The theory of electrons and its applications to the phenomena of light and radiant heat online

11







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-