of a small piece of white paper.
Then if the candle be replaced by a heated iron ball, and
the blackened face of the thermopile be placed where the
image was found, the galvanometer will at once show that
the thermopile is heated. The largest effect will be
obtained when the face of the pile is exactly at the focus
previously found. The reflection of the non-luminous
rays from the two mirrors takes place in the same manner

Fig. 34.

as that of the luminous rays, for they converge to the
same point. If the ball be heated to a dull red, the con-
vergence of the heat at the focus of the mirror may be
readily ascertained by the hand. The thermopile will
detect it when the ball has cooled to such an extent that
it may be held in the fingers.

In connection with this apparatus, attention may be
called to the fact that if the ball be replaced by a piece
of ice the current through the galvanometer will show
that the thermopile is cooled. The significance of this
fact will appear later.

It has been demonstrated by experiment that —
(1) Radiant heat is reflected copiously from metals in
the same manner as light.

RADIATION AND ABSORPTION.

107

(2) When radiant heat is reflected, either from glass or
polished metals, the variation of the intensity with the
angle of incidence follows the same law as that applying
to light ; that is, the percentage of the incident radiation
which is reflected increases with the angle of incidence.
Thus glass, which at normal incidence reflects only 4.3 per
cent, at 88° reflects 81.9 per cent of the incident radiation.

(3) Heat is diffusely reflected in the same manner as
light. Just as diffusion is selective for light, red flannel
for example appearing brilliantly red in the less refrangible
end of the spectrum and black in the green (I., 219), so,
as Melloni showed, diffusion is selective also for the non-
luminous radiations.

71. The Law of Inverse Squares. — Melloni was the
first to perform an ingenious experiment to demonstrate
that the thermal radiation received by any small area
varies inversely
as the square
of its distance
from the source.
BO (Fig. 35)
is a shallow box
filled with hot
water and hav-
ing its anterior
face covered
with lampblack.
A is a thermopile with a converging cone to concentrate
the radiations on its blackened face. Let it be placed
in the position A, and let the resulting deflection of
the galvanometer be noted ; then let the thermopile be
moved to double the distance from the box at A'. The

Fig. 35.

108 HEAT.

galvanometer will indicate the same current as before.
The radiating surfaces in the two cases are the bases of
the dotted cones. Their linear dimensions are as one to
two and their areas as one to four. Since the radiation
from a four-fold area produces the same effect at twice
the distance, the intensity of the radiation received from
any small area must vary inversely as the square of the
distance. Since the radiating surface increases as the
square of the distance, the intensity of the radiation must
diminish as the inverse square of the distance. In the
same way a uniformly red-hot surface, viewed by the eye
through a tube, appears equally bright at all distances, so
long as the surface fills the field of view through the tube.

72. Refraction of Radiant Heat (S., 196). — Herschel
made the observation that there are dark heat-radiations
in the solar spectrum beyond the red end. Their exist-
ence there demonstrates that they are emitted along
with radiations of shorter wave-length from a source of
high temperature like the sun.

After Melloni had discovered that rock salt transmits
all kinds of non-luminous radiations with nearly equal
facility, while every other substance absorbs them with
avidity, he made use of rock-salt lenses and prisms to
demonstrate that the radiation from a non-luminous source
' is capable of refraction. Glass is as opaque to radiation
from a non-luminous source as black glass is to the visual
rays. But by employing rock salt Melloni knew that the
radiation which he was studying was not stopped by the
substance of his prisms and lenses. He was thus able to
demonstrate that the radiation from a body at low tempera-
tures may be concentrated at the focus of a lens, and
may be refracted by a prism. The receiving apparatus

RADIATION AND ABSORPTION. 109

employed in this investigation was a sensitive thermopile,
and the source for obscure rays a blackened copper cube
filled with water at 100° C.

Forbes subsequently measured the index of refraction
from several sources of varying temperature, and demon-
strated that the refrangibilit}7" for non-luminous rays is less
than for luminous rays. The following are the indices of
refraction of rock salt for the several sources :

Mean luminous rays 1.602

Heat from incandescent platinum 1.572

Heat from a lamp without a chimney 1.571

Heat from brass at 370° C 1.568

The refrangibility therefore decreases with the tempera-
ture of the source, and the obscure rays are of smaller
refrangibility or longer wave-length than the visual rays.

73. Polarization of Heat (S., 200). — Another evi-
dence of the fundamental identity of radiant heat and
light is derived from experiments in polarization. Malus
and Berard first showed by reflection experiments, similar
to those applied to light (I., 228), that the radiant heat of
the sun is capable of polarization. Later Forbes showed
that whether the source were a lamp or brass heated below
luminosity, the radiation is polarized by transmission
through tourmaline, and suffers extinction in the same
manner as light when the two plates are crossed (I., 224).
By the use of mica plates split by heat and acting like a
bundle of plates, he demonstrated that dark heat is polar-
ized by reflection and refraction. Mica in this state is
nearly opaque to light, but transmits non-luminous radia-
tions quite freely. When two such plates are placed at
the proper angle with the beam, and with the one turned

110 HEAT.

90° around the beam with respect to the other, they were
found to stop a large portion of the incident heat, includ-
ing the radiation from a blackened vessel containing boiling
water.

From such facts as the foregoing it can be affirmed that
we have the most complete experimental evidence that
radiant heat and light are transmitted through the ether
by the same undulatory disturbance, whatever may be its
mechanism. Not only are Fraunhofer (absorption) lines
found in the visible solar spectrum, but the thermopile and
the bolometer1 reveal their presence in the infra-red end.
Rowland's photographs of the solar spectrum, extending
beyond the visible limit at the violet end, exhibit no dis-
tinctions which mark the boundaries of the visible portion.
That limit is imposed by the structure and physiology of
the eye. Langley has measured the energy of the radiation
from his bolometer at — 2° C. to a block of ice at — 20° C.
The analogy between radiant heat and light does not need
the support of any additional evidence.

74. Heat the Measure of Radiant Energy (M., 238).
— From all the facts at command we have reached the
conclusion that radiant heat, like light, is propagated as a
transverse undulation in the ether as a medium. If by
some means, such as transmission through a prism, the
radiations have been separated according to wave-length,
and if from them we select for examination those that will
excite vision when received into the eye, or initiate
chemical changes in the appropriate substance, or finally

1 The bolometer, invented by Professor S. P. Langley, is an instrument whose
operation depends on the change of electrical resistance with temperature. A
thin strip or grating of blackened metallic foil composes one arm of a Wheat-
stone's bridge. When it is exposed to radiation it is heated, and the heat-energy
can be measured by means of the deflection of a galvanometer.

RADIATION AND ABSORPTION. Ill

produce heat when absorbed by lampblack, then it will be
found that, as the intensity is changed, all of these effects
rise and fall together. It is therefore the same ethereal
disturbance which produces visual, actinic, or thermal
effects, according to the constitution of the absorbent
which determines its function.

But while these radiations produce three distinct effects,
only one of them can be taken as the measure of the
energy transmitted, viz.,v the heat generated when they
are completely absorbed. This is true, not only because
the visual or chemical impressions produced by different
kinds of radiations are not proportional to the energy in-
volved, but because they are specific effects depending
on wave-length. While the physiological effect of light
of a definite wave-length bears some relation to the energy
of the vibrations, yet neither in vision nor in photography
can the results be taken in any scientific sense as a
measure of the energy of the cause. Chemical changes
are doubtless initiated by light because of the co-vibra-
tional action, whereby the unstable molecular equilibrium
of certain chemical compounds is broken up and more
stable combinations follow as a result of molecular forces.
But the energy that topples over a brick at the top of a
building and initiates the downfall is not measured by the
effect produced by the brick in falling under the operation
of gravity.

On the other hand, when any radiation is completely
absorbed by lampblack, its energy has simply undergone a
transformation from the energy of ethereal vibrations into
the energy of molecular agitation, which is called heat.
An energy spectrum of the radiations from anjr source
may therefore be mapped out by means of appropriate
apparatus. This has been done by Professor Langley, not

112 HEAT.

only for the solar spectrum, but for the spectra of radia-
tions from blackened copper at several low temperatures.
One important conclusion reached by him is that when the
energy and wave-lengths are plotted as coordinates, the
maximum energy ordinate moves toward the shorter wave-
lengths as the temperature of the source rises.

75. Absorption of Radiation (S., 198; P., 464). —
We are familiar with what occurs when luminous radia-
tions are incident on a body. In general, one part is
reflected, another is transmitted, and a third is absorbed.
Thus, a piece of red glass reflects a portion of the incident
beam, transmits only light belonging near the red end of
the spectrum, and absorbs the rest, converting its energy
into heat. If the transmission is reduced to zero, the body
is opaque ; if the surface is composed of lampblack, the
reflected light is sensibly zero and the entire incident
beam is absorbed. The absorption which rejects the red
only is called selective absorption, while that of lamp-
black is general. Absorption may, however, be general
as contrasted with selective, without being total.

This division of incident radiation, either by general or
by selective absorption, is not peculiar to those radiations
that affect the eye. Bodies which transmit radiant heat are
said to be diathermanous, while those which absorb it are
called athermanous. A body transparent to light is not
therefore transparent also to non-luminous radiations.
Common glass is transparent even to vibrations somewhat
beyond the violet of the solar spectrum ; but it is very
athermanous to long heat-waves. Melloni showed that a
sheet of glass 2.6 mms. thick stops all the radiation from
blackened copper at 100° C, and all but 6 per cent from
copper at 390° C. If a sheet of glass be held between the

RADIATION AND ABSORPTION. 113

heated ball and the mirror in the experiment of Fig. 34,
little or no heat will be- detected at the focus of the distant
mirror. All glass exhibits selective absorption, but col-
ored glass has its range of absorption extended to some
portions of the visible spectrum.

Hard rubber in thin sheets is opaque to light, but quite
transparent to long heat-waves. Carbon disulphide trans-
mits in almost equal degree the luminous and the non-
luminous rays ; but if iodine be dissolved in it, more and
more light will be cut off as iodine is added, till at length
the solution becomes opaque. But heat is still freely
transmitted, or the solution is diathermanous. Tyndall
demonstrated that, by enclosing it in a hollow lens with
rock-salt faces, it transmits enough heat from an electric
arc light to raise platinum to incandescence at the focus.
These facts lead to the conclusion that selective absorp-
tion extends throughout the entire spectrum, visible and
invisible.

76. Two Characteristics of Absorption. — The radi-
ation from a hot body which has passed through one plate
is more easily able to pass through another of the same
substance. This is precisely similar to the fact that the
light which colored glass transmits is almost wholly trans-
mitted by a second piece of glass of the same kind.
Melloni found that a plate of alum which transmitted only
9 per cent of the radiation from a naked lamp transmitted
90 per cent of the heat coming through a plate of the same
material. A second plate of selenite transmits 91 per cent
of the radiation transmitted by a first one. It is possible
to find athermanous combinations, just as red and green
glass together are opaque to light. Tims alum and black
mica form a nearly athermanous combination.

114 HEAT.

The hypothesis to account for this fact applies the prin-
ciple of sympathetic vibration in Sound. Any resonant
body absorbs those vibrations which correspond with its
own vibration-rate (I., 151). So the molecules of every
substance are assumed to have vibration-rates of their own ;
and when the disturbances transmitted to them by the
associated ether have corresponding rates, the vibrations
are taken up by the body. Periodic disturbances of other
frequencies are rejected and pass through.

The second important general fact is that most sub-
stances, including those transparent to light, are nearly
opaque to radiations of long wave-length. It is much
easier to find transparent substances than diathermanous
ones. Rock salt is diathermanous in a remarkable degree,
but Balfour Stewart has shown that it absorbs those vibra-
tions of great wave-length which it radiates when heated ;
and Forbes has shown that the general index of refraction
of a beam of radiant heat is increased by transmission, indi-
cating that the percentage loss is the greater on the less
refrangible side. This rule is not without exceptions.
The solution of iodine in carbon disulphide is a case in
point ; and a piece of smoked rock salt stops most of the
light, but transmits heat.

77. Diathermancy of Liquids. — The diathermancy
of liquids was investigated by Melloni by enclosing them
in a glass cell, while the source of heat was an Argand
lamp with a glass chimney. For such radiations water is
exceedingly opaque. The solution of a salt rather in-
creases its diathermancy. A solution of alum is slightly
more diathermanous than pure water. This conclusion is
contrary to the common opinion, but it has lately been
confirmed by Shelford Bidwell. The old notion that a

RADIATION AND ABSORPTION. 115

strong solution of alum is more athermanous than water
was probably derived from the fact that a plate of alum is
highly athermanous ; but it is less so than rock candy or
ice, though the thermopile will readily reveal the heat
transmitted through a block of the latter substance. Water
and ice appear to be pervious and impervious to the same
radiations, so that one may be used as a sieve to secure
radiations that will pass through the other.

In Tyndall's experiments the liquids were contained in
a cell with rock-salt faces, and the source of heat was an
incandescent platinum spiral. The results are in substan-
tial agreement with those of Melloni.

78. Diathermancy of Gases (P., 470; Tyn., 274).' —
Experiments on the most elaborate scale by Tyndall failed
to show any appreciable absorption of heat by dry air.
They were conducted by passing radiant heat through a
tube filled with pure air and closed at both ends with
plates of rock salt.

The old opinion that other gases and vapors are equally
diathermanous proved not to be true. Ammonia, defiant
gas, sulphur dioxide, marsh gas, hydrogen disulphide, and
nitrous oxide were shown to absorb very perceptible por-
tions of the thermal flux through the tube.

When the temperature of the source is raised, the per-
centage of absorption diminishes. The diathermancy of
volatile liquids and that of their vapors appear to follow
nearly the same relative order. In the main, the molecules
retain their power as absorbers independently of the state
of aggregation. Since ice and water are very athermanous,
aqueous vapor may be expected to show marked absorption
of radiant heat. Tyndall's experiments lead to the con-

1 Tyndall's Contributions to Molecular Physics in the Domain of Radiant Heat.

116 HEAT.

elusion that this anticipation in regard to the opacity of
aqueous vapor is justified. But it has been contested by
Magnus, who found the effect of dry air to be precisely
the same as that of moist air, and " that the water present
in the atmosphere at 16° C. exercises no perceptible influ-
ence on the radiation."

79. Prevost's Theory of Exchanges (M., 240; S.,
204). — If a warm body, such as a thermometer, be hung
within an enclosure cooler than itself, it will lose heat by
radiation and convection till thermal equilibrium ensues.
Even in a vacuum the equilibrium will be attained by
radiation alone. The question arises, Does all radiation
cease when the body and the enclosure are at the same
temperature, and does it radiate no heat when surrounded
by bodies warmer than itself ? If a cold body were intro-
duced into the enclosure it would immediately begin to
receive heat by radiation ; but it can have no direct effect
on the radiation of other bodies within the enclosure.
Prevost therefore came to the conclusion that the radia-
tion continues all the time, and that its intensity has
no relation to the temperature of other bodies, but is a
function of the nature of its surface and of its temper-
ature. If the body radiates more than it receives, its tem-
perature falls ; but if it receives more than it radiates, its
temperature rises. " If two bodies, have the same tempera-
ture, the radiation emitted by the first and absorbed by the
second is equal in amount to the radiation emitted by the
second and absorbed by the first during the same time."

Prevost was probably led to this theory of exchanges, or
of a movable equilibrium of temperature, by the experi-
ment described in Art. 70, where the piece of ice at the
focus of one mirror caused a fall of temperature of the

RADIATION AND ABSORPTION. 117

thermopile at the focus of the other. Since cold is only
the absence of heat, it is inadmissible to suppose that cold
is radiated. Such a supposition is not only unscientific,
but unnecessary. The thermopile radiates toward the ice
exactly as it radiates toward the hot ball, but it receives
from the ice less than it expends by radiation, and its
temperature therefore falls.

The two processes of radiation and absorption are then
going on simultaneously and continuously, and a stationary
temperature is maintained only so long as the emission
and the absorption are exactly equal to each other. Pre-
vost's theory has been greatly extended at various times
by Leslie, Stewart, Kirchhoff, and others. It has not only
been verified by subsequent investigations, but it has
suggested new theories which have also received experi-
mental verification. It is necessary to prepare the way
before proceeding to the extension of Prevost's theory, by
a brief account of Leslie's experiment on radiation and
by some definitions.

80. Leslie's Experiment. — Leslie examined the radi-
ating power of different surfaces by means of a hollow
metal cube ; one side was polished, a second was roughened,
a third was covered with varnish or with white lead, and
the fourth with lampblack. When the cube was filled with
boiling water the relative radiations from the several sur-
faces were compared ; the roughened surface was found to
radiate more freely than the polished one, while it was sur-
passed by the third and fourth, which exhibited nearly
equal radiating power.

In a similar way Leslie investigated the reflection of
heat from surfaces of different character, and found that
the best reflectors are the poorest radiators. Taking a

118 HEAT.

polished brass surface as a standard of comparison, he
found the following relative reflecting powers :

. . . 100

. 60

... 90

Amalgamated tin . .

. 10

Tin .... .

... 80

. 10

Steel . . . .

... 70

The absolute reflecting power, that is, the percentage
of incident radiation which is reflected, has since been
measured for several substances, with the following re-
sults :

Silver . 0.97 Steel 0.82

Gold 0.95 Zinc 0.81

Brass 0.93 Iron 0.77

Platinum ...... 0.83 Cast iron 0.74

81. Definitions. — Lampblack is taken as the standard
with which to compare the absorption and radiation of other
surfaces because it reflects no sensible part of the radiation
incident on it, and because it radiates more freely than any
other substance. Emissive power and absorbing power
may then be defined with respect to lampblack as follows :

The emissive power, or emissivity, of a surface is the
ratio of the quantity of radiation which it emits to the
quantity which a lampblack surface of equal area emits at
the same temperature in the same time.

The absorbing power of a surface is the ratio of the
quantity of radiation which it absorbs to the amount which
a lampblack surface of equal area would absorb in the
same time.

Since a lampblack surface is assumed to absorb all the
radiation which falls on it, the absorbing power of a body
under given conditions may be more simply defined as the

RADIATION AND ABSORPTION.

119

fraction of the whole incident radiation which it absorbs
under those conditions.

These two quantities are connected by the simple re-
lation that the emissivity and absorbing power of any
surface at a given temperature are equal.

Tyndall found the following values by coating the faces
of a Leslie cube with powders of the different materials :

Substance.

Rock salt 0.319

Fluor spar 0.577

Red oxide of lead 0.741

Oxide of cobalt 0.732

Sulphate of iron 0.824

Absorbing power. Emissive power.

0.307

0.589
0.707
0.752
0.808

These numbers do not differ greatly, considering the diffi-
culty of an exact numerical determination.

A simple experi-
ment demonstrates
the equality be-
tween the absorbing
power and the emis-
sivity. Let AB and
CD (Fig. 36) be two
tin plates with the
front of one polished
and that of the other
covered with lamp-
black. To the back
of each is soldered a
piece of bismuth E
to form a thermo-
electric couple. The
polished and lampblack sides are arranged to face each
other, and between them is placed a Leslie cube L. The

A \

■\c.

■ ¥i

— .

k\l

Fig. 36.

120 HEAT.

side of the cube facing the polished plate is covered with
lampblack, while the side facing the lampblack is polished.
The wires at Gr lead to a galvanometer. If one of the
thermoelectric junctions be heated more than the other,
the differential electromotive force generated will produce
a current.

If now the cube be filled with boiling water and be
placed exactly midway between the plates, the galvanom-
eter will show no current. Hence the amount of heat
absorbed by the two plates must be the same. The black-
ened face of the cube radiates more than the polished face ;
but the polished plate absorbs only a fraction of the inci-
dent radiation, while the blackened one absorbs all the
radiation coming from the polished face of the cube pre-
sented to it. It follows that the fraction which the polished
plate absorbs is just equal to the fraction which the pol-
ished face radiates, both compared with lampblack, or the
emissive and absorbing powers of the polished plate are
the same.

82. Extension of Prevost's Theory (M., 243 ; S., 207 ;
P., 442). — The theory of exchanges may be shown to
be applicable to every distinction in the
quality of the radiation as well as to the
total amount of it. By quality of radia-
tion is meant any specific difference, such
as wave-length or plane of polarization,
which may affect absorption.

Imagine a thermometer T suspended
in a blackened chamber with which it is
in thermal equilibrium (Fig. 37). It will
Fig. 37. ke in equilibrium with the enclosure and

with everything in it in whatever part of the chamber it

RADIATION AND ABSORPTION. 121

may be placed. Suppose its bulb to be covered with lamp-
black; it then radiates and absorbs a maximum quantity
of heat, and its radiation equals its absorption because
its temperature remains constant. Another thermometer,
whose bulb is silvered, will indicate the same temperature ;
but it absorbs only about three per cent of the incident
radiation ; therefore to maintain its temperature unchanged,
it must radiate only the same small per cent as compared
with the blackened one. The same relation will hold
true for another thermometer covered with any other
substance.

If any part of the walls of the enclosure exhibits some
selective absorbing power, then the stream of radiation
from this part of the enclosure must remain unaltered
because the blackened thermometer maintains a constant
temperature ; therefore the wall ' must radiate specifically
what it absorbs, both in quantity and quality, so that the
emitted radiation added to the reflected radiation shall
equal lampblack radiation.

Suppose further that a thin plate of some substance,
which transmits radiations of certain definite wave-lengths
only, be suspended within the enclosure. This plate will
radiate just as much as it absorbs because its temperature
remains constant. But since the blackened thermometer
continues to receive the same radiation in amount and
quality from the direction of the plate, the latter must
emit on one side exactly the same quality of radiation
which it absorbs on the other, so that the transmitted plus
the emitted radiation shall remain equal both in quantity
and quality to the stream of radiant heat from that direc-
tion before the introduction of the plate.

It may thus be seen that the stream of radiation in such
an enclosure must be the same throughout in quantity and

124 HEAT.

within so as to be supported at the centre of the bomb.
After removal from the fire the apparatus is placed in the
dark. The light received by the eye, viewing the tourma-
line through the hole, then comes only from the tourmaline
itself, since no light enters the opposite hole and none is
transmitted from the iron. When examined by means of
a polariscope, this light is found to be polarized in a plane
at right angles to the light which the crystal transmits ;
or, in other words, the light emitted is polarized in the
same plane as the light absorbed.

84. Law of Cooling (S., 230; M., 246). — Newton's
law of cooling is that the rate of cooling of a heated body
is proportional to its excess of temperature over that of
the surrounding medium. This law holds only approx-
imately for small differences of temperature and fails
entirely when the excess is large.

The most elaborate investigations on this subject are
those of Dulong and Petit. They were conducted by the
use of a large thermometer within a spherical shell of
copper, blackened on the inside and exhausted of air.

The first conclusion reached was that, for a given excess
of temperature of the thermometer above that of the en-
closure, the rate of cooling in a vacuum increases in a
geometrical series when the temperature of the enclosure
increases in an arithmetical series, and the ratio of the
geometrical series is the same whatever be the excess of
temperature. Thus, if the excess of temperature be 200° C,
the rate of cooling for the enclosure at 0° was 7.40 ; at
20°, 8.58 ; at 40°, 10.01 ; at 60°, 11.64 ; at 80u, 13.45. The
average ratio of these successive numbers, and of others
found by the same experimenters, was 1.165, while the
temperature of the enclosure increased by equal steps of
20° C.

RADIATION AND ABSORPTION. lib

The formula of radiation obtained by Dulong and Petit,
which does not express the facts with great exactness, is

R = mot + &,
where R is the quantity ©f heat radiated in unit time from
unit area of the surface at the temperature t, m is a con-
stant depending on the substance and the nature of the
surface, a is a constant equal to 1.0077 fcr the Centigrade
scale, and k is a constant not yet determined.

From an examination of the data of Dulong and Petit,
Stefan concluded that the radiation emitted is proportional
to the fourth power of the absolute temperature, or

R = n (273 + 04,
where n is a constant and t is the temperature of the radi-
ating body. A similar expression holds for the rate of
cooling if the specific heat of mercury be assumed to be
constant. If t is the temperature of the enclosure and t'
the excess of temperature of the thermometer, then the
rate of cooling will be the difference between the radia-
tion of the thermometer and the counter radiation of the
walls of the enclosure, and we may write :

Rate of cooling = n (273 + t + t'y - n (273 + t)*.
This formula has been deduced theoretically by Boltzmann,
and is in better agreement with more recent experiments
than that of Dulong and Petit.

The rate of convective cooling in a gas was expressed
by Dulong and Petit as follows :

where a and b are constants for any given gas, p is the
pressure, and t the excess of temperature of the cooling
body over the gas. This rate is independent of the nature
and surface of the body, but varies with its form and
dimensions.

124 HEAT.

RADIATION AND ABSORPTION. 125

The formula of radiation obtained by Dulong and Petit,
which does not express the facts with great exactness, is

M = met + k,
where M is the quantity of heat radiated in unit time from
unit area of the surface at the temperature t, m is a con-
stant depending on the substance and the nature of the
surface, a is a constant equal to 1.0077 ftr the Centigrade
scale, and k is a constant not yet determined.

From an examination of the data of Dulong and Petit,
Stefan concluded that the radiation emitted is proportional
to the fourth power of the absolute temperature, or

R = n (273 + OS
where n is a constant and t is the temperature of the radi-
ating body. A similar expression holds for the rate of
cooling if the specific heat of mercury be assumed to be
constant. If t is the temperature of the enclosure and t'
the excess of temperature of the thermometer, then the
rate of cooling will be the difference between the radia-
tion of the thermometer and the counter radiation of the
walls of the enclosure, and we may write :

Rate of cooling = n (273 + t + tT)4 - n (273 + t)4.
This formula has been deduced theoretically by Boltzmann,
and is in better agreement with more recent experiments
than that of Dulong and Petit.

The rate of convective cooling in a gas was expressed
by Dulong and Petit as follows :

r = apH1-223,

where a and b are constants for any given gas, p is the
pressure, and t the excess of temperature of the cooling
body over the gas. This rate is independent of the natui e
and surface of the body, but varies with its form and
dimensions.

126 HEAT.

CHAPTER IX.

THERMODYNAMICS.

85. First Law of Thermodynamics. — A short account
of the experiments of Rum ford and Davy has already been
given in Chapter I. They go to show that heat implies
motion of the invisible particles of matter, and that heat
is the energy of this motion. The science of thermody-
namics is based on two fundamental laws relating to the
conversion of heat into work. The first law is the prin-
ciple of Conservation of Energy applied to heat. It
postulates the equivalence between heat and energy, and
may be expressed as follows:

When work is transformed into heat or heat into work,
the quantity of work is dynamically equivalent to the
quantity of heat.

It has also been expressed in this way :

"When equal quantities of mechanical effect are pro-
duced by any means whatever from purely thermal sources,
or are lost in purely thermal effects, equal quantities of
heat are put out of existence, or are generated " (Kelvin).

This law has been confirmed in a variety of ways :

1. The experiments of Joule, Rowland, and others in
generating heat by the expenditure of work.

2. The experiments of Hirn and others, showing that
when work is done by a heat-engine heat disappears. Hirn
made a fair calculation of the ratio between the two.

THERMOD YNAMICS.

127

3. Investigations on the specific heat of air and other
gases under the two conditions of constant pressure and
constant volume permit of the calculation of the ratio
between the units of heat and of work. This calculation
was first made by Dr. Julius Mayer in 1842.

The limits of this book will restrict the discussion to the
first of these investigations.

86. Joule's Experiments (P., 575). — The investiga-
tions of Joule to determine the dynamical equivalent of
heat, or the ratio between the
units of heat and of work, are
examples of the highest class
of experimental research. Rum-
ford made a rough calculation
of the mechanical work ex-
pended in heating a pound of
water one degree; Joule in-
vestigated this relation by a
long series of varied and elab-
orate experiments which left
little for subsequent investiga-
tors, except the refinement of
details and an increase in the
scale on which the experiments
were conducted. The results

of all his experiments were fairly concordant, and a brief
description of the latest one of 1878 must suffice here.

The plan was to heat water by churning it with paddles,
and to find the ratio between the work expended in turning
the paddles and the number of heat units generated.
Hence both the work done and the heat generated had to
be measured.

Fig. 38.

128 HEAT.

The former was accomplished by an arrangement devised
by Hirn. The calorimeter h (Fig. 38), containing the
water, was supported on a hollow cylindrical vessel w,
which floated in water in v. It was thus free to turn
around a vertical axis, and the pressure was taken off the
bearings. The paddles within the calorimeter were carried
on a vertical axis b, about which the calorimeter could also
turn. A piece of box-wood was inserted in the axis at o
to prevent the conduction of heat downward from the
bearing c. There was a horizontal fly-wheel at /, and
the paddles were turned by the hand-wheels d and e.

To prevent the turning of the calorimeter by the friction
of the water, two thin silk strings were wound in a groove
around it, and, passing over two light pulleys, carried
weights k, k. These weights were adjusted till they
remained stationary, while the shaft and paddles revolved
at a suitable uniform speed, which was recorded by the
counter g. The weights then gave the torque necessary
to keep the calorimeter at rest, or the moment of the force
exerted by the paddles on the water. To measure the
work transmitted, it was then only necessary to multiply
this moment by the angular velocity of the shaft.

Let w be the mass of each weight, r the radius of the
groove in the calorimeter, and n the number of rotations
per second. Then since the work done is the same as if
the axle and paddles were at rest, and the calorimeter was
made to turn n times per second by the fall of the weights,
the energy expended can be readily calculated. In one
turn the weights would descend a distance 2irr. Hence
in n turns the work is

lirr x n x 2wg = knrnrwg.
2irn is the angular velocity of the axle, and 2rwg is the
moment of the couple made by the two weights.

THERMODYNAMICS. 129

To measure the heat generated, let M be the mass of
water and m the water equivalent of the calorimeter and
paddles, and let t be the rise in temperature. Then the
heat generated is (M + m)t. . The ratio of the work done
to the heat generated is

^irnrwg
(M + m)t '

Corrections for radiation and other losses are required.
Joule's experiments proved that this ratio, which is the
work done to produce a unit of heat, is constant. It is
called Joule's equivalent, and is represented by the letter J. •
The fundamental equation expressing this law is

W = JIT,

where W is the number of units of work and H the num-
ber of units of heat.

Joule's final value for J in gravitational units was
1390.59 ft.-lbs. or 423.85 kilogramme-metres. That is,
the heat which will raise a kilogramme of water 1° C.
will, if applied mechanically, lift 423.85 kilogrammes 1
metre high at sea-level. Of course the gramme can be
substituted in this expression without other change.

87. Rowland's Experiments (P., 583). — In 1879
Rowland extended the work of Joule by a series of
exhaustive experiments which leave nothing to be desired.
His object was to reduce the temperatures to those of
the air thermometer, and to increase the rate at which
the work was done and the heat was generated.

Rowland's plan was the same in principle as Joule's, the
chief differences being that the paddles were turned from
below by power derived from a steam engine, and the
revolutions were recorded on a chronograph. On the

130 HEAT.

same chronograph were recorded the transits of the mer-
cury over the divisions of the thermometer. The rate at
which heat was generated in Rowland's apparatus was 50
times as great as in Joule's. Joule's rate of increase of
temperature was only 0°.62 C. per hour, while Rowland's
was 35°. The correction for radiation was thus reduced
in the inverse ratio of the rates.

For the sake of comparison, Rowland reduced Joule's
results to the air thermometer and the latitude of Bal-
timore, where his own experiments were conducted.
Combining the results, he deduced 426.75 from Joule's
experiments, and 427.52 gramme-metres from his own,
both at 14°.6 C. His series of experiments at different
temperatures shows that the specific heat of water is a
minimum at about 30° C.

To reduce Rowland's result to C.G.S. units, the above
quantity must be changed to gramme-centimetres and then
multiplied by the value of g at Baltimore, which is 980.05.
Hence

J= 427.52 x 100 x 980.05 = 4.19 x 107 ergs,
or one calorie is equivalent to 4.19 x 107 ergs.

88. The Relation between J and R. — The constant B,
in the equation for a perfect gas, pv = MT, is numerically
equal to the dynamical equivalent of the difference be-
tween the two specific heats of a gas (34). The demon-
stration is as follows : If v be the volume of unit mass of
the gas at absolute temperature T, then v/T is the increase
in volume, or the expansion, for one degree, and pv/T is the
work done by the gas during the expansion under pressure
p (I., 44). The specific heat at constant volume Sv is the
heat required to raise the temperature of unit mass one
degree when the volume is kept constant ; while the specific

T HER MOD YNAMICS. 131

heat under constant pressure Sp is the heat required to
raise the temperature of the same mass one degree when
the pressure is kept constant. Since there is no internal
work, the latter will exceed the former by the thermal
equivalent of the work done in expanding under constant
pressure. Hence we may write

J(SP-SV} = ^ = R.

R may be evaluated if the density is known. Let d be
the density of the gas ; then since v is the volume of unit
mass, dv = 1, and R = p/Td.

For air d = 0.001293 when p = 76 cms. of mercury —
1033.3 gms. per square cm. = 1033.3 x g dynes. Therefore

R = 1033.3 x# = 2,927 q.
0.001293 x 273 ' y

For any other gas the value of R may be found by
dividing the value of R for air by the relative density of
the gas.

Sp for air is 0.2374 (Art. 34) ; if the ratio between the
two specific heats be assumed to be 1.41, in accordance with
the best experimental results, then the above equation ex-
pressing the relation between J and R will give for J the
value of 42,420 gramme-centimetres, or 4.16 x 107 ergs.

89. Coefficient of Elasticity of a Gas (M., 106). —
Before proceeding to the second law of thermodynamics
it is desirable to introduce some topics subsidiary to it.
Since the working medium for the conversion of heat into
work is usually a gas or a vapor, a few propositions relat-
ing to them are necessary.

The coefficient of elasticity of a fluid is the ratio be-
tween any small increase of pressure and the resulting

132

HEAT.

voluminal compression. Let V be the initial volume and
v the diminution in volume due to an increment of press-
ure p. Then v/ V is the compression per unit of volume.
The quotient of the increment of pressure by this com-
pression is the coefficient of elasticity of volume ; or, in
symbols,

v V

r v

Since voluminal compression is only a ratio, the coefficient

of elasticity is a quantity of the same kind as a pressure.

Let volumes be rep-
resented by abscis-
sas and corresponding
pressures by ordinates
(Fig. 39). Then to
volume FP will cor-
respond pressure LP.
If now the pressure be
increased" to MQ, the
volume will decrease
to GQ. The coordi-
nates of the point P
represent the initial
and those of Q the
final condition of the

body with respect to volume and pressure, the temperature

remaining constant.

Join P and Q and produce the line to its intersection

E with the axis of pressures. Then will FE represent

the coefficient of elasticity. For

FEFP
RQ~RP'

THERMOD YNA MICS. 133

FP V

But RQ is the increment of pressure, and = — .

F RP v

Hence

FP V

If therefore the relation between the volume and pressure
of a gas under the condition of a constant temperature be
represented by a curve traced by the point P, then the
coefficient of elasticity for any point P may be found by
drawing PE tangent to the curve at P and a horizontal
line PF ; the portion FE of the axis of pressures included
between PE and PF will represent the coefficient of elas-
ticity on the same scale as the pressures.

If the temperature is not constant, but is increased by
the compression, the effect will be to increase the increment
of pressure for any given decrement of volume. Hence
the corresponding coefficient of elasticity will be increased.
It is therefore evident that a gaseous substance has two
coefficients, one corresponding to constant temperature and
the other to the case where no heat is allowed to escape or
to enter during compression or expansion. The first is ap-
plicable to long continued stresses ; the second to rapidly
changing or alternating forces, as in the vibrations consti-
tuting sound, in which there is insufficient time for the
equalization of temperature by conduction and radiation.
The ratio of these two elasticities is the same as that of the
two specific heats.

90. Isothermal Lines (M., 108; S., 438). — If the
ordinates of the curve traced by P represent pressures and
the abscissas volumes of a gas at constant temperature,
then the curve expresses the relation between p and v and

134

HEAT.

is called an isothermal line (Fig. 40). If the temperature
be increased to T + 1 and be kept at this value, another
isothermal line will be obtained lying wholly above the
one for T. In this way any number of isothermal lines
may be drawn corresponding to regular intervals of tem-
perature. From such a diagram it is evident that, when

two out of the
three quantities
p, v, T, are given,
the third may
be found graphi-
cally.

If the sub-
stance follows
Boyle's law,
then for a con-
stant tempera-
ture pv is a con-
stant, and this
product is rep-
resented in the
figure by the
area OFPL. If
this area is con-
stant the curve is known as a rectangular hyperbola.
The isothermal line corresponding to any temperature is
therefore a rectangular hyperbola.

It is a property of this hyperbola that if a tangent to the
curve be drawn through any point P till it meets Op in E,
then OF equals FE. But FE equals the coefficient of
elasticity of the gas and OF is the pressure. Hence the
coefficient of a perfect gas obeying Boyle's law is numeri-
cally equal to the pressure. This result was reached in
another way in the theory of sound (I., 118).

Fig. 40.

THERMODYNAMICS. 135

91. Adiabatic Lines. — It remains to consider the
properties of a gas under the condition that no heat enters
or leaves it during the expansion or compression. If the
point traces a line expressing the relation between volume
and pressure in this case, it is called an adiabatic line.
When adiabatic lines cross isothermal lines, they are always
inclined to the horizontal at a greater angle than the
isothermal lines, because as the gas expands the pressure
diminishes more rapidly than for an isothermal line, since
the temperature is reduced by the work done in expanding
under pressure.

The equation to an adiabatic line is

pyy=-d constant.'

1 Let dQ be the quantity of heat required to raise unit mass of a perfect gas
through the temperature difference dT under constant pressure p. This heat is
all expended in changing the temperature and doing external work. The quantity
required for the former purpose is S dT. If the volume increases by a quantity
dv under pressure p, the work done is pdv, and the heat required is {pdv)/ J.
Hence the whole heat necessary to effect the transformation is

When a gas expands adiabatically no heat enters or leaves it, and dQ=0.
Therefore

SvdT+Pd^. = 0.
Differentiating the equation pv = RT, we have
pdv + vdp = RdT.
Substituting in the last equation the value of dT obtained from this one, and
replacing R by its value J (S —S) from Art. 88, we have

Sppdv + Sfvdp=0.
If y denotes the ratio S/S^, then

ydv + dp^Q
v p
Integrating, y log v + log p = constant,

or, pvy « constant.

136

HEAT.

92. Carnot's Cycle (M., 138). — If a volume of gas vt
at pressure px and temperature Tx is allowed to expand
isothermally to the condition vx' and p/ represented by the
point B (Fig. 41), then work has been done against
external forces equal to the area ABvx'vx (I., 44). If now
the gas expands adiabatically from condition B at tempera-
ture Tx to condition C at temperature T,, then the gas
does work represented by the area BCv2'vx'.

Suppose now the
gas to be compressed
isothermally along
the line CD. Then"
the work is done on
the gas with loss of
heat, or is negative,
and it is represented
by the area ODv2v2'.
Lastly let the gas be
compressed adiabati-
cally from condition
J) to condition A.
Then the work done
on the gas raises its
temperature from T2 to Tx and equals the area DAvxv2.
The algebraic sum of the several parts of the work is
then the area ABCI), enclosed between the two isother-
mals and the two adiabatics.

The working substance has returned to its initial vol-
ume, pressure, and temperature, and has gone through
an operation called a cycle. It is known as Carnot's Cycle.
The advantage gained by supposing the working substance
carried through a complete cycle of operations is that there
is then no balance of work done by or against internal

Fig. 41.

THERMOD YNAMICS.

137

forces, as there might be if the substance were not left in
its initial state.

If Hi is the quantity of heat supplied at the higher
temperature Tx , and H2 the heat lost to surrounding bodies
at the lower temperature jP2, then

Heat utilized Hx — H2 Tx — T2 w

? -== = efficiency.

Heat supplied Hx

Fig. 42.

93. Carnot's Engine. — Carnot's engine is an ideal
one designed to embody the series of operations described
in the last ar-
ticle. Suppose
Z), the working
substance (Fig.
42), to be con-
tained in a
cylinder imper-
vious to heat
except through
its bottom,
which is as-
sumed to be a perfect conductor. Let A and B be two
stands, the temperatures of which are maintained at the
values Tv and T2 respectively. C is another stand the
top of which is supposed to be perfectly non-conducting.
Suppose the working substance D at the temperature of
the hot stand 2^, and that its volume and pressure are
represented by vx and px , the coordinates of the point A
on the isothermal line AB in the diagram of the last
article. Then we shall have the following operations:

First Operation. Place the cylinder containing the
working substance D on A and allow the piston to rise.
Heat flows in through the bottom of the cylinder to keep

138 HEAT.

the temperature of the working substance at the point T^
and the substance expands along the isothermal line AB
to the point B. During this operation the substance is
doing work by its pressure against the piston. It is
positive and is denoted by the area A Bv^vx . During this
operation a quantity of heat Hi has passed from* A into
the substance.

Second Operation. The cylinder is now transferred to
the non-conducting stand C and the substance is allowed
to expand adiabatically, thus losing heat till its tempera-
ture falls from Tx to T2 . Its expansion is represented by
the adiabatic line BC. The work done by the substance
during this process is equal to the area BCv/v^.

Third Operation. The cylinder is next placed on the cold
body B, and the piston is pressed down till the volume
and pressure are represented by the coordinates of D.
Heat passes out through the bottom of the cylinder, the
substance remaining at the temperature T2 . Its compres-
sion is represented by the isothermal line CD, and the
work done on it equals the area CI>v2v./; this work is nega-
tive. During this operation a quantity of heat H2 has
flowed from the working substance into the cold body B.

Fourth Operation. Finally place the cylinder on 0 and
force the piston down. The temperature rises and the
relation of the volume and the pressure will be represented
by the adiabatic line DA. Continue the operation till the
temperature has risen to that of the hot body Tx . Then
work equal to the area DAv^v-i is done on the substance,
and is negative.

The substance has thus passed through a series of opera-
tions by which it has finally been brought back in all
respects to its initial state. When the piston is rising the
substance is doing work ; this is the case in the first and

THERMODYNAMICS. 139

second operations. When the piston is sinking it is per-
forming work on the substance ; this is the case in the third
and fourth operations. The useful work done by the sub-
stance is the difference between the positive and negative
work, and is represented by the area AB CD.

The physical results at the end of the cycle are the
following :

(1) A quantity of heat Hi taken from A at the temper-
ature Tx during the first operation.

(2) A quantity of heat ff2 communicated by the
working substance to B at the temperature T2 during
the third operation.

(3) The performance by the substance of work equal
to the area ABOD.

94. Reversibility of Carnofs Engine (M., 149; S.,

351) Let us now suppose all the preceding operations

to be reversed, or that the engine is worked backwards, or
is reversed in all its physical and mechanical actions.

Beginning at the higher temperature and at volume vlt
let the cylinder be placed on C and let the substance ex-
pand along the adiabatic line AB, while the temperature
falls from Tx to T2 . Next place the engine on B and allow
the substance to expand isothermally along DC. During
this latter expansion heat BT2 will be taken from the colder
body B ; and by the two expansions the body has done
work denoted by the area ADCv2'vv .

Now place the engine on C and compress adiabatically
till the temperature rises from T2 to Tx. Then removing
it to A, compress the substance isothermally along BA till
it again returns to its initial volume and pressure. During
the last compression, heat Hx has been given out to A at
the higher temperature 2\, and work has been done in

140 HEAT.

compressing the substance adiabatically and isothermally
in the two compressions equal to the area CBAv^vJ.

In this reverse action of the engine more heat has been
given out to A at the higher temperature than has been
drawn from B at the lower temperature, and more work
has been done on the engine than by it equal to the area
ABCD. It is possible then to convey heat from a colder
body to a hotter one, but only at the expense of mechanical
work.

95. Carnot's Principle. — Heat may be transferred
from a hot body to a cold one either directly by conduc-
tion, or indirectly by means of an artificial engine, in such
a way that part of the heat is converted into mechanical
work ; but heat never flows from a cold body to a hot one,
and it can be thus transferred only by artificial means and
at the expense of mechanical work.

What is known as Carnot's principle, derived from a
consideration of his reversible engine, is as follows : " If a
given reversible engine, working between the upper tem-
perature T} and the lower temperature T<> , and receiving
a quantity Hi of heat at the upper temperature, produces
a quantity w of mechanical work, then no other engine,
whatever be its construction, can produce a greater quan-
tity of work when supplied by the same amount of heat
and working between the same temperatures."

Suppose an engine M to have a higher efficiency than
this reversible one. Let it be coupled to a reversible
engine N working backwards. Then since M converts a
larger portion of the heat Hx into mechanical work than N
requires to restore the heat Hi from the refrigerator to the
source, the two engines constitute an automatic arrange-
ment by which M, by the use of heat Hi , supplies to N

THERMODYNAMICS. 141

sufficient energy to enable it to restore to the source more
heat than iZi; or, in other words, the coupled engines
would run perpetually, transferring heat continuously
from colder bodies to hotter ones. Such an operation is
denied by experience, and is inadmissible. Therefore no
engine can be more efficient than the ideal reversible one
of Carnot.

96. The Second Law of Thermodynamics. — The
second law of thermodynamics expresses a conception
derived from Carnot's reversible engine, and is stated by
Clausius as follows :

" It is impossible for a self-acting machine, unaided by
any external agency, to convey heat from one body to
another at a higher temperature."

Lord Kelvin gives it in a slightly different form :

"It is impossible, by means of inanimate material
agency, to derive mechanical effect from any portion of
matter by cooling it below the temperature of the coldest
of the surrounding objects."

These statements apply only to the performance of
engines working in a complete cycle. Without this
limitation it is evident that the heat of a body, that of
a compressed gas for example, may be converted into
work by cooling it below surrounding objects.

Since the quantities of heat taken in and given out by
a reversible engine depend only on the temperatures of
the source and the cooler, the ratio of the two tempera-
tures may be made equal to that of the quantities of heat to
form a scale of temperature. Then HJH2= TJT2. Such
a scale agrees with that of a perfect gas thermometer.

142 HEAT.

CHAPTER X.

THE KINETIC THEORY OF GASES.

97. Molecular Hypotheses. — The comparative sim-
plicity of the laws relating to gases has stimulated inquiry
into a kinetic theory to account for them on simple
dynamical principles. The results are encouraging to the
extent that they exhibit satisfactory agreement between
the deductions from theory and the laws established by
experiment.

Certain preliminary hypotheses relating to molecular
motion in gases are assumed, though not without justifica-
tion. Since it cannot be assumed that all like molecules
even have the same velocity, the statistical or average
method is adopted, which applies the reasoning to certain
groups of molecules whose velocities do not differ by more
than a very small quantity from a mean value. It is then
possible to discover definite relations between the physical
properties of such a group without knowing anything
about the performance of individual molecules.

Some of the hypotheses are the following :

(1) Molecules of the same gas are alike, and are
separated by intervals which are very great compared
with the size of the molecules. This inference is drawn
from the fact that when a gas is heated so as to become lu-
minous the colors emitted are independent of the pressure ;
that is, the colors depend on the nature of the molecules

THE KINETIC THEORY OF GASES. 143

and not on the distance between them ; for if the molecu-
lar distances were relatively small, mutual action would
ensue, and this action would depend on the pressure
which changes the intervals between the molecules.

(2) The molecules of a gas move in straight lines
between mutual encounters. Their motion for any excur-
sion is uniform and rectilinear. The phenomena of diffu-
sion exhibit rectilinear motion.

(3) All molecules of the same gas have equal masses,
and the average kinetic energy is the same for all mole-
cules at the same temperature.

(4) When two sets of molecules of different kinds
are placed in the same enclosure, kinetic and thermal
equilibrium ensues. The average kinetic energy of trans-
lation for one set is then the same as for the other; this
statement may be extended to any number of sets. If
mx and m2 are the two molecular masses, then

1 , 1 ., mi vl
-171^= -m2v:,, or — = — ;

2 2 m-2 v\

\mv- is called the average kinetic energy of agitation of a
single molecule. The velocity v is " the square root of
the mean square " of all the molecules whose velocities
differ by only a small amount. The squares of the rates
of diffusion of different gases through small pores are in-
versely as their molecular masses. Thus, hydrogen dif-
fuses four times as fast as oxygen. This should be the
case if the two gases have the same molecular kinetic
energy at the same temperature as assumed.

98. Theory of the Pressure of a Gas (M., 319 ; P.,
69). — Let a molecule of mass m approach the side of a
cubical box of unit volume with a normal velocity u. If it
rebounds with the same velocity, the change in momentum

144 HEAT.

will be 2mu. If the molecule moves backwards and for-
wards between two opposite sides of the box with veloc-
ity w, it will strike each side \u times a second, since the
space traversed between two successive impacts on the
same wall is two linear units. Hence the total change
of momentum of the molecule per second with respect to
the wall is

2mu x - u = mu2.

If the unit cube contains n such molecules, then the press-
ure, which is the rate of change of momentum, will be
p = "2mu2 = rnLu2.
But if u2 is the mean of the squares of all
the velocities normal to the face of the
cube, then nu2 = Sm2, and
p =. mnu2.
In general a molecule may be moving in
any direction with a velocity v. If we
suppose that u is the velocity normal to
the plane between A and B (Fig. 43), and un w2, the
two other rectangular components, then
v2 = w2 + u\ + u\.
If now V2 denotes the mean of the squares of all the
molecular velocities of the different groups, with corre-
sponding meanings for Z72, U\, U22, then

v2= u2+ u~i+ u\.

But since the molecules do not accumulate in any part
of the enclosure, as many passing on the average across
the plane between A B in one direction as the other, the
pressure in all directions will be the same, or

U2= u\=u\ = \ v\

THE KINETIC THEORY OF GASES. 145

Therefore when the molecules are moving in all directions

within the cube, the pressure on each face of unit area

will be

p = — mn V2
r 3

While we may not know the absolute mass of each mole-
cule nor the number in unit volume, yet the product mn
of the two is the mass in unit volume, or the density.
Hence

* 3
The pressure is therefore one-third the product of the
density of the gas and the mean square of the molecular
velocity.

99. Mean Square of the Velocity of Hydrogen

From the preceding expression Joule calculated the square
root of the mean square of the velocity of hydrogen as
follows :

The data are, d = 0.0000896 ; p = 1033.3 x 980 dynes.
Hence

3 x 1033.3 x 980 -Ojl-00

— 0 QQ00896 — = 184'133 cms- Per second,

or in round numbers 184,000. This is the value for the
hydrogen molecule between impacts at 0° C. and 76 cms.
pressure.

100. Deduction of Boyle's Law. — If v denotes now
the volume containing unit mass of the gas, then

IP , 1 m

p = — — and pv = - V-.
r 3 v ^3

Since heat is energy of motion, the mean square V3 is a

'rV

146 HEAT.

function of the temperature of the gas only. Consequently
pv at anyone temperature is -a constant; this is Boyle's
law " raised from the rank of an experimental fact to that
of a deduction from the kinetic theory of gases."

101. Law of Gay-Lussac. — Consider two gases in
thermal equilibrium. Then for the two we have

px s= -w^ijvf and p2 — -min&\.
o o

If the pressures are equal

miHivl =s m2n2v\ .

But since they have the same temperature

mjV\ = m2vl,

for the mean kinetic energy of translation of the molecules
is the same for each gas at the same temperature.
Dividing the two equations member by member and

Mi = n2 ,

or equal volumes of all gases at the same temperature and
pressure contain the same number of molecules. This is
known as the law of Gay-Lussac or of Avogadro. While
this demonstration cannot be considered as stringent, it
shows that this hypothesis is entirely in harmony with the
kinetic theory of gases.

If we put dj = mxnx and d2 = m2n2, then since iti = n2 we
have

d2 m2

or the densities of two gases at the same temperature and
pressure are directly proportional to their molecular
masses or weights.

THE KINETIC THEORY OF GASES. 147

102. Total Molecular Energy. — The mean kinetic
energy of agitation of a molecule is \m V'1. But its energy-
may be due partly to the vibration of its parts and to
rotation. Clausius and others have assumed that the
energy of internal agitation tends toward a value having a
constant ratio to the energy of agitation of the molecule
as a whole. The whole energy will then be proportional
to the energy of agitation, and may be written

-BmV\
2

Then the total kinetic energy of the gas contained in
unit volume of n molecules is

K.=l/3mnVa

But sin ce p = ^mn V1 ,

The energy per unit mass may be found by multiplying
the energy per unit volume by the number of units of
volume containing unit mass, or

Km=^Pv.

103. Specific Heat at Constant Volume. — Since the
product pv is proportional to the absolute temperature, the
last equation shows that the energy per unit mass is also
proportional to the temperature on the absolute scale.
The specific heat at constant volume is the increase in the
energy of unit mass for one degree increase of tempera-
ture. Hence in dynamical units

2 T

148 HEAT.

that is, the entire energy divided by the number of degrees
of temperature gives the energy corresponding to one
degree.

Now since pv I T is a constant for gases obeying the
laws of Boyle and Charles, it follows that the specific heat
at constant volume must be constant for any gas, what-
ever its pressure and temperature. This conclusion
is in harmony with the experimental results of Reg-
nault (34).

For different gases the specific heat is directly propor-
tional to the volume v containing unit mass, or inversely
proportional to the density and directly proportional to /3.
Since /3 is nearly the same for several gases, the specific
heat of these gases is inversely as their densities, or
inversely as their molecular weights ; and therefore the
product of specific heat and molecular weight is the
same for all such gases. This is the law of Dulong and
Petit.

104. Ratio of the Two Specific Heats. — The thermal
capacity of any mass M of a gas at constant volume con-
sists of the energy of the molecular motion of translation
plus the energy of the internal motions of the molecules
for one degree of temperature. If U denotes this internal
energy for one degree, then in dynamical units

Also, since the work done in expanding unit mass of a
gas under constant pressure is pv / T for one degree rise
of temperature, we may write for the thermal capacity
under constant pressure,

THE KINETIC THEORY OF GASES. 149

From (100) pv = £ V2. Therefore,

2 T T Z T
Therefore, to find the ratio of the two specific heats,
1 M V2 „ IMF2 bMV2 ^
S,_ 2 T 3 T _6 T

2 T 2 T

But E is necessarily positive ; hence y must always be
less than f or 1.667, which would be its value if E were
zero. As E increases, y approaches unity. These con-
clusions are justified by experiment as shown by the fol-
lowing table :

7 7

Mercury .... 1.666 Chloroform . . . 1.200

Oxygen .... 1.404 Methyl ether . . 1.113

Nitrogen .... 1.410 Ethyl ether . . . 1.029
Ammonia . . . 1.300

The value of y approaches its upper limit only in the
case of mercury, which is the only monatomic gas exam-
ined. The simple constitution of such a gas would lead
to the anticipation that its internal molecular energy
is negligible as compared with the energy of molecular
translation. In all other gases the internal energy is very
appreciable, and it increases as the number of atoms in the
molecule increases. As the molecule becomes more com-
plex, its internal energy represents a larger fraction of the
heat applied to warm the gas.

150 ELECTRICITY AND MAGNETISM.

ELECTRICITY AND MAGNETISM.

CHAPTER XL

ELECTRIC CHARGES.

105. Electricity and Electrification. — The simple
elementary phenomenon that a piece of amber, rubbed
with a flannel cloth, acquires the property of attracting bits
of paper, pith, or other light bodies has been known since
about 600 B.C. But it appears not to have been known
for the following 2,200 years that any bodies except amber
and jet were capable of this kind of excitation. About
1600 Gilbert, an English physician, discovered that a large
number of substances possess the same property. These
bodies he styled electrics, but the word electricity to des-
ignate the invisible agent concerned in the phenomena
appears to have been introduced by Boyle in 1675.

Electrical phenomena are now well understood, but the
nature of electricity remains obscure. It was long supposed
to be a kind of subtle fluid; in later times philosophers
were disposed to consider it a form of energy transformable
into heat and light. But it is now quite certain that, while
it may be a form of attenuated matter, like the ether, it is
not energy. " It is quite true that electricity under press-
ure or in motion represents energy, but the same thing is
true of water or air, and we do not therefore deny them to
be forms of matter." When a body is electrically excited

ELECTRIC CHARGES. 151

it is said to be electrified, and electrification is always a
result of work done in charging with electricity. Electri-
fication, or electricity under pressure, is therefore a form
of potential energy, just as air under pressure and water
elevated above the earth represent potential energy. But
air and water on the one hand and electricity on the other
are not energy, but only, its vehicles or receptacles.

Electricity, like matter and energy, appears to be inde-
structible. Its distribution is subject to control; it may
be put under electric pressure, or be endowed with kinetic
activity; it may represent energy of stress or energy of
motion ; but when its energy has been spent in producing
physical effects, its quantity has suffered no diminution.
It has simply been strained and moved like matter. The
only way to charge a body is to pass to it electricity,
from outside ; none can be created or generated and none
destroyed.

106. Division of the Subject. — The study of electric
currents began near the close of the eighteenth century,
and the earlier observed phenomena relating to them were
widely differentiated from the older manifestations of
electrostatic charges. It has therefore long been customary
to divide the entire subject into three grand divisions, viz. :
static electricity, magnetism, and current electricity. But
since all the phenomena of electrostatics can be produced
by means of electricity set in motion and put under stress
by batteries or dynamo-electric machines, it is apparent
that electricity, however excited, is one and the same
agent. At the same time magnetism is inseparable from
electric currents and must be studied in connection with
them. While, therefore, the general phenomena and laws of
electrostatics, or electricity in equilibrium under pressure,

152 ELECTRICITY AND MAGNETISM.

are conveniently studied together, it should be clearly
perceived that this is merely a matter of convenience, and
that such a classification is not imposed by fundamental
differences. Electric charges and electric sparks may now
be produced as well from one source of electricity as
another; and magnetism may be evoked by electrostatic
discharges, by electric convection, and by electric currents.
Nevertheless it will be convenient to study first the facts
and principles applying especially to electrostatics, and
then those relating to electric currents and their magnetic
effects. A fourth division for purposes of classification
comprises the study of periodic or undulatory disturbances
propagated through the ether as waves of electromagnetic
origin. This subject is the most difficult one of all, but
possesses for us surpassing interest. It includes the
electromagnetic theory of light, elaborated by Maxwell
and confirmed experimentally by Hertz.

107. Attraction and Repulsion. — Support a small

pith-ball by a
silk fibre (Fig.
44) and present
to it a warm
glass tube ex-
cited by rubbing
with a piece of
silk. The pith-
ball will first be
attracted, but if
it be allowed to
come in contact
with the electri-
fied glass, it will

Fig. 44.

ELECTRIC CHARGES.

153

then be strongly repelled. If a stick of sealing-wax,
electrified by rubbing with flannel, be used instead of the
glass tube, the results will be exactly similar.

Two lacts are clearly exposed by this experiment : (1)
A body may be charged by contact with an electrified
body. (2) When one body is charged by contact with
another the two repel each other.

Boyle discovered that the attraction between the electri-
fied and the unelectrified body is mutual. Excite a glass
tube and lay it in a light wire stirrup supported by a silk
thread (Fig. 45). If the hand be presented to it, it may
be made to swing round by the attraction. Force, whatever
its origin, is of the nature of a stress in the medium, and
action and reaction are equal (I., 42).

108. Two Kinds of Electrification. — Not all electri-
fied bodies repel each other. If a second excited glass
tube be presented to the one hung in
the stirrup (Fig. 45), there will be
mutual repulsion between them. On
the contrary, an excited stick of
sealing-wax will attract the pith-ball
charged by contact with an electrified
glass tube ; and if the pith-ball be
charged by contact with the rubbed
sealing-wax, it will be repelled by
the sealing-wax, but attracted by the
glass tube rubbed with silk.

So if two or three pith-balls, hung by silk fibres (Fig.
46), be touched either with an excited glass tube or a
stick of electrified sealing-wax, they will fly apart by
mutual repulsion. It is, therefore, inferred that there are
two kinds of electrification, or that electricity manifests

Fig. 45.

154 ELECTRICITY AND MAGNETISM.

itself under two opposite aspects, analogous to the oppo-
site properties possessed by the two poles of a magnet.
The electricity excited by rubbing glass with silk Du Fay
called vitreous electricity; and the electricity excited on
such substances as. sealing-wax, resin, amber, shellac, and
hard rubber when rubbed with flannel, he called resinous

electricity. The former Frank-
lin called positive and the latter
negative electricity ; and this clas-
sification is better than Du Fay's,
since glass does not always show
positive nor resin negative elec-
trification. The result of friction
depends on the rubber as well as
on the material rubbed.

From such experiments as the
Flg' A6' foregoing is derived the first law

of electrostatics, viz., bodies similarly electrified repel and
those oppositely electrified attract one another.

The student should guard against the inference, from
the expression " two kinds of electrification," that there
are two kinds of electricity, called positive and negative,
respectively. Positive and negative forces constituting a
stress are not essentially different forces, nor are positive
and negative rotations different except in respect to alge-
braic sign. Yet in both cases the forces and motions may
annul each other, as equal quantities of positive and nega-
tive electricity neutralize each other. The terms positive
and negative are applied to electricity merely for the pur-
pose of enabling us to describe concisely, and, to that
extent, to explain certain electrical phenomena.

109. Conductors and Insulators. — Gilbert concluded

ELECTRIC CHARGES. 155

that some bodies were capable of electrical excitation and
others were not. To substances like metals which gave
no sign of electrification when held in the hand and rubbed
he gave the name "non-electrics." In 1729, however,
Stephen Gray discovered that Gilbert's " non-electrics "
convey away the " electric virtue " as fast as it is excited,
and therefore show no signs of electrification. If a metal
rod be held by a glass handle, it can be excited by rubbing
it with silk. Gray succeeded in conveying electric charges
a distance of seven hundred feet by means of a hempen
thread .suspended by silk loops, and Du Fay carried them
to nearly double this distance by means of moistened
thread. Ever since Gray's discovery bodies have been
divided with respect to their power of conveying elec-
tricity into conductors and non-conductors, or insulators.
The latter Faraday preferred to call dielectrics. It should
be noted, however, that all bodies can be arranged in a
graded series having the best conductors at one end and
the poorest at the other. None conduct perfectly and none
insulate perfectly. Pure copper, silver, and other metals
are the best conductors ; and the best insulators are silk,
shellac, glass, and quartz. More definite data on the
specific resistance of various conductors will be given
in treating of electric currents.

110. Electric Field and Lines of Force. — The old
mathematical notion of action at a distance has now been
abandoned ; and when there is attraction or repulsion be-
tween separated bodies, the action is conceived to take place
through the agency of the intervening medium. This con-
ception, developed by Faraday and elaborated by Maxwell
in its application to electricity, has been very fruitful in
discovery, and bears every mark of conforming to the truth.

156

ELECTRICITY AND MAGNETISM.

In harmony with this view, a region within which the
medium is under stress is said to be a field of force ; and
an electric field is one in which the forces acting are electric
in their origin. For concreteness the stress in the medium
is said to act along lines of force. An electric field may-
be completely specified by giving at every point in it the
direction and magnitude of the resultant electric force.
The direction of the force is best expressed by the device
of lines of force. A line of force must be conceived, so
drawn in the electric field that a tangent to it at any point
represents the direction of the electric intensity at the
point. For brevity the expression " force at a point " is
used to signify the intensity of the force sustained by unit
quantity of the active agent at the point, or the electric
intensity at the point.

Lines of electric force always spring from a
positively electrified surface and end in a nega-
tively electrified one. The stress along these
lines is a tension, tending to shorten them. It
is accompanied by a pressure at right angles to
the lines and tending to separate them.

When one electrified body attracts another,
the two are drawn together by these taut lines
of force stretching between them. When two
plates oppositely electrified face each other (Fig.
47), lines of electric force stretch across from
the positive to the negative, and the tension in
the medium tends to draw the plates together.

/*^j\

Fig. 47.

111. Equal Charges of Opposite Sign. — When a body
is electrically excited by friction, the body rubbed and the
rubber are equally electrified, but with charges of opposite
sign. The equality consists in the ability of the one

ELECTRIC CHARGES. 157

charge to exactly neutralize the other. If a stick of seal-
ing-wax, provided with a flannel cap with a silk cord
attached (Fig. 48), be excited by turning it around a few
times inside the cap, it will not attract a positively electri-
fied pith-ball if the cap be left on ; but if the cap be with-
drawn by the cord, the sealing-wax will attract the pith
and the cap will repel it.

The electrification of a body consists
in the separation of two equal charges
of opposite sign against their mutual at-
traction. Hence the medium between
them is strained by the operation, and
work is done. A positively charged
conductor, insulated by supports of glass,
shellac, silk, or other nonconductors, is connected to other
bodies by invisible lines of electric force, springing from the
positive charge and extending to the equal negative one on
surrounding bodies. The slightest charge of positive elec-
tricity at one point always means an equal charge of the
opposite sign as near to it as the conductivity of the
dielectrics permits.

Whatever operations of electrically exciting, discharg-
ing, and the like, may be carried on within an insulated
conducting chamber, no signs of excitation will be ex-
hibited without. The positive and negative excitations
exactly neutralize each other outside the chamber.

112. Electroscopes. — An electroscope is an instru-
ment for detecting electric charges. The simplest one,
which was employed by Gilbert, consists of a long straw,
turning freely on a sharp point, which must be insulated
from the earth. A pith-ball suspended by a silk thread is
also a convenient sensitive electroscope.

158

ELECTRICITY AND MAGNETISM.

The Gold-leaf Electroscope is still more sensitive.
Through the top of a glass jar passes a brass rod, terminat-
ing in a ball above, and bent at right angles below to
receive two strips of gold leaf (Fig. 49). The top of the
jar should be coated with shellac both within and without.
Two strips of tin foil are pasted inside the jar from the

bottom up to the lower level
of the gold leaves to prevent
the latter from sticking to
the glass when they are vio-
lently repelled.

If the knob be touched
with a positively electrified
glass tube the leaves will be
mutually repelled with +
charges. The approach of
any other charged body will
cause them to diverge more
widely if the charge pre-
sented is -f , and to approach
each other if it is — .

Fig. 49.

113. Charge External. — When a conductor is electri-
fied by friction or by electricity conveyed to it from some
external source, the charge always resides on the outside.
Biot devised a direct demonstratrmi by fitting to an insu-
lated copper ball two hemispherical copper shells. When
the whole was charged and the shells were then deftly
removed by glass handles, the charge was found to be
entirely removed with them. A simple demonstration
of the law is afforded by a hollow metal sphere with a
hole at the top and insulated on a glass stem (Fig. 50).
It may be tested by means of a proof-plane, which is

ELECTRIC CHARGES.

159

composed of a small metal disk with a shellac or ebonite
handle. If the proof-plane be applied to the outside of the
charged sphere, a small charge
may be removed and tested by an
electroscope. If the proof-plane
be passed through the hole in the
sphere and applied to the inner
surface, it will be found on with-
drawal to exhibit no trace of elec-
trification. The proof-plane may
be charged from the outside of
the sphere, and then be made to
touch the interior. It will lose
all its charge and will show none
on withdrawal.

Faraday constructed a cube 12
feet on each side and covered it
with tin foil. He went inside of ^
it with his electroscopes; but
while it was charged so that long Fig. 50.

flashes were given off from the
outside, he could detect no signs of electrification within.

114. Distribution of Charge. — The quantity of elec-
tricity (119) on unit surface of a conductor, or the ratio
of the quantity on any small area to the area itself, is
called the surface density. The distribution of an electric
charge is not such as to give uniform surface density over
an insulated conductor, except in the case of a sphere
remote from other conductors and electrified bodies. The
distribution on conductors of various shapes was investi-
gated by Coulomb by moans of the proof-plane and torsion
balance (116). The following is a summary of results:

160 ELECTRICITY AND MAGNETISM.

(1) On a cylinder with rounded ends the surface den-
sity is greatest at the ends.

(2) On a flat disk the density is much greater at the
edges than on the flat surfaces, but over the latter the dis-
tribution is fairly uniform Except near the edges.

(3) With two spheres in contact the charge is nothing at
the point of contact, increases rapidly between 30° and 60°
from that point, and becomes greatest at 180°. When the
spheres are of unequal size, the density at corresponding
points is greater on the small sphere than on the large one.

The density is greatest on those parts of a conductor
which project most and have the greatest convexity.
Hence at sharp points, such as that of a needle, the density
is very great, and as a consequence the charge escapes
rapidly from them. It is therefore necessary to round
off all edges of insulated conductors and to make them
smooth.

115. Redistribution of Charge. — Coulomb demon-
strated that when a charged conducting sphere is brought
into contact with an identical one in the neutral state,
each will then possess a quantity equal to half of the
original charge. If the second sphere, instead of being
neutral, is itself charged, the final charges are again equal.
Each of them is half the algebraic sum of the initial
charges, so that both spheres will be neutral if those
charges were equal and of opposite sign.

The result will be the same with two like conductors of
any form whatever when one touches the other, provided
they are symmetrical with respect to the point of contact.
If this condition of symmetry is not fulfilled, the charges
will divide unequally, but so that their algebraic sum
always equals that of the initial charges.

ELECTRIC CHARGES.

161

Since the charge resides on the outside, if a small charged
sphere be introduced into a larger hollow one, it will give
up its charge entirely to the larger sphere. By this means
a conductor may be charged by successive additions of
small quantities, or one can increase or decrease the electric
charge on the outside of a closed surface by introducing
within small positive or negative charges.

116. Coulomb's Torsion Balance. — The torsion balance
was invented by Coulomb for the pur-
pose of investigating the law of at-
traction and repulsion between two
charges of electricity. The instru-
ment is now obsolete, but it illus-
trates the meaning of the law of
inverse squares which was established
by Coulomb's elaborate experiments.

From a torsion head h (Fig. 51)
is suspended a very fine wire, carry-
ing at its lower end a light shellac
rod with a gilt pith-ball b. The shel-
lac rod swings inside a protecting glass
case, around which is a graduated
scale 8 at the level of the gilt ball. A
shellac rod, carrying another gilt ball
c, can be introduced through a hole in
the top of the case. The torsion head
is divided into degrees, and is pro-
vided with an index. The rod carrying the torsion wire
can be turned independently of the rest of the head, so that
the index can be held at zero, while the rod and wire are
turned till the movable ball just touches the fixed one
without any torsion of the wire. Calcium chloride, or

Fig. 51.

162 ELECTRICITY AND MAGNETISM.

some other drying agent, is placed in the case to keep
the air dry.

117. Law of Inverse Squares. — When the instrument
has been set as described, the vertical rod is removed, the
attached ball is charged, and is then replaced in the instru-
ment. It touches the ball b and divides its charge with it.
Repulsion follows, and the ball b moves away till the tor-
sion couple of the suspending wire equals the moment of
the force due to the mutual repulsion. The distance
between the balls is not sensibly different from the arc
of the circle separating them, if the balls are not many
degrees apart. The balls are now made to approach each
other by turning the torsion head and twisting the wire.
The two divided circles then give the whole angle of tor-
sion of the wire. The principle employed in comparing
the forces is that when a wire is twisted, the couple of
torsion is proportional to the angle through which the wire
is twisted. For example, if the moments of the couples
required to twist a wire through 10° and through 20° are
measured, the latter will be found to be twice as great as
the former.

The following data belong to one of Coulomb's experi-
ments: The first deflection of the movable ball was 36°.
To reduce it to 18° it was found necessary to turn the head
through 126°; and for a further reduction to 8°.5 an addi-
tional rotation of 441° was required. The several relative
distances of the balls were then about as 1 to \ to \ , and
the torsion of the wire was 36° for the first distance,
126 + 18 = 144° for the second, and 441 + 126 + 8.5 = 575°.5
for the third. But 144 is 4 x 36, and 575.5 is nearly 16 x 36 ;
so that as the distance is reduced successively to £ and \ ,
the force is increased to 4 and 16 times respectively.

ELECTRIC CHARGES. 163

The law of attraction was also investigated by a similar
method, and was found to hold within the same limits.
Also the dependence of the force on the charge was
examined by touching one of the balls with an insulated
one of the same size. Half of the charge was thus
removed, and the force was found to be reduced to one-
half. If the charge of either ball was reduced, the mutual
force was reduced in the same ratio.

118. Second Law of Electrostatics. — The second law
of electrostatic action, established by the experiments of
Coulomb, may be enunciated as follows : The force between
two charged bodies is directly proportional to the product of
the two charges, and inversely proportional to the square of
the distance between them.

The law of distance does not hold unless the dimensions
of the charged conductors are very small in comparison
with the distance between them. The charge on a sphere
acts as if it were collected at its centre (121) only when
the distribution of this charge is not affected by the
influence of neighboring charges. In Coulomb's experi-
ment the actual mean distance of the two charges when
the balls were brought as near together as 8°.5 was greater
than the distance between the centres of the spheres. The
force between two flat disks near each other does not vary
appreciably with a moderate change in the distance.

If the two quantities q and q' are on infinitesimal spheres,
the distance of whose centres is r, then the force between
them may be expressed by the formula

/oc + i^-

~" r*

The positive sign corresponds to similar charges, and there-
fore to repulsion, and the negative sign to attraction.

164 ELECTRICITY AND MAGNETISM.

119. The Unit of Quantity. — The definition of the
electrostatic unit of quantity is derived from the law of
attraction and repulsion. If the force in the foregoing
proportion is to become unity when the distance and the
charges are unity, unit quantity must be defined as fol-
lows: The electrostatic unit of electricity is that quantity
which repels an equal and similar quantity, at a distance of
one centimetre in air, with a force of one dyne.

Since the intensity of an electric force is the force
exerted on unit quantity, it follows that the electric inten-
sity at a point distant r centimetres in air from a charge q
is q I r*. The reason for inserting the expression " in air "
will appear later (165).

120. Indirect Proof of the Law of Inverse Squares.
— It has already been pointed out
that no electric force can be detected
inside a hollow conductor. This ex-
perimental fact furnishes the basis
of the most conclusive proof that
the force varies inversely as the
square of the distance.

The following may be considered
ng. 52. as an illustration of the principle

rather than a rigid mathematical dem-
onstration : Let P (Fig. 52) be any point within a charged
conducting sphere, and let a narrow cone of two sheets be
described with P as the apex, and cutting the sphere in
two areas s and s' at ah and a'V respectively. Then, since
the surface density is supposed to be uniform, the quantities
on the two areas are proportional to those areas; but the
areas are proportional to the squares of their respective
distances from P. To prove this latter relation, it must

ELECTRIC CHARGES. 165

be noted, first, that the two areas are sections of the cone
equally inclined to its axis. Let ab and a'b' (Fig. 53) be
oblique sections of a cone making the same angle with the
axis. Their linear dimensions are directly proportional to
the distances PA and PB ; and since the areas of similar
figures are proportional to the squares of their homologous
dimensions, the areas of the two sections are proportional
to the squares of PA and PB.

It follows that the two quantities on s and s'
are proportional to the squares of Pa and Pa'.
Hence the two forces acting on P are directly
proportional to the squares of Pa and Pa', and
inversely proportional to some function of these
distances. But since there is no force inside a
charged sphere, and since the whole surface may
be divided into a series of such pairs of sections
made by a cone, and what is true of the whole
is true of each pair, it follows that the forces due to the
charges on s and 8/ are equal to each other. But the only
function of the distances which will satisfy this condition
is the inverse square. The forces are proportional to the
acting quantities, which are directly proportional to the
squares of the distances ; the forces are also inversely pro-
portional to the squares of the same distances ; and, being
opposite in direction, the resultant is zero.

121. Force Outside a Charged Sphere. — The force or
electric intensity at any point outside a charged sphere,
over which the distribution is uniform, is the same as if
the entire charge were collected at its centre. This propo-
sition admits of simple demonstration.

Let P be the point at a distance D from the centre of
the sphere (Fig. 54). Let <r be the surface density, and

166

ELECTRICITY AND MAGNETISM.

let 8 be the area of a very small element of the surface at
the point B. The quantity on it is scr, and if p is the
distance PB, the force at P due to this element of the
charge is sa/p2. Since the entire surface of the sphere is

Fig. 54.

symmetrical with respect to the line PO, the resultant of

all the forces due to the several elements of the charge

must be along P 0. The component of the force sa/p- along

this line is Sa

f = _ cos a,

P~
where a is the angle OPB.

Draw BA, making the angle ABO equal
to a. Also let co be the solid angle* which
the area s subtends at A. The projection s'
of the area s at right angles to A B subtends
the same angle co at A. Since the angle be-
tween s and s' is a (Fig. 55), we may write
$'= cor2 = 8 cos a,

s =

cor~
cos a

Substituting this value of s in the expression for/, and
j, cor2

ELECTRIC CHARGES. 167

The triangles OB A and OBP are similar, and therefore

r_R

where B is the distance P 0. Hence by substituting above,

r R"

^ Z>2

This is the force due to a single element of the surface.
For the entire surface the force is the sum of the small
forces due to all such elements, or

The expression 2o> is the entire solid angle subtended by
the surface of the sphere at any point within it, and this is
4ir. Hence

But \nrR-<r is the product of surface of the sphere and the
surface density, or the whole charge on the sphere, and B
is the distance between the point P and the centre of the
sphere. Therefore the expression for F is precisely the
same as would be obtained for the force at P if the whole
charge were at the centre of the sphere. It is worth noting
that this demonstration applies equally well to the force of
gravity due to a thin shell of matter, when the shell is of
uniform thickness and density.

122. Force very near a Charged Sphere. — If the
point P in Fig. 54 is made to approach the sphere, the point
A also moves toward the surface to meet P ; and when P
is at the surface B equals R and

F= 47TO-,

or the electric intensity is independent of the size of the

168 ELECTRICITY AND MAGNETISM.

sphere, and is numerically equal to 4-7T times the surface
density. This result, which is known as Coulomb's Law,
requires modification when the sphere is surrounded by
some other dielectric than air. It applies to any charged
conductor. Since there is no force inside the sphere, the
change of force in passing from a point just outside to the
interior is 47rcr.

If a plane perpendicular to P 0 be drawn through A, it
will divide the spherical surface into two parts, each of
which subtends at A the same angle 2tt. Hence half the
force is due to the charge to the right of this dividing plane,
and the other half to the charge to the left of it. At the
surface of the sphere one of these charges is contained on
an infinitesimal area, and the other is the charge on all the
rest of the sphere. The force is then the same as that due
to a plane of indefinite extent, tangent at C and charged
on both sides.

123. Force near a Charged Plane Conductor (Th.,
262). — Imagine a plane of indefinite extent charged

positively on one side to
a density a. Let P be
the point at which the
force is to be determined
(Fig. 56), and PO the
normal to the plane. Let
8 be any small surface on
the plane, and go the solid
angle which it subtends
at P. It is the solid

Fig. 50

angle at the apex of the
cone made by drawing lines from the boundary of s to
the point P. The force at P due to the charge on this

ELECTRIC CHARGES. 169

element is sa/r\ and the component of this force along the
normal P 0 is

f = -p C0Sa>

where a is the angle between the normal and the axis of
the cone.

As, in Art. 121, the orthogonal section of the cone
tf = tor2 and

& = oar2 = 8 cos a.

Therefore « = •

cos a

Substituting in the equation for/, we have

Since the resultant of all the forces due to the elementary
charges is along the normal, the total intensity of the force
at P is

But 1(o is the solid angle subtended at P by a plane of
indefinite extent, and this is the angle subtended by a
whole hemisphere, or 2tt. Therefore

F=2tt<t.

In the C.G.S. system the force is in dynes.

If the plane is limited and the point P indefinitely near
it, the force is again 2ir<r.

Since the force on a + unit above the plane is directed
upward and below the plane downward, in passing through
the plane the force changes by the quantity 47r<r.

PROBLEMS.
1. Two equal small balls are chax*ged with -{-30 and — 6 units of
electricity respectively. Find the mutual force between them when
their centres are G cms. apart, before and after contact with each
other.

170 ELECTRICITY AND MAGNETISM.

2. A ehai-ge of 100 units is applied to a sphere of 10 cms. radius.
What is the surface density ?

3. In the last problem, what is the value of the electric intensity
at the surface ?

4. Two small balls, each one gm. in mass, are suspended from
the same point by silk fibres 490 cms. long. If g is 980 dynes, show
that the balls will diverge to a distance of one cm. if each is charged
with one unit of electricity.

5. Two small spheres 10 cms. apart are charged with -4-5 units
and — 5 units respectively. Find the direction and magnitude of the
force acting on a -j- unit at a distance of 10 cms. from both charges.

ELECTRIFICATION BY INFLUENCE. 171

CHAPTER XII.

ELECTRIFICATION BY INFLUENCE.

124. Fundamental Phenomena. — A charged con-
ductor exerts influence, or acts inductively, on all neigh-
boring bodies. If it be positively charged, lines of electric
force start from it and proceed to an equal negative quan-
tity on adjacent bodies. The influence is exerted along
these lines of force, or lines of tension.

Let an insulated' +

sphere A (Fig. 57), jg fcf+ __--

+ J

charged positively,
be placed near an in-
sulated cylindrical ^^&+
conductor B. Light Fig "'

pith-balls suspended by linen threads at either end of B
will diverge, and the nearer A approaches B the wider the
divergence, unless the charges on A and B unite by a
spark across the air-gap. If A and both ends of B be now
examined by means of a proof-plane and an electroscope,
it will be found that the charge on A has been redistributed,
so that the surface density on the side toward B is greater
than on the remote side ; also the end a of the cylinder will
be found to be negatively charged, the central portion will
be neutral, and the end b will be positively charged. The
density at b will be less than at a, and the neutral line will
be somewhat nearer a than b.

172 ELECTRICITY AND MAGNETISM.

When A is removed or discharged by connecting with
the earth, all signs of electrification on B disappear. The
separation of the positive and negative charges on B
through the influence of the charge on A is called electro-
static induction, or electrification by influence.

125. Charging by Influence. — If the conductor B be
connected with the earth while under the inductive influ-
ence of A, the repelled charge will pass off, leaving only
the attracted electricity. This latter charge is said to be
" bound " in distinction from the " free " charge which goes
to the earth. If now A be removed while B remains insu-
lated, the charge on the latter will be distributed over the
whole conductor, and B is said to have been charged by
influence or induction.

The electrification of B represents energy. Work has
been done in removing A against the attraction of the —
charge on B. If B uninsulated were to be brought up to
A from a distance, and then removed after insulating it, the
work done by mutual attraction during the approach would
be less than that done against the attraction during the
withdrawal, because the acting charge on B in the latter
movement remains constant, while during the approach of
B to A the charge on B increases from nothing to the
maximum. The working force is then less during the
approach than during the recession.

If when the — charge has been insulated on B the posi-
tive on A is discharged to earth, the electrification of B
still represents energy. The energy of the discharge of A
under these conditions is less than that required to charge
it when removed from inductive action on other bodies.
This will be better understood after studying the relation
between energy and potential.

ELECTRIFICATION BY INFLUENCE. 173

126. Electrification with like Charges by Influence.
— When a body is charged by influence as explained in
the last article, the repelled charge always becomes free,
and the conductor is charged so that the inducing and the
induced charges are of opposite sign. In this case pro-
vision must be made for drawing off the repelled charge.

It is quite possible, how-
ever, to provide for the re-
moval of the attracted charge,
so that the conductor under
influence shall remain charged
with electricity of the same —

sign as the influencing charge.
Imagine the conductor B pro- /"" "\
vided with a row of sharp [ i \
points at the end a (Fig. 58), V J
and let a circular glass plate Fi 58

be revolved with its edge be-
tween A and B. The attracted charge will then acquire
so great a density on the points that they will discharge it
on the revolving plate. If another row of points c, con-
nected with the earth, be placed opposite the same side of
the glass plate, but out of the inductive action of A, then
as the plate revolves it will give up to c the negative
charge acquired at «, and c will convey it to the earth. In
this way B is left with a + charge. Work is done in turn-
ing the glass plate against the attraction of the unlike
charges on it and A.

127. Attraction due to Induction. — The simple facts
of induction furnish an explanation of the attraction be-
tween electrified and unelectrified bodies. The induced
charge of opposite sign always accumulates on the part

174 ELECTRICITY AND MAGNETISM.

of the conductor nearest the inducing charge, while the re-
pelled charge retires to the most distant parts of the conduc-
tor, or goes to the earth if a conducting path is furnished.
If an excited glass rod (7 (Fig. 59) be
presented to an uncharged pith-ball sus-
pended by silk, negative electricity will be
induced on the pith-ball at a and positive
(fr y-\ g I at b. Since the former is nearer C than
Fi 59 the latter, the attraction will prevail over

the repulsion, and the pith-ball will on the
whole be attracted. If the pith-ball be touched while
under induction, the repelled + charge will go to earth
and the attraction will be increased.

If the pith-ball be slightly charged positively, then the
resultant action on it will be the algebraic sum of the
repulsion due to this charge, and the attraction due to
induction. Repulsion will generally be first observed as
the pith-ball is brought near C, but at smaller distances the
inductive attraction will prevail. Repulsion is therefore a
better test of an independent charge than attraction.

128. Relation between the Induced and the Inducing
Charges. — The charge on a conductor under
induction can never exceed the inducing
charge. It must be borne in mind that the
bound electricity is held by attraction exerted
along lines of force. If all the lines from
the inducing charge proceed to the induced
charge, the two will then be equal. Gen-
erally only a portion of the lines are common
to the two charges, while the remainder go to other bodies.
If a charged ball be nearly surrounded by a hollow con-
ductor (Fig. 60), all the lines of force from the ball A will

ELECTRIFICATION BY INFLUENCE.

175

end in the induced charge on the enclosure. No sensible
portion of them will escape through the small opening.
A — charge will then spread over the interior of B equal
in amount to the + charge on A.

This case furnishes an exception to the general law that
the charge is confined to the outside of a conductor; but
it is held on the inside by inductive action from A, or is a
bound charge. If B should be insulated while under
induction and A then removed without contact with B,
the — charge on B would become free and would spread
over the exterior.

ci +

129. Faraday's Ice-pail Experiment. — Faraday em-
ployed a pewter ice-pail as a convenient hollow conductor
to test the relative values of
the induced and inducing:
charges. A is a section of a
well-insulated pail (Fig. 61).
The outside is connected with
a gold-leaf electroscope E.
A charged ball C is let down
into the pail by means of a
silk thread. As soon as it
enters the pail the gold leaves
begin to diverge, and the di-
vergence increases till the ball
reaches a certain depth. Be-
yond this point the divergence

remains constant. Evidently the divergence increases up
to the point where all the lines of influence from the ball
run to the negative charge on the inside of the pail. With
the ball still lower, the distribution of the charge, both
on the inside and the outside of the pail, may be changed,
but the quantity remains the same.

Ft*. 61.

176 ELECTRICITT AND MAGNETISM.

If now the ball be allowed to touch the pail, not the
slightest change in the divergence of the gold leaves can
be detected. The meaning is that the free positive charge
on the outside of the pail, when the ball is acting induc-
tively on it, is exactly the same as the charge communi-
cated by the ball on making contact. The inducing and
the induced charges are therefore equal.

The experiment was varied by touching the pail while
under influence from th^ ball. The gold leaves collapsed.
On withdrawing the ball they again diverged to the same
extent as before, but with a negative charge. If then the
charged ball were replaced and made to touch the pail, all
signs of electrification disappeared, or the induced nega-
tive charge was exactly equal to the positive conveyed by
the ball.

Faraday extended these experiments by placing four
cylinders or ice-pails one within another, but all separately
insulated. The entrance of the ball caused a divergence
of the leaves of the electroscope connected with the outer
pail. No change in the divergence could be detected
when the cylinders, while remaining insulated from the
earth, were connected together one after another, showing
that the successive inductions resulted in separating equal
quantities of positive and negative electricity on each pail,
alternating with each other, — the inside of each pail being
charged negatively and the outside positively.

130. The Electrophorus. — The electrophorus is a
simple instrument, invented by Volta, for the purpose of
obtaining an indefinite number of small charges by in-
fluence from a single charge produced by friction. It con-
sists of a metal base or sole, a dielectric disk of resinous
material or vulcanite fitting the base, and a cover provided

ELECTRIFICATION BY INFLUENCE.

177

with an insulating handle (Fig. 62). The form shown in
the figure is so made that the handle can be screwed either
to the cover or the base. In the middle of the disk is a
brass stud screwed into the base and connecting the base
and cover when the latter is applied to the disk.

To use the electrophorus the dielec-
tric must first be electrified by striking
with a cat's skin. A chamois skin will
answer, but cat's fur is better. This
gives to the hard rubber disk a — charge,
and if it is warm and dry it will retain
its charge for some time. The cover is
then placed on the disk, touched with
the finger or to the sole, if the instru-
ment is not provided with the brass
stud to connect the two metal plates,
and is then lifted by the glass handle.
It will be found to be charged posi-
tively to such a degree that a spark
may be obtained from it by presenting
the knuckle. The operation may be repeated an indefinite
number of times without removing any appreciable part of
the original charge from the vulcanite, since the cover
touches it at a few points only.

The operation of the instrument is easily explained by
the principle of influence. When the cover is placed
"m the excited disk, it is really insulated from it and is
powerfully acted on inductively. A positive charge accu-
mulates on its lower surface and a free negative one on
the top. The latter is removed from the cover when
touched by the finger or to the base. When the cover is
lifted by the glass handle the positive charge on it is sepa-
rated from the negative on the disk and becomes free. No

fig. 62.

178 ELECTRICITY AND MAGNETISM.

part of the original charge has been removed; that re*
mains on the vulcanite disk to serve for the repetition
of the operation. It is slowly dissipated if the air is damp
or if the vulcanite is not dry.1

131. Energy of the Successive Charges. — Since the
successive charges on the cover in the normal use of the

electrophorus are not derived from

the disk, it is important to explain

the source of the energy repre-

'] j i "7 TTT sented by them ; for electrification

jl ^ is a form of energy and cannot be

produced without the expenditure
of energy in some other form.

When the cover is on the disk
and the — charge has been removed,
it is held down to the disk by the
lines of force running from the
positive on it to the negative on
the disk. A few lines also run _ CA

Fig. 64.

from the base to the disk, as shown

in Fig. 63. Now to lift the cover without discharging it,

1 It is possible to obtain six successive sparks from the electrophorus by
one application of the cover. For this purpose the base must be placed on an
insulating stand and the cover must not come into electric contact with it. The
several operations are as follows :

(1) Beat with cat's skin and remove the repelled — charge from the base.

(2) Apply the cover and remove from it the free — charge.

(3) The induction on the cover diminishes the influence of the — charge of
the disk on the base and releases part of the + charge. In other words, while
the chai'ge on the vulcanite is engaged in holding the + charge on the cover, it
lets go some of the positive on the base, which may be removed.

(4) The last operation allows greater induction on the cover. Bring cover
and base into contact and a spark will pass.

(5) Lift the cover. The minus charge on the disk again attracts positive on
the base and releases negative, which may be removed.

(6) Discharge the positive on the cover.

ELECTRIFICATION BY INFLUENCE.

179

these lines of force must be stretched and broken. As
the cover is withdrawn fewer lines run from it to the disk
and more come from the base,- as illustrated in Fig. 64.
Hence to lift the cover work must be done against the
force represented by the tension of these stretched lines,
in addition to the work done against gravity. This extra
work is equal to the energy of the charge.

132. Lord Kelvin's "Water-dropping" Accumulator.
— This interesting device illustrates the accumulation of
electric charges by influence, and serves as an introduction
to the continuous electrophorus, or influence machine,
about to be described.

A and B are two insulated hol-
low conductors electrically insu-
lated and called inductors (Fig.
65) ; A' and B' are two others
called receivers, all shown in sec-
tion. O and D are pipes from which
water issues in drops at the middle
of A and B. These conductors are
initially charged with very small
positive and negative charges.

The operation is as follows : As
drops issue from the two nozzles
they are influenced inductively,
since they are not completely sur-
rounded by the hollow inductors,
those in A have a — charge and

Fig. 65.

When the drops fall.

those in B a + one.
The two lower cylinders or receivers contain funnels
which receive the drops and their charges, thus increasing
the electrification of the two sets of conductors. The
effect is cumulative, and the electric density increases till

180 ELECTRICITY AND MAGNETISM.

sparks pass between parts of the apparatus, or the water-
drops are scattered about over the edges of the receivers.
It is essential that the two streams shall be discontinuous
or be broken into drops.

The energy of the charges is derived from the potential
energy of the falling water. The drops are attracted up-
wards and fall more gently than they would if free. Their
loss in potential energy in falling from inductors to re-
ceivers is therefore less than that corresponding to the
difference of levels, and this difference in energy is the
energy of the charges which they convey.

133. The Holtz Influence Machine (Th., 65 ; B., 584).

— It was long ago seen that if the principle of the elec-

Fig. 66.

trophorus could be made to act continuously by mechanical
means, an influence machine could be constructed which
would be superior to the old method of producing electrifi-
cation by friction. This has been accomplished by several

ELECTRIFICATION BY INFLUENCE.

181

inventors, and frictional machines have in consequence
gone out of use.

The first successful influence machine was the one made
by Holtz in 1865. Inasmuch as it has been superseded by-
others having the advantage of being self-exciting, a brief
description must suffice.

A thin vertical glass plate revolves very near another of
somewhat larger diameter and fixed (Fig. 66). The fixed
plate has two openings or windows cut through at the ends
of a horizontal diameter. Extending from these openings
on the back of the plate are two long sectors of paper, pro-
vided at the windows with tongues or notched edges point-
ing toward the back of the revolving plate. These sectors
constitute the field plates or armatures. They extend about
60°, and opposite their extreme ends in front are two metal
combs connected by a
diagonal neutralizing
rod, running along a
diameter. The two
other combs in front of
the rotating plate and
just opposite the win-
dows are collecting
combs connected with
the two discharge balls.

To explain the action
it is best to adopt the
diagrammatic method
of Bertin, in which the
two plates are shown as two concentric cylinders (Fig.
67). A and B are the field plates, g and h the neutraliz-
ing brushes, and A' and B' the collecting combs joined to
the balls N and P, between which the discharges take place.

Fig. 67.

182 ELECTRICITY AND MAGNETISM.

To start the machine N" and P are brought together,
and one paper sector, as A, is feebly excited positively
by contact with the charged cover of the electrophorus,
or by induction from excited vulcanite. As the glass disk
is revolved the induction between e and g causes the latter
to discharge negative on the front of the plate, while
positive is repelled to the other comb h and is there
discharged on the plate. When the negative comes
round to the window at B, it acts inductively on the paper
armature B and on the comb B'. Positive is discharged on
the plate from both of these, leaving B negatively excited.
At the same time the + discharge on the plate at h is carried
around to the window at A, where it attracts — from both
A and A', leaving both -}-. The continuation of this action
results in the intense excitation of the two armatures or
field plates. It will be observed that the arrangement is
such as to carry away in each case the attracted charge, or
the parts are charged by influence in such a way that the
inducing and the induced charges have the same sign.

Turning now to the combs A' and B', the balls JV and P
may be separated, and the induction from A and B and from
e and / keeps the upper half of the revolving plate, front
and back, charged with — and the lower half with +
electricity. The charges on the front are carried off by the
combs A' and B' and unite by means of a spark between N
and P. Small Leyden jars (152) are connected with one
or both of the discharge rods for the purpose of collecting
a greater quantity for each discharge.

134. The Toepler (Voss) Machine (Th., 59; B., 588).

— The only advantage possessed by this form of machine
is that it is self-exciting and will work in a damp atmos-
phere when the Holtz will not. There are no windows in

ELECTRIFICATION BY INFLUENCE.

183

the fixed plate, and underneath the paper armature c and
</ are three disks of tin foil connected by a narrow strip
of the same material, as shown in Fig. 68. To the front of
the revolving plate are pasted at equal distances six or
eight small tin-foil disks with a low metal button in the
centre of each. The tin-foil disks on the fixed plate are
electrically connected to bent metal rods, as shown at a

Fig. 68.

and a' These carry in front tinsel or fine wire brushes,
which touch the metal buttons on the revolving plate as
they pass under them. The diagonal neutralizing rod has
tinsel brushes in addition to the combs. The small disks
on the front plate are rotating carriers, and each is charged
inductively by being placed in momentary connection with
one under opposite electrical influence. At the same time
the points on the neutralizing rod discharge on the revolv-
ing plate, as in the Holtz machine.

184

ELECTBICITY AND MAGNETISM.

The action may be explained by the aid of the diagram
(Fig. 69). The neutralizing brushes are set so as to con-
nect the carriers, as b and e, just before they pass beyond
the influence of the armatures A and B. They thus acquire

by influence — and +
charges respectively.
Passing on to the po-
sitions c and /, they
are brought into mo-
mentary contact with
— b the armatures by a
and a', and deliver up
to them their small
charges. This action
is repeated by each
pair of carriers, how-
ever small may be the
initial excitation of
A and B. In this way A becomes more highly 4- and
B more highly — . When the carriers are highly charged
they do not give up their entire charges to the armatures,
and the collecting combs A' and B' receive the residue in
addition to the charges carried on the glass. There is
usually enough excitation by friction or by contact of dis-
similar substances to start the machine.

Fig. 69.

135. The Wimshurst Machine (Th., 61; B., 589). —

Wimshurst's influence machine is the simplest of all in
construction, and is very effective. Both glass plates ro-
tate, but in opposite directions. They are provided with a
number of narrow tin-foil sectors arranged radially on the
outer sides (Fig. 70). These strips act both as carriers
and as inductors. Across the front is fixed a diagonal

ELECTRIFICATION BY INFLUENCE.

185

conductor, armed at both ends with tinsel brushes. Across
the back is another rod at right angles to the one in front.
Its brushes touch the metal sectors on the back plate. Col-
lecting and discharging apparatus is added to utilize the
charges produced. These must be well insulated from

Fig. 70.

each other on the two sides of the machine. Leyden jars
may be used as in the other machines.

The action will be understood from the diagram (Fig.
71), in which again the two plates are represented as sec-
tions of concentric cylinders, after Bertin and Thompson.
The inner cylinder represents the front plate, and the
outer one the other. Suppose a back sector to receive a
slight charge. As a front sector a passes the outer charged
one, it is acted on inductively and an electric displacement

186

ELECTRICITY AND MAGNETISM.

takes place along the conductor, leaving a slightly charged
negatively, while b receives a corresponding + charge.
These small charges will be carried forward opposite c and
d. Here c and d are touched by the brushes at the back,
and at the same instant are under the influence of the —
and + charges on a and b respectively. They will, there-
fore, receive + and — charges, and will convey them in the
opposite direction to the motion of the front sectors. All
the sectors will thus become highly charged by the cumu-
lative effect of reciprocal
influence, the front sec-
tors on the upper half
carrying — charges from
left to right, and the
back sectors carrying +
p charges from right to
left. On the lower half
of the plates a similar
but inverse set of opera-
tions occurs. Each metal
sector is alternately un-
der influence and acting
as an inductor. By this
double action — charges are continually conveyed by both
plates to the right and + ones to the left. The collecting
combs draw off these charges and convey them to the dis-
charging balls.

In all influence machines the plates are turned in oppo-
sition to the attractions between unlike electrifications.
Hence, more work is done in turning the plates when the
machine is in operation than when it is not excited.

The stress between the fixed and movable parts, or
between parts moving in opposite directions, is an opposing

Fig. 71

ELECTRIFICATION BY INFLUENCE. 187

stress, or tends to turn the plates in the direction opposite
to their proper motion as a generator. All these machines
are therefore reversible, or may be rotated backwards as
motors, by communicating to their armatures a continuous
supply of electricity.

188 ELECTRICITY AND MAGNETISM.

CHAPTER XIII.

ELECTRICAL POTENTIAL.

136. Definition of Potential. — The term Potential was
introduced by George Green, of England, in "1828, but his
theorems connected with it remained unknown till most
of them had been rediscovered by Lord Kelvin, Clausius,
and others. This function plays a highly important r61e
in the study of electrical phenomena. It is intimately
connected with the law of Conservation of Energy, and
has had an important bearing on the progress of electrical
theory and practice.

Consider two similar electrical charges left to themselves.
The mutual repulsion between them will cause them to
move apart till they are beyond each other's influence.
The mutual potential energy of such a system in any given
position is the work done by their mutual repulsion in
separating them to an infinite distance, or in conveying one
of the charges to the boundary of the field produced by
the other.

The potential at any point, due to a given positive
charge, is the mutual potential energy between this charge
and unit quantity of positive electricity placed at the point.
It is the same as the work which must be done on a posi-
tive unit of electricity in bringing it up to the point from
an infinite distance, or from the boundary of the field of
force due to the given charge. If the potential is assumed

ELECTRICAL POTENTIAL. 189

to be zero at some place chosen as a standard of reference,
then any point will have a positive potential if work must
be done in bringing a positive charge from the zero point
to it, and negative if work is required to convey a positive
charge from it to the zero point. For convenience the
potential of the earth is usually taken to be the arbitrary
zero. Positive electricity, left to itself, tends to flow along
lines of force toward points where the potential is lower;
negative electricity travels toward higher potentials.

137. Difference of Potential. — Consider two points,
A and B, and let the potentials at these points be repre-
sented by V\ and V2 respectively. Then since work equal
to V\ is required to convey a unit of + electricity from an
infinite distance to the point A, and a quantity V2 from an
infinite distance to the point B, it is obvious that the work
done by the electrical forces in displacing a positive unit of
electricity from the one point to the other is V[— V2. The
work is independent of the path followed in going from A
to B ; otherwise it would be possible, by making a quantity
of electricity circulate between A and B by suitable paths,
to produce an infinite quantity of work without an equiva-
lent expenditure.

138. Bquipotential Surfaces. — An equipotential sur-
face is the analogue of a level surface. It is a surface per-
pendicular at every point to the direction of the force ; or,
in other words, all the lines of force which it encounters are
normal to it. There is then no component of force along
an equipotential surface, and no work is spent in displacing
any quantity of electricity on such a surface. The poten-
tial at all points of an equipotential surface is therefore the
same.

190

ELECTRICITY AND MAGNETISM.

Consider two such surfaces Sx and Sa, whose potentials
are Vx and V2 . The work which must be done in displac-
ing the unit quantity from the one surface to the other is
then the difference of the two potentials, or V[ — V2. It is
independent of the path travelled and of the position of
the point of departure and the point of arrival on the
two surfaces. If a quantity q units is conveyed from
one surface to the other, the work required is q times
as great as for one unit, or q(Vi— V7). The numerical
measure of the electrical work is therefore a product of two
factors, one of them a potential difference and the other a
quantity of electricity. If the potentials of the two sur-
faces differ by unity, then one erg of work must be spent
in conveying the unit quantity from one surface to the
other.

130. Expression for Force in Terms of Potential. —
Let there be two equipotential surfaces, S and S', very
near together (Fig. 72), and let their po-
tentials be V and F7. Let F be the con-
stant force along a normal between P
and P' equivalent to the variable one be-
tween the two surfaces. If n is the dis-
tance PP', the work done by the force
in conveying a unit quantity from one
surface to the other is F x n. We have
then

Fn= V- P,
V-V

The electric intensity along a line of force is therefore
the rate at which the potential diminishes per unit length
along that line.

ELECTRICAL POTENTIAL. 191

Reduced to limits, or to infinitesimal values,

dn
This expression is the strength of field at any point.
The minus sign indicates that the positive direction of the
force is the direction in which the potential diminishes.
In general the intensity of the force in any direction is the
rate of diminution of the potential in that direction.

140. Equilibrium of a Conductor. — When a charge
of electricity is imparted to a conductor, it at once distrib-
utes itself over the surface and comes to equilibrium.
The surface of the conductor is therefore an equipotential
surface. Moreover, since there is no force inside a con-
ductor, due to a charge on its surface, there is no difference
of potential throughout its entire volume, since force is the
rate of variation of potential. Hence all points of a charged
conductor have the same potential.

The surface of an insulated conductor under the influence
of a charged one is an equipotential surface, because there
is no electric flow along it. This equality of potential in
the presence of an influencing + charge is brought about
by the negative charge on the near end a (Fig. 57) and the
positive on the remote end b. The potential at a, due to the
+ charge on A, is higher than at the more distant point b ;
but the negative charge near a lowers the potential of the
nearer half of the cylinder, and the positive near b raises
the more distant half to the same level as a. If now the
cylinder be connected with the earth, it will be reduced to
the same potential as the earth, or to zero. The cylinder
will then remain charged negatively, but its potential will
be zero. The positive potential due to A and the negative
due to its own charge then everywhere equal eacli other,

192 ELECTRICITY AND MAGNETISM.

and the resultant is zero. It is evident that surface density
and potential are not in any sense the equivalents of each
other.

141. Potential equals £-• — Consider the potential
at A, at a distance r from an element q of the charge at 0

A A

O(fff-) ^, ? ? r? ? £f=lB

Fig. 73.

(Fig. 73). Let B be at a distance r1 from 0. Let the dis-
tance between A and B be divided into n very small ele-
ments, so that the points of division are distant n, r2, r3,
etc., from 0.

Then the force at r is q / r2, the force at rx is <? / r\ , etc.

If r and ^ are very nearly equal, we may put without
sensible error q I rrx as the equivalent force which will do
the same amount of work as the variable force between
the two adjacent points at r and rx . This force is smaller
than the first expression above and larger than the second
one.

Then to carry unit charge from rx to r work must be
spent equal to

TTi \r TxJ

Similarly the work between r.2 and rx is q ( _ — _ J .
From r1 to rn_! the work is . . . q | -J .

ELECTRICAL POTENTIAL. 193

The whole work done in transferring the unit quantity
from B to A is the sum of all these elements of work ; it
is evident that on adding, all the terms containing the r's
cancel out except the first and the last, or

Work from B to A

/l_l\
q\r r')'

Next suppose the point B moved off to an infinite dis-
tance. Then 1 / r1 becomes zero, and

Work from infinity to A = - •

But by definition this is the potential at A, since it is the
expression for the work spent in bringing unit quantity of
electricity from an infinite distance to the point. There
will be similar expressions for the several elements of the
charge, and the resulting potential at A will be the alge-
braic sum of the potentials due to the several elements, or

142. Potential of a Sphere. — Let the sphere have a
charge Q. Every element q of this charge is at a distance
r from the centre of the sphere ; and the potential at the
centre due to this element is q /r, where r is the radius.
The potential due to the entire charge is then

q 1 Q

^r r * r

But as all points of a conductor in equilibrium have the
same potential, the potential of every point of a sphere
due to a charge Q is Q/r.

Since a charge, uniformly distributed over a sphere, acts
on external points as if it were collected at its centre, the

194

ELECTRICITY AND MAGNETISM.

potential at any point outside of the sphere and distant d
units from its centre is Q/d.

143. Electrometers. — An electrometer is an instru-
ment designed to measure differences of electrostatic
potential. Its indications depend on the attraction be-
tween an electrified and an unelec-
trified plate, or on the action between
two conductors electrified to different
potentials. Sir Snow Harris was the
first to construct such an instrument.
It was made like a balance, with a
small pan P (Fig. 74) on one end
of the beam, and a small round disk
d on the other, just above a fixed
insulated plate a. When a was
electrified it attracted d, and the
attraction was counterbalanced by-
weights in the pan P. But the
plate d was not protected from in-
ductive influence, and no precise ab-
solute measurements involving the
dimensions of the disks could be made, because the surface
density was not uniform over the whole disk (see Fig. 47),
but was greatest at the edges, where the lines of force were
not parallel to one another, but curved outward. This
difficulty was overcome by Lord Kelvin, to whom we are
indebted for modern electrometers.

The essential addition of Lord Kelvin is the guard ring
shown in Fig. 75. The suspended disk C fits, without
contact, an aperture in the guard ring A, to which it is
electrically connected. The disk C is the only part of the
area utilized ; the surface density over it is uniform and
the lines of force between it and B are parallel.

Fig. 74.

ELECTRICAL POTENTIAL.

195

144. Attracted-disk Electrometer. — In the attracted-
disk electrometer the attraction between two parallel disks
at different potentials is counterbalanced by a weight D
(Fig. 75). The disk C, when in position, is adjusted so
that its lower face is as nearly as possible in the same plane
with the lower surface of the guard ring A. The lever L
is pivoted on a torsion wire stretched between two insulated
pillars EE. A lens G- is mounted so as to observe an
index hair at the end of the lever L relative to two dots
on the plate F. The
plate C is in posi-
tion when the hair
is between the two
dots. The disk B is
insulated and can
be raised or low-
ered by means of a
micrometer screw
not shown.

The counterpoise
D is such that when
B and C are at the same potential, the index hair rises above
the sighted position. The force required to bring the hair
down to the sighted position is determined by placing a
small weight on C and a " rider " on the arm L. But when
B and C are at different potentials, the attraction between
them draws C down ; the plate B is then adjusted in height
till the index hair comes to the sighted position. The
attraction between the plates is then equal to the force of
gravity on the weights previously determined.

Fig. 75.

145. Theory and Use of the Instrument. — Let V\
be the potential of the movable disk (7, which is charged

196 ELECTRICITY AND MAGNETISM.

positively to a surface density a ; and let V2 be the poten-
tial of the plate B. Since the lines of force between the
two plates are parallel, the surface densities of the plates
are of opposite sign and numerically equal. Then the
electric intensity, or the 'force on a positive unit, between
the plates is 47ro-, an attraction of lira- by the fixed plate,
and a repulsion of lira- by the movable plate. The two
plates are equipotential surfaces and V\ — V2 is the work
which must be done on a positive unit to convey it from
C to B. Therefore, since work equals the product of force
and distance,

jr- r2 = 47T(rA

where J) is the distance between the fixed and movable
plates.

The electric intensity at 0 due to the charge on B is
27rcr. If S is the area of the movable plate (7, the charge on
it is So: Therefore the normal mechanical force pulling
the plate downward is

F=2ir<rxS(r = 27r(T2S.

Whence ./ ^g

By substitution in the equation above we have

Now F is known from the weights previously applied, and

IsttF .

S can be measured ; \J — s~ is therefore the constant of

the instrument. If F is measured in dynes, S in square
centimetres, and D in centimetres, the measurement of D
determines the difference of potential in absolute measure.

ELECTRICAL POTENTIAL

197

Practically there is great difficulty in measuring D with
sufficient accuracy. Hence a different method of measure-
ment is adopted. The plate B is kept charged to a definite
potential, and the disk 0 is first connected to the earth,
whose potential is zero, and B is adjusted in height till C is
in the sighted position; a reading of the micrometer is
then taken. The conductor to be tested is then connected
with C and another adjustment of B is made and a reading
is taken. Let the distances between B and C for the two
adjustments be D and ZK Then we have for the potential
of 0

It is then necessary to determine the difference D—D1
only, and this can always be done with great accuracy.

In the most elaborate modern instruments the disk C is
suspended by small springs, and both are protected from
inductive influence by a cylindrical metal cover.

146. The Quadrant Electrometer (J. J. T., 98). —
The force F measured by the instrument just described

Fig. 7«.

varies as the square of the potential difference. When
this potential difference diminishes, the force falls off very
rapidly. For this reason the instrument is not suitable for

198

ELECTRICITY AND MAGNETISM.

the measurement of very small potential differences ; for
these Lord Kelvin devised the quadrant electrometer.

The most essential parts are the cage, or quadrants, and
the needle (Fig. 76). The needle, a thin oblong piece of
aluminium with broad rounded ends, shown in dotted out-
line in the figure, is suspended by a very fine wire or fibre

so as to turn in a horizontal plane
around a vertical axis. It swings
centrally within four quadrants, a,
5, <?, d, which together form a short
hollow cylinder with parallel ends.
Opposite quadrants, as a and c, and
b and d, are connected electrically.
The needle is supported on a stiff
wire carrying a mirror M (Fig. 77)
at the top, and connecting at the
bottom with the jar B by a fine
platinum wire dipping into sul-
phuric acid

Consider the needle charged posi-
tively. If all the quadrants are at
the same potential, the needle will
take a position depending only on
the torsion of the suspending fibre ; but if a and c, for ex-
ample, be at a higher potential than b and c?, the forces
acting on both ends of the needle form a couple which will
turn it opposite to watch-hands. If the potential of a and
c is lower .than that of the other pair of quadrants, the
needle will turn the other way ; it will come to rest when
there is equilibrium between the two couples, the one due
to the electrical forces, and the other to the torsion of the
suspending fibre.

Let V0 denote the potential of the needle, Vx and V2 the

Fig. 77.

ELECTRICAL POTENTIAL. 199

potentials of the two pairs of quadrants, and 6 the angular
deflection of the needle; then the equation of equilib-
rium is

0=C(V1-V2){K-$(iV1+V2)}, . . (a)

where C is a constant.1

If V0 be very large in comparison with the other poten-
tials, the term £ ( V\ + FT) may be neglected in comparison
with it, and

d=o{v1-v^ r0, . . . . (6)

or the deflection is proportional to the difference of poten-
tial to be measured. The sensibility is proportional to V0i
the potential of the needle.

When the needle is thus charged from a source inde-
pendent of the quadrants, the instrument is said to be used
heterostatically.

147. Quadrant Electrometer used Idiostatically. —
For the measurements of larger potential differences the
needle is connected with one pair of quadrants, so that
there is only one source of electrification, and this use of
the electrometer is called idiostatic. We may then put V0
equal to V\ , and equation (a) becomes

e=C(Vx-v,y,

or the deflection oi the needle is proportional to the square
of the potential difference of the quadrants. The physical
explanation is that doubling the potential doubles the
charges on the quadrants and the needle ; and since the
force is proportional to the product of these charges,
the force is quadrupled.

For measuring large potential differences the quadrant

1 J. J. Thomson's Elements of Electricity and Magnetism, p. 103.

200 ELECTRICITY AND MAGNETISM.

electrometer, or electrostatic voltmeter, may be used idio-
statically in a different way.1 If the suspension is provided
with a torsion head and a horizontal scale, graduated in
equal divisions, the charged needle may be brought back
to its initial or zero position by turning the torsion head
and twisting the suspending fibre. This adjustment is
made either by a telescope, or by means of a beam of light
reflected from the mirror M. The forces are then propor-
tional to the angular twist of the suspending fibre, and the
potential difference to the square root of this twist. In
this way potentials from 10 volts upwards may be readily
measured.

PROBLEMS.

1. What would be the potential difference between A and B
(Fig. 73) if O were charged with 100 units of -|- electricity, the dis-
tance r being 10 cms. and r' 15 cms. ?

2. Positive charges of 150, 424, and 300 units are placed at the
three corners A, B, C, of a square 30 cms. on a side. Calculate the
potential at the fourth corner D.

3. Positive charges of 50 units are placed at the three corners of
an equilateral triangle whose sides are 50 cms. Find the potential
at the centre of the circumscribing circle.

4. What would be the potential at the same point in the last prob-
lem if the charges were placed at the middle points of the three
sides?

5. Find the potential at the centre of the square in problem 2,
and the work to be done to bring a -[- unit from D to the centre.

6. A sphere 10 cms. in diameter is charged with 50 units of posi-
tive electricity. Find the potential at the surface of the sphere, and
at a point 20 cms. distant from its surface.

i Carhart »M Patterson's Electrical Measurements, p. 200.

CAPACITY AND CONDENSERS. 201

CHAPTER XIV.

CAPACITY AND CONDENSERS.

148. Definition of Capacity. — The electrical capacity
of a conductor is denned as the numerical value of its
charge when its potential is unity, all other conductors
within its field being at zero potential. Since the potential
of such a conductor is directly proportional to its charge,
the charge corresponding to unit potential, or its capacity,
may be found by dividing its total charge by the number
of units of potential to which it is raised ; or, in symbols,

C~V\ ■

where C denotes the capacity. Also

<2=(7Fand V=Sf.

149. Capacity of an Insulated Sphere. — The capacity
of a sphere at a great distance from all other conductors is
numerically equal to its radius in centimetres. For the
potential of such a sphere is Q /r.

Hence 0=$=Q—9.=r.

V r

The radius must be expressed in centimetres because the
centimetre is the unit of length employed in denning the
unit of quantity.

Two spheres of unequal radii when charged to the same

202 ELECTRICITY AND MAGNETISM.

potential have surface densities inversely as their radii.
For

Q CV V

4:7rr2 4:7rr2 4irr

Therefore <r varies inversely as r, or, for the same
potential,

If a small sphere is connected to a large one by a fine wire,
and if it is then supposed to diminish in size while its po-
tential remains unchanged, the surface density on it will
vary inversely as its radius. If it becomes indefinitely
small, its surface density becomes indefinitely great. The
electric intensity just at its surface increases therefore as its
diameter decreases. This relation explains the discharging
power of points.

150. Condensers. — Two conductors placed near to-
gether, but insulated from each other, form with the
dielectric a condenser. The effect of the additional con-
ductor is to increase the charge without any increase of
potential. In other words, the capacity of the one conductor
is greatly increased by the presence of the other. If the
charges are equal and opposite in sign, the charge on either
conductor when the potential difference between the two is
unity is called the capacity of the condenser.

Let a horizontal brass plate with rounded edges be
mounted on an insulating glass standard, and let a plate
of glass CD (Fig. 78), larger than the brass plate, be placed
on the latter. On this place another brass plate of the
same dimensions as the lower one. Connect one plate with
one electrode of an influence machine, and the other plate

CAPACITY AND CONDENSERS.

203

with the other electrode, and charge them. If now they
are disconnected from the machine and the upper one be
touched with the finger, the attached pith-balls, which must
be hung with linen threads, will
collapse. But if the upper plate
be lifted by its insulating stem,
the pith-balls will again diverge
and a small spark may be drawn
from the plate. The two metal
disks and the glass plate consti-
tute a condenser.

If the upper plate be charged
positively, its positive charge at-
tracts a nearly equal negative
charge on the lower plate, and the
two are bound so long as the
plates remain in position close to-
gether. The induction takes place
through the glass, better in fact
than through air.

Let the plates be again charged
as before. If then one end of a
bent wire be placed in contact
with one plate and the other end
be made to touch the other plate, a bright electric spark
will pass just before the second contact is made. The
charge of either plate is evidently greatly augmented by
the presence of the other. If one plate be connected to the
source of electrification and the other to the earth, then the
former is called the collecting plate and the latter the con-
densing plate ; the insulator between them is the dielectric,
or the medium through which the mutual electric action
between the plates takes place.

Fig. 78.

204

ELECTRICITY AND MAGNETISM.

151. Capacity of Two Concentric Spheres. — Let the
radius of the inner sphere be r and that of the inner surface
of the outer one r' (Fig. 79), and let the outer sphere be
connected to the earth. Then its potential
and that of all other neighboring bodies is
zero. Hence, since lines of force connect
only bodies of different potentials, all the
lines of force from the insulated charged
sphere A run to the outer sphere B. Their
charges are then equal and of opposite sign,
+ Q and - Q.

The potential at 0, the common centre
of the two spheres, is

Fig. 79.

Q Q

\r r1 J

But this is the potential of the inner sphere because the
potential inside a charged conductor is the same as at any
point on its surface. From the last equation

r — r
When V becomes unity the charge by definition is the

capacity, or .

C=.rr .
r1 — r

When r1 — r is very small, that is, when the two spherical

surfaces are very near together, the capacity becomes very

large. The expression for the capacity is then

rr1 _r (r + 0

T t '

where t is the thickness of the dielectric. When t is very
small compared with r, this expression becomes

t kirt bat '

CAPACITY AND CONDENSERS. 205

where S is the surface of the sphere. The capacity per
unit area in air is therefore 1 / Anr times the reciprocal of
the distance between the conductors.

If the outer sphere be supposed to expand indefinitely,
or to be removed, while the inner one is insulated, the
potential of the inner sphere will increase ; for

Now if r and r1 are very nearly the same, the potential for
a given charge Q may be small ; but as r1 increases,
(r'—r)/r/ approaches unity and finally V=Q/r. The
condensing plate decreases the potential, therefore, in the
ratio of r' — r to r7, the charge on the collecting plate
remaining the same. Or conversely, for the same potential,
the condensing plate increases the charge in the ratio of
r1 to r1 — r.

152. Capacity of Two Parallel Plates. — When the
plates are so close together that the curved lines at the
edges are negligible in comparison with the others, ail
the lines may be conceived as straight and at right angles
to the plates. The capacity is then easily calculated. If
t is the distance separating the plates, or the thickness of
the air film as the dielectric, then the electric intensity
between the plates is uniform, and the work done in con-
veying a unit charge from the plate of higher potential to
the other is

V=Ft,

where Via the potential difference between the plates and
F is the electric intensity.

The surface densities will be equal and of opposite sign,
+ o- on the one of higher potential and — <r on the other.
Then the electric intensity between the plates is 47r<r, half

206

ELECTRICITY AND MAGNETISM.

of this expression being due to the charge on one plate and
the other half to the other, as before explained. Therefore

F= 47TO-,

and from the last equation V— Airat.
If A is the area of each plate,

When V is unity the charge on one of the plates of area
A is A I Airt, and this by definition is the capacity. This
expression is the same as that for the capacity of two con-
centric spheres.

153. The Leyden Jar. — The Leyden jar was the earli-
est form of condenser. It was discovered accidentally by
Cuneus at Leyden in 1746 while attempting to collect
"electric fluid" in a bottle half filled with
water and held in the hand. The water was
connected with an electric machine. While
holding the bottle in one hand and attempting
to remove the connecting chain with the other
Cuneus received an unexpected shock, from
which it took two days to recover his mental
equilibrium. It is evident that the water in
the bottle served as the collecting plate and
the hand as the condensing plate, the glass
being the dielectric.

As now made a Leyden jar consists of a
wide-mouthed jar of thin flint glass, coated within and with-
out with tin foil for about three-fourths of its height (Fig.
80). The metal knob is connected to the inner coating by
a rod terminating in a short piece of chain. The jar may
be charged by holding it in the hand, touching the knob to

Fig. 80.

CAPACITY AND CONDENSERS. 207

one electrode of an influence machine, and bringing the
outer coating so near the other electrode that a series of
sparks will pass across. If charged too
highly it will discharge along the glass
over the top. A hissing, crackling sound
indicates a partial brush discharge over
the surface of the glass above the tin foil.
It may be safely discharged by a dis-
charger (Fig. 81) held by the glass
handles, one ball being brought into con-
tact with the outer coating and the other
with the knob.

If A is the area of the tin foil and t the thickness of the
glass, then if the space between the tin-foil coatings were
filled with air, the capacity would be approximately

since this case is practically the same as that of two parallel
plates, if t is small in comparison with the radius of the jar.
It will be explained shortly that the effect of interposing
the glass instead of air between the two coatings is to
increase the capacity by a factor K, so that

4irt

K is a constant depending on the kind of glass, and varies
from about 4 to 10 for different specimens.

154. Residual Charge. — If a Leyden jar be left
standing for a few minutes after it has been discharged, the
two coatings will gradually acquire a small potential differ-
ence and a small discharge can be again obtained from it.
This is called the residual charge. Several of them, of

208

ELECTRICITY AND MAGNETISM.

decreasing potentials, may sometimes be observed. The
magnitude of the residual charge depends upon the original
potential difference to which the jar was charged, the length
of time it is left charged, and the kind of glass of which
it is made.

155. Seat of the Charge. — The Leyden jar with re-
movable coatings is due to Franklin. By means of it he
showed that the charge resides on the surface of the glass.

Fig. 82.

A (Fig. 82) is the jar complete, B is the glass vessel, 0
the outer and D the inner metallic plate. If the jar be
charged in the usual way and be placed on an insulating
support, the inner plate may be removed by lifting it by
the curved rod ; then the outer plate may be removed from
the glass jar. The two plates are then completely dis-
charged. After putting the jar together again, it can be
discharged with a bright flash. The coatings serve as dis-
chargers for the glass. The charge on two small areas of
the glass may be made to unite with a crackling sound
by touching them at the same time with the ringers. " The
two conducting surfaces may therefore be regarded simply
as the boundaries of the intervening dielectric."

CAPACITY AND CONDENSERS. 209

156. Energy expended in charging a Condenser. —
The energy expended (138) in transferring Q units of
electricity through a potential difference of V — V0 is

Q (F"— Vo). If the charge Q is transferred from the earth,
whose potential is zero, to a conductor whose potential is

V, the work done is Q V. But in charging a condenser, or

any conductor, the potential is zero at the beginning of the
charging and J7" at the end. The mean potential to which
the charge is raised is then \ V. The work done in charg-
ing the condenser is therefore

• But since Q = CV,

The energy expended in charging a condenser to a poten-
tial difference F"is one-half the product of the capacity
and the square of the potential difference. Potential
corresponds to height when work is done against gravity.
Thus in building a brick tower of uniform cross-section A,
the mean height to which the bricks are lifted is half the
final height of the tower ; the work done in gravitational
units is one-half the product of the mass M and the height
h of the tower. If with half the cross-section the tower
is carried to twice the height, the work done is simply
doubled because the same mass is lifted to double the
mean height. If, however, the tower with the cross-section
A is built to twice the height 2A, the work is quadrupled
because both the mass lifted and the mean height are
doubled. The constant is the area A, corresponding to
capacity, and the work done varies as the square of the
height.

157. Energy lost in dividing a Charge. — Let Ox and
C-i be the capacities of two condensers, and let the first be

210 ELECTRICITY AND MAGNETISM.

charged with a quantity Q. The energy of the charge is

1 ■ 2 <7X

After the charge has been divided between the two con-
densers by connecting them in parallel, the potential differ-
ence has fallen to • „ . Hence the energy is then

1(C l c^( Q V- 1 Q*

It is less than the energy before the division so long as C2
has any value in comparison with Cx. If the two capacities
are equal, the energy after the division is one-half as great
as before it. The other half is represented by the energy
of the spark at the moment of the division. The lowering
of the potential by the increase of capacity diminishes the
energy represented by a given charge. Energy is always
lost by the division, whatever be the relative capacities of
the condensers, except when the potential differences of the
two are the same before they are joined in parallel ; but in
this case there is no electric flow and no lowering of the
potential difference.

158. Energy of Similar Condensers in Parallel. — If
n condensers of the same capacity
C are charged in parallel, for ex-
ample, n Leyden jars of the same
size with their outside coatings
connected together, and likewise
all their inside coatings (Fig. 83),
the capacity of the whole is n
times the capaoity of a single condenser, because the effect
is simply to increase the size of the coatings.

CAPACITY AND CONDENSERS. 211

The energy of discharge of a single condenser is %CV2,
and for n condensers of the same capacity it is

The energy is thus increased in proportion to the number
of similar condensers.

159. Energy of Condensers connected in Series. —
If several Leyden jars are insulated and the outside of the
first is connected to the inside of

the second, etc. (Fig. 84), they are

said to be connected in series or

in "cascade." The inside of the

first jar is one side of the compound

condenser, and the outside of the

last one is the other side. If the Fig. 84.

potential difference between the

two sides is V, then the energy of each of the n similar jars

V2
is \Q '_, ; and the energy of the charge in the n jars is
ri~

w= 1 Cir-.

Hence the energy of the charge of the n jars in series is 1/n
of the energy of one of the jars charged to the same poten-
tial difference between its two coatings.

160. Electric Strain. — The phenomenon of the residual
charge may be best explained by considering the dielectric
as the medium through whose agency the induction takes
place. The charging of a Leyden jar is accompanied by
the straining of the glass. If the potential difference is
raised to a sufficiently high value, the glass may be strained
beyond the elastic limit and may give way with a disruptive

212 ELECTRICITY AND MAGNETISM.

discharge of the jar. The glass is then shattered at the
point through which the discharge takes place. In the case
of air or other fluid dielectrics, such as insulating oils, the
dielectric may be broken down by a disruptive discharge,
but the damage is immediately repaired by the inflow of
the insulating fluid.

By subjecting plate glass to powerful electrostatic stress
and passing plane polarized light through it at right angles
to the lines of force, Kerr discovered that glass becomes
doubly refracting, and is strained as if it were com-
pressed along lines of force. Quartz behaves in the same
way, but resins and all insulating fluids examined, except
olive oil, act as if extended along lines of force. The
action requires about thirty seconds to reach its maximum
effect, and about the same time is required for complete
recovery. Kohlrausch and others have pointed out the
analogy between Kerr's discovery and the elastic fatigue of
solids after being subjected to a twisting strain. A fibre of
glass if twisted does not immediately regain its initial form
when released from the stress, but a slight set remains from
which it slowly recovers.

The glass of a charged jar is then greatly strained, and
it does not at once recover when the jar is discharged. Its
after-recovery from distortion sets free energy which is
represented by the residual charge. Hopkinson has shown
that it is possible to superpose several residual charges of
opposite signs. In the same way a glass fibre may be
twisted first in one direction and then the other, and the
residual twists will appear in reverse order. No residual
charges can be obtained from air condensers, nor from those
with quartz plates. Correspondingly, quartz shows no elas-
tic fatigue after being twisted.

Siemens has shown that the glass of a Leyden jar is

CAPACITY AND CONDENSERS. 213

sensibly warmed by rapid charging and discharging. The
distortion shows a lag behind the electric stress, a phenom-
enon known as hysteresis when applied to magnetic induc-
tion. The result in both cases is the absorption of energy
and the generation of heat. The quantity of heat generated
is proportional to the square of the potential difference to
which the condenser is subjected.

161. Dielectric Polarization (B., 573). — Faraday's
theory of induction was that it is an action between con-
tiguous parts of the dielectric, resulting in a certain
polarized state of its particles. In proof of this polariza-
tion he placed bits of dry
silk filaments in turpen-
tine contained in a long
rectangular glass vessel
with pointed conductors
entering from opposite
ends. When one of
these was connected with

the earth and the other with a frictional machine, the bits
of silk collected together along lines of induction, forming
long filaments of considerable tenacity. Matteucci de-
monstrated that the dielectric is polarized or charged by
contiguous particles throughout. He formed a condenser
of a large number of thin plates of mica compressed be-
tween two terminal metal plates (Fig. 85). After charg-
ing it and insulating, it was found on removing the mica
plates and examining them that each one was charged
positively on one side and negatively on the other, all the
positive sides being turned toward the positive electrode,
and all the negative ones in the opposite direction.

Maxwell explains the residual charge by assuming that

214 ELECTRICITY AND MAGNETISM.

the dielectric is not homogeneous, and that it therefore
becomes electrified at the surfaces separating the non-
homogeneous parts, like the electrification of the mica
plates. The reunion of these charges is gradual after the
first discharge, and their external effect is shown as a
residual charge. "According to this theory all charge is
the residual effect of the polarization of the dielectric "
(Maxwell). In the interior of the dielectric the polariza-
tion is neutralized by adjacent opposite charges ; " it is
only at the surface of the dielectric that the effects of the
charge become apparent."

162. Distinction between Conductors and Insulators
(Max., 156). — The potential difference between the
boundaries of a dielectric is the electromotive force acting
on it. If the dielectric is an imperfect insulator, the state
of constraint is continually giving way or being relaxed.
The medium yields to the electromotive force, and the
potential energy of its distortion is converted into heat.
In good insulators the rate at which this conversion takes
place is very slow.

In conductors the electric polarization gives way as fast
as it is produced by the electromotive force, with a steady
transfer of electricity; this transfer is called an electric
current. The external agency which maintains the cur-
rent is constantly doing work in restoring the polarization,
and the result of expending energy on the conductor is the
generation of heat. Non-conductors are capable of main-
taining the state of electric polarization ; in conductors
this polarization breaks down as fast as it is formed. The
application of an electromotive force to the former causes
a momentary transfer of electricity ; its application to the
latter produces an electric current.

CAPACITY AND CONDENSERS. 215

163. Electric Displacement. — Electricity exhibits the
properties of an incompressible fluid. Electric charges
reveal themselves only at the boundaries between conduc-
tors and the dielectric. Lines of induction run from the
positive charge at one boundary through the dielectric to
the negative at the other ; and if we conceive tubes
of induction bounded laterally by lines of induction,
every tube in a dielectric between two conductors has
equal charges on its two ends, or the induced and the
inducing charges are equal to each other. All cases of
electrification are cases of the transfer of electricity. Hence
Maxwell proposed his theory of electric displacement, which
supposes that when an electromotive force acts on a dielec-
tric, as in induction, electricity is transferred or displaced
along tubes of induction. The electromotive force not
only distorts the medium, but transfers electricity by
stretching the dielectric. If a dielectric, polarized by elec-
tric displacement, be left to itself, the elastic reaction pro-
duces a back electromotive force and a reverse electric
transfer to restore the equilibrium. This restoration to
equilibrium constitutes the discharge of the condenser. A
disruptive discharge means the rupture of the dielectric,
usually the air. If the discharge is abrupt, the sudden
release of the dielectric from strain is followed by rapid
electric displacements in opposite directions alternately,
till the energy of the charge is all wasted in heat. This
phenomenon is known as the oscillatory discharge of a
condenser. It was discovered by Joseph Henry in 1842.

164. Electric Transfer always in Closed Circuits. —
The electric-displacement theory of Maxwell has led to a
conception of the electric circuit which allows the contrast
between a conducting and a non-conducting circuit to be

216

ELECTRICITY AND MAGNETISM.

expressed in a simple manner. In a circuit made up
entirely of conductors the operation of an electromotive
force causes a continuous flow of electricity ; but if the
circuit is only partly conducting and in part composed of
a dielectric, then the action of an electromotive force in
the circuit produces a transient electric flow through the
conductor and an equivalent displacement through the di-
electric. The amount of the flow depends on the capacity
of the dielectric as a condenser and the magnitude of the
electromotive force, or Q= CV. Through the conductor
electricity is transferred by the process of conduction ;
through the dielectric it is transferred as a displacement,

Fig. 86.

that is, it is forced along by straining the medium. Dis-
placement always produces a reactive electromotive force
which counterbalances the direct electromotive force and
effects a discharge when the latter is removed.

Consider a circuit of water-pipes filled with water and
containing a pump P (Fig. 86). If there were no
obstructions in the pipes the motion of the pump would
cause a circulation of water through the system. This
arrangement corresponds to a conducting circuit. But
if we imagine elastic diaphragms stretched across the
enlarged pipe at many points, as a, b, c, c?, e, the rotation of

CAPACITY AND CONDENSERS.

217

the pump, so as to produce a flow in the direction of the
arrows, displaces water along the enlarged pipe by stretch-
ing the diaphragms, and causes a transient current through
the remainder of the system. The displacement ceases as
soon as the reaction of the diaphragms equals the force
applied to the pump. The same quantity of water is
transferred across every cross-section of the pipes through-
out the whole system, whether the diaphragms are present
or not. Without the diaphragms the flow would be con-
tinuous ; with them it continues only so long as they yield
to the stress of the water. If the force applied to the
pump be withdrawn, the reaction of the tense diaphragms
produces a counter-flow. The diaphragms represent the
dielectric and the unobstructed pipes the conductor. So
in charging a condenser the same quantity of electricity
is displaced through the dielectric as flows along the con-
ducting part of the circuit.

165. Specific Inductive Capacity. — Different dielec-
trics possess different powers B
of transmitting induction
across them. The density of
the charges at the surfaces of
the condensing plates, with a
given difference of potential
between them, depends not
only on the distance between
them, but also on the facility
with which the dielectric per- '—
mits electric displacement.

Let J., B, C (Fig. 87), be three insulated conducting
plates. To the back of A and C are suspended pith-balls.
Let B receive a positive charge and let A and C be charged

;-•'

- +

+ -

Fig. 87.

218 ELECTRICITY AND MAGNETISM.

negatively by induction. If they are touched with the
finger the pith-balls will collapse and remain in contact
with the plates. If now A, for example, be moved nearer
to B, both pith-balls will diverge, the one on A with a
+ charge and the one on C with a — one. The diminished
distance between A and B permits increased induction
which transfers + electricity to the back of A; but the
increased induction on the left of B diminishes it on the
right, and some of the — charge on C becomes free and
spreads over the back of the plate.

Replace A in the first position, with B charged as before
and the pith-balls not diverging. Interpose between A and
B, without touching them, a thick plate of glass or sulphur.
Both pith-balls will again diverge as if A had been moved
nearer B, showing that the effect is the same as the reduc-
tion of the thickness of the air between the plates. The
capacity of a condenser depends then on the nature of the
insulating medium between the two opposed conductors.
The specific inductive capacity, or dielectric constant, of a
substance is the ratio of the capacity of a condenser with
the substance as the dielectric to its capacity when the
dielectric is air. The dielectric constants of all gases are
nearly the same, but those of solids differ greatly.

166. Faraday's Experiments. — The first experiments
on specific inductive capacity were those of Cavendish, but
they were unknown till Faraday had made his discoveries
in the same subject. Faraday's experiments were made
with two exactly similar condensers shown in section in
Fig. 88. The metallic sphere A is supported by the rod M,
and both are insulated from the outer shell B by a plug
of shellac. The shell B is made in two halves which can
be detached from each other, so that the space between

CAPACITY AND CONDENSERS.

219

A and B can be filled either with a solid dielectric or with
a gas.

When both condensers were filled with dry air and one
of them was charged, it divided its
charge equally with the other on
joining them in parallel, its poten-
tial falling to one-half. The ca-
pacities of the two were therefore
the same. The space within one of
the condensers was then filled with
a solid, such as shellac, and the
above experiment was repeated.
The resulting potential was then
less than half the initial potential.

Let F"be the potential of the air
condenser before the division of the
charge, and C its capacity. If K
is the specific inductive capacity of
the dielectric in the second con-
denser, the capacity of this condenser will be KC. Let
V be the common potential of the two after the division of
the charge ; then

V- V

Fig. 88.

Whence

In this way Faraday obtained for sulphur, as compared
with air, the value 2.26, and for shellac, 2.0.

Faraday's discovery of this property of a dielectric led
him to adopt the view that the effects observed in an
electric field are to be ascribed to the action of the dielec-
tric between electrified bodies, and not to the action of an
electrified body on others at a distance.

220 ELECTRICITY AND MAGNETISM.

167. Recent Results. — The dielectric constant is
smaller in rapidly oscillating fields than in slowly changing
ones, because of the absorption of the charge which takes
place with the continued application of an electromotive
force in one direction. This fact explains to a certain
extent the great discrepancies which are found among the
results obtained by different observers. The following
table illustrates the difference between the values derived
from rapid and from slow methods :

Rapid. Slow.

Glass 3.013 to 3.243

" dense flint 7.37

" light flint 6.72

Ebonite 2.284 3.15

Gutta-percha 2.462

Paraffin (solid) 1.994 2.29

" (liquid) 1.92

Shellac 2.747

Sulphur 2.579 3.97

Mica 6.64

Turpentine 2.23

Distilled water 76.

Alcohol . 26.

Northrup has recently measured the specific inductive
capacity of paraffin and plate glass, both with rapidly
oscillating and slowly changing fields, with the following
results :

Rapid. Slow.

Paraffin 2.25 2.32

Plate glass 5.86 6.25

The following are the specific inductive capacities of
several gases:

Hydrogen 0.999674 Carbon monoxide . 1.001

Carbon dioxide . . . 1.000356 defiant gas . . . 1.000722
Sulphur dioxide . . . 1.0037

CAPACITY AND CONDENSERS. 221

168. Effect of the Dielectric on the Electric Intensity
(J. J. T., 116"). — Consider two parallel-plate condensers
A and B, the plates being at the same distance in the two,
but the dielectric of A being air and that of B a medium
whose specific inductive capacity is K. Let us suppose
the charge per unit area, or the surface density, on the
plates of A and B is the same. Then, since the potentials
are inversely as the capacities when the charges are the
same, and since the capacity of B is K times that of A, it
follows that the potential difference between the plates of
A is K times as great as that between the plates of B.
But in both cases the electric intensity in the dielectric is
the rate of variation of the potential per unit length. Now
as the thickness of the dielectric is the same in A as in
B, while the potential difference of A is K times as great
as in B, it follows that the electric intensity in the air
between the plates of A is K times as great as in the
dielectric of B, or the electric intensity is inversely as the
specific inductive capacity.

We have seen in Art. 152 that the electric intensity F
between two plates in air is 47ro-. Hence in a medium
whose dielectric constant is iT,

V 47TO-

F=~K-

Thus with given charges the forces in the field are dimin-
ished by introducing a medium with a large specific induc-
tive capacity.

169. Effect of the Dielectric on the Forces between
the Plates. — From the equation of the last article,

47T<T=KF=^E,

t
where t is the thickness of the dielectric.

222 ELECTRICITY AND MAGNETISM.

Whence <r = - — .

4tt£

The force on unit quantity on one of the plates, due to
the charge on the other, is JJF, and on unit area it is %Fa.
Hence the force on either plate per unit area is

F<r _ 2tto-2
2 K '

With a given charge, or given surface density, the force
between the plates is inversely as the specific inductive
capacity.

A . . Fa- V KV KV>

Again, since — - = -- . - — = - — ■ ,
B 2 2t 4:7rt 8irt-

it follows that, with a given potential difference, the force
between the plates is directly proportional to the specific
inductive capacity.

PROBLEMS.

1. Two Leyden jars are charged with quantities as 1 to 4. The
tin-foil surface of the second jar is twice as large as that of the first
and the glass is half as thick. Find the relative energy of the two
charges.

2. An insulated metal ball of 10 cms. radius, and removed from
all other conductors, is charged with 100 units of electricity. What
will be its potential if it be then surrounded by a smooth conducting
shell of 11 cms. radius, and connected to earth ?

3. If one of two insulated conducting spheres, 20 cms. in diam-
eter, be charged to a potential of 15 units, and then be connected with
the other sphere, by means of a long thin wire, find the energy of
the discharge between them.

4. Two Leyden jars of 200 sq. cms. tin-foil surface and glass
1 mm. thick, specific inductive capacity 6.283, are charged to poten-
tials of 100 and 10 units respectively. Find the energy lost in ergs
on connecting them in parallel.

5. Find the capacity of a spherical condenser, the radii of the

CAPACITY AND CONDENSERS. 223

opposed surfaces being 9 and 10 cms., and the dielectric paraffin,
whose specific inductive capacity is 2.3.

6. Two circular brass plates 30 cms. in diameter are separated
by glass 2 rams, thick and of specific inductive capacity 6. If they
are charged to a potential difference of 1,000 units, find the force of
attraction between them.

7. In the last problem, find the surface density on the boundary
between the glass and the plates.

224 ELECTRICITY AND MAGNETISM.

CHAPTER XV.

ATMOSPHERIC ELECTRICITY.

170. Lightning an Electrical Phenomenon. — While
some of the early philosophers surmised that the lightning
flash was an electrical discharge, yet this view obtained
but little currency till Franklin's suggestion in 1749 to
apply his discovery of the discharging power of points
to the investigation of the problem had actually been car-
ried into effect. In 1752 d'Alibard, acting on Franklin's
suggestion, erected an iron rod 40 feet high, but not con-
nected with the earth, and drew sparks from passing
clouds. About the same time (1752) Franklin sent up
his famous kite by means of a linen thread, during a pass-
ing storm, and held it by means of a silk ribbon between
his hand and a key attached to the thread. When the
thread had been wetted by the rain, sparks were drawn
from the key and a Leyden jar was charged. The next
year Richmann, of St. Petersburg, was killed by light-
ning while experimenting with a rod similar to that of
d'Alibard.

171. The High Potential of Thunder Clouds. — The
source of the electrification of the atmosphere and of clouds
remains as yet unsettled. But given ever so slight an
electrification of aqueous vapor, it is not difficult to ac-
count for the high potential exhibited by thunder clouds.1

1 Atmospheric Electricity, Carhart, Jour. Am. Elec. Soc, 1880.

ATMOSPHERIC ELECTRICITY. 225

Consider minute spherules of water condensing to large
drops in the formation of clouds and rain. Since the vol-
umes of spheres vary as the cubes of their radii, eight
small drops condense into one of double radius. There-
fore each of the larger drops contains eight times the
charge of the smaller ones. But since the capacity of a
sphere is numerically equal to its radius, the larger sphere
has only double the capacity of the smaller ones. There-
fore its potential, which is the quotient of the charge by
the capacity, is quadrupled. The potential then increases
as the square of the radius of the drops. If the potential
rises according to such a law, the inductive influence and
tendency to discharge from drop to drop through a cloudy
atmosphere rise in the same proportion.

172. Effect of Electrification on Condensation. — It
is a fact of common observation that a small ascending jet
of water is resolved into drops, which describe, divergent
trajectories. By reason of the different velocities and
directions of motion of the individual drops they come into
frequent collision and then rebound. The influence of
electrification on the recoil of the drops after collision is
most marked and interesting. The subject has been in-
vestigated by Lord Rayleigh ' with important results.

If the ascending jet is strongly electrified, the repulsion
between the drops scatters them and prevents collision ; but
with very feeble electrification, the drops coalesce on impact
and the stream is thus rendered much smoother. This coa-
lescence was demonstrated to be due to slightly different de-
grees of electrification in the impinging particles of water.
Their attraction and union appear to be due to induction,
the resulting force of which is always an attraction.

1 Proceeding! of the Royal Soc, Vol. LXXVIII., p. 406.

226 ELECTRICITY AND MAGNETISM.

The bearing of these results on precipitation of aqueous
vapor is obvious. Innumerable minute globules of water,
feebly charged to different potentials, collide and coalesce
into drops which descend by gravity. A slight amount
of electricity in the atmosphere is therefore favorable to
aqueous precipitation, while higher electrical excitation is
unfavorable to it.

It is an observed fact of frequent occurrence that a vivid
flash of lightning is quickly followed by a sudden and
heavy downpour of rain. It is clearly impossible to tell
which is antecedent to the other, the discharge or the con-
densation; for, while the flash reaches the observer first,
light travels from the place of condensation in negligible
time, and the discharge may therefore be subsequent to the
sudden condensation. If the condensation occurs before
the discharge, it is accompanied by a sudden rise of poten-
tial in the enlarged drops, leading to an electric discharge.

173. Lightning Flashes. — Lightning flashes are dis-
charges between oppositely charged conductors. They
may occur between two clouds or between a cloud and the
earth. The rise of potential in a cloud causes a corre-
sponding accumulation in the earth underneath ; and unless
this accumulated charge is carried off by the silent action
of points, when the stress in the air as the dielectric reaches
a certain limiting value, the air breaks down with a sudden
subsidence to equilibrium. J. J. Thomson estimates the
dielectric strength of the air under ordinary conditions of
pressure and temperature to be about 0.41 gm., or 398 dynes,
per square centimetre. When the electric tension along
lines of force is greater than this, a disruptive discharge
takes place. This limiting stress may be reached in two
different ways, which will now be described.

ATMOSPHERIC ELECTRICITY.

227

174. Discharge with Steady Strain. — When the
stress in the dielectric is gradually increased, the medium
is finally strained to the point of rupture, and a discharge
takes place. Under these circumstances a point will offer
protection and effect a silent discharge. This condition
Lodge has called the " steady strain," and has illustrated it
as follows : l A and B (Fig. 89) are the discharge termi-
nals of an influence ma-
chine, L is a Leyden jar,
T and T' two tin plates -
connected with the two
coatings of the jar. On
the lower plate are three
conductors terminating
as shown. Under these
conditions, as the jar is
charged the stress in-
creases gradually ; but the pointed conductor, even when
very low compared with the others, prevents a discharge
altogether. This is true even when a high liquid resist-
ance is interposed between it and the lower tray. If the
point be removed or covered, and the knobs be positive,
long flashes may be obtained, but always to the small knob
until it is about three times as far from the upper plate as
the large knob. The reason for these phenomena is that
the air breaks down at the weakest point, or where the
stress is greatest, and this is at the surface of greatest cur-
vature or smallest area. The high liquid resistance inter-
posed in the path of the discharger makes no difference in
the length of the spark, but does affect its noise and
violence.

1 r ,

Fig. 89.

1 Lodge's Lightning Conductors and Lightning Guards, p. 54.

228

ELECTRICITY AND MAGNETISM.

175. Discharge with Impulsive Rush. — In the last
article the potential difference between the plates increases
gradually till the limit of the dielectric strength of the air
is reached. Lodge has arranged a different experiment to
illustrate the very sudden development of a potential differ-
ence and a discharge with an " impulsive rush."

The two Leyden jars in series (Fig. 90) stand on the
. same wooden table.

-A0 ! — U — They charge gradually,

"" I the outer surfaces

811 ill

Fig. 90.

through the table, and
ultimately discharge at
A. This discharge be-
tween the inner coats
releases the charges on
the outer coats, a violent
rush takes place, produc-
ing a sudden stress in the medium between the plates, and
the conductors are struck. The small knob no longer pro-
tects the larger one, nor does the point exert any special
protective influence. All three terminals are equally liable
to be struck, if of the same height, and all three may be
struck at once. If a liquid resistance is interposed in the
path of either, it fails to protect the other two even if
taller than they. In this case the electric pressure is devel-
oped with such impulsive suddenness that the dielectric
appears to be as liable to break down at one point as
another. Such sudden rushes are liable to occur when two
clouds spark into each other, and then one overflows into
the earth. The highest and best conducting objects are
then struck irrespective of points and terminals. The
conditions determining the path of the discharge in the
case of these impulsive rushes are entirely different from

ATMOSPHERIC ELECTRICITY. 229

those of the steady strain, and points are incompetent to
afford protection by preventing them.

176, Lightning Protectors. — The revision of theory
and the results of experiment have left much of former
recommendations relating to lightning protectors of doubt-
ful value. Some of the reasons for this statement will
appear in treating current induction in a later chapter ;
enough has already been said to furnish a basis for a few
simple directions concerning protection from lightning.

For the condition of steady strain pointed conductors are
still advisable ; but it is not necessary to provide the
elaborate terminals formerly deemed essential. Nor is a
copper conductor of large section necessary or desirable.
It is far better to provide a number of paths for the
discharge down several different parts of a building, each
consisting of a large galvanized-iron wire sharpened at the
top, avoiding short bends and loops, and ending in a mass
of iron or charcoal buried in moist earth. Such a conduc-
tor may be fastened directly to the building without insula-
tors. It is probable that a number 4 or 6 iron wire, B.S.G.,
will safely carry off any discharge that is likely to traverse
it. The writer has known a much smaller iron wire to
conduct safely a discharge which converted smaller copper
wire into metallic vapor and did other damage. It is
not desirable that the lightning conductor should have a
very low resistance. If it is large enough to convey the
current without fusion, it will dispose of the energy of
the discharge at a safer rate than a larger conductor
would.

Tall chimneys may be adequately protected by three or
four iron wires ranged around the outside, not placed
together, but connected at frequent intervals, and all well

230 ELECTRICITY^ AND MAGNETISM.

grounded. Since the heated air of a chimney furnishes an
easy path for lightning, it is well to connect the iron wires
with a copper band just above the mouth of the chimney.

The expense of erecting such lightning guards is merely
nominal. When coal is burned they will need renewal
occasionally on chimneys ; the expense of such renewals is
inconsiderable, but the need of repairs is often overlooked
till the damage is done.

177. Method of measuring the Potential of the Air.
— The earth is almost always negative relative to the air,
and the potential of the latter increases with the elevation
above the surface. The quadrant electrometer has done
excellent service in these determinations. To put the
needle or one pair of quadrants in electrical equilibrium
with the air at any elevation, the slow match and the
water-dropping collector are the most effective. Both of
these, when insulated from the earth, furnish means of
electrical convection by the disengagement and release
of small particles. Each small mass carries with it an
electrical charge, and the potential of the conductor is
thereby quickly brought to that of the equipotential sur-
face" of the air passing through the point from which the
matter breaks away. The water-dropper is a well-insulated
reservoir from which a long tube extends through an
opening in the wall, so that the nozzle is in the open air.
In half a minute after turning the tap, the potential of the
system is reduced to that of the air at the point where the
jet of water ceases to be a continuous stream.

Mascart's method of using the quadrant electrometer for
this particular purpose is preferable to the older procedure.
The middle point of a large number of cells of battery,
or simple elements of zinq and copper in distilled water

ATMOSPHERIC ELECTRICITY. 231

(183), is put to earth, while one of its terminals is con-
nected to one pair of quadrants and the' other terminal to
the alternate pair. The water-dropping collector is con-
nected to the needle. The alternate quadrants are then
charged to equal potentials of opposite sign, and the
amount and direction of the deflection depends on the
value and sign of the charge conveyed to the needle.

178. Results of Observation. — Disruptive discharges
occur when the stress in the air exceeds the limit of its
dielectric strength (173). The needle of the electrometer
becomes very much agitated on the approach of a thunder
cloud ; and after various fluctuations it settles down to a
steadily increasing deflection in one direction until a flash
of lightning occurs, when the needle darts back to zero.
The lightning flash indicates a return of the strained
medium to equilibrium.

In clear weather the potential of the air is sometimes
nearly as high as during a storm, but shows smaller fluctua-
tions. The value of the potential gradient found by
McAdie at the Blue Hill Observatory,1 as the result of
over a thousand observations, was 540 volts (189) for a
difference of elevation of 138 metres. This is equivalent
to 3.91 volts per metre, or 0.00013 electrostatic units per
centimetre of elevation. On certain clear days the varia-
tion of potential with the elevation reached twice this
value, or about 8 volts per metre. During thunder storms
the potential gradient may amount to 35 volts per metre,
or 0.0012 electrostatic units per centimetre.

By means of kites McAdie has shown that the potential
difference in clear weather increases as the kite rises ; and,
further, that it is possible to obtain sparks from a perfectly

1 Annals of the Astron. Observ. of Harvard College, Vol. XL., Part I.

232 ELECTRICITY AND MAGNETISM.

cloudless sky, and generally at an elevation not exceeding
500 metres.

From a long series of observations at Washington, Pro-
fessor Mendenhall concludes that the electrical condition
of the atmosphere furnishes no reliable data for weather
forecasts.

179. The Aurora. — The aurora, or polar light, is due
to silent or brush discharges in the upper regions of the
atmosphere. In the arctic regions it occurs almost nightly,
but with varying intensity. Lemstrom has shown that the
illumination of the aurora is due to currents of positive
electricity passing from the higher regions of the atmos-
phere to the earth. In our latitude these silent discharges
in the rarefied atmosphere are infrequent. When they are
visible they are accompanied by great disturbances of the
earth's magnetism and by earth currents. In polar latitudes
the irregular motions of the magnetic needle indicate the
coming of auroral displays. These magnetic disturbances
are sometimes of simultaneous occurrence in widely sepa-
rated portions of the earth.

PRIMARY VELL8. 233

CHAPTER XVI.

PRIMARY CELLS.

180. Steady Currents. — It has been shown that a
Holtz influence machine, when rotated uniformly, is capa-
ble of producing an electric current; that is, a uni-
form as distinguished from a transient flux of electricity
through a conducting circuit. But the resistance which
the machine itself opposes to any transfer of electricity
reduces the current to a very small value.

To produce a uniform electric current through a con-
ductor, a constant potential diffarence must be maintained
between its terminals. The quantity which flows in unit
time along the conductor is called the strength or intensity
of the current. It is impracticable to effect this uniform
flow by an influence machine, and much more so by a
frictional machine. It may be done by the application of
chemical energy, as in the voltaic cell ; by the application
of heat, as in the thermal couple ; or by the application of
mechanical energy, as in the dynamo machine. In all
three cases the energy applied is converted, at least in
part, into the energy of the transport of electricity under
an electric pressure equal to the potential difference
established by the apparatus. These three methods of
maintaining a difference of electric potential will be taken
up in order.

181. Volta's Pile. — The modern electrical era dates
from Galvani's discovery, in 1786, that muscular contrac-

234

ELECTRICITY AND MAGNETISM.

tions are produced when a bimetallic arc of iron and
copper connects the lumbar nerve and the crural muscle of
a freshly killed frog. In the hands of Volta this observa-
tion ripened into the discovery that a potential difference
is established by the contact of two
different metals, such as zinc and
copper, especially if they are sepa-
rated, except at the point of contact,
by moist cloth. Volta constructed a
chain of elements to which in 1800
he gave the name artificial electric or-
gan, but which has since been known
as the voltaic pile.

It consisted of many disks of cop-
per and zinc, either placed in contact
or soldered together in pairs, and
piled up with interposed layers of
cloth moistened with water, or with a
solution of salt. The order of assem-
blage was zinc-copper-cloth, zinc-cop-
per-cloth, from bottom to top. ' Fig.
91 shows one of the early forms, with
zinc at the bottom and copper at the
top. The column was held in place
by glass rods. The bottom disk was
called the negative pole and the top
one the positive. A pile composed of from twenty to
forty pairs produced sensible physiological effects when
the experimenter grasped the two terminal wires n and
p with moistened hands, or placed them on the tongue.

Fig. 91.

182. The Dry Pile. — Behrens and Zamboni replaced
the cloth in Volta's pile with paper, and made what was

PRIMARY CELLS. 235

called a dry pile. It was made of gold and silver paper,
the former coated on one side with copper foil and the
latter with tin foil. Sheets of these papers were placed
together with their metallic sides outward, and small disks
cut from them were piled up to the number of many hun-
dreds or even thousands, in such a way that the copper of
all the pairs was turned in the same direction. Such dry
piles were capable of charging Leyden jars and of pro-
ducing shocks.

In the Clarendon laboratory at Oxford is an instrument
consisting of two dry piles connected at the top and ter-
minating at the bottom in two tiny bells close together,
and composing the positive and negative poles. A minute
ball is suspended between them by a silk thread. The
little ball gets a charge from one bell and conveys it to
the other. The electric field between the two bells is
strong enough to keep the ball swinging and to make a
soft but audible tinkle. It was set up in 1840, and has
been ringing ever since. The energy required is very
small, and is necessarily limited by the energy stored up
in the materials of the pile. Dry piles constitute a
transition device between a frictional machine and a vol-
taic cell.

183. Simple Voltaic Element. — If a strip of zinc
amalgamated with mercury be placed in sulphuric acid
diluted with about twenty times its volume of water,
bubbles of hydrogen will collect on the zinc, but the
chemical action will soon apparently cease. No change
will be produced by placing a strip of clean copper in the
same solution until the two metals are connected either
directly or by means of some good conductor (Fig. 92) •
The acid then attacks the zinc, hydrogen is freely liberated

236 ELECTRICITY AND MAGNETISM.

at the surface of the copper plate, and a dense solution of
zinc sulphate streams down from the zinc. The liquid
product of the chemical action appears at the zinc plate,
and the gaseous product at the copper. As soon as the
connection between the two metals is interrupted, chemical
action ceases and hydrogen is no longer disengaged.

If the two plates be connected to
/^""^T opposite sides of a quadrant electrom-

c|L zjJ eter, it will be found that the zinc

PIBESk is negative and the copper positive.
A potential difference is thus estab-
lished between the two plates by
^^-1| afc.. immersing them in the acid solution.
^feJEzJHII The copper strip is called the positive
^!g^a electrode, and the zinc the negative.

Such a system of two metals im-
mersed in a liquid which acts chemically on one of them
constitutes a simple voltaic cell or element. The negative
electrode is usually zinc, the positive one may be copper,
silver, or platinum ; while for the exciting liquid Volta
used water, salt water, sulphuric acid, hydrochloric acid,
or a caustic alkali.

When the plates are joined by a conductor a number of
new phenomena appear, which are ascribed to an electric
current flowing through the conductor from the copper to
the zinc, and through the liquid from the zinc to the
copper. The zinc wastes away, and the energy of its union
with the acid is in part given out by degrees as the energy
of the electric current, which may be made to do work or
to generate heat.

When a number of voltaic cells are joined together they
compose a voltaic battery.

PBIMARY CELLS. 237

184. Chemical Action in the Simple Voltaic Cell. —
The chain of elements in the cell is as follows :

Zn | HtSOi+aq. | R.SO^ aq. \ Ou.

The operation, which is repeated over and over, may be
indicated thus :

Zn | KSOi+aq.imSOt + aq. | Cu,

giving ZnSOA + aq. | HMO^ + aq. \ R2 \ Cu.
The arrow shows the direction of the current through the
cell. The zinc and hydrogen carry positive charges in one
direction, while the " sulphion," or S0t, carries a negative
charge in the opposite direction, and the sum of these two
kinds of charges carried per second is the value of the
current. The dissociated atoms or molecules, such as zinc
and jSOa, are called ions. All metals and hydrogen are
electro-positive ions ; that is, they travel with the current
and carry positive charges through the electrolyte, the liquid
solution through which a current passes. An electrolyte
conducts only by means of the migration of these ions, set
free by electrolytic dissociation. Molecules not decomposed
are electrically neutral. Only the dissociated molecules
are instrumental in conducting a current. Clausius sup-
posed that dissociation and recomposition of molecules in
a solution are going on continuously; but the view now
acquiring prominence is that conduction by an electrolyte
depends on permanent and not momentary dissociation of
the positive and negative ions. According to this view
the separated ions convey their electric charges with a
small but calculable velocity through the electrolyte,
instead of by a series of decompositions and exchanges as
illustrated above.

238 ELECTRICITY AND MAGNETISM.

A chemical system in which the changes of energy, asso-
ciated with the changes of matter, produce a difference
of electric potential is called a voltaic cell. A voltaic cell
must contain an electrolyte, either a solution in water or a
molten salt.

185. Electromotive Force. — Electromotive force
(E.M.F.) is the cause of an electric flow. It is often
expressed as an electric pressure, from its analogy to water
pressure. Volta supposed the origin of the electromotive
force of a voltaic cell to be at the contact of the zinc and
copper; but while there certainly is an E.M.F. of contact,
it is much too small to account for the observed E.M.F.
of a voltaic cell. It is more rational to suppose that the
seat of the E.M.F. is at the point where the transforma-
tion of the energy takes place ; that is, at the contact of the
zinc and acid. There is also an opposing E.M.F. at the
contact of the copper and the acid, but the former is
the larger, and the difference of the two is the effective
E.M.F. of the cell.

The E.M.F. of any form of voltaic cell depends on the
materials employed, and is entirely independent of the size
and shape of the plates. It is modified by their oxidation
and by the density of the solutions. Oxidation of the
copper plate increases the E.M.F., while oxidation of the
zinc plate diminishes it.

The E.M.F. of a cell is the measure of the work re-
quired to cause a unit quantity of electricity to flow round
the entire circuit. If the two poles of a cell be connected
with two parallel plates composing a condenser, then a
momentary transfer of electricity takes place through-
out the circuit, by conduction through the cell and the
conductors, and as an electric displacement through the

PRIMARY CELLS.

. 239

dielectric between the plates of the condenser. The plates
will then be maintained at a difference of potential, and
this potential difference is equal to the electromotive force
of the cell.

A voltaic cell is a device to produce E.M.F., or electric
pressure. It does not generate electricity, but generates
the E.M.F. which sets electricity flowing.

nffgffv

186. Electromotive Force and Potential Difference.
— The potential difference between the points A and B
(Fig. 93) is the work
which must be done in
the transfer of the unit
quantity of electricity
from A to B through
the external circuit R.
It is often called the
fall of potential from
A to B. It is the part
of the E.M.F. of the
cell necessary to drive "
the given current
through the external
resistance R. Work must also be done in carrying the
unit quantity from the negative terminal B through the
cell to the positive terminal A. The E.M.F. of the cell is
the total work expended in carrying the unit quantity
round the entire circuit. Electromotive force and poten-
tial difference must not be identified. The former should
be regarded as establishing the latter rather than the
reverse. It is quite possible to imagine conditions under
which a current may flow through a uniform conductor
without any potential difference between different points

Fig. 93.

240 ELECTRICITY AND MAGNETISM.

of it, but not without the existence of an E.M.F. The
potential difference between any two points of a circuit
is numerically equal to the E.M.F. producing the current
from the one point to the other when the circuit between
the points contains no source of E.M.F. The current then
flows from the point of higher to the point of lower poten-
tial. In the interior of the cell the current flows across
from the zinc to the liquid, or from lower to higher
potential. It is forced upward by the E.M.F. which has
its seat there. This E.M.F. may be compared to a pump
which sets water circulating through a system of hori-
zontal pipes against friction. In any portion of the system
the force producing the flow between two points is the
difference of water pressure between those points. The
force is all applied, however, at the pump, and this pro-
duces the pressure throughout the system. Electricity
stored up in a condenser is under pressure just as water
lifted against gravity is under pressure. In both cases a
flow will be produced by this pressure if the requisite
conditions are supplied.

187. Polarization. — If the circuit of a simple voltaic
element be closed the current will fall off rapidly in
intensity, and will at length almost cease to flow. The
hydrogen covering the copper plate as a film produces a
state known as the polarization of the cell. Polarization is
a counter E.M.F. set up by the tendency of the hydrogen
to oxidize. Hydrogen, like zinc, is an electro-positive
element, and produces an E.M.F. opposed to that due to
the union of the zinc and the acid.

Besides generating an E.M.F., the hydrogen film intro-
duces a resistance or obstruction to the flow of the current
from the liquid to the copper. This is an additional reason
for the weakening of the current.

PRIMARY CELLS. 241

188. Depolarization by Chemical Means. — Any
device that will prevent the liberation of hydrogen and
its deposit on the positive electrode will largely obviate
polarization. It will not, of course, prevent the falling off
in the current on account of the exhaustion of materials in
immediate contact with the plates. This defect may be
ascribed to the slowness with which the liquid contents
of the cell diffuse.

Let a cell be made by placing in a small glass jar enough
chemically clean mercury to cover the bottom, and filling
with a saturated solution of common salt. Hang a plate
of zinc in the liquid, and thrust into the mercury the
exposed end of a rubber-covered copper wire to serve as
the positive terminal. Close the circuit through some
simple current indicator, such as a common telegraph
sounder of a few ohms resistance. The armature will be
drawn down strongly at first ; but in the course of a few
minutes the magnet will release it, showing that the cell
has become polarized. The action of the released electro-
positive sodium on water at the surface of the mercury
produces sodium hydroxide and hydrogen.

Keeping the circuit closed, drop into the cell a very
small piece of mercuric chloride (HgCl^) no larger than the
head of a pin. The armature of the sounder will be sud-
denly drawn down, showing recovery of the cell from
polarization. The mercuric chloride furnishes chlorine
atoms which unite with the hydrogen on the surface of the
mercury, and so reduce the polarization. The chloride
will be exhausted in a few minutes, and polarization will
again ensue.1

189. The Daniell Cell.— The first cell practically free

* This experiment is due to D. II. Fitch.

242

ELECTRICITY AND MAGNETISM.

from polarization was the invention of Professor Daniell,
of London. In this cell the liberation of hydrogen is
entirely prevented by surrounding the copper plate with
a saturated solution of copper sulphate (CuSOi), so that
electro-positive copper instead of electro-
positive hydrogen is deposited on the
copper plate. A zinc bar Z (Fig. 94)
is immersed in the acidulated water in
an unglazed earthenware cup P ; the
copper plate C is a cylinder of sheet
copper surrounded with a saturated so-
lution of CuSOi. Some spare crystals
of this salt should be added to supply
the waste during the action of the cell.
The E.M.F. of a Daniell cell is a little
over one volt. The volt is the prac-

Figr. 94.

tical unit of E.M.F. (295).

190. Chemical Action in the Daniell Cell. — With
acidulated water the chemical processes may be represented
as follows :

Znx | HMO, | H2SO, \\ CuSO, \ CuSO, \ Cuy.

After the first step in the reaction this becomes
Znx_, | ZnSOA | H2tS04 \\ mSO, \ Cu80A \ Cuy+l.

The direction of the current through the cell, indicated
by the arrow, is the direction followed by the electro-
positive elements, Zn, IT, and Cu. They are said to migrate
from the negative toward the positive pole.

It is better to immerse the zinc in dilute zinc sulphate

PRIMARY CELLS. 243

than in acidulated water. The chain of elements is then

• Zn]~Znsbt | ZnS04 \\ CuSO, | CuS04 \ Cu.

Hydrogen then takes no part in the operation. In either
case, zinc enters into combination as ZnSOi and metallic
copper is liberated. The zinc sulphate increases in amount
and the copper sulphate decreases.

Advantage is often taken of the difference in density of
the two sulphate solutions to effect a separation between
them. The copper electrode is then placed in the bottom
of the jar with the CuSO^, and the zinc is suspended in the
lighter ZnSO^ near the top. Such an arrangement is known
as a gravity cell. It must be kept at work to prevent the
diffusion of the CuSOt upward as far as the zinc plates.

191. Theory of the Production of a Current. — A
brief summary of the modern electro-chemical theory
respecting a voltaic element may be reviewed with profit
without committing ourselves to its truth. When a metal
is immersed in a solvent, there is present an expansive force
tending to drive its molecules into solution. It is analogous
to the expansive force producing sublimation, and is called
" solution tension." Opposing this force is the pressure of
the dissolved atoms of the metal analogous to vapor press-
ure; this follows the laws of Boyle and Charles, and is
called "osmotic pressure." Besides^ all metal ions carry
positive charges. Hence when a metal, like zinc, is dipped
into acidulated water, containing free hydrogen and
sulphion ions, electro-positive zinc atoms are driven into
solution until the solution tension comes into equilibrium
with the osmotic pressure and the electrostatic repulsion
tending to drive these atoms out of the solution. It does
drive hydrogen out against the zinc. The same process

244 ELECTRICITY AND MAGNETISM.

goes on with copper, but its solution tension is less than
that of zinc.

When zinc is placed in zinc sulphate and copper in
copper sulphate, the two solutions being kept apart by a
porous diaphragm, zinc goes into solution by its solution
tension, and the resulting osmotic pressure throughout the
liquid drives copper atoms out of solution till there is equi-
librium, — on the copper side by the solution tension and
electrostatic repulsion between the positive charge, acquired
by the copper plate, and the electro-positive copper ions in
the one direction, and the osmotic pressure in the other.
It is assumed without apparent justification that the ions
have large electrostatic capacity.

If now the circuit be closed a transfer of electricity takes
place through the conductor, the equilibrium can no longer
be maintained, and there is a continuous solution of zinc
and a continuous reduction of copper, both these electro-
positive ions carrying positive charges and thus producing
an electric current. As the density of the zinc sulphate
increases, the number of free zinc ions increases, with a cor-
responding increase of osmotic pressure. If at the same
time the density of the copper sulphate decreases, the
osmotic pressure on the copper ions decreases. Both actions
weaken the electromotive force which drives the ions
across with their charges. It is easily seen that the current
consists of the existent charges which are only passed on
by the moving ions. As the copper ions are driven out,
the zinc ions take their places in combination with SO^.

192. Chemical Action in Relation to Energy. — It is
desirable to add to the theory outlined that the chemical
displacement involved is conditioned on the fact that the
energy of combination of ZnSOi is greater than that of

PRIMARY CELLS.

245

CuSOA. Hence the energy expended in decomposing
CuSOi is less than that evolved in the formation of an
equivalent quantity of ZnSO±. The heat of formation of
65 gms. of zinc to form ZnSOA is 242,000 calories, while
that of 63.4 gms. of copper, a chemically equivalent
weight, to form CuSOi is 191,400 calories. The difference
of 50,600 calories must be released as heat, or in the form
of the kinetic energy of an electric current. The mate-
rials in the cell represent potential energy, and potential
energy tends to become kinetic whenever the conditions
will permit of the transformation. The sole condition in
the Daniell cell is that the circuit shall be closed.

Fig. 95.

193. The Bunsen Cell. — A cleft cylinder of zinc is
immersed in dilute sulphuric acid, and within a porous

246 ELECTRICITY AND MAGNETISM.

cup is a prism of hard-baked carbon surrounded by strong
nitric acid (Fig. 95). When the electro-positive hydrogen
passes through the porous cup toward the positive elec-
trode it encounters the nitric acid. The acid acts as a
powerful depolarizer by oxidizing the hydrogen. Nitric
acid is a good conductor, the E.M.F. of the cell is nearly
twice as great as that of the Daniell, and a current of
several amperes may be taken from it.

Bunsen's cell is a modification of Grove's, and differs
from it only by the substitution of hard carbon for plati-
num as the positive electrode. The sole advantage of the
Bunsen is in point of economy.

A useful modification of this cell, in which the corrosive
nitric acid is avoided, is made by placing the zinc in the
porous cup, and several carbon rods, for example, electric-
light carbons, in a circle around the porous cup. The
liquid in which they are immersed is a saturated solution
of potassium nitrate, acidulated with about one-tenth its
volume of strong sulphuric acid. Sodium or ammonium
nitrate may be used instead of the potassium salt.

194. The Bichromate Cell. — This cell differs from
the Bunsen only in the character of the depolarizer. If
sodium (or potassium) bichromate in solution be treated
with sulphuric acid, chromic acid (CrOa) is formed. This
acid is rich in oxygen and gives it up readily to nascent
hydrogen. If the porous cup holding the carbon prism
be filled with a strongly acid solution of the bichromate,
the E.M.F. of the cell will be about the same as if nitric
acid were used. Since both liquids now contain sulphuric
acid, the porous cup may be dispensed with.

To prepare the solution, dissolve 200 gms. of sodium
bichromate in a litre of water and add 150 c.c. of strong

PRIMARY CELL 8.

247

sulphuric acid. When the solution begins to show signs

of exhaustion, add 25 to 30 c.c. of acid per litre. The

sodium salt is greatly to be preferred to

the potassium salt. It dissolves more

freely and without heat, and it does not

form double salts with chromium, which

crystallize out and are somewhat difficult

of removal. The E.M.F. is about the same

as that of the Grove or Bunsen.

Fig. 96 is a common form of bichromate
cell, in which the zinc plate Z can be lifted
out of the solution by the rod a when the
cell is not in use.

Fig. 96.

195. Local Action and Amalgamation.
— The zinc of commerce contains impu-
rities, such as bits of iron and carbon.
These form local closed circuits when the zinc is im-
mersed in an acid solution, and chemical action goes on
when the circuit is open, with constant waste of zinc.
This chemical action, which contributes nothing to the
current from the cell, is called local action. The chief
remedy against it is the amalgamation of the zinc by clean-
ing it with sulphuric acid and rubbing over the surface a
little mercury. The mercury readily alloys with the zinc
and forms an amalgam. Zincs used in an acid solution
should always be amalgamated.

The immunity of amalgamated zinc from attack is due
to the smooth amalgamated surface. The hydrogen is
given off from it less freely than from a rough unamalga-
mated surface. The solution tension of amalgamated zinc
is greater than that of common commercial zinc ; and the
former, opposed to the latter in an acid solution, forms a

248

ELECTRICITY AND MAGNETISM.

negative electrode. With amalgamated zinc in dilute acid,
the chemical action is soon arrested under atmospheric
pressure ; but if the pressure on the liquid be reduced by
an air-pump, hydrogen will be freely evolved and the zinc
will waste away. The liberation of hydrogen from zinc in
dilute sulphuric acid, or from sodium amalgam and salt
solutions, can be brought to a standstill by sufficient press-
ure.1 The amalgamation of the zinc reduces the pressure
necessary to arrest chemical action.

196. The Leclanche" Cell. — An-
other class of cells employs a solid
depolarizer. The most important of
these from a practical point of view
for working electric bells, telephone
transmitters, and other like purposes,
is the cell invented by Leclanche*.
It is a zinc-carbon couple, with a
nearly saturated solution of ammo-
nium chloride as the electrolyte,
and manganese dioxide (iHfw(92) as
the depolarizer. The carbon elec-
trode is packed in a porous cup
with the manganese dioxide in granules mixed with broken
carbon to increase the conductivity. The zinc is a rolled
rod about one centimetre in diameter. Fig. 97 shows a
cell complete. The porous cup in this particular form has
a flange resting on the top of the glass jar. This closes it
and prevents evaporation.

If the circuit be kept closed for several minutes, the
accumulation of hydrogen on the carbon plate produces
polarization ; but on opening the circuit again, the depo-

1Nernst's Theoretical Chemistry, Trans, by Palmer, p. 613.

Fig. 97.

PRIMARY CELLS. 249

iarizer slowly removes it with recovery of the E.M.F. No
serious local action takes place on open circuit. This cell
will stand without material waste for months or even years.
It is therefore well suited for domestic purposes.

197. Chemical Action in the Leclanch.6 Cell. — When
the circuit is closed zinc displaces ammonium from the
amnionic chloride, and the ammonium breaks up into
ammonia and hydrogen, the former escaping when the cell
is worked hard, and the latter being oxidized by the black
oxide of manganese. Zinc chloride is formed at the expense
of zinc and amnionic chloride. When a Leclanche* cell has
been left undisturbed for some time, it will be found that
the zinc is eaten away toward the surface of the liquid, or
is cone-shaped, with the large end at the bottom. This
coning is due to local action arising from a difference in
the composition of the liquid at the top and bottom. The
double chloride of zinc and ammonium settles down
towards the bottom of the cell; and zinc in ammonium
chloride is negative to zinc in this dense double salt, and
wastes away slowly as the negative electrode, the lower
end of the rod being the positive. There appears to be no
remedy for this kind of local action. It goes on with zinc
in a zinc salt if the density is greater at the bottom than at
the top.

Leclanche cells are sometimes made portable by filling
the space inside the cell with a spongy mass, consisting of
oxide of zinc, plaster of Paris, dextrine, starch, lime,
chloride of zinc, and ammonium chloride. The cell is
then known as a dry cell.

198. The Copper Oxide Cell. — In general, solid depo-
larizers are less effective than liquid ones. But there are

250

ELECTRICITY AND MAGNETISM.

two notable exceptions, oxide of copper and chloride of
silver. Both of them readily give up their electro-negative
ion to nascent hydrogen, and become excellent conductors
by reduction of the metal.

The copper oxide cell was invented by Lalande and
Chaperon. A spiral of zinc is immersed in a solution cf
caustic potash or soda, containing 30 to 40 parts of the
alkali to 100 of water. The posi-
tive electrode is either iron or cop-
per in contact with cupric oxide.
One of the early forms is shown in
Fig. 98, where D is the zinc spiral,
A an iron cup containing the cupric
oxide B, and 6r a caoutchouc tube
surrounding the zinc at the surface
of the. liquid. The liquid is covered
with a layer of heavy paraffin oil to
prevent access of the carbon diox-
ide of the air to the caustic alkali.

The zinc replaces hydrogen in
the alkali, forming sodium zincate
(Na.Zn 0.,} ; the ejected hydrogen,
migrating with the current, abstracts oxygen from the
cupric oxide, and metallic copper is reduced.

In the Edison-Lalande cell the copper oxide is employed
as a compressed plate held in a copper frame. Such a plate
may be made by mixing cupric oxide with five or ten per
cent of magnesium chloride and heating the thick mass in
an iron mould.

Fig. 98.

199. The Silver Chloride Cell. — The metallic ele-
ments are zinc and silver, and on the silver is cast silver
chloride as the depolarizer. The exciting liquid or elec-

PRIMARY CELLS.

251

Fig. 99.

trolyte is a dilute solution of amnionic chloride containing
23 gms. to the litre of distilled water. A denser solution
dissolves silver chloride. In this cell, as made by Warren
de la Rue, the silver wire and its chloride were surrounded
by a small cylinder
of parchment paper
to prevent internal
short-circuits. The
zinc rod and silver
wire were held in
a paraffin stopper,
and the cells were
connected in series
by wedging the sil-
ver wire of one cell
into the zinc rod of
the next (Fig. 99).
By joining 15,000 of these cells in series, de la Rue per-
formed many of the experiments usually conducted by
means of an influence machine. This cell polarizes but
slightly and recovers promptly, but it can be used for
small currents only.

200. The Clark Standard Cell. — The E.M.F. of the
Daniell cell is more nearly constant than that of any of
the others thus far described. The cell first made by
Latimer Clark, and since investigated by many physicists,
has a perfectly constant E.M.F., if set up and used in
accordance with specifications which have received national
approval.1 The cell has now been adopted as an interna-
tional standard of E.M.F.

The negative electrode is either pure zinc or a 10 per

1 Carhart and Patterson's Electrical Measurements, p. 176.

252

ELECTRICITY AND MAGNETISM.

cent amalgam in a neutral saturated solution of zinc
sulphate, and the positive electrode is pure mercury in
contact with a paste of mercurous sulphate. The cell
must contain zinc sulphate crystals in excess. A portable

form is shown in Fig. 100, in
which the contents are kept
from mixing by the asbestos
fibre and the form of the zinc.
Its E.M.F. is 1.434 volts at 15°
C. It diminishes by about 0.001
volt per degree rise of tempera-
ture between 10° and 25° C.

Von Helmholtz, in 1882, sug-
gested the substitution of the
chlorides of zinc and mercury
for the sulphates. The E.M.F.
is then lower, and may be made
exactly one volt by adjusting
the density of the zinc chloride
solution. The temperature coefficient is only about one-
eighth as large as that of the Clark cell containing excess
of zinc sulphate crystals.

Weston has modified the Clark cell by substituting
cadmium and cadmium sulphate for zinc and its sulphate.
The E.M.F. is then slightly above one volt, and the varia-
tion with temperature is very small.

Pt.wire — yt=^==Hzr^z-=.—

Fig. 100.

201. Data relating to Cells. — It is convenient to
collect in tabular form the following data relating to the
cells described :

PRIMARY CELLS.

253

Cell.

Negative
electrode.

Excitant.

Depolarizer.

Positive
electrode.

Approxi-
mate volts.

Volta

Zinc
Cadmium

Copper

Carbon

Platinum

Carbon

Copper

Silver

Mercury

1.0

Daniell ....
Bunsen ....

Zinc-carbon . .
Bichromate . .
Leclanche' . . .
Lalande ....
Silver chloride,

Calomel ....
Weston ....

ZnSOi+aq.

II2SOi+«q.

it u

NHiCl+aq.
NaOH
NH^l+aq.
ZnSOA+aq.
Zn Cl^-^-uq.
CdSOi \-aq.

CuSOi+aq.
HNOz

NaNO-^n^SO^

2faiCr201+ff„SOi

3In02

CuO

AgCl

ffff2S04

/Tg2Cl2

IJg2SOi

1
1
1
1
1
0
1
1
1
1

434

022

202. Effects of Heat on Voltaic Cells. — Two differ-
ent effects are produced by heating a voltaic cell. The
resistance of the liquid to the passage of the current is
lessened, and the E.M.F. suffers a small change, either an
increase or a decrease.

I
o

^— 1

50°

r.V

Fig. 101.

Professor Daniell found that a larger current was ob-
tained from his cell when he heated it to 100° C. This
result is due to the fact that the relative decrease in the
internal resistance of the cell is much larger than the rela-
tive diminution in the E.M.F. The curve in Fig. 101

254 ELECTRICITY AND MAGNETISM.

shows the relation between the internal resistance and the
temperature of a Daniell cell between 15° and 68° C. The
resistance is reduced to less than one-half its initial value.
The temperature coefficient of a Daniell cell is only
about 0.007 per cent; that is, the E.M.F. falls 0.007 volt
for 100° rise of temperature. The Clark standard has a
larger coefficient. Its E.M.F. at any temperature t may
be found from the formula,

.£=1.434 {1-0.00077 (*-15)} volts.

The temperature coefficient of a calomel (von Helm-
holtz) cell is positive, and only about 0.01 per cent for one
degree C.

ELECTROLYSIS. 255

CHAPTER XVII.

ELECTROLYSIS.

203. Electrolytes. — Metals, carbon, and some other
substances conduct electric currents without any percep-
tible effect on them except an elevation of temperature,
due to the resistance which they offer. But chemically
compound liquids conduct only by undergoing decompo-
sition. If, for example, a current passes between two
platinum plates immersed in dilute sulphuric acid, chemical
decomposition takes place, oxygen is liberated at the
platinum plate by which the current enters the solution
and hydrogen at the plate by which it leaves. This pro-
cess of decomposition by an electric current is called elec-
trolysis, and the substance undergoing decomposition or
dissociation is an electrolyte. Electrolytes may be solids,
liquids, or even gases. Iodide of silver is an example of a
solid electrolyte ; while dilute acids, solutions of metallic
salts and alkalis, and some fused solid compounds are
examples of liquid electrolytes.

The conductors by which the current enters and leaves
the liquid are called electrodes — the former the anode and
the latter the cathode.

The ions into which a substance is divided by the current
are called anions when they appear at the anode, and
cations at the cathode. Hydrogen and the metals always
appear at the cathode ; that is, they travel with the current
or are electro-positive.

l'.-.i;

ELECTRICITY AND MAGNETISM.

■a

]!*o

204. Electrolysis of Water. — Perfectly pure water
does not appear to conduct electricity at all. But if it be
acidulated with a small quantit}r of sulphuric acid, it is
decomposed as a secondary action. In
Hofmann's apparatus (Fig. 102) the
acidulated water is poured into the bulb
at the top, and the air escapes by the
glass taps till the tubes are filled. If
the taps are then closed and an electro-
motive force of three or more volts be
applied to the two pieces of platinum
foil at the bottom, bubbles of gas will
be seen to rise from them. The gases
collect in the two tubes and may be ex-
amined as they escape through the taps.
Oxygen will be found at the electrode
at which the current enters the appa-
ratus, and hydrogen at the other.

The volume of the hydrogen is not
exactly twice that of the oxygen, be-
cause the latter is more soluble in water than the former,
and about one per cent of it is evolved in the denser form
of ozone ; on the other hand, more hydrogen than oxygen
is absorbed or occluded by the platinum electrodes.

The chemical reactions may be represented, without
reference to the theory of the operation, as follows :

Fig. 102.

Cathode. Pt \ R2S04 \ EiSOi \ H20 \ Pt. Anode.

The primary electrolysis is that of sulphuric acid, while
the water is decomposed at the end of the chain by the
*S'04. As often as one atom of oxygen is set free at the
anode, two of hydrogen are liberated at the cathode.

ELECTROLYSIS.

257

205. Electrolysis of Copper Sulphate. — Copper sul-
phate presents one of the simplest cases of electrolysis.
Suppose the electrodes to be copper; the passage of the
current then simply transfers copper from the anode to
the cathode.

Anode. Cu j OuS04 \ OuSO^fOu. Cathode.

— e>

There is no change in the density of the whole solution ;
the copper ions migrate toward the cathode and the S04
ions toward the anode.

If platinum electrodes are used, copper will be deposited
on the cathode, and the S04 at the anode will decompose
water and release oxygen with the formation of IT2^Oi.

206. Electrolysis of Sodium Sulphate. — A salt of
one of the alkaline metals presents some secondary reac-
tions of interest. With a solution of sodium sulphate free
sodium cannot exist in water
at the cathode, but unites with
water, forming sodium hydrox-
ide and hydrogen ; at the anode
the #04 decomposes water, as
in the other cases, and liberates
oxygen.

Let the solution be placed in
a flat glass tank with a non-
conducting partition a extend-
ing nearly to the bottom (Fig.
103). Add to the solution a little extract of purple cab-
bage. When the current is passed the liquid will turn
red at the anode and green at the cathode, the former
color being due to the acid formed, and the latter to the

«,

T '

-~—^^

—

Fig. 103.

258 ELECTRICITY AND MAGNETISM.

alkali. Stop the flow of the current and mix the liquids
on the two sides of the partition; both the red and the
green colors will disappear with the restoration of the faint
purple, showing that the acid and the alkali were produced
in chemically equivalent quantities which neutralize each
other. The final result is the decomposition of water.

207. Electrolysis of Lead Acetate. — Place the solu-
tion, which may be made clear by the addition of a small
quantity of acetic acid, in a flat glass tank and electrolyze
between two lead wires. The lead separated from the
clear solution will be deposited on the cathode in the form
of shining crystals, which will grow rapidly, giving rise to
what is known as the " lead tree." If the process is not
conducted too rapidly, these crystals will assume very
beautiful forms. The lead goes into solution at one elec-
trode and comes out of solution at the other.

After a few minutes reverse the current ; the first crys-
talline deposit will gradually disappear, and another one
will form on the other wire. In this way the disappear-
ance of the lead at the one electrode and its appearance
at the other may both be observed at the same time. The
reaction is precisely like that of copper sulphate between
copper electrodes.

Cathode. PbJTb (C2HzO?)2 | P5(C25"302)2 | Pb. Anode.

208. Quantitative Laws of Electrolysis. — Faraday
showed that the masses of the ions separated are connected
by a very simple relation with the quantity of electricity
which passes through the electrolyte. This relation is
expressed by the following laws:

ELECTROL YSIS. 259

I. The mass of an electrolyte decomposed by the passage
of an electric current is directly proportional to the quantity
of electricity that passes through it.

If the current be kept constant, the mass of the ion
liberated in a given time will be directly proportional to
the strength of the current.

II. If the same quantity of electricity passes through
different electrolytes, the masses of the different ions liberated
at the electrodes are proportional to their chemical equiva-
lents.

Thus, if the same current passes through a series of elec-
trolytic cells, in which it liberates as ions hydrogen,
chlorine, copper, and silver, then for every gramme of
hydrogen set free, 35.46 gms. of chlorine, 31.7 of copper,
and 107.9 of silver will be separated.

The electro-chemical equivalent of an ion is the number
of grammes of it deposited by the passage of unit quantity
of electricity. When the current has unit strength, unit
quantity flows through any cross-section of the conductor
in one second of time. Faraday's laws may then be com-
bined in the statement that the number of grammes of an
ion deposited by the passage of a current through an
electrolyte is equal to the continued product of the strength
of the current, the time in seconds during which it flows,
and the electro-chemical equivalent of the ion.

209. Electro-chemical Equivalents. — The electro-
chemical equivalents of the several ions are proportional to
the relative masses of them which take part in equivalent
chemical reactions. The electro-chemical equivalents of
those ions which have a valency of one are proportional
to their atomic weights, and to half the atomic weights if
the ions have a valency of two. Elements which form two

260

ELECTRICITY AND MAGNETISM.

series of salts, such as copper in cupric and cuprous salts,
and mercury in mercuric and mercurous salts, have different
electro-chemical equivalents according as they are deposited
from solution of cupric or cuprous, mercuric or mercurous
salts.

The following table of electro-chemical equivalents is
based on the practical unit of quantity of electricity called
the coulomb, which is one-tenth the C.G.S. electromagnetic
unit of quantity (see Chapter XXI.) :

Ion.

Atomic weight.

Chemical
equivalent.

Electro- chemical

equivalent in

grammes per

coulomb.

1
23
39.03

107.92
63.4
63.4

199.8

199.8
55.9
55.9
58.6
65

15.96
35.46

126.85

1
23
39.03

107.92
31.7
63.4
99.9

199.8
18.64
27.95
29.3
32.5
7.98
35.46

126.85

0 000010362

Silver

0.0011180

" (mercurous) ....
Kickel

0.0003285
0.0006570
0.0010352
0.0020704
0.0001932
0.0002898
0.0003043

0.0003370

0.0000827

0.0000367

0.0013143

210. The Silver Voltameter. — If a neutral 15 per
cent solution of silver nitrate {AgNO^) is electrolyzed
between a silver anode and a platinum cathode, or between
two silver electrodes, silver is transferred with the current,
and is deposited on the cathode as adherent crystals if the
electrode be of sufficient size. Silver is removed from the
anode by the acid radical N0A as fast as it is deposited on
the cathode. The uniform results obtained in the electrol-

ELECTROLYSIS.

261

ysis of silver nitrate have led to its adoption as a standard
method for the measurement of a current. When applied
to this purpose an electrolytic apparatus is called a
voltameter.

A convenient form of
silver voltameter is
shown in Fig. 104. The
middle silver plate is the
cathode, and the two
outer ones together con-
stitute the anode. They
are attached by spring
clamps to terminals af-
fixed to an insulating
support ; the whole can
be removed from the
solution by loosening the
screw B. A rack and
pinion, worked by means

of the milled head P, allows the plates to be adjusted
to a greater or less depth of immersion. The anode
plates must be of pure silver.

The practical unit of current strength is the ampere.
Its electro-magnetic definition will be given later. A cur-
rent has the strength of one ampere when it deposits silver
at the rate of 0.001118 gm. per second, or 4.025 gms. per
hour. The mass of silver deposited in t seconds by a
current of I amperes is

m = Izt,
where z is the electro-chemical equivalent. Hence

Fig. 104.

J =

m_
zt

262 ELECTRICITY AND MAGNETISM.

The divisor in this expression may be 4.025 multiplied by
the time of deposit expressed in hours.

211. The Copper Voltameter. — The silver voltameter
is not employed for currents much larger than one ampere ;
for larger currents the copper voltameter is used. It con-
sists of smooth copper plates immersed in a solution of
copper sulphate acidulated with a few drops of sulphuric
acid. The electro-chemical equivalent of copper (cupric)
is less than one-third that of silver; for the same current
the weight of copper deposited in a given time is therefore
correspondingly less. The results are not so uniform as
those secured by silver nitrate, the practical electro-chemical
equivalent being a function of the temperature and the
density of the current at the cathode. By density of cur-
rent is meant the fraction of an ampere per square centi-
metre of cathode surface. It is commonly expressed
reciprocally as the number of square centimetres per
ampere.

212. Reversibility of the Daniell Cell. — When a cur-
rent flows from zinc to copper through a Daniell cell, zinc
is dissolved and copper is deposited. The E.M.F. of the
cell operating in this way as a generator is about 1.1 volts.
Suppose now an opposing E.M.F. greater than 1.1 volts be
applied to the terminals of the cell. The copper then
becomes the anode and the zinc the cathode, or the cell is
worked backwards. When the cell is worked forwards

'as a generator, the electro-positive ions travel toward the
copper plate, as represented in the upper diagram of Fig.
105 ; the cell is then giving out energy in the form of an
electric current, with a corresponding loss in its store of
potential energy. Suppose this process to continue till

ELECTROLYSIS.

263

one gramme-equivalent (65 gms.) of zinc has been dis-
solved, and one gramme-equivalent (63.4 gms.) of copper
has been deposited. Then let the cell be worked back-
wards with the reactions of the lower diagram of Fig. 105 ;
the cell is then receiving energy, and storing it up in the
increase of zinc and CuS04 at the expense of copper and
ZnSOk. When a gramme-equivalent of copper has been
removed from the copper plate
and a gramme-equivalent of
zinc has been deposited on the
zinc plate, the cell is in its in-
itial state. There has been no
loss of materials, and they are
in the same chemical condition
as at the outset. Except for
the small loss by heat due to
resistance, the energy given out
by the cell during the direct
action equals the energy stored
up during the reversal of its
functions. Hence during the direct action as a generator
there can be no counter E.M.F. to work against to prevent
the conversion of the potential energy of the cell into the
energy of an electric current. This cell is therefore com-
pletely reversible and does not polarize.

The simple voltaic element belongs to another class.
Suppose it to work forwards till an equivalent of hydrogen
has been given off at the copper and an equivalent of zinc
has gone into solution. Then let it be worked backwards
till an equivalent of copper has gone into solution and an
equivalent of hydrogen has been given off at the zinc.
At the end of the experiment an equivalent of both zinc
and copper has gone into solution and two equivalents of

Fig. 105.

264

ELECTRICITY AND MAGNETISM.

hydrogen have been set free. The cell does not return
to its initial state at the end of the experiment, and there
must be a compensation for the net chemical changes.1
This compensation is found in the counter E.M.F. of
polarization when the cell works forward as a generator.
The simple voltaic element is an example of a non-revers-
ible or polarizable cell. Only reversible elements work
with maximum efficiency.

213. Polarization of an Electrolytic Cell. — If the
two platinum electrodes of Hofmann's apparatus (Fig.
102) be connected to a sensitive gal-
vanometer immediately after they
have been used for the electrolysis of
sulphuric acid, it will be found that
energy has been stored up to some
extent and the cell will furnish a
current. The chemical and electrical
functions are now reversed; the hy-
drogen and oxygen on the electrodes
unite to form water, and a reverse
current flows through the cell. The
apparatus may be set up as in Fig.
106. B is the battery to furnish the
current to decompose the sulphuric
acid. Hydrogen accumulates in the
tube S and oxygen in the tube 0,
Let the two-point switch 8 be now turned so as to cut
off the battery and to join the electrolytic cell to the
galvanometer Gr. The needle will be sharply deflected by
the current from the Hofmann's apparatus. To determine
its direction, a thermal couple, consisting merely of a

'Nernst's Theoretical Chemistry, Trans, by Palmer, p. 597.

Fig. 106.

ELECTROL YSIS. 265

copper and an iron wire soldered together and placed in
the circuit of the galvanometer at T, is convenient. When
such a couple is slightly heated a current passes across
from Cu to Fe. It may be tried before charging the
electrolytic cell, and the direction of the deflection of the
galvanometer may be noted. It will then be found that
the current produced by the electrolytic cell will flow out
from A and in at C, or in the reverse direction to the
current which separates the gases, oxygen and hydrogen.
The E.M.F. of polarization is therefore a back or resisting
E.M.F.

214. Electrolysis with and without Polarization. —
When electrolysis takes place between two metallic plates
of the same kind, immersed in a salt of the same metal,
the polarization of the electrodes is very small. Thus,
with copper in copper sulphate, or zinc in zinc sulphate, or
silver in silver nitrate, the polarization is slight ; the small
counter E.M.F. exhibited is probably due to a difference
in the surface of the anode and cathode, and to a difference
in the density of the solutions in immediate proximity to
the plates. Zinc in zinc sulphate shows no appreciable polar-
ization.

But when the electrolysis effects a change in the chemical
composition of the electrolyte, polarization results. The
ions set free, such as hydrogen and oxygen, have a ten-
dency to reunite by means of a reverse current and a
reverse chain of molecular interchanges. In such cases
work is done during electrolysis, and potential energy is
stored up in the form of chemical separations.

In the first kind, where the metal is simply transferred
from one electrode to the other, a very weak E.M.F. is suf-
ficient to produce electrolysis ; in the second, the applied

260

ELECTRICITY AND MAGNETISM.

E.M.F. must exceed the counter E.M.F. of polarization
before visible separation of the ions is accomplished.

215. Grove's Gas Battery. — Grove's gas battery is

constructed on the basis of
the facts just described. The
platinum strips are fused
into the tops of the two
tubes (Fig. 107), which are
fitted into two necks of a
WouliFs bottle filled with
dilute sulphuric acid. After
the tube H has been nearly-
filled with hydrogen by elec-
trolysis, the terminals P and
N become respectively the
positive and negative of a
voltaic element. The sur-
faces of the platinum plates
are covered with platinum
black for the purpose of in-
creasing the surface of the
liquid in contact with plati-

Fig. 107.

num. The action may be represented thus :

H2 I H,SO< I H,SO< I 0.

After the first exchange of atoms this becomes

H,SO, | HjSOt | LT20.

Water is re-formed at the expense of the oxygen and hydro-
gen. The water or sulphuric acid voltameter is therefore

ELECTROLYSIS.

267

a reversible element. Similar results maybe obtained with
other gases, notably hydrogen and chlorine.

216. Plant's Storage Cell. — If the platinum elec-
trodes of the sulphuric acid voltameter be replaced by
lead, we have the Plante storage cell, which is the basis of
all modern storage batteries. Take two pieces of sheet
lead and solder to each a short length of copper wire as
a terminal. Attach the lead strips to opposite sides of a
block of dry wood, and immerse the
plates in dilute sulphuric acid (Fig.
108). Pass a current through the
cell for a few minutes. The oxygen
liberated at the anode will oxidize the
lead, forming a dark-brown coating ^
of the peroxide of lead. An ordinary
electric house-bell may be connected |\
with the cell by a switch, as in Fig.
106. When the switch is turned,
cutting off the charging battery and
connecting the lead electrolytic cell
with the bell, the latter will ring
vigorously for a few seconds. The operation may be
repeated, showing that energy is stored up in the cell by
the process of electrolysis. The E.M.F. of polarization
in this case is somewhat over two volts. Plante" subjected
his cells to repeated charging in opposite directions, so
that both plates should be modified to an appreciable
depth by alternate oxidation and reduction. This process
was called "forming" the plates.

In most modern storage cells the plates, cast or rolled in
the form of grids, are provided with lead oxides which
compose the "active material." These oxides are changed

Fig. 108.

268 ELECTRICITY AND MAGNETISM.

into peroxide at the anode, and reduced by hydrogen to
spongy lead at the cathode during the operation of charg-
ing. The chemical reactions of
the storage cell are very complex,
and are to some extent undeter-
mined. Sulphuric acid is formed
during the charging of the cell,
and disappears during the dis-
charge. Some sulphate of lead is
also formed during the discharge,
and is reduced by hydrogen with
slow charging. The electrode
which is the anode when charging
and the cathode when discharging
is called the positiye pole of the
cell. Fig. 109 represents a cell
of the " chloride accumulator."

217. Theory of Electrolysis.
— Many reasons have been ad-
duced which go to show that dissociation of acids and
salts takes place when they are dissolved in water. Hy-
drochloric acid, for example, is dissociated into positive
hydrogen and negative chlorine ; sulphuric acid into two
positive hydrogen atoms and the negative acid radical
SOi. This dissociation, if it actually occurs, is intimately
connected with the conduction of electricity by electro-
lytes. Clausius proposed the theory that momentary dis-
sociations occur with succeeding recombinations, a process
of intermolecular exchanges ; and that the electric current
determines only the direction in which such exchanges
shall take place. Such transient dissociation would suffice
to account for the observed conduction of very small cur-

ELECTROLYSIS. 269

rents by electrolytes without any visible separation of free
ions ; but it is incompetent to explain other facts of physi-
cal chemistry. This phenomenon of partial electrolysis
von Helmholtz called electrolytic convection, and assumed
that it takes place by the agency of the uncombined atoms
in the liquid. The modern theory makes all electrolytic
conduction depend upon these dissociated atoms.

If gaseous hydrochloric acid be introduced between
platinum electrodes connected with a voltaic battery, no
appreciable transfer of electricity occurs ; neither does
'pure water conduct electricity ; but if the hydrochloric
acid be dissolved in water, the solution becomes conduct-
ing, with the electrolytic separation of hydrogen and
chlorine. The inference is justifiable that the acid must
have undergone an important molecular change by solution
in water, because after solution it conducts electricity, and
before solution it does not. The same inference does not
apply to the solvent, because it does not suffer electrolytic
decomposition. The molecular change which the acid
undergoes by solution is dissociation into electro-positive
and electro-negative ions, thus :

+ —
HCl = H+ 01.

The capacity of a dissolved substance to conduct electricity
therefore presupposes a molecular cleavage into positively
and negatively charged atoms. The larger the number of
such dissociated molecules in a solution, the better it con-
ducts. It is not necessary that all the molecules of the
substance be dissociated by the solvent. Those that are
not decomposed remain electrically neutral and take no
part in the transfer of electricity.

Let the cell in Fig. 110 contain a water solution of
hydrochloric acid with platinum electrodes. These elec-

270

ELECTRICITY AND MAGNETISM.

trodes are charged as shown by connection with a battery,
which maintains a constant potential difference between

them. The solution contains pos-
itively charged hydrogen atoms
and negatively charged chlorine
atoms, besides neutral molecules
which have not been decomposed.
Then the positive charge on the
anode attracts the negative chlo-
rine atoms and repels the positive
hydrogen atoms, while the reverse
actions occur at the negatively
charged cathode. All these forces combine to produce a
simultaneous and equal procession of hydrogen atoms from
anode to cathode, and of chlorine atoms from cathode to
anode. This double procession of free ions with their
electric charges represents the passage of an electric cur-
rent through an electrolyte.

0-©-G

-0 0-0-

©♦0 ©

Fig. MO.

218. Electrolysis in the Arts. — Electrolysis is now
employed on a large scale for a number of distinct pur-
poses in the arts and industries. These may be classed
under four heads, viz., the reduction of metals from their
ores or solutions ; the copying of types, casts, woodcuts,
and metal work ; the covering of objects in base metals
with gold, silver, or nickel ; and the manufacture of various
chemicals, such as caustic soda, bleaching liquors, and
chlorate of potassium. The first three of these are in-
cluded under the general term of electro-metallurgy.

Pure copper is now produced on an enormous scale by
electro-deposition. After a second process of reduction in
a blast furnace the " blister " copper, containing small
quantities of gold, silver, oxide of iron, and sulphides, is

ELECTROLYSIS. 271

cast into slabs which serve as the anode plates in the elec-
trolytic bath of copper sulphate. Several plants are now
in operation in the United States, with a capacity of from
50 to 100 tons of pure copper dailyr

Aluminium is reduced in .large quantities from a fused
mixture of electrolytes. Cryolite, a double fluoride of
aluminium and sodium, is first fused by the passage of a
very large current between huge carbon electrodes. To
this fused mass is added bauxite, a ferruginous hydrate 0f
aluminium, and this is dissolved by the fused cryolite.
The cryolite serves as the bath and the aluminium oxide
is electrolized. Jts solution produces a marked reduction
in the resistance of the bath. Only a small per cent of the
cryolite is decomposed.

If copper is deposited on any surface, such as coins,
ornaments, and stereotype plates, an exact impression is
obtained in reverse relief. If a mould in plaster or wax
be taken of any object, and be covered with a conducting
film of plumbago or finely powdered bronze, the mould
can be coated with a deposit of copper. When this is
filled with type metal, an exact reproduction of the origi-
nal is obtained. This process is largely employed to
reproduce repousse* and other works of art in facsimile,
and to multiply copies of woodcuts or other engravings
for printing. The electrolytic solution is acidulated
copper sulphate.

The art of electro-plating was invented early in the
present century. The objects to be covered with a thin
deposit of gold, silver, or nickel must first be made chemi-
cally clean ; they are then hung in the bath as the cathode.
For gold and silver plating the solution is cyanide of gold
or silver dissolved in cyanide of potassium ; for nickel
it is a double sulphate of nickel and ammonium. The

272 ELECTRICITY AND MAGNETISM.

anode in each case must be a plate of the same metal as
the one to be deposited at the cathode. The solution
then continues to have the same density.

PROBLEMS.

1. The weight of a cathode silver plate was 30.3726 gms. before
the deposit on it and 30.4685 gms. after deposition, which lasted
half an hour. Find the average current in amperes.

2. The following data are taken from a copper voltameter meas-
urement :

Weight of cathode before deposit .... 83.4925 gms.

" after " .... 84.4475 "
Time of deposit, 30 min.
Find the mean current.

3. The silver deposited in a silver voltameter in 45 min. was
2.8095 gms. Find the average current.

4. A current of 1 ampere is sent through three electrolytic cells
in series for 30 min. The first contains cyanide of silver dissolved
in cyanide of potassium ; the second, zinc sulphate ; the third, nickel
sulphate. Find the weight of metal deposited in each.

5. If one litre of hydrogen under standard conditions weighs
0.08987 gm., how many amperes will liberate 250 c.c. of hydrogen
in 10 m. 22 s.?

OHM'S LAW AND ITS APPLICATIONS. 273

CHAPTER XVIII.

'OHM'S LAW AND ITS APPLICATIONS. •

219. Ohm's Law. — The relation between the electro-
motive force and the current was first enunciated by Dr.
G. S. Ohm, of Berlin, in 1827. It has since been known
as Ohm's Law.

If E be the E.M.F. between two points of a conductor
and / the current flowing through it, then if suitable
units be chosen,

E=RI,

where R is a quantity called the resistance of the con-
ductor; it is independent of the value and direction of the
current flowing, and depends only on the material of
the conductor, its length and sectional area, its tempera-
ture and state of strain.

The above equation is an expression of Ohm's law ; it is
usually written in the equivalent form,

If the practical units now adopted internationally be
employed, this law may be expressed by saying that the
number of amperes flowing through a circuit is equal to
the number of volts of electromotive force divided by the
number of ohms of resistance.

When this formula is applied to the entire circuit, which
may contain several sources of E.M.F. of different signs,

274 ELECTRICITY AND MAGNETISM.

and both metallic and electrolytic resistances, it is not
quite so simple to apply. There are then several electro-
motive forces, some tending to produce a flow in one
direction and some in the other ; and a number of different
resistances each obstructing the flow, whether it takes place
in one direction or the other. Then

j -E\ + E> -f- E-,\ + • • • — -E
Mi + R-2 + R-i + 2R

Each E.M.F. must be taken with its proper sign. Resist-
ance is not a directed quantity. If, for example, there are
several voltaic cells in the circuit, some of them may be
connected in the wrong direction so that they oppose
the current ; or the circuit may include electrolytic or
storage cells or motors, which offer resistance in the form
of a counter E.M.F. All such electromotive forces must
be reckoned as negative.

220. Resistance. — Resistance is that property of a
conductor in virtue of which the energy of a current is
converted into heat. It is independent of the direction
of the current, and the transformation into heat occasioned
by it is an irreversible one ; that is, there is no tendency for
the heat-energy to revert to the energy of an electric
current.

The practical unit of resistance is the ohm. It is repre-
sented by the resistance offered to an unvarying electric
current by a column of mercury at the temperature of
melting ice, 14.4521 grammes in mass, of a constant cross-
sectional area and of a length of 106.3 centimetres. This
statement is equivalent to a cross-sectional area of one
square millimetre.

OHM'S LAW AND ITS APPLICATIONS. 275

221. Laws of Resistance. — The resistances of diverse
conductors are found to conform to the following laws :

(1) The resistance of a uniform conductor is directly
proportional to its length.

(2) The resistance of a uniform conductor is inversely
proportional to its cross-sectional area. The resistances of
round wires are therefore inversely proportional to the
squares of their diameters.

(3) The resistance of a uniform conductor of given
length and cross-section depends upon the material of
which it consists. This property is called its specific re-
sistance.

222. Specific Resistance. — A definite meaning may
be given to specific resistance by conceiving the material
to be in the form of a centi-
metre cube (Fig. Ill), a
cube whose edges are 1 cm.
iu length. The specific re-
sistance is the resistance
which this cube opposes to
the passage of a current
from one face a to the opposite one b. If the conductor
is a cylinder 1 cm. long with parallel ends of one square
centimetre area, the resistance from a to b is the same
as that of the cube. The specific resistance may be rep-
resented by s. Then the following formula expresses all
the laws of resistance :

Is
r = _,

where I is the length of the conductor in centimetres, and
a its sectional area in square centimetres. A table of
specific resistances is given in the Appendix, Table IV.

276 ELECTRICITY AND MAGNETISM.

223. Conductivity. — The inverse of a resistance is
called conductivity, or sometimes conductance. A conductor
whose resistance is r ohms has a conductivity equal to 1/r.
When a number of conductors are joined in parallel with

one another, the conductivity of the whole is the sum of
their several conductivities. Let two conductors of resist-
ances rx and r2 be joined in parallel between the points A
and B (Fig. 112). Let Vx and V2 be the potentials of
A and B respectively. Then since Vi — V2 equals the
E.M.F., we have by Ohm's law

r rx r2

The first member of this equation is the total current,
which is equal to the sum of the currents through the two
branches ; and r is the combined resistance of the two con-
ductors in parallel. Hence

1=1+1 .

r rx r2

Similar reasoning applies to any number of parallel con-
ductors.

From the last equation,

fir2
ri + tt

224. Effect of Heat on Resistance. — The resistance
of metallic conductors in general increases when the tem-

OHM'S LAW AND ITS APPLICATIONS. 277

perature rises. If Mo is the resistance of a conductor at
0° C. and Rt at t° C, then the equation

expresses the relation between the two through a consid-
erable range of temperature. The constant a is called the
temperature coefficient. For most pure metals it is about
0.4 per cent for one degree C, or 40 per cent for a range
of 100 degrees of temperature. The temperature coeffi-
cient for pure copper between 20° and 250° C. was found
by Kennelly and Fessenden to be 0.00406. Dewar and
Fleming have measured the resistances of pure metals in
liquid oxygen at temperatures of —182° and —197? C,
and have shown that the resistance of all of them decreases
with fall of temperature as if it would become zero at
— 273° C, the zero of the absolute scale. They would
then offer no obstruction to the passage of a current, how-
ever great. Pure copper is the best known conductor, but
it is only slightly better than silver.

The temperature coefficient of alloys is smaller than
that of pure metals. German silver has a coefficient only
about one-tenth as great as that of copper ; while that of
platinoid is only one-half as great as that of German silver.
Manganin, an alloy of manganese, copper, and nickel, has at
certain temperatures a small negative temperature coeffi-
cient; that is, its resistance diminishes slightly as the
temperature rises.

The resistance of carbon and of electrolytes decreases
when the temperature rises. Thus, the resistance of an
incandescent lamp filament is only about half as great at
normal incandescence as when cold. Solutions of ZtiSOt
and of CuSOA have a temperature coefficient somewhat
over 0.02, or 2 per cenb for one degree C.

278 ELECTRICITY AND MAGNETISM.

225. Loss of Potential proportional to Resistance.
— If Vi and Vs are the potentials of two points A and B
on a conductor, then by Ohm's law

It is obvious from this equation that the potential differ-
ence between any two points on a conductor through
which a constant current is flowing is proportional to the
resistance between them, provided the conductor is not
the seat of an E.M.F. Even when electromotive forces
are encountered, the loss of potential, when a given cur-
rent flows through a resistance,
is proportional to that resistance.
If another point be taken be-
tween A and B so situated that
the resistance between it and B
is one-half the resistance be-
tween A and B, then the poten-
tial difference between this point
and B is also reduced in the

Fig 113.

same ratio.
Let the distances measured along Or represent resist-
ances (Fig. 113), and those along Ov, potentials. Then
AP equals V\ and BQ, V2\ also PQ stands for the resist-
ance B between the points A and B on the conductor.
Join A and B and let BO be drawn parallel to Or; then
will AC be equal to Vx— V2, the potential difference
between the points A and B. The slope of the line AB
represents the rate at which the potential drops along the
resistance B. Moreover, since

tan <f> = AC/BC=B/R = 7,
it is evident that the tangent of the angle of slope equals
the strength of the current.

<p\>

(

OnM'S LAW AND ITS APPLICATIONS.

279

The principle that the loss of potential is proportional
to the resistance passed over, when the current is constant,
is one of very frequent application in electrical measure-
ments.

226. Wheatstone's Bridge. — The instrument known
as a Wheatstone's bridge illustrates the use made of the
principle of the last article. It is a combination of resist-
ances more commonly used than any other method for
the comparison of two of them. It consists of six conduc-
tors connecting four points ; in one of these conductors is
a source of E.M.F., and in another branch is a galvanom-
eter, or sensitive current detector.

Let 4, B, C, B (Fig. 114),
be the four points, B' the bat-
tery, and Gr the galvanometer.
Then since the fall of poten-
tial between A and B is the
same by the path ABB as by
ACB, there must be a point
B on the former which has
the same potential as the
point C on the latter. If the
circuit through the galvanometer is made to connect these
two equipotential points, no current will flow through it.

Let 7] be current through Rt ; it will also be the current
through RA , because none flows through the galvanometer,
and the same quantity of electricit}1- flows toward B as
away from it. Also, let Z, be the current through the
branch ACB. Then, the potential difference between A
and B being the same as that between A and (7, we have
by Ohm's law (219)

BJ^BJi (a)

Fig. 114.

280 ELECTRICITY AND MAGNETISM.

Similarly, RJ, = RJ, (6)

Dividing (a) by (5), f» = §.

It 3 Mi

This equation may be written

R\ R%

R-2 Rt
or Ri : R-2 : : R3 : R± .

When therefore the resistances are so adjusted that no
current flows through the galvanometer, the four form a
proportion. In practice three of the resistances are fixed,
and the adjustment for a balance is made by varying the
fourth. It is necessary to know only the ratio RA / Ru for
example ; then the equation gives the relation between
Ri and R2.

227. Cells joined in Series. — Let there be n similar
voltaic cells, each having an electromotive force E and an
internal resistance between the terminals of the cell equal
to r. Then if R is the external resistance, by Ohm's law

1-R + r

The n cells may be joined in series by connecting the
negative of the first with the positive of the second ; the
negative of the second with the positive of the third, and
so on. Then the total E.M.F. between the positive of the
first and the negative of the last will be nE, and the entire
internal resistance will be nr. Hence

j_ nE
R + nr

If R is small in comparison with r, then 1= E/r nearly, or

0773/' S LAW AND ITS APPLICATIONS.

281

the current is no greater than could be obtained from one
cell. But if R is large in comparison with r, or even wr,
then the current is nearly n times as great as one cell alone
will yield.

228. Graphical Representation of Potentials for
Cells in Series. — Let there be three cells in series ; and
let AB (Fig. 115) represent 3r, the internal resistance of
the three. Also let
BO equal the exter-
nal resistance B on
the same scale. Be-
ginning at A, erect a
perpendicular Ab
equal to JS, the E.M.
F. of one of the cells.
Suppose the E.M.F.
to originate at the
surface of the zinc.
Then as the current flows across through the liquids over
the resistance r there will be a fall of potential represented
by the sloping line be. At c, the zinc of the second cell, there
is a sudden rise of potential cd, equal to Ab, and then a fall
from d to e; at e there is a third rise, represented by ef ;
then another drop from /to g over the internal resistance
of the last cell. The potential difference between the ter-
minals of the battery is then Bg, and this is the loss of
potential over the external resistance R.

The line AD represents 3-27, and DF is the loss of poten-
tial in the three cells on account of their internal resist-
ance. Then

3J57 E'

tan <f> =

R+Zr R

= ~ = I.

282 ELECTRICITY AND MAGNETISM.

Since the tangent of the angle of slope is the numerical
value of the strength of current, it is evident that the
lines be, de, and/# must slope at the same angle as DC, or
must be parallel to one another, because the current has
the same value in every part of the circuit.

If the external resistance were made infinite by opening
the circuit, the line DC would become horizontal, and the
current zero. Also, with any given external resistance,
the less the internal resistance the less the difference
between SU and E'.

229. Cells joined in Parallel. — A battery is said to
be connected in parallel, or in multiple, when all the pos-
itive terminals are joined together, and likewise all the
negatives. The chief object aimed at is the reduction
of the internal resistance. In the case of storage cells,
which have a very low resistance, they may be joined in
parallel when it is desired to use a larger current than the
normal discharge current for one cell. With several cells
in parallel, the current through the external circuit is
divided among them.

If n similar cells are connected in parallel, the E.M.F.
is the same as for a single cell, but there are n internal
paths of equal resistance through the cells, and the result-
ant internal resistance is r/n. Hence

n
In case R is small in comparison with r, the reduction of
the internal resistance secured by joining the n cells in
parallel results in a larger current, but no such result
follows for a large external resistance. For the latter
condition the cells should be in series.

ohm's law and its applications. 283

230. Cells in Multiple Series. — Let there be m series
of n cells each, the m series beiDg joined in parallel. The
whole number of cells is then ran. The current will be

nE E

R + ™ X+L

m n m

To find the condition for a maximum current it may be
remarked that the product of the two terms in the denomi-
nator of the last expression is Rr/nm, a constant. R and
r are assumed to be constant, and nm is the whole number
of cells. But when the product of two terms is a con-
stant, their sum is least when they are equal to each other,
or when R/n = rim. For this condition

R=nr..

But R is the external resistance and nr/m is the internal
resistance. For the greatest steady current, therefore, the
cells should be so arranged that the resulting internal
resistance shall be equal to the external resistance. The
efficiency may then be said to be 50 per cent, since half
the energy is wasted internally and half may be utilized
externally. This relation does not hold if there is a
counter E.M.F. in the circuit.

231. Variation of Internal Resistance with Current.
— The internal resistance of a given cell is not a fixed
quantity. It changes with the operation of the cell, on
account of the chemical changes going on which alter the
composition of the liquids. It is also dependent on the
current drawn from the cell. The larger the current,
the smaller is the measured internal resistance. The

284

ELECTRICITY AND MAGNETISM.

curves of Fig. 116 represent graphically the relation be-
tween the internal resistance and the current for two
particular cells. The lower curve was made from obser-

°j \

•^ \

O r.

Ami

teres

.02

.04 .06

.10 .12 .H
Fig. 116.

.16

.18 .20

.22 .24

vations on an old " dry cell," and the upper one from
observations on a Daniell cell. The scale for the internal
resistance of the latter is twice as large as for the former.
The dry cell showed a most remarkable fall in the resist-
ance as the current increased.

PROBLEMS.

1. Three Daniell cells are connected in series; the E.M F. of
each cell is 1.1 volts and the internal resistance 2 ohms; if the
external resistance is 5 ohms, find the current.

2. Two Leclanche cells are joined in parallel ; each has an E.M.F.
of 1.5 volts and an internal resistance of 4 ohms. If the external
resistance consists of two parallel conductors of 2 and 3 ohms
respectively, find the current through each branch-

OHM'S LAW AND ITS APPLICATIONS. 285

3. Deduce the formula for the resistance of three conductors in
parallel.

4. Three Bunsen cells are connected in series with one another
and with one copper oxide cell, the latter with its poles set the wrong
way round. If the internal resistance of the Bunsens is 0.5 ohm
each and that of the other cell 0.2, find the current through an
external resistance of 3 ohms (201).

5. Two equal masses of copper are drawn into wire, one 10
metres long and the other 15 metres. If the resistance of the shorter
piece is 0.4 ohm, find that of the longer.

6. Three wires are joined in parallel ; their resistances are 30,
20, and 60 ohms. Find the resultant resistance.

7. The resistance between two points A and B of a circuit is
25 ohms ; on joining another wire in parallel between A and B the
resistance becomes 20 ohms. Find the resistance of the second wire.

8. The terminals of a battery of five Grove cells in series, the
total E.M.F. of which is 9.5 volts, are connected by three wires,
each of 12 ohms resistance. If the current through each wire is
one-third of an ampere, find the internal resistance of each cell.

9. Given 24 cells, each of 1 volt E.M.F. and 0.5 ohm internal
resistance. How should they be connected to give a maximum cur-
rent through an external resistance of 3 ohms? What will be the
current ?

10. What is the resistance of a column of mercury 212.6 cms.
long and 0.5 of a square millimetre in cross-section, at a tempera-
ture of 25° C. ? Temperature coefficient of mercury, 0.072 per cent
per degree C.

286 ELECTRICITY AND MAGNETISM.

CHAPTER XIX.

THERMAL RELATIONS.

232. Conversion of Electric Energy into Heat. — ■
Electric energy is readily convertible into other forms.
If an electric current encounters a back E.M.F. anywhere
in the circuit, work will be done by the passage of the
current against this opposing E.M.F. Such is the case in
electrolysis and in the storage battery. All the energy of
an electric current not so converted, or stored up in some
form of stress, is dissipated as heat. Heat appears
wherever the circuit offers resistance to the current. In a
simple circuit containing no devices for transforming and
storing energy, all of it is frittered away as heat. Part
of it disappears in heating the battery or other generator,
and the remainder in heating the external circuit.

The heat evolved by dissolving 33 gms. of zinc in sul-
phuric acid Favre found to be 18,682 calories. When the
same weight of zinc was consumed in a Smee cell, the heat
evolved in the entire circuit was 18,674 calories. These
operations were conducted by introducing the vessel con-
taining the zinc and acid in the first case, and the Smee cell
and its circuit in the second case, into a large calorimeter.
The two quantities are nearly identical, or the heat evolved
is the same whether the solution of the zinc produces a
current or not. When the electric current was employed
to do work in lifting a weight, the heat generated in the
circuit was diminished by the exact thermal equivalent of

THERMAL RELATIONS.

287

the work done. When, therefore, a definite amount of
chemical action takes place in a battery and no work is
done, the distribution of the heat is altered, but not its
amount.

i\WS

dis-

233. Laws of the Development of Heat. — The 1
of the development of heat in an electric circuit were
covered experimentally by Joule and
Lenz. The latter experimented with a
simple calorimeter represented in Fig. 117.
A thin platinum wire, joined to two stout
conductors, was enclosed in a wide-
mouthed bottle containing alcohol. A
thermometer t was passed through a hole
in the insulating stopper of the bottle.
The resistance of the fine wire was known,
and the observations consisted in measur-
ing the current and noting the rise of
temperature. Joule found that the num-
ber of units of heat generated in a con-
ductor is proportional —

(1) To its resistance.

(2) To the square of the strength of the current.

(3) To the length of time the current flows.

234. The Heat Equivalent of a Current. — Let the
potentials of two points A and B of a conductor be \\ and
V% ; and let Q units of electricity be transferred from A to
B in the time t. Then the work done, expressed in ergs,
will be .

W=Q(Vi-V.}.

If all this work is converted into heat, W= JH, by (86) ;
if the current of strength i" flows for' time t, the quantity

Fig. 117.

288 ELECTRICITY AND MAGNETISM.

Q = It, since the strength of current is the quantity passing
any section of the conductor in one second ; also Vx — V-2 =
RI. Substituting,

JH= PRt,
, „- PRt PRt

and ^-J^OlbOUT

The current strength and the resistance are expressed in
C.G.S. electromagnetic units (294). An ampere is 10-1
C.G.S. unit ; an ohm, 10'J. If the measurements are made
in amperes and ohms, then for I2 must be substituted
J2x 10~2, and for R, Rx 109. The equation then becomes

H= -^X]Z t = J2Rt x °-24-
• 4.19 xlO7

The energy expended per second is the product of the cur-
rent strength and the electromotive force. If I be measured
in amperes and E in volts (a volt is 108 C.G.S. units), then

W= IE x 10-1 x 108 = IEx 107 ergs per second,
or IE watts (I., 43). But

H— I-R x 0.24 = IE x 0.24 calories per second.
Therefore one watt is equivalent to 0.24 calorie per second.

235. Counter B.M.P. in a Circuit. — The total activ-
ity, or rate at which a generator is supplying energy to
the circuit, is represented in part by the heat evolved in
accordance with Joule's law and in part by work done, such
as chemical decomposition by electrolysis, the mechanical
work of a motor, etc. In every case of doing work the
energy absorbed? is proportional to the current strength in-
stead of its square. We may therefore write for the whole
energy transformed in time t

IEt = PRt + Alt.

THERMAL RELATIONS. ' 289

The first term of the second member of this equation is the
waste in heat ; the second, the work done ; A is a constant.
Dividing through by It and transposing,

j_E-A

R '

R is the entire resistance of the circuit. It is evident from
the form of the equation that the quantity A is of the
nature of an E.M.F. Since it is affected by the negative
sign it is a counter E.M.F. The effective E.M.F. produc>
ing the current is the applied E.M.F. less the back E.M.F.
This counter E.M.F. is a necessary phenomenon in every
case in which work is done by an electric current.

236. Division of the Energy in a Circuit. — If the
counter E.M.F. be represented b}r E', the equation for the
current by Ohm's law is

E—FJ
R '
But the heat waste in watts is

PR = I (E-EO = IE- IE'.

Now IE is the total activity in the portion of the circuit
considered. The heat generated in this same portion of
the circuit of resistance R is less than the entire activity
by IE' watts. Hence the energy spent per second in doing
work is the product of the current strength and the counter
E.M.F.

The ratio of the work done to the heat waste is
IE' E'

I^E-E') E-E''
The efficiency with which electric energy is converted into
work increases therefore with the counter E.M.F.

290 ELECTRICITY AND MAGNETISM.

237. Applications of the Heating Effect of a Current.
— Of the various applications of heating, the following are
some of the more important:

1. Electric Cautery. A thin platinum wire heated to
incandescence is sometimes employed in surgery instead
of a knife. Platinum is used because it is infusible, except
at a high temperature, and is not corrosive.

2. Safety Fuses. Advantage is taken of the low tem-
perature of fusion of some alloys, in which lead is a large
constituent, for the purpose of automatically interrupting
the circuit when for any reason the current becomes ex-
cessive. Some of these alloys, notably those containing
zinc, may oxidize on heating ; and if the current be in-
creased slowly the fused metal may become encased in
the oxide as an envelope, and be heated to redness with-
out breaking the circuit. Safety fuses should be mounted
on non-combustible bases; their length should be pro-
portioned to the voltage employed on the circuit in which
they are placed. Provision is sometimes made for an
automatic blast, produced by the explosive vaporization
of the metal, to blow out the arc which is formed between
the terminals when the fuse " blows."

3. Electric Heating. Electric street-cars are sometimes
heated by a current through suitable iron-wire resistance
embedded in cement, asbestos, or enamel. Similar de-
vices for cooking have now become articles of commerce.
Small furnaces for fusing, vulcanizing, and enameling
in the operations of dentistry are also in use. For such
purposes electric heating offers a wide field of appli-
cation.

4. Electric Welding. If the abutting ends of two rods
are pressed together while a large current passes through
them, enough heat is generated at the junction where the

THERMAL RELATIONS.

291

resistance is greatest to soften and weld them. This
method has been perfected by Elihu Thomson, who em-
ploys several hundred amperes in some
cases, but under a low electric pressure.
Fig. 118 shows three small welded
joints.

Similar devices are now employed for
the local annealing of armor plates ; the
metal is in this way softened at points
where it is to be drilled.

238. The Electric Arc. — In 1800
Sir Humphrey Davy discovered that if
two pieces of charcoal, connected by
suitable conducting wires to a powerful
voltaic battery, be brought into contact
and be then separated a slight dis-
tance, brilliant sparks will pass be-
tween them. But no mention was made of the electric
arc till 1808. In 1810 Davy exhibited the arc light at the
Royal Institution.

With a battery of 2,000 simple elements, when the car-
bons were drawn apart to a distance of several inches, the
carbon was apparently volatilized, and the current was con-
ducted across in the form of a curved flame or arc. A
brilliant light was emitted at the same time by the white-
hot carbon electrodes, which rapidly burned away, unless
they were enclosed in a vacuum. Foucault surmounted
this difficulty in 1844 by making use of the dense carbon
from a gas retort in place of the wood charcoal pencils.

When the carbon points are separated the heat due to
the current volatilizes some of the carbon, or the volatile
constituents not expelled by previous baking, and this

Fig. 118.

292

ELECTRICITY AND MAGNETISM.

carbon vapor conducts the current across. The passage of
the current heats the carbons to vivid incandescence. Since
gases are poor radiators, the dazzling light is emitted
chiefly by the carbon electrodes and especially by the posi-
tive one. In it is formed a small cavity by the transport
of carbon across to the negative. According to Violle, the
temperature of this cup-shaped depression, or crater, is
about 3,500° C. It is the temperature at which carbon
volatilizes. The positive carbon wastes away about twice

as fast as the negative. The
appearance of the two carbon
pencils is shown in Fig. 119.

The resistance of the elec-
tric arc may be only a fraction
of an ohm. It is not large
enough to account for all the
heat developed; but the crater
in the positive appears to be
the seat of a counter E.M.F.
of about 39 volts for a quiet
arc. Hence a potential dif-
ference of from 40 to 45 volts
is necessary to maintain a
steady arc without hissing.
The large quantity of heat generated is due to the fact
that the current encounters an opposing E.M.F. at the
arc, and energy is in con3equence transformed into heat.

Fig. 119.

239. The Carbon Filament. — In the incandescent
system of electric lighting the heat is due to the simple
resistance of a thin carbon filament. Carbon is the only
substance thus far found to be available, because it does
not fuse and has a high radiating power.

THERMAL RELATIONS. 293

The filament is made of a variety of materials, including
certain vegetable fibres, silk, and parchmentized cotton
thread. After preliminary treatment it is carbonized by
raising to a cherry-red heat out of contact with the air. It
is then surrounded by an atmosphere of rarefied hydro-
carbon vapor, and is raised to a white heat by a current.
The heat decomposes the vapor, and the carbon residue is
deposited in a dense form on the filament. By this treat-
ment it acquires a hard, steel-gray surface and greater
uniformity. Its durability is thereby greatly increased.

The filament is finally mounted, in an exhausted glass
globe and provided with convenient external terminals.
The vacuum prevents oxidation and loss of energy by heat
convection.

The temperature to which the carbon filament can be
raised is limited by volatilization, and by a tendency of the
carbon to disintegrate at high temperatures. This disin-
tegration rapidly reduces the thickness of the filament and
blackens the glass bulb.

A 100-volt, 16-candle-power lamp has a resistance hot of
about 200 ohms. The current is then half an ampere, and
each lamp transforms into heat 50 watts, or three and one-
eighth watts per candle. A 50-volt lamp of the same candle
power has only one-quarter of the resistance and takes
double the current for the same candle power.

240. Thermal Electricity. — When heat is applied to
the junction of two dissimilar substances an E.M.F. is pro-
duced, which will cause a current to flow across the junction
from the substance of lower potential to the one of higher
if there is a closed circuit. This phenomenon is the con-
verse of the generation of heat by a current. It was
discovered by Seebeck in 1821 or 1822. If a circuit be

294 ELECTRICITY AND MAGNETISM.

formed of an iron and a copper wire, and if the tempera-
ture of one of the junctions be raised above that of the
other, a current will flow across the warmer junction from
copper to iron.

The heated junction is the seat of an E.M.F. of such
direction that the iron is at a higher potential than the
copper. A current therefore flows around through the cir-
cuit from the warmer iron across the cooler junction and
back to the warmer copper. Across the warmer junction
the current flows from lower to higher potential.

The dissimilar substances composing a thermo-electric
pair may be either two metals, a metal and a liquid, two
liquids, or even two pieces of the same metal at different
temperatures or in different physical states.

241. Neutral Temperature. — The E.M.F. of a thermal
element is small, and depends not only on the temperature-
difference of the two contacts, but on tne absolute values
of their temperatures. Every combination of two metals
appears to have what is called a neutral temperature. It
is the mean of the temperatures of the two junctions when
the electromotive forces at the two are equal and in oppo-
site directions round the circuit. For this neutral tem-
perature there is therefore no current. For silver and iron
the neutral temperature is 223°.5 C. ; for copper and
iron it is 274°.5 C. When the mean temperature is above
the neutral temperature for the two substances, the current
is reversed. If tv and t% are the temperatures of the two
junctions, there is no current when tx equals t>, and none
when £ (£, + Q equals the neutral temperature.

If an iron and a copper wire be twisted together and
their free ends connected to a galvanometer, moderate
heating of the twisted junction will cause a current to flow

THERMAL RELATIONS.

295

across it from copper to iron ; but if the junction be heated
to a dull red, the galvanometer will indicate a reversal of
the current.

242. Variation of Thermal Electromotive Force with
Temperature. — If one junction of a thermal couple, such
as iron and copper, be kept at a fixed temperature, while
that of the other is gradually raised, the E.M.F. increases
to a maximum, then diminishes,
at length vanishes, and is finally
reversed. With most pairs of
metals, if the temperatures be
plotted as abscissas and the
electromotive forces as ordi-
nates, the result will be a para-
bola with its axis vertical (Fig.
120). If, therefore, e denotes
the E.M.F. and t the tempera-
ture, and if E and T denote
the E.M.F. and temperature corresponding to the vertex
of the parabola, we obtain

E-e = b<iT-ty,
where b is a constant. This equation expresses the prop-
erty of a parabola that the square of the distance of any
point from the axis is proportional to the distance of the
same point from the tangent through the vertex. The
curve in the figure is drawn for the case where the tem-
perature of the one junction is zero. If it be above zero,
the parabola corresponding will have the same axis as this
one, but will lie below it. The temperature corresponding
to the maximum ordinate will be the same. It is the
neutral point for the given pair of metals.

In particular cases the curve is a straight line ; in others

296

ELECTRICITY AND MAGNETISM.

it is made up of parts of parabolas, with their axes parallel,
but with their vertices turned alternately in opposite direc-
tions (Peddie).

243. Thermo-electric Diagram. — The relation be-
tween E.M.F. and temperature just described led Lord
Kelvin and Professor Tait to adopt an elegant method
of constructing a thermo-electric diagram. The thermo-

+15
+10

+ 5
0

-5
-10
-15

■y$£>

Suiss-

*^i?

COPfj

B_ ;

LEAD

rt\

-£i4T

!*y*L_

""""-^

X)°

x>°

K)°

X)°

10°

electric power of any couple is the E.M.F. corresponding to
a temperature difference of one degree between the two
junctions. It is, in other words, the rate of variation of
the E.M.F. with temperature. By a simple application
of the Differential Calculus to the equation of the last
article, we obtain for this rate of variation,

at
This expression represents the thermo-electric power, and
it is the equation of a straight line. If then this line for

THERMAL RELATIONS. 297

some standard metal be made to coincide with the axis of
temperature, the lines obtained from observations on couples
of other metals with it will in general be straight lines ;
taken together, these lines form a thermo-electric diagram.
The point of intersection of any pair of lines corresponds
with the temperature of maximum E.M.F. for this pair of
metals. Thus the copper-iron lines cross at 274°.5 ; this is
therefore the temperature at which the thermo-electric
power of these metals becomes zero. It is also the neutral
temperature for the pair. Fig. 121 is the thermo-electric
diagram for several metals compared with lead. The inter-
sections of some of these lines lie beyond the limits of
Tait's experimental diagram. The palladium-copper lines
if produced would meet at — 170° C. Dewar and Fleming
have found, by means of the low temperature obtained by
liquid oxygen, that thermo-electric inversion for this pair
does occur at about — 170°.

244. Electromotive Force in the Thermo-electric
Diagram. — From the manner in which a thermo-electric
diagram is constructed, it follows that the E.M.F. between
any pair of metals between two temperatures is equal to
the area of the figure included between the ordinates cor-
responding to those temperatures and the thermo-electric
lines of the metals. Thus, if the cooler junction of a
copper-iron couple be at 100° and the warmer at 200°,
the effective E.M.F. in the circuit will be represented by
the area abed ; but if the warmer junction be at 400°, the
E.M.F. will be equal to the difference of the areas abn and
&d'n. If the triangle above the intersection n be larger
than the one below n, the E.M.F. will be reversed.

The ordinates represent thermo-electric powers. But
de/dt = Thermo-electric power,
and therefore de = Thermo-electric power x dt.

298 ELECTRICITY AND MAGNETISM.

Now de is the small E.M.F. corresponding to a small tem-
perature difference dt, and the second member of the last
equation is a small area whose length is a line ab and whose
width is an element of temperature measured at right angles
to ab. The E.M.F. for any finite temperature-difference
is therefore an area such as abed, which is made up of a
number of small areas corresponding to minute temperature-
differences.

245. Thermo-electric Series. — A thermo-electric
series is a table of metals showing their thermo-electric
relation to one another. Since the thermo-electric power
depends on the absolute temperature of the junctions, such
a list is good only for some definite mean temperature.
The following series gives the E.M.F. in microvolts
(millionths of a volt) between each metal and lead, with a
difference of one degree between the junctions when their
mean temperature is 20° C. :

Bismuth -89 Silver + 3.0

Cobalt ....
German silver . .
Mercury ....

Lead

Tin

Platinum . . .
Gold

-22

+ 3.7

-11.75

+ 3.8

- 0.418

, + 17.5

0.0

Antimony, axial . ,

, + 22.6

+ 0.1

Antimony, equatorial .

+ 26.4

+ 0.9

Tellurium . . . .

+ 502

+ 1-2

Selenium . . . .

+ 807

When a junction of any pair of these metals is moderately
heated, the current flows across it from the metal standing
higher in the list toward the one standing lower. For the
smaller values of the thermo-electric powers, the results
obtained by different observers are not very concordant.

246. The Thermopile. — The E.M.F. of a single ther-
mal element is very small; to get a larger E.M.F. a

THERMAL RELATIONS.

299

Fig. 122.

number of similar couples may be joined in series. With

n such couples in series the potential difference between

the extreme terminals is n times that of a single couple,

and the internal resistance of the

series is still very low. Fig. 122

shows the method of connecting in

series. If the bars A are antimony

and B bismuth, then heating the

junctions c, c, e, will cause a current

to flow through the circuit in the

direction of the arrow ; but if these

junctions be cooled, or the alternate ones d, d, be heated,

the current will circulate in the other direction.

When a number of bars of antimony and bismuth are

soldered together in this way, and packed together in the

form of a cube, with insulating material between adjacent
bars, so that opposite faces of the cube
form alternate junctions, the instrument is
called a thermopile (Fig. 123). If a face
of such a pile be blackened with lamp-
black and be provided with a reflecting
cone, the instrument becomes a sensitive
detector of radiant heat (69).

247. The Peltier Effect. — In 1834 Peltier discovered
the phenomenon which bears his name ; it is an extension
of the discovery of Seebeck. If a bismuth-antimony junc-
tion be heated, the current flows across from the former to
the latter. Peltier discovered that if a current from an
external E.M.F. be sent through such a compound bar from
bismuth B to antimony A (Fig. 124), the junction will be
cooled ; but if it be sent the other way, the junction will
be heated.

300

ELECTRICITY AND MAGNETISM.

The long arrow shows the direction of the current sent
through ; the small arrows at a and b indicate the direction
of the E.M.F. at the junctions. At a the thermal E.M.F.
is in the direction in which the current is flowing. Hence

A -

•»-

Fig. 124.

at this junction work is done on the current, and the heat
of the metals is converted into the energy of the current.
At b the thermal E.M.F. opposes the current, which there-
fore does work on the junction and heats it.

The thermal effect at a junction of dissimilar substances
differs greatly from the thermal effect due to simple resist-
ance. The Peltier effect is reversible, the current heating
or cooling the junction according to its direction, while
the quantity of heat evolved or absorbed varies simply
as the current; the heat due to resistance is independent
of the direction of the current, and is proportional to the
square of its strength.

248.

Experiment to show the Peltier Effect. — Con-
nect one or two Leclanche" cells with
a thermopile, as in Fig. 125. S is
a two-point switch. When it is
turned in the direction of the full
line, the battery circuit through the
thermopile is closed and the galva-
nometer circuit is open. When it
stands in the direction of the dotted
line, the battery is cut off and the
thermopile is connected with the

THERMAL RELATIONS. 301

galvanometer. In order to show that the current given
by the thermopile P is opposite in direction to the cur-
rent through it from the battery, insert in the circuit
of the galvanometer at I7 a copper-iron junction. With
the switch at J, the current produced by heating this junc-
tion Hows from Cu to Fe, and the direction of the gal-
vanometer deflection may be noted. Turn the switch for
a moment to a and then back again to b. The galvanom-
eter will show a current coming from the thermopile,
and the direction of the deflection will be the same as
when the junction T was warmed. Hence B must be the
positive and A the negative of the thermopile as a gen-
erator. But the current from the battery enters the pile
at B and leaves it at A. The thermal effects produced by
the current through the pile are such as to generate a
counter E.M.F.

249. The Thomson Effect. — For the
purpose of explaining electric inversion
in such couples as iron and copper, Lord
Kelvin assumed that the Peltier effect be-
comes zero at the neutral temperature.
No heat is then absorbed or evolved at a
junction at this temperature, but heat is
generated at the other junction, since the current there
meets a counter E.M.F. If in Fig. 126 the junction J is
at the neutral temperature T, and J' at a lower temperature
£, the current will flow in the direction of the arrows. At </',
therefore, it flows from Fe to Cm, and heat is generated by
the Peltier effect. There is then no conversion of thermal
into electrical energy at the junctions ; but since there is
no other possible source of the energy of the current except
heat, Lord Kelvin was led to predict that heat is absorbed

302

ELECTRICITY AND MAGNETISM.

at parts of the circuit other than the junctions. This pre-
diction he subsequently verified by experiment.

In copper heat is absorbed when the current passes from
cold parts to hot parts ; in iron it is absorbed when the
current passes from hot parts to cold parts. This phenom-
enon is called the Thomson Effect, or the Electric Convec-
tion of Heat.

Consider a metallic bar ABO (Fig. 127) heated at the
middle B and cooled at the ends A and 0. Then the dis-
tribution of heat may be repre-
sented by the curve abc. But if
a current be passed from A to
(7, then in metals like copper the
curve of the distribution of heat
becomes somewhat like a'bc/.
Since a current in copper absorbs
heat as a liquid does in flowing
from the cold to the hot parts of a tube, electricity is some-
times said to have specific heat. It is positive in metals
like copper and negative in metals like iron. In lead the
Thomson effect is nearly or quite zero ; it is for this reason
that lead is chosen as the zero line of the thermo-electric
diagram.

250. Thermo-electromotive Force between Metals
and Liquids. — The thermo-electromotive forces origi-
nating at metal-liquid contacts have special interest because
of their relation* to the temperature coefficient of voltaic
cells. These electromotive forces are larger than most of
those between metals. Thus, the thermo-electric power
of Zn — ZnSO is 0.00076 volt for a mean temperature
of 18°.5 C. j that of Ou— OuSO is 0.00069 volt for about
the same temperature. In microvolts these are 760 and 690

THERMAL RELATIONS. 303

respectively. Since the metal is positive to the liquid in
both cases, and there is no appreciable E.M.F. at the con-
tact of the two liquids, the temperature coefficient of a
Daniell cell is the difference of the above two thermo-
electric powers, or 0.00007 volt per degree C. It is, more-
over, negative because the thermal E.M.F. on the zinc side
is greater than on the copper side. This conclusion has
been fully verified by experiment.

The author has applied the same method of analysis to
other cells, such as the Clark without zinc-sulphate crystals,
and the calomel cell ; the results with all of them show
that the temperature coefficient is determined by the super-
position of the several thermal electromotive forces at the
contacts of the dissimilar substances in the cell, whenever
this coefficient is not complicated by the solution and re-
crystallization of salts. Whether the resultant temperature
coefficient shall be positive or negative depends on the
relative values .and signs of the thermal electromotive
forces on the two sides of the cell.

PROBLEMS.

1. The poles of a voltaic cell are joined by two wires in parallel
alike in every respect, except that one is twice as long as the other.
What are the relative quantities of heat generated in the two?

2. The E.M.F. of a battery is 20 volts and its internal resistance
2 ohms. The potential difference between its poles when connected
by a wire A is 16 volts; it falls to 14 volts when A is replaced by
another wire B. Calculate the number of calories of heat generated
in the external circuit in 3 min. in the two cases.

8. A current of 10 amperes passes through a resistance of 2 ohms
for 14 sec. Find the number of calories of heat generated.

4. The resistances of two wires are as 3 to 4. Find the relative
quantities of heat produced in the same time, — (1) when they are
joined in series, (2) when connected in parallel between the poles
of a voltaic cell.

304 ELECTRICITY AND MAGNETISM.

5. A battery has an E.M.F. of 8.5 volts ; the total resistance in
the circuit is 20 ohms, including an electrolytic cell. The heat gen-
erated per second in a 5.12-ohm coil included in the circuit is 0.12
calorie. What is the counter E.M.F. of the electrolytic cell ?

6. If one junction of an antimony-bismuth pair be at 20° and the
other at 65° C, what will be the E.M.F. ?

7. A ring is made partly of copper and partly of iron wire.
Compare the E.M.F. if one junction be kept at 0° and the other at
100° C. with the E.M.F. obtained by keeping one junction at 175° and
the other at 275° C.

PROPERTIES OF MAGNETS. 305

CHAPTER XX.

PROPERTIES OF MAGNETS.

251. Relation to Electricity. — The most important
properties of an electric circuit are its magnetic relations.
Magnetism is more readily and conveniently evoked by
electric currents than by any other means. In fact, von
Siemens said that " the electric current, or generally elec-
tricity in motion, is the only known source of all magnet-
ism." But the magnetic properties of an electric current
must be studied by means of magnets; it is, therefore,
necessary that some preliminary study of the properties of
a magnet should precede the study of the magnetic rela-
tions and effects of electric currents.

252. Fundamental Phenomena. — Black oxide of iron,
known as magnetite, is widely distributed, and is some-
times found to possess the property of
attracting iron. If a piece of it be sus-
pended by an untwisted thread (Fig. 128)
its longer dimension will point not far
from north and south. Such bodies are
called magnets. The property of orienta-
tion has been utilized in navigation for
several centuries, and from this fact the

magnet in early times acquired the name of lode at one, or
leading stone.

306

ELECTRICITY AND MAGNETISM.

A/WUfr,

Fig. 129.

E*^P«

253. Artificial Magnets. — If a piece of hard iron or
steel be stroked with a lodestone it will acquire the same
magnetic properties ; fine iron flings will cling to it, and
if suspended it will point north and south. The end which
points northward is called the north-seeking pole and the

other end the south-
i!j|||||||& seeking pole; the
magnet is said to
possess polarity. If
a bar magnet be dipped into iron filings they will cling to
it in tufts near the ends (Fig. 129), but there will be few
or none near the middle. This region is called the equator.
If a long thin rod be magnetized longitudinally the ends act
as centres of force or poles, and the imaginary line joining
these poles is the magnetic axis. The remainder of the mag-
net is apparently nearly devoid of magnetic properties. In
short thick magnets the poles are less definitely defined.

A thin pointed bar of magnetized steel, provided with a
cap having hard steel or agate set in it, so that it may
turn freely on a sharp steel point around a vertical axis, is
called a magnetic needle (Fig. 130).

254. First Law of Magnetic Force. — • If the S-
seeking pole of a bar
magnet be presented to
the N-«eeking pole of a
magnetic needle (Fig.
130), they will mutually
attract each other; but
if the N-seeking pole be
brought near the same
pole of the needle, re-
pulsion will be observed.

Fig. 130.

The law of attraction and repul«

PROPERTIES OF MAGNETS. 307

sion is accordingly formulated as follows : Like magnetic
poles repel and unlike poles attract each other.

255. Magnetic Substances. — A magnetic substance
is one capable of being affected by a magnet. A piece of
soft iron will attract either pole of a magnetic needle, but
it does not itself retain the property of attracting other
masses of iron, and does not possess the power of orienta-
tion when freely suspended horizontally. It has no fixed
poles and no equator.

Other substances attracted by a magnet are nickel, cobalt,
manganese, chromium, and cerium. Only nickel and cobalt
show decided magnetic properties comparable with iron.
Some gases are feebly magnetic, and liquid oxygen exhibits
conspicuous magnetic properties.

Another class of substances are apparently repelled by a
magnet. These are called diamagnetic to distinguish them
from paramagnetic bodies like iron and nickel. Among
them are bismuth, antimony, tin, copper, and some others
in a less marked degree. Paramagnetic bodies are often
designated simply by the word " magnetic."

256. Magnetic Induction. — When a magnet attracts
a piece of soft iron, the iron first be-
comes a temporary magnet by induction.
Magnetic induction is analogous to
electrostatic induction, and takes place
along lines of induction or lines of
magnetic force. When one piece of
iron has been attached to the pole of
a magnet, it may in turn act inductively
on a second one, and so on in a series F'*- l3'-

of temporary magnets of decreasing strength (Fig. 131).

%£

308 ELECTRICITY AND MAGNETISM.

But if the magnet be detached from the first piece and be

slowly withdrawn, all the small
iron cylinders will fall apart, and
they will not again attract one
-^y Cj^ \ another till they are once more

^ — ^ brought under the inductive in-

fluence of a magnet. A bar of iron
near a magnet is attracted because it becomes a temporary
magnet by induction, with the pole nearest to the pole of
the inducing magnet of the opposite kind or sign (Fig.
132). Induction thus precedes attraction.

257. Permanent and Temporary Magnets. — Per-
manent and temporary magnets differ only in degree. The
softest iron retains a small amount of magnetism after it
has been brought under the action of a magnetizing force,
while hardened steel retains a large proportion of it. The
latter loses some of its magnetism as soon as the magnet-
izing force is withdrawn, while the former loses the larger
part. A much larger magnetizing force is required to
magnetize hard steel than soft iron to the same magnetic
strength. The relation between the part lost and the part
retained depends on the quality and hardness of the iron
and on the after treatment which it receives. Cast-iron
retains an appreciable fraction of the magnetism induced in
it, and this property is utilized in starting the excitation of
dynamo machines. The property of resisting magnetiza-
tion or demagnetization is called retentivity. The reten-
tivity of hardened steel is much greater than that of soft
iron.

258. Magnetic Field. — Magnetic induction, like elec-
trostatic induction, is exerted through the agency of the

PROPERTIES OF MAGNETS.

309

surrounding medium. Evidence in support of this asser-
tion will accumulate as we advance in tlie study of the
subject. It would be unphilosophical to imagine an inde-
pendent medium or ether for every kind of action propa-
gated through space ; it is therefore assumed that the ether
concerned in magnetic induction is the same as that essential
to the phenomena of light and electrostatics. The ether
about a magnet is under magnetic stress, since the space
there is traversed by magnetic forces. Such a region, in
which a magnetic pole tends to move in a definite direction,
is a magnetic field.

Lines of magnetic force, or magnetic induction, are lines
along which a single ideal magnetic pole would tend to
move. The positive direction along a line of force is the

Fig. 133.

direction toward which a free N-seeking pole is urged. If
an observer stands with his back to a N-seeking pole, he is
looking in the positive direction of the lines of force coming
from that pole.

Paramagnetic substances like iron tend to move from the
weak to the strong parts of a magnetic field, while dia-
magnetic substances like bismuth tend to move from the
strong to the weak parts of the field.

310

ELECTRICITY AND MAGNETISM.

259. Magnetic Figures. — Magnetic figures, or a map
of the lines of magnetic force about a magnet, have been
known from early times. Fig. 133 shows the forms assumed

Fig. 134.

by iron filings sifted on a glass plate over a bar magnet.
When the plate is gently tapped the filings arrange them-
selves in curved lines running between the N and S poles.
Since the field is symmetrical about the magnetic axis, such
a figure may be obtained in any plane passing through the

Fig. 135.

axis. Each particle of iron is magnetized by induction
and sets itself along a line of force. The whole field about
such a magnet is therefore pervaded by lines of force.

PROPERTIES OF MAGNETS. 311

They form closed curves ; through the magnet they run from
the S to the N pole, while they complete their circuit in the
air from the N around to the S pole.

Fig. 134 was taken from the unlike poles of two similar
magnets. The lines of force stretch across from one to the
other. Now, lines of force show a tendency to shorten.
They act like stretched elastic cords mutually repelling one
another. Hence these two poles of opposite sign are drawn
together.

Fig. 135 was made from two like poles. No lines extend
across from one to the other. Moreover, the elasticity or
resiliency of these lines under distortion is plainly such
as to force the magnets apart, so that the lines may recover
their normal distribution about each pole.

260. Magnetic Shielding (J.J.T., 261). — Magnetic
attraction and repulsion, and magnetic induction take place
through all non-magnetic substances as if nothing were
interposed. Suspend a small piece of magnetized watch-
spring by a silk fibre inside a glass bottle or a large test-
tube. It is affected by

external iron or mag- -^fTzf^^

nets as if the glass were "i^-'/^===^*\^L

not present. The free- ~i~_~_~-~_~_~S.fL £ V-V-"

dom thus secured from IV J J

drafts of air makes this Z^

a good magnetoscope. -^^^C

Magnetic forces act

° Fig. 136.

across all substances,

except iron or other magnetic materials if of sufficient thick-
ness. A conductor is a perfect screen from electrostatic
action for bodies within it. A magnetic needle in a hollow
iron ball is screened in like manner from another system of

312 ELECTRICITY AND MAGNETISM.

magnetic forces, but only imperfectly. Consider a magnetic
needle inside an iron shell placed in a uniform magnetic
field; that is, a field consisting of a system of parallel equi-
distant lines of force. When the ball is introduced into
this field it is no longer uniform, but the lines pass through
the iron in preference to the air. Thus in Fig. 136 let P
be the needle within the shell. The lines of force crowd
into the iron. They are thus deflected toward the iron
within and without. A few will still traverse the hollow
space, but the number of these may be made indefinitely
small with a sufficient thickness of soft iron. If the inner
radius of the shell is one-half the outer, it may easily be
that the magnetic force inside is not more than ^shr of that
outside. The ratio depends on the quality of the iron, or
on what may here be called its specific conductivity for
lines of force (309).

261. Consequent Poles. — A bar of steel may be mag-
netized in such a way that it will have a succession of

poles alternating
V» 8 n s\ in sign. Thus in

Fig. 137 there are
Fig- ,37, north poles at N,

iV^ and south poles at £, S. The lines of force do not run
entirely through the length of the magnet, but the iV's are
centres from which they emerge from the magnet and the
*$".$ are centres to which they converge. A consequent
pole forms a part of two magnetic circuits. Such poles
are often used in dynamo-electric machines.

A ring may be magnetized either so as to present con-
sequent poles, or in such a way that it will exhibit no
external magnetic effects. Fig. 138 shows the lines of
force about a ring with consequent poles at 1 and 3. In

PROPERTIES OF MAGNETS.

313

Fig. 139 there are no poles ; that is, there are no points

at which the

lines of force

pass from the

iron into the air.

This ring con-
stitutes a closed

magnetic cir-
cuit, or one in

which the lines

of force are

wholly in the

iron. Such a

ring has no ex-
ternal magnetic

effect, so long as

there is no

change in its

magnetism, because there are no external lines of force.

Closed magnetic circuits are more
retentive of magnetism than open
ones.

Iljllj

^P*^*tt

BfilP

||gg&£- ;>"-^3

3^^^^^

IMP

wKwwrit

§*t|llllli
ilpfillii

WKm

Fig. 138.

262. Effects of Heat on Mag-
netism. — If a permanent mag-
net be heated to a bright-red
heat, all signs of magnetism dis-
appear. Up to 680° C. iron shows
but a slight change in its mag-
netic properties ; above this a rapid decrease in magnetic
susceptibility takes place, so that at about 750° C. it ceases
entirely to be magnetic and is quite indifferent toward a
magnet. Iron has therefore a magnetic limit, determined

Fig. 139.

314

ELECTRICITY AND MAGNETISM.

by temperature, and beyond this limit it is not affected by
magnetism. Nickel loses its magnetic properties at about
350° C. Chromium ceases to be magnetic at about 500°.
The temperature at which magnetic susceptibility reap-
pears when the temperature is reduced is lower than the

critical temperature at which it
disappears when the temperature is
raised.

Manganese is magnetic only at
temperatures near 0° C. Accord-
ing to Dewar, when iron is cooled
to about —200° C. in liquid oxygen
its susceptibility is twice as great
as at 0° C.

The loss of magnetization by
heat in the case of nickel is beau-
tifully shown by the simple appa-
ratus of Fig. 140, designed by
Bidwell. A thin tongue of nickel
is soldered to a copper disk and the
whole is blackened and suspended
by silk threads. A permanent mag-
net M is held in such a position that
it retains the nickel tongue just over
the flame of the alcohol lamp. When
the nickel is heated to the proper temperature the magnet
releases it and the nickel-copper bob swings as a pendulum.
During one or two vibrations it loses sufficient heat by
radiation and convection to recover its magnetism; it is
then attracted again and held by the magnet. This opera-
tion is repeated as soon as the nickel is again heated by
the lamp.

Fig. 140.

PROPERTIES OF MAGNETS. 315

263. Strength of Pole. — The strength of pole, or
degree of magnetization, of a magnet is defined by means
of its effect on another magnet. Thus, if at the same
distance the N pole of magnet A repels the N pole of
magnet B with a force/, and magnet O repels B with a
force 2/, then O is said to have twice the strength of pole
of A. Strength of pole is denoted by the letter m.

264. Unit Pole. — Consider two long, slender, uni-
formly magnetized needles with their similar poles A and

N B placed at a distance of one centimetre in air, the other
poles being so far away that they exert no appreciable
influence in the neighborhood of A and B. Then if A and
B are equal poles and the force between them is one dyne,
both A and B are poles of unit strength. A unit pole
repels an equal and similar pole at a distance of one centi-
metre in air with a force of one dyne. It is necessary to add
the qualifying phrase " in air," because the force would
not be one dyne if a magnetic substance intervened.

A pole of strength 2 would repel a unit pole at unit
distance with a force of two dynes. Hence if m and m' are
the strengths of two poles, the distance between them being
unity, the repulsion between the two is mm' dynes. If the
poles are of opposite signs mm' is negative, or a negative
force means an attraction.

The strength or intensity of a magnetic field at any point
is the force exerted on unit pole placed at the point, the
introduction of this pole not being supposed to influence
the field. Strength of field, or the flux of magnetic force
per square centimetre, is conventionally denoted by the
number of lines of force passing through one square centi-
metre at right angles to the direction of the field. It is
designated by the letter 8$.

316

ELECTRICITY AND MAGNETISM.

Imagine a sphere of unit radius described about a unit
pole as a centre. Then the intensity of the field at every
point on the surface of this sphere is unity, or one line
passes through every square centimetre. Therefore the
number of lines belonging to unit pole is 47r, since the
surface of the sphere is 4tt square centimetres ; and for a
pole of strength m the number of lines radiating is 4?rm.

265. Magnetic Moment. — The moment of a magnet
is the product of the strength of its poles and the distance
between them, or

Let the dotted lines (Fig. 141) be the direction of the field
of unit strength, and let ns be a magnet whose strength of
pole is m. Then the force on either
pole is m and the two forces consti-
tute a couple. The moment of this
couple when the magnetic axis of ns
is perpendicular to the lines of force
of the field is ml, and this is the mag-
netic moment.
Y
m,

266. Intensity of Magnetization.

Intensity of magnetization is the
magnetic moment per unit of volume
of the magnet. It must be regarded as having not only
magnitude but direction, its direction being that of the
axis of the magnet. If s is the sectional area of a long
uniform rod and I its length, then

n_ml _ m
Intensity of magnetism is the pole-strength per unit of area.

Fig. 141.

PROPERTIES OF MAGNETS. 317

267. Second Law of Magnetic Force. — The first
law (254) is qualitative. Coulomb, by means of his tor-
sion balance applied to magnetic poles instead of to electric
charges, gave quantitative expression to the law of mag-
netic force as affected by the distance between the poles :

The force between two magnetic poles is proportional to
the product of their strengths and inversely proportional
to the square of the distance between them.

This distance must be so great that the poles may be
regarded as mere points. Then from the definition of unit
pole we may write

268. Theory of Magnetic Figures. — The law of
inverse squares can now be applied to elucidate the form
of the curves developed about a magnet by means of iron
filings. Let NS (Fig. 142) be a long thin magnet, and
let P be a N-pointing pole in the field of NS. It will be
attracted by S and repelled by N along the lines PS and
PN respectively. The forces will be inversely as the
squares of these distances, and may be represented by
the lines PA and PB. Both forces act on the same pole.
Complete the par-
allelogram, and /">c _ ^
the diagonal PC
is the resultant
force. Since an
equal and oppo-
site force acts on
the south pole of
the same small magnet represented by a short iron fil-
ing, the two forces compose a couple tending to set the
particle of iron or other small magnet along the diagonal

Fig. 142.

318

ELECTRICITY AND MAGNETISM.

of the parallelogram. This line is therefore tangent to
the curved line of force passing through P.

If another point P' be chosen, equidistant from N and $,
the two forces of attraction and repulsion on either pole at
P' are equal and the diagonal is parallel to the axis of NS.
Continuing in this way, the direction of the intensity of the
field may be found at many points, and the directions com-
bined as tangents to a curve will map out lines of force.

269. Magnetic Forces by Method of Deflections. —
Two methods of making magnetic measurements are
worthy of discussion here. In the first a magnetic deflect-
ing force is compared with the intensity of the field in
which the magnet is placed by observing the angle of
deflection. If a magnetic needle be poised on a sharp
point or be suspended by a fine fibre, and if it be de-
flected by any means from the magnetic meridian, the
forces tending to bring it back consti-
tute a couple; and for equilibrium this
couple must be equal to the one pro-
►SFwducing the deflection.

Let NS (Fig. 143) be the direction
of the magnetic field, and let the magnet
be deflected by some force 8- at right an-
gles to the field of force. Then the forces
acting on the pole of the magnet are 88m
in the direction of the field and Shn at right
angles to the field. The moment of the
first force tending to replace the magnet
in the direction of the field is 8Bml sin 0,
where &8 is the intensity of the field, m is the strength
of pole of the needle, I is the half-length of the needle, and
I sin 0 is the lever arm AB. The moment of Efan is

Fig. 143.

PROPERTIES OF MAGNETS.

319

Efm x BO — S-ml cos 6. Equating the two moments
and

96ml sin 6 = &ml cos 0

or &= 96 tan 6.

The magnetic force producing a deflection is equal to
the product of the strength of field and the tangent of the
angle of deflection.

270. Method of Oscillations. — When a suspended
magnetic needle is disturbed from its position of equilib-
rium it describes a series of oscillations like a pendulum.
If the angular deflection be small the vibrations will all
be accomplished in the same period. The law of the
vibration of such a needle is the same as that of the pen-
dulum (I., 71), since the restoring couple is proportional
to the sine of the angle of deflection 6 (Fig.
144) ; and when this angle is small the mo- Nj
tion is simple harmonic. I.

We may therefore write for the period of a
complete vibration

where K is the moment of inertia of the needle,
96 the intensity of the field, and dlb is the
product ml corresponding to Mh in the case
of the pendulum.
From this equation

*"*\li

m96 = —=irsKn't

or the intensity of the field is proportional to the square of
the vibration-frequency.

The fields at two places may be compared by observing

320 ELECTRICITY AND MAGNETISM.

the number of vibrations made by the same magnet in
equal times, first at the one place and then at the other.
Then

86 _ T'-_n2
- d6'~¥~nn'

271. Comparison of Pole-strengths by Oscillations.
— Let one of the magnets to be compared be placed in the
same magnetic meridian with the oscillating needle, and
let the field produced by it at the needle be hy. Then

m(Jh+my = ftM£n{=Anl .... (6)

If the first magnet be replaced by the second one at the
same distance from the needle, then

37o(h2+&6) = Anl . . . . (V)

Subtract (a) from (b~) and (c) and

dTBh^Ainl-n^,

gWh, = A(nl-n2').

Whence £ = 4^-

h-i n-2 — n-

This equation gives the ratio of the pole-strengths of the
two magnets which produce fields hi and h. at the needle if
the distance be constant.

The law of inverse squares can be demonstrated in a
similar way by observing the oscillations of a needle first in
the earth's field alone, and then in the earth's field plus
that of the influencing magnet placed at successive distances
from the needle.

272. Magnetization and Mechanical Stress. — Joule
* observed that an iron rod increases in length when

PROPERTIES OF MAGNETS. 321

magnetized, but that no change of volume takes place.
Hence the rod must contract in sectional area. He con-
cluded that if a rod be magnetized circularly, that is, so
that the lines of magnetization are circles around the axis
of the rod, it should contract in length. This conclusion
he verified by experiment.

Bidwell l has extended Joule's observations by showing
that at a certain magnetization the elongation reaches a
maximum, and that for magnetizing forces beyond that the
elongation is less and less until the magnet finally remains
unaffected; any increase of the magnetizing force beyond
this point causes the rod to shorten. Effects of the same
kind occur in rings forming closed magnetic circuits ; the
diameter is increased by small magnetizing forces and is
decreased with larger ones.

The mechanical extension of a wire produces increase of
magnetization with small magnetizing forces ; but Villari
found that when the field is sufficiently intense, extension
causes a decrease of magnetization. This effect is called
the Villari reversal. Compression produces the opposite
effects to extension.

A circularly magnetized iron wire, when twisted, becomes
magnetized longitudinally ; and, conversely, torsion in weak
fields diminishes longitudinal magnetization and produces
circular magnetization. We may therefore conclude that
the superposition of both circular and longitudinal mag-
netizations will cause torsional strain. Wiedemann has
demonstrated this to be true in the case of iron. With
small magnetizing forces the twist is in one direction, but
when the magnetizing forces are large there is a reversal of
the direction of the twist. Nickel also exhibits a Villari
critical point and reversal for its residual magnetism ; but

lProc. Roy. Soc, XL., pp. 109, 267.

322 ELECTRICITY AND MAGNETISM.

for large magnetizing forces extension diminishes its mag-
netization and compression increases it.

273. Magnetism Molecular. — Numerous facts point
to the conclusion that magnetism is a molecular phenome-
non. If a piece of magnetized watch-spring be broken in
two, each half will be a magnet with its poles pointing in
the same direction as in the original magnet. Smaller
subdivision of the watch-spring simply increases the number
of poles without destroying the magnetism. It is therefore
inferred that the ultimate particles or molecules of steel
and iron are magnets, and that they are naturally and
permanently such. If a glass tube be filled with fine iron
filings, it may be magnetized ; if it be then shaken so as to
rearrange the particles, all signs of magnetization disappear.
The demagnetization produced by vibrating an- iron bar is
a phenomenon of similar character. If iron be cast in an
intense magnetic field it is found to be strongly magnetized.
Beetz deposited iron electrolytically in a thin line on silver
parallel to the lines of force in a strong magnetic field.
The iron was found to be so highly magnetized that no
more permanent magnetism could be induced in it.

Weber's hypothesis is that the molecules of iron and
other paramagnetic substances are natural magnets, but in
the unmagnetized state of the mass their axes lie in all
directions indifferently; when subjected to a magnetizing
force the magnetic axes of the molecules turn round more
or less in the direction of the axis of magnetization.
When they have all been turned in this direction the iron
is saturated and its magnetization can receive no further
increase. .As soon as the magnetizing force is withdrawn,
the molecules spring back partly toward their former posi-
tions ; thus, some of the magnetism is temporary, or the

PROPERTIES OF MAGNETS. 323

magnet has been supersaturated. In soft iron the mole-
cules offer less resistance to this molecular motion or rear-
rangement than in steel. Hence hardened steel possesses
greater coercive force and greater retentivity. To Weber's
theory Maxwell made the addition that the magnetized
molecules are rotating around their longer axes.

274. E wing's Theory of Magnetism. — Instead of sup-
posing that in the unmagnetized state the molecular mag-
netic axes are turned criss-cross, Ewing has shown that
the particles are arranged so as to form closed magnetic
circuits, or, at least, stable configurations under the action
of their mutual forces. A group of such molecules will
arrange themselves so as to satisfy their relative attractions
and repulsions. To illustrate his theory Ewing constructed
a model, consisting of short lozenge-shaped magnets piv-
oted on points and arranged at equal distances in a hori-
zontal plane. Any small number of these may group
themselves in several stable configurations. When they
are simply agitated they settle down into groups of equi-
librium. With a small external magnetizing force these
needles turn through a small angle only ; when the force
reaches a larger value, some of the needles suddenly turn
round and new groupings result, with most of the needles
pointing in the direction of the magnetizing force ; any
further increase of the magnetizing force produces but
little effect. These three stages correspond to three similar
ones often observed in magnetizing iron (316).

275. The Earth a Magnet. — Since a suspended mag-
netic needle tends to set itself in a definite direction, it
follows that the space about the earth is a magnetic field.
A small magnet shows that a couple acts on it to bring its

324 ELECTRICITY AND MAGNETISM.

axis into a definite azimuth, but no force tends to produce
motion of translation. This relation is due to the fact that
the magnetic pole of the earth is so far distant in compari-
son with the length of the small magnet that the forces
on the two poles of the latter are rigorously equal and in
opposite directions. The same condition may be described
by saying that the magnetic field due to the earth in the
vicinity of the magnet is uniform.

Take a piece of gas-pipe a metre long and carefully freed
from magnetism. If it be held horizontally east and west,
either end of it will attract both the N-seeking and the
S-seeking pole of a magnetic needle. Gradually tilt it
into a vertical position. Its lower end will become a N
pole and will repel the N pole of the needle. Reverse it
and the lower end is again a N pole and the upper end a
S pole. Hold it vertically, or, better still, in the meridian
and inclined about 75° below the horizontal toward the
north, and strike it a sharp blow on the upper end with a
hammer. It has now acquired permanent magnetism with
the N pole at the lower end. This fact can be demon-
strated by holding the pipe horizontally east and west.
By reversing it and striking it on the other end the polar-
ity may be reversed, and by graduating the strength of the
blow the pipe may be nearly or quite demagnetized.

The earth acts inductively on the pipe, as any other
magnet does on a piece of iron, putting it under magnetic
stress. The vibration due to the blow gives a certain free-
dom of motion to the molecules, and they arrange them-
selves to some slight extent under the influence of the
earth's magnetic stress. With the molecules so arranged
the pipe becomes a magnet. Bars of iron or steel in a
vertical or in a horizontal north-and-south position acquire
magnetism by induction from the earth. This is especially

PROPERTIES OF MAGNETS. 325

true if they are subjected to frequent jarring. Drills, rail-
way iron, beams, and posts are illustrations.

Since opposite poles attract, it is evident that the north-
ern hemisphere of the earth has the polarity corresponding
to the S-seeking pole of a magnet. This south magnetic
pole does not correspond with the geographical pole of the
northern hemisphere. Sir J. C. Ross, in 1831, found it to
be situated in Boothia Felix, just within the Arctic Circle, in
latitude 70° 5' N., and longitude 96° 46' W. of Greenwich.
Schwatke concluded in 1879, from his observations, that
the pole had shifted to longitude 99° 35' W. The magnetic
pole in the southern hemisphere has never been reached.

276. Magnetic Decimation (B., 682). — The magnetic
meridian is the vertical plane coinciding in direction with
the earth's field and containing, therefore, the axis of a
suspended magnetic needle. This meridian does not in
general coincide with the geographical meridian. The
angle between the two is called the magnetic declination.
The declination is east or west according as the N-seeking
pole of the needle points to the east or to the west of the
geographical meridian. The existence of magnetic decli-
nation was not known in Europe till the thirteenth century
and was first distinctly delineated on a map in 1436. To
Columbus belongs the undisputed discovery that the decli-
nation is different at different points of the earth's surface.
In 1492 he discovered a place of no declination in the
Atlantic Ocean north of the Azores.

Lines connecting points of equal declination are called
isogonic lines, and the line of no declination is an agonic
line. According to a chart constructed by the United States
Coast and Geodetic Survey, the agonic line in 1890
entered the United States from the Atlantic Ocean at

326 ELECTRICITY AND MAGNETISM.

Charleston, passed in a northwesterly direction through
Columbus, Ohio, about centrally through the lower penin-
sula of Michigan, across Grand Traverse Bay, Lake Michi-
gan, the upper peninsula, and Lake Superior.

The declination on the most easterly border of Maine is
now (1896) about 20° W., and on the extreme north-
western boundary of the State of Washington it is 23° E.
These values will not change much by the year 1900.

277. Variations in Declination. — The earliest re-
corded declination is that of London in 1580. It was then
11° 18' E. In 1657 it was zero at the same place. A
westerly declination then set in and attained a maximum
value of 24° 27' W. about 1816 ; since then it has been
slowly diminishing to its present value (1896) of about 16°
43' W. The needle will again point true north in London
about 1976, thus completing a half-cycle of changes in a
period of some 320 years.

Similar variations are in progress in other parts of the
earth. This change of long period is called the secular
variation of the declination. Besides it, there are the
diurnal and annual variations. In high latitudes the former
may reach 1°, but in middle latitudes it has a mean value
of about 7 \'. The annual variation is small, and is subject
to a periodicity corresponding apparently with the sun-spot
period of about eleven years.

Besides these variations, magnetic perturbations occur
during earthquakes, volcanic eruptions, and particularly
during auroral displays. The perturbations due to this last
cause sometimes reach a value as large as one or two
degrees. They are felt over wide areas, and are called
magnetic storms.

PROPERTIES OF MAGNETS. 327

278. Inclination or Dip. — If a magnetic needle be
carefully balanced on an axis through its centre of gravity
before magnetization, its N-seeking pole after magnetization
will incline below the horizontal in the northern hemisphere
by an angle ranging from 0° to 90°. This angle is called
the inclination or dip. Norman, a London instrument-
maker who first measured it in 1576, constructed a dipping
needle, which is a magnetic needle free to turn about a
horizontal axis in a vertical plane, and is provided with a
graduated vertical circle. The dip in London in 1576 was
71° 50'. It undergoes secular changes like those of the
declination. The dip in London for 1900 will be 67° 9' and
in Washington, 70° 18'. It reached its maximum value in
London in 1720 and has since been slowly diminishing. At
the magnetic pole in the northern hemisphere the needle
points vertically downwards.

279. Isoclinic Lines. — Lines connecting points of
equal inclination on the earth's surface are isoclinic lines.
In the vicinity of the equator is a line of no inclination,
called the magnetic equator. It is a somewhat irregular
line and crosses the earth's equator at two
points, in longitude near 2° E. and 170°
W. It veers as far south as lat. 16° and
as far north as lat. 10°. The isoclinic
of 72° passes near Princeton, Pittsburgh, ~
Fort Wayne, Michigan City, Iowa City,
Helena, and Vancouver Island on the Pacific
coast.

280. Relations between Declination,
Inclination, and Total Intensity. — If 8 denotes the
angle of dip, then the total intensity of terrestrial mag-

328 ELECTRICITY AND MAGNETISM.

netism may be resolved into a vertical and a horizontal
component (Fig. 145) as follows:

<?= JsinS,

3g=Jcos8.

Hence tan S = -— •

Lines connecting places where the horizontal component
of terrestrial magnetic intensity is the same are called
isodynamic lines.

PROBLEMS.

1. A magnet whose strength of pole is 150 is placed in a mag-
netic field whose intensity is 0.18. What forces act on its poles ?

2. A bar magnet, 10 X 2 X 0.25 cms., was magnetized to a
strength of pole of 50. What was the intensity of magnetization

3. The horizontal component of the earth's magnetism at station
A was found to be 0.183; a magnet was oscillated at station A and
station B, and made 60 oscillations in 11 m. 24 s. at the former and
in 12 m. 12 s. at the latter. Find the horizontal intensity at B.

4. A rectangular magnet, whose length was 15 cms. and strength
of pole 50, was set oscillating in a field whose horizontal intensity
was 0.18. It made 80 complete vibrations in 15 m. 4 s. Find its
moment of inertia.

5. When the magnet of problem 4 was made to oscillate at equal
distances from two magnets A and B successively, it made 80 com-
plete vibrations in 9 m. 4 s. and 10 m. 40s. respectively. Compare
the strength of pole of A and B.

MAGNETIC RELATIONS OF A CURRENT. 329

CHAPTER XXI.

MAGNETIC RELATIONS OF A CURRENT.

281. Oersted's Discovery. — The discovery by Oersted
at Copenhagen in 1819 was one of prime importance. He
observed that when a magnetic needle is brought near a
long straight wire conveying a current, the needle tends to
set itself at right angles both to the wire and to a perpen-
dicular drawn to it from the centre of the needle ; also that
the direction in which the needle turns depends on the

+ _

Fig. 146.

direction of the current through the conductor. A current
through a conductor therefore produces a magnetic field
about it. At this point the analogy between an electric
current and a stream of water flowing through a pipe fails,
for such a stream produces no effect in the region sur-
rounding the pipe.

Let a current flow through the conductor above the
needle NS from north to south as indicated (Fig. 146).
The N pole will turn toward the east. If the current be
reversed through this conductor, or if it pass from north to
south through the conductor under the needle, the N pole

330

ELECTRICITY AND MAGNETISM.

will turn toward the west. A current upward through a
vertical wire near the N pole of the needle will deflect it

in the direction of the arrows ;
that is, the N pole turns toward
the east.

If the wire be carried round
the needle in a rectangular
loop (Fig. 147), both branches
of it will contribute to the force
of deflection, and the N-seeking
pole at the left will turn to-
ward the east.

Fig. 147.

282. Ampere's Rule. — All the movements of a mag-
netic needle under the influence of a current may be
summed up in one rule. That of Ampere is the follow-
ing: Conceive a man swimming with the electric current
through a conductor and facing the needle; then the
N-seeking pole will always be deflected in the direction
of his left hand. Since the action between the current
and the needle, like all others, is mutual, the conductor
will be. urged toward his right.

A somewhat more convenient " rule of thumb " for most
cases is the following: Conceive the current flowing in
the direction of the extended fingers of the outstretched
right hand, with the palm turned toward the needle ; then
the N pole will be acted on by a magnetic force in the
direction of the extended thumb.

283. Maxwell's Rule The rule suggested by Max-
well has the advantage that it expresses reciprocally the
relation between the direction of the current and the direc-
tion of the deflection. Consider a right-handed screw ; if

MAGNETIC RELATIONS OF A CURRENT.

331

Fig. 148.

the direction of the current be that of the forward motion
of the screw as it enters the nut, the positive direction of
the magnetic force is the di-
rection in which the screw
turns (Fig. 148). The same
relation is represented by the
circle and the arrow in Fig.
149. If the current flows in
the direction of the long arrow, the resulting magnetic
force is with the arrows around the circle ; conversely, if
the current flows around the circle
with watch hands, the positive direc-
tion of the magnetic force, or the di-
rection in which a N-seeking pole is
urged, is along the axis of the circle
away from tne observer. If the fingers
of the closed right hand represent the
circle with the current flowing out at
the finger tips, the outstretched thumb
points in the direction of the lines of force.

284. Magnetic Field about a "Wire. — A
little consideration will show that if an observer
identifies himself with the conductor, the current
running from foot to head (Fig. 150), a single
isolated N-seeking pole would be urged round
him in a circle from right to left. The lines of
force due to a current are therefore concentric
circles about the conductor as a centre. Fig. 151
is made from the curves developed by iron filings
on a sheet of cardboard whose plane was per-
pendicular to the wire. The wire is seen end-on
in the figure. On gently tapping the paper the filings
arrange themselves in circles.

Fig. 149.

Fig. 150.

332

ELECTRICITY AND MAGNETISM.

This figure is a representation of what exists in any

plane perpendicular to
the wire. The entire
region about a con-
ductor conveying a
current is therefore
filled with these circu-
lar magnetic whirls.
They show that the
ether is under stress,
and therefore pos-
sesses potential en-
It is rather
important to
the attention

■Ktf4>< ••• :

Fig. 151.

ergy
more
direct

to these magnetic
effects in the ether about the current than to what goes
on within the conductor itself.

285. Magnetic Field about a Current through a Cir-
cular Conductor. — If the conductor be bent into a circle
or a loop (Fig. 152), the
space within it possesses
magnetic properties. All
the lines of force pass
through the loop so as to
urge the N-seeking pole of
a magnet downwards. The
current is flowing round
the loop, viewed from
above, in the direction of
watch hands (compare Fig.
149). Such a loop carrying a current acts like a magnetic

Fig. 152.

MAGNETIC RELATIONS OF A CURRENT. 333

shell ; that is, one side of it attracts the N-seeking pole of
a magnet and the other repels it. A magnetic shell is
equivalent to a thin sheet made up of short bar magnets
placed side by side with their N poles forming one surface
of the plane or shell, and their S poles the other. It is
known as the lamellar distribution of magnetism. An
electric circuit is in every case equivalent to a magnetic
shell whose contour coincides with the circuit. The shell
is of such strength that the number of lines of force
coming from it is the same as the number due to the cur-
rent in the loop ; that is, the magnetic shell and the closed
circuit have in their vicinity identical magnetic fields.
The difference between them is that the shell is impervious
while the circuit is not.

286. Intensity of Field at Centre of Circular Coil.
— The intensity of the magnetic field at any point is the
force acting on a unit pole placed at the point. Faraday
showed that the magnetic intensity produced by a current
is proportional to the current, and Biot and Savart demon-
strated experimentally that for a current of indefinite
extent it is inversely proportional to the distance between
the conductor and the point. Laplace proved that this
latter result follows from the law of inverse squares as
applied to the mutual action between an element of the
circuit and the pole, thus confirming the law of Ampere.
Hence the intensity due to the current in an element ds of
the conductor, at a point P on a perpendicular from the
element, is

~. kids

where d is the distance between the current-element and
the point P, k is the force on unit pole due to unit current
at unit distance, and I is the strength of the current.

334 ELECTRICITY AND MAGNETISM.

If now P is at the centre of the circle around which the
current is flowing, then the intensity at the centre due to
the current in the entire circumference will be

- JcIZds 2irrkl 2irkl

&~ — = — -J- =

v r1 r

If the unit current is so defined as to make k equal to unity,
we have

287. The Electromagnetic Unit of Current. — The
electromagnetic system of electrical units in common use
is based on the magnetic effects of a current. If an element
of a conductor one centimetre long be bent into an arc of one
centimetre radius, then the current through it will have unit
strength when it exerts a force of one dyne on a unit pole at
the centre of the arc. This definition is equivalent to mak-
ing k equal to unity in the last article. If the field due
to unit current in unit length of the conductor is unity,
the field due to the whole circumference will be 2tt ; and
if the current is J, it will be 2irl. If, further, the radius is
not unity, but r, the circumference will be 27rr, and then

_. 2irrl 2ttI

tr = — 5- =

r" r

The ampere is one-tenth of this absolute or C.G.S. unit of
current. The unit of quantity in the electromagnetic
system is the quantity which passes any cross-section of a
conductor in one second when the current through it has
unit strength. The practical unit of quantity is the
coulomb ; it corresponds with the ampere, and is one-tenth
of an absolute unit.

MAGNETIC RELATIONS OF A CURRENT. 335

288. Galvanometers. — We are now prepared to con-
sider in an elementary way several types of galvanometers
or instruments for measuring electric currents. When
their scales are graduated so as to read directly in amperes,
or when the readings reduce to amperes by the application
of a simple formula, galvanometers are called ammeters.

There are three general types of galvanometers : (1)
those in which the current flowing through a fixed coil of
wire causes the deflection of a suspended magnetic needle,
usually at the centre of the coil ; (2) those in which the
coil is movable around a vertical axis between the poles of
a fixed magnet. (3) These two kinds of galvanometers
are applicable to direct currents only. For both direct
and alternating currents another kind is employed, in which
both the fixed and the movable parts are coils. These are
called electrodynamometers.

289. The Tangent Galvanometer. — The tangent gal-
vanometer consists of a short magnetic needle poised at
the centre of a large vertical coil with its plane in the
magnetic meridian. The radius of the coil must be large
in comparison with the length of the needle,
which turns about a vertical axis.

The magnetic field produced by the cur-
rent through the large coil is nearly uniform
near its centre, and is perpendicular to the
plane of the coil. For a short needle, there-
fore, the deflecting force is perpendicular
to the horizontal component of the earth's
magnetism, and its motion round a vertical
axis will not carry its poles into a magnetic
field of different strength. For equilibrium the moments
of these two forces are equal.

336 ELECTRICITY AND MAGNETISM.

Let NS (Fig. 153) be the magnetic meridian, and the
trace of the plane of the coil with its centre at 0. Then
the two forces acting on the pole at A of strength m are
cfSm in the magnetic meridian and 27rml/r at right angles
to it ; r is the radius of the coil consisting of a single turn.
If I is the half-length of the needle A 0, and 0 the angle
of deflection, then

96mxAB=27T^xOB,
r

or 96ml sin 0 = ^^1 1 cos 0.

Both m and I cancel out, and the deflection is independent
of the pole-strength. From this equation

I = 88 ~ km<9.
Ztt

For n turns of wire, where n is only a small number and
the n turns may be considered coincident,

I=ggJL tan0.

The fraction 27r/r, or 2trn/r^ is called the constant of the
galvanometer. It equals the strength of field produced at
the centre by unit current through the coil. If this con-
stant is denoted by 6r, the equation for the current may
be written simply

Gr
I is here expressed in C.G.S. units. In amperes,

1= 10?? tan 0.

For a uniform magnetic field the current is proportional to
the tangent of the angle of deflection. The chief objection

MAGNETIC RELATIONS OF A CURRENT. 337

to the use of this form of galvanometer is the variability
of 96.

290. Nobili's Astatic Pair. — For greater sensibility
the controlling couple of the earth's field on the movable
magnetic system must be reduced. This may be done by
means of a weak compensating magnet, either above or
below the movable magnetic needle, with its N-seeking pole
turned toward the north. The field produced by it at the
needle is then opposed to the earth's field. .

Nobili's astatic pair is another method ^

in common use. It consists of a pair of
needles (Fig. 154) mounted in the same ""
vertical plane, but with their similar poles
turned in opposite directions. If their

. Fig. 154.

magnetic axes were rigorously in the same
plane, their lengths equal, and their poles of the same
strength, such a system would stand indifferently in any
azimuth. In practice neither condition is exactly met, but
the system has a small directive force tending to set it in
the plane of the magnetic meridian.

If both needles are surrounded with coils so connected
that the current flows round them in opposite directions,
the two forces of deflection will turn the system in the
same direction, while the opposing controlling force is re-
duced to a small value.

291. The Astatic Mirror Galvanometer. — In Fig. 155
the coils are swung open to expose to view the astatic sys-
tem. It consists of minute pieces of magnetized watch-
spring at the centres of the coils above and below. They
are mounted on an aluminium wire, and midway between
them is a small round mirror to reflect a beam of light

338

ELECTRICITY AND MAGNETISM.

which serves as a long pointer. The lower set of magnets
has a slightly greater magnetic moment than the upper one.

The four coils are
so joined in series
that the current
through them oper-
ates to turn the
whole system in
the same direction.
The control mag-
net in this particu-
lar instrument is
under the base. It
is movable around
a vertical axis, and
its effective mag-
netic moment can
be varied by turn-
ing the milled
screw S. The
screw turns two
soft-iron nuts
_. .„ threaded on the

Fig. 155.

magnet so as to
partly close its magnetic circuit, and thus alter its external
field of force.

292. The d'Arsonval Galvanometer. — It is imma-
terial from a magnetic point of view whether the coil or
the magnet of a galvanometer is made movable, since the
action between them is reciprocal. The great advantage
of the d'Arsonval galvanometer is that it has a strong field
of its own, which is only slightly affected by the earth's

MAGNETIC RELATIONS OF A CURRENT.

339

magnetism or. by iron or other magnetic materials in its
neighborhood. It is also possible to so shape the pole-
pieces of the permanent magnet in it that the deflection
shall be strictly proportional to
the current through a wide
range. The well-known Weston
instruments are of this type. .
Fig. 156 is a d'Arsonval gal-
vanometer of the ordinary pat-
tern. The coil swings in the
strong field between the poles
of the upright magnet and the
cylindrical sofMron core inside
of it. It is suspended by a fine
wire or thin phosphor-bronze
strip, through which the cur-
rent enters the coil, while a
straight wire or a helix con-
veys it out at the bottom.

The Ayrton-Mather form of
this galvanometer has a single ring magnet with only a

narrow vertical opening between
its poles (Fig. 157). In this open-
ing is placed a tube containing a
long narrow coil without any iron
core. It is suspended as in the
other form. Its plane must be
parallel to the lines of force in the
narrow gap in which it hangs.

293. Potential Galvanome-
ters. — Galvanometers used for the purpose of determining
the potential difference between two points of a circuit

Fig. 156

340 ELECTRICITY AND MAGNETISM.

must be of high resistance. If they are graduated to read
in volts they are called voltmeters. They are always con-
nected as a shunt. Thus, if the galvanometer Gr (Fig. 158)
is connected to the points A and B as a shunt to the resist-
ance s, and if its resistance is high in comparison with 8,
so that no appreciable part of the whole current passes
through the galvanometer, then the small current that
does pass through it is strictly de-
pendent on the potential difference
between A and B. Any sensitive
galvanometer may be used as a volt-
meter by adding a sufficiently large
resistance in series with it. Unless
the resistance of a voltmeter is high,
the application of its terminals to two
points of a circuit, so as to put it in parallel with a resist-
ance through which a current is flowing, will diminish
the potential difference which it is desired to measure.

If the galvanometer resistance is 99 times that of the
shunted resistance s, then one per cent of the current goes
through the galvanometer, and the potential difference
between the terminals of s is reduced one per cent.

294. Electromagnetic Units. — It will be convenient
for reference to bring together the several electrical units
expressed in magnetic measure in the C.G.S. system.

Unit Strength of Current. A current has unit strength
when a length of one centimetre of its circuit, bent into an
arc of one centimetre radius, exerts a force of one dyne on
a unit magnetic pole (264) at its centre (287).

Unit Quantity of Electricity. It is the quantity conveyed
by unit current in one second.

Unit Potential Difference. Unit potential difference, or

MAGNETIC RELATIONS OF A CURRENT. 341

unit electromotive force, exists between two points when
the transfer of unit quantity of electricity from one point
to the other requires the expenditure of one erg of work.

Unit Resistance. A conductor offers unit resistance when
unit potential difference between its ends causes unit
current to flow through it.

Unit Capacity. A conductor has unit capacity when unit
quantity charges it to unit potential.

295. Practical Electrical Units. — Several of the ab-
solute electromagnetic units are inconveniently small and
others inconveniently large for practical use. Hence the
following multiples and sub-multiples of them have been
generally adopted as practical units :

Current. The ampere, equal to 10-1 C.G.S. units ; it is
practically represented by the current which will deposit
silver from silver nitrate at the rate of 0.001118 gm. per
second (210).

Quantity. The coulomb, equal to 10_1 C.G.S. units of
quantity ; it is the quantity conveyed by a current of one
ampere in one second.

Electromotive Force. The volt, equal to 10s C.G.S. units ;
it is 1000/1434 of the E.M.F. of a standard Clark cell at
15° C. (200).

Resistance. The ohm, equal to 109 C.G.S. units ; a volt
produces an ampere through a resistance of an ohm ; prac-
tically represented by the resistance of a uniform column of
mercury 106.3 cms. in length and 14.4521 gms. mass at 0°
C. (220).

Capacity. The farad, equal to 10-9 C.G.S. units ; it is
the capacity of a condenser which is charged to a potential
of one volt by one coulomb. The microfarad, chiefly used
in practice, is one-millionth of a farad, or 10-15 C.G.S. units.

342 ELECTRICITY AND MAGNETISM.

Work. The joule, equal to 107 ergs ; it is represented by
the energy expended per second by one ampere in one
ohm.

Power. The watt, equal to 107 ergs per second ; it is
equivalent to the power of a current of one ampere flowing
under an electric pressure of one volt, or one joule per
second ; approximately -^^ of a horse power.

Induction. The henry, equal to 109 C.G.S. units ; it is
the induction in a circuit when the electromotive force
induced in this circuit is one volt, while the inducing
current varies at the rate of one ampere per second (338).

The prefixes kilo- and milli- combined with any of the
preceding units signify a thousand and a thousandth respec-
tively. Thus a kilowatt is a thousand watts, and a millivolt
is a thousandth of a volt. The prefixes mega- and micro-
signify a million and a millionth respectively. Thus, a
megohm is a million ohms, and a microfarad is a millionth
of a farad.

ELECTRODYNAMICS. 343

CHAPTER XXII.

ELECTRODYNAMICS.

296. Electrodynamics. — The term electrodynamics is
applied to that part of the science of electricity which
is concerned with the force exerted by one current on
another. The reciprocal action between conductors con-
veying currents was discovered by Ampere in 1821, shortly
after Oersted's discovery of the reciprocal action between
a current and a magnet. So far as demonstrated, the forces
are between the conductors conveying the currents rather
than between the currents themselves. Every conductor
through which a current is flowing is surrounded by a
magnetic field, and the magnetic fields of two such con-
ductors react on each other.

297- Magnetic Fields about Parallel Currents (Th.,
385). — The reciprocal action between conductors carry-
ing currents is purely magnetic, and may be accounted for
by the stresses set up in the surrounding medium. The
magnetic field about a single conductor is composed of
concentric circles (284) ; but when the fields of two con-
ductors are in part superposed, the composite magnetic
figures will be those due to the resultant of the two sets
of forces in every part of the field. Moreover, these figures
will exhibit attraction or repulsion between the conductors
according to the relative directions of the currents through
them.

344

ELECTRICITY AND MAGNETISM.

Fig. 159 is the field developed by iron filings about two
parallel wires passing through the two holes and with the
currents flowing in the same direction. In addition to
the distortion of the small circles immediately about the
conductors, showing that they are crowded together on
the outward sides and elongated between the wires, there
are continuous curves enclosing both circuits. These are
due to the coalescence of a number of circles belonging
to the two currents. The conductors are drawn together
by the tension along these lines of force. Midway between
the two is a region where the magnetic forces represented
by the circles are in opposite directions, and here the field
is neutral.

Wj^SS^S^Nik*

v^&siffiu

flap

IllliPi'

iS&^&JL

Fig. 159.

Fig. 160.

Fig. 160 is the field about two parallel conductors with
the currents flowing in opposite directions. It is the same
as the field through the centre of a circular conductor and
perpendicular to its plane. Midway between the two wires
the lines of force have the same direction in space, and
produce over a small area a uniform field. This is the
field utilized in the tangent galvanometer. The circles
about the wires are all excentric, but there are no lines
common to the two conductors ; the resiliency of these
lines, or their tendency to recover from the distortion,
forces the conductors apart.

EL ECTR 0 D YNA MICS.

345

298. Laws of Parallel and Oblique Currents. —

I. Parallel conductors conveying currents in the same
direction attract each other ; if the currents are in opposite
directions they repel each other.

This law is true for two portions of the same circuit or
for two independent circuits. It depends on the relation
of the two magnetic fields and not on their independent
origin.

II. Two conductors crossing obliquely attract each other
if the currents in them both flow toward the point of crossing
or away from it; but they repel if one flows toward and the
other away from this point.

The motion always tends to make the
conductors not only parallel, but coinci-
dent. If two flat spirals, like the one in
Fig. 161, be suspended by long wires so
that their planes are parallel, or make a
small angle with each other, they will ex-
hibit mutual attraction and repulsion in a
marked manner.

III. The force between two parallel conductors is propor-
tional to the product of the current strengths, to the length of

the portions taken, and inversely as the
distance between them.

299. Ampere's Stand. — For the
purpose of demonstrating the fore-
going laws, Ampere designed a stand
to hold a movable frame carrying
a current (Fig. 162). At a and b
are mercury cups into which dip the
terminals of the balanced frame.
Another conductor placed parallel to either side of this

Fig. 161.

Fig. 162.

346

ELECTRICITY AND MAGNETISM.

Fig. 163.

rectangle, or obliquely to it, will show attraction or repul-
sion ; the same apparatus will serve to show the reaction
of a magnet on a current.

Such a circuit as the one shown in the
figure tends to set its plane at right angles
to the magnetic meridian, with the current
flowing down on the east and up on the
west side of it. The direction of its own
field will then coincide with that of the
horizontal component of the earth's field.
Fig. 163 is an example of an astatic cir-
cuit that is not affected by terrestrial mag-
netism. The left side constitutes a south
pole and the right side a north pole ; that
is, the lines of force on the right of the
figure are directed toward an observer looking
at the figure, and away from him on the left.
Therefore the right side is repelled by the N-
seeking pole of a magnet and the left side is
attracted.

300. Electromagnetic Rotations. — A
large number of different devices have been
designed for the purpose of showing that
continuous rotations may be produced by the
action between a magnet and a circuit, or
between two parts of the same circuit. In the
earlier apparatus one part of the circuit was
made movable, and the circuit was kept closed
by making connection with a liquid conductor
like mercury. Fig. 164 is one of the forms
designed by Faraday ; a copper wire is hung by a hook at
the top, and the lower end dips into a cup of mercury M

EL ECTR OD YNAMICS.

347

surrounding the pole of a magnet. If the current flows
down through the wire, the lower end will rotate around the
pole clockwise-
Barlow's wheel (Fig.
165) is another device to
secure continuous rotation
by the action between a
magnet and a current. Con-
tact is made by mercury in
the trough C, and the ac-
tion of the magnetic field

is on the radial current from the mercury to the axis A of
the copper wheel.

301. Electrodynamometers. — The
electrodynamometer is an instrument
designed originally by Weber to meas-
ure the strength of a current by the
electrodynamic action between two
coils of wire, one fixed and the other
movable about a vertical axis through
its own plane. The coils are set with
their magnetic axes at right angles
(Fig. 166), and the free coil moves in
a direction to make their axes coincide.
Let AB be a single convolution of
the fixed coil, and CD one of the sus-
pended coil. The ends a and b of the
latter dip into mercury cups and the
two coils are in series, as shown by
the arrows. The movable coil is sus-
pended by silk threads, or on a point resting in a jewel, and
a helix is rigidly connected with it and with the torsion

Fig. 166.

348

ELECTRICITY AND MAGNETISM.

head T above. The movable conductor is then subjected
to a system of forces tending to turn it in the direction
indicated.

When the coil CD is deflected by sending a current
through the instrument, the torsion head is turned by hand
so as to bring the coil back to its zero or initial position.

The couple due to the action be-
tween the coils is then offset by
the couple of torsion of the helix.
Now the couple of torsion is pro-
portional to the angle of torsion by
Hooke's law, the forces of resti-
tution being, within elastic limits,
proportional to the distortion itself.
The electrodynamic action between
the coils is proportional to the
square of the current,
since doubling the current
doubles it through both
coils, and therefore quad-
ruples the force. The
square of the current is
then proportional to the
angle through which the counterbalancing helix is twisted, or

I2 = A2D,

I=AVD.

A is a constant depending on the windings and the helix.
Since this expression is the common equation of a parabola,
if the currents and twists of the helix are plotted as coor-
dinates, the resulting curve will be a parabola. The twist
D may be expressed in any convenient divisions of a circle
into equal parts.

Fig. 167.

ELECTROB YNAMICS.

349

Fig. 167 is one form of the complete instrument, showing
the coils, the helix, and the scale at the top with the
pointers. The movable coil is raised so that the suspend-
ing point is lifted out of the jewel bearing.

The fixed coil may be considered as furnishing a mag-
netic field corresponding to that of the permanent magnet
in the d'Arsonval galvanometer; but in this instrument
the field reverses with the current, and therefore the deflec-
tion is in the same direction whether the current goes in
one direction through the instrument or the other. It may
therefore be used with alternating or reversing currents as
well as with direct ones.

Fig. 168.

302. Kelvin Balances. — The justly celebrated instru-
ments of Lord Kelvin for measuring currents operate by
means of the electrodynamic action between parallel fixed
and movable coils. This action is counterbalanced by
adjustable weights or sliders instead of the torsion of a
helix. They are therefore dependent on the force of
gravity.

The coils are ring-shaped and horizontal. The movable
rings E and F (Fig. 168) are attached to the ends of
a horizontal balance beam, which is supported by two

350 ELECTRICITY AND MAGNETISM.

trunnions a and b, each hung by an elastic ligament of
fine copper wires. These are utilized to pass the current
into and out of the movable coils. A, B and (7, D are two
pairs of fixed coils, connected as shown, so that the mov-
able ring on either side is attracted by one fixed ring and
repelled by the other. When a current passes through the
six coils in series, the beam tends to rise at F and sink at
E. It is brought back to zero by sliding a weight to the
right along a graduated horizontal arm attached to the
beam of the balance. The weights are so adjusted that
the readings on this arm give the current either directly
or else by means of a table of double square roots. The
current is proportional to the square root of the reading
on a scale of equal parts. These balances, like the elec-
trodynamometer, may be used for alternating as well as for
direct currents.

303. Convection Currents. — Two concurring parallel
currents attract and two like electric charges repel each
other. According to Maxwell, the electrodynamic attraction
should exactly equal the electrostatic repulsion when the
electrical charges move with the velocity of light. Ac-
cording to Faraday, a stream of particles carrying electric
charges has a magnetic effect like a current of electricity.
This was demonstrated to be true by Rowland in 1876, who
found that a charged disk, when rapidly rotated, had a
feeble magnetic effect equivalent to a circular current.
Conversely convection currents are acted on by magnets.
The electric arc behaves like a flexible conductor. It may
even be ruptured by the deflecting influence of a powerful
magnet. Elihu Thomson has utilized this effect to extin-
guish an arc started by lightning on an electric lighting
circuit.

ELECTRODYNAMICS. 351

304. The Hall Effect. — In 1880 Hall discovered that
when a current flows through a very thin strip of metal
in a powerful magnetic field, with its plane perpendicular
to the lines of force, an E.M.F. appears to be developed in
the strip at right angles both to the field and to the direc-
tion of the current. The result is that the lines connecting
equipotential points are no longer at right angles to the
lines of flow, or the equipotential lines and the current
lines are both displaced. The displacement is in one direc-
tion in gold and bismuth, and in the other in iron and tel-
lurium. S. P. Thompson has shown that bismuth, which
exhibits the Hall effect in a marked degree, undergoes a
change of resistance in a magnetic field. The increase of
resistance shown by bismuth is so marked that this prop-
erty is utilized to measure the strength of the magnetic
field in which it is placed.

352 ELECTRICITY AND MAGNETISM.

CHAPTER XXIII.

ELECTROMAGNBTISM.

305. Solenoids. — Since a circular current is equiva-
lent to a plane magnetic shell, if we build up a cylinder
of such equal circular currents, all parallel to one another
and with their similar faces all turned in the same direc-
tion, we shall have the equivalent of a cylindrical magnet.
Such a system of circular currents
ft(W\?W^ constitutes a solenoid (Fig. 169).
The practical solenoid is simply a

Fig. 169. J

helix of a large number of flat turns
close together. Each turn of the helix may be resolved
into a plane circular current, ABC (Fig. 170), and a linear
current AC perpendicular to the plane of the
circle. The entire helix of n turns is there-
fore equivalent to n circular currents and a
linear current along the axis of the helix. If
the current returns along the axis, as in the
figure, the external field is due to the circular
elements only.

If such a solenoid be suspended on an Ampere's stand
it will set its axis in the magnetic meridian when a cur-
rent is passed through it. It is therefore equivalent to a
magnet, and its poles can be determined by Maxwell's rule
(283). Its poles will be attracted and repelled by a magnet
like a magnetic needle. The direction of the current is
with or against watch hands according as its S-seeking or
N-seeking pole is presented to the observer.

ELECTROMA GNETIS^f.

353

306. Effect of introducing Iron. — When iron is placed
in a magnetic field it becomes magnetized by induction
(256). If, therefore, a bar of iron be introduced into a
solenoid conveying a current, it will be magnetized by the
magnetic force along the axis of the helix. The presence
of the iron not only confines the lines of induction more
closely to the helix, but it greatly increases the number of

them, as represented in the solenoids of Fig. 171. These
solenoids are left-handed, but their poles may be deter-
mined in the usual way by the application of the " rule of
thumb" (282).

307. Electromagnets. — Directly after Oersted's dis-
covery Arago and Davy independently discovered that iron
and steel may be magnetized by inserting bars or strips of
them into a coil of insulated wire through which an electric
current circulates. If the bar be of soft iron it will exhibit
notable magnetic effects only so long as the current flows
through the magnetizing coil. The loss of magnetization
is not complete when the current is interrupted ; the small
amount remaining is called residual magnetism.

354

ELECTRICITY AND MAGNETISM.

Such temporary magnets produced by the magnetic in-
duction within a solenoid or magnetizing helix are called
electromagnets. When properly proportioned they are much
more powerful than permanent magnets. The polarity and
the relation of the poles to the direction of the current
may be determined by one of the usual rules.

Fig. 172

308. Horseshoe Mag-net. — The most common form
of electromagnet is the horseshoe type (Fig. 172). The
windings on the two iron cylinders or
cores must be in a direction to make
the two poles of opposite signs. It
is the same as if the two cores were
straightened out and the bar wound
continuously from end to end.

The armature, not shown in the
figure, consists of a flat bar like the yoke at the other
end, and extending across from pole to pole. Its cross-
section should be equal to that of
the cores. As a rule, the cores, the
yoke, and the armature should form
a nearly closed magnetic circuit
(261). If a ring be wound contin-
uously with a right-handed helix
so as to form a closed circuit, and
if connection be made with it at two
points diametrically opposite (Fig.
173), and a divided current be sent
through, there will be a consequent
south pole where the current enters
and a north pole where it leaves the
ring. The lines of force about it are those of Fig. 138.
The poles are consequent because they belong to two

Fig. 173.

ELECTROMAGNETISM. 355

magnetic circuits, or to a divided circuit through the

iron.

309. Magnetic Permeability. — The effect of placing
iron in a magnetic field is to increase greatly the number
of lines of induction running through the space occupied
by the iron. When these lines of magnetic induction
traverse the iron it is magnetized. The increase in the
number of lines due to the iron may amount to several
thousand per square centimetre.

Let 58 stand for the induction, or the number of lines
per square centimetre, through the iron. Then the ratio
between ££ and 96 is called the permeability, or

li = €B/&6,

where /* stands for the permeability. It expresses the fact
that iron transmits the inductive effect better than air, or
is more permeable. Magnetic induction is /* times the
magnetic force.

310. Magnetic Susceptibility. — The intensity of mag-
netization is the pole strength per unit area of the polar
surface (266). Magnetic susceptibility is the ratio between
the intensity of magnetization and the strength of the
field, or in symbols,

k = 3/86.

The conception involved in permeability rather than the
one in susceptibility is the modern one derived from
Faraday.

311. Relation between fi and k. — Let*^ be the num-
ber of lines of magnetic force existing in the air before the
introduction of the iron. Then the iron adds to these the

356 ELECTRICITY AND MAGNETISM.

lines due to a magnet of pole strength m. Hence, if * is
the sectional area of the uniformly magnetized bar,

ggs = &6s + 4ttw,

or m=^88 + 4Tr- = 9e+47rJ.

Wherefore, *®_i , a- A. ,

and /x = 1 + 47r/c.

Susceptibility may be negative ; but while permeability may

be less than unity, it is never negative.

312. Paramagnetic and Diamagnetic Substances
compared (J. J. T., 257). — The concept involved in
permeability permits a clear distinction to be drawn be-
tween paramagnetic and diamagnetic substances. Para-
magnetic substances are those whose permeability is greater
than unity ; and since the permeability of air is practically
unity, paramagnetic substances are those more permeable

to lines of magnetic in-
duction than air. On the
other hand, diamagnetic
substances have a permea-
bility less than unity, or
are less permeable than
air. Permeability ex-
Fig- '74. presses the number of
magnetic lines in the medium for every line of magnet-
izing force applied to produce them.

Paramagnetic substances concentrate the magnetic lines
and diamagnetic substances diffuse them. If iron be placed
in a magnetic field, it will cause more lines of induction to
pass through than through air; but if bismuth be placed

ELECTROMAGNETISM.

357

there instead of iron, fewer lines will pass through it than
through the air previous to its introduction.

If an iron sphere be placed in a uniform magnetic field
(Fig. 174) the effort of the lines will be to run as much as
possible through the sphere.
This action proceeds on the
principle that the potential
energy of a system always
tends to as small a value as
possible; for when the same
number of lines pass through
iron as through air they have
less energy in unit volume of iron.

If the sphere in Fig. 175 be bismuth the effort of the
magnetic lines will be to avoid it. There are fewer lines
of induction in it than in air. For the same number of
lines the energy per unit volume is greater in bismuth than
in air.

When the lines of force pass from air to a paramagnetic
substance they are bent away from the normal in the sub-
stance ; but when they pass from air to a diamagnetic
substance they are bent toward the normal.

Fig. 175.

313. Movement of Paramagnetic and Diamagnetic
Bodies in a Magnetic Field. — Faraday examined the
magnetic behavior of a large num-
ber of bodies in the intense field
between the pointed poles of a
powerful electromagnet. A small
bar of iron suspended between the
poles (Fig. 176) turns in the axial
direction ab, while a bar of bismuth
sets its longer axis in the equatorial direction cd. If the

Fig. 176.

c58 ELECTRICITY AND MAGNETISM.

bismuth is in the form of a cube or lump it is repelled
toward one side. Iron moves into the stronger parts of the
field and bismuth into the weaker. They are examples of
the two classes into which bodies are divided with respect
to the action of magnetism on them.

These movements may be satisfactorily explained by the
relative permeability of the body and the medium in which
it is suspended. Feebly magnetic bodies behave as if they
were diamagnetic when surrounded by a more highly mag-
netic medium. A small glass tube containing a weak solu-
tion of ferric chloride is paramagnetic in air; but when
suspended in a stronger solution of ferric chloride, it takes
a cross-position like a diamagnetic body. When, therefore,
any substance assumes the equatorial position, the only
inference which can justly be drawn from this behavior is
that its permeability is less than that of the air or other
medium surrounding it.

In general, liquids are diamagnetic ; liquid oxygen and
solutions of salts of the magnetic metals are exceptions.

314. Magrne-crystallic Action. — In crystalline bodies
the permeability may vary with the direction. Such a
substance is said to be ceolotropie. Tyndall found that the
magnetic axis or line of greatest permeability in a crystal
is in general an axis of greatest density, and it is this axis
that tends to place itself either along the magnetic field
or across it according as the crystal is paramagnetic or
diamagnetic.

Directions of unequal induction or permeability can
be produced artificially by pressure. Thus, a small roll
of powdered bismuth, made adhesive by mixing with
gum-water, will set itself across the field between the
poles of the excited magnet ; but if it be squeezed flat by

ELECTROMAGNETISM. 359

mechanical pressure, it will then turn in the axial position.
The lines of pressure transverse to the thickness are then
the lines of closest proximity of the particles and the lines
of most powerful induction.

315. Magnetic Induction and Magnetic Force. — If
a long iron bar be placed in a uniform magnetic field par-
allel to the lines of force, the lines of force in the bar are
called lines of magnetic induction. They will be parallel to
the axis of the bar in the portions distant from the ends.
If a narrow crevasse perpendicular to the lines of induc-
tion be made in the bar, the flux of force in the crevasse
continues as a flux of induction in the iron. In the air the
flux may be considered indifferently as induction or force. '
Lines of induction are consequently continuous throughout
the magnetic circuit. Near the ends of the bar the lines
of induction have not the same direction as the lines of
force of the uniform field. The poles induced in the bar
produce lines of force running counter to the lines of
induction in the iron. In some seolotropic substances the
axis of magnetization does not coincide with the lines of
force of the impressed magnetic field.

316. Curves of Magnetization. — When an iron bar or
ring is subjected to a gradually increasing magnetizing
force, the flux of induction through it increases at first
slowly, then very rapidly, and after this very slowly. The
ratio between 6B and 86 decreases toward a constant quan-
tity, which equals unity in the limit. If the magnetizing
force 86 be plotted horizontally and the induction £B ver-
tically, the resulting curve represents the successive stages
in the magnetization of iron. In Fig. 177 a is the curve
for mild steel, b for wrought iron, and c for cast iron.

360

ELECTRICITY AND MAGNETISM.

If the ratio of <£B to 96 were constant, the curve of mag-
netization would be a straight line. Since the curve is
concave toward the horizontal axis, except for very small
values of 96, fi is not a constant, but decreases with
increase of induction.

1X000

10000

8000

MOO

■1

4ae

Fig. 177.

Beyond the bend of the curve the iron is said to J?e
approaching saturation. For good soft iron this stage is
reached when £B equals from 16,000 to 18,000 lines per
square centimetre, with 96 from 50 to 200.

317. Hysteresis. — If the magnetization is carried
through a complete cycle by increasing the magnetizing
force by successive steps from zero to some definite value,
decreasing it from that value by small steps through zero
to an equal value in the other direction, and then again
reducing it to zero and completing the cycle, the curve

ELECTROMA GNETISM.

361

connecting 68 and 96 will not be the same with decreasing
values of 96 as with increasing ones (Fig. 178). The
induction 68 lags behind the magnetizing force. Thus,
when 96 is reduced to zero from its maximum positive
value, 68 has the value Ob, and 96 must be given a nega-

(li

6000

Fig. 178.

tive value equal to Oc before the induction becomes zero.
So when 96 returns from its maximum value in the other
direction to zero, the induction decreases only to the value
Oe. 1 his phenomenon of the lag of the induction behind
the magnetizing force Ewing has called magnetic hysteresis.
The result of plotting the corresponding values of 68 and
96 through a complete cycle is a curve enclosing an area,
and this, area represents the heat lost per cubic centimetre
in the iron in carrying it through a single cycle.1

* Ewiug's Mag. Ind. in Iron and other Metals.

3ti2 ELECTRICITY AND MAGNETISM.

318. Remanence and Coercive Force. — A cyclic mag-
netization curve, exhibiting hysteresis, serves among other
things to give definiteness to the terms remanence or reten-
tivity and coercive force. The residual value of £B when
BS is reduced to zero is Ob (Fig. 178). This value is the
remanence. It depends on the quality of the iron and the
limit to which 68 has been pushed. The figure relates to
a closed magnetic circuit consisting of a ring. The value
of &6 required to reduce this residual induction to zero,
viz., Oc, is the measure of the coercive force. Mechanical
vibration applied by external forces has the effect of dimin-
ishing residual magnetism, coercive force, and hysteresis.
If the iron in thin plates be carried rapidly through suc-
cessive cycles of magnetization by alternating currents, a
vibration will be set up in the plates unless they are rigidly
clamped together. Any vibration resulting from this
cause absorbs energy and increases the area of the hys-
teresis curve.

319. Law of the Magnetic Circuit. — The idea of a
magnetic circuit in a vague form is older than that of an
electric circuit, for it appears to go back to Euler in 1761.
Later Joule l asserted that the resistance to induction is
proportional to the length of a closed magnetic circuit ;
and Faraday insisted that the lines of magnetic force are
always closed curves. He also made the very apt com-
parison of an electromagnet with open magnetic circuit to
a voltaic cell immersed in an electrolyte of poor conduc-
tivity. The low permeability of the air corresponds to the
low conductivity of the electrolyte.

Maxwell gave mathematical expression to Faraday's

1 Reprint of Sci. Papers, Vol. I., p. 34.

ELECTR0MAGNET1SM. 363

ideas. He says: "In isotropic media the magnetic induc-
tion depends on the magnetic force in a manner which
exactly corresponds with that in which the electric current
depends on the electromotive force." 1

But the first definite expression of the law of the mag-
netic circuit in the form of an equation, like the equation
expressing Ohm's law, was given by Rowland in 1873 ; he
says expressly that it " is similar to the law of Ohm." 2

In 1883 Bosanquet introduced the term " magnetomotive

force," corresponding to electromotive force in the electric

circuit. We may then write

-, Jt a Magnetomotive force

Magnetic flux = — -s _ — . .

Magnetic reluctance

Before attempting to write a more definite equation for
the magnetic circuit, it is necessary to introduce certain
general propositions which determine the magnetomotive
force.

320. Rotation of a Closed Circuit in a Magnetic

Field Conceive a current of I C.G.S. units flowing

through the half circle ^ S^

abed (Fig. 179), and let / r/i|\

there be a unit magnetic / / j H \

pole at the centre P. 4 ". H ' . **

Then the field produced

at P by the current urges the pole in a direction normal
to the plane of the ring. The circuit is urged by an
equal force in the opposite direction. Let be be unit
length of the curve. Then by Ampere's law of the recip-
rocal mechanical action between a magnet and a current,
which has been experimentally demonstrated, we have

1 Elec. and Mag., Vol. II., p. 51.

i Phil. Mag., Vol. XL VI., August, 1873.

364 ELECTRICITY AND MAGNETISM.

(286) the force at P due to the current I in the length be
of the conductor equal to I/r2. Hence, the work done in
rotating the arc be through 360° about the axis ad against
this force is fxbex 2irr'. But this is / times the area of
that portion of the spherical surface generated by be during
the rotation. Hence, the entire work done against the
magnetic reaction between the whole semi-circumference
and the unit pole at the centre, for one revolution, is the
product of / and the surface of the sphere whose radius
is r, or

W=f X 47rr = - - X 47rr2 = knl.
r

Since 4-7T lines of force radiate from unit pole, and all of
these are cut by the semi-circle during one rotation around
the axis ad, it follows that the work done is the product
of the whole number of lines cut by the conductor and the
strength of the current flowing through it.

Suppose further that the rotation takes place in a period
t, that R is the resistance of the conductor between the
points a and d, and E the potential difference between
the same points. Then from the law of conservation of
energy the whole electrical work done is the sum of the
energy spent in heat and the work done in rotating the
conductor in the magnetic field. We may therefore write

Hit = PBt + 4ttJ
as the energy equation.

Therefore, E=IR+—,

E-±L

and j _ t

R~'

This is an expression for the current in the form of Ohm's

ELECTROMA GNETISM.

365

law. It shows that there is generated by the rotation an
E.M.F. equal to ±ir/t. But this fraction is the rate at
which the 4-7r lines of force from the unit pole are cut
by the conductor. The E.M.F. generated by a conductor
cutting across lines of magnetic force is, therefore, the rate
at which they are cut.

These two propositions we have derived from Ampere's
law and the conservation of energy applied to a particular
case. While the method is not a perfectly general one,
the results are of general application. In estimating the
number of lines cut or the rate of cutting them, attention
must be paid to the direction in which they are cut, and the
algebraic sum must be taken in all cases.

321. Force at a Point due to a Straight Current of
Indefinite Length (Th., 335). — Let ah (Fig. 180) be a
portion of the straight conductor conveying a b
current of strength i, and let P be a point at a
distance r from it. Then if we imagine a unit
pole at P, and if the conductor be carried round
it at the constant distance r, or the pole round
the conductor at the same distance, all the lines
of force from the pole will be cut once. Hence,
the work done will be Airl. If the field pro-
duced by the current at the point P is eft?, the
work done is the product of the field intensity
and the distance 2irr, or 2irr 96- Hence

2irr&8 = 4irJ,
98 = 2I/r.

Fig. 180.

If the current is in amperes, then the force in dynes at the
point is

a?=2J/10r.

366 ELECTRICITY AND MAGNETISM.

322. Force within a Helix. — Let AB (Fig. 181)
represent a section through the axis of a long helix, and

let unit pole be at the point P.
Let there be n turns of wire in
a length of one centimetre par-
allel to the axis of the helix, each
turn carrying a current I. Then
if the unit pole be carried along
the axis from P to P\ a distance of one centimetre, each
of the 47r lines of force from this pole will be cut by n
turns of wire. Hence, the whole number of lines cut will
be 4-7m, and the work done \nrnl. Since the distance
moved is one centimetre, the force is numerically equal
to the work, or

QS = ±TrnL

If the current is in amperes,

a? = 4™// 10.

This is the value of the field at points distant from the
ends of the helix. At the ends only half as many lines
would be cut by a movement of one centimetre, and the
field is only 2ttiiI/10.

If the helix or solenoid forms a closed curve, so that
there are no ends to the helix, the field along the magnetic
axis will be everywhere the same.

323. Magnetomotive Force. — The electromotive force
in a circuit is the work required to carry unit quantity of
electricity entirely round the circuit (186). So the mag-
netomotive force is the work done in carrying a unit pole
once round the magnetic circuit. If L is the length of
the solenoid, the work done will be L times the strength
of field or 47raiZ, if the field is uniform. If it be not

EL ECTROMA GNETISM.

367

uniform, then the magnetomotive force is the "line-inte-
gral" of the field intensity round the whole magnetic
circuit. Now nL is the entire number of turns of wire in
the solenoid. Let this be denoted by N; then the mag-
netomotive force is

# = 47riV7/10,

if the current is expressed in amperes. NI is called the
ampere-turns. The magnetomotive force in a long solenoid
is, therefore, 1.257 times the ampere-turns.

• 324. Reluctance (Th., 369). — The magnetic reluc-
tance of a bar of iron is "its resistance to lines of force."
It may be calculated from its length, its sectional area, and
its permeability, just as the electrical
resistance of a conductor may be cal-
culated from its length, its cross-sec-
tion, and its specific conductivity.
Let the length of the bar be I cms.,
its section S square cms., and its
permeability fi. Then its reluctance is

Let us apply this formula to the case
of the closed circuit of an electro-
magnet (Fig. 182). It is made up of
two parts, the core and the armature
sections, and permeabilities be denoted by Zi, and Z2, Si and
1%, and /*i and fi2 respectively. Then the reluctance of
the whole circuit is

&>= -V-V

Fig. 182.

Let the lengths,

368

ELECTRICITY AND MAGNETISM.

325. Law of the Magnetic Circuit applied. — When
the magnetic circuit is not closed, the lines of induction
must be forced across the air-gap be-
tween the faces of the iron parts of the
circuit. Suppose the armature removed
a short distance l8 from the poles (Fig.
183). Then the length of the circuit
is thereby increased 2l3 cms., and ad-
ditional reluctance is introduced equal
to 21J S3, where $5 is the cross-section
of the air traversed by the induction.
The permeability of the air is unity,
and does not appear in the expression.
We may therefore write for the flux

Fig. 183.

of magnetic induction
10

AttNI

'1 1 '2

where 7" is expressed in amperes.

While this expression is simple in theory it is rendered
difficult of application because /*, unlike specific conduc-
tivity, is not a constant, but is a function of the magneti-
zation or induction in the iron. In applying the formula
to any particular magnetic circuit it is necessary to know
the curve of magnetization or the quality of iron used, and
to ascertain from it or from tables the values of fi corre-
sponding to the degree of saturation which it is desired to
use. When this has been determined the formula gives
the number of ampere-turns of excitation required. For
open magnetic circuits an allowance must be made for
leakage of lines of force through the air between parts of
the magnet. This leakage requires excitation, but con-
tributes nothing to the purpose for which the magnet is

ELECTROMAGNETISM. 369

designed. The allowance for it must be estimated from
experience with the particular form of magnet employed.
The electromagnets of dynamos are' designed by a process
similar to this.

326. Motion in Electromagnetic Systems. — When-
ever any part of an electromagnetic system is movable, for
example, the armature of an electromagnet, the tendency
is always to move in a direction
to reduce the magnetic reluc-
tance and so to increase the mag-
netic flux. When the armature
approaches the poles, the air-gap
is shortened, the reluctance is
diminished, and more lines of
induction traverse the magnetic
circuit. So when any change
tends to occur in the configura-
tion of the parts of an electromagnetic system, it is always
such as to make the magnetic flux a maximum.

The same law may be applied to the dynamic action
between conductors conveying currents. Their relative
movements are in a direction to make the flux of magnetic
lines around them a maximum. Hence, two circuits tend
to move toward coincidence. Each is urged to a position
that will make the lines of force common to the two as
numerous as possible. Similar statements hold with respect
to a magnet and a circuit. When a bar magnet and a
helix come into a relative position where the middle point
of the former coincides with the mean plane of the latter,
the lines of force of the two are identical in direction
through the helix, and the position is one of stable
equilibrium (Fig. 184).

370 ELECTRICITY AND MAGNETISM.

327. Superficial Magnetization by Electric Dis-
charges.— Steel needles or small steel rods may be mag-
netized by the passage of an electric discharge around
them, or even at right angles to their length. It has long
been known that lightning flashes sometimes magnetize
hard steel. If a Leyden jar be discharged through a strip
of tin foil across which lies a sewing-needle, the needle
will be magnetized by the discharge. Better results will
be obtained by surrounding the needle with an open helix
of rubber-covered wire and discharging through it. It was
with simple means like these that Joseph Henry discovered
the oscillatory character of the Leyden-jar discharge.

Anomalous results have sometimes been observed in the
relation of the poles to the direction of the discharge around
the needles or rods, the poles being turned in the direction
opposite to what the rule would lead one to expect. This
result is due to the oscillatory discharge combined with the
superficial character of the magnetism imparted. If small
steel rods, magnetized by electric discharges, be examined
by removing the external portions with acid, it will be
found that the magnetized part is confined to a thin shell,
the underlying parts remaining ur magnetized. If a second
discharge succeeds the first in the opposite direction, it will
reduce the external magnetism to zero if the magnetism of
half the shell is reversed. Two shells of equal magnetic
moment will then be superposed in opposite senses. If
therefore the reverse discharge have more than half the
magnetizing effect of the first, the resultant magnetism
will be apparently " anomalous ; " but it is accounted for
by the direct and reverse discharges, and does not con-
stitute an exception to the law of magnetization.

Fig. 185 contains the curves obtained from two glass-
hard steel rods, 6 cms. long and 1.8 mms. in diameter, mag-
netized by ten successive discharges of a small Leyden jar

EL ECTR OMA GNETISM.

371

Fig. 185.

all in the same direction.1 The relation of the two magnet-
izing coils was such that the first reverse oscillation was
more powerful with B than with
A. The data for these curves
were obtained by removing suc-
cessive portions of the outside
with acid and measuring the 10
magnetic moments after each re-
moval. Moments are plotted as
ordinates, and decreasing weights 6
as abscissas. The moment of B
at first increases to a maximum,
and then decreases parallel to the
.A-curve. B had a thin external
shell magnetized in a sense op-
posite to that of the underlying portions. When this had
all been removed, the magnetic moment was a maximum.

PROBLEMS.

1. An iron bar 50 cms. long and 3 cms. in diameter was magnet-
ized to 15,780 lines per square centimetre, when \i equaled 800.
Find the reluctance and the total induction through the bar.

2. A ring of soft iron 20 cms. in diameter and 3 sq. cms. sectional
area is wound uniformly with a magnetizing helix. Find the number
of ampere-turns required to magnetize to 13,640 lines per square
centimetre, with// equal to 2,200 ; what will be the total induction ?

3. A straight wire carries a current of 10 amperes ; find the
force in dynes on a pole of strength 20 at a distance of 5 cms.
from the wire.

4. A conductor is bent into a circle of 15 cms. I'adius; find the
current through it which will deflect a short magnet at its centre 45°
if the horizontal intensity of the earth's field is 0.25.

5. An electric motor is wound with 128 wires on the outside of
the armature; the total magnetic flux through it is 1,250,000 lines;
find the work done in ergs in one revolution when a current of 50
amperes flows through each wire ; also find the power in kilowatts
when thei'e are 9G0 revolutions per minute.

• Atner. Jour. Sci., XXXI., April, 1886.

372 ELECTRICITY AND MAGNETISM.

CHAPTER XXIV.

ELECTROMAGNETIC INDUCTION.

328. Faraday's Discovery. — It has been seen that
Oersted's discovery led speedily to the discovery of mag-
netization by electric currents, and to the mechanical action
between conductors conveying them. Faraday completed
this correlated group of electromagnetic phenomena by
discovering in 1831 the laws of the electromagnetic induc-
tion of currents, or the laws under which induced currents
are produced by means of other currents or by magnets.1
These discoveries are of great interest, and it is of the
utmost importance that the student should familiarize
himself with the laws of induced currents, and should
connect them with the phenomena and laws developed in
the last three chapters.

Induced electromotive forces and currents are those
produced by the action of magnets and other currents.
Strictly only electromotive forces are induced; currents
flow as a consequence when the circuit in which the elec-
tromotive force is generated is closed. But the electromo-
tive force may still be induced whether the circuit is closed
or not.

All modern methods of producing large currents for
commercial purposes by dynamo machines, and all induc-
tion coils and alternate current transformers, are based on
electromagnetic induction.

1 Maxwell's Elec. and Mag., Vol. II., p. 163.

ELECTROMAGNETIC INDUCTION.

373

Fig. 186

329. Induction by Magnets. — Let a coil of insulated
wire of many turns be connected to a sensitive galva-
nometer (Fig. 186), and thrust
into it the pole B of a bar mag-
net. The galvanometer will in-
dicate a transient current, which
will continue to flow only dur-
ing the motion of the magnet.
If the magnet be withdrawn
from the coil a transient in-
duced current will flow in the
reverse direction.

When the magnet enters the
coil it carries with it its lines of
force, and they are therefore cut
across by the spirals of the coil. Now it will be seen in
Art. 320 that the reasoning there employed is independent

of the electro-
motive force E.
Hence, this may
be made equal
to zero, and the
conclusion still
holds that the
E. JVf. F. gener-
ated by cutting
across lines of
force is equal to the rate at which they are cut by the con-
ductor. For most cases it is better to express the E.M.F.
induced as the rate of change of the magnetic induction
through an electric circuit. .

If a coil of fine wire be wound around the armature of
a magnet (Fig. 187), then when the armature is in contact

Fig. 187.

374

ELECTRICITY AND MAGNETISM.

with the poles the flux of induction through the coils is
a maximum. When it is pulled away the magnetic flux
through the armature and the coil decreases rapidly, and a
direct E.M.F. is generated. This experiment illustrates
Faraday's method of producing electric currents by the
aid of magnetism.

330. Direction and Value of an Induced Electromo-
tive Force. — The numerical value of an induced elec-
tromotive force in C.G.S. units may be expressed as
follows :

The E.M.F. induced

is equal to the rate of

. change of the number of

^ lines of force threading

through the circuit.
-^ If d<& is the change
in the magnetic flux
through the circuit tak-
ing place iri the short
time dt, the induced
E.M.F. is
E=-d$/dt.
The minus sign indicates that a direct E.M.F. corresponds
to a decrease in the flux of induction. It is to be noted
that number of lines of force, magnetic induction, and
magnetic flux are all equivalent expressions.

The direction of the induced E.M.F. Faraday deter-
mined by experiment, but it can be deduced from con-
siderations with which we are already familiar. Let the
magnet NS (Fig. 188) be thrust into the helix. Then if
an E.M.F. is generated and a current circulates through
the coil, the energy of the current must be derived from the

Fig. 188.

ELECTROMAGNETIC INDUCTION.

375

work done in moving the magnet. There must therefore
be a resistance opposing this movement ; this resistance is
due to the helix considered as a magnetic shell, and the cur-
rent must flow around it in a direction to make a iVpole of
the side entered by the i^pole of the magnet. Its direction
is therefore against the motion of watch-hands as indicated
by the arrows. If the observer looks along the positive
direction of the lines of force, a current flowing with
watch-hands is said to be direct; if opposite to watch-
hands, it is indirect. Therefore we have the following law
relating to the direction of the induced E.M.F. :

An increase in the number of lines of force threading
through a helix produces an indirect E.M.F. , while a decrease
in the number of lines produces a direct E.M. F.

The minus sign in the expression above corresponds to
this statement.

331. Induction by Cur-
rents (J. J. T., 374). — Since
a current through a solenoid
produces a magnetic field
equivalent to that of a mag-
net, the same induction effects
will be produced by inserting
a helix conveying a current
into the long coil (Fig. 189)
as by thrusting in the magnet.
Let the circuit P include a
battery and a key, and the cir-
cuit S a galvanometer. The
former is called the primary,
and the latter the secondary.

If the current through P is kept constant while the
coil is moved about, then when P approaches S an E.M.F.

Fig. 189.

376 ELECTRICITY AND MAGNETISM.

is generated in iS tending to send a current in the opposite
direction to that round P ; while if P is moved away from
S, the E.M.F. induced in S is in the direction of the cur-
rent round P. These electromotive forces in S act only
so long as P is moving. If P is kept fixed while jS is
moved, the results are the same.

Next, let P be in a fixed position near jS with the key
open. Then on closing the key in P the galvanometer
needle will be deflected. This deflection is not a perma-
nent one, but the needle oscillates and finally returns to
its initial position of rest, indicating the passage of a
sudden discharge through the galvanometer. The direc-
tion of this momentary current is opposite to that through
P. On opening the key another similar momentary cur-
rent passes through S, but in the same direction as through
P. Thus the starting or stopping of a current in P is
accompanied by the induction of another current in a
neighboring circuit S.

The sudden increase of the current in P produces an
opposite current in S, and the sudden decrease of the
current in P produces a current through 8 in the same
direction as through P.

If while P remains inside of S, or coaxial with it, a bar
of soft iron is placed within it, there is an increase of mag-
netic flux through both P and S, and the E.M.F. generated
in S is in the same direction as that produced by closing
the key in P, moving P toward S, or increasing the cur-
rent through P. The withdrawal of the iron produces
the opposite effects to its insertion in the coil.

The law of the direction and magnitude of the E.M.F.
generated inductively by another current is the same as
that given in the last article. When the magnetic flux
changes, an E.M.F. is produced equal to the rate of change

ELECTROMAGNETIC INDUCTION. 377

in the magnetic flux passing through the circuit. The
positive direction of the E.M.F. and of the flux through
the circuit are related to each other as are the rotation and
translation of a right-handed screw.

332. Faraday's Ring. — Faraday wound upon an iron
ring two coils of wire P and S (Fig. 190). When a bat-
tery and a key were included in the circuit P and a gal-
vanometer in S, whenever the
circuit of P was closed or opened
a momentary current was pro-
duced in the closed circuit 3.
In this experiment the iron is
the medium through which the

° Fig. 190.

induction between P and S takes

place. The current through P magnetizes the iron ring as
a closed magnetic circuit. The starting of the current in
the circuit P sends magnetic lines through S and produces
in it an inverse current ; the stopping of the primary cur-
rent withdraws lines and produces a direct current through
the secondary. A larger deflection of the galvanometer
will be produced by the first closing of the primary cir-
cuit than by opening it, or by closing it a second time
unless the current be reversed. The reason is that the
ring forms a closed magnetic circuit, and its retentivity or
remanence is so great that only a small part of the lines
of force drop out when the magnetizing current ceases to
flow. But if the current through the primary be reversed,
all the lines will be taken out and will be put in again the
other way round. Hence, a large induction will take
place in S. A closed magnetic circuit is not well adapted,
therefore, to produce induction effects by merely opening
and closing the primary circuit.

378 ELECTRICITY AND MAGNETISM.

The relation between P and S is a mutual one. If S is
made the primary, induced electromotive forces will be
generated in P as the secondary. The Faraday ring with
its two coils of wire is the type of the modern transformer
for alternating currents.

333. An Inductive System a Conservative System.
— It will be instructive to look at a system of two circuits,
or of a circuit and a magnet, as a conservative system.
The action between the parts of the system always tends
to maintain unchanged the number of lines of force
threading through the circuits. Thus in Fig. 188 the
approach of the magnet to the coil increases the mag-
netic flux through it, and the induced current is in a
direction to send a counter flux through the coil so as to
keep the magnetic induction through it constant. In Fig.
190 the primary current produces a magnetic flux in the
ring, and the current induced in the secondary produces
a magnetic flux in the other direction round the ring ; that
is, the induced current opposes the change in the flux.

After the magnetizing current has produced a steady
magnetic flux through the iron, the opening of the primary
induces a secondary current in the same direction round
the ring as the primary, and this tends to maintain the
flux of induction through the iron unchanged. The same
principle may be applied to two coils without iron. There
is no exception to the law that the induced currents are
always in a direction to conserve the magnetic flux through
the circuit in which the induction takes place. This law
means that the magnetic flux through a circuit does not
change abruptly — a property of the magnetic field anal-
ogous to inertia in matter.

ELECTROMAGNETIC INDUCTION. 379

334. Lenz's Law (J. J. T., 432; Max., II., 177).—
When induced currents are produced by the motion of a
conductor in a magnetic field, the circuit is acted on by
a mechanical force. Lenz's law is that the direction of
this force always tends to stop the motion which gives rise
to it. Lenz's law is a particular case of the property of
conservation described in the last article. Every action
on an electromagnetic system, which involves a transfor-
mation of energy, sets up reactions tending to preserve
unchanged the state of the system.

Let E be the E.M.F. generated, I the induced current,
X the mechanical force parallel to the axis of x, and u the
velocity of the circuit in the direction x; then the work
done on the circuit is Xu, and this is represented by the
electrical activity, or the product of the current and the
E.M.F. ; hence

Xu^EL

An example of Lenz's law is afforded by a coil revolving
in a magnetic field. The mechanical action of the field on
the current induced in the coil produces a couple tending
to stop the rotation. The oscillations of the coil of a
d'Arsonval galvanometer (292) subside quickly when the
coil is short-circuited. The galvanometer is then a mag-
neto-electric machine, and the currents induced in the
closed coil bring it to rest.

335. Arago's Rotations. — When a magnet is suspended
horizontally over a copper disk and the disk is rotated,
induced currents are produced in it. These give rise to
a force opposing the rotation. Since the force between
the disk and the magnet is a mutual one, a couple acts on
the magnet and turns it, if it is free to move, in the same

380 ELECTRICITY AND MAGNETISM.

direction as the disk. Or if the magnet is spun round a
vertical axis and the disk is movable, it is dragged after
the magnet. These motions are called Arago's rotations;
they were discovered by Arago, but
were first explained by Faraday. In-
duced currents flow in closed circuits
through the disk, and the action be-
tween them and the magnet tends to
stop the disk ; or if the magnet oscil-
lates, the induced currents damp its
motions. Thus in Fig. 191, if the
needle ab oscillates over the disk as it
moves in the direction of the arrows, a current is induced
on the M side which repels the needle, and one on the N
side attracting it ; or the current under it flows from the
centre toward the circumference if a is a N-seeking pole.

336. Other Examples of Lenz's Law. — Let a
copper cube or cylinder be suspended between the
pointed poles of a powerful electromagnet (Fig.
192). The cube may be set rotating by twisting
the thread and releasing it. When the electro-
magnet is excited the cube is instantly brought
to rest; it begins to spin again as soon as the
current is cut off, and is again arrested on closing
the circuit. This resistance to motion in a mag-
netic field is sometimes called magnetic friction.

In another experiment a disk of copper is made
to rotate rapidly between the poles of an electro- f
magnet (Fig. 193). When the magnet is excited
the disk appears to meet with a sudden resistance. Fou-
cault found that if it is forced to rotate it is heated by
the induced currents flowing in it. These induced currents

ELECT RC MAGNETIC INDUCTION.

381

Fig. 193.

in masses of metal are often called Foucault currents.
There is a pair of eddy currents in the part of the disk
passing the poles; and these currents, as in Arago's rota-
tions, hold the disk back.

The drag due to eddy currents is
proportional to the speed and to the
square of the magnetic field ; for the
force is proportional to the field and
the current, and the current is pro-
portional to the field and the speed.
When the field is constant the force
is therefore proportional to the speed
of rotation.

The principle is employed to produce damping in rota-
tory meters. A copper disk, attached to the shaft to which
is connected a dial train, rotates between the poles of fixed
magnets. The drag on the copper disk keeps the speed
proportional to the torque.

337. Coefficient of Mutual Induction. — The preced-
ing examples of induction by currents all belong to the
class of mutual induction between two circuits. If we cal-
culate the E.M.F. generated by mutual induction, we shall
find that it contains a factor depending on the relative
position of the two circuits, the number of turns of wire
in each, and the reluctance of their common magnetic
circuit.

For definiteness take the case of Faraday's ring (Fig.
190). Let Nx and N3 be the number of turns of wire on
P and S respectively. By Art. 322 the magnetic flux
through the helix P is

4>=47riVy/10£&,

where I is the current in amperes and <9& is the reluctance

382 ELECTRICITY AND MAGNETISM.

of the iron ring (324). When the current is passed
through the coil P, if all the magnetic lines run through
S, the total number of lines cut by the iV2 turns in the
secondary will be

The quantity 4arNiNfi 'cfb, due to the passage of one C.G.S.
electromagnetic unit of current through the primary coil
.P, is called the coefficient of mutual induction M. Usually
the magnetic flux through the secondary is somewhat less
than that through the primary. The coefficient M then
denotes the number of lines of force common to the two
coils due to one C.G.S. unit of current through the pri-
mary, multiplied by the number of turns of wire in the
secondary. The practical unit of mutual inductance is
the henry. It is equal to 109 C.G.S. units as calculated
above. Applied to mutual induction, it is the induction
in the secondary when the E.M.F. induced is one volt,
while the inducing current in the primary varies at the
rate of one ampere per second.

Let dl/dt be the rate at which the current varies in the
primary. Then

E=-M-dI/dt.

E will be in volts if M is in henrys, I in amperes, and t in
seconds.

338. Self-induction. — Joseph Henry discovered that
a current through a helix with parallel spirals of wire acts
inductively on its own circuit, producing what he called
the extra current. No spark is produced when such a
circuit is closed, but a bright spark breaks across the gap
when the circuit is opened. The effects are not very
marked unless the helix contains an iron core.

Even a single circuit is a conservative system as regards

ELECTROMAGNETIC INDUCTION. 383

the magnetic flux through it. When the current magnet-
izes the core, the effect is the same as if a magnet had
been plunged into the helix ; that is, the induced E.M.F.
is a counter E.M.F. tending to prevent the flux of mag-
netic induction through the circuit. The result is that
the current in such a circuit does not reach its maximum
value abruptly, but only after a short interval depending
on the value of the coefficient of self-induction, or simply
the inductance. When the circuit is opened the induced
E.M.F. is direct and tends to prolong the current, or to
resist the diminution in the magnetic flux.
Let there be JV turns of wire on the coil. Then

3> = 47riVrZ710£&.

The total cutting by the N spirals, if all the lines pass
through them, is

This expression divided by the interval required for the
change of flux to take place is the E.M.F. of self-induc-
tion. The lines cut when 10 amperes (one C.G.S. unit)
pass through the coil is the value of the inductance L, or

The value of the induced E.M.F. depends on the rate
of change of magnetic flux ; and since the self-induction
prevents the current from reaching its steady value at
once, during this variable state the rate of increase is not
uniform ; it is better therefore to express the inductance
differently. If di/dt is the rate at which the current
changes value, where i is its instantaneous value, then the
induced E.M.F. is

e = —L ' di/dt.

The unit of inductance, the henry, is the inductance in a

384 ELECTRICITY AND MAGNETISM.

circuit when the E.M.F. induced in this circuit is one volt,
while the inducing current varies at the rate of one ampere
per second.

339. Growth of Current in Inductive Circuits. —
When a constant E.M.F. is impressed on a circuit possess-
ing self-induction, the current does not attain its permanent
value instantly. During the variable stage its value is not
given by the simple application of Ohm's law ; the induc-
tance is another quality of the circuit, besides its resistance,
which determines the instantaneous value of the current.
This inductance is a property of a circuit in virtue of
which the passage of a current through it is accompanied
by the absorption of energy in the form of a magnetic
field. If no other work is done, part of the energy flow-
ing out from the source is converted into heat, and the
rest is stored in the ether as the potential energy of the
magnetic field. This storage of energy goes on while
the current is rising from nothing to its steady value.
The energy so stored is equal to \LI\ where I is the
steady value of the current given by Ohm's law.1 The
work represented by this energy is done by the current
against the E.M.F. of self-induction. A circuit has large
self-induction, therefore, when a relatively large quantity
of energy is stored in its field while the current is rising to
its final value.

The student should note that the inductance X is a con-

1The induced E.M.F. is L - , and the work done in the element of time dt is
dt

L —idt, or Lidi. If this expression is integrated between the limits 0 and 7, the
dt

whole work done, or the energy stored in the magnetic field when the current

reaches its greatest value /, is

fLidi = \LV.

ELECTROMAGNETIC INDUCTION. 385

stant for any given form of circuit only when this circuit
consists of non-magnetic material and is surrounded by a
non-magnetic medium. If it contains iron, then L changes
with the value of the magnetic flux, because the reluctance
is dependent on the permeability, and the permeability
changes with the degree of magnetization of the iron.

340. Helmholtz's Equation. — The equation first given
by von Helmholtz expresses the value of the current in
an inductive circuit at any time t after a constant E.M.F.
has been applied to it. If E is the impressed E.M.F., E
the resistance of the circuit, t the value of the current at
any time t after closing the circuit, then the effective
E.M.F. required to produce the current i is Ri, and we
have the equation

E=Ri + L^.

The impressed E.M.F. is equal to the sum of the induced
and effective electromotive forces. The solution 1 of this
equation, if L is constant, is

i=T{l-e-Rt/Ly

i Divide the equation of electromotive forces by R and
E_.,Ldi
-R—l + Rdt'

or I-i = T*i (where T= L/R).

Whence S--A-

T I-i

Integrating, = log (/ — »') + constant (= — log /, for when t is zero i is

zero, and log / + constant = 0) .

Hence -1 — log (I-i) -log /= log 1=1,

and l=i = ra,L,

386

ELECTRICITY AND MAGNETISM.

After t seconds, therefore, the current falls short of its
maximum value by a quantity Ie . The quotient of

the inductance by the resistance L/R'm called the "time-
constant " of the circuit. It is the time required for the
current to reach 0.632 of its final value ; for when T
(or L/R) equals £, Rt/L becomes unity. Then

■Rt/L i . - 1 € — 1

1-i

= 1-,

Substitute for e its value 2.7183 and the expression equals
0.632. If, for example, L were 2 henrys and R 1 ohm,
the time-constant would be two seconds ; or in two seconds
the current would rise to 0.632 of its final value. This
retardation in the growth of the current is due to the fact
that it has to create magnetic fields. Energy is stored up
in these fields, and the resistance to the work done on
them is manifested as an opposing electromotive force.
As this opposition dies away, the ef-
fective electromotive force increases,
and the current rises to the value
given by Ohm's law.

341. Energy stored in a Mag-
netic Field. — Let M (Fig. 194) be
a large electromagnet, B a storage
battery, L an incandescent lamp of a
normal voltage equal to that of the
battery, and K a circuit-breaker.
The circuit is divided between the
electromagnet and the lamp; and
since the former is of very low resistance in comparison
with the latter, when the current reaches its steady state
most of it will go through the coils of the magnet. The
lamp is non-inductive; on closing the circuit, the self-

Fig. 194.

ELECTROMAGNETIC INDUCTION. 387

induction of the electromagnet acts against the current,
like a large resistance, and sends most of it round through
the lamp. It accordingly lights up at first, hut quickly
becomes dim, as the current grows to its steady value
through M.

On breaking the circuit and cutting off the battery
entirely, the lamp again flashes up brightly. The lamp
and the electromagnet are then together on a closed
circuit. The energy stored in the magnetic field, as a
strain in the ether about it, is converted into electric
energy, and a reverse current through the lamp lights it
for an instant. This example illustrates not only self-
induction, but the storage of energy in the ether about an
electromagnet.

i'TOTOWs

Fig. 195.

While the iron core in a helix greatly increases the self-
induction, it would be a mistake to assume that the induc-
tion may not be very appreciable without it. A steady
direct current was sent through an electrodynamo meter E
and a coil of wire AB without an iron core (Fig. 195).
The current was such that the potential difference between
the terminals AB was 27 volts. The direct current was
then replaced by an alternating current of the same mean
square value as indicated by the electrodynamometer (301).
The energy expended on the coil AB was then the same as
before, since none was absorbed by iron as heat by reason of
hysteresis. The energy lost was all converted into heat, and
was equal to PR or 27 / watts. But with an alternating
current the potential difference between the terminals of

388 ELECTRICITY AND MAGNETISM.

AB was 100 volts. A pressure of 100 volts was necessary
to force the same current through a coil that required
only 27 volts for a direct current. The difference must
be ascribed to the self-induction of the coil.

342. The Induction Coil. — An induction coil is com-
monly employed to obtain transient electric flashes of high
E.M.F. in rapid succession. In modern terms it is a step-
up transformer, with open magnetic circuit. About an
iron core, consisting of a bundle of fine iron wires to avoid
the production of induced currents through the mass of
metal in the core, is wound a primary coil of comparatively
few turns of stout wire ; outside of this, and as carefully
insulated from it as possible, is the secondary composed of a
very large number of turns of fine wire. In Spottiswood's
great coil, which gave a 42^-inch spark, the secondary con-
tained 280 miles of wire wound in 340,000 turns.

The primary must be provided with a circuit-breaker if
the coil is to be used with direct currents. It is commonly
made automatic by a vibrating device actuated by the core
and similar to that of a vibrating electric bell.

In large coils the secondary is wound in flat spirals, and
these are slipped on over the primary and separated from
one another by insulating rings. The difference of poten-
tial between adjacent turns of wire is then not so large as
when the entire coil is wound in layers from end to end,
and it is easier to maintain the insulation. The ratio of
the transformation of the electromotive force is nearly the
same as the ratio between the number of turns of wire on
the primary and secondary.

343. Action of the Coil. — While the quantity of elec-
tricity flowing through the closed secondary coil is the

ELECTROMAGNETIC INDUCTION.

389

same at " make " and " break " of the primary, still the
E.M.F. induced in breaking the circuit is so much higher
than in making it that the inductive effects are chiefly
those belonging to the former. This result is brought
about largely by the condenser.

On closing the primary circuit the counter E.M.F. due to
self-induction reduces the time rate of change of the cur-
rent on which the E.M.F. induced in the secondary
depends; but on breaking the primary circuit the self-

Fig. 196.

induction of the primary generates a direct E.M.F. which
tends to prolong the current and prevent its abrupt fall to
zero by sparking across the break. The condenser is added
for the purpose of suppressing this spark and aiding in
the abrupt descent of the primary current to zero. Fig.
196 represents the essential parts of an induction coil ;
PP is the primary and SS the secondary. The circuit is
automatically opened at the point b by the attraction of
the core on the small mass of soft iron F, which is mounted
on a spring.

The condenser C is joined to the points h and m on
opposite sides of the break. When the primary circuit

390 ELECTRICITY AND MAGNETISM.

is opened at 5, the extra current flows into the condenser ;
but as there is a complete discharge circuit for the con-
denser back through the primary in the opposite direction
to the battery current, the condenser discharge thus aids
in demagnetizing the core, or in rapidly reducing the mag-
netic flux by actually producing a negative one.

Lord Rayleigh has shown1 that the best results are
secured when the capacity of the condenser is just great
enough to absorb a charge at a rate equal to the full deliv-
ery of the primary current during the time the break-
points are separating beyond the residual sparking-distance.
The condenser then causes an electric recoil in the current
and returns the charge stored up as an equal current in the
reverse direction through the primary, thus doubling the
change in the flux and doubling the induced E.M.F. and
current in the secondary ; for the removal of all the lines
of force in one direction and the insertion of them in the
reverse direction amounts to a double diminution of them.
The conditions are then those described by the word
resonance. The current through the primary is rendered
oscillatory by means of the condenser. Instead of absorb-
ing the energy represented by the spark when no con-
denser is used, the condenser stores the energy momentarily
and then returns it to the primary, and by mutual induc-
tion to the secondary, to be expended there as a longer
spark or a greater current.

344. Discharges in Partial Vacua. — Many remark-
able luminous effects, which are but imperfectly under-
stood, are produced when the discharges from an induction
coil pass through residual gases under a low pressure in
glass vessels. Such discharges will pass through the air

i Phil. Mag., 1870, p. 428; Fleming's Alter. Current Trans., Vol. I., p. 383.

ELECTROMAGNETIC INDUCTION.

391

left in receivers exhausted by a good mechanical air-pump,
but the best results are obtained with tubes exhausted by
a mercurial air-pump to a pressure of about 2 mms. of mer-
cury and permanently sealed. Platinum electrodes are
melted into the glass at the two ends. The celebrity of
the tubes made by Geissler gave to them the name " Geiss-
ler tubes." Some of the patterns are shown in Fig. 197.

The luminous effects are more intense in the narrow
connecting tubes than in the larger bulbs. The cathode
exhibits a bluish or violet
glow, while the light at the
anode is of smaller extent,
but brighter. The colors
given by a gas depend on
its nature. The narrow por-
tions of a tube containing
hydrogen glow with a bril-
liant crimson. Vapor of
water gives the same color, indicating the dissociation
of the vapor by the discharge. Tubes containing carbon
dioxide emit a pale gray light, but show splendid stratifica-
tions. The glow when examined by the spectroscope gives
the lines characteristic of the gas in the bulb.

Fluorescent materials in Geissler tubes are beautifully
luminous. Uranium glass, and solutions of quinine, ses-
culin, and naphthaline-red in tubes surrounding the ex-
hausted one, are among the best examples of fluorescent
bodies. Kerosene oil also shows marked fluorescence.
• The stria? or stratifications of the tube consist of portions
of greater luminosity separated by darker intervals. They
originate apparently at the positive and become more
numerous up to a definite point of the exhaustion, after
which they broaden out and diminish in number. J. J.

Fig. 197.

392 ELECTRICITY AND MAGNETISM.

Thomson has produced strise throughout a tube 50 feet
long, except near the cathode. They present a peculiar
flickering unstable motion, similar to that sometimes
observed during auroral displays. The striae are hotter
than the darker spaces between them.

345. Discharges in High Vacua. — When the ex-
haustion of a bulb is carried to a millionth of an atmos-
phere, the phenomena of the electric discharge entirely
change character. Such tubes can scarcely be said to con-
duct at all, apparently because of some difficulty which
the discharge encounters at the electrodes, for J. J. Thom-
son has shown that high vacua are good conductors.1
These tubes have been investigated by Crookes with great
skill, and they are therefore called " Crookes tubes."

When the exhaustion has been
carried to a millionth of an atmos-
phere, the mean free path of the
molecules is increased a million
fold and becomes comparable with
the dimensions of the containing:

Fig. 198. tit •

vessel, lhe disorderly motions
of the molecules of the residual gas may then be directed
by electrical or thermal means along definite paths.
The characteristic light of a Geissler tube then almost
entirely disappears by the broadening of the dark space
about the cathode till it reaches the opposite wall
of the bulb. The residual electrified gas is projected
entirely across the bulb in radiant streams, and the bom-
barded walls of the tube exhibit remarkable phosphorescent
effects, the color depending on the kind of glass and on

lElectridan, June 7, 1895.

ELECTROMAGNETIC INDUCTION.

393

the substances, such as diamond, ruby, or various sul-
phides, subjected to this molecular cannonade (Fig 198).

Evidence is abundant that the projected molecules of
the residual gas move in straight lines, except as they are
deflected by a magnet or by mutual repulsion. The dis-
charge in a Geissler tube is acted on by a magnet like a

Fig. 199.

flexible conductor conveying a current ; but the stream of
radiant matter, as Crookes calls it, when once deflected
by a magnet does not recover its former direction of
motion after passing the magnet (Fig. 199).

Fig. 200.

Any obstructions placed in the path of these " cathode
rays " appears to stop them and casts a shadow by protect-
ing the wall of the tube behind it from the bombardment
(Fig. 200). If such obstruction consists of delicately

394 ELECTRICITY AND MAGNETISM.

poised vanes, they are set moving by this molecular wind.

If the cathode is made concave, the paths of the molecules
cross at the focus, and glass or even plati-
num may be fused at this point (Fig.
201).

Two parallel streams of such flying
molecules are deflected by a magnet, but
repel each other like charges of the same
sign. Hence their velocity is probably
less than that of light, for at this speed
they would act like two currents; but
their electrostatic repulsion is not offset
fc by their electrodynamic attraction as par-
allel currents.

Fig. 201.

346. Cathode Rays. — The projection
of electrified molecules of the residual gas from the cathode
plate of a Crookes tube is not the only action going on at
that electrode. Hertz discovered that the emanations or
" rays " from the cathode are not transmitted through mica,
glass, or other transparent substances, but that they do pass
through metallic foil. By means of vacuum tubes with
a small window of aluminium foil at one end, Lenard
demonstrated that the " rays " from the cathode pass
through aluminium into the air, where they retain the
remarkable property of exciting phosphorescence. Ap-
parently these rays can be produced only in a good
vacuum ; but when they have passed through a medium
pervious to them into the air they retain their character-
istic properties. Professor Rcintgen, of Wiirzburg, has just
discovered that these cathode rays, or some unknown radia-
tions from the phosphorescent glass, pass through opaque
bodies like wood, paper, hard rubber, aluminium, etc., and

ELECTROMAGNETIC INDUCTION. . 395

that they affect a sensitized photographic plate. In this
way it has been found possible to photograph objects
entirely concealed from view, such as the bones of the
living hand "or coins in a leather purse. These pictures
are silhouettes or shadows. The unknown rays producing
this effect seem not to be refrangible, and as far as now
known are not reflected. Rontgen says that they originate
at the part of the tube which exhibits bright phosphores-
cence ; if so, they are not the cathode rays of Lenard,
from which they are differentiated in several ways. The
Lenard rays are deflected by a magnet, while the Rontgen
rays are not; the former are quickly quenched in air at
atmospheric pressure, while the latter can be detected at a
distance of two metres from the source ; the former do not
pass through glass, while the latter do. Aluminium is
permeable to both. Rontgen has shown that his unknown
rays will pass through 200 times as thick a sheet of alumin-
ium as of platinum.

It has long been suspected that there are longitudinal as
well as transverse vibrations in the ether ; some physicists
have contended that they must exist. Rontgen is inclined
to ascribe the remarkable phenomenon that he has discov-
ered to such longitudinal disturbances in the ether.

347. The Telephone. — The transmitter and the re-
ceiver for the electric transmission of speech may be
identical instruments, but in practice they are usually
different. The transmission is commonly effected by
having a rounded platinum pin pressed lightly by a deli-
cate spring against a polished carbon surface and mounted
in contact with an elastic diaphragm. This platinum-
carbon contact forms part of a local electric circuit. The
contact resistance is varied by the vibrations of the dia-

396

ELECTRICITY AND MAGNETISM.

phragm, so that the strength of the current is modified in
accordance with the aereal movements constituting sound
in the neighborhood of the mouthpiece of the instrument.
The current, thus moulded by the voice, passes through
the primary of a small induction coil, while the secondary
pulses are sent to the transmitting line.

The receiver (Fig. 202) consists of a thin iron dia-
phragm D held in close proximity to the pole of a small
electromagnet BB, which in turn is mounted on the end
of a permanent magnet M. The electric pulses coming

through the line actu-
ate the electromagnet
and so vary the mag-
netic field at the pole.
When the current
runs in one direction
the attraction be-
tween the magnet and
the disk is increased; when it flows in the other direction
it is diminished. The disk is thus forced to repeat the
vibrations of the diaphragm in the transmitter, and it
throws the air in contact with it into similar vibrations
and reproduces the sounds.

The receiver may also be used as a transmitter. The
to-and-fro motion of the iron disk, in conforming to the
sound-waves impinging on it, varies the magnetic induc-
tion between it and the pole. A movement of the lines of
force in the field near the end of the magnet is thus
brought about ; and this variation in the magnetic flux
through the coil produces induced currents in it, which
are transmitted to the distant station, where they actuate
the receiver in the manner described.

Fig. 202.

DYNAMOS AND MOTORS.

397

CHAPTER XXV.

DYNAMOS AND MOTORS.

348. Ideal Simple Dynamo. — A dynamo is a machine
for converting the energy of mechanical motion into the
energy of an electric current. It is a generator of electro-
motive force, and is based on the principles of electromag-
netic induction discovered by Faraday. It consists of a
system of conductors, called an armature, revolving in

Fig. 203.

a magnetic field in such a way as to vary continuously
the magnetic flux through them.

Suppose a single loop of wire to revolve in a uniform
magnetic field between the poles NS of a magnet (Fig.
203) around a horizontal axis in the direction of the
arrow. The loop of wire in the position in the figure
encloses the maximum magnetic flux. When it has re-
volved through an angle 6 the flux through it will be
reduced to 3> cos 0, where <P is the maximum ; for the pro-

398 ELECTRICITY AND MAGNETISM.

jection of the loop on the plane perpendicular to the field
varies as the cosine of the angle of displacement from that
plane. After a quarter turn the loop does not enclose
any lines of force; as it revolves further they thread
through in the opposite direction, and this is equivalent
to a continued diminution of the magnetic flux through
the loop. During the second half-revolution the opposite
changes take place ; when the loop has revolved through
360° it returns to its initial relation to the magnetic field.
The magnetic flux through the loop varies therefore as
the cosine of the angle of displacement 6. During the
first half-revolution a direct current flows around the loop
in the direction of the arrows ; during the second half it is
reversed. The E.M.F. therefore changes sign twice every
revolution. Such a loop, or a coil composed of a number
of parallel turns, generates an alternating electromotive
force.

349. Law of the Electromotive Force. — The induced
electromotive force is not equal to the total magnetic flux
through the circuit, but to the rate of change of that flux.
Now the total flux varies as the cosine of the angle de-
fining the position of the loop ; and when the flux is a
maximum, its rate of change is a minimum and conversely.
Hence when 6 is zero or 180° the E.M.F. generated is
zero ; while for the positions 90° and 270° the E.M.F. is a
maximum. The trigonometrical function that is related
in this way to the cosine is the sine.1 Hence the law of
the variation of the electromotive force, generated by the
revolution of the loop in a uniform magnetic field, is
the same as the variation in the value of the sine of the
angle of position. If, therefore, we plot uniform distances

1 The differential of the cosine is minus the sine.

DYNAMOS AND MOTORS.

399

along a straight line to represent equal increments of 0,
and erect perpendiculars to denote the values of the cor-
responding sines of 0, the curve connecting the extremities
of the ordinates will be a sine curve. In Fig. 204 the
heavy line / is the cosine curve, representing the changes

Fig. 204.

in the magnetic flux ; the light line II is the sine curve,
whose ordinates denote the rate of change of the flux, or
the induced E.M.F. Their maximum values differ by
90°, or a quarter of a period. When the magnetic flux
decreases through its zero value at B, its rate of change is
greatest and there the E.M.F. is a maximum.

The current in such a loop is an alter-
nating one, having alternately numeri-
cally equal values in opposite directions
through the loop. To make it uni-
directional in the external circuit a two-
part commutator must be used (Fig.
205). The two parts of the split tube,
insulated from each other and mounted
on the shaft, are connected with the two terminals of the
rotating coil. The brushes leading to the external circuit
are so placed that they exchange contacts with the two
commutator segments in passing through the positions
where the current changes its direction through the coil.
The pulses are then all in one direction in the external

Fig. 205

400 ELECTRICITY AND MAGNETISM.

circuit. Alternate loops of the sine curve are thus re-
versed, so that all of them lie on the same side of the
zero line.

350. The Drum Armature. — The modern Drum
Armature for direct currents (Fig. 206) is an evolution
from the shuttle armature with a single coil. If a second
coil be wound around an iron core with its plane at right
angles to the first, it will generate electromotive forces
differing in phase from the first by a quarter of a period.

Fig. 206.

When the two are rectified i» the external circuit they
combine to give a fluctuating current of twice the fre-
quency of either, superposed on a current of constant
value, so that the resulting current never drops to zero.
By increasing the number of sections of the coils wound
at equal angular distances around the outside of the arma-
ture core, the E.M.F. and current are rendered nearly
constant in the external circuit. The sections are all
joined in series and the junctions between them are con-
nected to the commutator bars, which are insulated from
one another. When the brushes bear on opposite bars, it
will be readily seen that the current has two paths through
the armature ; so that one brush is constantly positive and
the other negative. By this arrangement the potential
difference between the brushes is kept up to the highest
value given by half the coils in series. The brushes must

DYNAMOS AND MOTORS.

401

be placed near that part of the field where the E.M.F. in
any coil passes through zero and reverses.

351. The Field-Magnet. — The magnetic field in
direct current machines is produced by a. large electro-
magnet excited by a cur-
rent from the armature.
The residual magnetism of
the cores is sufficient to
start the induced current ;
and when the entire cur-
rent is carried around the
coils of the field-magnet,
the dynamo is connected in
series. The circuit of a
series dynamo is shown in
Fig. 207. Such a machine
is adapted to furnish con-
stant currents only; it is
employed in arc-lighting.

The field may also be ex-
cited by a shunt winding,
consisting of many turns of wire connected as a shunt to the
external circuit. In Fig. 208 this shunt circuit is shown
connected to the brushes. It is employed on circuits
requiring a constant potential difference between the main
conductors. When the current changes as a result of a
change in the external resistance, the excitation of the
field-magnet remains nearly the same and the E.M.F.
generated is therefore nearly constant.

A compound-wound dynamo consists of a combination of
shunt and series coils on the field-magnet. It is designed
to maintain the potential difference more nearly constant

\ MAIN CIRCUIT_

Fig. 207.

402

ELECTRICITY AND MAGNETISM.

than is possible with a simple shunt machine. When a
current flows through the armature, there is in conse-
quence of its resistance a loss of potential difference be-
tween the brushes. This
loss occasions a further
loss of voltage by reduc-
ing the exciting current
through the field-magnet.
Hence by carrying the
whole current around the
field-magnet in a series
coil of a few turns, the
increased excitation thus
produced makes up for
the loss of potential in
the armature and main-
tains a constant potential
difference between the
brushes. If the armature
were without resistance,
compounding would not be necessary to keep the potential
difference constant at the brushes, except for the demag-
netizing effect of the armature considered as an electro-
magnet.

An over-compounded machine has enough series turns to
more than compensate for the loss of potential when a
larger current flows through the armature. Hence the
potential difference between the brushes will increase with
an increase of load. The object is to compensate for a
further loss of potential in the mains, so as to maintain
the potential difference constant at some distant centre of
distribution.

Fig. 208.

DYNAMOS AND MOTOBS. 403

352. The Gramme Ring. — The Gramme ring is a
different type of armature. It is a laminated iron ring
wound continuously with a large number of turns of wire,
all coiled in one direction and joined in series. Fig. 209
shows diagrammatically the relation of the several parts
of the machine. The eight coils are wound right-handedly,
and each junction between coils is joined to a commutator
bar. In this figure the upper brush is the positive, and the
current flows from it around the external circuit back to
the lower brush.
When a coil is in \

the highest posi-
tion in the fig-
ure, the maxi-
mum flux passes
through it ; as
the ring rotates
the flux through
the coil de-
creases, and after a quarter revolution all the lines are
taken out ; they then begin to thread through the other
way. The current through each coil reverses twice dur-
ing each revolution, and there are two circuits through
the armature, exactly as in the drum type. In both cases
the iron, which is used to increase the magnetic flux
through the armature coils, must be laminated by planes
at right angles to the axis of rotation, for the purpose of
preventing induction currents in the iron. These cur-
rents would heat it and waste energy.

353. Reactions in the Field of a Dynamo and a
Motor. — An electric motor for direct currents is con-
structed in the same manner as a generator. The study

Fig. 209.

404

ELECTRICITY AND MAGNETISM.

of a magnetic field through which a current is passing
throws much light on the interactions between the field
and the armature. Fig. 210 is the field between unlike
poles distorted by a current through the loop of wire which
came up through one hole and went down through the
other. The lines of force are so distorted that some of
them thread through the loop. Now if we conceive this
loop to rotate counter-clockwise around an axis perpen-
dicular to the plane of the paper, then it is clear that

Fig. 210.

mechanical force must be applied to keep up the motion,
because the tension along the lines of force drags the loop
back. The armature therefore turns against the magnetic
forces or torque of the field acting on it. When used as
a generator, the field of the machine is distorted in the
direction of the rotation.

If, on the contrary, we conceive this loop of wire to
rotate as an armature under the action of the magnetic
stress on it, then the relative density of the lines in differ-
ent parts of the field remains the same and the armature
reverses its motion.

When the machine is used as a generator mechanical

DYNAMOS AND MOTORS. 405

power is converted into electrical energy, because the
rotation of the armature is kept up against the internal
magnetic actions in . the field. Work is then done on the
machine as a generator. When it is used as a motor
electrical energy is converted into mechanical energy,
because the rotation takes place in the direction of the
magnetic effort between the field and the armature. Work
is then done by the machine as a motor.

354. Direction of Rotation as a Motor. — A series
machine when used as a motor runs in the opposite direc-
tion to its motion as a generator. Its rotation will be in
the same direction whether the current goes through it
one way or the other, since it is reversed through the
armature when it is reversed through the field.

A shunt machine runs in the same direction as a motor
and as a generator. If in Fig. 208 the current from an
external source enters by the lower brush, as in the figure,
its direction through the armature remains unchanged ;
but it goes through the field coils in the opposite direction
to the arrows, and the armature and the field are now iji
parallel with reference to the external source ; when used
as a generator the external circuit and the field are in
parallel with respect to the armature as the source of
the electric pressure. The field is therefore reversed.
But as a motor the machine runs with the magnetic torque,
and as a generator against it; so that running with the
torque when the field only is reversed is the same as run-
ning against it before the field is reversed. It is clear
then that the shunt machine runs in the same direction
whether it is used as a motor or a generator.

The same is true of a compound-wound machine so long
as the ampere-turns of the shunt coil overbalance those of

406

ELECTRICITY AND MAGNETISM.

the series ; the two coils act against each other when the
machine runs as a motor.

355. Counter Electromotive Force in a Motor. —
The armature of an electric motor revolves in a magnetic
field and generates an E.M.F. A little consideration will
show that this E.M.F. must be an opposing one tending to
reduce the current through it. In Fig. 211 a generator
and a motor are connected together. The direction of

GENERATOR

Fig. 211.

rotation in the two machines is the same. The direction
of the electromotive forces generated in both armatures
is shown by the arrows. They are toward the lower
brush in both, because both armatures revolve in the
same direction in similar fields. But in the generator
the current runs in the same direction as the E.M.F.
generated in its armature, while in the motor it runs
against this generated E.M.F. Its own E.M.F. therefore
opposes the current.

If the motor is provided with a fly-wheel to keep up its
speed when the current from the generator is cut off, a
voltmeter placed across its terminals, as V in the figure,
will show only a slightly diminished E.M.F. immediately
after the circuit is broken, if there is no load on the motor

DYNAMOS AND MOTORS. 407

to produce a quick slackening of the speed. The volt-
meter shows no reversal of the current when the generator
is cut off. This fact shows that the positive brush of the
generator is connected to the positive of the motor, or that
the E.M.F. of the motor is a back E.M.F. The voltmeter
may be replaced by an incandescent lamp ; it will glow
nearly as brightly for a few seconds directly after the main
circuit is opened as before.

356. Work done by a Motor. — We have seen in
Articles 235 and 320 that the work done against an
opposing E.M.F. is measured electrically by the product
of this E.M.F. and the current. Now the total work done
on the motor is the product of the E.M.F. applied to its
terminals and the current, or EI. The difference between
the two, EI—E'I, is the energy converted into heat
(236). With an electrically perfect motor, therefore, the
work done by it is the difference between the whole energy
applied to it and the waste in heat, or the work done
against its counter E.M.F., E'l.

The two factors of the power, measured mechanically,
are the torque and the speed. The torque is the moment
of the couple producing the rotation; it is proportional
both to the strength of the field and the current in the
armature. If the field is kept constant, the torque is
proportional to the current, and the E.M.F. to the speed.
Hence we may write

IE' = AnT,
where T is the torque, n the number of rotations per
second, and A a constant.

When the motor is working under a fixed load, an
increase of the field increases the torque and therefore
decreases the speed n; weakening the field on the other

408 ELECTRICITY AM) MAGNETISM.

hand diminishes the torque and increases the speed. Both
these conclusions follow from the constancy of the product
nT under the assumed condition of a fixed load. In both
cases the speed changes till the counter E.M.F. acquires
the same value that it had before the change was made in
the field.

357. Electrical Efficiency of a Motor. — If IT and IP
denote the power expended on the motor and the power
given out by the motor respectively, then the electrical
efficiency, or conversion-factor, is

W= ^IE'^E'
W e IE E"
or the ratio of the counter E.M.F. to the applied E.M.F.
If the applied E.M.F. is a constant, the efficiency increases
with the counter E.M.F. Now the effective E.M.F. pro-
ducing the current is E— E', and the larger E' the smaller
is this difference and the smaller the current. When the
current is small work is done at a slow rate, but a larger
fraction of the power applied is spent in useful work. It
is necessary to point out that this relation assumes an
electrically perfect motor. Since a certain current is
needed to make the motor run at the required speed
without doing any useful work, the useful current is the
difference between the whole current and the current
required to run the motor up to speed without load. It is
therefore evident that a practical motor does not have its
highest commercial efficiency when working under the
smallest loads, for then a large fraction of the current does
not contribute to the useful work done.

The work done by a motor per second is
_ v,E-E>

E'I=E

DYNAMOS AND MOTORS. 409

Since R is constant the work done will be a maximum
when the product E{(E—E'~) is a maximum. Now the
sum of the two factors of this product is the applied
E.M.F., E; and when the sum of two factors is a con-
stant their product is greatest when they are equal to
each other. The condition for maximum activity is then

E' = E-E\ otE' = %E.
A motor does work at the greatest rate when the current
is reduced by the counter electromotive force to half the
value it would have if the motor were standing still. The
efficiency is then only 50 per cent.

358. Efficiency of Transmission. — When power is
transmitted to a distance electrically, high efficiency re-
quires high electromotive force. This is equally true
whether the energy is used for lighting or for power. The
energy lost in the line as heat is PR watts, where R is
the resistance of the line. To keep this waste small while
the power transmitted is increased, the voltage must be
raised. The current depends on the difference between
the applied and the counter electromotive forces E — E',
while the power put into the circuit is IE watts and
the power given out by the motor IE' watts. If the
difference E—E' is kept constant, the current and the
waste in heat will remain constant, while the power trans-
mitted will be proportional to the applied E.M.F. The
factor that determines the heat waste is controlled by
keeping the current small; while the other factor that
enters into the measure of the power transmitted, that is,
the electromotive force, is raised. The other way of re-
ducing the energy lost in the line is to reduce the resist-
ance ; but this method involves the use of a quantity of
copper the cost of which is prohibitive.

410 ELECTRICITY AND MAGNETISM.

359. Alternators. — The armatures already described
generate alternating electromotive forces that follow the
law of variation of a sine curve more or less closely. A
complete series of changes in the electromotive force or
current represented by this curve is called a period, and
the number of periods in a second is the frequency of
the alternations. In two-pole machines the frequency
is the same as the number of revolutions per second.

When the alternating cur-
rent is utilized in the exter-
nal circuit, the frequency
is restricted to a lower
limit of about 25 and a

2_D higher one of about 150.

Fig. 212.

B If the frequency is less than
25 per second the eye can
detect the variations in the
brightness of an incandes-
cent lamp ; while for fre-
quencies much above 130
or 140 the effects of self-induction are greatly exaggerated.
Within the above limits multipolar machines must be used
to avoid excessive speed of revolution. The frequency n
is then the speed of rotation multiplied by the number of
pairs of poles.

The circuit through the armature of an alternator is of
the simplest kind. The field is separately excited so that
the polarity of the poles remains fixed. It will readily be
seen that the successive armature coils must be so con-
nected that the circuit reverses in direction around the
coils from one to the next (Fig. 212). For high voltage
they are all joined in series. A complete period is the
time required for a coil to pass from one pole to the next
one of the same sign.

DYNAMOS AND MOTORS. 411

360. Lag of Current behind the Electromotive Force.
— When an alternating electromotive force is applied to a
circuit possessing inductance one of the novel and essential
facts is that the current reaches its maximum value later
than the electromotive force ; and, as a consequence, Ohm's
law is no longer adequate to give its value. The effect of
self-induction is not only to introduce an additional electro-
motive force, but to produce a lag of the current in phase
behind the electromotive force impressed on the circuit by
the generator.

Let an alternating current, following the simple har-
monic law, be represented by the heavy sine curve I of
Fig. 204. Then, since the induced electromotive force is
proportional to the rate of
change of the current when
there is no iron in or about
the circuit, the induced E.
M.F. curve may be repre-
sented by the light line II
This is also a sine curve,
since the differential coef- a
ficient of a sine function
is itself a sine function. But the latter curve reaches
its maximum value a quarter of a period later than the
former. When the current is a maximum at A its rate
of change is zero, and when it diminishes through its zero at
B its rate of change is a maximum. The induced electro-
motive force and the current are said to be in quadrature.

The effective electromotive force producing the cur-
rent by Ohm's law must correspond in phase with the
current itself. The maximum induced and effective elec-
tromotive forces may therefore be represented by the two
adjacent sides of a right triangle (Fig. 213), where be

Leal

412 ELECTRICITY AND MAGNETISM.

is the induced E.M.F. and ab the effective E.M.F. ; the
hypotenuse ac is therefore the maximum impressed E.M.F.
(I., 31). But the current agrees in phase with ab ; it
therefore lags behind the impressed electromotive force by
the angle <f>. In the absence of capacity in the circuit, this
angle becomes zero only when the inductance is zero.

The instantaneous values of the several electromotive
forces may be found by revolving the triangle around a as
a centre, and projecting the three sides upon some straight
line through a, as in Part I., Fig. 18.

361. Value of an Alternating Current. — The instan-
taneous value of an alternating current following the law

of sines is

i = I sin 6 = 1 sin cot,

where I is its maximum value and a> the angular velocity
2<7m (I., 33).

If the induced electromotive force is proportional to the
change-rate of the current (338), then

L • di/dt = Lml cos cot,
since the rate of change of the sine is the cosine. This is
the expression for the instantaneous value of the induced
electromotive force. Its maximum value is Lool, the
maximum value of the cosine of an angle being unity.
Therefore in the triangle of electromotive forces (Fig.
213), the side be equals Lwl. Also ab equals RI, because
it is the effective electromotive force, and by Ohm's law it
is the product of the resistance and the current. There-
fore ac equals I (i22 + ZV)^ ; but the hypotenuse is the
maximum impressed electromotive force. Then

E=I(&+ JGV)*,

and ^~/fc>„ , T, ,.■•

(ir + La)-)h

DYNAMOS AND MOTORS. 413

The expression (722 + iV)^ is called the impedance. The
impedance shows that the effect of inductance on the
value of the current is equivalent to additional resistance.
Also from the figure

tan (f> = -—- .

It is evident, therefore, that the angle of lag increases with
the coefficient of self-induction L and with the frequency
(a> = 2ttw). In these equations / and E denote the max-
imum current and impressed electromotive force. The
current lags as if the angle in the auxiliary circle of refer-
ence were <ot — <f> instead of cot. We may therefore write
for the instantaneous current

i = (ff+'LV)'Si°('"^)l
where the term (f> is added to show that the current lags
behind the electromotive force E.

The effect of capacity in series is to produce a lead
instead of a lag of the current, and the one offsets the
other when La> = l/Cco.1

362. Virtual Volts and Amperes. — All practical in-
struments for measuring alternating currents and pressures
take account of the " square root of the mean square "
values and not the arithmetical mean. Thus the electro-
dynamometer (301), the Kelvin balances (302), and the
electrostatic voltmeter (147) all integrate the forces oper-
ating them, and these are proportional to the squares of
the current and of the electric pressure. If the current
and the electromotive force follow the sine law, the mean
given by these instruments is 0.707 of the maximum

1 Carhart and Patterson's Electrical Measurements, p. 239.

414 ELECTRICITY AND MAGNETISM.

values. When a voltmeter on an alternating circuit reads
70.7, the voltage alternately rises to + 100 and sinks to
— 100 as positive and negative maxima. The values
given by these instruments are virtual volts and virtual
amperes.

The virtual values exceed the arithmetical mean values
by 10 per cent.1 A continuous current and an alternating
current of equal virtual value have the same heating
effect ; but a continuous current equal to the arithmetical
mean of the alternating one will have a smaller heating
effect in the ratio of 1 to 1.23 (or .637"2 to ,707s).

363. Choking Coils. — Consider a circuit with small
resistance and large inductance. The current will then
depend largely on the latter ; or, if H is negligible,

1= U/Lto.
This formula holds either for maximum or for virtual
values. Coils with a divided iron core, having small
resistance and large self-induction, are called choking coih.
Thus if n were 134, L 100 henrys, and E 1,000 volts, the
current through the coil of negligible resistance would be
only 0.012 ampere. A current of about this value flows
through the primary of a transformer on a thousand-volt
circuit when the secondary is open. It is approximately
independent of the resistance.

364. "Wattmeters. — The measurement of power in
circuits conveying alternating currents cannot be made
in the same way as when continuous currents are employed,

i The mean of the squares of the sines throughout a half-period is 1/2. The
square root of the mean square value is therefore l/<v/2 of the maximum, or
0.707. The mean value of the sines throughout a half-period, on the contrary, is
2/n-, or 0.637.

DYNAMOS AND MOTORS. 415

where the energy spent on any part of the circuit is
measured by finding the current through it and the poten-
tial difference between its extreme points ; for the potential
difference and the alternating current are not in step
unless the circuit is non-inductive. Thus in the example
of Art. 341, the energy expended on the coil with the
alternating current was apparently 100 I watts, while in
.reality it was only 27 / watts. When the electromotive
force and current differ in phase, one of them is sometimes
positive while the other is negative ; hence a part of their
instantaneous products are positive and part negative.
During that part of the period when this product is nega-
tive the circuit is restoring power to the source. The
integrated difference between the two products is the
work done.

Power on alternating circuits may be measured by a
wattmeter. If the movable coil of an electrodynamometer,
consisting of several turns of wire, be disconnected from
the field coil and be connected in series with sufficient non-
inductive resistance as a shunt to the circuit in which the
power is to be measured, while the fixed coil is connected
in series with this circuit, the indications of the instrument
will be proportional to the integrated sum of the instan-
taneous products of the electric pressure and the current.
When the instrument, which is then called a wattmeter,
has been properly calibrated, it measures the power ex-
pended in watts. It is of course equally applicable to
continuous currents.

365. Transformers (J. J. T., 405). — A transformer
is an induction coil with a primary of many turns, a second-
ary of a smaller number, and a closed magnetic circuit.
It is employed with alternating currents as a " step-down "

416 ELECTRICITY AND MAGNETISM.

instrument for the purpose of reducing the high electro-
motive force on the transmitting line to a low electromotive
force for lighting and power. It is entirely reversible and
can be used equally well for the " step-up " process with
alternating currents.

The primary and secondary coils are wound round an

iron core (Fig. 214), but are insulated from each other as

^-^ perfectly as possible. In

/\ /*^\ practical transformers the

K //:r^^---\N *lon encl°ses tfle vrifQ
/ "fs ^ y s rather than the reverse.

1/ Jj The iron serves as a path

p A \) ,'V A-—-— for the flux of magnetic

\~^V^-r \/ induction. The student

VSV^ 1^0*0 should notice that the re-

v — ' lation of the current and

Pig- 214. .

the flux is a reciprocal
one, so that they may always exchange places.' With
either relative arrangement of the iron and the coils, nearly
all the lines of induction produced by the primary pass
also through the secondary, and vice versa.

When the secondary is open the transformer acts simply
as a " choking coil ; " the current passing through the
primary is then only the very small one required to mag-
netize the iron for the generation of the counter E.M.F.,
which is then nearty equal to the impressed E.M.F. When
the secondary is closed the currents in the primary and
secondary are nearly in the inverse ratio of the turns of
wire on the two, or NJNx, where Nx denotes the turns on
the primary and N2 the number on the secondary. The
electromotive forces generated in them, when there is no
magnetic leakage, is directly as the ratio of transformation
NjN2 . The energy in the secondary circuit is therefore

DYNAMOS AND MOTORS.

417

nearly the same as that expended on the primary. The
small difference is chargeable to loss in the copper of the
primary and to losses in heating the iron on account of
hysteresis and Foacault currents.

The secondary current is nearly opposite in phase to the
primary, and causes a diminution in the apparent self-in-
duction of the primary coil, so that the larger the second-
ary current the larger the primary. The transformer is
therefore nearly self-governing. The power absorbed by
the primary increases as the resistance of the secondary
decreases; but it reaches a maximum for a particular
value of the secondary resistance, below which the energy
absorbed by the transformer decreases. This critical value
of the resistance is larger the higher the frequency.

Fig. 215.

366. Polyphase Currents. — It has long been known
that two or more alternating currents of the same frequency,
but differing in phase by any desired quantity, may be

418

ELECTRICITY AND MAGNETISM.

obtained from one generator. If, instead of a commutator,
four insulated rings on the shaft be connected to four
equidistant points of either a drum armature or a Gramme

ring, the currents in
a b a. the externally sepa-

)( rate circuits will differ
in phase by a quarter
of a period. In the
small laboratory ma-
chine of Fig. 215 the
exciting current flows
through the revolving
field-magnet by way
of the brushes bearing on the two rings. The armature
is a stationary ring wound continuously on a laminated
iron core, with four con-
ductors leading from
points 90° apart. Each
pair, 180° apart, compose
an alternating circuit.
It is obvious that one
current passes through
its maximum at the same
instant that the other
passes through its mini-
mum value (Fig. 216). In a similar way three-phase cur-
rents will pass through conductors 120° apart. If there
are but three conductors, each one serves as a return for
the other two, since the algebraic sum of either two cur-
rents is at any instant equal to the third (Fig. 217).

367. The Rotatory Field. — When an alternating cur-
rent passes through a coil of wire without iron it produces

6° .

1 6JO*

\ ,2<?y

180°

\2+o7

3<?0"

i 3607

r
i

Fig. 217.

DYNAMOS AND MOTORS.

419

an alternating magnetic field along its axis. If the current
follows the sine law, the magnetic flux will follow the sine
law also. Let two such coils be set with their axes at
right angles, and let the equal alter-
nating currents through them differ
in phase by a quarter of a period.
Two simple harmonic motions of
equal amplitude, at right angles,
and differing in phase by a quarter
of a period, combine to produce
uniform circular motion (I., 29).
Hence the two coils, AA and BB
(Fig. 218), will produce in a simi-
lar way a rotatory magnetic field near their common centre.
Ferraris (1888) mounted within them a hollow copper

cylinder on pivots at top and
bottom. When the two-phase
currents from the small machine
(Fig. 215) are sent through the
Ferraris apparatus, the copper
cylinder is set rotating. The
rotation of the field produces
currents in the copper, as in
Arago's rotations. By Lenz's
law the motion of the cylinder
is in a direction to check the
action going on ; hence the cyl-
inder is dragged around in the same direction as the
rotation of the field ; for, if the speed of the cylinder were
the same as that of the field, no current would be induced.
If one current is reversed with respect to the other, that is,
if its phase is changed by 180°, the direction of rotation of
both the field and the cylinder is reversed. The cylinder

Fig. 219.

420

ELECTRICITY AND MAGNETISM.

tends to run up to synchronism with the field, but never
reaches it ; the difference in their speeds is just sufficient
to produce currents to supply the requisite torque. If
the rotation of the field produces a direct E.M.F., t he-
rotation of the cylinder, which is equivalent to the rota-
tion of the field in the other direction, produces a counter
E.M.F., and the latter is always smaller than the former.

368. Induction Motor. — A rotation of the field may
also be produced by winding the coils of the two circuits

on an iron ring

(Fig. 219). The
coils A and A'
are wound so as
to make conse-
quent poles at B
and B', while the
coils B and B'
produce conse-
quent poles at A
and A'. When
one of these cur-
rents is a maxi-
mum, the poles in
the ring are con-
centrated as in
Fig. 220, which
was made from a
photograph. Fig. 221 shows the field an eighth of a
period later, when the two currents have the same instan-
taneous value. Both poles have spread out uniformly a
quarter of the way around the ring in the direction of
the rotation. As the first current diminishes further

Fig. 220.

DYNAMOS AND MOTORS.
t

toward zero, these broad poles contract

Fig. 221.

nated iron cylinder with heavy ..
conductors embedded in its --
periphery and running parallel
with its axis of rotation. They
are connected together at the
ends of the cylinder so as to
form a " squirrel-cage " of cop-
per. The induced currents
through this cage produce a
torque which drags the cylin-
der after the rotating field.
Three-phase induction motors
are constructed on a similar
plan (Fig. 222).

421

their posterior
ends ; and, after
a quarter of a
period, are again
concentrated at
points 90° in ad-
vance of the
starting-point.
The poles thus
move round the
ring by a motion
which may be
compared to that
of a " measuring
worm."

Inside the ring
is mounted a
" rotor," consist-
ing of a lami-

422 ELECTRICITY AND MAGNETISM.

CHAPTER XXVI.

ELECTRIC OSCILLATIONS AND WAVES.

369. Oscillatory Discharges. — Allusion has already
been made to the oscillatory character of the discharge of
a Leyden jar. It was discovered by Joseph Henry in 1842
by studying the singular phenomena of the magnetic effects
produced by it in small steel needles, which were not
always found to be magnetized in the expected direction.
In 1853 Lord Kelvin gave the mathematical theory of
electric oscillations, and in 1858 Fedderson analyzed the
spark of a small discharge into a number of images by a
revolving mirror. Such a discharge consists of electric
surges first in one direction and then the other. The
charge deports itself as if it possessed inertia ; when the
condenser is suddenly discharged through a low resistance,
the first rush surges beyond the condition of equilibrium,
and the condenser is charged in the opposite sense ; a
reverse discharge follows, and so on, — each successive
oscillation being weaker than the preceding, till after a
few surges the oscillations cease. That such is the char-
acter of the discharge of a Leyden jar has been abundantly
demonstrated by experiment.

When the coatings are connected by a discharger of self-
induction L and negligible resistance, the electrostatic
energy, %Q2/ C, disappears and becomes the electromagnetic
energy of the discharge current, § LP. This in turn is re-
converted into the electrostatic energy of a reverse charge

ELECTRIC OSCILLATIONS AND WAVES. 423

of the jar; a second conversion into the electromagnetic
form follows, and so on. Each conversion of the energy
from the potential form to the kinetic or the reverse is ac-
companied by a loss of heat, till the energy is all expended.
The oscillations of a small Leyden jar, charged by con-
necting its two coatings with the secondary terminals of
an induction coil, can be readily exhibited to a large
number of persons. It is convenient, though not essential,
to close and open the primary circuit by means of a seconds
pendulum. A pointed strip of tin foil must be brought
over from the inner coating of the jar so as to leave a small
spark gap between it and a point connected with the outer
coating. At every break of the primary circuit a spark
will leap across this gap if the adjustments are properly
made. If it is viewed in a four-square mirror rotating
with moderate speed, it is found to consist of from about
four to twelve successive images. A single observer may
view it by a telescope after reflection from a mirror on the
end of a tuning-fork making about 100 vibrations a second.
The rate of oscillation in this case is comparatively slow
on account of the large self-induction of the secondary
coil, but the whole series of oscillations takes place in the
"incredibly short space of time occupied by a spark."

370. Period of an Oscillation. — Whether a discharge
is oscillatory or only intermittent depends on the relation
between the resistance and self-induction of the discharge
circuit and the capacity of the condenser.

If R denotes the resistance in ohms, L the self-induction
in henrys, and 0 the capacity in farads, the discharge will
be oscillatory when

r < */njo.1

1 Phil. Mag. (4) 5, p. 393.

424

ELECTRICITY AND MAGNETISM.

When R is small the period of the oscillations is

T= 2W~CL.
This formula corresponds with the condition required
for capacity to neutralize self-induction (361), when
La = 1/ Ceo. Since eo = 2"7m and T— 1/n, if we solve the
equation Leo = 1/ Ceo for T, we obtain the expression
above for the period, 2tt/vCL.

When the jar is discharged through a low resistance,
oscillations take place because the choking reactions due
to self-induction are neutralized by the capacity. The
oscillations then continue, like the vibrations of a tuning-
fork, till their energy is expended partly in heat and partly
in a manner to be described presently.

371. Electrical Resonance. — If the period of oscilla-
tion of a Leyden jar is determined by its capacity and

self-induction, it should be pos-
sible to apply to the phenomenon
the principle of resonance in
Sound (I., 151), provided the
inductive effects of discharge
currents are conveyed to other
condensers. This has been done.
The oscillatory character of a
condenser discharge is demon-
strated by its power of evoking
oscillations of the same period
in neighboring condensers. The
following instructive experiment
is due to Lodge : x Two similar

Fig. 223. , b

Leyden jars are connected to
discharge circuits of equal size (Fig. 223) ; but while that

1 Modern Views of Electricity, p. 338.

ELECTRIC OSCILLATIONS AND WAVES. 425

of A is interrupted by a spark gap, that of B is complete
and is adjustable by means of the slider S.

If now the coatings of A are connected to the two elec-
trodes of an influence machine, this jar discharges across
the gap, and the oscillations at every discharge disturb the
circuit of B, exciting in it feebler oscillations of the same
period. By tuning the two circuits to unison by moving
the slider, the oscillations in B become sufficiently violent
to make it overflow through the tin-foil strip c, which
comes over from the inner coating and nearly touches the
border of the outer one. This provides an easy overflow
path, so that when the jars are near together and the two
discharge circuits are parallel, every discharge of A is
accompanied by a bright spark at the air gap c.

372. Electromagnetic Waves. — When a current is
established through a conductor a magnetic field is set up
about it, and when the current is increased the magnetic
field is increased also ; the magnetic lines enlarge and
new ones push out from the conductor. When the circuit
is opened or reversed, these lines close in on the conduc-
tor and restore to it the energy stored in the ether
through an E.M.F. of self-induction. But when the cur-
rent oscillates with extreme rapidity, part of the energy
radiates into space, or electromagnetic waves are set up in
the surrounding medium. With the slow alternations
employed commercially the loss by electromagnetic radia-
tion is inappreciable, but such is no longer the case when
the rate equals a million or more a second, as in the oscil-
lations of a Leyden jar.

Joseph Henry appears to have been the first to detect
electromagnetic waves radiating from a circuit running
round a room when an inch spark from the prime con-

426 ELECTRICITY AND MAGNETISM.

ductor of a frictional machine was thrown on to the end
of the circuit. Sewing-needles were magnetized in a
parallel circuit thirty feet below, with two floors and
ceilings intervening. He says : " The diffusion of motion
in this case is almost comparable with that of a spark
from a flint and steel in the case of light." Thanks to
the remarkable researches of Hertz, we now know it to be
the same. The magnetic field produced by the discharge
through the one conductor spread with the velocity of
light to the closed circuit below, where a part of its
energy was absorbed by cutting through the circuit, and
produced an electric flow sufficient to magnetize the
needles placed in the helix.

The energy stored in a Leyden jar is not all dissipated
in the heat of the spark, but some of it is radiated into
space in the form of electric and magnetic waves.

373. Maxwell's Electromagnetic Theory of Light. —
The foundation of Maxwell's theory was laid by the
experiments of Faraday in electrostatic and electromag-
netic induction. These premise a medium as the agency
through which induction takes place. When, therefore,
a periodic disturbance, like the discharge of a Leyden jar,
induces similar disturbances in conductors about it, they
do not occur simultaneously with the initial one, but later
and later in proportion as the conductors in which they
are induced are more and more remote from the source.
In other words, the inductive action proceeds from the
source in the form of electric and magnetic waves.

Maxwell saw that it is not philosophical to fill all space
with a new medium whenever any new phenomenon is to
be explained, and that the evidence for the existence of
such a medium will be strengthened if it can be shown

ELECTRIC OSCILLATIONS AND WAVES. 427

that the properties which must be attributed to it to
account for electromagnetic phenomena are of the same
kind as those which we attribute to the luminiferous
ether. He therefore proposed the theory that waves of
light are not mere mechanical motions, but consist of
undulations partly electrical and partly magnetic ; oscillat-
ing electric displacements are accompanied by oscillating
magnetic forces at right angles to them ; both lie in the
plane of the wave, or are at right angles to its direction of
propagation.

Maxwell arrived at the conclusion that the propagation
of an electromagnetic disturbance through the ether takes
place in accordance with the laws governing the transfer
of motion through an elastic solid, and that the speed of
propagation is

where /x and K are the permeability and the dielectric
constant respectively. For all transparent media /* is
nearly unity. Hence the speed of light through two
transparent media should be inversely as the square roots
of their dielectric constants. If the velocity of light in a
vacuum be taken as unity, the absolute index of refraction
(I., 187) equals 1/v. Therefore the square of the index
of refraction of any substance should equal K, if the
electromagnetic theory is true. The agreement between
the two is not very close except for waves of great length.
The index of refraction corresponding to waves of longest
period should be selected, because these are the only ones
whose motion can be compared with the slow processes by
which the capacity of the dielectric is determined.

According to the same theory the velocity of propa-
gation should be the number of electrostatic units of
quantity in one electromagnetic unit. Michelson's value

428

ELECTRICITY AND MAGNETISM.

for the speed of light (1882) is 2.9985 x 1010, and Row-
land's determination of the ratio v is 2.9815 x 1010, both
in centimetres per second. So far, therefore, the prob-
abilities favored Maxwell's theory, but no decisive test
had been applied.

374. Hertz's Researches.1 — To Hertz belongs the
credit of having put the theory of electromagnetic waves
to the test of experiment, and of demonstrating the truth

Fig. 224.

of Maxwell's theory of light. The simplicity of his appli-
ances is no less remarkable than the magnitude of the
results derived from them. With the insight of genius he
seized on the only available means of producing electric
waves short enough to be measurable, viz., the disturb-
ances propagated outward from the discharge of a con-
denser of small capacity.

Hertz's apparatus to serve as the source of the waves he
called an oscillator (Fig. 224). It consisted of two metal-
lic plates A and B 40 cms. square and mounted 60 cms.

1 Hertz's Electric Waves, Trans, by D. E. Jones.

ELECTRIC OSCILLATIONS AND WAVES. 429

apart. The balls at the spark gap were kept brightly
polished. The receiver, or resonator, was a circle 70 cms.
in diameter, and its spark gap was adjustable by means
of a micrometer screw. The oscillator was connected to
the induction coil. The plates formed a condenser of
small capacity with air as the dielectric, and the discharge
across from ball to ball was oscillatory. This oscillation
had a definite period, and hence a succession of electro-
static and electromagnetic waves of equal period were
emitted by it. The half period was 1/100,000,000 of a
second.

The finite speed of the wave was demonstrated by plac-
ing a large sheet of zinc on a distant wall of the room and
observing the sparks produced at the small break in the
resonator in different positions along the dotted base line.
The metal acted as a reflector, so that stationary waves
were produced by interference between the direct and
reflected waves precisely as in Sound. The nodes and
antinodes were detected with considerable precision. The
distance between them determined the wave-length, and
the product of the wave-length and the frequency of the
oscillation gave the velocity. This was found to be of the
same order of magnitude as the known velocity of light,
though the data for calculating the period are somewhat
uncertain. Professor Trowbridge has since measured the
velocity of electric waves by a direct method, with a result
agreeing very well with the velocity of light.

By the aid of large parabolic zinc reflectors Hertz
demonstrated that electric waves are reflected to a focus
in the same manner as light. He also constructed a huge
prism of asphaltum and measured its index of refraction.
Gratings consisting of parallel conducting bars exhibited
polarization effects.

430 ELECTRICITY AND MAGNETISM.

Thus Hertz demonstrated that the waves radiating from
an oscillatory discharge spark and the associated condenser
are capable of reflection, refraction, and polarization the
same as light. They possess all the characteristics of light,
and are light except in point of wave-length. Maxwell's
theory does not replace the undulatory theory of light, but
supplies the mechanism of the undulations.

375. Faraday's Magneto-optic Rotations. — The first
definite relation between light and magnetism was estab-
lished by Faraday in 1845. A beam of plane polarized
light is transmitted through a transparent diamagnetic
medium. When a magnetic force is made to act in the
direction of the rays of light within the medium, the
plane of polarization is rotated in the direction in which
the current must circulate around the beam to produce
the given magnetic field.

Let a beam of light, polarized by transmission through
a Nicol's prism (I., 229), pass through a prism of heavy
glass (borosilicate of lead), with parallel polished ends
and placed in a powerful magnetic field, whose direc-
tion coincides with that of the beam of light. A second
Nicol's prism as an analyzer receives the beam, and is
turned so as to cut off all the light. The glass can be
conveniently placed in the magnetic field by boring holes
through the pole pieces attached to a large electromagnet.
The holes and the glass prism are all arranged in line for
the transmission of the polarized light.

When the magnet is excited light passes through the
analyzer. It may be extinguished by rotating it through a
small angle, but it will not be possible to produce complete
extinction ; colors will appear, showing that the angle of
rotation is a function of the wave-length. It is nearly

ELECTRIC OSCILLATIONS AND WAVES. 431

inversely as the square of the wave-length. If the elec-
tromagnet is large, it will be evident that time is required
to magnetize it, inasmuch as the transmitted light grows
sensibly in intensity for a second or more after closing the
circuit through the coils. On the other hand, Professor
Lodge has shown that the rotation of the beam of light,
first in one direction and then in the other, follows the
oscillations of the discharge of a Leyden jar through the
coils producing the field without iron.

376. Verdet's Constant. — The angle through which
the plane of polarization is turned depends on the fol-
lowing:

(1) It is proportional to the distance which the beam
travels within the medium. The direction of the plane of
polarization therefore changes continuously from incidence
to emergence.

(2) It depends on the nature of the medium. In
some paramagnetic substances it is opposite in direction
to the current producing the magnetization.

(3) It is proportional to the resolved part of the mag-
netic field in the direction of the beam.

This last fact was discovered by Verdet. The three
laws may be combined in one formula,

6 = wldS cos a,

where w is Verdet's constant determined by the nature
of the substance. &S cos a is the component of the field
in the direction of the beam, and I is the distance between
the points of incidence and emergence. The expression
IdS cos a is the difference in magnetic potential between
the point where the beam of light enters aud leaves the
medium. Lord Rayleigh found for carbon bisulphide at

432 ELECTRICITY AND MAGNETISM.

18° C. the constant 0.04202 in minutes of arc for a mag-
netic potential difference of one C.G.S. unit.

377. Explanation of Magneto-optic Rotation. — A
raj of plane polarized light may be resolved into two cir-
cularly polarized rays of the same period, each of half the
amplitude of the plane rectilinear vibration, and with the
motions in opposite directions round the circles (I., 32).
If now one of these circular vibrations be accelerated the
plane of the resultant rectilinear harmonic motion will be
rotated in the direction of the accelerated circular com-
ponent, since the resulting motion always lies in the plane
of symmetry. The circular vibration in the direction of
the rotation performs a larger number of vibrations within
the transparent medium than the other one. This mode
of stating what has taken place is independent of any
theory of light, and depends only on facts ascertained by
experiment.

The direction of the rotation in space is the same
whether the light passes one way or the other through the
magnetic field. Hence the effect may be increased by
passing the same beam back and forth by reflection along
the same magnetic field.

Magnetism consists of something in the ether analogous
to a whirl. This whirl apparently increases one of the
circular components of the plane polarized beam and so
rotates the plane of polarization.

APPENDIX.

TABLE I.
Absolute Dilatation of Mercury (S., 51).

Temp, by
air ther-
mometer.

Dilatation from
0« to t* U.

Mean coefficient

between

0° and t° C.

Coefficient

referred to vol.

at 0«.

True
coefficient.

.00017905

.001792

.00017925

.00017950

.00017922

.003590

.00017951

.00018001

.00017938

.005393

.00017976

.00018051

.00017955

.007201

.00018002

.00018102

.00017972

.009013

.00018027

.00018152

.00017989

.010831

.00018052

.00018203

.00018006

.012655

.00018078

.00018253

.00018024

.014482

.00018102

.00018304

.00018041

.016315

.00018128

.00018354

.00018059

100

.018153

.00018153

.00018405

.00018076

.019996

.00018178

.00018455

.00018092

120

.021844

.00018203

.00018505

.00018109

130

.023697

.00018228

.00018556

.00018125

140

.025555

.00018254

.00018606

.00018142

150

.027419

.00018279

.00018657

.00018159

160

.029287

.00018304

.00018707

.00018175

170

.031160

.00018329

.00018758

.00018190

180

.033039

.00018355

.00018808

.00018206

190

.034922

.00018380

.00018859

.00018221

200

.036811

.00018405

.00018909

.00018237

210

.038704

.00018430

.00018959

.00018252

220

.040603

.00018456

.00019010

.00018267

230

.042506

.00018481

.00019061

.00018282

240

.044415

.00018506

.00019111

.00018297

250

.046329

.00018531

.00019161

.00018313

260

.048247

.00018557

.00019212

.00018327

270

.050171

.00018582

.00019262

.00018341

280

.052100

.00018607

.00019313

.00018355

290

.054034

.00018632

.00019363

.00018370

300

.055973

.00018658

.00019413

.00018384

310

.057917

.00018683

.00019464

.00018398

320

.059866

.00018708

.00019515

.00018412

330

.061820

.00018733

.00019565

.00018426

840

.063778

.00018758

.00019616

.00018440

850

.065743

.00018784

.00019666

.00018453

(433)

434

ELECTRICITY AND MAGNETISM.

TABLE II.

Volume and Density of Distilled Water after Kosetti (S., 54).

Tempera-
ture.

Volume.

Density.

Tempera-
ture.

Volume.

Density.

— 10°

1.001858

.998145

14°

1.000701

.999299

— 9

1.001575

.998427

1.000841

.999160

— 8

1.001317

.998685

1.000999

.999002

— 7

1.001089

.998911

1.001160

.998841

— 6

1.000883

.999118

1.001348

.998654

- 5

1.000702

.999298

1.001542

.998460

— 4

1000545

.999455

1.001744

.998259

— 3

1.000410

.999590

1.001957

.998047

— 2

1.000297

.999703

1.002177

.997826

— 1

1.000203

.999797

1.002405

.997601

1 .000129

.999871

1.002641

.997367

1.000072

.999928

1.002888

.997120

1.000031

.999969

1.003144

.996866

1.000009

.999991

1.003408

.996603

1.000000

1.003682

.996331

1.000010

.999990

1.003965

.996051

1.000030

.999970

1.004253

.995765

1.000067

.999933

1.00770

.99235

1.000114

.999886

1.01195

.98820

1.000176

.999824

1.01691

.98338

1.000253

.999747

1.02256

.97794

1.000345

.999655

1.02887

.97194

1.000451

.999549

1.03567

.96556

1.000570

.999430

100

1.04312

.95865

APPENDIX.

435

TABLE in.

Pressure of Aqueous Vapor in Mms. of Mercury (G., 130).

t'C.

Mms.

t'C.

Mm*.

t'C.

Mms.

%*C.

Atmos.

— 10

2.08

13.54

525.39

100

1.0

— 9

2.26

14.42

633.69

110

1.4

— 8

2.46

15.36

733.21

120

1.96

— 7

2.67

16.35

99.1

735.85

130

2.67

— 6

2.89

17.39

99.2

738.50

140

3.57

— 5

3.13

18.50

99.3

741.16

150

4.7

— 4

3.39

19.66

99.4

743.83

160

6.1

— 3

3.66

20.89

99.5

746.50

170

7.8

— 2

3.96

22.18

99.6

749.18

180

9.9

— 1

4.27

23.55

99.7

751.87

190

12.4

4.60

24.99

99.8

754.57

200

15.4

4.94

26.51

99.9

757.28

210

18.8

28.10

100

760.00

220

22.9

5.69

29.78

100.1

762.73

230

27.5

6.10

31.55

100.2

765.46

6.53

41.83

100.3

768.20

7.00

54.91

100.4

771.95

7.49

71.39

100.5

773.71

8.02

91.98

100.6

776.48

8.57

117.48

100.7

779.26

9.17

148.79

100.8

782.04

9.79

186.94

100.9

784.83

10.46

233.08

101

787.59

11.16

288.50

105

906.41

11.91

354.62

110

1075.37

12.70

433.00

436

ELECTRICITY AND MAGNETISM.

TABLE IV.

Specific Resistances in C.6.S. Units at 0* C-'

Metals.

Platinum

Gold

Palladium

Silver

Copper

Aluminium 99 #

Iron

Nickel

Tin

Magnesium

Zinc

Cadmium

Lead

Thallium

Alloys,

Platinum-Silver

Pt, 33; Ag, 66.

Platinum-Iridium ....
Pt, 80; Ir, 20.
Platinum-Rhodium . . .
Pt, 90; Rd, 10.

Gold-Silver

Au, 90; Ag,10.

Aluminium-Silver ....
Al, 94; Ag,6.

Aluminium-Copper . . .
Al, 94; Cu, 6.

Copper-Aluminium . . .
Cu, 97; Al, 3.

Manganin

Cu, 84; Mn, 12; Ni, 4.

German Silver

Platinoid

Spec. Resist.

j emp. uoer. oetween
0" and 100* C.

10,917

0.00367

2,197

0.00377

10,219

0.00354

1,468

0.00400

1,561

0.00428

2,563

0.00423

9,065

0.00625

12,323

0.00622

13,048

0.00440

4,355

0.00381

5,751

0.00406

10,023

0.00419

20,380

0.00411

17,633

0.00398

Spec. Resist.

Temp. Coef. at 15<> C.

31,582

0.000243

30,896

0.000822

21,142

0.00143

6,280

0.00124

4,641

0.00238

2,904

0.00381

8,847

0.000897

46,678

0.0000

29,982

0.000273

41,731

0.00031

I Dewar and Fleming, Phil. Mag., Vol. XXXVI., p. 271.

INDEX.

Absorption, of radiation, 112 ; two
characteristics of, 113.

Accumulator, Kelvin's water-drop-
ping, 179.

Adiabatic lines, 135.

Agonic lines, 325.

Air thermometer, constant volume,
39 ; method of measuring poten-
tial of the, 230.

Alcohol thermometer, 18.

Alternators, 410.

Amalgam, 247.

Ampere, 334; the, 341.

Ampere's rule, 330; stand, 345,
363.

Andrews, 76, 85.

Anions, 255.

Arago, 353 ; rotations, 379.

Arc, electric, 291.

Armature, 354; drum, 400; the
Gramme, 403.

Arts, electrolysis in the, 270.

Astatic, pair, Nobili's, 337 ; mirror
galvanometer, 337.

Athermanous, 112.

Atomic heat, 50.

Attraction, and repulsion, 152;
due to induction, 173.

Aurora, the, 232.

Ayrton-Mather, 339.

Bacon, 2.

Numbers refer to pages.

Balance, Coulomb's torsion, 161 ;

Kelvin, 349.
Barlow's wheel, 347.
Battery, Grove's gas, 266; voltaic,

236.
Bichromate cell, 246.
Bidwell, 314, 321.
Boiling point, 15, 69 ; effect of

pressure on, 69.
Bosanquet, 363.
Bottomley's experiment on rege

lation, 60.
Boutigny, 72.
Boyle, 2 ; thermometer, 19 ; law,

37 ; and Charles' laws combined,

38 ; laws, deduction of, 145, 153.
Bridge,. Wheatstone's, 279.
Budde, 72.
Bunsen, 59 ; cell, 245.

Caloric, 2.

Calorie, 42.

Calorimetry, 42.

Capacity, definition of, 201; of

insulated sphere, 201 ; of two

concentric spheres, 204 ; of two

parallel plates, 205.
Carbon, filament, 292; specific

heat of, 48.
Carnot's cycle, 136; reversibility

of engine, 139.
Cathode, 255 ; rays, 394.

438

INDEX.

Cations, 255.

Cautery, electric, 290.

Cavendish, 218.

Cell, bichromate, 246; Bunsen,
245 ; chemical action in Daniell,
242; Clark standard, 251; cop-
per oxide, 249; Daniell, 241,
253 ; data relating to, 252 ; grav-
ity, 243; Leclanche, 248;
Plante's storage, 267; reversi-
bility of Daniell, 262; silver
chloride, 250 ; effect of heat on,
253.

Cells, in multiple series, 283; in
parallel, 282 ; in series, 280.

Celsius, 16.

Change of volume during fusion,
56.

Charge, distribution of, 159 ; ex-
ternal, 158; redistribution of,
160; residual of Leyden jar,
207.

Charged sphere, force outside of,
165.

Charges, equal and of opposite
sign, 156.

Charles, law of, 20.

Chemical action in relation to
energy, 244.

Choking coils, 414, 416.

Circular coil, intensity of field at
centre of, 333.

Clark standard cell, 252.

Clausius, 141, 147, 268.

Coefficient, of elasticity of a gas,
131 ; of thermal conductivity,
94.

Coefficients, of dilatation and
pressure, table of, 37 ; of length
and volume, relation between,
26.

Coil, the induction, 388.

Coils, choking, 414, 416.

Cold due to evaporation, 76.

Comparator, interferential, 29.

Concentric spheres, capacity of
two, 204.

Condensation, effect of electrifica-
tion on, 225.

Condensers, 202 ; capacity of, 202 ;
connected in series, energy of,
211 ; energy expended in charg-
ing, 209.

Conduction, by gases, 98; by
liquids, 99; by solids, 91; in
wood and crystals, 97.

Conductivities, comparison of
thermal and electrical, 95 ; table
of, 96.

Conductivity, coefficient of ther-
mal, 94 ; electrical, 276.

Conductor, equilibrium of a, 191.

Conductors and insulators, 154;
distinction between, 214.

Consequent poles, 312.

Constant, Verdet's, 431.

Convection, by hydrogen, 102 ; cur-
rents, 350; electric, of heat,
302 ; electrolytic, 269 ; in gases,
100 ; in liquids, 99.

Cooling, Newton's law of, 124.

Copper, oxide cell, 249; voltam-
eter, 262.

Cores, 354.

Coulomb, 160; the, 341.

Coulomb's law, 168; torsion bal-
ance, 161.

Counter E.M.F. in a circuit, 288.

Critical temperature, 85.

Crookes tubes, 392.

Cryolite, 271.

Cubical dilatation of solids, 23.

INDEX.

439

Cuneus, 206.

Current, electromagnetic unit of,
334 ; heating effect of, 290 ; in-
tensity of, 233; la£ of, behind
E.M.F., 411; magnetic relations
of, 329 ; through a circular con-
ductor, magnetic field about,
332.

Currents, convection, 350; poly-
phase, 417; steady, 233; theory
of production of, 243 ; value of
alternating, 412.

Curves of magnetization, 359.

Cycle, Carnot's, 136.

D'Alibard, 224.

Daniell cell, 224; chemical action
in, 242.

D'Arsonval galvanometer, 338,
349 ; Ayrton-Mather form of,
339.

Davy, experiment, 5, 291, 353.

Declination, magnetic, 325 ; varia-
tions in, 326.

Definition of capacity, 201.

Definitions, 118.

Deflections, magnetic forces by
method of, 317.

De la Tour, 84.

Depolarization by chemical means,
241.

Depretz, 56.

Dew, 80 ; point, 79.

Dewar, 86, 297, 314.

Diathermancy, of gases, 115; of
liquids, 114.

Diathermanous, 112.

Dielectric, effect on electric in-
tensity, 221 ; on the forces be-
tween the plates, 221.

Dielectric polarization, 213.

Dielectrics, 155.

Dilatation, of gases, 35; of liquids,
30 ; of solids, the cubical, 23 ; of
water, 33.

Dip, magnetic, 327.

Dipping needle, 327.

Discharge, with impulsive rush,
228 ; with Steady strain, 227.

Discharges, in high vacua, 390;
oscillatory, 422.

Discovery, Faraday's, 372.

Displacement, electric, 215.

Distribution of charge, 159.

Dry pile, 234.

Dulong and Petit, 50, 124 ; experi-
ments, table of, 47.

Dynamo, and motor, reactions in
field of, 403; compound- wound,
401; ideal simple, 397; over-
compounded, 402.

Earth a magnet, 323.

Ebullition, 65, 67.

Effect, Hall, 351; Peltier, 299;
Thomson, 301.

Efficiency, electrical, of a motor,
408 ; of transmission, 409.

Electric, arc, 291; cautery, 290;
displacement, 215; energy, con-
version of into heat, 286; field
and lines of force, 155 ; heating,
290 ; intensity, effect of the die-
lectric on, 168; pressure, 288;
strain, 211; transfer in closed
circuits, 215; welding, 291.

Electrical, efficiency of a motor,
408; potential, 188; resonance,
424; units, 341.

Electricity, and electrification,
150; thermal, 293; three divis-
ions of, 151.

440

INDEX.

Electrification, by influence, 171 ;
effect of, on condensation, 225;
two kinds of, 153; with like
charges by influence, 173.

Electro-chemical equivalents, 250.

Electrode, positive, 236 ; negative,
236.

Electrodes, 255.

Electrodynamics, 343.

Electrodynamometer, 347, 413.

Electrolysis, 255 ; in the arts, 270 ;
quantitative laws of, 258; of
copper sulphate, 257; of lead
acetate, 258 ; of sodium sul-
phate, 257; of water, 256;
theory of, 268 ; with and with-
out polarization, 265.

Electrolyte, 237.

Electrolytes, 255.

Electrolytic cell, polarization of,
264.

Electromagnetic, rotations, 346 ;
systems, motion in, 369 ; theory
of light, Maxwell's, 426 ; waves,
425.

Electromagnets, 353.

Electrometer, 194 ; attracted disk,
195 ; theory and use of, 195 ;
quadrant, 197.

Electromotive force, 238 ; and po-
tential difference, 237 ; counter,
in motor, 406; direction and
value of induced, 374 ; law of,
375, 398.

Electrophorus, 176.

Electroscope, 157; gold-leaf, 158.

Electrostatics, second law of, 163.

Element, voltaic, 236.

Energy, chemical action in rela-
tion to, 244 ; expended in charg-
ing condenser, 209 ; heat a form

of, 1 ; in a current, division of
the, 289 ; lost in dividing a
charge, 209 ; of similar con-
densers in parallel, 210; of suc-
cessive charges, 178; stored in
magnetic field, 386; total molec-
ular, 147.

Equal charges of opposite sign,
156.

Equation, Helmholtz's, 385.

Equator, magnetic, 327.

Equilibrium of a conductor, 191.

Equipotential surfaces, 189.

Equivalents, e 1 e c t r o - chemical,
259.

Erman, 57.

Euler, 362.

Evaporation, 65 ; cold due to, 76 ;
in a closed space, 66.

Ewing's theory of magnetism,
323.

Exchanges, Prevost's theory of,
116.

Expansion, 11; linear, 25; of
liquids and gases, 12.

Expression for force in terms of
potential, 190.

External charge, 158.

Extra current, 382.

Farad, 341.

Faraday, 83, 155, 159, 218, 258,
350, 362, 426; discovery, 372;
experiment, 175 ; ice-pail experi-
ment, 175; magneto-optic rota-
tions, 430 ; ring, 377.

Fahrenheit, 56.

Field, electric, 155 ; magnet, 401 ;
rotatory, 418.

Filament, the carbon, 292.

Fizeau, 24.

INDEX.

441

Fleming, 297.

Forbes, 109, 114.

Force, electromotive, 238; ex-
pression for, in terms of poten-
tial, 190; magnetomotive, 366;
near a charged plane conductor,
168 ; outside a charged sphere,
165 ; very near a charged sphere,
167 ; within a helix, 366.

Franklin, 208, 224.

Fraunhofer lines, 110.

Fusing point, 54.

Fusion, 54; change of volume
during, 56; latent heat of,
61.

Galileo, 19.

Galvanometer, 335; astatic mir-
ror, 337; d'Arsonval, 338; po-
tential, 339; tangent, 335.

Gas, and vapor, distinction be-
tween, 86; battery, 266; coeffi-
cient of elasticity of, 131 ;
volume proportional to absolute
temperature, 38.

Gases, dilatation of, 35; law of,
146; liquefaction of, 83; spe-
cific heat of, 52 ; theory of the
pressure of, 143.

Gay-Lussac, 56; law of, 146.

Geissler tubes, 391 ; stria? in,
391.

Gilbert, 155 ; electroscope, 157.

Gramme ring, 403.

Gravity cell, 243.

Gray, 155.

Grove's gas battery, 266.

Growth of current in inductive
circuits, 384.

Hall effect, 351.

Heat absorbed in solution, 63
effect of on resistance, 276
equivalent of a current, 287
laws of development of, 287
modes of transmission, 90; po-
larization of, 109.

Helix, force within, 366.

Helmholtz, 252 ; equation, 385.

Henry, 215, 342, 382, 425 ; the, 342.

Hertz, 394, 426 ; researches, 428.

Holtz influence machine, 180.

Hooke, 14.

Hopkinson, 212.

Horse-shoe magnet, 353.

Humidity, relative, 78.

Hydrogen, convection in, 102;
mean square of the velocity of,
145.

Hygrometer, Regnault's, 80.

Hysteresis, 213, 360.

Ice-pail experiment, Faraday's,
175.

Inclination, magnetic, 327.

Induced, and inducing charges,
relation between, 174; electro-
motive force, 372.

Inductance, 383.

Induction, attraction due to, 173 ;
by magnets, 373 ; coefficient of
mutual, 381 ; coefficient of self,
383; coil, 388; magnetic, and
magnetic force, 359; motor,
420 ; self, 382.

Inductive, capacity, specific, 217;
circuits, growth of current in,
384 ; system, 378.

Influence, charging by, 172 ; elec-
trification by, 171 ; electrification
with like charges by, 173;
machine, the Holtz, 180.

442

INDEX.

Ingenhausz, experiment of, 92.
Instability, condition of, 55.
Insulators and conductors, 154;

distinction between, 214.
Intensity of field at centre of

circular coil, 333.
Inverse squares, law of, 107, 162 ;

proof of law, 164.
Ions, 237.
Iron, effect of introducing, 353 ;

hysteresis in, 360.
Isoclinic lines, 327.
Isodynamic lines, 327.
Isogonic lines, 325.
Isothermal lines, 133.

J and R, relation between, 130.
Jolly, 39.
Joule, 341, 362.
Joule's experiment, 127.

Kelvin, 141, 194, 198; balances,
349 ; water-dropping apparatus,
179.

Kerr, 212.

Kohlrausch, 212.

Lag, of current behind E.M.F.,
411 ; of induction behind mag-
netizing force, 361.

Langley, 7, 110, 111.

Latent heat, of fusion, 61 ; of
vaporization, 74.

Law, Lenz's, 379, 380; of devel-
opment of heat, 287 ; of inverse
squares, 107; of magnetic cir-
cuit, 368; of magnetic force,
306, 316; of resistance, 275;
Ohm's, 273.

Laws of Boyle and Charles com-
bined, 38.

Leclanche" cell, 248; chemical
action in, 249.

Lemstrom, 232.

Lenard, 395.

Lenz's law, 379, 380.

Leslie's experiment, 117.

Leyden jar, 206, 211, 228, 370;
residual charge of, 207 ; seat of
the charge in, 208.

Light, Maxwell's electromagnetic
theory of, 426.

Lightning, an electrical phenom-
enon, 224; flashes, 226; pro-
tectors, 229.

Linear expansion, 25 ; measure-
ment of, 28.

Lines, isoclinic, 327; isodynamic,
327 ; isogonic, 325 ; isothermal,
133 ; of force and electric field,
155.

Liquefaction of gases, 82; of oxy-
gen and nitrogen, 87.

Liquid and gaseous states, con-
tinuity of, 84.

Liquids, convection in, 99 ; dia-
thermancy of, 114 ; dilatation of,
30.

Local action and amalgamation,
247.

Machine, Holtz influence, 180;
Toepler, 182; Wimshurst, 184.

Magne-crystallic action, 358.

Magnetic, circuit, law of, 362, 368 ;
equator, 327; field, 308; field
about a wire, 331 ; field about
current through circular con-
ductor, 332 ; field, energy stored
in, 386 ; fields about parallel cur-
rents, 343; figures, 310; force
due to straight current, 365;

INDEX.

443

force, first law of, 306; second
law of, 317; forces by method
of deflections, 318; by method
of oscillations, 319; inclination,
326; induction, 307; induction
and magnetic force, 359; mo-
ment, 31G ; permeability, 355 ;
shielding, 311; substances, 31 1^
susceptibility, 355.

Magnetic relations, Ampere's rule,
330 ; Moreland's rule, 330 ; Max-
well's rule, "330; of a current,
329.

Magnetism, and mechanical stress,
320; effects of heat on, 313;
Ewing's theory of, 323; molec-
ular, 322.

Magnetization, by electric dis-
charges, 370; curves of, 359.

Magneto-optic rotations, 430; ex-
planation of, 432.

Magnets, 305; artificial, 306; horse-
shoe, 354; induction by, 313;
permanent and temporary, 308.

Mascart, 230.

Material bodies, heat in, 2.

Maxwell, 166, 214, 350, 362, 426.

Maxwell's rule, 330.

McAdie, 231.

Mechanical stress and magnetism,
320.

Melloni, 108, 112, 114.

Melting point, influence of press-
ure on, 58.

Mendenhall, 232.

Mercury, 13, 346, 347.

Metals and liquids, thermo-electro-
motive force between, 302.

Michelson, 28, 427.

Molecular hypothesis, 142.

Moreland's rule, 330.

Morley's comparator, 29.

Motor, counter E.M.F. in, 406;
direction of rotation as a, 405;
electrical efficiency of, 408; in-
duction, 420; work done by,
407.

Motors and dynamos, 397.

Negative, electricity, 154; elec-
trode, 236.
Neutral temperature, 294.
Newton, 114.

Newton's law of cooling, 124.
Nobili's astatic pair, 337.

Oersted's discovery, 329.

Ohm, 274, 341.

Ohm's law, 273.

Oscillation, period of, 423.

Oscillations, comparison of pole-
strengths by, 320; magnetic
forces by method of, 318.

Oscillator, Hertz's, 428.

Oscillatory discharges, 422.

Osmotic pressure, 243.

Oxygen and nitrogen, liquefaction
of, 87.

Parallel, and oblique currents, 345 ;

plates, capacity of two, 205.
Paramagnetic and diamagnetic

substances compared, 356.
Peltier effect, 299; experiment to

show, 300.
Person, 55.
Petit and Dulong, experiment, 47,

50, 124.
Phosphorus, dilatation of, 68.
Pile, the dry, 234; Volta's, 233.
Plane, conductor, force near a

charged, 168.

444

INDEX.

Plant6's storage cell, 267.

Polarization, 240; dielectric, 213;
of electrolytic cell, 264 ; of heat,
109.

Pole, unit, 315.

Poles, consequent, 312; strength
of, 315.

Pole-strengths by oscillations,
comparison of, 320.

Polyphase currents, 417.

Positive, electricity, 154; elec-
trode, 236.

Potential, difference of, 189; dif-
ference of, and E.M.F., 239;
electrical, definition of, 188 ; ex-
pression of force in terras of,
190; high, of thunder clouds,
224 ; loss of, proportional to re-
sistance, 278 ; of the air, method
of measuring, 230 ; of a sphere?
193; results of observation, 231.

Practical electrical units, 341.

Pressure, influence of on boiling
point, 69; influence of on melt-
ing point, 58; of a gas, theory
of, 143.

Preston, 47.

Prevost's theory of exchanges,
116; extension of, 120.

Primary cells, 233.

Production of a current, theory
of, 243.

Quadrant electrometer, 197; used
heterostatically, 199 ; used idio-
statically, 199.

Quantity, unit, of electricity, 164 ;
of heat, 42.

Radiant energy, heat the measure
of, 110.

Radiant heat, 6; and light identi-
cal, 7 ; refraction of, 108.

Radiation, 6 ; appliances for the
study of, 104 ; invisible, reflected
like light, 105.

Ratio of the two specific heats,
148.

Rayleigh, 225, 390, 431.

Rays, cathode, 394; Rontgen,
394.

Reactions in field of dynamo and
motor, 403.

Redistribution of charge, 160.

Refraction of radiant heat, 108.

Regelation, 59.

Regnault, 33, 37; conclusions re-
specting specific heat of gases,
52 ; hygrometer, 80.

Relation between induced and in-
ducing charges, 174; between
fi and k, 355.

Reluctance, 367.

Remanence and coercive force,
362.

Researches, Hertz's, 428.

Residual, charge of Leyden jar,
207; magnetism, 353.

Resinous electricity, 154.

Resistance, 274 ; effect of heat on,
276; laws of, 275; specific, 275;
variation of internal, with cur-
rent, 283.

Resonance, electrical, 424.

Resonator, Hertz's, 429.

Reversibility of Daniell cell, 262.

Richmann, 224.

Ring, Faraday's, 377.

Rontgen, 394.

Rogers' comparator, 28.

Ross, 324.

Rotation, direction of as a motor,

INDEX.

445

405; Faraday's magneto-optic,

430, 432; in a magnetic field,

363.
Rotations, Arago's, 379.
Rotatory field, 418.
Rowland, 50, 363; experiments,

129, 350.
Rumford's experiment, 3.

Safety fuses, 290.

Scale, centigrade, 16; Fahren-
heit's, 16; Reaumur's, 16.

Schwatke, 324.

Second law of electrostatics, 163.

Seebeck, 293, 299.

Self-induction, 382.

Siemens, 212.

Silver, chloride cell, 250 ; voltame-
ter, 260.

Similar condensers in parallel,
210.

Simple, bodies, atomic heat of,
50; voltaic element, 235, 237.

Solenoids, 352 ; effect of introduc-
ing iron in, 353.

Solids, conduction by, 91.

Solution, heat absorbed in, 63;
tension, 243.

Specific heat, at constant volume
147; by method of mixtures,
44; of carbon, 48; of gases, 51;
of water, 49 ; ratio of the two,
148 ; table of Dulong and Petit,
47 ; variation of, with tempera-
ture, 47.

Speed, 407.

Sphere, capacity of an insulated,
201 ; capacity of two concentric,
204; force outside a charged,
165 ; force very near a charged,
167 ; potential of, 193,

Spheroidal state, 65, 72.

Steady strain, discharge with, 227.

Stewart, 114, 122.

Strength of pole, 315.

Sublimation, 65, 73.

Successive charges, energy of,

178.
Surface density, 159.
Surfaces, equipotential, 189.
System, the inductive, 378.

Tait, 96.

Telephone, the, 395.

Temperature, definition of, 9 ; neu-
tral, 294.

Theory, of electrolysis, 268; of
magnetism, 322.

Thermal, capacity, 45 ; electricity,
293; E.M.F., variation with
temperature, 295.

Thermodynamics, first law of,
126; second law of, 141.

Thermo-electric, diagram, 296;
series, 298.

Thermo E.M.F., between metals
and liquids, 302 ; in diagram, 297.

Thermometer, air, 19 ; alcohol,
18 ; Boyle's, 19 ; constant vol-
ume air, 39 ; fixed points on, 14 ;
Galileo's, 19; mercurial, 13;
scales, 15, 16.

Thermopile, 105, 298.

Thomson, 58, 350, 392 ; effect, 301-

Thunder clouds, high potential,
224.

Toepler machine, 182.

Torque, 407.

Torsion balance, 161.

Total molecular energy, 147.

Transfer, electric, in closed cir-
cuits, 215.

INDEX.

Transformers, 415.
Transmission, efliciency of elec-
tric, 409; of heat, modes of, 90.
Trowbridge, 429.
Tyndall, 115, 119, 358.

Unit, capacity, 341; electromag-
netic, 340; magnetic pole, 314;
of current, the electromagnet^
334; of electromagnetic quan-
tity, 340; of electrostatic quan-
tity, 164; potential difference,
340 ; practical electrical, 341 ;
quantity of heat, 42 ; resistance,
341 ; strength of current, 340.

Vacua, discharges in high, 392 ;

discharges in partial, 390.
Value, of alternating current, 412.
Vaporization, 65 ; latent heat of,

74.
Variation of internal resistance

with current, 283.
Velocity, of hydrogen, mean square

of, 145.
Verdet's constant, 431.

Villari, critical point, 321; rever-
sal, 321.

Virtual volts and amperes, 413.

Vitreous electricity, l.">4.

Volt, 341; virtual, 413.

Volta, 176 ; pile, 233.

Voltaic cells, effect of heat on,
253; element, simple, 2;;.">.

Voltameter, copper, 262; silver,
260.

Volume of gas proportional to
temperature, 38.

Water, electrolysis of, 256; spe-
cific heat of, 49.
Watt, 342.
Wattmeters, 414.
Waves, electromagnetic, 425.
Weber, 322.

Wells' explanation of dew, 80.
Wheatstone's bridge, 279.
Wiedemann, 321.
Wimshurst machine, 184.
Wollaston's cryophorus, 77.

Zero, absolute, 20; change of, 17.

SCIENCE. fi8

Walker's Anatomy, Physiology, and Hygiene.

New Edition, revised and enlarged. By JEROME WALKER, M.D.
i2mo, cloth, 495 pages. Price, $1.20.

THIS is a thorough and careful revision by the author of the
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The principle which has been kept consistently in mind in the
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A great deal of valuable matter has been collected for the ap-
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58 SCIENCE.

Electrical Measurements.

By Professor HENRY S. CARHART and Asst. Professor G. W. PATTER
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TN this book are presented a graded series of experiments for
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SCIENCE. 69

The Elements of Chemistry.

By Professor Paul C. Freer, University of Michigan. i2mo, cloth,
294 pages. Price, #1.00.

IN the preparation of this book an attempt has been made to
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Elements of Chemical Physics.

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THIS volume furnishes a full development of the principles
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60 SCIENCE.

Descriptive Inorganic General Chemistry.

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IT aims to give a systematic course of Chemistry by stating
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Principles of Chemical Philosophy.

By Josiah Parsons Cooke, late Professor of Chemistry, Harvard
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THE object of this book is to present the philosophy of chem-
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