zotero-db/storage/9IZRBMHE/.zotero-ft-cache

THE CORPUSCULAR
THEORY OF MATTER
J. J. THOMSON, M.A. F.R.S. D.Sc. LL.D. Ph.D.
PROFESSOR OF EXPERIMENTAL PHYSICS, CAMBRIDGE, AND PROFESSOR OF NATURAL PHILOSOPHY AT THE ROYAL INSTITUTION, LONDON.
LONDON
ARCHIBALD CONSTABLE & CO. LTD.
10 ORANGE STREET LEICESTER SQUARE W.C.
1907 3)

%^^l^ 3f
BRADBURY, AftNEW, & CO. LD.^PRINTERS, LONDON AND TONBRIDGK.

PREFACE
This book is an expansion of a course of lectures given at the Eoyal Institution in the Spring of 1906. It contains a description of the properties of corpuscles and their application to the explanation of some physical phenomena. In the earlier chapters a considerable amount of attention
is devoted to the consideration of the theory that many
o' the properties of metals are due to the motion of corpuscles diffused throughout the metal. This theory has received strong support from the investigations of Drude and Lorentz ; the former has shown that the theory gives an approximately correct value for the ratio of the thermal and electrical conductivities of pure metals and the latter that it accounts for the long-wave radiation from hot bodies. I give reasons for thinking that the theory in its
usual form requires the presence of so many corpuscles
that their specific heat would exceed the actual specific heat of the metal. I have proposed a modification of the theory which is not open to this objection and which makes the ratio of the conductivities and the long-wave radiation of the right magnitude.
The later chapters contain a discussion of the properties of an atom built up of corpuscles and of positive electricity,
the positive electricity being supposed to occupy a much
larger volume than the corpuscles. The properties of an
atom of this kind are shown to resemble in many respects
those of the atoms of the chemical elements. I think that a theory which enables us to picture a kind of model atom and to interpret chemical and physical results in terms of

vi

PEEFACE.

such model may be useful even though the models are crude, for if we picture to ourselves how the model atom,
must be behaving in some particular physical or chemical process, we not only gain a very vivid conception of the process, but also often suggestions that the process under consideration must be connected with other processes, and thus further investigations are promoted by this method ; it also has the advantage of emphasising the unity of chemical and electrical action.
In Chapter VII. I give reasons for thinking that the number of corpuscles in an atom of an element is not greatly in excess of the atomic weight of the element, thus in particular that the number of corpuscles in an atom of
hydrogen is not large. Some writers seem to think that this makes the conception of the model atom more difficult.
I am unable to follow this view ; it seems to me to make
the conception easier, since it makes the number of possible atoms much more nearly equal to the number of
the chemical elements. It has, however, an important bearing on our conception of the origin of the mass of the atom, as if the number of corpuscles in the atom is of the same order as the atomic weight we cannot regard the mass of an atom as mainly or even appreciably due to the mass of the corpuscles.
I am indebted to Mr. G. W. C. Kave for assisting in
revising the proof sheets.

Cambridge, July 1 5, 1907.

J. J. Thomson.

CONTENTS

— I. Introduction Coepuscles in Vacuum Tubes .

.

1

II. The Origin of the Mass of the Corpuscle .

.

28

III. Properties of a Corpuscle

43

IV. Corpuscular Theory of Metallic Conduction

.

49

V. The Second Theory of Electrical Conduction . 86

VI. The Arbangement of Corpuscles in the Atoii . 103

VII. On the Number of Corpuscles in an Atom . . 142

INDEX

169

THE
CORPUSCULAR THEORY OF MATTER

CHAPTEE I.

The theory of the constitution of matter which I propose

to discuss in these lectures, is one which supposes that the

various properties of matter may be regarded as arising

from electrical effects. The basis of the theory is electricity,

and its object is to construct a model atom, made up of

specified arrangements of positive and negative electricity,

which shall imitate as far as possible the properties of the

We real atom.

shall postulate that the attractions and

repulsions between the electrical charges in the atom follow

the familiar law of the inverse square of the distance,

though, of course, we have only direct experimental proof

of this law when the magnitude of the charges and the

distances between them are enormously greater than those

which can occur in the atom. "We shall not attempt to go

behind these forces and discuss the mechanism by which

they might be produced. The theory is not an ultimate one ; its object is physical rather than metaphysical. From

the point of view of the physicist, a theory of matter is a

policy rather than a creed; its object is to connect or

co-ordinate apparently diverse phenomena, and above all

to suggest, stimulate and direct experiment. It ought to

furnish a compass which, if followed, will lead the observer

further and further into previously unexplored regions.

T.M.

B

2 THE COEPUSCULAK THEOEY OF MATTEE.

Whether these regions will be barren or fertile experience alone will decide ; but, at any rate, one who is guided in this way will travel onward in a definite direction, and will not wander aimlessly to and fro.
The corpuscular theory of matter with its assumptions of electrical charges and the forces between them is not nearly so fundamental as the vortex atom theory of matter, in which all that is postulated is an incompressible, friction-

less liquid possessing inertia and capable of transmitting
pressure. On this theory the difference between matter
and non-matter and between one kind of matter and

another is a difference between the kinds of motion in the

incompressible liquid at various places, matter being those

portions of the liquid in which there is vortex motion.

The simplicity of the assumptions of the vortex atom theory

are, however, somewhat dearly purchased at the cost of the

mathematical difficulties which are met with in its develop-

ment ;

and for many purposes a theory whose consequences

are easily followed is preferable to one which is more

fundamental but also more unwieldy. We shall, however,

often have occasion to avail ourselves of the analogy which

exists between the properties of lines of electric force in the electric field and lines of vortex motion in an incompressible

fluid.

To return to the corpuscular theory. This theory, as I

have said, supposes that the atom is made up of positive

A and negative electricity.

distinctive feature of this

— — theory the one from which it derives its name is the

peculiar way in which the negative electricity occurs both in
the atom and when free from matter. We suppose that the

negative electricity always occurs as exceedingly fine par-

ticles called corpuscles, and that all these corpuscles, when-

ever they occur, are always of the same size and always carry

the same quantity of electricity. Whatever may prove to

be the constitution of the atom, we have direct experi-

mental proof of the existence of these corpuscles, and I will

begin the discussion of the corpuscular theory with a

description of the discovery and properties of corpuscles.

;

COEPUSCLES IN VACUUM TUBES.

3

Corpuscles in Vacuum Tubes.
The first place in which corpuscles were detected was a highly exhausted tube through which an electric discharge
was passing. When I send an electric discharge through
this highly exhausted tube you will notice that the sides of the tube glow with a vivid green phosphorescence. That this is due to something proceeding in straight lines from
— the cathode the electrode where the negative electricity — enters the tube can be shown in the following way :
the experiment is one made many years ago by Sir William
Crookes. A Maltese cross made of thin mica is placed
between the cathode and the walls of the tube. You will notice that when I send the discharge through the tube, the green phosphorescence does not now extend all over the end of the tube as it did in the tube without the cross. There is a well-defined cross in which there is no ]3hosphorescence at the end of the tube ; the mica cross has thrown a shadow on the tube, and the shape of the shadow
proves that the phosphorescence is due to something, travelling from the cathode in straight lines, which is stopped by a thin plate of mica. The green phosphorescence is caused by cathode rays, and at one time there was a keen
controversy as to the nature of these rays. Two views
were prevalent, one, which was chiefly supported by English physicists, was that the rays are negatively electrified bodies shot off from the cathode with great velocity the other view, which was held by the great majority of German physicists, was that the rays are some kind of
ethereal vibrations or waves.
The arguments in favour of the rays being negatively charged particles are (1) that they are deflected by a magnet in just the same way as moving negatively
electrified particles. We know that such particles when
a magnet is placed near them are acted upon by a force whose direction is at right angles to the magnetic force, and also at right angles to the direction in which the particles are moving. Thus, if the particles are moving
b2

4 THE COEPUSCULAR THEORY OF MATTER'.
horizontally from east to west, and the magnetic force is horizontal and from north to south, the force acting on the negatively electrified particles will be vertical and downwards.
When the magnet is placed so that the magnetic force is
along the direction in which the particle is moving the
latter will not be affected by the magnet. By placing the
magnet in suitable positions I can show you that the cathode particles move in the way indicated by the theory. The observations that can be made in lecture are neces-
sarily very rough and incomplete ; but I may add that elaborate and accurate measurements of the movement of
^
FIG. 1.
cathode rays under magnetic forces have shown that in this respect the rays behave exactly as if they were moving
electrified particles.
The next step made in the proof that the rays are negatively charged particles, was to show that when they are caught in a metal vessel they give up to it a charge of negative electricity. This was first done by Perrin. I have here a modification of his experiment. ^ is a metal
cylinder with a hole in it. It is placed so as to be out of
the way of the rays coming from C, unless they are deflected by a magnet, and is connected with an electroscope. You see that when the rays do not pass through the hole in the
cylinder the electroscope does not receive a charge. I now, by means of a magnet, deflect the rays so that they pass
through the hole in the cylinder. You see by the divergence

COEPUSCLES IN VACUUM TUBES.

5

of the gold-leaves that the electroscope is charged, and on testing the sign of the charge we find that it is negative.

Deflection op the Eats by a Chaeged Body.
If the rays are charged with negative electricity they ought to be deflected by an electrified body as well as by a
magnet. In the earlier experiments made on this point no such deflection was observed. The reason of this has been shown to be that when the cathode rays pass through a gas they make it a conductor of electricity, so that if there is any appreciable quantity of gas in the vessel through

FIG. I.
which the rays are jDassing, this gas will become a conductor of electricity, and the rays will be surrounded by a conductor which will screen them from the effects of electric force just as the metal covering of an electroscope
screens off all external electric effects. By exhausting the vacuum tube until there was only an exceedingly small quantity of air left in to be made a conductor, I was able
to get rid of this effect and to obtain the electric deflection of the cathode rays. The arrangement I used for this purpose is shown in Fig. 2. The rays on their way through
A the tube pass between two parallel plates, , B, which can be
connected with the poles of a battery of storage cells. The pressure in the tube is very low. You will notice that the rays are very considerably deflected when I connect the plates with the poles of the battery, and that the direction

6 THE COEPUSCULAE THEOEY OF MATTEE.
of the deflection shows that the rays are negatively charged.
We can also show the effect of magnetic and electric force
on these rays if we avail ourselves of the discovery made by Wehnelt, that lime when raised to a red heat emits when
negatively charged large quantities of cathode rays. I have here a tube whose cathode is a strip of platinum on which there is a speck of lime. "When the piece of platinum is
made very hot, a potential difference of 100 volts or so is sufficient to make a stream of cathode rays start from this speck you will be able to trace the course of the rays by
;
the luminosity they produce as they pass through the gas.
PIG. 3.
You can see the rays as a thin line of bluish light coming from a point on the cathode ; on bringing a magnet near it the line becomes curved, and I can bend it into a circle or a spiral, and make it turn round and go right behind the cathode from which it started. This arrangement shows in a very striking way the magnetic deflection of the rays. To show the electrostatic deflection I use the tube shown in
B Fig. 3. I charge up the plate negatively so that it repels
the pencil of rays which approach it from the spot of lime on the cathode, C. You see that the pencil of rays is deflected from the plate and pursues a curved path whose distance from the plate I can increase or diminish by increasing or diminishing the negative charge on the plate.

COEPUSCLES IN VACUUM TUBES.

7

We have seen that the cathode rays behave under every
test that we have api^Ued as if they are negatively electrified particles ; we have seen that they carry a negative charge of electricity and are deflected by electric and magnetic forces just as negatively electrified particles would be.
Hertz showed, however, that the cathode particles possess another property which seemed inconsistent with the idea that they are particles of matter, for he found that they were able to penetrate very thin sheets of metal, for example, pieces of gold-leaf placed between them and the glass, and produce appreciable luminosity on the glass after doing so. The idea of particles as large as the molecules of a gas passing through a solid plate was a somewhat startling

riG. 4.

— one in an age which knew not radium which does project
particles of this size through jjieces of metal much thicker
— than gold-leaf and this led me to investigate more closely

the nature of the j)articles which form the cathode rays.
The principle of the method used is as follows : When a

particle carrying a charge e is moving with the velocity v

across the lines of force in a magnetic field, placed so that

the lines of magnetic force are at right angles to the motion

H of the particle, then if

is the magnetic force, the

moving particle will be acted on by a force equal to He r.

This force acts in the direction which is at right angles to

the magnetic force and to the direction of motion of the

particle, so that if the jJarticle is moving horizontally as in

the figure and the magnetic force is at right angles to the

plane of the paper and towards the reader, then the negatively

8 THE COEPUSCULAE THEOEY OF MATTEE.

electrified particle will be acted on by a vertical and upward force. The pencil of rays will therefore be deflected upwards

and with it the patch of green phosphorescence where it strikes the walls of the tube. Let now the two parallel plates
B A and (Fig. 2) between which the pencil of rays is moving
be charged with electricity so that the upper plate is nega-

tively and the lower plate positively electrified, the cathode rays will be repelled from the upper plate with a force Xe

where A' is the electric force between the plates. Thus, if the

plates are charged when the magnetic field is acting on the

rays, the magnetic force will tend to send the rays upwards,

while the charge on the plates will tend to send them down-

We wards.

can adjust the electric and magnetic forces

until they balance and the pencil of rays passes horizon-

tally in a straight line between the plates, the green patch

of phosphorescence being undisturbed. "When this is the

case, the force He v due to the magnetic field is equal to
— — Xe the force due to the electric field and we have

He V = Xe

ov v= -X

Thus, if we measure, as we can without difficulty, the
X H values of and when the rays are not deflected, we can
determine the value of r, the velocity of the particles. The velocity of the rays found in this way is very great ; it
varies largely with the pressure of the gas left in the tube.
In a very highly exhausted tube it may be 1/3 the velocity of
light or about 60,000 miles per second ; in tubes not so
highly exhausted it may not be more than 5,000 miles per second, but in all cases when the cathode rays are produced in tubes their velocity is much greater than the velocity of
any other moving body with which we are acquainted. It
is, for example, many thousand times the average velocity
with which the molecules of hydrogen are moving at
ordinary temperatures, or indeed at any temperature yet
realised.

COEPUSCLES IN VACUUM TUBES.

9

Determination of e/vH.

Having found the velocity of the rays, let us in the pre-

ceding experiment take away the magnetic force and leave

the rays to the action of the electric force alone. Then the

particles forming the rays are acted upon by a constant

vertical downward force and the problem is practically that

of a bullet projected horizontally with a velocity v and fall-

We ing under gravity.

know that in time t the body will

fall a depth equal to ^ g t"^ where g is the vertical acceleration ; in our case the vertical acceleration is equal to A' e/m
m where is the mass of the particle, the time it is falling

is l/v where I is the length of path measured horizontally,

and V the velocity of projection. Thus, the depth the

particle has fallen when it reaches the glass, i.e., the down-

ward displacement of the patch of phosphorescence where

the rays strike the glass, is equal to

1 Xe l^

m 2"

v^

We can easily measure d the distance the phosphorescent
X patch is lowered, and as we have found v and and I are
easily measured, we can find ejiii from the equation :

X m

e-

The results of the determinations of the values of ejm

made by this method are very interesting, for it is found

that however the cathode rays are produced we always

get the same value of ejm for all the particles in the

We rays.

may, for example, by altering the shape of the

discharge tube and the pressure of the gas in the tube, pro-

duce great changes in the velocity of the particles, but unless

the velocity of the jparticles becomes so great that they are

moving nearly as fast as light, when, as we shall see, other

considerations have to be taken into account, the value of

ejm is constant. The value of ejin is not merely inde-

pendent of the velocity. What is even more remarkable is

that it is independent of the kind of electrodes we use and

;
10 THE COEPUSCULAE THEOEY OF MATTEE.
also of the kind of gas in the tube. The particles which form the cathode rays must come either from the gas in the tube or from the electrodes ; we may, however, use any kind of substance we please for the electrodes and fill the tube with gas of any kind, and yet the value of ejin will remain unaltered.
This constant value is, when we measure e/m in the
C. G. S. system of magnetic units, equal to about 1"7 x 10''. If we compare this with the value of the ratio of the mass to the charge of electricity carried by any system previously
known, we find that it is of quite a different order of magni-
tude. Before the cathode rays were investigated the charged
atom of hydrogen met with in the electrolysis of liquids was the system which had the greatest known value for ejm, and in this case the value is only 10*; hence for the
corpuscle in the cathode rays the value of e/in is 1,700 times the value of the corresponding quantity for the charged
hydrogen atom. This discrepancy must arise in one or other of two ways, either the mass of the corpuscle must be very small compared with that of the atom of hydrogen, which until quite recently was the smallest mass recognised in physics, or else the charge on the corpuscle must be very
much greater than that on the hydrogen atom. Now it has
been shown by a method which I shall shortly describe that the electric charge is practically the same in the two cases hence we are driven to the conclusion that the mass of the corpuscle is only about 1/1700 of that of the hydrogen atom. Thus the atom is not the ultimate limit to the sub-
division of matter ; we may go further and get to the
corpuscle, and at this stage the corpuscle is the same from
whatever source it may be derived.
COHPUSCLES VERY WIDELY DISTRIBUTED.
It is not only from what may be regarded as a somewhat
artificial and sophisticated source, viz., cathode rays, that we can obtain corpuscles. "When once they had been discovered it was found that they were of very general occurrence. They are given out by metals when raised to

CORPUSCLES IN VACUUM TUBES.

11

a red heat : you have already seen what a copious supply is given out by hot lime. Any substance when heated gives out corpuscles to some extent; indeed, we can detect the emission of them from some substances, such as rubidium and the alloy of sodium and potassium, even when they are cold; and it is perhaps allowable to suppose that there is some emission by all substances, though our instruments
are not at present sufficiently delicate to detect it unless it is unusually large.
Corpuscles are also given out by metals and other bodies, but esjjecially by the alkali metals, when these are exposed to light. They are being continually given out in large quantities, and with very great velocities by radio-active substances such as uranium and radium ; they are pro-
duced in large quantities when salts are put into flames,
and there is good reason to suppose that corpuscles reach us from the sun.
The corpuscle is thus very widely distributed, but whereever it is found it preserves its individuality, e/iii being
always equal to a certain constant value. The corpuscle appears to form a part of all kinds of
matter under the most diverse conditions ; it seems natural, therefore, to regard it as one of the bricks of which atoms
are built up.

Magnitude of the Electric Charge carried by the
Corpuscle.

I shall now return to the proof that the very large value

of ejin for the corpuscle as compared with that for the atom

of hydrogen is due to the smahness of m the mass, and not

We to the greatness of e the charge.

can do this by

actually measuring the value of e, availing ourselves for

this purpose of a discovery by C. T. E. Wilson, that a

charged jDarticle acts as a nucleus round which water

vapour condenses, and forms drops of water. If we have air

saturated with water vapour and cool it so that it would be

supersaturated if there were no deposition of moisture, we

know that if any dust is present, the particles of dust act

12 THE COEPUSCULAR THEORY OF MATTER.
as nuclei round which the water condenses and we get the too famihar phenomena of fog and rain. If the air is quite dust-free we can, however, cool it very considerably without any deposition of moisture taking place. If there is no dust, C. T. E. Wilson has shown that the cloud does not form until the temperature has been lowered to such a point that the supersaturation is about eightfold. When,
however, this temperature is reached, a thick fog forms,
even in dust-free air. When charged particles are present
FIG. O.
in the gas, Wilson showed tbat a much smaller amount of
cooling is sufficient to produce the fog, a fourfold super-
saturation being all that is required when the charged particles are those which occur in a gas when it is in the state in which it conducts electricity. Each of the charged particles becomes the centre round which a drop of water forms ; the drops form a cloud, and thus the charged particles, however small to begin with, now become visible and can be observed. The effect of the charged particles on the formation of a cloud can be shown very distinctly by the

COEPUSCLES IN VACUUM TUBES.

13

following experiment. The vessel A, which is in contact

with water, is saturated with moisture at the temperature

of the room. This vessel is in communication with B, a

cylinder in which a large piston, C, slides up and down the ;
piston, to begin with, is at the top of its travel ; then by

suddenly exhausting the air from below the piston, the

pressure of the air above it will force it down with great

A rapidity, and the air in the vessel

will expand very

quickly. When, however, air expands it gets cool ; thus the
air in A gets colder, and as it was saturated with moisture

before cooling, it is now supersaturated. If there is no

dust present, no deposition of moisture will take place
A unless the air in is cooled to such a low temperature that

the amount of moisture required to saturate it is only

about 1/8 of that actually present. Now the amount of

cooling, and therefore of supersataration, depends upon the

travel of the piston ; the greater the travel the greater the

cooling. I can regulate this travel so that the super-

saturation is less than eightfold, and greater than four-

We fold.

now free the air from dust by forming cloud after

cloud in the dusty air, as the clouds fall they carry the

dust down with them, just as in nature the air is cleared by

showers. We find at last that when we make the expansion

We no cloud is visible.

now put the gas in a conducting

A state by bringing a little radium near the vessel ; this fills
the gas with large quantities of both positively and nega-
tively electrified particles. On making the expansion now,

an exceedingly dense cloud is formed. That this is due to

the electrification in the gas can be shown by the following

experiment: Along the inside walls of the vessel A we have two

vertical insulated plates which can be electrified; if these

plates are electrified they will drag the charged particles out

of the gas as fast as they are formed, so that by electrifying

the plates we can get rid of, or at any rate largely reduce,

the number of electrified particles in the gas. I now repeat

the experiment, electrifying the plates before bringing up

the radium. You see that the presence of the radium hardly

increases the small amount of cloud. I now discharge the

—

14 THE COEPUSCULAE THEOEY OP MATTEE.

plates, and on making the expansion the clond is so dense
as to be quite opaque.
We can use the drops to find the charge on the particles,
for when we know the travel of the piston we can deduce the amount of supersaturation, and hence the amount of water deposited when the cloud forms. The water is deposited in the form of a number of small drops all of the same size ; thus the number of drops will be the volume of the water deposited divided by the volume of one of the drops. Hence, if we find the volume of one of the drops we can find the number of drops which are formed round
the charged particles. If the particles are not too numerous, each will have a drop round it, and we can thus find the number of electrified particles.
If we observe the rate at which the drops slowly fall down we can determine the size of the drops. In consequence of
the viscosity or friction of the air small bodies do not fall with a constantly accelerated velocity, but soon reach a speed which remains tiniform for the rest of the fall ; the smaller the body the slower this speed, and Sir George Stokes has shown that v, the speed at which a drop of rain falls, is given by the formula

2 g a-

~ ^

9 H-

where a is the radius of the drop, g the acceleration due to gravity, and /a the co-efiicient of viscosity of the air. If we substitute the values of g and fx., we get

V = 1-28 X 10^ a^

Hence, if we measure v we can determine a, the radius of

We the drop.

can, in this way, find the volume of a drop,

and may therefore, as explained above, calculate the number

of drops, and therefore the number of electrified particles.

It is a simple matter to find, by electrical methods, the total

quantity of electricity on these particles; and hence, as we

know the number of particles, we can deduce at once the

charge on each particle.

—

COEPUSCLES IN VACUUM TUBES.

15

This was the method by which I first determined the charge on the particle. H. A. AVilson has since used a simpler method founded on the following principles. C. T. E. Wilson has shown that the drops of water condense more easily on negatively electrified particles than on positively electrified ones. Thus, by adjusting the expansion, it is possible to get drops of water round the negative f)articles and not round the positive ; with this expansion, therefore, all the drops are negatively electrified. The size of these drops, and therefore their weight, can, as before, be determined by measuring the speed at which they fall under gravity. Suppose now, that we hold above the drops
a positively electrified body, then since the drops are negatively electrified they will be attracted towards the
positive electricity and thus the downward force on the drops will be diminished, and they will not fall so rapidly as they did when free from electrical attraction. If we adjust the electrical attraction so that the upward force on
each drop is equal to the weight of the drojJ, the drojps will not fall at all, but will, like Mahomet's coffin, remain sus-
pended between heaven and earth. If, then, we adjust the electrical force until the drops are in equilibrium and neither fall nor rise, we know that the ujDward force on the drop is equal to the weight of the drop, which we have already determined by measuring the rate of fall when the drop was not exposed to any electrical force. If Xis the electrical force, e the charge on the drop, and iv its weight, we have, when there is equilibrium

X = e

IV.

X Since can easily be measured, and iv is known, we can
use this relation to determine e, the charge on the drop.
The value of e found by these methods is 3"1 X 10"^° electro-
static units, or 10"^" electromagnetic units. This value is
the same as that of the charge carried by a hydrogen atom in the electrolysis of dilute solutions, an approximate value of which has long been known.
It might be objected that the charge measured in the

16 THE COEPUSCULAR THEOEY OF MATTEE.

preceding experiments is the charge on a naolecule or

collection of molecules of the gas, and not the charge on

a corpuscle. This objection does not, however, apply to

another form in -which I tried the experiment, where the

charges on the particles were got, not by exposing the gas

to the effects of radium, but by allowing ultra-violet light to

fall on a metal plate in contact with the gas. In this case,
as experiments made in a very high vacuum show, the

electrification which is entirely negative escapes from the
metal in the form of corpuscles. When a gas is present*

the corpuscles strike against the molecules of the gas and

stick to them. Thus, though it is the molecules which are

charged, the charge on a molecule is equal to the charge on

a corpuscle, and when we determine the charge on the

molecules by the methods I have just described, we deter-

mine the charge carried by the corpuscle. The value of the

charge when the electrification is produced by ultra-violet

light is the same as when the electrification is produced by

radium.

,

We have just seen that e, the charge on the corpuscle, is

in electromagnetic units, equal to lO"^, and we have pre-
m viously found that elm., being the mass of a corpuscle, is
= equal to 1"7 X 10^, hence 7h 6 X 10"^^ grammes.
We can realise more easily wBat this means if we express

the mass of the corpuscle in terms of the mass of the atom

of hydrogen. We have seen that for the corpuscle

— e/?)i

Vl

X

10'' ;

while

if

25

is

the

charge carried

by

an

atom of hydrogen in the electrolysis of dilute solutions, and

M = the mass of the hydrogen atom, E\M

10*; hence

= We e\tn

1700 Fj\M.

have already stated that the

value of e found by the preceding methods agrees well

with the value of H, which has long been approximately

known. Townsend has used a method in which the value

of e/-E is directly measured and has showed in this way also
= that e is equal to -E ; hence, since elm 1700 EIM, we have
M = 1700 )/(, i.e., the mass of a corpuscle is only about

1/1700 j)art of the mass of the hydrogen atom.

In all known cases in which negative electricity occurs in

CORPUSCLES IN VACUUM TUBES.

17

gases at very low pressures it occurs in the form of corpuscles, small bodies with an invariable charge and mass. The case is entirely different with positive electricity.

The Caekiers of Positive Elbctbicity.
We get examples of positively charged particles in various
phenomena. One of the first cases to be investigated was that of the " Canalstrahlen " discovered by Goldstein. I have here a highly exhausted tube with a cathode, through
which a large number of holes has been bored. When I
send a discharge through this tube you will see the cathode rays shooting out in front of the cathode. In addition to these, you see other rays streaming through the holes in the cathode, and travelling through the gas at the back of

J
^yK.
FIG. 6.

the cathode. These are called " Canalstrahlen." You notice that, like the cathode rays, they make the gas luminous as

they pass through it, but the colour of the luminosity due to the canalstrahlen is not the same as that due to the cathode rays. The distinction is exceptionally well marked

in helium, where the luminosity due to the canalstrahlen is tawny, and that due to the cathode rays bluish. The
luminosity, too, produced when the rays strike against a

solid is also of quite a different character. This is well
shown by allowing both cathode rays and canalstrahlen to strike against lithium chloride. Under the cathode rays

the salt gives out a steely blue light, and the spectrum is a

continuous one ; under the canalstrahlen the salt gives out

a brilliant red light, and the spectrum shows the lithium

line. It is a very interesting fact that the lines in the
spectra of the alkali metals are very much more easily

T.M.

c

18 THE COEPUSCULAE THEOEY OF MATTEE.
obtained when the canalstrahlen fall on salts of the metal than when they fall on the metal itself. Thus when a pool of the liquid alloy of sodium and potassium is bombarded by canalstrahlen the specks of oxide on the surface shine with
a bright yellow light, while the untarnished part of the surface is quite dark.
The canalstrahlen are deflected by a magnet, though not to anything like the same extent as the cathode rays. Their deflection, too, is in the opposite direction, showing that
they are positively charged.
Value of e/m foe the Particles in the Canalstrahlen.
W. Wien has applied the methods described in connection
with the cathode rays to determine the value of e/vi for the particles in the canalstrahlen. The contrast between the results obtained for the two rays is very interesting. In the case of the cathode rays the velocity of different rays
in the same tube may be different, but the value of e/m for
these rays is independent of the velocity as well as of the nature of the gas and the electrodes. In the case of the canalstrahlen we get in the same pencil of rays not merely variations in the velocity, but also variations in the value
of e/m. The difference between the values of e/m for the cathode rays and the canalstrahlen is also very remarkable. For the cathode rays e/m always equal to l"7XlO^; while
for canalstrahlen the greatest value ever observed is 10*, which is also the value of e/m for the hydrogen ions in the
electrolysis of dilute solutions. When the canalstrahlen
pass through hydrogen the value of e/m for a large portion of the rays is 10*. There are, however, some rays present
even in hydrogen, for which e/m is much less than 10*, and
which are but slightly deflected even by very intense
magnetic fields. When the canalstrahlen pass through
very pure oxygen, Wien found that the value of e/m for the most conspicuous rays was about 750, which is not far from what it would be if the charge were the same as for the canalstrahlen in hydrogen, while the mass was greater in the proportion of the mass of an atom of oxygen to that

CORPUSCLES IN VACUUM TUBES.

19

of an atom of hydrogen. Along with these rays in oxygen there were others having still smaller values ol^ejin, and some having ejm equal to 10*.
As the canalstrahlen or rays of positive electricity are a very promising field for investigations on the nature of positive electricity, I have recently made a series of experiments on these rays in different gases, measuring the deflections they experience when exposed to electric and magnetic forces and thus deducing the values of «/m and V. I find, when the pressure of the gas is not too low.

FIG. 7.
The portions with the cross shading is the deflection under both electric and magnetic force ; the portion with vertical shading the deflection under magnetic force ; that with the horizontal shading under electric force alone.
that the bright spot produced by the impact of these rays on the phosphorescent screen is deflected by electric and "magnetic forces into a continuous band extending, as shown in Fig. 7, on both sides of the undeflected portion,
the portion on one side {cc) is very much fainter than that on the other, and also somewhat shorter. The direction
of the deflection of the band cc shows that it is produced by particles charged with negative electricity, while the brighter band hh is due to particles charged with positive electricity. The negatively charged particles which produce the band cc are not corpuscles, for from the deflections in this band we can find the value of ej'm ; as this value
c2

'
20 THE COEPUSCULAR THEOEY OF MATTEE.
comes out of the order 10*, we see thpt the mass of the carrier is comj)arable with that of an atom, and therefore
immensely greater than that of a corpuscle. When the
pressure is very low the portion of the phosphorescence deflected in the negative direction disappears and the phosphorescent spot, instead of being stretched by the electric and magnetic forces into a continuous band, is broken up into two patches, as in the curved parts of Figs. 8 and 9. Fig. 8 is the appearance at exceedingly low pressures. Fig. 9 that at a somewhat higher jpressure. For one of these patches
the maximum value of e/m is about 10*, and for the other about 5 X 10^. The appearance of the patches and the values
of e/m at these very low jDressures are the same whether

o
oo
FIG. 8.
The curved patches represent the deflection under both electric and magnetic force.

FIG. 9.

the tube is filled originally with air, hydrogen, or helium.

Another experiment I tried was to exhaust the tube until

the pressure was too low for the discharge to pass, and then

to introduce into the tube a very small quantity of gas, this

increases the pressure and the discharge is able to pass

through the tube. The following gases were admitted into

the tube : air, carbonic oxide, oxygen, hydrogen, helium,

argon and neon, but whatever the gas the appearance of the

phosphorescence was the same. In every case there were

= = two 23atches, one having e/m

10*, the other ejm

X 5

lO''.

At these very low pressures the intensity of the electric

field in the discharge tube is very great.
When the 23ressure in the tube is not very low the nature

of the positive rays depends to a very considerable extent

—

COEPUSCLES IN VACUUM TUBES.

21

upon the kind of gas with which the tube is filled. Thus,
for example, in air at these pressures the phosphorescent spot is stretched out into a straight band as in Fig. 7 ; the
maximum value of e/m for this band is 10*. In hydrogen
at suitable pressures we get the spot stretched out into two
bands as in Fig. 10 ; for one of these bands the maximum

FIG. 10.
X value of e/m is 10*, while for the other it is 5 10^. In
helium we also get two bands as in Fig. 11, but while the
maximum value of e/m in one of these bands is 10*, the
same as for the corresponding band in hydrogen, the
maximum value of e/ni in the other band is only 2'5 X 10^. We see from this that the ratio of the masses of the carriers
FIG. 11.
in the two bands is equal to the ratio of the masses of the atoms of hydrogen and helium.
At some pressures we get three bands in helium, the
value of e/m being respectively 10*, 5 X 10^, and 2'5 X 10'^
The continuous band into which the bright phosphorescent spot is stretched out when the pressure is not exceedingly low can be explained as follows :
The rays on their way to the screen have to pass through gas which is ionised by the passage through it of

22 THE COEPUSCULAE THEOEY OF MATTEE.
the rays ; this gas will therefore contain free corpuscles.
The particles which constitute the rays start with a charge of positive electricity ; some of these in their journey
through the gas may attract a corpuscle, the negative
charge on which will neutralise the positive charge on the
particle. The particles when in this neutral state may be
ionised by collision and reacquire a positive charge, or by
attracting another corpuscle they may become negatively charged, and this process may be repeated several times in
their journey to the screen. Thus, some of the particles, instead of being positively charged for the whole of the
time they are exposed to electric and magnetic forces, may
be for a part of that time without a charge or even have a
negative charge. Now the deflection of a particle will be
proportional to the average value of its charge while under the action of electric and magnetic forces ; if the particle is without charge for a 23art of the time, its deflection will be less than that of a jDarticle which has retained its positive charge for the whole of the journey, while the
small number of particles, which have a negative charge
for a longer time than they have a positive, will be deflected in the opposite direction and produce the faint tail of phosphorescence which is deflected in the opposite direction to the main portion.
It is remarkable and suggestive that even when great care is taken to eliminate hydrogen from the tube, we get at all pressures a large quantity of rays for which e/m is equal to IC, the value for the hydrogen atom ; and in
manj^ cases this is the only definite value of ejin to be
observed, for the continuous band in which we have all values of e/in is due, as we have seen, not to changes in m, but to changes in the average value of e.
If the presence of rays for which e/m = 10"^ was entirely
due to hydrogen present as an impurity in the gas with which the tube is filled, the positive particles being hydrogen ionised by the corpuscles projected from the cathode, we should have expected, since the ionisation consists in the detachment of a corpuscle from the molecule, that the

COEPUSCLES IN VACUUM TUBES.

23

positively charged particles would be molecules and n-ot atoms of hydrogen.
Again, at very low pressures, when the electric field is very intense, we get the same two types of carriers whatever kind of gas is in the tube. For one of these types
e/m = 10* and for the other ejm = 5 X 10^ ; the second value
corresponds to the positive particles which are given out by radio-active substances. The most obvious interjjretation of this result is that under the conditions existing in the discharge tube at these very low |)ressures all gases give off positive particles which resemble corpuscles, in so far as they are indej)endent of the nature of the gas from which they are derived, but which difi'er from the corpuscles in having masses comjDarable with the mass of an atom of hydrogen, while the mass of a corpuscle is only 1/1700 of this mass. One type of positive jDarticle has a mass equal to that of an atom of hydrogen, the other type has a mass double this ; and the experiments I have just described indicate that when the |3ressure is very low and the electric field very intense, all the positively electrified particles are of one or other of these types.
We have seen that for the positively charged particles
in the canalstrahlen the value of ejm dejpends, when the pressure is not too low, on the kind of gas in the tube, and
is such that the least value of vi is comparable with the. mass of an atom of hydrogen, and is thus always immensely greater than the carriers of the negative charge in the
cathode rays. We know of no case where the mass of the
positively charged particle is less than that of an atom of hydrogen.
Positive Ions from Hot Wires.
When a metallic wire is raised to a red heat it gives out
positively electrified particles. I have investigated the values of e/?ft for these particles, and find that they show the same peculiarities as the positively charged particles in the canalstrahlen. The particles given off by the wire are
not all alike. Some have one value of ejm, others another,

;

24 THE COEPUSCULAE THEOEY OF MATTEE.

but the greatest value I found in my experiments where
the wire was surrounded by air at a low pressure was 720,
and there were many particles for which ejm was very much smaller, and which were hardly affected even by very
strong magnetic fields.

Positive Ions feom Eadio-activb Substances.

The various radio-active substances, such as radium,

polonium, uranium, and actinium, shoot out with^great
S velocity positively electrified particles which are called rays.

The values of ejm for these particles have been measured by

Eutherford, Des Coudres, Mackenzie, and Huff, and for all
— the substances hitherto examined radium and its trans— formation products, polonium, and actinium the value of

e/m is the same and equal to 5 X 10^ the same as for one

type of ray in the vacuum tube. The velocity with which

the particles move varies considerably from one substance

to another. As these substances all give off helium, there

is prima jacie evidence that the a particles are helium.

For a helium atom with a single charge, e\m is 2"5 X 10^,

hence if the a particles are helium atoms they must carry ,
a double charge ; the large value of e\m shows that the

carriers of the positive charge must be atoms, or molecules

of some substance with a small atomic weight. Hydrogen

and helium are the only substances with an atomic weight

small enough to be compatible with so large a value of e\m

as 5,000, and of these, helium is known to be given off

by radio-active substances, whereas we have as yet no

evidence that there is any evolution of hydrogen.

= Positive particles having e/»i

5 X 10^ are found,

as we have seen, in all vacuum tubes carrying an electric

discharge when the pressure in the tube is very low

the velocity of these particles is very much less than

that of the a particles. From the researches of

Bragg, Kleeman, and Eutherford, it ajDpears that the a

particles lose their power of ionisation and of producing

phosphorescence when their velocity is reduced by passing

through absorbing substances to about 10' cm/sec. The

COEPUSCLES IN VACUUM TUBES.

25

interesting point about this result is that the positively electrified particles in a discharge tube can produce ionisa-
tion and phosphorescence when their velocity is very much
smaller than this.
This may possibly be due to the « particles being much fewer in number than the positively charged particles in a
discharge tube ; and that as the a particles are so few and far between, a particle in its attempts at ionisation or at producing phosphorescence receives no assistance from its companions. Thus, if ionisation or phosphorescence requires a certain amount of energy to be communicated to a system, all that energy has to come from one particle. When, however, as in a discharge tube, the stream of
particles is much more concentrated, the energy required by the system may be derived from more than one jsarticle,
the energy given to the system by one particle not having been entirely lost before additional energy is supplied by another particle. Thus the effects produced by the particles might be cumulative and the system might ultimately receive the required amount of energy by contributions from several particles. Thus, although the contribution from any one particle might be insufficient to produce ionisation or phosphorescence, the cumulative effects of several might
be able to do so.
Another way in which the sudden loss of ionising power might occur is that the power of producing ionisation may be dependent on the possession of an electric charge by the particle, and that when the velocity of the particle falls below a certain value, the particle is no longer able to escape from a negatively charged corj)uscle when it passes
close to it, but retains the corpuscle as a kind of satellite, the two forming an electrically neutral system, and that inasmuch as the chance of ionisation by collision diminishes
as the velocity increases, when the velocity exceeds a certain
value, such a neutral system is not so likely to be ionised
and again acquire a charge of electricity as the more slowly moving j)articles in a discharge tube.
These investigations on the properties of the carriers of

;

26 THE COEPUSCULAE THEOEY OF MATTEE.

positive electricity prove: (1) that whereas in gases at very

low pressures the carriers of negative electricity have an ex-

ceedingly small mass, oiily about 1/1700 of that of the

hydrogen atom, the mass of the carriers of positive elec-

tricity is never less than that of the hydrogen atom

(2) that while the carrier of negative electricity, the cor-
puscle, has the same mass from whatever source it may be derived, the mass of the carrier of the positive charge may

be variable : thus in hydrogen the smallest of the positive

particles seems to be the hydrogen atom, while in helium,

at not too low a pressure, the carrier of the positive electricity

is partly, at any rate, the helium atom. All the evidence

at our disposal shows that even in gases at the lowest pres-

sures the positive electricity is always carried by bodies at

least as large as atoms ; the negative electricity, on the other

hand, is under the same circumstances carried by corpuscles,

bodies with a constant and exceedingly small mass.

The simplest interpretation of these results is that the

positive ions are the atoms or groups of atoms of various

elements from which one or more corpuscles have been

removed. That, in fact, the corpuscles are the vehicles by

which electricity is carried from one body to another, a

positively electrified body differing from the same body

when unelectrified in having lost some of its corpuscles while

the negative electrified body is one with more corpuscles

than the unelectrified one.

In the old one-fluid theory of electricity, positive or

negative electrification was due to an excess or deficiency

of an " electric fluid." On the view we are considering

positive or negative electrification is due to a defect or

excess in the number of corpuscles. The two views have

much in common if we

suppose

that the

" electric

" fluid

is built up of corpuscles.

In the corpuscular theory of matter we suppose that the

atoms of the elements are made uj) of positive and negative

electricity, the negative electricity occurring in the form of
corpuscles. In an unelectrified atom there are as many units

of positive electricity as there are of negative ; an atom with

COEPUSCLES IN VACUUM TUBES.

27

a unit positive charge is a neutral atom which has lost one corpuscle, while an atom with a unit negative charge is a neutral atom to which an additional corpuscle has been attached. No positively electrified body has yet been found
with a mass less than that of a hydrogen atom. We cannot,
however, without further investigation infer from this that the mass of the unit charge of positive electricity is equal
to the mass of the hydrogen atom, for all we know about the
electrified system is, that the positive electricity is in excess by one unit over the negative electricity ; any system containing n units of positive electricity and {u - 1) corpuscles would satisfy this condition whatever might be the 'value of n. Before we can deduce any conclusions as to the mass
of the unit of positive electricity we must know something
about the number of corpuscles in the system. We shall
give, later on, methods by which we can obtain this information we may, however, state here that these methods
;
indicate that the number of corpuscles in an atom of any
element is proportional to the atomic weight of the element
—it is a multiple, and not a large one, of the atomic weight of
the element. If this result is right, there cannot be a large
number of corpuscles and therefore of units of positive electricity in an atom of hydrogen, and as the mass of a corpuscle is very small compared with that of an atom of hydrogen, it follows that only a small fraction of the mass of the atom can be due to the corpuscle. The bulk of the mass must be due to the positive electricity, and therefore the mass of unit positive charge must be large compared
— 'with that of the corjDuscle the unit negative charge.
From the experiments described on p. 19 we conclude that positive electricity is made up of units, which are inde-
pendent of the nature of the substance which is the seat of
the electrification.

CHAPTEE II.

THE ORIGIN OF THE MASS OF THE CORPUSCLE.

The origin of the mass of the corpuscle is very interesting,

for it has been shown that this mass arises entirely from

We the charge of electricity on the corpuscle.

can see

how this comes about in the following way. If I take
M an uncharged body of mass at rest and set it moving

with the velocity V, the work I shall have to do on

the body is equal to the kinetic energy it has acquired,

i.e., to ^ MV^. If, however, the body is charged with

electricity I shall have to do more work to set it moving

with the same velocity, for a moving charged body pro-

duces magnetic force, it is surrounded by a magnetic field

and this field contains energy; thus when I set the body

in motion I have to supply the energy for this magnetic as

well as for the kinetic energy of the body. If the charged

body is moving along the line OX, the magnetic force at a

P POX point is at right angles to the plane

; thus the lines

OX of magnetic forces are circles having

for their axis. The

P magnitude of the force at is equal to ^ ^ ^2"' where 6

denotes the angle POX. Now in a magnetic field the energy
per unit volume at any place where the magnetic force is

OEIGIN OF THE MASS OF THE CORPUSCLE. 29

H equal to is H'^/Stt. Thus the energy per unit volume at
P arising from the magnetic force produced by the moving

charge is

/ ," , and by taking the sum of the

energy throughout the volume surrounding the charge, we

find the amount of energy in the magnetic field. If the

moving body is a conducting sphere of radius a, a simple

calculation shows that the energy in the magnetic field is

1 g2 T72

equal to

. The energy which has to be supplied to

set the sphere in motion is this energy plus the kinetic energy of the sphere, i.e., it is equal to

'A

3a

or

A 1
+ TT

[/ m

2V

, '

-2 —] V~
3 «/

Thus the energy is the same as if it were the kinetic
— 2 e^
energy of a sphere with a mass ni -\- - instead of m.
o ct
Thus the apparent mass of the electrified body is not in but
m — 4- - . The seat of this increase in mass is not in the
3a electrified body itself but in the space around it, just as if
the ether in that space were set in motion by the passage through it of the lines of force proceeding from the charged body, and that the increase in the mass of the charged body arose from the mass of the ether set in motion by the
lines of electric force. It may make the consideration of
this increase in mass clearer if we take a case which is not electrical but in which an increase in the apjDarent mass occurs from causes which are easily understood. Suppose
M that we start a sphere of mass with a velocity V in a
vacuum, the work which has to be done on the sphere is
M I V^. Let us now immerse the sphere in water : the
work required to start the sphere with the same velocity will evidently be greater than when it was in the vacuum, for the motion of the sphere will set the water around it in

30 THE COEPUSCULAE THEOEY OF MATTEE.

motion. The water will have kinetic energy, and this, as

well as the kinetic energy of the sphere, has to be supplied

when the sphere is moved. It has been shown by Sir

George Stokes that the energy in the water is equal to

^ Ml F^ where Mi is the mass of half the volume of the

water displaced by the sphere. Thus the energy required

+ to start the sphere is ^ (M
M + behaves as if its mass were

Mi) F^ and the sphere
Mi and not M, and for

many purposes we could neglect the effect of the water if

we supposed the mass of the sphere to be increased in the

way indicated. If we suppose the lines of electric force

proceeding from the charged body to set the ether in

motion and assume the ether has mass, then the origin of the

increase of mass arising from electrification would be very

analogous to the case just considered. The increase in

mass due to the charge is -

— ;

thus for

a given charge

the intrease in the mass is greater for a small body than for
a large one. Now for bodies of ordinary size this increase
of mass due to electrification is for any realisable charges quite insignificant in com23arison with the ordinary mass. But since this addition to the mass increases rapidly as the body gets smaller, the question arises, whether in the case of these charged and exceedingly small corpuscles
the electrical mass, as we may call it, may not be quite
appreciable in comparison with the other (mechanical) mass.
We shall now show that this is the case ; indeed for
corpuscles there is no other mass : all the mass is
electrical.
The method by which this result has been arrived at is as follows : The distribution of magnetic force near a moving electrified particle depends upon the velocity of the particle, and when the velocity approaches that of light, is' of quite a different character from that near a slowly moving particle. Perhaps the clearest way of seeing this is to follow the changes which occur in the distribution of the electric force round a charged body as its velocity is
gradually increased. When the body is at rest the electric

OEIGIN OF THE MASS OF THE COEPUSCLE. 31

force is uniformly distributed round the body, i.e., as long

as we keep at the same distance from the charged body

the electric force remains the same whether we are to the

east, west,

north

or

south

of

the

particle ;

the lines

of

force

which come from the body spread out uniformly in all

directions. When the body is moving this is no longer the

OA case, for if the body is moving along the line

(Fig. 13),

the lines of electric force tend to leave the regions in the
neighbourhood of OA and OB, which we shall call the

23olar regions, and crowd towards a plane drawn through
O at right angles to OA ; the regions in the neighbourhood

riG. 13
of this plane we shall call the equatorial regions. This crowding of the lines of force is exceedingly slight when the velocity of the body is only a small fraction of that of light, but it becomes very marked when the velocity of the body is nearly equal to that velocity ; and when the body moves at the same speed as light all the lines of force leave the
region round OA and crowd into the plane through at
right angles to OA, i.e., the lines of force have swung round until they are all at right angles to the direction in which the particle is moving. The effect of this crowding of the lines of force towards the equatorial j^lane is to weaken the magnetic force in the polar and increase it in the equatorial

;

32 THE COEPUSCULAE THEOEY OF MATTEE.

regions. The polar regions are those where the magnetic force was originally weak, the equatorial regions those where it was strong. Thus the effect of the crowding is
to increase relatively the strength of the field in the strong
parts of the field and to weaken it in the weak parts. This makes the energy in the field greater than if there were no

— crowding, in which case the energy is

where e is

»

3a

the charge, v the velocity and a the radius of the sphere.

When we allow for the crowding, the energy will be

1— a

e^

V'
-, where a is

a

quantity which will be equal to

3

a

unity when v is small compared with c the velocity of

light, but becomes very large when v approaches c. The

part of the mass arising

from

the

charge

—2
is

a

—e^ ,

thus

3a

since a depends upon r — the velocity of the — particle the

electrical mass will dei^end upon v, and thus this part of

the mass has the peculiarity that it is not constant but

depends upon the velocity of the particle. Thus if an

appreciable part of the mass of the corpuscle is electrical

in origin, the mass of rapidly moving corpuscles will be

greater than that of slow ones, while if the mass were in

the main mechanical, it would be independent of the

velocity. Eadium gives out corpuscles which move with

velocities comparable with that of light and which are

therefore very suitable for testing whether or not this

increase in the mass of a corpuscle with its velocity takes

place. This test has been apjplied by Kaufpaann, who has

measured the value of mje for the various corpuscles moving

We with different velocities given out by radium.

can

— calculate the value of the coefficient a the quantity which

exjDresses the effect of the velocity on the mass. The value of this quantity depends to some extent on the view we take as to the distribution of electricity on the corpuscle we get slightly different values according as we suppose

the electricity to be distributed over the surface of a conducting sphere of radius a, or rigidly distributed over the

OKIGIN OF THE MASS OF THE COEPUSCLE. 33
surface of a non-conducting sphere of the same radius, or uniformly distributed throughout the volume of such a sphere. In calculating these differences we have to suppose the charge on the sphere divided up into smaller parts and that each of these small parts obeys the ordinary laws of electrostatics. If we suppose that the charge on the corpuscles is the unit of negative electricity, it is not permissible to assume that smaller portions will obey the
ordinary laws of electrostatic attraction.
Perhaps the simplest assumption we can make is that the energy is the same as that outside a sphere of radius a
moving with the velocity V and with a charge e at its
centre. I have calculated the value of a on this supposition; the results are given in the following Table. The first column of the Table contains the velocity of the corpuscles, which were the object of Kaufmann's experiments ; the second column, the values found by Kaufmann for the ratio of the mass of corpuscles moving with this velocity to the mass of a slowly moving corpuscle, and the third column the value of a calculated on the preceding
hypothesis.
Velocity of Corpuscle.

.

34 THE COEPUSCULAE THEOEY OF MATTEE.

and is not resident in the corpuscle itself ; hence, from our
point of view, each corpuscle may be said to extend through-
out the whole universe, a result which is interesting in connection with the dogma that two bodies cannot occupy
the same space.
From the result that the whole of the mass is electrical
m we are able to deduce the size of the corpuscle, for if
is the mass,
m = —2 —e2 3a
= Now we have seen that e/ni 1'7 X 10^, and that in = electromagnetic measure e 10^°. Substituting these values = we find that a the radius of the corpuscle 10"^^ cm. The
radius of the atom is usually taken as about 10"' cm., hence the radius of a corpuscle is only about the one-hundredthousandth part of the radius of the atom. The potential

energy due to the charge is -

V , if is the velocity of

light; this potential energy is about the same in amount as the kinetic energy possessed by an « particle moving with a velocity about one-fiftieth that of light.

Evidence of the Existence of Corpuscles afforded by THE Zeeman Effect.
The existence of corpuscles is confirmed in a very striking way by the effect produced by a magnetic field on the lines of the spectrum and known as the Zeeman effect. Zeeman found that when the luminous body giving out the spectrum is placed in a strong magnetic
field, many of the lines which are single before the
application of the field are resolved into three or more components. The simplest case is when a line originally single is resolved into three components, the luminous body
being looked at in a direction at right angles to the lines of magnetic force ; the middle line of the three occupies its
old position, and the side lines are separated from it by an amount proportional to the magnetic force. All the lines

ORIGIN OF THE MASS OF THE CORPUSCLE. 35
are plane polarised, the plane of polarisation of the middle line being at right angles to that of the side lines. If the same line is looked at in the direction of the magnetic force, the middle line is absent and the two side lines are circularly polarised in opposite senses.
The theory of this simple case, which was first given by Lorentz, is as follows : Let us assume that the vibrating system giving out the line is a charged body, and that it is vibrating under the action of a force whose magnitude is directly proportional to the distance of the vibrating bo(^ from a fixed f)oint, and whose direction always passes through the point. Suppose that is the fixed point and
P the electrified body, and let us suppose that the latter is m describing a circular orbit round ; let be the mass of

FIG. 14.

the body, fj-OP the force acting upon it ; then the radial

acceleration towards

is equal to r^jOP, v being the

velocity of the body. But the product of the mass and the

radial acceleration is equal to the radial force f-OP, hence

Jg = ,.0P

— If 0) is the angular velocity, v m.OP, hence

V = m o,^

ii or

= .0

/m^

The time of vibration is the time OP takes to make a

complete

revolution

or

Stt/oj thus ;

w,

which

is

called

the

frequency of the vibration, is proportional to the number of

vibrations per second. In this case the frequency of vibra-
tion will evidently be the same whether P goes round in

the direction of the hands of a watch or in the opposite
d2

36 THE COEPUSCULAE THEOEY OP MATTEE.

direction. Let now a magnetic force at right angles to the

plane of the paper and downwards act upon the charged

body. As we have had occasion to remark before, when a

charged body moves in a magnetic field it is acted upon by

a force which is at right angles to its direction of motion

and also to the magnetic force, and equal to Hev sin
H where is the magnetic force, e the charge on the body, v H its velocity, and 6 the angle between the directions of

and V.

Let now the charged particle be describing a circle in the

direction indicated by the arrow round 0, the magnetic

force being at right angles to the plane of the paper and

downwards. The force due to the magnetic field will be radial

and in this case directed inwards, and equal to Hev; hence,

in addition to the radial force i^.OP, we have the force

Hev ; equating the product of the mass and the radial acceleration to the radial force we have

m^ + = iJ^-OP He. V

(1)

= and since v w X OP

.=1^+ ^ ^2

It
m

_^

Hej^
m

or

'Am

Jjt+a
^ m 4,m

thus CO is greater than before, and if /i/m is large compared
with Hejm and equal to u)\ we have

= + CO

He -1
,
(Oq

m 2

approximately ; wg is the frequency without the magnetic

— — 1 He

field, thus the change in the frequency is

, and in

this case it is an increase.

P Suppose, however, that

were describing the circle

in the opposite direction, then, since the direction of

motion is reversed the force produced by the magnetic

field will be reversed and the force will now be outwards

instead of inwards ; thus, instead of equation (1) we have
"^ = l^OP-Hev.

OEIGIN OF THE MASS OF THE COEPUSCLE. 37

and this treated in the same way as equation (1) leads to

the result

=

(J)

(Ufi

1m
2m

Thus the frequency of vibrations in this direction is diminished by an amount equal to that by which the frequency in the opposite direction is increased. Thus the charged body will go round faster in one direction than

FIG. 15.
in the opposite. I have here an experiment to illustrate a
similar effect in a mechanical system. A conical pendulum
has for the bob a flywheel which can be caused to rotate about its axis of rotation. The rotating fly wheel causes a force to act on the bob of the pendulum ; this force is at right angles to the direction of motion of the bob, and is proportional to its velocity. It is thus analogous to the force acting on the charged particle due to the magnetic field. The radial force on the electrified particle

38 THE COEPUSCULAE THEOEY OF MATTEE.

is represented by the component of gravity at right angles

to the axis of the pendulum. I set this pendulum swinging

as a conical pendulum with the fly wheel not in rotation.

As you would naturally suppose, it goes round just as fast in

the direction of the hands of a watch as in the opposite

direction. I now set the fly wheel in rapid rotation and

repeat the experiment. You see that now the pendulum

goes round distinctly more rapidly in one direction than in

the oi^posite, and the direction in which the rotation is most

rapid is that in which the rotation of the pendulum is in the

same direction as that of the flj' wheel.
We see from these considerations that a corpuscle which,

when free from magnetic force, would vibrate with the same

frequency in wliatever direction it might be displaced wOl

no longer do so when placed in a magnetic field. If the

corpuscle is displaced so as to move along the lines of

magnetic force, the force on the corpuscle due to the

magnetic field will vanish, since it is proportional to

the sine of the angle between the magnetic force and the

direction of motion of the particle ; and in this case the

frequency will be the same as without the field. "When,

however, the corpuscle vibrates in the plane at right angles

+ to the lines of magnetic force the frequency will be w

— — — i

if it goes round in one direction, and m ^

if

it goes round in the other. Thus in the magnetic field

the corpuscles will vibrate with the three frequencies o>,

+ — — —; 0)

i

,m

"

III

i
"

m

one of these being the same as

when it was undisturbed. Thus, in the spectroscope

there will be three lines instead of one, the middle line

being in the undisturbed position. If, however, we look at

the corpuscle in the direction of the magnetic force, since

the vibrations corresponding to the undisturbed position of

the lines are those in which the vibrations are along the

lines of magnetic force, and since a vibrating electrified

particle does not send out any light along the line of its

vibration, no light will come from the corpuscle to an eye

OKIGIN OP THE MASS OF THE COEPUSCLE. 39

situated along a line of magnetic force passing through the

corpuscle, so that in this case the central line will be absent,

while the two side lines which correspond to circular orbits

described by the corpuscle in opposite directions will give
rise to circularly polarised light. By finding the sense of

rotation of the light in the line whose frequency is greater

than the undisturbed light, it has been shown that the
light is due to a negatively electrified body. By measuring

the displacement of the lines we can determine the change

— H J-fp

in frequency, i.e., ^

, so that if

is known, ejm can be

determined. In this way Zeeman has found the value of

ejm to be of the order 10^, the same as that deduced by the

direct methods previously described. The values of ejm

got in this way are not the same for all lines of the spectra,

but when the lines are divided up into series, as in Paschen

and Eunge's method, the diiiferent lines in the same series

all give the same value of ejm.

The displacement of the lines produced by the magnetic

field is proportional to ejm, and thus for light due to the

oscillations of a corpuscle the displacement will be more

than a thousand times greater than that due to the vibra-

tion of any positive ion with which we are acquainted. It

requires very delicate apparatus to detect the displacement

when ejm is

10''
:

a displacement one-thousandth part of

this would be quite inappreciable by any means at present at our disposal, hence we may conclude that the light in

any lines which show the Zeeman effect (and in line

spectra as distinct from band spectra, all lines do show this

effect to some extent) is due to the vibrations of corpuscles

and not of atoms.

The Zeeman effect is so important a method of finding

out something about the structure of the atom and the

nature of the vibrating systems in a luminous gas, that it is desirable to consider a little more in detail the nature of the conclusions to be drawn from this effect. In the first place it is only a special type of vibration that will show
the Zeeman effect. The simple case we considered was

40 THE COEPUSCULAE THEOEY OF MATTEE.

when the corpuscle was attracted to (Fig. 14) by a force

proportional to OP; this force is the same in all directions,

so that if the corpuscle is displaced from and then let go it

will vibrate in the same period in whatever direction it may

be displaced : such a corpuscle shows the Zeeman effect.

P If, however, the force on

were different in different

directions so that the times of vibration of the corpuscle

depended on the direction in which it was displaced,

then the vibrations would not have shown this effect.

The influence of the magnetic force would have been of a
A lower order altogether than in the preceding case. single

particle placed in a field of force of the most general

character might vibrate with three different periods and

thus give out a spectrum containing three lines, but if such

a particle were placed in a magnetic field these lines would

not show the Zeeman effect; all that the magnetic force

0---0 o
FIG. 16.
could do would be to slightly alter the periods by an amount infinitesimal in comparison with that observed in the Zeeman effect. There could be no resolution of the lines into triplets ; it is only in the special case when the periods
all become the same that the Zeeman effect occurs. We can
easily imagine cases in which some lines might show the Zeeman effect, while others would not do so. Take the case
B of two corpuscles A and attracted to a point (Fig. 16)
and repelling each other, they will settle into a position of equilibrium when the repulsion between them balances the attraction exerted by 0. In the most general case there would be six different frequencies of vibration (each corpuscle contributing three) and none of these would show the Zeeman effect. In the special case where the force exerted by is the same in all directions, three of these frequencies coincide, two others vanish, and there is one remaining isolated. The spectrum is reduced to two

;

ORIGIN OF THE MASS OF THE CORPUSCLE. 41

lines ;

one of

these (that corresponding to

the

coalescence

of the three lines) would show the normal Zeeman effect

while the other would not show it at all. With more com-

plicated systems we might have several lines showing the

Zeeman effect accompanied by others which do not show it.
When more lines than one show the Zeeman effect, the

magnitude of the effect may differ from line to line. Thus,

take the case of four corpuscles mutually repelling each

other and attracted towards a point 0. In the most general

case this system would have twelve different frequencies, three

for each corpuscle, and as long as these remained different

none of them would show the Zeeman effect. If, however,

the force exerted by is the same in all directions, two sets

of three of these frequencies become equal, three frequencies

vanish, two others coincide, and one remains isolated ; the twelve frequencies are now reduced to four, the two lines cor-

responding to the sets of three frequencies which had coalesced

will both show the Zeeman effect, but not to the same extent,

the alteration in frequency for one line being the normal

—He

amount i

while for the other line it is only half that

III

amount. The other lines do not show the Zeeman effect.

The reader who is interested in this subject is referred for

other instances of systems illustrating this effect to a

paper by the writer in the Proceedings of the Cambridge

Philosophical Society, vol. xiii., p. 39.

It is remarkable that, as far as our knowledge extends, all
the lines in a line spectrum show the Zeeman effect. This

might arise from the vibrating systems being single

corpuscles, only influenced slightly, if at all, by neigh-

bouring corpuscles, or it might arise from the vibrations of

more complicated systems, provided the radiation corre-

sponding to frequencies which on the theory would not show

the Zeeman effect, has great difficulty in leaving the
vibrating system. We have an example of the second

condition in the case of two corpuscles shown in Fig. 16 the vibration which does not show the Zeeman effect is the
B one when the middle point of A and remains at rest

—

42 THE COEPUSCULAE THEOEY OF MATTER.

B and A and are approaching or receding from it with

equal velocities ; thus the charged corpuscles are moving

with equal velocities in opposite directions and their effects,
at a distance from large compared with OA and OB will

neutralise each other. On the other hand, the vibrations
B which show the Zeeman effect are those in which A and

are moving in the same direction, so that the effects due to

one will supplement those due to the other, and thus the

intensity of the radiation from this vibration will greatly

exceed that from the other ; thus this vibration might give

rise to visible radiation while the other did not. The

vibration of a system of corpuscles which produces the

greatest effect at a distance, is the one where all the

corpuscles move with the same speed and in the same

direction ; it can be easily shown that for this case the

effect of a magnetic field is to increase or diminish all the
He
frequencies by the normal amount A m

A case in which the Zeeman effect might be abnormally
— large is the following : Suppose we have two corpuscles
B A and moving round the circumference of a circle with

constant angular velocity co, always keeping at opposite ends

of a diameter, then the frequency of the optical or magnetic

effect produced by this system is not co but 2 &),for each particle has only to go half way round the circumference to make the

state of the system recur. If now we place the system in a

magnetic field where the magnetic force is perpendicular to

+ — the circle the angular velocity w will become <« i

^^^

+ — the frequency of the system 2 «

, thus the change in

—m the frequency is , which is twice the normal effect.

CHAPTER ni.
PEOPEEIIES OF A CORPUSCLE.
Haying demonstrated the existence of corpuscles, it v^ill be convenient for pui-poses of reference to summarise their
properties.
Magnetic Fobce due to Corpuscles.
A moving corpuscle produces around it a magnetic field.
If the corpuscle is moving in a straight line with a uniform velocity v, -which is small compared with the velocity of

FIG. 17.

light, it produces a magnetic field in which the Hnes of magnetic force are chcles having the line along which the corpuscle is moving for their axis; the magni-
P tude of the force at a point is equal to -—-.^ sin 6, where

e is the charge on the moving particle 0, and 6 the
— angle between OP and OX the line along which the

P corpuscle is moving. The direction of the force at

POX (Fig. 17) is at right angles to the plane

and down-

wards from the plane of the paper if the negatively charged

particle is moving in the direction OX. The magnetic

force thus vanishes along the line of motion of the particle

and is greatest in the plane thi-ough at right angles to

44 THE COEPUSCULAE THEOEY OF MATTEE.

the direction of motion ; the distribution of force is symmetrical with respect to this plane.
If the velocity of the uniformly moving particle is so great that it is comparable with c the velocity of light, the
P intensity of the magnetic force at is represented by the
more complicated expression

e V sm 9

g

c-^:)- ,.2 /]

V2

.9

X2

The direction of the force is the same as before. The effect

of the greater velocity is to make the magnetic force

OX relatively weaker in the parts of the field near

and

stronger in those near the equatorial plane, until when

the speed of the corpuscle is equal to that of light the

magnetic force is zero everywhere except in the equatorial

plane, where it is infinite.

Electric Field eound the Moving Corpuscle.

P The direction of the electric force at is aloiig OP, and

whatever be the speed at which the corpuscle is moving,
E H the electric force and the magnetic force are connected

by the relation

H E c^

=^ V

sin 6 ;

thus when the corpuscle is moving slowly

E = i4

the same value as when the particle is at rest (remembering that e is measured in electro-magnetic units).
"When the corpuscle is moving more rapidly we have

E - — {c^

v^

2
- (l -2 sill" ej

and in this case the electric force is no longer uniforinly distributed, but is more intense towards the equatorial
regions than in the polar regions near OX. When the

PROPEETIES OF A CORPUSCLE.

45

corpuscle moves with the velocity of light all the lines of electric force are in the plane through at right angles to
OX.
When the corpuscle is moving uniformly the lines of force
are carried along as if they were rigidly attached to it, but
when the velocity of the corpuscle changes this is no longer
the case, and some very interesting phenomena occur. We
can illustrate this by considering what happens if a cor-

FIG. 18.
puscle which has been moving uniformly is suddenly stopped. Let us take the case when the velocity with which the particle is moving before it is stopped is small compared with the velocity of light; then before the stoppage the lines of force were uniformly distributed and were moving
forward with the velocity v. When the corpuscle is stopped,
the ends of the lines of force on the corpuscle will be stopped also ; but fixing one end will not at once stop the whole of the line of force, for the impulse which stops the tube travels along the line of force with the velocity of light, and thus takes a finite time to reach the outlying

46 THE COEPUSCULAK THEORY OF MATTER.

parts of the tube. Hence when a time t has elapsed after

the stoppage, it is only the parts of the lines of force

which are inside a sphere whose radius is ct which have

been stopped. The lines of force outside this sphere

will be in the same position as if the corpuscle had

not been stopped, i.e., they will pass through 0', the

position the corpuscle would have occupied at the time t if

the stoppage had not taken place. Thus the line of force

which, when the corpuscle was stopped was in the position
OQ will, at the time t be distorted in the way shown in

Fig 18. Inside the sphere of radius ct the line of force will
be at rest along OQ ; outside the sphere it will be moving
forward with the velocity v, and will pass through 0', the

point would have reached at the time t if it had not been

stopped. Since the line of force remains intact it must be

bent round at the surface of the sphere so that the portion
inside the sphere may be in connection with that outside.

Since the lines of force along the surface are tangential

there will be, over the surface of the sphere, a tangential

electric force. This tangential force will be on the surface

of a sphere of radius ct and will travel outwards with the

velocity of light. If the stoppage of the sphere took a

short time tt, then the tangential part of the lines of force

will be in the spherical shell between the spheres whose
— radii are ct and c (t ir), t being the time which has elapsed
— since the stoppage began, and t tt since it was completed.

This shell of thickness cv, filled with tangential lines of

electric force, travels outwards with the velocity of light.

The electric force in the shell is very large compared with the

We force in the same region before the shell is stopped.

can

P prove that the magnitude of the force at a point in the shell

— 18 equal c c v sxi^x— , where S is the thickness of the shell, UL ' o

and 6 the angle POX. Before the corpuscle was stopped

the force was -—5, thus the ratio of the force after the
OP''

—OP V

stoppage to the force before is equal to

j- «i« ^- As S

;

PEOPEETIES OF A COEPUSCLE.

47

is very small compared with OP, this ratio is very large
thus the stoppage of the corpuscle causes a thin shell of intense electric force to travel outwards with the velocity of light. These pulses of intense electric force constitute, I
think, Eontgen rays, which are produced when cathode rays are suddenly stojDped by striking against a solid obstacle. The electric force in the pulse is accompanied
by a magnetic force equal in magnitude to ^'^^'^ and at

right angles to the plane POX. The energy in the pulse
due to this distribution of magnetic and electric force is
— — equal to | - ; it is thus greater when the thickness of the o
pulse is small than when it is large. The thickness of the
pulse is, however, proportional to the abruptness with which the corpuscle is stopped ; and as the energy in the
pulse is radiated away it follows that the more abruptly the corpuscles are stopped the greater the amount of energy radiated away as Eontgen rays. If the corjDuscle is stopped
so abruptly that the thickness of the pulse is reduced to the diameter of the corpuscle the whole of the energy iu the magnetic field round the corpuscle is radiated away.
If the corpuscle is stopped more slowly only a fraction of this energy escapes as Eontgen rays.
Inside the shell, i.e., in the space bounded on the out-
side by the sphere of radius OP = { ct), there is no magOP netic force, while outside the sphere whose radius is
the magnet force is the same as it would be if the particle
had not been stopped, i.e., at the point Q it is equal to

6V

%,,

,, sin

where
<t>,

0'

is where

would have been if the

corpuscle had gone on moving uniformly, and i^ is the angle QO' X. The pulse in its outward passage wipes out, as it were, the magnetic force from each place as it passes
over it.
T\'e have seen that when the corpuscle is stopped there is a pulse of strong electric and magnetic force produced which carries energy away. It is not necessary that the

—

48 THE COEPUSCULAE THEOEY OF MATTEE.

corpuscle should be reduced to rest for this pulse to be

formed ;

any change

in

the

velocity will

produce a similar

pulse, though the forces in the pulse will not be so intense

as when the stoppage is complete. Since any change in

the velocity produces this tangential electric field, such a

field is a necessary accompaniment of a corpuscle whose

motion is accelerated, and we can show that if when at the

particle has an acceleration/ along OX, then after a time t has

P elapsed there will be at a point

distant ct from

a

tangential electric force equal to & "^I SViZ and a magnetic

force at right angles both to

P and the electric force,

& f SXTt

equal to L

. The rate at which energy is being

radiated from the corpuscle has been shown by Larmor to

be equal to f

—- ; thus a corpuscle whose velocity is

changing loses energy by radiation.

CHAPTEE IV.

COBPUSCULAE THBOEY OF METALLIC CONDUCTION.

We now proceed to apply these properties of corpuscles
to the explanation of some physical phenomena ; the

first case we shall take is that of conduction of electricity

by metals.
On the corpuscular theory of electric conduction through

metals the electric current is carried by the drifting of

negatively electrified corpuscles against the current. Since

the corpuscles and not the atoms of the metal carry the

current, the passage of the current through the metal does

not imply the existence of any transport of these atoms

along the current ; this transport has often been looked for
but never detected. We shall consider two methods by

which this transport might be brought about.

In the first method we suppose that all the corpuscles

which take part in the conduction of electricity have got
into what may be called temperature equilibrium with their surroundings, i.e., that they have made so many

collisions that their mean kinetic energy has become equal

to that of a molecule of a gas at the temperature of the

metal. This implies that the corpuscles are free not merely

at the instant the current is passing but that at this time

they have already been free for a time sufficiently long to
allow them to have made enough collisions to have got into

temperature equilibrium with the metal in which they are moving. The corpuscles we consider are thus those
whose freedom is of long duration. On this view the drift
of the corpuscles which forms the current is brought about by the direct action of the electric field on the free

corpuscles.
— Second Method. It is easy to see, however, that a

T.M.

E

50 THE COEPUSCULAR THEOEY OF MATTEE.

current could be carried through the metal by corpuscles

which went straight out of one atom and lodged at their

first impact in another ; such corpuscles would not be free

in the sense in which the word was previously used and

would have no opportunities of getting into temperature

equilibrium with their surroundings. To see how conduc-

tion could be brought about by such corpuscles, we notice

that the liberation of corpuscles from the atoms must be

brought about by some process which depends upon the

We proximity of the metallic atoms.

see this because the

ratio of the conductivity of a metal in a state of vapour to

the conductivity of the same metal when in the solid state

is exceedingly small compared with the ratio of the densities

in the two states. Some interesting experiments on this

point have been made by Strutt, who found that when mercury

was heated in a vessel to a red heat, so that the pressure

and density must have been exceedingly large, the con-

ductivity of the vapour was only about one-ten millionth of

the conductivity of solid mercury. If, however, corpuscles

readily leave one atom and pass into another when the atoms

of the metal are closely packed together, we can see how the

electricity could pass without any accumulation of free

corpuscles. For, to fix our ideas, imagine that the atoms

of the metal act on each other as if each atom were an

electric doublet, i.e., as if it had positive electricity on
one side and negative on another. A collection of such

atoms if pressed close together would exert considerable
force on each other, and the force exerted by an atom A

on another B, might cause a corpuscle to be torn out of B.

If this got free and knocked about for a considerable time

it would form one of the class of corpuscles previously con-
B A sidered, but even if it went straight from into it might

still help to carry the current. If the atoms were arranged

without any order, then, though there might be interchange

of corpuscles between neighbouring atoms, there would be no

flux of corpuscles in one direction rather than another, and

therefore no current. Suppose, however, that the atoms

get polarised under the action of an electric force, which

;

THEOEY OP METALLIC CONDUCTION. 51

force is, say, horizontal and from left to right, then the

atoms will have a tendency to arrange themselves so that

the negative ends are to the left, the positive ones to the

B right. Consider two neighbouring atoms A and (Fig. 19) :

A B if a corpuscle is dragged out of into it will start from

B the negative end of A and go to the positive end of

there
;

will thus be more corpuscles going from right to left than

in

any other direction ;

this

will give rise

to a current from

left to right, i.e., in the direction of the electric force.
We shall develop the consequences of each of these

theories so as to get material by which they can be tested.

A piece of metal on the first of these theories contains a

large number of free corpuscles disposed through its volume.

These corpuscles can move freely between the atoms of the

metal just as the molecules of air move freely about in the

0® 0© GX+) e±) 0© 0©

FIG. 19.
interstices of a porous body. The corpuscles come into collision with the atoms of the metal and with each other and at
these impacts suffer changes in velocity and momentum
in fact, these collisions play just the same part as the collisions between molecules do in the kinetic theory of gases. In that theory it is shown that the result of such collisions is to produce a steady state in which the mean kinetic energy of a molecule depends only upon the absolute temperature : it is independent of the pressure or the nature
We of the gas, thus it is the same for hydrogen as for air.
may regard the corpuscles as being a very light gas, so that the mean kinetic energy of the corpuscles will only depend upon the temperature and will be the same as the mean
kinetic energy of a molecule of hydrogen at that temperature. As, however, the mass of a corpuscle is only about 1/1700 of that of an atom of hydrogen, and therefore only
about 1/3400 of that of a molecule of hydrogen, the mean
B2

52 THE COEPUSCULAE THEOEY OF MATTEE.
value of the square of the velocity of a corpuscle must be 3400 times that of the same quantity for the molecule of hydrogen at the same temperature. Thus the average velocity of the corpuscle must be about 58 times that of a molecule of hydrogen at the temperature of the metal in which the molecules are situated. At 0° C. the mean velocity of the
hydrogen molecule is about 1'7 X 10^ cm/sec, hence the
average velocity of the corpuscles in a metal at this temperature is about 10^ cm/sec, or approximately 60 miles
per sec. Though these corpuscles are charged, yet since as many are moving in one direction as in the opposite, there will be on the average no flow of electricity in the metal. The case is, however, altered when an electric force acts throughout the metal. Although the change produced in the velocity of the corpuscles by this force is, in general, very small compared with the average velocity of translation of the corpuscles, yet it is in the same direction for all of them, and produces a kind of wind causing the corpuscles to flow in the opposite direction to the electric force (since the charge on the corpuscle is negative) , the velocity of the wind being the velocity imparted to the corpuscles by the electric force. If u is this velocity and n the number of corpuscles per unit volume of the metal, the number of corpuscles which in one second cross a unit area drawn at right angles to the electric force is n n, and if e is the charge on a corpuscle, the quantity of electricity carried through this area per second is n u e ; this
quantity is the intensity of electric current in the metal ; if
we denote it by i, we have the equation i = n u e. We now
X proceed to find u in terms of the electric force in the
metal. While the corpuscle is moving in a free path in the interval between two collisions, the electric force acts upon it and tends to make it move in the opposite direction to itself. When, however, a collision occurs, the shock is so violent
that the corpuscle moves off in much the same way, and with much the same velocity, as if it had not been
under the electric field. Thus the effect of the electric field is, so to speak, undone at each collision ; after the collision the electric force has to begin again, and the

THEOEY OF METALLIC CONDUCTION. 53

velocity communicated by the electric field to the corpuscle will be that which it gives to it during its free path. Jeans has shown that there is a slight persistence of an effect produced on a molecule after an encounter with another molecule, that each collision does not, as it were, entirely wipe out all the effects of the previous history of the molecule. To calculate the amount of this persistence we have to know the nature of the effect we call a collision ; in
m our case the effect is not of importance. If is the mass
of the corpuscle, the velocity the corpuscle owes to the action of the electric force increases uniformly from zero at

X — the beginning of the free path to

t at the end, t

lit

being the time between two collisions; hence the mean

X~ velocity due to the force is -

t, and this is the velocity

given to the particles by the electric force. If we care

to take into account the persistence of the impression

produced by the electric force we can do so by introduc-

ing a factor p into the expression and saying that

— the average velocity u due to the electric field is - /3

t.

2

in

Unless, however, we have a knowledge of the nature of the

collision between a corpuscle and the atom, all that we can

determine about /3 is that it is a quantity somewhat greater

= m ^ than unity. Since it

-^
2

t, and i

n u e, we have

2m

Now unless the electric force is enormously large the

change in the velocity of the corpuscle due to the electric force
— will be quite insignificant in comparison with v the average

We velocity of translation of the corpuscle.

may therefore put

= t A./r, where A is the mean free path of a corpuscle, hence

= 1= /3n
2

m

V

-/3n

—.

2

m v~

Now //( v^ is twice the average kinetic energy of a

—

54 THE COEPUSCULAE THEOEY OP MATTEE.

corpuscle, and therefore twice the kinetic energy of a
molecule of hydrogen at the same temperature; m y^ is
thus equal to 2 a ^ where is the absolute temperature and
= 2a 7-2 X 10-"/273.
From the relation

.
= — t

=z

1 -

a«

71

Xe'^kv
s-

m 2

v^

1/3 n e2^,kv^X
4 a.e

we see that the specific conductivity of the metal is equal to Pne^k v/4 aB; thus the specific conductivity on this theory is independent of the electric force X, so that Ohm's law is

true.
If the electric force were so large that the velocity generated in a corpuscle during its free path were large
.
compared with the average velocity of a corpuscle, the relation between current and electric force would take a different form. In this case the velocity of the particle is generated by the field, so that if iv is this velocity then

—Xe — I- miv^
2

\ or iv

/ m >V

the average velocity
;

J^-^, is one-half of this, i.e.,

and the current

=ne V i

m tJ^. Thus in this case the current, instead of
'2

being proportional to the electric force, would be propor-

tional to the square root of it, so that Ohm's law would no

longer hold. This state of affairs would, however, only

occur when the electric force was exceedingly large, too

large to be realised by any means we have at present at our
X command. For it requires e A. to be large compared with

the mean kinetic energy of a corpuscle, which at 0° C.
X is equal to 3-6 X lO-^*. Now e is IQ-^", thus A. must be large
compared with 3-6 x 10". We do not know the free path

of a corpuscle in a metal, but as the free path in air

whose density at atmospheric pressure is only -0015 is only

10"^ cms., the free path in a metal can hardly be greater

than 10-^ cms. Thus the value of A' necessary to give to

—

THEORY OP METALLIC CONDUCTION. 55

the corpuscle an amount of kinetic energy large compared
with that it possesses in virtue of the temperature of the metal, must be of the order 10^*, i.e., a million volts per
centimetre. We have no experimental evidence as to how
a conductor would behave under forces of this magnitude. If we assume that A, is of the order 10^' we can get an
— estimate of n the number of corpuscles in a cubic centi-
metre of the metal. Let us take for example silver, whose specific conductivity is 1/1600 at 0° C. ; we have, using the expression we have found for the conductivity

_ 1 ~ 1600

§

n

e^Xv .

4 ae '

= = = = = if we put e

lO^^o^ x

10"^ r

10'', (3

1, -2 a.

— 7-2 X 10"" we find n

X 9

10^3.

Now, in a cubic centimetre of silver there are about

1"6 X 10^ atoms of silver, and thus from this very rough

estimate we conclude that even in a good conductor like

silver the number of corpuscles is a quantity comparable

with the number of atoms.

If the carriers instead of being corpuscles were bodies

with a greater mass the number of carriers would be greater

than that just found. For we see from the preceding

formula that if the carriers are in temperature equilibrium
with the metal n \t must be constant if the conductivity is given. Hence if the mass of the carriers were much greater than that of a corpuscle and therefore r and X much smaller,

71 would have to be much larger, that is, the number of

carriers in silver would have to be much greater than the

number of atoms of silver, a result which shows that the

mass of a carrier cannot be comparable with that of an atom.

COMPAEISON OP THE ThBEMAL WITH THE ElECTEICAL
Conductivity.
If one part of the metal is at a higher temperature than another, the average kinetic energy of the corpuscles in the hot parts will be greater than that in the cold. In
consequence of the collisions which they make with the atoms

—

—

— —;

56 THE COEPUSCULAR THEOEY OP MATTER.

of the metal, resulting in alterations in the energy, the cor-
puscles will carry heat from the hot to the cold parts of the metal ; thus a part at least of the conduction of heat through the metal will be due to the corpuscles. If we assume that the whole of the conduction arises in this way, we can find an expression for the thermal conductivity in terms of the quantities which express the electrical conductivity. It is proved in treatises on the kinetic theory of gases that k the thermal conductivity of a gas is given by the expression

k = ^n\Va

(see Jean's " Kinetic Theory of Gases," p. 259). Here k is

measured in mechanical units, and the effect of persistence

of the velocities after the collisions has been neglected.

Hence to compare k with c the electrical conductivity we
= must in the expression for the latter quantity put /? 1

doing this, we obtain

nXv (i^

hence

— 1 1 ~ "? '^'''

4 . a^ ^ 3

Thus neither n nor A, the quantities which vary from

metal to metal, appears in the expression for cjk, so that the

theory of corpuscular conduction leads to the conclusion

that the ratio of the electrical to the thermal conductivity

should be the same for all metals and should vary inversely

as the absolute temperature of the metals.
We can calculate the numerical value of the ratio of the
two conductivities on the preceding theory as follows : lip is the pressure of a gas in which there are n molecules per

cubic centimetre,

the absolute temperature, then
^ p ^ aO . n\

hence

_ a 6

^p

e

1n e

Now e is the charge on an atom of hydrogen, and if n is
the number of hydrogen molecules in a cubic centimetre of gas at a pressure of one atmosphere {i.e., 10" dynes), and

—— —
.

THEOEY OF METALLIC CONDUCTION. 57

at 0° C, we have, since one electromagnetic unit of electricity liberates 1"2 cubic centimetres of hydrogen at

this pressure and temperature

C— hence at 0°

= 2-4 ne

1;

— = 3-6 X lOe,
e

so that at this temperature

= J °-
'"-'

= 6-3 X IQio in absolute measure,

c 3 e2 273

The following are the values of kjc for a large number of metals as determined by Jaeger and Diesselhorst in their most valuable paper on this subject :

Material.

Tliermal conductivity. Temperature coefficient

Electrical conductivity.

of tliis ratio.

Copper, commercial Copper (1), pure Copper (2), pure Silver, pure Gold(l) Gold (2), pure ... Nickel Zinc (1) Zinc (2), pure ... Cadmium, pure . . Lead, pure Tin, pure Aluminium Platinum (1) Platinum (2), pure Palladium Iron (1) Iron (2)
Steel
Bismuth ... Constantan (60 Cu 40 Ni) Manganine
(84 Cu 4 Ni 12 Mn)

At 18° 0.
6-76 X 10"'

6-65 X 101"

6-71 X 10'°

6-86 X IQi"

X 7-27

101°

7-09 X 1010

6-99 X IQi"

7-05 X IQi"

6-72 X IQi"

7-06 X IQi"

7-15 X 101"

7-35 X IQi"

6-36 X 101°

7-76 X IQi"

7-53 X 101"

7-54 X 101°

8-02 X 101°

8-38 X 101°

9-03 X 101°

9-64 X 101°

11-06 X 101°

9-14 X 101'

Per cent.
0-39 0-39 0-37 0-36 0-37 0-39 0-38 0-38 0-37 0-40 0-34 0-43
0-46 0-46 0-43 0-44 0-35 0-15 0-23
0-27

1
58 THE CORPUSCULAR THEORY OP MATTER.
It will be seen that the observed values of the ratios of the
thermal and electrical conductivities of many metals agree
closely with the result deduced from theory, while others show considerable deviations. Again, the temperature
coefficient of this ratio is for many metals in agreement with the theory. On the theory the ratio is proportional
to the absolute temperature ; this gives a temperature
coefficient of "366 per cent., and we see that for many
metals the temperature coefficient is of this order. In the case of alloys the ratio of the thermal to the
0.08
\ 07

:3
to
g 4x10'

O

-i

3x10
"i;

<i

<j

ft)

-

I xib'

THEOEY OF METALLIC CONDUCTION. 59

although there are some exceptions, the ratio of the thermal

to the electrical conductivity is larger for alloys than for
pure metals. This and many other properties of conduc-

tion of electricitj' through alloj's can be explained by some

considerations given by Lord Eayleigh (Xature, LIV., p. 154, " Collected Works," vol. iv., p. 232). Lord Eayleigh points

out that in the case of a mixtm-e of metals there is,

owing to their thermo-electric properties, a source of

something which cannot be distinguished by experiments

from resistance, which is absent when the metals are pure.

To see this, let us suppose that the mixed metals are

arranged in thin layers, the adjacent layers being of

different metals, and that the current passes through the
body at right angles to the faces of the layer. Now when

a cm-rent of electricity passes across the junction of two

metals Peltier showed that the junction was heated if the

current passed one way, cooled if it passed the opposite

way, and that the rate of heat production or absorption was

proportional to the current passing across the junction.

Thus, where the current passes through the system of

alternate layers of the two metals, one face of each layer

will be cooled and the other heated, and thus in the pile

of layers differences of temperature proportional to the

current will be established.

These will set up a

thermoelectric force, tending to oppose the current,

proportional to the intensity of the current. Such

a force would produce exactly the same effect as a

resistance. Thus in a mixture of metals there is, in

addition to the resistance, a ' false resistance' due to thermo-

electric causes which is absent in the case of pure metals.

This false resistance being superposed on the other

resistance makes the electrical resistance of alloys greater

than the value indicated by the preceding theory. This

result gives an explanation of the fact that the ratio of the

thermal to the electrical resistance is greater for alloys than

it is for pure metals.

The experiments of Dewar and Fleming on the effect of

very low temperatures on the resistance of pure metals and

—

—

60 THE COEPUSCULAE THEOEY OF MATTEE.

alloys show that there is a fundamental difference between

the resistances of pure metals and mixtures, for while the

resistance of pure metals diminishes uniformly as the

temperature diminishes and would apparently vanish not

far from the absolute zero of temperature, the resistance of

alloys gives no indication of disappearing at these very low

temperatures, but apparently tends to a finite limit.

The electrical conductivity of a metal is proportional to n

the number of free corpuscles per unit volume. Now, since

a free corpuscle will continually be getting caught by and

attached to an atom, the corpuscles, when the metal is in a

steady state, must be in statical equilibrium ; the number
of fresh corpuscles produced in unit time being equal to the

number which disappear by re-combination with the atoms
in the same time. We should expect the number of

re-combinations in unit time to be proportional to the

number of collisions in that time, i.e., to «/t; where r is

the interval between two collisions ; t is equal to Xjv where A. is the free path and r the velocity of the corpuscle;

Hence the number of re-combinations in unit time will be

^n V

equal to y

where y represents the proportion between

the number of collisions which result in re-combination and the whole number of collisions. If q is the number of corpuscles produced per cubic centimetre per second, we have when there is statical equilibrium

2 = 7^n .r

Thus c the electrical conductivity of a metal is expressed by the equation

= C

- 1 /? g A.S e^

-

i

.

4 y a. 6

For most pure metals the conductivity is inversely proportional to the absolute temperature 0, hence we conclude
that q A2 must be independent of the temperature. Now
we should not expect X to vary more rapidly with the
temperature than the distance between two molecules, a

THEOEY OF METALLIC CONDUCTION. 61
quantity whose variation with the temperature is of the same order as that of the Hnear dimensions of the body, and therefore represented by the coefficient of thermal expansion, a very small quantity ; thus, since q A.^ is independent of the temperature, and X^ only varies slowly with the temperature, the variations of q with temperature can only be slight, hence we conclude that the dissociation of the atom which produces the corpuscles cannot to any considerable extent be the effect of temperature.
We should expect to have fewer free corpuscles and
therefore smaller conductivity in a salt of the metal than in the metal itself. For in the salt the atoms of the metal are all positively electrified and have already lost corpuscles,
which have found a permanent home on the atoms of the electro-negative element. From the positively electrified
metal atoms corpuscles will find it difficult to escape, and
the rate of production of free corpuscles will be very much
lower than in the pure metal, where in addition to positively electrified atoms neutral and negatively electrified atoms of the metal are present.
LoRBNTz Thboey of Eadiation.
Radiation of heat may be produced by the impact of corpuscles. When a corpuscle comes into collision with an
atom it experiences rapid changes in its velocity, and therefore will, as explained on p. 46, emit pulses of intense electric and magnetic force ; the thickness of these pulses will be the distance traversed by light during the time occupied by a collision. Thus, if we consider any atom of the metal, it will be from time to time, as the corpuscles strike against it, the centre of pulses of intense electric and magnetic force. These forces at a point near the atom
will vary in a very abrui3t manner. A pulse of intense
electric force, lasting for a very short time, will pass over the point, then there will be an interval in which the electric force disappears, and again, after the space of time between two collisions, another intense pulse will pass over

—

;

62 THE COEPUSCULAR THEOEY OF MATTEE.

the point. Now though the electric force jumps about in
this abrupt way, we know by the theorem due to Fourier that it can be represented as the sum of a number of terms, each of which is of the form cos (pt + e) where t represents the time. Each of these terms represents a harmonic wave of electric force, and by the electro-magnetic theory of light a harmonic wave of electric force is a wave of light or radiant heat. Thus we can represent the irregular, jerky electric

field j)roduced by the collision as arising from the superposition of a number of waves of light or radiant heat, and if we can calculate the amplitude of vibration of the disturbance of any period, we can calculate at once the energy in the light of this period emitted by one molecule, and therefore, by summation, by the metal.

Of the whole group of waves which represent the electric
field due to the collisions, Lorentz has shown how to calculate the amplitudes of those whose wave length is very large indeed compared with the free path of the corpuscles, and has shown that the energy in the vibrations whose frequency is between q and Sq given out per second per unit of area of a plate where thickness is A is equal to

—dq q^ £:>. -2

4: TT e^ n X V

;

c represents the velocity of light, e the charge on the
corpuscle, X the mean free path of a corpuscle, and v its mean velocity of translation. This represents the rate at which the body emits energy. To find the amount of energy of this frequency present in the body when the radiation is in a steady state, we must take into account the
absorption of this energy in its course through the body. For imagine a body built up of piles of parallel plates then if there were no absorption the energy emitted by the most distant portions would reach any point Q, and if the size of the body were infinite the amount of energy per unit
volume at Q would be infinite also. If, however, there was
strong absorption, so that the radiation was practically all absorbed in the space of one millimetre, then it is evident

——

——

THEOEY OF METALLIC CONDUCTION. 63

that the portions of the body whose distance from Q is more

than one millimetre will not send any energy to Q, and how-

ever large the body may be the energy at Q will be finite.

When the energy in the body has settled down into a steady

state, the energy given out by any portion must be equal to

the amount acquired by absorption. This principle enables

us to find the amount of energy per unit volume of the body

when the radiation is in a steady state. The absorption of

these very long waves in a conductor is due to the same

cause as the production of heat in the conductor when an

electric current passes through it, since these waves are made
X up of electric and magnetic forces. When an electric force

acts on a conductor and produces an electric current whose

intensity is i, the rate at which energy is absorbed per unit

volume is A'(, or if o- is the specific resistance of the con-

ductor the rate at which energy is absorbed is equal to X^/a-,

We E since (ri = X.

must express this in terms of the energy

per unit volume in the conductor. One half of this energy

is due to the electric field, the other half to the magnetic

field which accompanies it ; the energy per unit volume due

to the electric field is - ^, c being the velocity of light

E = through the medium, hence

X^

-.

„, and X^ = 4 tt c^ £,

hence X^ja- the rate at which energy is absorbed per unit

volume is equal to

E 4:7rC^

O"

and the rate per unit area of a plate of thickness A is

£ 4 TT c^

A

(T

Now in a steady state the energy emitted is equal to the
energy absorbed ; the expression for the rate at which energy is emitted is given on p. 62 ; equating this to the rate at which the energy is absorbed, we have

cr

O tt'' C

——

——

64 THE COEPUSCULAE THEOEY OF MATTEE.

but (see p. 66)

1 e^kn V

0-

Aad

when 6 is the absolute temperature.

value for l/o-, equation (1) becomes

Substituting this

^ 2

X (e^

n t-)

d q'^

ao

,

(2)

A a. 6

Q-n^ C

The quantities n and A, which differentiate one substance from another occur in the same form on both sides of the
equation : one side expresses the absorption, the other the
radiation, and we see that the ratio of the two is independent of the nature of the substance. Hence this view of radiation would explain Kirchhoff's law that good radiators are also good absorbers. Dividing out the common factors from equation (2), we get

-n

A O. 6

97

or if A. is the wave length of the vibration whose frequency is 2 we have, since
5 = 2.-,
E ^^^dX,

and this is the expression for the amount of energy per

unit volume whose wave length is between X and d X when

the absolute temperature is 0. This expression does not

involve any constant which depends upon the nature of

the body, hence it would be the same at the same tempera-

E ture for all bodies. The expression for

is of the type

— / {X 6)

/ g- , where (A. 6) denotes a function of X and 6.
A

The

researches of Wien have shown that it is only a formula of

this type which fits in with the values of the radiation

observed by him and others in experiments with bodies at

different temperatures. The preceding expression is of the

type suggested by Lord Eayleigh (Phil. Mag., June, 1900).

Since a. d represents the mean kinetic energy of any gas

—

—

THEOEY OF METALLIC CONDUCTION. 6S
at the absolute temperature 6, we can calculate the value of a, and thus arrive at a numerical estimate of the amount of radiation given by the preceding expression. If we find this coincides with the observed amount it will be a strong
confirmation of the theory.
By the kinetic theory of gases, if j> is the pressure,
N the number of molecules per unit volume of the gas

o

hence \ mi^, the mean kinetic energy of a particle, is equal

= to 3 _2;/2 N, but J 711 ifi

a6, hence

2iV

Now at the pressure of 760 millimetres of mercury and

N= a temperature of 0° C, jj = 10", e = 273, and

4 X 10^^

hence a = 1-32 X 10"'l Assuming that the radiation is

expressed by equation (1), we can use the equation if we

know the amount of radiation to find a, and Lorentz finds

from the experiments made by Lummer and Pringsheim and

Kurlbaum on the amount of radiation given out by hot bodies

= that a

X 1'2

10"'^.

Thus the a_rguinent between theory

and the results of experiment is very satisfactory and gives

us considerable confidence in the truth of the theory. It

ought, however, to be pointed out that we should get the
same expression for the radiant energy E, whatever may

be the mass or charge of the moving electrified bodies,

which are supposed to generate this energy by their col-

lisions and absorb it by their motion in the electric field,
provided that the mean kinetic energy of these bodies had the

same value as that we have assumed for the corpuscles.

The energy calculated in this way by Lorentz is only a

part of the energy radiated in consequence of the collisions.

It is that part which, when the electric forces produced by

the collisions is exjDressed by Fourier's method as the sum

of a number of harmonic comjDonents, corresponds to the

part of the disturbance which can be. expressed by the

T.M.

F

66 THE COEPUSCULAK THEORY OF MATTEE.
terms with exceedingly long wave lengths. But the disturbance, as we have seen, consists in a succession of
exceedingly thin pulses, the thickness of the pulse being comparable with the distance passed over by light in the time occupied by a collision, while the part calculated by Lorentz is only the part which can be represented by harmonic terms whose wave length is long compared with the distance passed over by light, not in the short space
occupied by a collision, but in the much longer interval
which elapses between two collisions. It is evident that
Lorentz's investigation leaves out of consideration a large part of the radiation, and that this part, arising from the
accumulation of a number of thin pulses, will be analogous
— to the Eontgen rays that, in fact, they will be Eontgen
rays, mainly of a very absorbable type, since the corpuscles
which produce them are moving much more slowly than
the cathode rays in the ordinary Eontgen ray bulb. In fact, a mathematical investigation leads us to the conclusion that, of the energy radiated at a collision, there will be more of this type than the long wave type calculated by Lorentz. The character of the radiation will depend upon the time taken by a collision between the corpuscle and a molecule, if this time is so short that the distance travelled by light during the collision is very small compared with the wave length of light in the visible part of the spectrum, then the resulting radiation will be of the Eontgen ray type and not visible light. If, however, the time of collision is so prolonged that light during this time can travel over a distance comparable with the wave length of light in the visible part of the spectrum, then the
resulting radiation will be visible light, and the maximum
intensity of this light will be in that part of the spectrum where the wave length is comparable with the distance
travelled by light during a collision, i.e., when the period
of vibration of the light is comparable with the time of a collision. The intensity of light having smaller wave lengths than this will rapidly fall off as the wave length diminishes. Thus in the case of these prolonged collisions

THEOEY OF METALLIC CONDUCTION. 67

the radiation would be ordinary light, the intensity rising
to a maximum at a particular part of the spectrum and
then diminishing rapidly in the region of smaller wave lengths. These are characteristic properties of the radiation
emitted by a black body. We know, however, the character
of the radiation from such a body depends only upon the temperature and not at all upon the nature of the body, thus the colour of the light at which the intensity of the
radiation is a maximum depends only on the temperature
moving towards the blue end of the spectrum as the
temperature is increased. On the theory that this radiation
arises from the collision of corpuscles the wave length
where the intensity of the radiation is a maximum depends
on the duration of the collision ; hence, if the radiation from hot substances arises in the way we have sup>posed, the duration of a collision between a corpuscle and a molecule of the substance must be independent of the nature of the substance and depend only upon the temperature, and the higher the temperature the shorter must be the duration of the collision.
By the application of the Second Law of Thermodynamics it has been shown that when the body is at the absolute
temperature 6 the amount of energy in the part of the spectrum comprised between wave lengths X and X. -\- d \ must be of the form \-^ 4, (\ 0) d \; where <^ is a function which cannot be determined by thermodynamical principles alone. The mathematical theory of the production of radiation by colhsions shows that this energy is given by

F T an expression of the form \-^

\jy-r,] d X where

is

the duration of the collision F the velocity of light and
F represents a function whose form depends upon the
nature of the forces exerted during the collision. Comparing
T these two expressions we see that must be conversely
proportional to 6, that is, inversely proportional to the
square of the velocity of the corpuscles. The velocity of corpuscles at 0° C. when in temperature equilibrium with their surroundings is about 10' cm./sec, the wave length at
F2

—

68 THE CORPUSCULAE THEOEY OF MATTEE.

which the intensity is greatest at 0° C. is about 10~^ cm. In a Eontgen ray bulb giving out hard rays the velocity of
the corpuscles may be about 10^° cm./sec, or 10^ times the
velocity of those in the metal ; hence, if the law of duration of impacts is true, the radiation produced by the impact of
the corpuscles in the tube should be a maximum for a wave
length of 10~^/10'' or 10"^ cm., as this is of the same order as the thickness of a pulse of very penetrating Eontgen radiation ; this test, as far as it goes, confirms the law of
the duration of collisions.

The Effect of a Magnetic Field on the Flow of an Electeic Cukeent : The "Hall Effect."

Hall found that the lines of flow of an electric current

through a metallic conductor are distorted when the con-

ductor is placed in a magnetic field. The distortion is of

the character which would be produced if an additional

electromotive force ys'ere to act at right angles to the

original one producing the current, and also at right

angles to the magnetic force. Thus if a horizontal

electromotive force producing a current from right to left

acts on a thin piece of metal in the plane of the paper, if

the plate is placed in a magnetic field whose lines of force

are at right angles to the plane of the paper and down-

wards, the current is distorted as if a small vertical electro-

motive force in the plane of the paper acted upon the
— metal. In some metals for example, bismuth and silver

this force would be vertically upwards ; in others, such as

iron, cobalt, and tellurium, the force would be vertically

downwards. In some alloys it is said that the force is in

one direction for small magnetic forces and in the opposite
direction for large ones. In many cases it is not propor-

tional to the magnetic force. The theory of electric conduction we have been considering would indicate a

distortion of the lines of flow of a current by a magnetic

fie.ld, as the following considerations will show.

Suppose a current of electricity flows from right to

left through the plate.

This, on the view of the

THEORY OF METALLIC CONDUCTION. 69

current j)reviously taken, indicates that the negative cor-

puscles have, on the average, a finite velocity from left to

right. Let the average value of this velocity of drift of the

negative coriDuscles be u. If a magnetic force downwards

at right angles to the plate acts on these corpuscles, they

will be acted on by a vertically upward force in the plane

of the paper, equal numerically to Heit, where e is the

H magnitude of the charge on the corpuscle, and

is the

intensity of the magnetic force. The force on the corpuscle

is the same as if there were an electromotive force acting

vertically downwards in the plane of the paper. Thus, there

would be a distortion of the lines of flow of the same sign

and character as the Hall effect in bismuth. If, however,

this were a complete representation of the action of the

magnetic field on the current, the Hall effect would be of
— — the same sign the sign it has for bismuth in all metals,

and would always be proportional to the magnetic force ;
neither of these statements is true. Inasmuch as the Hall

effect would be of the opposite sign, if the carriers of the

electricity through the metal were positively charged par-

ticles instead of negatively charged ones, some physicists,

in order to explain the existence of Hall effects of opposite

signs, have assumed that electricity is carried through metals

by two types of carriers, one positively the other negatively

electrified ;

in

some metals

the negative carriers

are

pre-

dominant, in others the positive. There are, I think, two

very serious objections to this assumption. In the first

place we have no evidence of the existence of positively

electrified particles able to thread their way with facility

through metals, and in the second place the assumption

does not explain the various phenomena connected with the

Hall effect. It would indeed exjolain the existence of Hall

effects of different signs, but on this hypothesis the amount

of the Hall effect would be proportional to the magnetic

force, which is by no means the case for all substances.

The complexity of the laws of the Hall effect suggests

that it is due to several causes, but we can, without calling

in the aid of positively charged carriers of electricity, see

70 THE CORPUSCULAE THEORY OF MATTER.
other sources for the variation in sign, and the failure to be directly proportional to the magnetic force. In the preceding investigation we have considered merely the effect of the magnetic force on the particle during its free path, and have neglected any influence of the magnetic force on the collisions between the corpuscles and the molecules.
We can, however, easily see how a magnetic field might
make suitable molecules arrange themselves so that they produce a rotatory effect on the motion of a corpuscle when the corpuscle came into collision with the molecule, and that the sign of this effect might in some cases be the same
as, in others opposite to, the rotation produced by the
magnetic field when the corpuscle was travelling over its
— — free path. Thus to take a simple instance imagine a body
whose molecules are little magnets ; then if the body is placed in a magnetic field such that the lines of force are vertical and downwards, the molecules of the body will arrange themselves so that their axes tend to be vertical, the negative poles being at the top, the positive at the bottom. Then close to the magnet, in the region between its poles, the lines of force due to the magnet will be in th« opposite direction to those due to the magnetic field, and the intensity of the force close in to the magnet
may be very much greater than that of the external field. In this case when the corpuscle came into collision with a
molecule the velocity would be rotated in the opposite direction to its rotation by the magnetic field before it came into collision with the magnet, i.e., while it was travelling
over its mean free path. In this case the expression for
the Hall effect would consist of two terms, one arising from the free path, the other from the collisions, and these terms would be of opjjosite signs. If the molecules were small portions of a diamagnetic substance it is easy to see that the effect due to the collisions would be of the same sign as that due to the free path. It is perhaps worthy of note that, with the exception of tellurium, which has quite an abnormal value, the substance for which the Hall effect has the largest negative value, calling the free path effect

THEORY OF METALLIC CONDUCTION. 71
positive, is iron. It would be interesting to see if in
exceedingly strong magnetic fields, much stronger than those
required to saturate the iron, the Hall effect would change
sign.
We must, however, I think, be careful not to import from
the kinetic theory of gases ideas about the free paths of
corpuscles which may not be applicable in the case of
metals. The study of metals by means of micro-photography has shown that their structure is extremely complex. This is illustrated by Fig. 21, which represents the appearance under the microscope of a piece of cadmiun
PIG. 21.
when polished and stained. A piece of metal apparently
consists of an assemblage of a vast number of small crystals, and the appearance of the metal when strained past the limit of perfect elasticity shows that under strain these crystals can slip past each other. The structure of a piece of metal is thus quite distinct, from that of a gas, where the particles are distributed at equally spaced intervals. In a metal, on the other hand, it would seem that the molecules of the
metal are collected in clusters, each cluster containing several molecules, and that the metal is built up of aggregates of such clusters. The collisions which determine the
free path of a corpuscle may be with these clusters and not

——

'

;

72 THE COEPUSCULAE THEOEY OF MATTEE.

with the individual molecules, and if this were so, large

variations in the free path might be brought about by

variations in the number of molecules in each cluster with-

out any variation of corresponding magnitude in the density

of the metal. Thus, to take a simple ease, suppose that the

clusters are little spheres, and let us compare the free paths of

a corpuscle (1) when there are n spheres of radius a per unit

m volume ; and (2) when there are spheres of radius b, the

amount of matter per unit volume being the same in the

— m two cases, so that na^

b^. If ^i and A^ are respectively

the free paths in the two cases, then

= A.1

\ m 5 and

=-
TT b^

= m and since ?i a^

b^ we have

= A.i/A.2

ajb.

So that in this case the free path would be proportional

to the radius of the cluster. Thus the bigger the cluster

the longer the free path. It follows that if a rise in tem-

perature caused the clusters to break up to some extent and

become smaller, it would produce a considerable diminution

in the free path of a corpuscle without any marked change

in the density, whereas in a gas a rise in temperature unac-

companied by a change in density would, if the collisions

between the molecules of a gas were like those between

hard elastic spheres, produce no change in the free path.

If the theory of conduction of electricity by corpuscles in

temperature equilibrium with their surroundings is true, we

must, I think, suppose that there is large variation of the

free path with the temperature and with the nature of the

We metal.

shall see from the consideration of the Peltier

effect that the number of free corpuscles per unit volume

does not, in general, vary greatly from one metal to another

so that the very large variations in the electrical resistance
of metals must arise much more from variations in the free

paths of the corpuscles than from variations in the number

of corpuscles. Hence the ratio of the free paths of the

THEOEY OF METALLIC CONDUCTION. 73
corpuscles will be of the same order as the ratio of their conductivities for electricity. Now, if the free paths of the corpuscles in the metal were determined by the same considerations as in a gas, i.e., if X were to be equal ioMj-nnf,
N being the number of molecules per unit volume, and
a the radius of the molecules, we can show that the variations in A. would not be nearly large enough to explain the variation in the electrical conductivity. For we can deter-
N mine by dividing the density of the metal by its atomic
weight, and we can get some information as to the value of a^ from the values of the refractive indices of compounds of the different metals. Doing this, we find that the variations
in 1/A' TT a" are not nearly so large as the variations in the electric conductivity, and that there is little, if any, correspondence between these quantities. Moreover, if
the theory we are discussing is correct there must not merely be large variations in the value of A, for the different metals, but even in the same metal at different temperatures. This follows from the consideration of the Thomson effect, i.e., the convection of heat by an electric current flowing along an unequally heated conductor.

Pbltieb Difference of Potential between Metals.

Suppose that we place two metals A and B, which are at

the same temperature, in contact, and that the pressure of

N m N the corpuscles (i.e., \

v" where is the number of

corpuscles in unit volume, m the mass, v the mean velocity

of the corpuscles) in A is greater than that in B. Then corB puscles will flow from A to ; but as these corpuscles are
B negatively charged, the flow of corpuscles will charge
A negatively and positively. The attraction of the positive

electricity in A will tend to prevent the corpuscles 'escaping

from it, and the flow will cease when the attraction of the

A positive electricity in and the repulsion of the negative in

B just balances the effect of the difference in pressure. The

A B positive electrification in

and the negative in

will be

close to the surface of separation, and these two electrifications

——

—

;

74 THE COEPUSCULAE THEOEY OF MATTEE.

will produce a difference in electric potential between A and

B, which we can calculate in the following way.

Let us suppose that there is a thin layer between the

B substances A B, in which the transition from A to takes

N place gradually. Let

be the number of corpuscles

per unit volume at a point distant x from one of the

boundaries of this layer, p the pressure of the corpuscles
X at this point, and the electric force. Then if e is the

charge on a corpuscle, the force acting on the corpuscles
X per unit volume is Ne. This, when there is equilibrium,

must be balanced by the force arising from the variation

in pressure as we pass from one side of the layer to the

other. The force due to the pressure is j^, hence

^^XNe.
aX
— But if 6 is the absolute temperature

^3

hence, if the temperature is constant across the layer, we

have

A 2 . 1 d

,,

3 N dx

Integrating both sides of this equation across the layer,

we get

2 ae,

A"i

„

3T^°"iV.= ^'

where V is the difference in potential between the two sides

of the layer and A"! and A2 are the numbers of corpuscles per
B unit volume in A and respectively. Thus in crossing

the junction of two metals there will, unless the number of

corpuscles in the two metals is the same, be a finite change

in potential. Now f a 61e =2)1 Ne, and since it is the same

for all gases we may take the case of hydrogen at 0° C.

and atmospheric pressure for which p = 10^, and Ne = '41

= - — thus at 0° C. f a eje = 2-5 x 10", so that in volts—

F

log -^

40 273 ^ A2

(1)'
^

THEOEY OF METALLIC CONDUCTION. 75
The potential differences which arise in this way are not comparable with the volta differences of potential between
metals in contact, for to produce a potential difference of
one volt, log Ni/N, = 40, or NJN, = 2-36 X 10"—a result
quite incompatible with the comparative values of the
resistances of two such metals as copper and zinc. Comparatively small variations in the number of corpuscles would, however, produce potential differences quite comparable with those measured by the Peltier effect, i.e., the heating or cooling of the junction of two metals when an
electric current passes across them. Thus, to take a ease where the Peltier effect is exceptionally large, that of
antimony and bismuth, whose V at 0° C. is about 1/30 of
a volt, we see from equation (1) that for these metals
log (NJN,) = 1-33, or N.jN, = 3-8. Thus, if the number of
corpuscles in the unit volume of antimony were about four times that in bismuth we should, on this theory, get Peltier effects of about the right amount. Since the Peltier
effect for antimony and bismuth is very much larger than that for most pairs of metals, we see that the theory indicates that in general the number of free corpuscles per unit volume does not vary much from one metal to another. From the Peltier effects of each metal with a standard metal we can get the ratio of the number of corpuscles in these metals to the number in the standard
metal. Having done this, since at the same temperature
the conductivity of the metals is proportional to the pro-
duct of the number of corpuscles per unit volume and the free path of a corpuscle in the metal, we can get the ratio of the free paths in the different metals, and we can then see whether the free paths obtained in this way can be reconciled with the other properties of the metals. The result of such a comparison leads, I think, to the conclusion that the mechanism by which we have supposed the electric current to be conveyed through a conductor is at most only a part and not the whole of the process of metallic conduction. One reason for this conclusion is the large changes which take place in the electrical resistance

76 THE CORPUSCULAR THEORY OF MATTER.

of some metals at fusion, changes which do not seem to be

accompanied by any corresponding change in their thermo-

electric quality. Thus the conductivities of tin, zinc and

lead at their melting points are, when the metals are in the

solid state, about twice what they are in the liquid. These

metals all contract on solidification, so that the average

distance between the molecules is greater in the liquid

than in the solid state. The electrical conductivity varies
N as the product of the number of corpuscles per unit

volume, and A, the free path of a corpuscle. Since the

distance between the molecules is greater in the liquid

than in the solid state, we should expect the free path of

K the corpuscles to be greater, but if Xi A^ and N^

are

respectively the values of TV X in the solid and liquid states,
= N'l Ai 2 X-2 A2, and since A^ is greater than Aj, A'j must be

greater than 2 N^. A reference to equation (1) will show

that this involves a Peltier effect between the solid and the

liquid metal of about half the magnitude of that between

bismuth and antimonj', and thus, as these effects go,
exceedingly large. Now Fitzgerald, Minarelli and Ober-

meyer, as quoted by G. Wiedemann, " Elektricitat," ii., p. 289, could detect no sudden change in thermo-electric circuits

with these metals when they passed from the solid to the

liquid state, whereas if the number of free corpuscles had

diminished to one half, the effect would have been very

conspicuous. There is thus a discrepancy between the

results of the determination of the relative number of

corpuscles in the two states by data derived (1) from thermo-electric phenomena; (2) from their electricresistance. This discrepancy is so large that it is impossible to suppose

it is due to any errors in the data derived from experiment.

The Thomson Effect.
Lord Kelvin showed that in some metals an electric current carries heat from the hot to the cold parts of the
metal, while in other metals the transference of heat is in the opposite direction. Let us calculate what this trans-
ference of heat would be on the theory we are discussing.

—— — —

THEOKY OF METALLIC CONDUCTION. 77

B Let A

he a. bar of metal, and let the temperature

increase from A to B. If the pressure of the corpuscles

depends upon the temperature there must be electromotive

forces along the bar to keep the corpuscles from drifting

under these pressure differences. If p is the pressure of

the corpuscle at a point distant x from the end A, then the

force acting on the corpuscles included between two planes
+ at distances x, x ^x, from A, is, per unit area of these

-— planes, equal to A a;

and acts from right to left.

To

Cv CC

X balance this we must have an electromotive force tending

to move the corpuscles from left to right, determined by

the equation

aX

or-

= Xe 1 'iP, n aX

where n is the number of corpuscles per unit volume at a distance x from A. If 6 is the absolute temperature of the
bar at A we have (see page 65)

hence

= — ^V

n a. a.

3

^ A X« =

(a 110).

3% a X

Hence a corpuscle in travelling from x -\- S x to ,r will
abstract from the metal an amount of heat whose mechanical
X equivalent is e 8 x, or

— - - - (a n 6) a X.
6naX

The corpuscle when at x-]-d x has an amount of kinetic

^dx\ energy equal to a (6 -{-

while at a; its kinetic energy

is reduced to a 6, hence the corpuscle will communicate to the metal between x and x-\-dx an amount of heat equal

—

—

—

-

78 THE COEPUSCULAE THEOEY OF MATTEE.

d6
to «T— dx; thus the total amount of heat communicated by

the corpuscle to the metal is

lid, ~ de

„] ,

-1— aX

T.
d

n

;— dX

{"-n 6) r
i

dx,

or-

L-^l±iane))d6.

\

d nd&

I

If the current i is flowing in the direction in which x increases, the number of corpuscles which cross, unit area in unit time, in the opposite direction to the current is i\e, and the mechanical equivalent of the heat they communicate to the metal between the places where the temperatures of the metal are respectively 6 and 6 -\- dQ\s, equal to

e\

3 ndd

I

But if o- is the " specific heat of electricity in the metal," this amount of heat is by definition equal to
— i<TdQ

the minus sign being inserted because the current is

flowing from the cold to the hot part of the circuit;
— hence

— 0-= — - (a— - - (a n S) \

e\

6 n d6

J

3e I

de ^ ]

^ _ 2 a

(2)

The term -3— in the expression for o- is the same for all
metals, and since the electro -motive force round a thermoelectric circuit consisting of two metals only involves the diference of the specific heats of electricity in the metals, this term will not affect the electromotive force round the

THEOEY OF METALLIC CONDUCTION. 79

circuit. It will, however, affect the amount of heat developed in the conductor, and we shall find that unless

this term is very nearly balanced by the term -^

Tfl-^'^S '^j

the amount of heat developed by the flow of a current

through an unequally heated conductor would be far

greater than the amount actually observed.

For a/Qe is about 0'45 X 10*, so that the amount of

heat expressed by the first term in equation (1) developed by

a unit current in flowing between two places where the

temperature differed by 1° C. would equal "45 X 10V4-2 X

X 10'', or 1'07

10""' calories per second.

The metal in which this heat effect is largest is, as far as

our present knowledge extends, bismuth, and for this

metal the observed effect is only about "3 X 10"* calories,

or about 1/3 of. the amount expressed by the term a/3 e, and the effect in bismuth is very much greater than in any

other metal ; hence since o- is small compared with a/3 e, we

have by equation (1)

log 11 ^= - log 9 -\- a, constant

approximately, so that approximately n will vary as

e'^, i.e., the number of free corpuscles will vary approxi-

mately as the square root of the absolute temperature. If

the specific heat of electricity is positive the number of

free corpuscles will vary a little more rapidly than this

with the temperature. If the specific heat is negative it

will vary a little less rapidly. This variation of the

number of free corpuscles with the temperature involves a

still more rapid variation of the mean free path. For

(see p. 54) we have seen that the electrical conductivity is

nXn proportional to

j $. Now v is proportional to 6^ and n,

as we have just seen, varies approximately according to the

same law, hence the electrical conductivity is ajDproximately

proportional to A. the free path of the corpuscles in the

metal. But for many pure metals the electrical con-

ductivity varies approximately as the reciprocal of the

:

80 THE COEPUSCULAE THEOEY OF MATTER

absolute temperature ; hence for these metals the mean

free path must also vary with the temperature in the same

way, i.e., be inversely proportional to the absolute tempera-

ture. This rapid variation of the free path with the

temperature would not be possible if the structure of the

metal were analogous to that of a ^as compressed so that

the distances between the molecules were all diminished in

We the same proportion.

have seen that if the metal

consisted of aggregations of molecules which broke up to

some extent as the temperature rose, we might get a rapid variation of the mean free path, with the tempera-

ture. Since the free path, according to this theory,

varies approximately as the reciprocal of the absolute

temperature, the free paths at the low temperatures

which can be obtained by the use of liquid air or liquid

hydrogen ought to be much greater than at ordinary

laboratory temperatures. Thus the effects which depend

on the free path, such as the effect of magnetic force on electrical resistance, or the absorption of light by the metal (which should vary greatly according as the time of vibration of the light is greater or less than the time occupied by a corpuscle to describe its free path), would be

greatly affected by the lowering of the temperature

experiments on these points would be valuable tests of the

theory. If X varies as I/O, A./?- the time occupied by a corpuscle in describing its free path will vary as IjOi. The

velocity acquired by a corpuscle under a constant electric force will also vary as 1/^i, and will thus diminish rapidly

as the temperature increases.

The Number of Ekbe Coepusclbs in Unit Volume
OF THE Metal.
We can determine from the amount of heat absorbed or
developed when a current of electricity passes across the junction of two metals, the ratio of the number of corpuscles in unit volume of the two metals, and from the Thomson effect we can determine the change in this number for any one metal with the temperature. Hence,

—

THEOEY or METALLIC CONDUCTION. 81

if we can determine the number of corpuscles per unit

volume in any one metal at any one temperature, we can

deduce the number in any other metal at any temperature.
We shall now pass on to the consideration of methods

to determine the absolute number of corpuscles per unit

volume ; since the electrical conductivity gives us the value

of n \, a method of determining A will also lead to the
determination of n. We shall begin with those methods

which lead to the direct determination of n.

One of the simplest of these in principle is founded on the

consideration of what takes place when a charge of

electricity is communicated to a piece of metal. Let us, to

fix our ideas, suppose that the charge is a negative one and

that it is carried by free corpuscles. These corpuscles must

occupy a layer of finite thickness at the surface of the

metal, for if the layer were reduced to infinitesimal thick-

ness the pressure exerted by these corpuscles would be vastly

greater than the pressure exerted by the eorjDuscles in the

interior of the metal, and the consequence would be that

corpuscles would diffuse from the layer into the interior

of the metal. The corpuscles will diffuse until the electric

force exerted by their charges is just able to balance the

forces arising from the difference of pressure between the

We surface and the interior.

can calculate the thickness of

the layer occupied by the negative charge in the following
way : Let A be the face of .a flat piece of metal having a

negative charge; let n be the number of corpuscles per

unit volume before the charge was communicated to the
+ metal, n f the number at a point at a distance x from the

surface of the plate after the charge was communicated,
X p the pressure of the corpuscles at this distance, and the

electric force tending to stop the corpuscles from moving

from left to right. Then when the corpuscles have got

into a steady state

= but p

% a- (n -\- i) 6, where a 5 is the mean kinetic

T.M.

G-

——

—

82 THE COEPUSCULAE THEOEY OF MATTEE.

energy of a corpuscle at the absolute temperature 0, and
— since n does not depend upon x, we have, assuming that t
is small compared with n

~ — aO . = Xen;

'6

ax

but

dX = -J—

.

.

A TT $ e,

aX

if e is measured in electrostatic units, hence

or—

— i A€-''='

= —— where p^

A

2

—^

and ^ is a constant. To find A we have

4a 6

^ ^ Jq ^ d X Q, ii Q is the charge per unit area ; hence

= e A
substituting for i,

Q, or

e

Thus the value of i is appreciable until x is large com-

pared with 1/^; we may thus take 1/p or (a d/Q x e^ «)* as

the measure of the thickness of the layer occupied by the

X electricity ; substituting for a.6 and e the values 3"6

10"^*

and 3 X IQ-i", we find that, at 0° C,

Now since we have—

pi I

X d = d X

4 TT e f

X = Q 4 tt

""^
e

^"•1-

/''Xdx=^-^.

This is the difference in potential between the surface and a point in the interior, hence we see that if we communicate a charge of electricity to a hollow conductor whose surface

THEORY OF METALLIC CONDUCTION. 83

is kept at zero potential, the interior of that conductor will

not, as is usually assumed in electrostatics, remain at zero

potential, but will change by 4 tt Q/p where Q is the charge
per unit area of the conductor. Hence, if we measure the

change produced by a known charge we shall determine p
= and hence n by the equation 15 ir n 10^ p^. If the number

of corpuscles is comparable with the number of molecules

of the metal, which we may take as between 10^^ and 10^^,

p will be comparable with 10^, and so the thickness of the

layer through which the electricity is distributed will be of the order of 10"^ cm. In this case the change in the

potential of the interior produced by any feasible charge

will be small, but not perhaps too small to be measurable.

If the conductor were exposed to air at atmospheric pressure

the greatest value oi i-n- Q possible without sparking would be 100 in electrostatic measure. By embedding the con-

ductor in a solid dielectric, such as paraffin, we could

probably increase 4 tt Q to 1000 without discharge. Q If 4 ir
= is 10^ and p 10^, the change in potential would be 10"^ in

electrostatic measure, or 3 X 10"^ of a volt, and this ought

to be capable of measurement.

Experiments have been made by Bose and others to see if

the electrical resistance would be altered by giving a charge

of electricity to a very thin conductor ; so far these have led

We to negative results.

might at first sight expect that if

we increased the supply of negative corpuscles by com-

municating a charge of negative electricity to the strip of

metal we should increase the conductivity ; but this need

not necessarily be the case, for suppose the surface instead

of being flat were corrugated, then the charge would be all

at the tops of the corrugations ; but this would be quite out

of the way of a current flowing through the film, which would

take the short circuit through the base of the corrugations.

As the electricity only penetrates a distance comparable with

the size of a molecule, it is impossible to avoid an effect of

this kind, however carefully the surface is polished.
We can, however, find both lower and upper limits to the

number of free corpuscles, and as these limits lead to

G2

84 THE COEPUSCULAR THEOEY OF MATTER.

contradiction we shall, after investigating them, proceed to

the consideration of the question whether the other view

of the function and disposition of the corpuscles alluded to

on page 49 is less open to objection.
We can obtain a lower limit to the number of free

corpuscles per unit volume of a metal by the consideration

of the results of the experiments of Rubens and Hagen on

the reflection of long waves from the surface of metals. It

follows from these experiments that the electrical con-

ductivity of metals when waves whose length equals 25 jx,

/A being 10"^ cm., pass through them is the same as the

conductivity under steady electrical forces, and that even

when the waves are as short as 4 /x the electrical conductivity

is within about 20 per cent, of that for steady forces. "We

can easily show that if k is the conductivity under steady

forces ; then when the forces vary as sin n t the conductivity

— T SZTl 71 [T
will be proportional to k ^ ,^ , where 2

is the interval

between two collisions. Thus, unless this interval be small compared with the period of the electric force the con-
T ductivity will be very materially reduced. Thus if were
as great as one quarter of the period of the force, so that

T = «

g, the conductivity would be reduced to l/(ir/2)^, or -4

of its steady value. As the diminution of the conductivity

for light waves whose length is 4 ju. is less than this, we

conclude that the interval between two collisions is less

than one-quarter the period of this light, or less than
X 3'3 10"^^ sec. Hence %i, the velocity under unit electric

— 1 e

force, since

it

is

equal
^

to

k
2 7rt

T,

will be less than

— X ^ 3-3

10"^^ , and since k the conductivity is n e u, n

m k 10^^
will be greater than kjeii, i.e., than -..n 2 •

= For silver k is about 5 X 10"*, and since e/m

X 1-7

10''

= and e 10"^", we see that n for this metal must be greater

X than 1-8

10=*.

THEOEY OF METALLIC CONDUCTION. 85

It is this result which leads to the difficulty to which we

have alluded, for if there were this number of corpuscles

per unit volume, then, since the energy possessed by each

corpuscle at the temperature 6 is ad, the energy required

to raise the temperature of the corpuscles in unit volume

= of the metal by 1° C. is n a, and since a

1'5 X 10"^^

(see page 65), the energy which would have to be communi-

cated to unit volume of the silver to raise the temperature

of the corpuscles alone would be greater than 1'3 X 1'8

X 10^ ergs., or about 6 gram calories. But to raise the

temperature of a cubic centimetre of silver one degree, only

requires about 0'6 calories, and this includes the energy

required to raise the temperature of the atoms of the metal

We as well as that of the corpuscles.

thus get to a con-

tradiction. The value of the specific heats of the metals

shows that the corpuscles cannot exceed a certain number,

but this number is far too small to produce the observed

conductivities if the intervals between the collisions are as

small as is required by the behaviour of the metals in

Rubens' experiments.

CHAPTEK V.
THE SECOND THEORY OF ELECTRICAL CONDUCTION.
We shall now proceed to develop the second theory of
electrical conductivity and see whether it is as successful in explaining the relation between the thermal and electrical conductivities as the other one, and whether or not it is open to the same objections.
On this theory the corpuscles are supposed to be pulled
out of the atoms of the metal by the action of the surrounding atoms. In order to get a sufficiently definite idea of this process to enable us to calculate the amount of electrical
0© 0© e© 0© 0© 0©
FIG. 22.
conductivity which it would produce^ we shall suppose that in the metal there is a large number of doublets, formed by the union of a positively electrified atom with a negatively electrified one, and that the interchange of corpuscles takes place by a corpuscle leaving the negative component of one of these doublets and going to the positive constituent of the other. Under the action of the electric force theSe doublets tend to arrange themselves along that line in the way indicated in Fig. 22, much in the same way as the Grotthus chains in the old theory of electrolysis. The corpuscles moving in the direction of the arrows will give rise to a
drift of negative electricity against the direction of the
electric force or a current of positive electricity in the same
direction as the force.
We now proceed to calculate the magnitude of the current

—

THEORY OF ELECTRICAL CONDUCTION. 87

produced in this way. Consider a doublet formed by a
+ — charge of electricity e, connected with another charge e,

and placed in an electric field where the intensity of the

electric force is X. The potential energy of the doublet,

when its axis (the line joining the negative to the positive

charge) makes an angle with the direction of the electric

— X force, is

e d cos &, where d is the distance between the

charges in the doublet. If the doublets distribute them-

selves as they would in a gas in which the distribution of

potential energy follows Maxwell's law, the number pos-
sessing potential energy V will be proportional to «~^,.
= where l/h f a 6, ad being as before the mean kinetic of a,
molecule at the absolute temperature 6. Then the number
of doublets whose axes make an angle between 6 and -\- d6

with the direction of X, is proportional to e''^'"''^'''* sin 6 d 6,

and the average value of cos 6 for these doublets is equal to

r ^xedmse ^Qg g shied 6

Xe Now

dh will, unless the electric force greatly exceeds

the value it has in any ordinary case of metallic conduction,

be exceedingly small, for the potential difference through

which the charge e must fall in order to acquire the energy

possessed by a molecule at the temperature 0° C, is about

1/25 of a volt, and h is proportional to the reciprocal of

this energy, thus unless the electric field is so strong that

there is in the space between the two components of the

Xed doublet a fall of potential comparable with this, h

will

be small. But when this is. so

and—

de— /" ^^xedeose ggg gj,j e

^h Xed

•' o

3

de= /" e''^»<*"«« sin 6

2,

X — hence the mean value of cos 6 is tt /t e d, or — ^.
If each doublet discharges acorpusclejjtimes a second, then

—

—

88 THE COEPUSCULAE THEOEY OP MATTEE.

in consequence of the polarisation we have just investigated,

the resultant flow of corpuscles will be the same as if each

doublet discharged a corpuscle parallel but in the opposite

— X direction to the electric force p

^

>r times per second.

Hence, if n is the number of doublets per unit volume, b the

distance between the centres of the doublets, the current

through unit area will be equal to

X 2 e-
^ 9

d pnb

If we assume that the orientation of the axes of the

doublets in a metal follows the same law as in a gas, this

will be the expression for the current through the metal,

hence c the electrical conductivity will be given by the

expression

~ 2 e^ d p n b

^

9

^Te

Thermal Conductivity.
If we suppose that the kinetic enei-gy of the corpuscle in
a doublet is proportional to the kinetic energy, i.e., to the temperature of the doublet, the interchange of corpuscles will carry heat from the hot parts of the metal to the cold,
and will thus give rise to the conduction of heat. Let us suppose that the kinetic energy of a corpuscle when in a doublet at temperature 6 i^ a B. If the corpuscle goes
+ from a doublet where the temperature is 6 8 6 to one where
the temperature is 0, it will, when the latter doublet has lost a corpuscle to make way for the one coming, have
caused a transference of heat equal to a 8 5. Consider
now the transference of heat across a plane at right angles to the temperature gradient. The number of corpuscles
crossing this plane in unit time is equal to ^ n b . j). If the difference of temperature between the adjacent doublets
is 8 6, this will transfer
— lib p a h 6
o

—

——

THEOEY OF ELECTRICAL CONDUCTION. 89

units of heat across the plane in unit time, but as b is the

= distance between the doublets 8 6

cl 6
-j— b, where x is

measured in the direction of the flow of heat. Hence k the thermal conductivity is given by the equation

^ K

—1

n

,0 0'

p

a,

o

Thus on this theory k/c, the ratio of the thermal to the
electrical conductivity is equal to

3 ho?e
2 cle"-'

On the theory discussed before this ratio was equal to
^4 '
3 e"

In a substance in which the doublets are so numerous as to be almost in contact, d and b will be very nearly equal to each other, and in this case the ratio of the conductivities on the new theory would be to that on the old in the pro-
portion of 9 to 8. When the doublets are more sparsely
disseminated b will be greater than d and the ratio of the conductivities given by the new theory will be greater than that given by the old. The agreement between theory and the results of experiment is at least as good in the new theory as in the old, for the new theory gives for good conductors results of the right, order of magnitude, while the presence of the factor hjd indicates that the ratio is not an absolute constant for all substances but varies within small limits for good conductors and wider ones for bad ones. All this is in agreement with experience.

Theory op Connection betweesj Eadiant Energy and the
Temperature.
We have seen (p. 61) that Lorentz has shown that
the long wave radiation can be regarded as a part of the

;

90 THE COEPUSCULAE THEOEY OP MATTER

electromagnetic pulses emitted when the moving cor-

puscles come into collision with the atoms of the substance

through which they are moving, and he has given an

expression for the amount of the energy calculated on

this principle, which agrees well with that found by

experiment. But in the new theory, as in the old, we have

the sudden starting and stopping of charged corpuscles and

therefore the incessant production of electromagnetic pulses

these when resolved by the aid of Fourier's theorem will

be represented by a series of waves, having all possible

We wave lengths from zero to infinity.

must see if the

energy in the long wave length radiation at a given

temperature would on the new theory be approximately

equal to that on the old.

It will be necessary to examine a little more closely than

we have hitherto done the theory of the radiation from metals

due to the stopping and starting of electrified systems

We inside the metal.

have already (see p. 64) quoted an

expression due to Lorentz for the amount of the very long

wave length radiation due to the stopping of corpuscles.
We can, however, by the following method, obtain an

expression for the energy corresponding to any wave lengths

emitted by unit volume of the metal. In the case of very

long waves this expression coincides with that given by

Lorentz.
We have seen that when the motion of an electrified

particle is accelerated it gives off pulses of electric and

magnetic force. If /is the acceleration of a charged body

P 0, at the time t, the magnetic force at a point at a time

+ — t

c

— , is
c

O—P—- where d is the angle OP makes the

direction of the acceleration, and c the velocity of light.
P The energy per unit volume at due to this magnetic field

— H = is equal to -- where O TT

OP ^ ' ^/" , and the amount of C

this energy, which flows out radially through unit area at

H^ P, is c

/8 -T.

Integrating over the surface of the sphere

with centre and radius OP we find that the flow of energy

—

THEOEY OF ELECTRICAL CONDUCTION. 91

-^. due to the magnetic field is, in unit time -

There is

an equal flow of energy due to the electric field, hence the rate at which the charged body is radiating energy is
—2 e—^^f^^, a result first given by Larmor.
OG
The total amount of energy radiated is

When we know / as a function of t we can find the total
amount of energy radiated. If we wish to find how much

of this energy corresponds to light between assigned limits

of wave length we must express, / by Fourier's theorem, in

terms of an harmonic function of the time.

Let us take the following case as representing the

stopping and starting of a charged particle in a solid. The

particle starts from rest, for a time ii has a uniform

acceleration ji, at the end of this time it has got up speed

and now moves for a time t^ with uniform velocity, at

the end of this time it comes into collision, and we suppose
— that now an acceleration fi acts for a time tx and reduces

it to rest again,

Thus /when expressed as a fimction of the time, if the

= time i

is taken as the time when it is at the middle

of its free path, has the following values

= = — = — + /

from i

00 to t

Til

I)

f^P from t = -

(«i

+

- = to i
I)

I

= — = /'

= from t

^ to i ^

2

2

/= -^fromi=|to< = fi+|

J ^ from i = (i-}-itoi=oo

—

92 THE COEPUSCULAR THEOEY OF MATTER.

Now by Fourier's theorem we have, if <p (t) is a function

of t,

= r — dq ^ (f)

1 /+ TT J n

J " cj> (m) cos q (u
— rr.

t)

du

applying this to onr case, and performing the integrations,

we find

= — r
/

^.

o
p-

r^ +— I
I

ti
sm sin a -^

.

q (-f^2—^ h-)

sm -'a

'I

d .

,

qt.

q.

Now Lord Eayleigh has shown {Philosophical Maqazine,
June, 1889, p. 466) that if

W = /" <^

-

/i (g) sin qt . d q

= /+" (^(t))^rfi l f1{fx{qy)'dq,

hence

— /3

16 P

^it

I

U 1"

/

sill-

q

~ .

sm^

q

-
q-

:^
d q^

The energy radiated from the charged body is equal to

3 7-^-J '^'

J ' '-'

?

'^^'

hence if there are s collisions per unit volume per second the energy radiated from unit volume per second is

a.c J

-,

dq,

and the energy corresponding to waves "which have a frequency between q and q -^ d q is equal to

•9

fi

.o

Ui n~ ^2)

THEORY OF ELECTRICAL CONDUCTION. 93

In the case considered by Lorentz the waves are very

+ long, i.e., q is small compared with 1/ii, or l/(fi

h) and

= s

"- ; in this case the preceding expression reduces to

A

'lll^lB^t.Hh + krq'dq.

{B)

X. d ir C

= Now /3

vjh, and if h, i.e., the time occupied by the

collisions is small compared with h the time spent in
— describing the free path, X v t^, so that the preceding

expressions become
— — n V 2 «^ A^ q'2 a7 q.
O TT C

Now — /c, the electric conductivity,

np A "1 ?^

^

^, so that the

energy radiated from unit volume in unit time is

k (f dq.

We can get an expression for the stream of radiant
energy by using the princijple that when things have got
into a steady state, the amount of energy absorbed by unit
volume in unit time is equal to the energy radiated from
E that volume in the same time. If is the electric force in
the stream of radiant energy i the intensity of the current,
E the energy absorbed in unit volume per unit time is i,
~ or, k E^ since i k E. Now W, the energy per unit
K— K E^
volume, is equal to -r where is the specific inductive

capacity in electromagnetic units ; hence the rate at which

—^ energy is absorbed is

W, and this, when things are

in a steady state, must be equal to the energy radiated, hence we have

W—~
K 3^c

k

q^

d

q,

— :
94 THE COEPUSCULAE THEOEY OF MATTEE.
the energy in the stream of radiant energy due to waves having a frequency between q and q -\- d q is equal to

If /A is the refractive index of the substance
K = ^vc^
hence the density of the stream of radiant energy is

a result which Lorentz has shown agrees well with the actual
determinations of the radiation. We must remember that

this result only holds when the frequency of the waves is

very small, not merely because it is only in this case that

the expression A reduces to B, but also because when the

frequency is large the conductivity k will not have the

value we have assigned to it.
To return to the expression A for the amount of energy radiated. We see that the maximum amount of the energy

for a given difference of frequency will be when the fre-

+ quency is such that qh is small and q {h

^2) finite, i.e.,

when the time of vibration of the light is comparable with

the time occupied in running over the free path

the energy in the light with this frequency is greater

than in the light whose frequency is very small ; we can,

however, easily show that, as we should expect, the

greatest amount of energy is in the waves whose time of

vibration is comparable with ti, the time occupied by a

collision.
We — can see this as follows ; since the rate of radiation of

U energy is - -^!—, then

the amount radiated by one

oc

corpuscle in the ease we have considered is

|lV^*i+|^-^^*. or ^-'L^^.,,

3c

3c

3c