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GIFT OF

Professor Whitten

B. G. TEUBNERS SAMMLUNG VON LEERBUCHERN

AUF DEM GEBIETE DER

MATHEMATIS CHEN WIS SEN SCHAFTEN

MIT EINSCHLUSS IHRER ANWENDUNGEN

BAND XXIX

THE THEORY OF ELECTRONS

AND ITS APPLICATIONS TO THE PHENOMENA

OF LIGHT AND KADIANT HEAT

A COURSE OF LECTURES DELIVERED IN COLUMBIA

UNIVERSITY, NEW YORK, IN MARCH AND APRIL 1906

BY

H. A. LORENTZ

PROFESSOR IN THE UNIVERSITY OF LEIDEN

LECTURER IN MATHEMATICAL PHYSICS

IN COLUMBIA UNIVERSITY FOR 1905-1906

SECOND EDITION

LEIPZIG: B. G. TEUBNER

1916

NEW YORK: G. E. STECHERT & Co., 129-133 WEST 20 STREET

\

*.

SOHUTZFOBMEI, FOE DIE VEBEINIGTBN STAATEN VON AMERTKA:

COPYRIOHT 1916 BY B. O. TEUBNEE IN LEIPZIG.

ALLE RECHTE, EINSOHLEESSLICH DES OBEESETZUNOSRECHTS, VOBB EH ALTEN.

PEEFAOE.

The publication of these lectures, which I delivered in Columbia

University in the spring of 1906, has been unduly delayed, chiefly

on account of my wish to give some further development to the sub-

ject, so as to present it in a connected and fairly complete form;

for this reason I have not refrained from making numerous additions.

Nevertheless there are several highly interesting questions, more or

less belonging to the theory of electrons, which I could but slightly

touch upon. I could no more than allude in a note to Voigt's

Treatise on magneto-optical phenomena, and neither Planck's views

on radiation, nor Einstein's principle of relativity have received an

adequate treatment.

In one other respect this book will, I fear, be found very deficient.

No space could be spared for a discussion of the different ways in

which the fundamental principles may be established, so that, for in-

stance, there was no opportunity to mention the important share that

has been taken in the development of the theory by L arm or and

Wiechert.

It is with great pleasure that I express my thanks to Professor

A. P. Wills for his kindness in reading part of the proofs, and to

the publisher for the care he has bestowed on my work.

Leiden, January 1909.

H. A. Lorentz.

In this new edition the text has been left nearly unchanged.

I have confined myself to a small number of alterations and additions

in the foot-notes and the appendix.

Haarlem, December 1915.

H. A. L.

M44286

CONTENTS.

Chapter Page

I. General principles. Theory of free electrons . i

II. Emission and absorption of heat. . . 68

HI. Theory of the Zeeman-effect ...:.'... . . . .,; ,,.. .... . 9$

IV. Propagation of light in a body composed of molecules. Theory of the

inverse Zeeman-efFect . 132

V. Optical phenomena in moving bodies ............... 168

Notes. ....... .... ... . . .'. . . . .V".' . ! J . . .* . . 234

Index : : \ . .- ' : -. .-.V'.'- . : - V - i - -'. . .-'^ .:..-. .-'340

CHAPTER I.

GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

The theory of electrons, on which I shall have the honor to

lecture before you, already forms so vast a subject, that it will be

impossible for me to treat it quite completely. Even if I confine

myself to a general review of this youngest branch of the science

of electricity, to its more important applications in the domain

of light and radiant heat, and to the discussion of some of the

difficulties that still remain, I shall have to express myself as con-

cisely as possible, and to use to the best advantage the time at our

disposal.

In this, as in every other chapter of mathematical physics, we

may distinguish on the one hand the general ideas and hypotheses

of a physical nature involved, and on the other the array of

mathematical formulae and developments by which these ideas and

hypotheses are expressed and worked out. I shall try to throw a

clear light on the former part of the subject, leaving the latter part

somewhat in the background and omitting all lengthy calculations,

which indeed may better be presented in a book than in a lecture. 1 )

1. As to its physical basis, the theory of electrons is an off-

spring of the great theory of electricity to which the names of

Faraday and Maxwell will be for ever attached.

You all know this theory of Maxwell, which we may call the

general theory of the electromagnetic field, and in which we con-

stantly have in view the state of the matter or the medium by which

the field is occupied. While speaking of this state, I must immediately

call your attention to the curious fact that, although we never lose

sight of it, we need by no means go far in attempting to form an

image of it and, in fact, we cannot say much about it. It is true

that we may represent to ourselves internal stresses existing in the

1) In this volume such calculations as I have only briefly indicated in iny

lectures are given at full length in the appendix at the end.

Lorentz. Theory of electron*. 2<l Kd. 1

2 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

medium surrounding an electrified body or a magnet, that we may

think of electricity as of some substance or fluid, free to move in

a conductor and bound to positions of equilibrium in a dielectric,

an 4. that w.e t mjay ;also conceive a magnetic field V as the seat of

certify :invisiblel Smbtions, rotations for example around the lines of

force. t . .AIL. this has;. been done by many physicists and Maxwell

jdmsefcias* s4ti 4ne* exa'mple. Yet, it must not be considered as

really necessary; we can develop the theory to a large extent and

elucidate a great number of phenomena, without entering upon

speculations of this kind. Indeed, on account of the difficulties into

which they lead us, there has of late years been a tendency to avoid

them altogether and to establish the theory on a few assumptions

of a more general nature.

The first of these is, that in an electric field there is a certain

state of things which gives rise to a force acting on an electrified

body and which may therefore be symbolically represented by the

force acting on such a body per unit of charge. This is what we

call the electric farce, the symbol for a state in the medium about

whose nature we shall not venture any further statement. The second

assumption relates to a magnetic field. Without thinking of those

hidden rotations of which I have just spoken, we can define this by

the so called magnetic force, i. e. the force acting on a pole of unit

strength.

After having introduced these two fundamental quantities, we

try to express their mutual connexions by a set of equations which

are then to be applied to a large variety of phenomena. The mathe-

matical relations have thus come to take a very prominent place,

so that Hertz even went so far as to say that, after all, the theory

of Maxwell is best defined as the system of Maxwell's equations.

We shall not use these formulae in the rather complicated form

in which they can be found in Maxwell's treatise, but in the clearer

and more condensed form that has been given them by Heaviside

and Hertz. In order to simplify matters as much as possible, I shall

further introduce units 1 ) of such a kind that we get rid of the larger

part of such factors as ATI and l/inr, by which the formulae were

originally encumbered. As you well know, it was Heaviside who

most strongly advocated the banishing of these superfluous factors and

it will be well, I think, to follow his advice. Our unit of electricity

will therefore be ]/4jr times smaller than the usual electrostatic unit.

1) The units and the notation of these lectures (with the exception of the

letters serving to indicate vectors) have also been used in my articles on

Maxwell's. Theory and the Theory of Electrons, in the ,,Encyklopadie der

mathematischen Wissenschaften", Vol. V, 13 and 14.

MATHEMATICAL NOTATION. 3

This choice haying been made, we have at the same time fixed for

every case the number by which the electric force is to be represented.

As to the magnetic force, we continue to understand by it the force

acting on a north pole of unit strength; the latter however is like-

wise ]/4 JT times smaller than the unit commonly used.

2. Before passing on to the electromagnetic equations, it will be

necessary to say a few words about the choice of the axes of coor-

dinates and about our mathematical notation. In the first place, we

shall always represent a line by s, a surface by 6 and a space by S,

and we shall write ds, de, dS respectively for an element of a line,

a surface, or a space. In the case of a surface, we shall often have

to consider the normal to it; this will be denoted by n. It is always

to be drawn towards a definite side and we shall agree to draw it

towards the outside, if we have to do with a closed surface.

The normal may be used for indicating the direction of a

rotation in the surface. We shall say that the direction of a rotation

in a plane and that of a normal to the plane correspond to each

other, if an ordinary or right-handed screw turned in the direction

of the rotation advances in that of the normal. This being agreed

upon, we may add that the axes of coordinates will be chosen in

such a manner that OZ corresponds to a rotation of 90 from OX

towards OY.

We shall further find it convenient to use a simple kind of

vector analysis and to distinguish vectors and scalar quantities by

different sorts of letters. Conforming to general usage, I shall denote

scalars by ordinary Latin or Greek letters. As to the vectors, I have,

in some former publications, represented them by German letters.

On the present occasion however, it seems to me that Latin letters,

either capital or small ones, of the so called Clarendon type, e. g.

A, P, C etc. are to be preferred. I shall denote by A A the component

of a vector A in the direction \ by A x , A y , A, its components parallel

to the axes of coordinates, by A, the component in the direction of

a line s and finally by A n that along the normal to a surface.

The magnitude of a vector A will be represented by A | . For

its square however we shall simply write A 2 .

Of the notions that have been introduced into vector analysis,

I must recall to your minds those of the sum and of the difference

of vectors, and those of the scalar product and the vector product of

two vectors A and B. The first of these ,,products", for which we

shall use the symbol

(A B),

is the scalar quantity defined by the formula

(A B) - | A| | B| cos (A, B) - AA + A,B, + A,B,

i*

4 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

The vector product, for which we shall write

[A-B],

is a vector perpendicular to the plane through A and B, whose

direction corresponds to a rotation by less than 180 from the direc-

tion of A towards that of B, and whose magnitude is given by the

area of the parallelogram described with A and B as sides. Its

components are

[A BJ. - A,B, - A,B y , [A - B], = AA - A X B,,

[A,B].-Mr-*A

In many cases we have to consider a scalar quantity <p or a

vector A which is given at every point of a certain space. If go is a

continuous function of the coordinates, we can introduce the vector

having for its components

d<p dtp dq>

fo> Wy J Ts'

This can easily be shown to be perpendicular to the surface

<p = const.

and we may call it the gradient of qp, which, in our formulae, we

shall shorten to ,,grad qp".

A space at every point of which a vector A has a definite

direction and a definite magnitude may be called a vector field, and

the lines which at every point indicate the direction of A may be

spoken of as vector- or direction-lines. In such a vector field, if

A x , A , A 4 are continuous functions of the coordinates, we can intro-

duce for every point a certain scalar quantity and a certain new

vector, both depending on the way in which A changes from point

to point, and both having the property of being independent of the

choice of the axes of coordinates. The scalar quantity is called the

divergence of A and defined by the formula

The vector is called the rotation or the curl of A; its com-

ponents are

_ _ .

3y "" dz > dz ' dx> dx " dy >

and it will be represented by the symbol ,,rot A".

If the divergence of a vector is at all points, its distribution

over space is said to be solenoidcd. On the other hand, we shall

speak of an irrotational distribution, if at all points we have

rot A = 0.

FUNDAMENTAL EQUATIONS FOR THE ETHER. 5

In order to complete our list of notations, I have only to add

that the symbol A is an abbreviation for

and that not only scalars but also vectors may be differentiated with

o

respect to the coordinates or the time. For example, ~ means a

vector whose components are

dk x Zk y dk,

3x> dx> dx>

and -TTT has a similar meaning. A differentiation with respect to the

time t will be often represented by a dot, a repeated differentiation

of the same kind by two dots, etc.

3. We are now prepared to write down the fundamental equa-

tions for the electromagnetic field in the form which they take for

the ether. We shall denote by d the electric force, the same symbol

serving for the dielectric displacement, because in the ether this has

the same direction and, on account of the choice of our units, the

same numerical magnitude as the electric force. We shall further

represent by h the magnetic force and by c a constant depending on

the properties of the ether. A third vector is the current C, which

now consists only of the displacement current of Maxwell. It exists

wherever the dielectric displacement d is a function of the time, and

is given by the formula

c - d. (i)

In the form of differential equations, the formulae of the electro-

magnetic field may now be written as follows:

div d = 0, (2)

div h = 0, (3)

; ' " roth = |c = |d, " (4)

rotd = -yh. (5)

The third equation, conjointly with the second, determines the

magnetic field that is produced by a given distribution of the

current C. As to the last equation, it expresses the law according

to which electric forces are called into play in a system with a

variable magnetic field, i. e. the law of what is ordinarily called

electromagnetic induction. The formulae (1), (4) and (5) are vector

equations and may each be replaced by three scalar equations relating

to the separate axes of coordinates.

I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

Thus (1) is equivalent to

and (4) to

r etc

^ - t?l/V.

J

*-\ ."\ ~~" ^ -

oy oz c ot

The state of things that is represented by our fundamental

equations consists, generally speaking, in a propagation with a velo-

city c. Indeed, of the six quantities d x , d y , d,, h,,., h y , h^, five may

be eliminated 1 ), and we then find for the remaining one # an equation

of the form

This is the typical differential equation for a disturbance of the

state of equilibrium, travelling onwards with the speed c.

Though all the solutions of our equations have this general

character, yet there are a very large variety of them. The simplest

corresponds to a system of polarized plane waves. For waves of this

kind, we may have for example

d y = a cos n (t - -], h z = acoswu - -1, (7)

all other components of d and h being 0.

I need not point out to you that really, in the state represented

by these formulae, the values of d y and h s , which for a certain value

of t exist at a point with the coordinate x, will after a lapse of

time dt be found in a point whose coordinate is x + cdt. The

constant a is the amplitude and n is the frequency, i. e. the number

of vibrations in a time 2n. If n is high enough, we have to do

with a beam of plane polarized light, in which, as you know already,

the electric and the magnetic vibrations are perpendicular to the ray

as well as to each other.

Similar, though perhaps much more complicated formulae may

serve to represent the propagation of Hertzian waves or the radiation

which, as a rule, goes forth from any electromagnetic system that is

not in a steady state. If we add the proper boundary conditions,

such phenomena as the diffraction of light by narrow openings or

its scattering by small obstacles may likewise be made to fall under

our system of equations.

The formulae for the ether constitute the part of electromagnetic

theory that is most firmly established. Though perhaps the way in

which they are deduced will be changed in future years, it is

1) See Note 1 (Appendix).

GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. 7

hardly conceivable that the equations themselves will have to be

altered. It is only when we come to consider the phenomena in

ponderable bodies, that we are led into uncertainties and doubts.

4. There is one way of treating these phenomena that is compa-

ratively safe and, for many purposes, very satisfactory. In following

it, we simply start from certain relations that may be considered as

expressing, in a condensed form, the more important results of electro-

magnetic experiments. We have now to fix our attention on four

vectors, the electric force E, the magnetic force H, the current of

electricity C and the magnetic induction B. These are connected by

the following fundamental equations:

div C = 0, (8)

div B = 0, (9)

rotH-~C, (10)

rotE = -{B, (11)

presenting the same form as the formulae we have used for the ether.

In the present case however, we have to add the relation between

E and C on the one hand, and that between H and B on the other.

Confining ourselves to isotropic bodies, we can often describe the

phenomena with sufficient accuracy by writing for the dielectric dis-

placement

D - < E, (12)

a vector equation which expresses that the displacement has the same

direction as the electric force and is proportional to it. The current

in this case is again Maxwell's displacement current

C = D. (13)

In conducting bodies on the other hand, we have to do with a

current of conduction, given by

J - tf E, (14)

where a is a new constant. This vector is the only current and

therefore identical to what we have called C, if the body has only

the properties of a conductor. In some cases however, one has been

led to consider bodies endowed with the properties of both conductors

and dielectrics. If, in a substance of this kind, an electric force is

supposed to produce a dielectric displacement as well as a current

of conduction, we may apply at the same time (12) and (14), writing

for the total current

C = D -f J = E 4- <*E. (15)

8 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

Finally, the simplest assumption we C 4 an make as to the relation

between the magnetic force and the magnetic induction is expressed

by the formula

B - f*H, (16)

in which is a new constant.

5. Though the equations (12), (14) and (16) are useful for the

treatment of many problems, they cannot be said to be applicable to

all cases. Moreover, even if they were so, this general theory, in

which we express the peculiar properties of different ponderable

bodies by simply ascribing to each of them particular values of the

dielectric constant f, the conductivity 6 and the magnetic permeabi-

lity p, can no longer be considered as satisfactory, when we wish to

obtain a deeper insight into the nature of the phenomena. If we

want to understand the way in which electric and magnetic properties

depend on the temperature, the density, the chemical constitution or

the crystalline state of substances, we cannot be satisfied with simply

introducing for each substance these coefficients, whose values are

to be determined by experiment; we shall be obliged to have recourse

to some hypothesis about the mechanism that is at the bottom of

the phenomena.

It is by this necessity, that one has been led to the conception

of electrons, i. e. of extremely small particles, charged with electricity,

which are present in immense numbers in all ponderable bodies, and

by whose distribution and motions we endeavor to explain all electric

and optical phenomena that are not confined to the free ether. My

task will be to treat some of these phenomena in detail, but I may

at once say that, according to our modern views, the electrons in

a conducting body, or at least a certain part of them, are supposed

to be in a free state, so that they can obey an electric force by

which the positive particles are driven in one, and the negative

electrons in the opposite direction. In the case of a non-conducting

substance, on the contrary, we shall assume that the electrons are

bound to certain positions of equilibrium. If, in a metallic wire, the

electrons of one kind, say the negative ones, are travelling in one

direction, and perhaps those of the opposite kind in the opposite

direction, we have to do with a current of conduction, such as may

lead to a state in which a body connected to one end of the wire

has an excess of either positive or negative electrons. This excess,

the charge of the body as a whole, will, in the state of equilibrium

and if the body consists of a conducting substance, be found in a

very thin layer at its surface.

In a ponderable dielectric there can likewise be a motion of the

ELECTRONS. 9

electrons. Indeed, though we shall think of each of them as having

a definite position of equilibrium, we shall not suppose them to be

wholly immovable. They can be displaced by an electric force exerted

by the ether, which we conceive to penetrate all ponderable matter,

a point to which we shall soon have to revert. Now, however, the

displacement will immediately give rise to a new force by which the

particle is pulled back towards its original position, and which we may

therefore appropriately distinguish by the name of elastic force. The

motion of the electrons in non-conducting bodies, such as glass and

sulphur, kept by the elastic force within certain bounds, together

with the change of the dielectric displacement in the ether itself,

now constitutes what Maxwell called the displacement current.

A substance in which the electrons are shifted to new positions is

said to be electrically polarized.

Again, under the influence of the elastic forces, the electrons can

vibrate about their positions of equilibrium. In doing so, and perhaps

also on account of other more irregular motions, they become the

centres of waves that travel outwards in the surrounding ether and

can be observed as light if the frequency is high enough. In this

manner we can account for the emission of light and heat. As to

the opposite phenomenon, that of absorption, this is explained by

considering the vibrations that are communicated to the electrons

by the periodic forces existing in an incident beam of light. If the

motion of the electrons thus set vibrating does not go on undisturbed,

but is converted in one way or another into the irregular agitation

which we call heat, it is clear that part of the incident energy will

be stored up in the body, in other terms that there is a certain ab-

sorption. Nor is it the absorption alone that can be accounted for

by a communication of motion to the electrons. This optical resonance,

as it may in many cases be termed, can likewise make itself felt

even if there is no resistance at all, so that the body is perfectly

transparent. In this case also, the electrons contained within the

molecules will be set in motion, and though no vibratory energy is

lost, the oscillating particles will exert an influence on the velocity

with which the vibrations are propagated through the body. By

taking account of this reaction of the electrons we are enabled to

establish an electromagnetic theory of the refrangibility of light, in

its relation to the wave-length and the state of the matter, and to

form a mental picture of the beautiful and varied phenomena of

double refraction and circular polarization.

On the other hand, the theory of the motion of electrons in

metallic bodies has been developed to a considerable extent. Though

here also much remains to be done, new questions arising as we

proceed, we can already mention the important results that have

10 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

been reached by Riecke, Drude and J. J. Thomson. 1 ) The funda-

mental idea of the modern theory of the thermic and electric pro-

perties of metals is, that the free electrons in these bodies partake

of the heat-motion of the molecules of ordinary matter, travelling in

all directions with such velocities that the mean kinetic energy of

each of them is equal to that of a gaseous molecule at the same

temperature. If we further suppose the electrons to strike over and

over again against metallic atoms, so that they describe irregular

zigzag-lines, we can make clear to ourselves the reason that

metals are at the same time good conductors of heat and of electri-

GIFT OF

Professor Whitten

B. G. TEUBNERS SAMMLUNG VON LEERBUCHERN

AUF DEM GEBIETE DER

MATHEMATIS CHEN WIS SEN SCHAFTEN

MIT EINSCHLUSS IHRER ANWENDUNGEN

BAND XXIX

THE THEORY OF ELECTRONS

AND ITS APPLICATIONS TO THE PHENOMENA

OF LIGHT AND KADIANT HEAT

A COURSE OF LECTURES DELIVERED IN COLUMBIA

UNIVERSITY, NEW YORK, IN MARCH AND APRIL 1906

BY

H. A. LORENTZ

PROFESSOR IN THE UNIVERSITY OF LEIDEN

LECTURER IN MATHEMATICAL PHYSICS

IN COLUMBIA UNIVERSITY FOR 1905-1906

SECOND EDITION

LEIPZIG: B. G. TEUBNER

1916

NEW YORK: G. E. STECHERT & Co., 129-133 WEST 20 STREET

\

*.

SOHUTZFOBMEI, FOE DIE VEBEINIGTBN STAATEN VON AMERTKA:

COPYRIOHT 1916 BY B. O. TEUBNEE IN LEIPZIG.

ALLE RECHTE, EINSOHLEESSLICH DES OBEESETZUNOSRECHTS, VOBB EH ALTEN.

PEEFAOE.

The publication of these lectures, which I delivered in Columbia

University in the spring of 1906, has been unduly delayed, chiefly

on account of my wish to give some further development to the sub-

ject, so as to present it in a connected and fairly complete form;

for this reason I have not refrained from making numerous additions.

Nevertheless there are several highly interesting questions, more or

less belonging to the theory of electrons, which I could but slightly

touch upon. I could no more than allude in a note to Voigt's

Treatise on magneto-optical phenomena, and neither Planck's views

on radiation, nor Einstein's principle of relativity have received an

adequate treatment.

In one other respect this book will, I fear, be found very deficient.

No space could be spared for a discussion of the different ways in

which the fundamental principles may be established, so that, for in-

stance, there was no opportunity to mention the important share that

has been taken in the development of the theory by L arm or and

Wiechert.

It is with great pleasure that I express my thanks to Professor

A. P. Wills for his kindness in reading part of the proofs, and to

the publisher for the care he has bestowed on my work.

Leiden, January 1909.

H. A. Lorentz.

In this new edition the text has been left nearly unchanged.

I have confined myself to a small number of alterations and additions

in the foot-notes and the appendix.

Haarlem, December 1915.

H. A. L.

M44286

CONTENTS.

Chapter Page

I. General principles. Theory of free electrons . i

II. Emission and absorption of heat. . . 68

HI. Theory of the Zeeman-effect ...:.'... . . . .,; ,,.. .... . 9$

IV. Propagation of light in a body composed of molecules. Theory of the

inverse Zeeman-efFect . 132

V. Optical phenomena in moving bodies ............... 168

Notes. ....... .... ... . . .'. . . . .V".' . ! J . . .* . . 234

Index : : \ . .- ' : -. .-.V'.'- . : - V - i - -'. . .-'^ .:..-. .-'340

CHAPTER I.

GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

The theory of electrons, on which I shall have the honor to

lecture before you, already forms so vast a subject, that it will be

impossible for me to treat it quite completely. Even if I confine

myself to a general review of this youngest branch of the science

of electricity, to its more important applications in the domain

of light and radiant heat, and to the discussion of some of the

difficulties that still remain, I shall have to express myself as con-

cisely as possible, and to use to the best advantage the time at our

disposal.

In this, as in every other chapter of mathematical physics, we

may distinguish on the one hand the general ideas and hypotheses

of a physical nature involved, and on the other the array of

mathematical formulae and developments by which these ideas and

hypotheses are expressed and worked out. I shall try to throw a

clear light on the former part of the subject, leaving the latter part

somewhat in the background and omitting all lengthy calculations,

which indeed may better be presented in a book than in a lecture. 1 )

1. As to its physical basis, the theory of electrons is an off-

spring of the great theory of electricity to which the names of

Faraday and Maxwell will be for ever attached.

You all know this theory of Maxwell, which we may call the

general theory of the electromagnetic field, and in which we con-

stantly have in view the state of the matter or the medium by which

the field is occupied. While speaking of this state, I must immediately

call your attention to the curious fact that, although we never lose

sight of it, we need by no means go far in attempting to form an

image of it and, in fact, we cannot say much about it. It is true

that we may represent to ourselves internal stresses existing in the

1) In this volume such calculations as I have only briefly indicated in iny

lectures are given at full length in the appendix at the end.

Lorentz. Theory of electron*. 2<l Kd. 1

2 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

medium surrounding an electrified body or a magnet, that we may

think of electricity as of some substance or fluid, free to move in

a conductor and bound to positions of equilibrium in a dielectric,

an 4. that w.e t mjay ;also conceive a magnetic field V as the seat of

certify :invisiblel Smbtions, rotations for example around the lines of

force. t . .AIL. this has;. been done by many physicists and Maxwell

jdmsefcias* s4ti 4ne* exa'mple. Yet, it must not be considered as

really necessary; we can develop the theory to a large extent and

elucidate a great number of phenomena, without entering upon

speculations of this kind. Indeed, on account of the difficulties into

which they lead us, there has of late years been a tendency to avoid

them altogether and to establish the theory on a few assumptions

of a more general nature.

The first of these is, that in an electric field there is a certain

state of things which gives rise to a force acting on an electrified

body and which may therefore be symbolically represented by the

force acting on such a body per unit of charge. This is what we

call the electric farce, the symbol for a state in the medium about

whose nature we shall not venture any further statement. The second

assumption relates to a magnetic field. Without thinking of those

hidden rotations of which I have just spoken, we can define this by

the so called magnetic force, i. e. the force acting on a pole of unit

strength.

After having introduced these two fundamental quantities, we

try to express their mutual connexions by a set of equations which

are then to be applied to a large variety of phenomena. The mathe-

matical relations have thus come to take a very prominent place,

so that Hertz even went so far as to say that, after all, the theory

of Maxwell is best defined as the system of Maxwell's equations.

We shall not use these formulae in the rather complicated form

in which they can be found in Maxwell's treatise, but in the clearer

and more condensed form that has been given them by Heaviside

and Hertz. In order to simplify matters as much as possible, I shall

further introduce units 1 ) of such a kind that we get rid of the larger

part of such factors as ATI and l/inr, by which the formulae were

originally encumbered. As you well know, it was Heaviside who

most strongly advocated the banishing of these superfluous factors and

it will be well, I think, to follow his advice. Our unit of electricity

will therefore be ]/4jr times smaller than the usual electrostatic unit.

1) The units and the notation of these lectures (with the exception of the

letters serving to indicate vectors) have also been used in my articles on

Maxwell's. Theory and the Theory of Electrons, in the ,,Encyklopadie der

mathematischen Wissenschaften", Vol. V, 13 and 14.

MATHEMATICAL NOTATION. 3

This choice haying been made, we have at the same time fixed for

every case the number by which the electric force is to be represented.

As to the magnetic force, we continue to understand by it the force

acting on a north pole of unit strength; the latter however is like-

wise ]/4 JT times smaller than the unit commonly used.

2. Before passing on to the electromagnetic equations, it will be

necessary to say a few words about the choice of the axes of coor-

dinates and about our mathematical notation. In the first place, we

shall always represent a line by s, a surface by 6 and a space by S,

and we shall write ds, de, dS respectively for an element of a line,

a surface, or a space. In the case of a surface, we shall often have

to consider the normal to it; this will be denoted by n. It is always

to be drawn towards a definite side and we shall agree to draw it

towards the outside, if we have to do with a closed surface.

The normal may be used for indicating the direction of a

rotation in the surface. We shall say that the direction of a rotation

in a plane and that of a normal to the plane correspond to each

other, if an ordinary or right-handed screw turned in the direction

of the rotation advances in that of the normal. This being agreed

upon, we may add that the axes of coordinates will be chosen in

such a manner that OZ corresponds to a rotation of 90 from OX

towards OY.

We shall further find it convenient to use a simple kind of

vector analysis and to distinguish vectors and scalar quantities by

different sorts of letters. Conforming to general usage, I shall denote

scalars by ordinary Latin or Greek letters. As to the vectors, I have,

in some former publications, represented them by German letters.

On the present occasion however, it seems to me that Latin letters,

either capital or small ones, of the so called Clarendon type, e. g.

A, P, C etc. are to be preferred. I shall denote by A A the component

of a vector A in the direction \ by A x , A y , A, its components parallel

to the axes of coordinates, by A, the component in the direction of

a line s and finally by A n that along the normal to a surface.

The magnitude of a vector A will be represented by A | . For

its square however we shall simply write A 2 .

Of the notions that have been introduced into vector analysis,

I must recall to your minds those of the sum and of the difference

of vectors, and those of the scalar product and the vector product of

two vectors A and B. The first of these ,,products", for which we

shall use the symbol

(A B),

is the scalar quantity defined by the formula

(A B) - | A| | B| cos (A, B) - AA + A,B, + A,B,

i*

4 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

The vector product, for which we shall write

[A-B],

is a vector perpendicular to the plane through A and B, whose

direction corresponds to a rotation by less than 180 from the direc-

tion of A towards that of B, and whose magnitude is given by the

area of the parallelogram described with A and B as sides. Its

components are

[A BJ. - A,B, - A,B y , [A - B], = AA - A X B,,

[A,B].-Mr-*A

In many cases we have to consider a scalar quantity <p or a

vector A which is given at every point of a certain space. If go is a

continuous function of the coordinates, we can introduce the vector

having for its components

d<p dtp dq>

fo> Wy J Ts'

This can easily be shown to be perpendicular to the surface

<p = const.

and we may call it the gradient of qp, which, in our formulae, we

shall shorten to ,,grad qp".

A space at every point of which a vector A has a definite

direction and a definite magnitude may be called a vector field, and

the lines which at every point indicate the direction of A may be

spoken of as vector- or direction-lines. In such a vector field, if

A x , A , A 4 are continuous functions of the coordinates, we can intro-

duce for every point a certain scalar quantity and a certain new

vector, both depending on the way in which A changes from point

to point, and both having the property of being independent of the

choice of the axes of coordinates. The scalar quantity is called the

divergence of A and defined by the formula

The vector is called the rotation or the curl of A; its com-

ponents are

_ _ .

3y "" dz > dz ' dx> dx " dy >

and it will be represented by the symbol ,,rot A".

If the divergence of a vector is at all points, its distribution

over space is said to be solenoidcd. On the other hand, we shall

speak of an irrotational distribution, if at all points we have

rot A = 0.

FUNDAMENTAL EQUATIONS FOR THE ETHER. 5

In order to complete our list of notations, I have only to add

that the symbol A is an abbreviation for

and that not only scalars but also vectors may be differentiated with

o

respect to the coordinates or the time. For example, ~ means a

vector whose components are

dk x Zk y dk,

3x> dx> dx>

and -TTT has a similar meaning. A differentiation with respect to the

time t will be often represented by a dot, a repeated differentiation

of the same kind by two dots, etc.

3. We are now prepared to write down the fundamental equa-

tions for the electromagnetic field in the form which they take for

the ether. We shall denote by d the electric force, the same symbol

serving for the dielectric displacement, because in the ether this has

the same direction and, on account of the choice of our units, the

same numerical magnitude as the electric force. We shall further

represent by h the magnetic force and by c a constant depending on

the properties of the ether. A third vector is the current C, which

now consists only of the displacement current of Maxwell. It exists

wherever the dielectric displacement d is a function of the time, and

is given by the formula

c - d. (i)

In the form of differential equations, the formulae of the electro-

magnetic field may now be written as follows:

div d = 0, (2)

div h = 0, (3)

; ' " roth = |c = |d, " (4)

rotd = -yh. (5)

The third equation, conjointly with the second, determines the

magnetic field that is produced by a given distribution of the

current C. As to the last equation, it expresses the law according

to which electric forces are called into play in a system with a

variable magnetic field, i. e. the law of what is ordinarily called

electromagnetic induction. The formulae (1), (4) and (5) are vector

equations and may each be replaced by three scalar equations relating

to the separate axes of coordinates.

I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

Thus (1) is equivalent to

and (4) to

r etc

^ - t?l/V.

J

*-\ ."\ ~~" ^ -

oy oz c ot

The state of things that is represented by our fundamental

equations consists, generally speaking, in a propagation with a velo-

city c. Indeed, of the six quantities d x , d y , d,, h,,., h y , h^, five may

be eliminated 1 ), and we then find for the remaining one # an equation

of the form

This is the typical differential equation for a disturbance of the

state of equilibrium, travelling onwards with the speed c.

Though all the solutions of our equations have this general

character, yet there are a very large variety of them. The simplest

corresponds to a system of polarized plane waves. For waves of this

kind, we may have for example

d y = a cos n (t - -], h z = acoswu - -1, (7)

all other components of d and h being 0.

I need not point out to you that really, in the state represented

by these formulae, the values of d y and h s , which for a certain value

of t exist at a point with the coordinate x, will after a lapse of

time dt be found in a point whose coordinate is x + cdt. The

constant a is the amplitude and n is the frequency, i. e. the number

of vibrations in a time 2n. If n is high enough, we have to do

with a beam of plane polarized light, in which, as you know already,

the electric and the magnetic vibrations are perpendicular to the ray

as well as to each other.

Similar, though perhaps much more complicated formulae may

serve to represent the propagation of Hertzian waves or the radiation

which, as a rule, goes forth from any electromagnetic system that is

not in a steady state. If we add the proper boundary conditions,

such phenomena as the diffraction of light by narrow openings or

its scattering by small obstacles may likewise be made to fall under

our system of equations.

The formulae for the ether constitute the part of electromagnetic

theory that is most firmly established. Though perhaps the way in

which they are deduced will be changed in future years, it is

1) See Note 1 (Appendix).

GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. 7

hardly conceivable that the equations themselves will have to be

altered. It is only when we come to consider the phenomena in

ponderable bodies, that we are led into uncertainties and doubts.

4. There is one way of treating these phenomena that is compa-

ratively safe and, for many purposes, very satisfactory. In following

it, we simply start from certain relations that may be considered as

expressing, in a condensed form, the more important results of electro-

magnetic experiments. We have now to fix our attention on four

vectors, the electric force E, the magnetic force H, the current of

electricity C and the magnetic induction B. These are connected by

the following fundamental equations:

div C = 0, (8)

div B = 0, (9)

rotH-~C, (10)

rotE = -{B, (11)

presenting the same form as the formulae we have used for the ether.

In the present case however, we have to add the relation between

E and C on the one hand, and that between H and B on the other.

Confining ourselves to isotropic bodies, we can often describe the

phenomena with sufficient accuracy by writing for the dielectric dis-

placement

D - < E, (12)

a vector equation which expresses that the displacement has the same

direction as the electric force and is proportional to it. The current

in this case is again Maxwell's displacement current

C = D. (13)

In conducting bodies on the other hand, we have to do with a

current of conduction, given by

J - tf E, (14)

where a is a new constant. This vector is the only current and

therefore identical to what we have called C, if the body has only

the properties of a conductor. In some cases however, one has been

led to consider bodies endowed with the properties of both conductors

and dielectrics. If, in a substance of this kind, an electric force is

supposed to produce a dielectric displacement as well as a current

of conduction, we may apply at the same time (12) and (14), writing

for the total current

C = D -f J = E 4- <*E. (15)

8 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

Finally, the simplest assumption we C 4 an make as to the relation

between the magnetic force and the magnetic induction is expressed

by the formula

B - f*H, (16)

in which is a new constant.

5. Though the equations (12), (14) and (16) are useful for the

treatment of many problems, they cannot be said to be applicable to

all cases. Moreover, even if they were so, this general theory, in

which we express the peculiar properties of different ponderable

bodies by simply ascribing to each of them particular values of the

dielectric constant f, the conductivity 6 and the magnetic permeabi-

lity p, can no longer be considered as satisfactory, when we wish to

obtain a deeper insight into the nature of the phenomena. If we

want to understand the way in which electric and magnetic properties

depend on the temperature, the density, the chemical constitution or

the crystalline state of substances, we cannot be satisfied with simply

introducing for each substance these coefficients, whose values are

to be determined by experiment; we shall be obliged to have recourse

to some hypothesis about the mechanism that is at the bottom of

the phenomena.

It is by this necessity, that one has been led to the conception

of electrons, i. e. of extremely small particles, charged with electricity,

which are present in immense numbers in all ponderable bodies, and

by whose distribution and motions we endeavor to explain all electric

and optical phenomena that are not confined to the free ether. My

task will be to treat some of these phenomena in detail, but I may

at once say that, according to our modern views, the electrons in

a conducting body, or at least a certain part of them, are supposed

to be in a free state, so that they can obey an electric force by

which the positive particles are driven in one, and the negative

electrons in the opposite direction. In the case of a non-conducting

substance, on the contrary, we shall assume that the electrons are

bound to certain positions of equilibrium. If, in a metallic wire, the

electrons of one kind, say the negative ones, are travelling in one

direction, and perhaps those of the opposite kind in the opposite

direction, we have to do with a current of conduction, such as may

lead to a state in which a body connected to one end of the wire

has an excess of either positive or negative electrons. This excess,

the charge of the body as a whole, will, in the state of equilibrium

and if the body consists of a conducting substance, be found in a

very thin layer at its surface.

In a ponderable dielectric there can likewise be a motion of the

ELECTRONS. 9

electrons. Indeed, though we shall think of each of them as having

a definite position of equilibrium, we shall not suppose them to be

wholly immovable. They can be displaced by an electric force exerted

by the ether, which we conceive to penetrate all ponderable matter,

a point to which we shall soon have to revert. Now, however, the

displacement will immediately give rise to a new force by which the

particle is pulled back towards its original position, and which we may

therefore appropriately distinguish by the name of elastic force. The

motion of the electrons in non-conducting bodies, such as glass and

sulphur, kept by the elastic force within certain bounds, together

with the change of the dielectric displacement in the ether itself,

now constitutes what Maxwell called the displacement current.

A substance in which the electrons are shifted to new positions is

said to be electrically polarized.

Again, under the influence of the elastic forces, the electrons can

vibrate about their positions of equilibrium. In doing so, and perhaps

also on account of other more irregular motions, they become the

centres of waves that travel outwards in the surrounding ether and

can be observed as light if the frequency is high enough. In this

manner we can account for the emission of light and heat. As to

the opposite phenomenon, that of absorption, this is explained by

considering the vibrations that are communicated to the electrons

by the periodic forces existing in an incident beam of light. If the

motion of the electrons thus set vibrating does not go on undisturbed,

but is converted in one way or another into the irregular agitation

which we call heat, it is clear that part of the incident energy will

be stored up in the body, in other terms that there is a certain ab-

sorption. Nor is it the absorption alone that can be accounted for

by a communication of motion to the electrons. This optical resonance,

as it may in many cases be termed, can likewise make itself felt

even if there is no resistance at all, so that the body is perfectly

transparent. In this case also, the electrons contained within the

molecules will be set in motion, and though no vibratory energy is

lost, the oscillating particles will exert an influence on the velocity

with which the vibrations are propagated through the body. By

taking account of this reaction of the electrons we are enabled to

establish an electromagnetic theory of the refrangibility of light, in

its relation to the wave-length and the state of the matter, and to

form a mental picture of the beautiful and varied phenomena of

double refraction and circular polarization.

On the other hand, the theory of the motion of electrons in

metallic bodies has been developed to a considerable extent. Though

here also much remains to be done, new questions arising as we

proceed, we can already mention the important results that have

10 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.

been reached by Riecke, Drude and J. J. Thomson. 1 ) The funda-

mental idea of the modern theory of the thermic and electric pro-

perties of metals is, that the free electrons in these bodies partake

of the heat-motion of the molecules of ordinary matter, travelling in

all directions with such velocities that the mean kinetic energy of

each of them is equal to that of a gaseous molecule at the same

temperature. If we further suppose the electrons to strike over and

over again against metallic atoms, so that they describe irregular

zigzag-lines, we can make clear to ourselves the reason that

metals are at the same time good conductors of heat and of electri-

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