L
COLUMBIA UNIVEESITY IN THE CITY OF NK\Y YORK
PUBLICATION NUMBER TWO OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH
ESTABLISHED DECEMBER 17TH, 1904
THE THEOKY OF ELECTRONS
AND ITS APPLICATIONS TO THE PHENOMENA OF LIGHT AND RADIANT 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 LKCTUKKR IN MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1905-1906
NEW YORK
THE COLUMBIA UNIVERSITY PRESS
THE MACMILLAN CO., AGENTS
LONDON: MACMILLAN & CO., LTD.
1909
On the seventeenth day of December, nineteen hundred and four, Edward Dean Adams, of New York, established in Columbia University "The Ernest Kempt on Adams Fnnd for Physical Research" as a memorial to his son, Ernest Kempion Adams, who received the degrees of Electrical Engineering in 1897 and Master of Arts in 1X98, and who devoted his life to scientific re- search. The income of this fund is, by the terms of the deed of gift, to be devoted to the maintenance of a research fellowship and to the publication and distribution of the results of scientific re- search on the part of the fellow. A generous interpretation of the terms of the deed on the part of Mr. Adams and of the Trus- tees of the University has made it possible to issue and distribute a limited number of copies of the following lectures as a publi- cation of the Ernest Kempton Adams Fund :
X
B. G. TEUBNERS SAMMLUNG VON LEHRBUCHERN AUF DEM GEBIETE DER
MATHEMATISCHEN WISSENSCHAFTEN
MIT EINSCHLUSS IHRER ANWENDUNGEN. BAND XXIX.
THE THEORY OE ELECTRONS
AND ITS APPLICATIONS TO THE PHENOMENA OF LIGHT AND RADIANT 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
LEIPZIG: B. G. TEUBNER 1909
LONDON: DAVID NUTT, 57-59 LONG ACRE - WILLIAMS & NORGATE, w.c.u HENRIETTA STREET NEW YORK: G. E. STECHERT & Co., 129-133 WEST 201S STREET
ALL BIGHTS RESERVED. PRINTED BY B.G.TEUBNER, LEIPZIG.
PREFACE.
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.
CONTENTS.
Chapter Page
I. General principles. Theory of free electrons 1
II. Emission and absorption of heat 68
III. Theory of the Zeeinan-effect 98
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 231
Index . 330
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 my lectures are given at full length in the appendix at the end.
Lorentz, Theory of electrons. 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, and that we may also conceive a magnetic field as the seat of certain invisible motions, rotations for example around the lines of force. All this has been done by many physicists and Maxwell himself has set the example. 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 force, 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 units1) of sucli a kind that we get rid of the larger part of such factors as 4n and ]/4jr, 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 |/4sr 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 having 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 }/4jr 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 <J and a space by S, and we shall write ds, da, 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 0 Y.
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 AA the component of a vector A in the direction It, by Ax, Ay, As its components parallel to the axes of coordinates, by A, the component in the direction of a line s and finally by AB 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 A2.
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) - ASB, + AyBy + A.B,.
i*
4 I. GENERAL PRINCIPLES. THEORY OP 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 B], = AyBs - A,By, [A B], - A^ - A,B,, [A B], = AxBy - AyB,.
In many cases we have to consider a scalar quantity q> or a vector A which is given at every point of a certain space. If <p is a continuous function of the coordinates, we can introduce the vector having for its components
d<p 8y> dy
Wx> 8y' Jz '
This can easily be shown to be perpendicular to the surface
tp = const.
and we may call it the gradient of q>, which, in our formulae, we shall shorten to ,,grad go".
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 Ax, Ay, Az 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
div A = ? A«. + ^ + 2 A< •
8x 8y 8z
The vector is called the rotation or the curl of A; its com- ponents are
9A. _ d A,, > ^ _ 0A, 3Ay _ aA, cy ~ - ' de ~ dx' dx " 8y '
and it will be represented by the symbol ,,rot A".
If the divergence of a vector is 0 at all points, its distribution over space is said to be solenoidal. On the other hand, we shall speak of an irrotational distribution, if at all points we have rot A = 0.
FUNDAMENTAL EQUATIONS FOR THE ETHKR. 5
In order to complete our list of notations, I have only to add that the symbol A is an abbreviation for
d^ , J**, os and that not only scalars but also vectors may be differentiated with
n A
respect to the coordinates or the time. For example, ? means a
o sc
vector whose components are
8x > dx>
O »
and ---- 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 (I 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)
rot d = * h. (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.
6 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
Thus (1) is equivalent to
C* = ~W> C ~8T> C* == ~W' and (4) to
U U ;'.!
On, ony _ I oux ,
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^., dy, d2, h,,., hy, hz, five may be eliminated1), and we then find for the remaining one $ an equation of the form
-4!S = (X (6)
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;/ = a cos n (t — -*-) , hz = a cos n (t - ^-) , (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 dy and h2, which for a certain value of t exist at a point with the coordinate x, will after a lapse of time St be found in a point whose coordinate is x + cSt. The constant a is the amplitude and n is the frequency, i. e. the number of vibrations in a time 2it. 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)
rot H =^-C, (10)
rotE= --'ft, (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 = B 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 = aE, (14)
where & 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 + J = £E + tfE. (15)
8 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
Finally, the simplest assumption we can make as to the relation between the magnetic force and the magnetic induction is expressed by the formula
B = ftH, (16)
in which i 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 s, the conductivity 0 and the magnetic permeabi- lity ft, 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- city, and that, as a general rule, in the series of the metals, the two conductivities change in nearly the same ratio. The larger the number of free electrons, and the longer the time that elapses between two successive encounters, the greater will be the conductivity for heat as well as that for electricity.
6. This rapid review will suffice to show you that the theory of electrons is to be regarded as an extension to the domain of electricity of the molecular and atomistic theories that have proved of so much use in many branches of physics and chemistry. Like these, it is apt to be viewed unfavourably by some physicists, who prefer to push their way into new and unexplored regions by follow- ing those great highways of science which we possess in the laws of thermodynamics, or who arrive at important and beautiful results, simply by describing the phenomena and their mutual relations by means of a system of suitable equations. No one can deny that these methods have a charm of their own, and that, in following them, we have the feeling of treading on firm ground, whereas in the molecular theories the too adventurous physicist often runs the risk of losing his way and of being deluded by some false prospect of success. We must not forget, however, that these molecular hypo- theses can boast of some results that could never have been attained by pure thermodynamics, or by means of the equations of the electro- magnetic field in their most general form, results that are well known to all who have studied the kinetic theory of gases, the theories of
1) E. Riecke, Zur Theorie des Galvanismus und der Wiirme, Ann. Phys. Chem. 66 (1898), p. 353, 545, 1199; Ober das Verhaltnis der Leitfahigkeiten der Metalle fur Waime und fur Elektrizitat , Ann. Phys. 2 (1900), p. 835. P. Drude, Zur Elektronentheorie der Metalle, Ann. Phys. 1 (1900), p. 566; 3 (1900), p. 369. J. J. Thomson, Indications relatives a la constitution de la matiere fournies par les recherches recentes sur le passage de I'electricite" a travers les gaz, Rapports du Congres de physique de 1900, Paris, 3, p. 138. See also H. A. Lorentz, The motion of electrons in metallic bodies, Amsterdam Proc. 1904-1905, p. 438, 588, 684.
ELECTRONS AND ETHEE. H
dilute solutions, of electrolysis and of the genesis of electric currents by the motion of ions. Nor can the fruitfulness of these hypotheses be denied by those who have followed the splendid researches on the conduction of electricity through gases of J. J. Thomson1) and his fellow workers.
7. I have now to make you acquainted with the equations forming the foundation of the mathematical theory of electrons. Permit me to introduce them by some preliminary remarks.
In the first place, we shall ascribe to each electron certain finite dimensions, however small they may be, and we shall fix our attention not only on the exterior field, but also on the interior space, in which there is room for many elements of volume and in which the state of things may vary from one point to another. As to this state, we shall suppose it to be of the same kind as at outside points. Indeed, one of the most important of our fundamental assumptions must be that the ether not only occupies all space between molecules, atoms or electrons, but that it pervades all these particles. We shall add the hypothesis that, though the particles may move, the ether alivays remains at rest. We can reconcile ourselves with this, at first sight, somewhat startling idea, by thinking of the particles of matter as of some local modifications in the state of the ether. These modifications may of course very well travel onward while the volume- elements of the medium in which they exist remain at rest.
Now, if within an electron there is ether, there can also be an electromagnetic field, and all we have got to do is to establish a system of equations that may be applied as well to the parts of the ether where there is an electric charge, i. e. to the electrons, as to those where there is none. As to the distribution of the charge, we are free to make any assumption we like. For the sake of convenience we shall suppose it to be distributed over a certain space, say over the whole volume occupied by the electron, and we shall consider the volume-density Q as a continuous function of the coordinates, so that the charged particle has no sharp boundary, but is surrounded by a thin layer in which the density gradually sinks from the value it has within the electron to 0. Thanks to this hypothesis of the continuity of p, which we shall extend to all other quantities occurring in our equations, we have never to trouble ourselves about surfaces of discontinuity, nor to encumber the theory by separate equations relating to these. Moreover, if we suppose the difference between the ether within and without the electrons to be caused, at least so
1) J. J. Thomson, Conduction of electricity through gases, Cambridge, 1903.
12 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
far as we are concerned with it, only by the existence of the volume- density in the interior, the equations for the external field must be got from those for the internal one by simply putting Q = 0? so that we have only to write down one system of differential equations.
Of course, these must be obtained by a suitable modification, in which the influence of the charge is expressed, of the equations (2) — (5) which we have established for the free, i. e. for the uncharged ether. It has been found that we can attain our object by the slightest modification imaginable, and that we can assume the following system
div d = Q, (17)
div h = 0, (18)
roth = |c = --J-(d + pV), (19)
rot d = ~h, (20)
in which the first and the third formula are the only ones that have been altered.
In order to justify these modifications, I must in the first place recall to your minds the general relation existing in Maxwell's theory between the dielectric displacement across a closed surface and the amount of charge e contained within it. It is expressed by the equation
f *.*« = * (21)
in which the integral relates to the closed surface, each element da of it being multiplied by the component of d along the normal n, which, as we have already said, is drawn towards the outside. Using a well known form of speech and comparing the state of things with one in which there would be no dielectric displacement at all, we may say that the total quantity of electricity that has been displaced across the surface (a quantity that has been shifted in an outward direction being reckoned as positive), is equal to the charge e. Now, if we apply this to an element of space dxdydz, taken at a point where there is a volume-density p, we have
e = Q dxdydz
and, since the integral in (21) reduces to
div d • dx dy dz,
we are at once led to the formula (17).
In the second place, we must observe that a moving charge constitutes what is called a convection current and produces the same magnetic effects as a common current of conduction; this was
FUNDAMENTAL EQUATIONS. 13
first shown by Rowland's celebrated and well known experiment. Now, if v is the velocity of the charge, it is natural to write pV for the convection current; indeed, the three components QVZ, pV , pV, represent the amounts of charge, reckoned per unit of area and unit of time, which are carried across elements of surface perpendicular to the axes of coordinates. On the other hand, if in the interior of an electron there is an electromagnetic field, there will also be a displacement current d. We are therefore led to assume as the expression for the total current
C = d + pV, (22)
and to use the equation (19) in order to determine the magnetic field. Of course, this is again a vector equation. In applying it to special problems, it is often found convenient to replace it by the three scalar differential equations
r *> ~w " -fo == -<r
_
3x dy ~
You see that by putting p = 0, in the formulae (17) and (19), we are led back to our former equations (2) and (4).
8. There is one more equation to be added, in fact one that is of equal importance with (17) — (20). It will have been noticed that I have carefully abstained from saying anything about the nature of the electric charge represented by p. Speculations on this point, or attempts to reduce the idea of a charge to others of a different kind, are entirely without the scope of the present theory; we do not pretend to say more than this, that p is a quantity, belonging to a certain point in the ether and connected with the distribution of the dielectric displacement in the neighbourhood of that point by the equation (17). We may say that the ether can be the seat of a certain state, determined by the vector d which we call the dielectric displacement, that in general this vector is solenoidally distributed, but that there are some places which form an exception to this rule, the divergence of d having a certain value p, different from 0. In such a case, we speak of an electric charge and under- stand by its density the value of div d.
As to the statement that the charges can move through the ether, the medium itself remaining at rest, if reduced to its utmost simplicity, it only means that the value of div d which at one moment exists at a point P, will the next moment be found at another place P'.
14 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
Yet, in order to explain electromagnetic phenomena, we are obliged to go somewhat further. It is not quite sufficient to con- sider Q as merely the symbol for a certain state of the ether. On the contrary, we must invest the charges with a certain degree of substantiality, so far at least that we recognize the possibility of forces acting on them and producing or modifying their motion. The word ,,force" is here taken in the ordinary sense it has in dynamics, and we should easily become accustomed to the idea of forces acting on the charges, if we conceived these latter as fixed to what we are accustomed to call matter, or as being a property of this matter. This is the idea underlying the name of ,,charged particle" which we have already used and shall occasionally use again for an electron. We shall see later on that, in some cases at least, the fitness of the name is somewhat questionable.
However this may be, we must certainly speak of such a thing as the force acting on a charge, or on an electron, on charged matter, whichever appellation you prefer. Now, in accordance with the general principles of Maxwell's theory, we shall consider this force as caused by the state of the ether, and even, since this medium pervades the electrons, as exerted by the ether on all internal points of these particles where there is a charge. If we divide the whole electron into elements of volume, there will be a force acting on each element and determined by the state of the ether existing within it. We shall suppose that this force is pro- portional to the charge of the element, so that we only want to know the force acting per unit charge. This is what we can now properly call the electric force. We shall represent it by f. The formula by which it is determined, and which is the one we still have to add to (17) — (20), is as follows:
f = d + |[v-h]. (23)
Like our former equations, it is got by generalizing the results of electromagnetic experiments. The first term represents the force acting on an electron in an electrostatic field; indeed, in this case, the force per unit of charge must be wholly determined by the dielectric displacement. On the other hand, the part of the force expressed by the second term may be derived from the law according to which an element of a wire carrying a current is acted on by a magnetic field with a force perpendicular to itself and the lines of force, an action, which in our units may be represented in vector notation by
FORCE ACTING ON UNIT CHARGE. 15
where i is the intensity of the current considered as a vector, and s the length of the element. According to the theory of electrons, F is made up of all the forces with which the field h acts on the separate electrons moving in the wire. Now, simplifying the question by the assumption of only one kind of moving electrons with equal charges e and a common velocity V, we may write
si = NeV, if N is the whole number of these particles in the element s. Hence
so that, dividing by Ne, we find for the force per unit charge
As an interesting and simple application of this result, I may mention the explanation it affords of the induction current that is produced in a wire moving across the magnetic lines of force. The two kinds of electrons having the velocity V of the wire, are in this case driven in opposite directions by forces which are determined by our formula.
9. After having been led in one particular case to the existence of the force d, and in another to that of the force — [V • h], we now
combine the two in the way shown in the equation (23), going beyond the direct result of experiments by the assumption that in general the two forces exist at the same time. If, for example, an electron were moving in a space traversed by Hertzian waves, we could calculate the action of the field on it by means of the values of d and h, such as they are at the point of the field occupied by the particle.
Of course, in cases like this, in which we want to know the force exerted by an external field, we need not distinguish the directions and magnitudes of f at different points of the electron, at least if there is no rotation of the particle; the velocity V will be the same for all its points and the external field may be taken as homogeneous on account of the smallness of the electron. If however, for an electron having some variable motion, we are required to calculate the force that is due to its own field, our analysis must be pushed further. The field is now far from homogeneous, and after having divided the particle into elements of volume, we must determine the action of the field on each of them. Finally, if the electron is treated as a rigid body, we shall have to calculate in the ordinary way the resultant force and the resultant couple.
16 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
10. While I ani speaking so boldly of what goes on in the interior of an electron, as if I had been able to look into these small particles, I fear one will feel inclined to think I had better not try to enter into all these details. My excuse must be that one can scarcely refrain from doing so, if one wishes to have a perfectly definite system of equations; moreover, as we shall see later on, our experiments can really teach us something about the dimensions of the electrons. In the second place, it may be observed that in those cases in which the internal state of the electrons can make itself felt, speculations like those we have now entered upon, are at all events interesting, be they right or wrong, whereas they are harm- less as soon as we may consider the internal state as a matter of little importance.
It must also be noticed that our assumptions by no means exclude the possibility of certain distributions of charge which we have not at first mentioned. By indefinitely diminishing the thickness of the transition layer in which Q passes from a finite value to 0, we can get as a limiting case that of an electron with a sharp boundary. We can also conceive the charge to be present, not throughout the whole extent of the particle, but only in a certain layer at its surface, whose thickness may be made as small as we like, so that after all we can speak of a surface -charge. Indeed, in some of our formulae we shall have in view this special case.
11. Since our equations form the real foundation-stones of the structure we are going to build, it will be well to examine them somewhat more closely, so that we may be sure that they are con- sistent with each other. They are easily shown to be so, provided only the charge of an element of volume remain constant during its motion.1) If we regard the electrons as rigid bodies, as we shall almost always do, this of course means that Q is constant at every point of a particle. However, we might also suppose the electrons to change their shape and volume; only, in this case, the value of p for an element of volume ought to be considered as varying in the inverse ratio as the magnitude of the element.
It is also important to remark that our formulae are applicable to a system in which the charges, instead of being concentrated in certain small particles, are spread over larger spaces in any way you like. We may even go a step further and imagine any number of charges with the densities QI} p2 etc., which are capable of penetrating each other and therefore of occupying the same part of space, and which move, each with its own velocity. This would require us to
1) Note 2.
DETERMINATION OF THE FIELD. 17
replace the terms p and (>v in (17) and (19) by ft + ft H ---- and d vi + ft V2 H ---- , the vectors V17 V2, ... being the velocities of the separate charges. An assumption of this kind, artificial though it may seem, will be found of use in one of the problems we shall have to examine.
12. I have now to call your attention to some of the many beautiful results that may be derived from our fundamental equations, in the first place to the way in which the electromagnetic field is determined by the formulae (17)— (20), if the distribution and the motion of the charges are supposed to be given. The possibility of this determination is due to the fact that we can eliminate five of the six quantities dz, dy, dz, h^, hy, bs, exactly as we could do, when we treated the equations for the free ether, and to the remarkable form in which the final equation presents itself.1) We have, for example, three equations for the components of d, which we may combine into the vector formula
, (24)
and the similar condition for the magnetic force
Ah-|2ti = -irotfov). (25)
It will not be necessary to write down the six scalar equations for the separate components; we can confine ourselves to the formulae for d.,. and h,,., viz.
Ad — d - 8f- '-4- - 8-^V*> (26}
«*°. c*ax — x + c* t '
Ah h -
An.-^n,- --c-g-
In order to express myself more clearly, it will be proper to introduce a name for the left-hand sides of these equations. The result of the operation A, applied to a quantity i/> that is a function of the coordinates x, y, z, has been called the Laplacian of #.
i 82 Similarly, the result of the operation A -- 1 --^ may be given the name
of the Dalembertian of the original quantity, in commemoration of the fact that the mathematician d'Alembert was the first to solve a partial differential equation, occurring in the theory of a vibrating string, which contains this operation, or rather the operation
og .« 03
5-3 --- i o,,, which is a special case of it. Of course, since vectors can be differentiated with respect to time and place, we may as well
1) Note 3.
Lorentz, Theory of electrons.
18 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
speak of the Dalembertian of a vector as of that of a scalar quan- tity. Accordingly, since, for a given distribution and motion of the charges, the right-hand members of our last equations are known functions of x, y, z, t, we see that the vectors (I and h, as well as each of their components, are determined by the values of their Dalembertians. We have therefore to look into the question, what will be the value of a quantity ty whose Dalembertian has a given value co. This is a problem which admits a simple solution. In the ordinary theory of the potential it is proved that a function «/> whose Laplacian has a given value w, may be found by the formula
where r is the distance from an element of volume dS to the point P for which we want to calculate ty, to the value of the Laplaciau in this element, and where we have to integrate over all parts of space in which ra is different from 0.
Now, it is very remarkable that a function ^ satisfying the equation
may be found by a calculation very like that indicated in (28) 1). The only difference is that, if we are asked to determine the value of V at the point P for the instant t, we must take for to the value of
this function existing in the element dS at the time t- — We shall henceforth include in square brackets quantities whose values
1'
must be taken, not for the time t, but for the previous time t - Using this notation, we may say that the function
is a solution of the differential equation (29). It should be observed that this also holds when to is a vector quantity; [ra] and *-=d8
will then be so likewise, and the integration in (30) is to be under- stood as the addition of an infinite number of infinitely small vectors. For purposes of actual computation, the vector equation may again be split up into three scalar ones, containing the components of o, and giving us those of ip.
1) Note 4.
POTENTIALS. 19
13. The above method of calculation might be applied to the equations (24) and (25) or (26) and (27). Since, however, the second members of these formulae are somewhat complicated, we prefer not directly to determine d and h, but to calculate in the first place certain auxiliary functions on which the electric and magnetic forces may be made to depend, and which are called potentials. The first is a scalar quantity, which I shall denote by <p, the second a vector for which I shall write a.
If the potentials are subjected to the relations
-?*--* (31)
and
Aa--ia = -l(,v, (32)
one can show1), by means of (17) — (20), that the dielectric displacement is given by
d = - -*- a - grad cp (33)
and the magnetic force by
h = rot a. (34)
You see that the equations (31) and (32) are again of the form (29), so that the two potentials are determined by the condition that their
Dalembertians must have the simple values — p and • — pV. Therefore, on account of (30), we may write
and
By these equations, combined with (33) and (34), our problem is solved. They show that, in order to calculate the field, we have to proceed as follows: Let P be the point for which we wish to determine the potentials at the time t. We must divide the whole surrounding space into elements of volume, any one of which is called dS. Let it be situated at the point Q and let the distance QP be denoted by r. In this element of space there may or may not be a part of an electron at a certain time. We are only concerned
with the question whether it contains a charge at the time t - Indeed, the brackets serve to remind us that we are to understand
1) Note 5.
2*
20 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
by Q the density existing in dS at the particular instant t -
and by pv the product of this density and the velocity of the charge within dS at that same instant. These values [p] and [pY] must be multiplied by dS and divided by r. Finally, we have to do for all elements what we have done for the one dS and to add all the results. Of course there may be many elements which do not con- tribute anything to the integrals, viz. all those which at the time
t — - did not contain any charge. c
14. What has been said calls forth some further remarks. In the first place, you see that the factor , which we have been so
anxious to get rid of, has again appeared. We cannot prevent it from doing so, but fortunately it is now confined to a few of our equations. In the second place, it is especially important to observe that the values of p and pV existing in a certain point Q at the
time t— - do not make themselves felt at the point P at the same
moment t - ' , but at the later time t. We may therefore really
speak of a propagation taking place with the velocity c. The parts of q> and a which are due to the several elements dS correspond to states existing in these elements at times which are the more remote, the farther these elements are situated from the point P considered.
On account of this special feature of our result, the potentials cp and a, given by (35) and (3(5), are often called retarded potentials.
I must add that the function (30) is not the most general solution of (29), and that for this reason the values of (33) and (34) derived from (35) and (36) are not the only ones satisfying the fundamental equations. We need not however speak of other solutions, if we assume that an electromagnetic field in the ether is never pro- duced by any other causes than the presence and motion of electrons.1)
15. The case of a single electron furnishes a good example for the application of our general formulae. Let us suppose in the first place that the particle never has had nor will have any motion. Then we have a = 0, and since p is the same at all instants, the scalar potential is given by
.
r
The equations (33) and (34) becoming
d = — grad cp, h = 0, we fall back on the ordinary formulae of electrostatics.
1) Note 6.
FIELD OF A MOVING ELECTRON. 21
We shall next consider an electron having (from t — — oo until t = + oo) a translation with constant velocity w along a straight line. Let P and P' be two points in such positions that the lino PP' is in the direction of the motion of the particle. It is easily seen that, if we wish to calculate <p, a, (I and h, first for the point P
and the time t, and then for the point P' and the time t + ' ' ''
w '
we shall have to repeat exactly the same calculations. If, for example, dS is an element of space contributing a part to the integrals (35) and (36) in the first problem, the corresponding integrals in the second will contain equal parts due to an element dS' which may be got by shifting dS in the direction of translation over a distance equal to PP'.
It appears from this that the electron is continually surrounded by the same field, which it may therefore be said to carry along with it. As to the nature of this field, one can easily deduce from (33)— (36) that, in the case of a spherical electron with a charge symmetrically distributed around the centre, if s is the path of the centre, the electric lines of force are curves situated in planes passing through s, and the magnetic lines circles having s as axis.1) The field is distinguished from that of an electron without translation, not only by the presence of the magnetic force, but also by an alteration in the distribution of the dielectric displacement.
We shall finally take a somewhat less simple case. Let us suppose that, from t = — oo until a certain instant tL, the electron is at rest in a position A, and that, in a short interval of time beginning at ti} it acquires a velocity w which remains constant in magnitude and direction until after some time, in a short interval ending at the instant t%, the motion is stopped. Let B be the final position in which the electron remains for ever afterwards.
If P is any point in the surrounding ether, we can consider two distances ^ and ?2, the first of which is the shortest of all the lines drawn from P to the points of the electron in the position A, and the second the longest of all the lines joining P to the electron in the position B. We shall suppose the interval tt — ^ to be so long
that «, + A- >«, + i~
It will be clear that in performing the calculation of <p and a, for the point P and for an instant previous to t^ + — , we shall get
a result wholly independent of the motion of the electron. This motion can by no means make itself felt in P during this first period, which will therefore be characterized by the field belonging
1) Note 7.
22 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
to the immovable electron. A similar field will exist in P after the time t% + -'-, every influence that has been emitted by the particle
while moving, having already, in its outward progress, passed over the point considered.
Between tt + — and (L -}-•*• the field at P will be due to the c c
moving electron. If we suppose the dimensions of the particle to be very small in comparison with the distances llf 12, and the velocity w to be acquired and lost in intervals of time much shorter than ts — t1, we may be sure that during the larger part of the interval between
ti H — L and <2 4- - the field at P will be what it would have been,
C C
had a constant velocity w existed for ever. Of course, immediately after t, -+- ' an(i shortly before t2 + it will be otherwise; then,
C C
there will be a gradual transition from one state of things to the other. It is clear also that these periods of transition, taken for different points P, will not be found to coincide. If Slt S2, S3 are parts of space at different distances from the line AB, Sl being the most remote and <S3 the nearest, it may very well be that, at some particular instant, /S, is occupied by the field belonging to the elec- tron while at rest in the position A, St by the field of the moving electron, and Ss by the final field.
16. Thus far we have only used the equations (17) — (20). Adding to these the formula (23) for the electric force, and supposing the forces of any other nature which may act on the electrons to be given, we have the means of determining, not only the field, but also the motion of the charges. For our purpose however, it is not necessary to enter here into special problems of this kind. We shall concentrate our attention on one or two general theorems holding for any system of moving electrons.
In the first place, suitable transformations of the fundamental formulae lead to an equation expressing the law of conservation of energy.1) If we confine ourselves to the part of the system lying within a certain closed surface <?, this equation has the form
cf[A • h].rftf = 0, (37)
which we shall now try to interpret. Since f is the force with which the ether acts on unit charge, gfdS will be the force acting on the element dS of the charge, and
1) Note 8.
ELECTRIC AND MAGNETIC ENERGY. PLOW OF ENEIKiY. 23
its work per unit of time. The first integral in (37) is thus seen to represent the work done by the ether on the electrons per unit of time. Combined with the work of other forces to which the electrons may be subjected, this term will therefore enable us to calculate the change of the kinetic energy of the electrons.
Of course, if the ether does work on the electrons, it must lose an equivalent amount of energy, a loss for which a supply of energy from the part of the system outside the surface 0 may make up, or which may be accompanied by a transfer of energy to that part. We must therefore consider
Hd' + hV-S (38)
as the expression for the energy contained within an element of volume of the ether, and
C/[d - hLrftf (39)
as that for an amount of energy that is lost by the system within the surface and gained by the surrounding ether.
The two parts into which (38) can be divided may properly be called the electric and the magnetic energy of the ether. Reckoned per unit of volume the former is seen to be
(40)
and the latter
^ = lh2- (41)
These values are equivalent to those that were given long ago by Maxwell. That the coefficients are \ and not 2it or something of the kind, is due to the choice of our new units and will certainly serve to recommend them.
As to the transfer of energy represented by (39), it must necessarily take place at the points of the surface a itself, because our theory leaves no room for any action at a distance. Further, we are naturally led to suppose that the actions by which it is brought about are such that, for each element da, the quantity c[d • h]ndd may be said to represent the amount of energy that is transmitted across this particular element. In this way we come to the conception, first formulated by Poynting1), of a current or flow of energy. It is determined by the vector product of (I and h, mul- tiplied by the constant c, so that we can write for it
8-c[d-h], (42)
1) J. H. Poynting, On the transfer of energy in the electromagnetic field, London Trans. 175 (1884), p. 343.
24 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
the meaning being that, for any element d6, the amount of energy by which it is traversed, is given for unit of time and unit of area by the component sn of the vector 8 along the normal to the element.
17. It is interesting to apply the above results to the beam of polarized light represented by our equations (7). We find for the energy which it contains per unit of space
and for the flow of energy across a plane perpendicular to the axis of x
cdyhz = ca2 cos2 n (t - — J •
The mean values of these expressions for a full period are
^ and
%ca\
Indeed, by a well known theorem, the mean value of cos2w(tf — — \
• 1 is •£.
It is easily seen that the expression \ca? may also be used for calculating the flow of energy during any lapse of time that is very long compared with a period.
If the beam of light is laterally limited by a cylindrical surface whose generating lines are parallel to OX, as it may be if we neglect diffraction phenomena, and if a normal section has the area 27, the flow of energy across a section is given by -J-ca227. It is equal for any two sections and must indeed be so, because the amount of energy in the part of the beam between them remains constant.
The case of a single electron having a uniform translation like- wise affords a .good illustration of what has been said about the flow of energy. After having determined the internal and the external field by means of the formulae (33) — (36), we can deduce the total electromagnetic energy from (40) and (41). I shall later on have occasion to mention the result. For the present I shall only say that, considering the course of the electric and the magnetic lines of force, which intersect each other at right angles, we must conclude that there is a current of energy, whose general direction is that of the translation of the electron. This should have been expected, since the moving electron is constantly surrounded by the same field. The energy of this field may be said to accompany the particle in its motion.
FLOW OF ENERGY. 25
Other examples might likewise show us how Poynting's theorem throws a clear light on many questions. Indeed, its importance can hardly be overestimated, and it is now difficult to recall the state of electromagnetic theory of some thirty years ago, when we had to do without this beautiful theorem.
18. Before leaving this subject I will, with your permission, call attention to the question, as to how far we can attach a definite meaning to a flow of energy. It must, I believe, be admitted that, as soon as we know the mutual action between two particles or ele- ments of volume, we shall be able to make a definite statement as to the energy given by one of them to the other. Hence, a theory which explains things by making definite assumptions as to the mutual action of the parts of a system, must at the same time admit a transfer of energy, concerning whose intensity there can be no doubt. Yet, even if this be granted, we can easily see that in general it will not be possible to trace the paths of parts or elements of energy in the same sense in which we can follow in their course the ultimate particles of which matter is made up.
In order to show this, I shall understand by P a particle or an element of volume and by A, B, C, . . . , A', B', C', . . . a certain number of other particles or elements, between which and P there is some action resulting in a transfer of energy and, in accordance with what has just been said, I shall suppose these actions to be so far known that we can distinctly state what amount of energy is interchanged between any two particles. Let, for example, P receive from A, B, C, . . . the quantities a, b, c, . . . of energy, and let it give to A', fi', C', . . . the quantities a, V, c, ... , gaining for itself a certain amount p. Then we shall have the equation
a + b + c+---^p + a' + V + c'-\ .
Now, though in our imaginary case each term in this equation would be known, we should have no means for determining in what way the quantities of energy contained in a, b, c, . . ., say the individual units of energy, are distributed among p, a, V, c, .... If, for example, there are only two terms on each side of the equation, all of the same value, so that it takes the form
a + b = a' -f- V,
we can neither conclude that a is the same energy as a and V the same as b, nor that a is identical to b, and b' to a. There would be no means of deciding between these two views and others that likewise suggest themselves.
For this reason, the flow of energy can, in my opinion, never have quite the same distinct meaning as a flow of material particles,
26 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
which, by our imagination at least, we can distinguish from each other and follow in their motion. It might even be questioned whether, in electromagnetic phenomena, the transfer of energy really takes place in the way indicated by Poynting's law, whether, for example, the heat developed in the wire of an incandescent lamp is really due to energy which it receives from the surrounding medium, as the theorem teaches us, and not to a flow of energy along the wire itself. In fact, all depends upon the hypotheses which we make concerning the internal forces in the system, and it may very well be, that a change in these hypotheses would materially alter our ideas about the path along which the energy is carried from one part of the system to another. It must be observed however that there is no longer room for any doubt, so soon as we admit that the pheno- mena going on in some part of the ether are entirely determined by the electric and magnetic force existing in that part. No one will deny that there is a flow of energy in a beam of light; therefore, if all depends on the electric and magnetic force, there must also be one near the surface of a wire carrying a current, because here, as well as in the beam of light, the two forces exist at the same time and are perpendicular to each other.
19. Results hardly less important than the equation of energy, and of the same general character, are obtained when we consider the resultant of all the forces exerted by the ether on the electrons of a system. For this system we can take a ponderable body which is in a peculiar electromagnetic state or in which electromagnetic phenomena are going on. In our theory the ponderomotive force exerted on a charged conductor, a magnet or a wire carrying a current, is made up of all the forces with which the ether acts on the electrons of the body.
Let 6 again be a closed surface, and F the resultant force oil all the electrons contained within it. Then, on account of (23), we may write
F = / Q { d + - [v • h] ) dS, (43)
J c
extending the integral to all the electrons, or as we may do as well (p being 0 in the space between the particles), to the whole space S. Now, by the application of the equations (17j — (20)1), this force F may be shown to be equal to the sum of two vectors
F = F! + F,, (44)
which are determined by the equations
1) Note 9.
and
STRESSES IN THE ETHER. 27
^ix ™? i I { 2d.rdn — d2 cos (w, a;) } da
+ -J- 1 { 2 h.,. hB — h2 cos (n, x) } da etc. (45)
F3= -if MS. (46)
The first part of the force is represented by an integral over the surface ff, its components, of which only one is given here, being determined by the values of d^, dy, dz, h^, etc. at the surface. The second part of the force, on the contrary, presents itself as an integral over the space S, not only over those parts of it where there is an electric charge, but also over those where there is none.
20. In discussing the above result we must distinguish several cases.
a) In all phenomena in which the system is in a stationary state, the force Fa, for which we may write
•9 == ^ ~3~t
2
disappears, and the whole force F is reduced to an integral over the surface 6. In other terms, the ponderomotive action can be regarded as the sum of certain infinitely small parts, each of which belongs to one of the surface-elements do and depends on the state existing at that element. A very natural way of interpreting this is to speak of each of these parts as of a stress in the ether, acting on the element considered.
The stress depends on the orientation of the element. If this is determined by the normal n, and if, using a common notation, we write Xn, Yn, Zn for the components of the force per unit area, exerted by the part of the medium on the positive side of the sur- face on the part lying on the negative side, we shall have
- X. = 1 { 2dxdn- d2 cos («, «)} + {• { 2hA - h2 cos (n, x) } etc. (48)
From these formulae we can easily deduce the components Xx> Yx> Zx, X, etc. of the stresses acting on elements whose normal is parallel to one of the axes of coordinates. We find
X, = \ (d,2 - dy2 - d/) + \ (h,2 - hy2 - h/) etc., (49)
x,= r, = d,d, + M, etc., (50)
precisely the values of the stresses by which Maxwell long ago accounted for the ponderomotive forces observed in electric and magnetic fields.
28 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
This method of calculating the resultant force is often very convenient, the more so because we can take for a any surface surrounding the body for which we have to solve the problem.
b) We are led to similar conclusions if we consider a system that is the seat of periodical phenomena, confining ourselves to the mean value of the force taken for a complete period T. The mean value being given by
the last term in (44) disappears. Indeed, by (47) the time integral of F2 is equal to the difference of the values of
- ^fsdS
for t = 0 and t — T, and these values are equal on account of the periodicity of the changes.
Hence, in this case also, the resultant force is reduced to surface- integrals, or, as we may say, to stresses in the ether.
It can easily be shown that the mean value of F (and of periodically changing quantities in general) during a lapse of time that is very much longer than a period T, is equal to the mean value during a period, even though the interval considered is not exactly a multiple of T.
21. An interesting example is furnished by the pressure of radiation. Let (Fig. 1) AB be a plane disk, receiving in a normal
direction a beam of light L, which, taking OX in the direction shown in the diagram, we can represent by our formulae (7). Let us take for a the surface of the flat cylin- drical box CDEF, whose plane sides lie before and behind the disk and are parallel to it. Then, if the plate is perfectly opaque, we have only to consider the stress on CD. Kg. i. Moreover, if the disk is supposed to be
perfectly black, so that there is no reflected
beam, there is only the electromagnetic field represented by the equations (7). Hence, since a normal to the plane CD, drawn towards the outside of the box CDEF, has a direction opposite to that of OX, the force acting on the absorbing body in the direc- tion of OX per unit area is given by
J.
RADIATION PRESSURE. 29
-X.-W + «-.•)> and its mean value by
i«2-
Comparing this with the value of the energy and attending to the direction of the force, we conclude that the beam of light produces a normal pressure on the absorbing body, the intensity of the pressure per unit of surface being numerically equal to the electromagnetic energy which the beam contains per unit of volume.
The same method can be applied to a body which transmits and reflects a certain amount of light, and to a disk on which a beam of light falls in an oblique direction. In all cases in which there is no light behind the disk, the force in the direction of the normal will be a pressure — Xx on the illuminated side, if the axis of x is directed as stated above.
We shall apply this to a homogeneous and isotropic state of radiation, existing in a certain space that is enclosed by perfectly reflecting walls. By homogeneous and isotropic we mean that the space is traversed by rays of light or heat of various directions, in such a manner that the radiation is of equal intensity in different parts of the space and in all directions, and that all directions of (I and h are equally represented in it. It can easily be shown that in this case there is no tangential stress on an element ds of the wall. As to the normal pressure, which is represented by — Xx, if the axis of x is made to coincide with the normal, we may write for it
p
where the horizontal bars are intended to indicate the mean values, over the space considered, of the several terms.1) But, on account of our assumptions regarding the state of radiation,
Each of these quantities is therefore equal to one third of their sum, i. e. to -J-d2. Similarly
v-v-v^fP-
Hence, if the formulae (40) and (41) are taken into account,
In this case, the pressure on the walls per unit of surface is equal to one third of the electromagnetic energy per unit of volume.
Later on, the problem of radiation pressure will be treated by a different method.
1) Note 10.
30 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
22. Thus far we have simplified the equation (44) by supposing the last term to vanish. In general, however, this term may not be omitted, and the force F cannot be accounted for by a system of stresses acting on the surface 6.
This conclusion takes a remarkable form, if the surface <? is supposed to enclose no electrons at all. Of course, the total force F must be 0 in this case, as may be seen from the original expression (43). Nevertheless, the force due to the stresses is not generally 0, having the value
(51)
It is worthy of notice that this last equation is quite independent of the theory of electrons, being a consequence of the fundamental equations for the case p = 0, i. e. of the equations for the free ether. It has indeed been known for a long time.1)
In the mind of Maxwell and of many writers on the theory there seems to have been no doubt whatever as to the real existence of the ether stresses determined by the formulae (49) and (50). Considered from this point of view, the equation (51) tells us that in general the resultant force Ft of all the stresses acting on a part of the ether will not be 0. This was first pointed out by Helm- holtz.2) He inferred from it that the ether cannot remain at rest, and established a system of equations by which its motion can be determined. I shall not enter upon these, because no experiment has ever shown us any trace of a motion of the ether in an electro- magnetic field.
We may sum up by saying that a theory which admits the existence of Maxwell's stresses leads to the following conclusions:
1. A portion of the ether is not in equilibrium under the stresses acting on its surface.
2. The stresses acting on the elements of a surface which surrounds a ponderable body will, in general, produce a resultant force different from the force acting on the electrons of the body according to our theory.
23. Having got thus far, we may take two different courses. In the first place, bearing in mind that the ether is undoubtedly widely different from all ordinary matter, we may make the assump- tion that this medium, which is the receptacle of electromagnetic energy and the vehicle for many and perhaps for all the forces acting on ponderable matter, is, by its very nature, never put in motion,
1) Note 11.
2) Helmholtz, Folgerungen aus Maxwell's Theorie fiber die Bewegungen des reiuen Athers, Ann. Phys. Chem. 53 (1894), p. 135.
IMMOBILITY OF THE ETHEK. 31
that it has neither velocity nor acceleration, so that we have no reason to speak of its mass or of forces that are applied to it. From this point of view, the action on an electron must be con- sidered as primarily determined by the state of the ether in the interior of each of its elements of volume, and the equation (43) as the direct and immediate expression for it. There is no reason at all why the force should be due to pressures or stresses in the uni- versal medium. If we exclude the idea of forces acting on the ether, we cannot even speak of these stresses, because they would be forces exerted by one part of the ether on the other.
I should add that, while thus denying the real existence of ether stresses, we can still avail ourselves of all the mathematical transformations by which the application of the formula (43) may be made easier. We need not refrain from reducing the force to a surface- integral, and for convenience's sake we may continue to apply to the quantities occurring in this integral the name of stresses. Only, we must be aware that they are only imaginary ones, nothing else than auxiliary mathematical quantities.
Perhaps all this that has now been said about the absolute immobility of the ether and the non-existence of the stresses, may seem somewhat startling. If it is thought too much so, one may have recourse to the other conception to which I have alluded. In choosing this, we recognize the real existence of Maxwell's internal forces, and we regard the ether as only approximately immovable.
Let us admit that between adjacent parts of the ether there is an action determined by the equations (48), so that an element of volume of the free ether experiences a force
and let us suppose the medium to move in such a way that it has a momentum
(52)
or -^8 per unit of volume. Let us further imagine that the density
of the ether is so great that only a very small velocity, too small to be detected by any means at our disposal, is required for the momentum (52). Then, the formula (51) which, applied to an element of the ether, takes the form
tells us that the assumed state of motion can really exist. This is clear because for very small velocities the resultant force acting on
32 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
the ether contained in a fixed element of volume may be said to be equal to the rate of change of the momentum that is found within that element.1)
On the other hand, in the case of an element dS occupied by a charge, the formula
may be interpreted as follows. The ether within the element is subject to a force F1? due to the stresses on the surface. Of this force, the part
goes to produce the change of momentum of the ether, the remaining part F being transferred to the charge.
You will readily perceive that, after all, the difference between the two modes of view consists mainly in the different interpretations given to the same equations.
24. Whatever may be our opinion about the questions we have now touched upon, our discussion shows the importance of the vector
which has a definite direction and magnitude for every element of volume, and of the vector
(53)
that may be derived from it by integration. Abraham2) of Gottingen has applied to these quantities the name of electromagnetic momentum. We may term them so, even if we do not wish to convey the idea that they represent a real momentum, as they would according to the second of the two lines of thought we have just followed.
The way in which the conception of electromagnetic momentum may be of use for the elucidation of electromagnetic phenomena comes out most clearly if, in dealing with a system of finite dimen- sions, as the systems in our experiments actually are, we make the enclosing surface 6 recede on all sides to an infinite distance. It may be shown that the surface -integrals in (45) then become 0, so that, if the integration is extended to all space, we shall have
F — g,
1) Note 12.
2) M. Abraham, Prinzipien der Dynamik des Elektrons, Ann. Phys. 10 (1903), p. 105.
RADIATION PRESSURE. 33
or in words: the force exerted by the ether on a system of electrons, or, as we may say, on the ponderable matter containing these elec- trons, is equal and opposite to the change per unit of time of the electromagnetic momentum. Now, since the action tends to produce a change equal to the force itself in the momentum (in the ordinary sense of the word) of the ponderable matter, we see that the sum of this momentum and the electromagnetic one will not be altered by the actions exerted by the ether.
Before passing on to one or two applications, I must call your attention to the intimate connexion between the momentum and the flow of energy 8. The equation (53) at once shows us that every part of space in which there is a flow of energy contributes its part to the vector G; hence, in order to form an idea of this vector and of its changes, we have in the first place to fix our attention on the radiation existing in different parts of space. If, in course of time, the flow of energy reaches new parts of space or leaves parts in which it was at first found, this will cause the vector G to change from one moment to another.
It must also be kept in mind that (53) is a vector equation and that (54) may be decomposed into three formulae giving us the components Fx, Fy, fe of the resultant force.
25. Very interesting illustrations of the preceding theory may be taken from the phenomena of radiation pressure, to which I shall therefore return for a moment. Let us consider, for example, a source of light sending out its rays in a single direction, which may be brought about by suitable arrangements, and let us suppose this radiation to have begun at a certain instant, so that we can speak of the first wave or of the front of the train of waves that have been emitted. This front is a plane at right angles to the beam and advancing with the velocity c. Hence, if £ is the normal section of the beam, the volume occupied by the radiation increases by c'E per unit of time. As we have seen, the flow of energy has the direction of the beam. In making the following calculation, we shall reason as if, at every point, the flow were constantly equal to the mean flow 8 taken for a full period. If the magnitude of this mean flow, which relates to unit of area, is | 8 , we shall find that of the electro- magnetic momentum, whose direction is likewise that of the beam, if we multiply -,- . 8 by the volume occupied by the light. It appears
,
C
from this that the change of G per unit of time is
consequently, since this vector has the direction of the rays, there will be a force on the source of light of the same intensity and in
Lorentz, Theory of electrons.
34 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
a direction opposite to that in which the rays are emitted. This force of recoil, which, however, is extremely small, may be compared with the reaction that would exist if the rays of light consisted of a stream of material particles. By similar reasoning we can deter- mine the pressure on the black disk we have formerly considered. But, in this case, it is best to imagine the radiation of the source to have been stopped at a certain moment, so that there is a plane which we may call the rear of the progression of waves. It approaches the black disk with the velocity c, and if 2 and | s | have the same meaning as just now, the magnitude of the electromagnetic momentum will diminish by
per unit of time. Consequently, there will be a normal pressure of this intensity acting on the disk. The result agrees with what we have deduced from the value of the stress in the ether, the quantity | 8 being related to the amplitude a by the equation
It is easy to extend these results to a more general case. Let a plane disk receive, from any direction we like, a beam of parallel rays, and let one part of these be reflected, another absorbed and the remaining part transmitted. Let the vectors 8, 8' and s" be the flows of energy per unit of area in the incident, reflected and trans- mitted beams, S, s', s" the mean flows taken for a full period, 2, £', E" the normal sections of the beams. Then, if we imagine the space occupied by the light to be limited by two fronts, one in the reflected and one in the transmitted beam, and by a rear plane in the incident one, all these planes travelling onward with the velocity c, the change of electromagnetic momentum will be given by the vector expression
and the force on the plate by
It must here be mentioned that the radiation pressure has been ob- served by Lebedew1) and by E. F. Nichols and Hull2), and that the theoretical predictions as to its intensity have been verified to within one percent by the measurements of the last named physicists.
1) P. Lebedew, Untersuchungen uber die Druckkrilfte des Licbtes, Ann. Phys. 6 (1901), p. 433.
2) E. F. Nichols and G. F. Hull, The pressure due to radiation, Astrophysical Journ. 17 (1903), p. 315; also Ann. Phys. 12 (1903), p. 225.
FIELD OF MOVING ELECTRONS. 35
26. The theory of electromagnetic momentum, which we have found of so much use in the case of beams of light that are emitted, reflected or absorbed by a body, is also applicable to the widely different case of a moving electron. We may therefore, without too abrupt a transition, turn once more to some questions belonging to what we can call the dynamics of an electron, and in which we are concerned with the field the particle produces and the force exerted on it by the ether. We shall in this way be led to the important subject of the electromagnetic mass of the electrons.
To begin with, I shall say some words about the field of a system of electrons or of charges distributed in any way, having a constant velocity of translation w, say in the direction of the axis of x, smaller than the speed of light c. We shall introduce axes of coordinates moving with the system, and we shall simplify our for- mulae by putting
-f = ft. (55)
Now, we have already seen that the field is carried along by the system. The same may be said of the potentials <p and a, which serve to determine it, and it may easily be inferred from this1) that
the values of -£ and -r - in a fixed point of space are given by
(} f 0 v
(jcp dn
Similarly
Thus the equation (31) takes the form
a/»q\ O (p 0 £p G (p /K.r*\
— o ) ""r~ ~T~ ~~" — O (OO J
whereas (32) may be replaced by the formula
the components ay and az being both 0, as is seen directly from (36). Comparing (56) and (57), we conclude that
a* = ftg>,
so that we have only to determine the scalar potential.
This can be effected by a suitable change of independent variables. If a new variable x' is defined by
xr = (i - p)- v* *, (5g)
1) Note 18.
3*
36 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
(56) becomes
^-L^l3p4.^__0 8x* + a*/8 + 8z* Q>
having the well known form of Poisson's equation. Since this equation occurs in the determination of the field for charges that are at rest, the problem is hereby reduced to an ordinary problem of electrostatics. Only, the value of cp in our moving system S is connected with the potential, not of the same system when at rest, but of a system in which all the coordinates parallel to OX have been changed in the ratio determined by (58).1 )
The result may be expressed as follows. Let S' be a system having no translation, and which we obtain by enlarging the dimen- sions of S in the direction of OX in the ratio of 1 to (1 — /32)~ . Then, if a point with the coordinates x, y, z in S and a point with the coordinates x, y, z in S' are said to correspond to each other, if the charges of corresponding elements of volume are supposed to be equal, and if <p' is the potential in S', the scalar potential in the moving system is given by
^(i-^-'V- (so)
Let us now take for the moving system a single electron, to which we shall ascribe the form of a sphere with radius H and a uniformly distributed surface -charge e. The corresponding system S' is an elongated ellipsoid of revolution, and its charge happens to be distri- buted according to the law that holds for a conductor of the same form. Therefore, the field of the moving spherical particle and all the quantities belonging to it, can be found by means of the ordinary theory of a charged ellipsoid that is given in many treatises. I sh;ill only mention the results obtained for the more important quantities. The total electric energy is given by
3 — 138 , l-f-|3
-
(61) i_ r r -•
and the magnetic energy by
T - g^j [-+ P- log ii-J - 2] • (62)
As to the electromagnetic momentum, this has the direction of the translation, as may at once be deduced from (53), because we know already that the general direction of the flow of energy coincides with that of the motion of the particle. The formula for the magni- tude of the electromagnetic momentum, calculated for the first time by Abraham, is
G
1) Note 14.
FORCE ON AN ELECTRON, DUE TO ITS OWN FIELD. 37
All these values U, T, j G | increase when the velocity is augmented. They become infinite for ft = 1, i. e. when the electron reaches a velocity equal to that of light.1)
27. According to our fundamental assumptions, each element of volume of an electron experiences a force due to the field produced by the particle itself, and the question now arises whether there will be any resultant force acting on the electron as a whole. The con- sideration of the electromagnetic momentum will enable us to decide this question.
If the velocity w is constant in magnitude and direction, as it has been supposed to be in what precedes, the vector G will likewise be constant and there will be no resultant force. This is very im- portant; it shows that, if free from all external forces, an electron, just like a material point, will move with constant velocity, notwith- standing the presence of the surrounding ether. In all other cases however there is an action of the medium.
It must be observed that, in the case of a variable velocity, the above formulae for U, T and G do not, strictly speaking, hold. However, if the variation of the state of motion is so slow that the
-D
change taking place in a time — may be neglected, one may apply the
formula (63) for every moment, and use it to determine the change 6 of the momentum per unit of time.8) As the result depends on the acceleration of the electron, the force exerted by the ether is like- wise determined by the acceleration.
Let us first take the case of a rectilinear translation with variable velocity w. The vector G is directed along the line of motion, and its magnitude 'is given by
dt die f, dp
Putting
•» ' ** 1 Jt \ O [
id4fi = m', (64)
dw c dp
we conclude that there is a force acting on the electron, opposite to its acceleration and equal to the product of the latter and the coefficient m.
In the second place, I shall consider an electron having a velocity W of constant magnitude, but of varying direction. The acceleration is then normal to the path and it is convenient to use vector equations. Let W be the velocity, W the acceleration, and
1) Note 15. 2) See 8 37.
38 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
let us take into account that in this case there is a constant ratio between G and w for which I shall write
We have also
G = m"w,
and the force exerted by the ether is given by
-G = -m"w.
It is opposite in direction to the normal acceleration W and has an intensity equal to the product of this acceleration with the coeffi- cient m".
In the most general case the acceleration j will be directed neither along the path nor normally to it. If we decompose it into two components, the one j' in the line of motion and the other j" at right angles to it, we shall have, for the force on the electron due to its own electromagnetic field, in vector notation1)
- m'\' - m"\". (66)
28. The way in which these formulae are usually interpreted will become clear to us, if we suppose the electron to have a certain mass m0 in the ordinary sense of the word, and to be acted on, not only by the force that is due to its own field, but also by a force K of any other kind. The total force being
K — m'\ — m"\",
the equation of motion, expressed in the language of vector analysis, will be
K-ro'J'-»»T==»»o Instead of this we can write
from which it appears that the electron moves, as if it had two different masses m0 + m' and m0 + m" , the first of which comes into play when we are concerned with an acceleration in the line of motion, and the second when we consider the normal acceleration. By measuring the force K and the accelerations j' and j" in different cases, we can determine both these coefficients. We shall call them the effective masses, m0 the material mass, and m', m" the electromagnetic masses. In order to distinguish m and m", we can apply the name of longitudinal electromagnetic mass to the first, and
1) Note 16.
ELECTROMAGNETIC MASS OF AN ELECTRON. 39
that of transverse electromagnetic mass to the second of these coefficients.1)
From what has been said one finds the following formulae for m and m":
m - -
or, expanded in series,
For small velocities the two masses have the same value
whereas for larger velocities the longitudinal mass always surpasses the transverse one. Both increase with /3, until for /3 = 1 , i. e. for a velocity equal to the speed of light, they become infinite.
If, for a moment, we confine ourselves to a rectilinear motion of an electron, the notion of electromagnetic mass can be derived from that of electromagnetic energy. Indeed, this latter is larger for a moving electron than for one that is at rest. Therefore, if we are to put the particle in motion by an external force K, we must not only produce the ordinary kinetic energy ^m0w2 but, in addition to this, the part of the electromagnetic energy that is due to the velo- city. The effect of the field will therefore be that a larger amount of work is required than if we had to do with an ordinary material particle #»0; it will be just the same as if the mass were larger than m0.
By reasoning of this kind we can also easily verify the for- mula (68). If the velocity is changing very slowly, we may at every instant apply the formulae (61) and (62). Since the total energy T -\- U is a function of the velocity tv, its rate of change is given by
LW= ^ ^J- — w. (73)
dw d$ c
This must be equal to the work done per unit of time by the moving force, or rather by the part of it that is required on account of the
1) The notion of (longitudinal) electromagnetic mass was introduced for the first time by J. J. Thomson in his paper ,,0n the electric and magnetic effects produced by the motion of electrified bodies", Phil. Mag. (6) 11 (1881), p. 227. The result of his calculation is, however, somewhat different from that to which one is led in the modern theory of electrons.
40 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
electromagnetic field. Consequently, dividing (73) by w, we find the intensity of this part, and if next we divide by w, the acceleration, the result must be the longitudinal electromagnetic mass. If one
calculates
, 1 d(T+U)_ _l_d_(r+_Z7)
~ cw ap /Jcs~ <//*
by means of the formulae (61) and (62), one really finds exactly the value (68).
29. A close analogy to this question of electromagnetic mass is furnished by a simple hydrodynamical problem. A solid, per- fectly smooth sphere, moving with the velocity W in an incom- pressible perfect fluid which extends on all sides to infinite distance, produces in this fluid a state of motion characterized by a kinetic energy for which we may write
if a is a constant, depending on the radius of the ball and on the density of the fluid. Under the influence of an external force applied to the ball in the direction of the translation, its velocity will change as if it had, not only its true mass m0, but besides this an apparent mass m ', whose value is given by
'_ * dT
~ W djwl " tt>
•A formula corresponding to the last equation of § 28.
We could have obtained the same result if we had first calculated the momentum of the fluid. We should have found for it
an expression from which we can also infer that the transverse apparent mass has the same value a as the longitudinal one. This is shown by the equation
" G
m =kr
3O. If, in the case of the ball moving in the perfect fluid, we were obliged to confine ourselves to experiments in which we measure the external forces applied to the body and the accelerations produced by them, we should be able to determine the effective mass »wc + m (or ma + m"}, but it would be impossible to find the values of m0 and m (or m"} separately. Now, it is very important that, in the experimental investigation of the motion of an electron, we can go a step farther. This is due to the fact that the electromagnetic mass is not a constant, but increases with the velocity.
RATIO OF THE CHAEGE TO THE MASS OF AN ELECTRON. 41
Suppose we can make experiments for two different known velocities of an electron, and that by this means we can find the ratio fc between the effective transverse masses which come into play in the two cases. Let x be the ratio between the electromagnetic transverse masses, calculated, as can really be done, by the formula (69). Then, distinguishing by the indices / and II the quantities relating to the two cases, we shall have the formulae
I f J
»w0 + mn mn
and the ratio between the true mass m0 and the electromagnetic mass m'i will be given by
w0 x — k
If the experimental ratio Ic differed very little from the ratio >c that is given by the formula (69), m0 would come out much smaller than m'j and we should even have to put m0 = 0, if k were exactly equal to x.
I have spoken here of the transverse electromagnetic mass, because this is the one with which we are concerned in the experiments I shall now have to mention.
31. You all know that the cathode rays and the /?-rays of radio-active bodies are streams of negative electrons, and that Gold- stein's canal rays and the «-rays consist of similar streams of positively charged particles. In all these cases it has been found possible to determine the ratio between the numerical values of the charge of a particle and its transverse effective mass. The chief method by which this has been achieved is based on the measure- ment, for the same kind of rays, of the deflections from their recti- linear course that are produced by known external electric and magnetic forces.
The theory of the method is very simple. If, in the first place, an electron having a charge e and an effective mass TO, moves in an electric field d, with a velocity w perpendicular to the lines of force,
the acceleration is given by - -; hence, if r is the radius of curvature
of the path,
w^ <• i d ] r m '
so that, if | d | and r have been measured, we can calculate the value of
42 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
Let us consider in the second place an electron moving in a magnetic field h, and let us suppose the velocity w to be perpendicular to the
magnetic force. Then, the field will exert on the particle a force - — ,
as is seen from the last term of (23). This force being perpendicular to the velocity, we shall have, writing r for the radius of curvature
of the path,
w* ew h
r cm
The determination of | h and r can therefore lead to a knowledge of the expression
and, by combining this with (74), we shall be enabled to find both w
and — m
32. I shall not speak of the large number of determinations of this kind that have been made by several physicists, and will only say a few words relative to the important work of Kaufmann1) on the /?- rays of radium. These rays appear to contain negative electrons with widely different velocities, so that it is possible to examine the question
whether — is a function of the velocity or a constant. Kaufmann's m
experiments were arranged in such a manner that the electric and the magnetic deflection, belonging to the same electrons, could be
measured, so that the values both of w and of — could be deduced from them. Now, it was found that, while the velocity w ranges from
Q
about 0,5 to more than 0,9 of the velocity of light, the value of diminishes considerably. If we suppose the charge to be equal for all the negative electrons constituting the rays, this diminution of —
must be due to an increase of the mass m. This proves that at all events the electromagnetic mass has an appreciable influence. It must even greatly predominate. Indeed, Kaufmann's numbers show no trace of an influence of the material mass m0, his ratio k of effective masses for two different velocities (a ratio which is the inverse of
that of the values of — ) agreeing within the limits of experimental
errors with the ratio x between the electromagnetic masses, as deduced from Abraham's formula (69).
1) W. Kaufmann, Uber die Konstitution des Elektrons, Ann. Phys. 19 (1906), p. 487.
MASS OF A NEGATIVE ELECTRON WHOLLY ELECTROMAGNETIC. 43
Of course, we are free to believe, if we like, that there is some small material mass attached to the electron, say equal to one hundredth part of the electromagnetic one, but with a view to simplicity, it will be best to admit Kaufmann's conclusion, or hypo- thesis, if we prefer so to call it, that the negative electrons have no material mass at all.
This is certainly one of the most important results of modem physics, and I may therefore be allowed to dwell upon it for a short time and to mention two other ways in which it can be expressed. We may say that, in the case of a moving negative electron, there is no energy of the ordinary form l-m0tv2, but merely the electro- magnetic energy T + U, which may be calculated by means of the formulae (61) and (62). For high velocities this energy is a rather complicated function of the velocity, and it is only for velocities very small compared with that of light, that the part of it which depends on the motion, can be represented by the expression \m'w3, where m has the value given by (72). This is found by expanding T+ U in a series similar to (70) and (71).
We obtain another remarkable form of our result, if in the equation of motion (67), which for m0 = 0 becomes
K - m'\ - m"\" = 0,
we attach to the two last terms their original meaning of forces exerted by the ether. The equation tells us that the total force acting on the particle is always 0. An electron, for example, which has an initial velocity in an external electromagnetic field, will move in such a manner that the force due to the external field is exactly counterbalanced by the force that is called forth by the electron's own field, or, what amounts to the same thing, that the force exerted by the resulting field is 0.
After all, by our negation of the existence of material mass, the negative electron has lost much of its substantiality. We must make it preserve just so much of it, that we can speak of forces acting on its parts, and that we can consider it as maintaining its form and magnitude. This must be regarded as an inherent property, in virtue of which the parts of the electron cannot be torn asunder by the electric forces acting on them (or by their mutual repulsions, as we may say).
33. In our preceding reasoning we have admitted the equality of the charges of all the negative electrons given off by the radium salt that has been used in Kaufmann's experiments. We shall now pass on to a wide generalization of this hypothesis.
44 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
As is well known, Faraday's law of electrolysis proves that all monovalent electrolytic ions have exactly equal charges, and that, if this is denoted by e, the charges of bivalent, trivalent ions etc. are 2e, 3e etc. Thus the conception has arisen that this e, say the charge of an ion of hydrogen, is the smallest quantity of electricity that ever occurs in physical phenomena, an atom of electricity, as we may call it, which can only present itself in whole numbers. Experimental determinations by J. J. Thomson1) of the charges carried by the ions in conducting gases, and certain speculations about the electrons which are vibrating in a body traversed by a beam of light, have made it highly probable that this same amount of charge e occurs in these cases, that it is, so to say, a real natural unit of electricity, and that all charged particles, all electrons and ions carry one such unit or a multiple of it. The negative electrons which constitute the /3-rays and the cathode rays are undoubtedly the simplest of all these charged particles, and there are good reasons for supposing their charge to be equal to one unit of electricity, i. e. to the charge of an ion of hydrogen.
Leaving aside the case of multiple charges, and ascribing to all electrons or ions, whether they be positive or negative, the same amount of electricity, we can say that the masses m of different particles are inversely proportional to the values that have been found
f e for — m
Now, for the negative electrons of the cathode rays and of the /3-rays, this latter value is, for small velocities2)
1,8
For an ion of hydrogen, the corresponding number can be drawn from the electrochemical equivalent of the gas. It is found to be
9650.6/4*:,
nearly 1900 times -smaller than the number for the free negative electrons. Hence, the mass of a negative electron is about the 1900th part of that of an atom of hydrogen.
1) See J. J. Thomson, Conduction of electricity through gases, and The corpuscular theory of matter, London, 1907, by the same author.
2) I write it in this form in order to show that the number is 1,8- 10", if the ordinary electromagnetic units are used. It may be mentioned here that Simon's measurements on cathode rays [Ann. Phys. Chem. 69 (1899), p. 589] lead to the value 1,878 • 10', and that Kaufmann, calculating his results by means of Abraham's formulae, finds 1,823 • 107. Later experiments on j3-rays by Bestelmeyer [Ann. Phys. 22 (1907), p. 429], however, have given the number 1,72 • 10'.
ELECTROMAGNETIC THEORY OF MATTER. 45
It must be noticed especially that the values of e obtained for different negative electrons are approximately equal. This lends a strong support to the view that all negative electrons are equal to each other. On the contrary, there are great differences between the positive electrons, such as we find in the canal rays and the a- rays
of radio-active substances. The values of -- belonging to these rays
are widely divergent. They are however all of the same order of magnitude as the values holding for electrolytic ions. Consequently, the masses of the positive electrons must be comparable with those of chemical atoms. We can therefore imagine the free electrons to be the product of a disintegration of atoms, of a division into a positively and a negatively charged particle, the first having nearly the whole mass of the atom, and the second only a very small part of it.
34. Of late the question has been much discussed, as to whether the idea that there is no material but only electromagnetic mass, which, in the case of negative electrons, is so strongly supported by Kauf- mann's results, may not be extended to positive electrons and to matter in general. On this subject of an electromagnetic theory of matter we might observe that, if we suppose atoms to contain negative electrons, of which one or more may be given off under certain circumstances, as they undoubtedly are, and if the part that remains after the loss of a negative particle is called a positive electron, then certainly all matter may be said to be made up of electrons. But this would be mere words. What we really want to know is, whether the mass of the positive electron can be calculated from the distribution of its charge in the same way as we can determine the mass of a negative particle. This remains, I believe, an open question, about which we shall do well to speak with some reserve.
In a more general sense, I for one should be quite willing to adopt an electromagnetic theory of matter and of the forces between material particles. As regards matter, many arguments point to the conclusion that its ultimate particles always carry electric charges and that these are not merely accessory but very essential. We should introduce what seems to me an unnecessary dualism, if we considered these charges and what else there may be in the particles as wholly distinct from each other.
On the other hand, I believe every physicist feels inclined to the view that all the forces exerted by one particle on another, all molecular actions and gravity itself, are transmitted in some way by the ether, so that the tension of a stretched rope and the elasticity of an iron bar must find their explanation in what goes on in the
46 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
ether between the molecules. Therefore, since we can hardly admit that one and the same medium is capable of transmitting two or more actions by wholly different mechanisms, all forces may be re- garded as connected more or less intimately with those which we study in electromagnetism.
For the present, however, the nature of this connexion is entirely unknown to us and we must continue to speak of many kinds of forces without in the least being able to account for their origin. We shall even be obliged to subject the negative electrons to certain forces, about whose mode of action we are in the dark. Such are, for example, the forces by which the electrons in a ponderable dielectric are driven back to their positions of equilibrium, and the forces that come into play when an electron moving in a piece of metal has its course changed by an impact against a metallic atom.
35. The universal unit of electricity of which we have spoken can be evaluated as soon as we have formed an estimate of the mass of the chemical atoms. This has been done fairly well in different ways, and we shall not be far from the truth if we take
1,3- 10- 24 gramm
for the mass of an atom of hydrogen. Combining this with the electrochemical equivalent of this element, which in our units is
0,0001036 — •?=—, we nnd lor the charge or an ion of hydrogen
This number must also represent the charge of a negative electron. Therefore, the value of - (for small velocities) being
1,8-
we find
m = 7 • 10~28 gramm.
Now, this must be the mass given by the formula (72). Substituting also the value of e, we get the following number for the radius
R= 1,5-10-13 cm.
We may compare this with the estimates that have been formed in the kinetic theory of gases. The distance of neighboring molecules in the atmospheric air is probably about
3 • 10-' cm
and the diameter of a molecule of hydrogen may be taken to be
2 • 10- 8 cm.
ELECTROMAGNETIC MASS OF A SYSTEM OP ELECTRONS. 47
You see that, compared with these lengths, the electron is quite microscopical. Probably it is even much smaller than a single atom, so that, if this contains a certain number of negative electrons, these may be likened to spheres placed at distances from each other that are high multiples of their diameters.
36. Before closing our discussion on the subject of electro- magnetic mass, I must call your attention to the question as to whether, in a system composed of a certain number of electrons, the electro- magnetic mass is the sum of the electromagnetic masses of the separate particles, or, as I shall rather put it, whether, if the system moves with a common velocity of translation, the electromagnetic energy, in so far as it depends on the motion, can be made up of parts, each belonging to one electron, so that, for small velocities, it can be represented by
This will, of course, be the case, if the electrons are so far apart that their fields may be said not to overlap. If, however, two electrons were brought into immediate contact, the total energy could not be found by an addition, for the simple reason, that, being a quadratic function of d and h, the energy due to the superposition of two fields is not equal to the sum of the energies which would be present in each of the two, if it existed by itself.
We have now to bethink ourselves of the extreme smallness of the electrons. It is clear that the larger part of the electromagnetic energy belonging to a particle will be found in a very small part of the field lying quite near it, within a distance from the centre that is a moderate multiple of the radius. Therefore, it may very well be that a number of electrons are so widely dispersed that the effective parts of their fields lie completely outside each other. In such a case the system may be said to have an electromagnetic mass equal to the sum of the masses of the individual electrons.
Yet there are important cases in which we are not warranted in asserting this. In order to make this clear, I shall call f\ the part of the field of an electron which lies nearest to the particle, and Fa the more distant part, the surface of separation between the two being a sphere whose radius is rather large in comparison with that of the electron. Then, if the electron is taken by itself, the part El of the energy contained within I\ far surpasses the energy E% which has its seat in I\. Now, if we have N electrons at such distances from each other that their fields F^ do not overlap, we shall have to add to each other the amounts of energy Ev The quantities Ea on the contrary must not be simply added, for the remoter fields F2 will certainly cover, partly at least, the same space 8. If, in this space,
48 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
the dielectric displacements or the magnetic forces due to the indi- vidual elestrons have directions mating rather small angles with each other, all the fields Fs, feeble though they are, may very well pro- duce a resulting field of appreciable energy. We have an example of this in the electric field of a charged conductor, and in the magnetic field around a wire carrying a current. The energy of this magnetic field may be shown to be, in very common cases, considerably larger than the sum of all the amounts of energy which I have called Elf at least in as much as these depend on the motion of the electrons. The possibility of this will be readily understood, if one thinks of the extreme case that, at a point of the space S, all electrons produce a magnetic force in exactly the same direction. Then, if each of these forces has the magnitude h j , the resultant magnetic force has the magnitude N \ h ! , so that the magnetic energy per unit of volume becomes ^2V3h2. This is proportional to the square of the number N which we shall suppose to be very large. On the other hand, the sum of the quantities Ev may be reckoned to be proportional to the first power of N.
This digression was necessary in order to point out the connexion between the electromagnetic mass of electrons and the phenomena of self-induction. In these latter it is the magnetic energy due to the overlapping of the feeble fields _F2 that makes itself felt. In dealing with effects of induction we can very well speak of the electro- magnetic inertia of the current, or of the electromagnetic mass of the electrons moving in it, but we must keep in mind that this mass is very much larger than the sum of those we should associate with the separate particles. This large value is brought about (as are all effects of the current) by the cooperation of an immense number of electrons of the same kind moving in the same direction.1)
37. In our treatment of the electromagnetic mass of electrons we have started from the expression (66) for the force to which an electron is subjected on account of its own field. However, this ex- pression is not quite exact. It is based on the assumption that the equation (63) may be applied to a case of non-uniform motion, and we observed already that this may be done only if the state of motion changes very little during the time an electromagnetic disturbance would take to travel over a distance equal to the dimensions of the electron. This amounts to saying that, if I is one of these dimen- sions, and r a time during which the state of motion is sensibly altered, the quantity
must be very small. 1) Note 17.
(75)
CT
RESISTANCE TO THE MOTION. 49
In reality, the force (66) is only the first term of a series in which, compared with the preceding one, each term is of the order of magnitude (75).
In some phenomena the next term of the series makes itself felt; it is therefore necessary to indicate its value. By a somewhat la- borious calculation it is found to be
i^V, (76)
where the vector V is twice differentiated with respect to the time. I may mention by the way that this formula holds for any distri- bution of the electric charge e.1)
In many cases the new force represented by (76) may be termed a resistance to the motion. This is seen, if we calculate the work of the force during an interval of time extending from I = ^ to t = t3. The result is
Here the first term disappears if, in the case of a periodic motion, the integration is extended to a full period; also, if at the instants tl and t2 either the velocity or the acceleration is 0. We have an example of the latter case in those phenomena in which an electron strikes against a ponderable body and is thrown back by it.
Whenever the above formula reduces to the last term, the work of the force is seen to be negative, so that the name of resistance is then justly applied. This is also confirmed by the form our formula takes for an electron having a simple harmonic motion. The velocity being given by
V = bcosnt,
where n is a constant, we may write V = — w2 V, and, instead of (76),
so that, in this particular case, the force is opposite to the velocity and proportional to it.
The work of (77) during a full period T is
'/'
t.
v*<«= ^VF. (78)
38. In all cases in which the work of the force (76) is nega- tive, the energy of the electron (if not kept at a constant value by
1) Note 18.
LOT fnt z, Theory of electrons.
50 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
the action of some other cause) must diminish, and that of the ether must increase. This means that there is a continous radiation from the particle outwards, such as cannot be said to exist when the velo- city is constant and the electron simply carries its field along with it.
For the purpose of getting a clear idea of the radiation, it is well to consider the field at a very large distance from the particle. We shall see that, if the distance is large enough, the radiation field gets, so to say, disentangled from the field we have formerly consi- dered, which is carried along by the moving particle.
In order to determine the field at a large distance, we can avail ourselves of the following formulae for the scalar and the vector potential, which hold for all points whose distance from the electron is very large compared with its dimensions:
(79)
Here, the square brackets have a meaning similar to that which we gave them in the general equations (35) and (36). If one wishes to determine the potentials at a point P for the time t, one must first seek the position M of the electron, which satisfies the condition that, if it is reached at the time t0, previous to t,
MP-c(t-t0).
The distance M P is denoted by r, and [v] means the velocity in the position M, Vr its component in the direction MP.
The formulae have been deduced from (35) and (36); the vector
1 - ^f- in the denominators shows, however, that the problem is not c
quite so simple as might be expected at first sight. A complication arises from the circumstance that we must not integrate over the space occupied by the electron at the particular instant which we have denoted by t0. On the contrary, according to the meaning of (35) and (36), we must fix our attention on the different points of the electron and choose for each of them, among all its succes- sive positions, the one M' which is determined by the condition, that, if it is reached at the time t^,
The time to is slightly different for the different points of the electron and therefore the space over which we have to integrate (which contains all the points M' ) cannot be said to coincide with the space occupied by the electron at any particular instant.1)
1) Note If).
RADIATION FROM AN ELECTRON. 51
39. Leaving aside these rather complicated calculations, I pro- ceed to the determination of the field at very large distances. The formulae (33) and (34) which we must use for this purpose require us to differentiate y and a. In doing so I shall omit all terms in which the square and the higher powers of the distance r appear in the denominator. I shall therefore treat as a constant the factor r in the denominators of (79), so that only Vr has to be differentiated in the expression for qp and, if we also neglect terms in which a component of the velocity is multiplied by one of the acceleration, only [v] in the second formula. Performing all operations and denoting by x, y, z the coordinates of P with respect to the point M as origin, and by j the acceleration of the electron in the position M, I find1)
The three formulae for d can be interpreted as follows. If the acce- leration j is decomposed into \r in the direction of M P and \p per- pendicular to it, the dielectric displacement in P is parallel to \p and its magnitude is given by
In order to see the meaning of the equations for h, we can introduce a vector k of unit length in the direction from M towards P. The
components of this vector being — , — , — , we have
The magnitude of h is therefore
_« !
I TT /' ' .' Jfl
equal to that of the dielectric displacement. Further, the magnetic force is seen to be perpendicular both to the line M P and to the dielectric displacement. Consequently there is a flow of energy along MP. It is easily seen that this flow is directed away from the position M of the electron, and that its intensity is given by
if & is the angle between MP and the acceleration j.2)
1) Note 20. 2) Note 21.
52 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
The result may be applied to any point of a spherical surface <j described around the centre M with r as radius. The total outward flow of energy across this sphere is given by
The reason for my former assertion that, at very large distances from the electron, the radiation field predominates over the field con- sidered in § 26, lies in the fact that, in the latter, d and h diminish
as 8 and in the radiation field only as
We can sum up the preceding considerations by saying that an electron does not emit energy so long as it has a uniform rectilinear motion, but that it does as soon as its velocity changes either in magnitude or in direction.
40. The theory of the production of Rontgen rays, first pro- posed by Wiechert and Stokes, and worked out by J. J. Thomson1), aifords a very interesting application of our result. According to it, these rays consist of a rapid and irregular succession of sharp electro- magnetic impulses, each of which is due to the change of velocity which an electron of the cathode rays undergoes when it impinges against the anti-cathode. I cannot however dwell upon this subject, having too much to say about the emission of light -vibrations with which we shall be often concerned.
If (an electron has a simple harmonic motion, the velocity is continually changing, and, by what has been said, there must be a continous emission of energy. It will also be clear that, at each point of the surrounding field, the state is periodically changing, keeping time with the electron itself, so that we shall have a radiation of homogeneous light. Before going into some further details, I shall first consider the total amount of energy emitted during a full period.
Let us choose the position of equilibrium as origin of coordinates and let the vibration take place along the axis of x, the displacement at the time t being given by
x = acos(nt + p). Then the acceleration is
- aw2 cos (nt + p).
I) E. Wiechert, Die Theorie iler Elektrodynamik und die Rontgen'sche Entdeckung, Abh. d. Phys.-okon. Ges. zu Konigsberg i. Pr. (1896), p. 1; Ober die Grundlagen der Elektrodynatnik, Ann. Phya. Chem. 59 (1896), p. 283; G. G. Stokes, On the nature of the Rontgen rays, Manch. Memoirs 41 (1897), Mem. 15; J. J. Thomson, A theory of the connexion between cathode and Rontgen rays, Phil. Mag. (5) 45 (1898), p. 172.
FIELD OF A VIBRATING ELECTRON. 53
If the amplitude a is very small, the sphere of which we have spoken in the preceding paragraph may be considered as having its centre, not in M, one of the positions of the electron, but in the origin 0, and we may understand by j the acceleration of the electron at the
time t - — , r being the distance from 0. Therefore, on account c
of (82), the flow of energy across the sphere will be per unit of time
Integrating this over a full period T we get
Now, if the amplitude is to remain constant, the electron must be acted on by an external force equal and opposite to the resistance (77). The work of this force is given by (78) with the sign reversed. Since the amplitude of the velocity is equal to the amplitude a of the elongation, multiplied by n, the work of the force corresponds exactly to the amount of energy (83) that is emitted.1)
41. For the sake of further examining the field produced by an electron having a simple harmonic motion, we shall go back to the formulae (79). Let us first only suppose that the motion of the electron is confined to a certain very small space S, one point of which is chosen as origin of coordinates. Let x, y, z be the coordinates of the electron, x, y, z its velocities and x, y, z the components of its acceleration. We shall consider all these quan- tities as infinitely small of the first order, and neglect all terms containing the product of any two of them. We shall further denote by x, y, S the coordinates of the point P for which we wish to determine the field, and by r0 its distance from the origin. Now, if M is the position of the electron of which we have spoken in our explanation of the equations (79), the distance MP = r will be in- finitely near the distance r0, and the time t0 infinitely near the time
£ _ -a - . The changes in the position and the velocity of the electron
C
in an infinitely small time being quantities of the second order, we may therefore understand by M the position at the instant t - ° , and by V the velocity at that time. Further:
T == 77 " -fa (77) M - gy (TT) [y] - -fc (TT) M , 1) Note 22.
54 I. GENERAL PRINCIPLES. THEORY Ol1 FREE ELECTRONS.
because, as is easily seen, the change in the distance between 0 and P, due to a shifting of the first point towards M, is equal to the change that would take place, if 0 remained where it was, but P were given a displacement — x, — y, — z. The square brackets now serve to in- dicate the values at the time t — — ; they will have this meaning in
c
all formulae that are now to be developed. Substituting the above value of — , and
where Vr may be considered as the component along OP, we find for the scalar potential
Having got thus far, we can omit the index o, so that r now means the distance from the origin 0 to a point with the coordinates x, y, z. As regards the last term, we can use the transformation
1 y rv i
c r LvjJ
, ., ' 3y »r
the last step in which will be clear, if we attend to the meaning of - - etc. The symbol [x] represents the value of x at the time t —
which we shall, for a moment, denote by t'. This time t' depends in its turn on the distance r, which again is a function of the coordi- nates x, y, z of the external point. Hence
1^ etc
'
._.
8x ' ' tit' Sr dx c r
Finally, the scalar potential becomes
._£_(! 3 [x] _a_fr]_ . AW). (84)
~ 4* I r ~dx r dy r 8z r \
The expression for the vector potential is even more simple, viz.
The radiation field, which predominates at large distances, and in which we find the flow of energy of which we have already spoken, is determined by the three last terms of 9 and by the vector potential. At smaller distances it is superposed on the field represented by the first term of <p, which is the same that would surround the electron if it were at rest.
FIKLD OF A 1'OLAUIZKU PARTK'I.K 55
42. By a slight change in the circumstances of the case, we can do away with the electrostatic field altogether. Let us suppose the electron to perform its vibrations in the interior of an atom or a molecule of matter, to which we shall now give the name of particle and which occupies the small space S. If the particle as a whole is not charged, it must contain, besides our movable electron, a charge — e, either in the form of one or more electrons, or distri- buted in any other manner. We shall suppose that this complementary charge — e remains at rest, and that, if the electron e did so likewise, in a determinate position, which we shall take as origin of coordinates, there would be no external field at all, at least not at a distance that is large in comparison with the dimensions of S. This being admitted, the immovable charge — e must produce a scalar potential equal and opposite to the first term in (84), so that, if we consider the field of the whole particle, this term will be cancelled. Our assumption amounts to this, that the charge — e is equivalent to a single electron — e at the point 0, so that, if the electron + e has the coordinates x, y, z, things will be as if we had two equal and opposite charges at a small distance from each other. We express this by saying that the particle is electrically polarized, and we define its electric moment by the equation
P - «r, (86)
where r is the vector drawn from 0 towards the position of the movable electron. The components of p are
P* = ex> P., = ey> P, = ez; (87)
and from (84) and (85) we find the following expressions for the potentials in the field surrounding the polarized particle
MAM -AM + AMl, (88)
4x\9« r h 0y r ' ciz r I'
(89)
These relations also hold in the case of a polarized particle whose state is somewhat more complicated. Let us imagine that it contains a certain number of electrons, any part of which may be movable. We shall find the potentials by calculating (84) and (85) for the separate electrons and adding the results. Using the sign 2 for this last operation, and keeping in mind that
Ze = 0, (90)
we shall again find the formulae (88) and (89), if we define the moment of the particle by the formula
p = Zer, (91)
56 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS. or its components by
(92)
It is even unnecessary that the charges should be concentrated in separate electrons. We can as well suppose them to be continuously distributed, but of course capable of moving or fluctuating in one way or another. Then the sums in the last formulae must be replaced by integrals. We shall have
flfdS - 0 (93)
and for the components of the moment
(94)
the integration being extended over the space S occupied by the particle. It must be noticed that on account of (90) and (93), the vectors (91) and (!'4) are independent of the choice of the point 0.
43. The formulae (88) and (89) show that the particle is a centre of radiation whenever the moment p is changing, and that it emits regular vibrations if p is a periodic function of the time.
We shall suppose for example that
p, = b cos (nt + p), py = 0, p, = 0, i, w and p being constants. Then we have
JU_i „(„(,_ .;) + ,),
and the field is easily determined by means of (88) and (89).
I shall not write down the general formulae but only those which hold for values of r that are very large compared with the wave-length, and which are obtained by the omission of all terms of
the order -j • They are as follows:
4«c'r
xy
inc'r "r
f cosjw^- j)4-J»}i
n!6 z I /. r\ , }
n,, = -r — 5 --- cos { n [ t ---- 1 + P 1 » 4jtc!r r \ \ c) ' ^ }
n*b y
(95)
corresponding to (80) and (81).1) 1) Note 23.
FUNDAMENTAL EQUATIONS FOR A MOVING SYSTKM. 57
I must add that our formulae for the field around a particle whose state of polarization is periodically changing, agree with those by which Hertz represented the state of the field around his vibrator.1)
44. We shall now pass on to certain equations that will be of use to us when we come to speak of the influence of the Earth's translation on optical phenomena. They relate to the electro- magnetic phenomena in a system of bodies having a common uniform translation, whose velocity we shall denote by w, and are derived from our original equations by a change of variables. Indeed, it is very natural to refer the phenomena in a moving system, not to a system of axes of coordinates that is at rest, but to one that is fixed to the system and shares its translation; these new coordinates will be represented by x', y', z'. They are given by
x' = x — wxt, y' = y-wytf, g' = e-vt,t. (96)
It will also be found useful to fix our attention on the velocity u of the charges relatively to the moving axes, so that in our funda- mental equations we have to put
v = w + u.
Now it has been found that in those cases in which the velocity of
w! translation W is so small that its square W2, or rather the fraction , ,
may be neglected, the differential equations referred to the moving axes take almost the same form as the original formulae, if, instead of t, we introduce a new independent variable f, and if, at the same time, the dielectric displacement and the magnetic force are replaced by certain other vectors which we shall call d' and h'. The variable t' is defined by the equation
*' = «_-I.(wX + w,y' + wX), (97)
and the vectors d' and h' by
T c L J?
We can regard t' as the time reckoned from the instant — (WX + W/ + Ws/),
1) H. Hertz, Die Krafte elektrischer Schwingungen , behandclt nach der Maxwell'schen Theorie, Ann. Phys. Chem. 36 (1888), p. 1.
58 I- GENERAL PRINCIPLES. THEORY OP PREE ELECTRONS.
which changes from one point to the other. This variable is there- fore properly called the local time, in order to distinguish it from the universal time t.
As to the vectors d' and h', the difference between them and d,
I ny 1
h is but small, since the fraction L -! is so. Even if we have to do
with the translation of the Earth, the value of W is no more than one ten -thousandth part of the velocity of light.
Neglecting terms with the square of '— , as has already been
c
said, one finds the following system of transformed equations:
, (100)
div h' = 0, (101)
roth'= J(d' + pU), (102)
rot d' = - 1 h'. (103)
The dot means a differentiation with respect to f, and the symbols div and rot (and, in the next paragraph, grad) serve to indicate differentiations with respect to x, y, z in exactly the same manner as they formerly indicated differentiations with respect to x, y, z. Rot h', for example, now means a vector whose components are i h; dVy a hi dV t h', dtix
_ _
dy' " da' ' W" 8x' ' dx' dy*"
You see that the formulae have nearly, but not quite, the same form
fw • u\ as (17) — (20), the difference consisting in the term -— -,-— in the first
equation.1)
45. Starting from the new system of equations, we can now repeat much of what has been said in connexion with the original one. For a given distribution and motion of the charges, the field is entirely determined, and here again the problem can be considerably simplified by the introduction of two potentials, a scalar and a vector one. These are given by the equations
(104)
^*"*/ '
and
where however the symbols [p] and [pu] require some explanation. If we want to calculate <p' and a' for a point P, for the moment at
1) Note 24.
RADIATION FROM A MOVING PARTICLE. 59
which the local time of this point has a definite value t', we must, for each element dS situated at a distance r from P, take the values of Q and <iii such as they are at the instant at which the local time of the element is
Finally, we have the following formulae for the determination of the field by means of the potentials1):
'), (106)
h' = rot a'. (107)
Here again, if we compare with (33) and (34), we notice a slight difference. In (33) there is no term corresponding to the last one in (106).
Notwithstanding the two differences I have pointed out, there is a large variety of cases, in which a state of things in a system at rest has its exact analogue in the same system with a translation. I shall give two examples that are of interest.
In the first place, the values of d' and h' produced by a particle moving with the velocity W, and having a variable electric moment, are given by formulae similar to those we formerly found for the radiation from a particle without a translation, und which I therefore need not even write down.
If the moment of a particle placed at the origin of coordinates is represented by
p, = &cos(»rf+j»), py = 0, p, = 0, (108)
all we have to do is to replace, in (95), d, h, x, y, z, t by d', h', x', y, z, t'.*)
In order to show the meaning of this result, I shall consider the
field at a point situated on the positive axis of y. It is determined by
all other components being 0. Since, neglecting terms of the second order, we may write
d = d'-i[w.h']
instead of (98), we have
dx = d;-^h;,
from which it appears that the dielectric displacement takes the form
d. — tf 608 j*^-- - in which a is a constant.
1) Note 25. 2) Note 26.
60 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
By substitution of the value of the local time and of the value (96) for y, this becomes
Thus we see that, at a fixed point of space, i. e. for a definite value of y, the frequency of the vibrations is given by
(i + -?)•
If the radiating particle has a positive velocity Wf/, i. e. one that is directed towards the point considered, this frequency is higher than that of the particle itself, which, as is shown by (108), still has the value n. This is the well known change of frequency which, according to Doppler's principle, is caused by a motion of the source of light.
46. Our second example relates to the reflexion of a beam of light by a perfectly reflecting mirror, for instance by one that con- sists of a perfectly conducting substance. We shall suppose the incidence to be normal, and begin with the case of a mirror having no translation, so that we have to use the original equations. Let the beam of light be represented by (7) and let the surface of the mirror coincide with the plane YOZ. Then, the reflected beam, which we shall distinguish by the suffix (r), is given by
dy(r) = - « cos n (< + y), h,(r) = a cos n(t + -J-) •
Indeed, these values satisfy the condition that, at the mirror, there be no dielectric displacement along its surface. If we put x = 0, we really find
«., + *,„ = 0.
The case of reflexion by a mirror moving with the velocity W^., in the direction of the axis of x, i. e. in the direction of its normal, can be treated by the same formulae, provided only we change x, t, d, h into x', t', d', h'.1) Therefore, if the incident beam is now represented by
Ay' = a cos n(t'- — V h.' = a cos nit' - — \, we shall have for the reflected light
d W) = - a cos n (f + -c) ' h'« W - a cos n (f + -7) '
Let us now examine the values of dy, ht, dy(r) and hz(r) in this case. The only component of W being wz, we find
1) Note 27.
UWLEXION BY A MOVING MIRROR. 01
so that the incident rays are given by
h, = a l + cos W *'- and the reflected rays by
In these formulae we shall now express t' and x' in terms of £ and x. The value of the local time is
and
z' = a; — Hence
The formulae are simplified if we put
Continuing to neglect the square of --*-, we infer from this
c
so that the final formulae for the incident rays are
d,, = a cosn^— *), h, = a cos and those for the reflected light
.^-^) («+:•).
These equations show that both the frequency and the amplitude of the reflected beam are changed by the motion of the mirror. The
62 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
(2 IV \ 1 - cx\, smaller than n, if the mirror recedes
from the source. These changes might have been predicted on the ground of Doppler's principle. As to the amplitude, it is changed in exactly the same ratio as the frequency, so that the reflected in- tensity is diminished by a motion in one direction and increased by a motion in the other direction.-
It is interesting to verify these results by considering the energy of the system. This may easily be done, if we fix our attention, not on the fluctuations of the electromagnetic energy, but on its mean value, so that, at every point of the beam, we and wm (§ 16) are considered as constants. Let the rays occupy a cylinder whose generating lines are parallel to OX and whose normal section is 27. Let P be a plane perpendicular to OX at some distance before the mirror and having a fixed position in the ether. If w.,. is positive, so that the mirror recedes, the space between the mirror and the plane P increases by WX27 per unit of time, so that the energy contained in that space increases by
Cw.4-tfjw.27.
Again, if p is the pressure on the mirror, the work done by the field will be
Consequently, if S is the current of energy towards the mirror per unit of area of the plane P, we must have
S27 = (we + wj W,27 + p w,27. (109)
We can easily calculate the quantities occurring in this equation. In the incident beam there is a flow of energy (§ 17)
towards the mirror, and in the reflected light a flow
K(I-^)'.-^(I-^:-).
away from it, so that
S = 2ttsW,. (110)
As to we, wm and p, we may take for them the values that would hold if the mirror where at rest, because these quantities have to be multiplied by w.,.. Therefore, since the value of we-\- wm consists of two equal parts, one belonging to the incident and the other to the reflected light1),
we+wm = tf. (Ill)
1) Note 28.
MOTION OF ELECTRICITY IN METALS. 63
Finally, by what has been said in § 25,
p = a2. (112)
The values (110), (111) and (112) really satisfy the condition (109).
47. I shall close this chapter by a short account of the appli- cation of the theory of electrons to the motion of electricity in metallic bodies. In my introductory remarks, I have already alluded to the researches of Kiecke, Drude and J. .1. Thomson. I now wish especially to call your attention to the views that have been put forward by the second of these physicists.
In his theory, every metal is supposed to contain a large number of free electrons, which are conceived to partake of the heat-motion of the ordinary atoms and molecules. Further, a well known theorem of the kinetic theory of matter, according to which, at a given tem- perature, the mean kinetic energy is the same for all kinds of par- ticles, leads to the assumption that the mean kinetic energy of an electron is equal to that of a molecule of a gas taken at the same temperature. Though the velocity required for this is very conside- rable, yet the electrons are not free to move away in a short time to a large distance from their original positions. They are prevented from doing so by their impacts against the atoms of the metal itself.
For the sake of simplicity we shall assume only one kind of free electrons, the opposite kind being supposed to be fixed to the ponderable matter. Now, if the metal is not subjected to an electric force, the particles are moving indiscriminately towards all sides; there is no transfer of electricity in a definite direction. This changes however as soon as an electric force is applied. The velocities of the electrons towards one side are increased, those towards the other side diminished, so that an electric current is set up, the intensity of which can be calculated by theoretical considerations. The formula to which one is led, of course contains the electric force, the number N of electrons per unit of volume, the charge e and the mass m of each of them. In the first place, the force acting on an electron is found if we multiply by e the electric force. Next dividing by m, we shall find the velocity given to the electron per unit of time. The velocity acquired by the electrons will further depend on the time during which they are exposed to the undisturbed action of the electric force, a time for which we may take the interval that elapses between two successive impacts against a metallic atom. During this
interval, the length of which is given by — , if I is the path between the two encounters, and u the velocity of the electron, the electric
64 I- GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
force produces a certain velocity which we can take to be lost again at the next collision.
These considerations will suffice for the explanation of the formula
(H3)
which Drude has established for the electric conductivity of the metal, and in which we must understand by n the mean velocity of the electrons in their irregular heat-motion, and by I their mean length of free path. Now, as I said already, the mean kinetic energy of an electron, for which we may write ^mu2, is supposed to be equal to the mean kinetic energy of a gaseous molecule. The latter is proportional to the absolute temperature T, and may therefore be represented by
ccT,
where K is a universal constant. If we use this notation, (113) takes the form
e*Nlu
48. Iii order to show you all the beauty of Drude's theory, I must also say a few words about the conductivity for heat. This can be calculated in a manner much resembling that in which it is determined in the kinetic theory of gases. Indeed, a bar of metal whose ends are maintained at different temperatures, may be likened to a colum of a gas, placed, for example, in a vertical position, and having a higher temperature at its top than at its base. The process by which the gas conducts heat consists, as you know, in a kind of diffusion between the upper part of the column, in which we find larger, and the lower one in which there are smaller molecular velo- cities; the amount of this diffusion, and the intensity of the flow of heat that results from it, depend on the mean distance over which a molecule travels between two successive encounters. In Drude's theory of metals, the conduction of heat goes on in a way that is exactly similar. Only, the carriers by which the heat is transferred from the hotter towards the colder parts of the body, now are the free electrons, and the length of their free paths is limited, not, as in the case of a gas, by the mutual encounters, but by the impacts against the metallic atoms, which we may suppose to remain at rest on account of their large masses.
Working out these ideas, Drude finds for the coefficient of con- ductivity for heat
(115)
CONDUCTIVITY FOR HEAT AND FOR ELECTRICITY. 65
49. It is highly interesting to compare the two conductivities, that for heat and that for electricity. Dividing (115) by (114), we get
which shows that the ratio must be equal for all metals. As a rough approximation this is actually the case.
We see therefore that Drude has been able to account for the important fact that, as a general rule, the metals which present the greatest conductivity for heat are also the best conductors of elec- tricity.
Going somewhat deeper into details, I can point out to you two important verifications of the equation (116).
In the first place, measurements by Jaeger and Diessel-
horst1) have shown that the ratio between the two conductivities
varies approximately as the absolute temperature, the ratio between
k the values of for 100° and 18° ranging, for the different metals,
between 1,25 and 1,12, whereas the ratio between the absolute temperatures is 1,28.
In the second place, the right-hand member of (116) can be calculated by means of data taken from other phenomena.2) In order to see this, we shall consider an amount of hydrogen, equal to an electrochemical equivalent of this substance, and we shall suppose this quantity to occupy, at the temperature T, a volume of one cubic centimetre. It will then exert a pressure that can easily be calculated, and which I shall denote by p.
We have already seen that the charge e which occurs in the formula (116), may be reckoned to be equal to the charge of an atom of hydrogen in an electrolytic solution. Therefore, the number
of atoms in one electrochemical equivalent of hydrogen is - -• The gas
€
being diatomic, the number of molecules is , and the total kinetic energy of their progressive motion is
aT 2e per cubic centimetre.
By the fundamental formula of the kinetic theory of gases the
1) W. Jaeger und H. Diesselhorst, Warmeleitung, Elektrizitatsleitung, Warmekapazitat und Thermokraft einiger Metalle, Sitzungsber. Berlin 1899, p. 719.
2) See M. Reinganum, Theoretische Bestimmung des VerhiVltnisses von Wiirme- und Elektrizitatsleitung der Metalle aus der Drudeschen Elektronen- theorie, Ann. Phys. 2 (1900), p. 398.
Lorentz, Theory of electrons. 5
66 I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS.
pressure per unit area is numerically equal to two thirds of this,
so that
«T
The equation (116) therefore takes the form
* 7
or
(117)
This relation between the conductivities of a metal and other quantities derived from phenomena which, at first sight, have no con- nexion at all, neither with the conduction of heat, nor with that of electricity, has been verified in a very satisfactory way.
The electrochemical equivalent of hydrogen being
0,000104
in our units, and the mass of a cubic centimetre of the gas at 0° and under a pressure of 76 cm of mercury being 0,0000896 gramra, one finds for the temperature of 18° (T = 273 + 18),
12,5xl06
!' = --,.--• (118)
cyiit
On the other hand, expressing <J in the ordinary electromagnetic units, Jaeger and Diesselhorst have found for silver at 18°
- - 686 X 108. a
In our units this becomes
fc 686 x 108
by which we find for the quantity on the right hand side of (117)
12.9X106
showing a very close agreement with the value we have just cal- culated for p.
50. I must add, however, that the numerical agreement becomes somewhat less satisfactory, if, instead of Drude's formulae for the con- ductivities, one takes the equations to which I have been led by cal- culations that seem to me somewhat more rigorous than his. Taking into account that the electrons in a piece of metal have unequal velo-
CONDUCTIVITY FOR HEAT AND FOR ELECTRICITY. 67
cities, and assuming Maxwell's law for the distribution of these among the particles, I find, instead of (114) and (115)1),
IT e'INu
3* K']' '
and
fc = *l/a uNlu. (120)
' f OJl
In these equations, u is a velocity of such a magnitude that its square is equal to the mean square of the velocities which the electrons have in their heat-motion, and / represents a certain mean length of free path.
The ratio of the two conductivities now becomes
a * \ e) '
it is still proportional to the absolute temperature, but it is only two thirds of the value given by Drude. On account of this we must replace (117) by the equation
J» —
whose right-hand side, in the example chosen in § 49, has the value
oJ/4*
This is rather different from (118).
If we prefer the formulae (119) and (120) to (114) and (115), as I think we are entitled to do, the agreement found in the prece- ding paragraph must be considered as produced by a fortuitous coin- cidence. Nevertheless, even the agreement we have now found, cer- tainly warrants the conclusion that, in Drude's theory, a fair start has been made towards the understanding of the electric and thermal properties of metals. It is especially important to notice that our calculations rest on the assumption that the free electrons in a metal have charges equal to those of the ions of hydrogen.
1) Note 29.
CHAPTER II.
EMISSION AND ABSORPTION OF HEAT.
51. The subject of this and my next lecture will be the radia- tion and absorption of heat, especially the radiation by what is called a perfectly black body, considered with regard to the way in which these phenomena depend on the temperature and the wave-length. I shall first recall to your minds the important theoretical laws which Kirchhoff, Boltzmann and Wien have found by an application of thermodynamic principles. After that, we shall have to examine how far the theory of electrons can give us a clue to the mechanism of the phenomena.
We must begin by clearly defining what is meant by the absorb- ing power and the emissivity of a body. Let ra and to' be two in- finitely small planes perpendicular to the line r joining their centres, and let M be a body of the temperature T, placed so that it can receive a beam of rays going through a' and w. We shall suppose this beam to consist of homogeneous rays whose wave-length is A, and to be plane -polarized, 'the electrical vibrations having a certain direction h, perpendicular to the line r. Part of the incident rays will be reflected at the front surface of the body, part of them will penetrate into its interior, and of these some will again leave the body, either directly or after one or more internal reflexions. How- ever this may be, the body M, if it be not perfectly transparent, will retain a certain amount of energy, an amount that is converted into heat, because we shall exclude from our considerations all other changes that might be produced.
The coefficient of absorption A is defined as the fraction indi- cating what part of the incident energy is spent in heating the body M.
On the other hand, of the whole radiation emitted by M, a certain portion will travel outwards through the two elements to and a'. We shall decompose this radiation into rays of different wave-lengths, and we shall fix our attention on those whose wave-length lies be- tween two limits infinitely near each other, I and I -\- ill. We shall also decompose the electrical vibrations of these rays into a com-
EMISSIV1TY AND ABSORBING I'OWKK. 69
ponent along the line h of which I have just spoken, and a second component perpendicular both to it and to the direction of the beam itself. It can easily be shown that the amount of energy emitted by the body per unit of time through the two elements of surface, so far as it belongs to rays of the wave-lengths that have been specified, and to vibrations of the direction h, is proportional to «, o', dk, and inversely proportional to the square of r. It can there- fore be represented by
The coefficient E is called the emissivity of the body M. It is a quantity depending on the nature of M, its position with respect to the line r, the wave-length A, the temperature T and the direction h which we have chosen for the vibrations.
Starting from the thermodynamic principle that in a system of bodies having all the same temperature, the equilibrium is not dis- turbed by their mutual radiation, and using a train of reasoning which I shall not repeat, Kirchhoff1) fiuds that the ratio
between the ernissivity and the absorbing power is independent, both of the direction we have chosen for h, and of the position and the peculiar properties of the body M. It will not be altered if we change the position of M, or replace it by an altogether different body of the same temperature. The ratio between the emissivity and the coefficient of absorption is a function of the temperature and the wave-length alone.
52. I shall now point out to you two other meanings that may be attached to this function. In the first place, following the example of Kirchhoff, we can conceive a perfectly black body, or, as we shall simply say, a black body, i. e. one that has the power of retaining for itself the total radiating energy which falls upon it. Its coeffi- cient of absorption is therefore 1, and if we denote its emissivity by Eb, the symbols A and E relating to any other body, we shall have
f = ^V (122)
We may notice in passing that Kirchhoff s law requires all black bodies, whatever be their nature, to have exactly the same emissivity.
1) G. Kirchhoff, tlber das Verhiiltnis zwischen dem Emissionsvermogen und dem Absorptions vermb'gen der Korper fiir Wilrme und Licht, Ann. Phys. Chem. 109 (I860), p. 275.
70 n. EMISSION AND ABSORPTION OF HEAT.
10
The equation (122) expresses one of the two meanings of
to which I have alluded. The other will become apparent, if we fix our attention on the state existing in the ether in the neighborhood of radiating bodies.
We shall consider a space void of all ponderable matter and surrounded on all sides by a perfectly black envelop, which is kept at a fixed temperature T. The ether within this space is traversed in all directions by rays of heat. Let to be an element of a plane situated at any point P of the space, and having any direction we like. We shall consider the quantity of energy by which this ele- ment is traversed per unit of time in the direction of its normal n, or rather in directions lying within an infinitely narrow cone, whose solid angle we shall denote by s, and whose axis coincides with the normal n, always confining ourselves to wave-lengths between I. and I + dl., and to a particular direction h of the electrical vibrations. By this I mean that all vibrations of the rays within the cone are decomposed along lines h and k that are perpendicular as well to each other as to the axis of the cone, and that we shall only con- sider the components having the first named direction.
Let P' be a point on the normal n, at a distance r from the point P, and let us place at f an element of surface perpendicular to r, and whose magnitude is given by
«' =»•*£. (123)
It is clear that, instead of speaking of the rays whose direction lies within the cone e, we may as well speak of those that are propagated through the elements o and «'.
The quantity we wish to determine is therefore the flow of energy through the two small planes, issuing from the part of the enclosing wall behind w. In virtue of the formula (121), it is given by
J$btam'dl
for which, on account of (123), we may write
(124)
Having got thus far, we need no longer consider the element w'; we have only to think of the element ra and the cone e.
Now, what is most remarkable in our result, is the fact that it is wholly independent of the position of the point P, the direction of the element K> and the directions h and k, in which we have de- composed the vibrations. The radiation field within the ether is a
ENERGY OF RADIATION PER UNIT OF VOLUME. 71
truly isotropic one, i. e. the propagation takes place in exactly the same manner in all directions, and electrical vibrations of all different directions occur with the same intensity.
We shall now calculate the amount of energy in this radiation field per unit of volume. In the case of a beam of rays of a definite direction the quantity of energy that is carried per unit of time through a plane co perpendicular to the rays, is equal to the amount existing at one and the same moment in a cylinder whose generating lines are parallel to the rays, and which has oj for its base and a height equal to the velocity of light e; it is CK> times the energy existing per unit of volume. Hence, the energy per unit of volume, belonging to the rays to which the expression (124) relates, is found if we divide that expression by cw; its value is
We must now keep in mind that we have all along considered only the rays whose direction lies within the cone e, and only those com- ponents of their vibrations which have the direction /(. If we wish to include all rays, whatever be their direction and that of their vibrations, we must make two changes. In the first place we must multiply by 2, because the vibrations of the direction it have the same intensity as those we have till now considered, and in the se- cond place we must replace £ by 4re, because equal quantities of energy belong to rays whose directions lie within different cones of equal solid angles. The final result for the amount of energy present in unit volume of our radiation field, the ,,density" of the energy, BO far as it is due to rays whose wave-lengths lie between the limits i and i + di, is
We shall write for tbis energy
so that, if we also take into account the relation (122), we have
(125)
V
This equation, which expresses the relation between J ' and the density of energy, shows us the other meaning that may be given
63. One word more may be said about the state of radiation characterized by the function F(\, T). For the existence of this
72 II. EMISSION AND ABSORPTION Ol1 HEAT.
state it is not at all necessary that the walls of the enclosure should be perfectly black. We may just as well suppose that they are per- fectly reflecting on the inside, and that the rays are produced by a body placed somewhere between them. Nor need this body be per- fectly black. Whatever be its nature, if it is maintained at the tem- perature T we have chosen once for all, it can always be in equili- brium with a state of radiation in which each element of volume contains the energy we have been considering. We may add that not only will it be in equilibrium with this state, but that it will actually produce it, provided only the body have some emitting power, however small it may be, for all wave-lengths occurring in the radiation of a black body of the same temperature. If this con- dition is fulfilled, the radiation in the ether will be independent of the nature of the matter in which it originates; it will be determined by the temperature alone.
54. Kirchhoff has already laid stress on the importance of the function I1' (I, T), which must be independent of the peculiar pro- perties of any body, and indeed the problem of determining this function is of paramount interest in modern theoretical physics. Boltzmann1) and Wien2) have gone as far towards the solution as can be done by thermodynamic principles, combined with general results of electromagnetic theory, if one leaves aside all speculations concerning the constitution of the radiating and absorbing matter.
Boltzmann's law shows us in what way the total energy existing per unit of volume in the radiation field we have spoken of, I mean the energy for the rays of all wave-lengths taken together, depends on the temperature. It is proportional to the fourth power of the absolute temperature, a result that had already been established as an empirical rule by Stefan.
In his demonstration, Boltzmann introduces the fact that there is a radiation pressure of the amount which we have formerly cal- culated.
Let us consider a closed envelop, perfectly reflecting on the in- side, and containing a body M to which heat may be given or from which heat may be taken, in one way or another. The remaining part of the space contains only ether, and the walls are supposed to be movable, so that the enclosed volume can be altered.
1) L. Boltzmann, Ableitung des Stefan'schen Gesetzes, betreffend die Abhangigkeit der Warmestrahlung von der Temperatur aus der elektromagnR- tischen Lichttheorie, Ann. Phys. Chem. 22 (1884), p. 291.
2) W. Wien, Eine neue Beziehung der Strahlung sckwarzer Korper zum zweiten Hauptsatz der Wiirmetheorie, Berlin. Sitzungsber. 1893, p. 55.
BOLTZMANN'S LAW. 73
The system we have obtained in this manner is similar in many respects to a gas contained in a vessel of variable capacity. It is the seat of a certain energy, and like a gas it exerts a pressure on the bounding walls; only, we have now to do, not with the collisions of moving molecules, but with the pressure of radiation. If the walls move outwards, the system does a certain amount of work on them. Hence, a supply of heat is required, if we wish to maintain a con- stant temperature, and the temperature is lowered by the expansion, if the process is adiabatic. You will easily see that the system may be made to undergo a cycle of operations, two of which are isother- mic and two adiabatic changes, and to which we may apply the well known law of Carnot.
Instead of imagining a cycle of this kind, I shall use a small calculation that will lead us to the same result. In all cases in which the state of a system is determined by the temperature T and the volume v, and in which the only force exerted by the system is a normal pressure p uniformly distributed over the surface, there is a simple thermodynamic relation by which we can learn something about the internal energy e. If we choose v and T as independent variables, the equation has the form
This may be applied to our envelop filled with rays, as well as to a gas; in a certain sense the case of the radiation is even the more simple of the two. The reason for this is, that the density of the energy depends solely on the temperature, so that, in an isothermic expansion, the new part that is added to the volume is immediately filled with an amount of energy proportional to its extent. The energy contained in the space that was already occupied by the radia- tion, remains unchanged, and the same may be said of the energy contained within the body M. In order to see this, we must keep in mind that, by what has been found in § 21, the pressure is equal to one third of the electromagnetic energy per unit of volume, so that the body remains exposed to the same pressure and, the tem- perature being likewise constant, will undergo no change at all.
Let us denote by K the electromagnetic energy per unit of volume, which, as we must take together all wave-lengths, may be represented by
00
K-fF(H,F)dl.
0
Then we shall have
74 II. EMISSION AND ABSORPTION OF HEAT.
and
dt jf. o~ = A,
8v
because, if the volume is increased by dv, the energy augments by Kdv. Substituting in the formula (126), we find
dK _ . dT K ~~ l T '
from which we deduce by integration
where C is a constant. The total energy per unit of volume, or as, in virtue of (125), we may also say, the total emissivity of a black body must be proportional to the fourth power of the temperature.
55. Passing on now to Wien's law, I shall first state the form in which it may be put if we avail ourselves of that of Boltzinann. Wien has not succeeded in determining the form of the function, which indeed cannot be done by thermodynarnic reasoning anil electromagnetic principles alone; he has however shown us how, as soon as the form of the function is known for one temperature, it may be found from this for any other temperature.
This may be expressed as follows. If T and T' are two diffe- rent temperatures, A and A' two wave-lengths, such that
i:l'=T':T, (127)
we shall have
F(l, T) : F(i', T) = A'5 : i5. (128)
If we put this in the form
we see that really F(Ji', 7") can be determined for all values of A', if we know F(l, T) for all values of L
We can also infer from (127) and (128) that if, while varying I and T, we keep the product /IT constant, the function isF(Ji, T) must also remain unchanged. Therefore, this last expression must be some function f(kT) of the product of wave-length and tempera- ture, so that our original function must be of the form
(129)
WIEN'S LAW.
75
The relation between the forms of the function .F(A, 71) for different temperatures comes out very beautifully. If, for a definite temperature T, we plot the values of l'\i, 7'), taking i as abscissae and F as ordinates, we shall obtain a certain curve, which may be said to represent the distribution of energy in the spectrum of a black body of the temperature T. From this we can get the corre- sponding curve for the temperature T' by changing all abscissae in the ratio of T to T, and all ordinates in the ratio of T5 to T*.
The form of the curve has been determined with considerable accuracy by the measurements of Lummer and Pringsheim.1) The accompanying figure will give an idea of it. It shows that, as could have been expected, the intensity is small for very short and very long waves, reaching a maximum for a definite wave-length which is represented by OA, and which I shall call lm. Now, if the curve undergoes the change of shape of which I have just spoken, this maximum will be shifted towards the right if T' is lower than T, and towards the left in the opposite case, the value of Am being in fact inversely proportional to the temperature. It is for this reason that Wien's law is often called the displacement- law (Verschiebungsgesetz).
The diagram may also be used for showing that Boltzmann's law is included in the formulae (127) and (128). The value of K is given by the total area included between the curve and the axis of abscissae, and this area changes in the ratio of T* to T'4, when the abscissae and ordinates are changed as has been stated.
56. It would take too much of our time, if I were to give you a complete account of the theoretical deductions by which Wien found his law. Just as in Boltzmann's reasoning, we can distinguish two parts in it, one that is based on the equations of the electro- magnetic field, and a second that is purely thermodynamic.
We have already seen that, for every temperature T, there is a perfectly definite state of radiation in the ether, which has the pro- perty that it can be in equilibrium with ponderable bodies of the temperature T. For the sake of brevity I shall call this the natural
1) 0. Lummer u. E. Pringsheim, Die Strahlung eines schwarzen Korpers zwischen 100 und 1300° C, Ann. Phys. Chem. 63 (1897), p. 395; Die Verteilung der Energie im Spektrum des schwarzen Korpers, Verh. d. deutscken phys. Ges. 1 (1899), p. 23.
76 U- EMISSION AND ABSORPTION OF HEAT.
state of radiation for the temperature T. It is characterized by a definite amount K of energy per unit of volume, proportional to I*4, and which may therefore be used, instead of T itself, for defining the state of the ether. If we speak of a natural state of radiation with the energy -density K, we shall know perfectly what we mean.
In this natural state the total energy is distributed in a definite manner over the various wave-lengths, a distribution that is expressed by the function F(i, T). Now, we can of course imagine other states having the same density of energy K, but differing from the natural one by the way in which the energy is distributed over the wave-lengths; it might be, for example, that the energy of the long waves were somewhat smaller, and that of the short ones somewhat greater than it is in the natural state.
Wien takes the case of a closed envelop perfectly reflecting on the inside, and containing only ether. He supposes this ether to be the seat of a natural state of radiation A with the energy-density K; this may have been produced by a body of the temperature T that has been temporarily lodged in the enclosure, and has been removed by some artifice. Of course, this operation would require a super- human experimental skill and especially great quickness, but we can suppose it to be succesfully performed. If then we leave the vessel to itself, the radiations that are imprisoned within it, will continue to exist for ever, the rays being over and over again reflected by the walls, without any change in their wave-lengths and their intensity.
At this point, Wien introduces an imaginary experiment by which the state of things can be altered. It consists in giving to the walls a slow motion by which the interior volume is increased or diminished. We have already seen (§ 46) that, if a mirror struck normally by a beam of rays is made to recede, this will have a twofold influence on the reflected rays; their frequency is lowered, so that the wave-length becomes larger, and their amplitude is diminished. The same will be true, though in a less degree, if the incidence of the rays is not normal but oblique, and in this case also the effect can be easily calculated.
In order to fix .our ideas we shall suppose the walls of our vessel to expand. Then, every time a ray is reflected by them, it has its amplitude diminished and its wave-length increased, so that, after a certain time, we shall have got a new state of radiation B, differing from the original one by its energy per unit of volume and by the distribution of the energy over the wave-lengths. The den- sity of energy will have a certain value K', smaller than the original value K, and the distribution over the wave-lengths will have been somewhat altered in favour of the larger wave-lengths.
WIEN'S LAW. 77
Of course, K' can have different values, because the expansion by which the new state is produced may be a large or a small one. Since, however, the changes in the amplitudes and those in the wave- lengths are closely connected, it is clear that the distribution of the energy over the wave-lengths must be quite determinate if we know K', so that it was possible for Wien to calculate it. His result may be expressed as follows.1) If
<f>(X)dl (130)
is the part of the original energy per unit of volume that is due to the rays with wave-lengths between 1 and Jl + di, the amount of en- ergy corresponding to the same interval in the new state B is given by
57. I hope I have given you a sufficiently clear idea of one part of Wien's demonstration. As to the second part, the thermo- dynamic one, its object is to show that the new state S, in which there is a density of energy K', cannot be different from a natural state of radiation having the same K', that it must therefore itself be a natural state. If it were not, we could place our vessel con- taining the state _B against a second vessel containing a natural state A with the same value K', the two states being at first separated by the walls of the two vessels. Then we could make an opening in these walls, and close it immediately by means of a very thin plate of some transparent substance. Such a plate will transmit part of the rays by which it is struck, and, on account of the well-known phenomena of interference, the coefficient of transmission will not be the same for different kinds of rays. Let us suppose it to be some- what greater for the long waves than for the short ones, and let us also assume that the state J3 contains more of the long waves than the state A', and less of the short waves. Then, it is easily seen that, in the first instants after communication has been established between the two vessels, more energy will pass from S towards A' than in the inverse direction, so that the energy of the two states will not remain equal. This can be shown to be in contradiction with the second law of thermodynamics.
Our conclusion must therefore be that, by means of the ex- pression (131), we can calculate the distribution of energy in a natural state characterized by K', as soon as we know the distribution, re- presented by (130), for a natural state characterized by K. Now, both states being natural ones, we shall have, if we write T and T'
1) Note 30.
78 II. EMISSION AND ABSORPTION OF HEAT.
for the temperatures to which they correspond,
K:K'=T*:T'1.
Therefore, (131) becomes
by which we are led to Wien's law in the form in which I have stated it.
58. Though Boltzmann and Wien have gone far towards determining the function F(3., T\ the precise form of the curve in Fig. 2 remains to be found, and since the means of thermodynamics are exhausted, we can only hope to attain this object, if we succeed in forming some adequate mental picture of the processes which mani- fest themselves in the phenomena of radiation and absorption.
The importance of the problem will be understood, if one takes into account that the curve in Fig. 2 requires for its determination at least two constants. Calling im the abscissa OA for which the ordinate is a maximum, we have by Wien's law
and if, as before, the total area included between the curve and the axis of abscissae is denoted by K, we shall have
Of the two constants a and b, the first determines, for a given tem- perature T, the position of the point A, and the second relates to the values of the ordinates, because the larger these are, the greater will be the area K. Now, if the state of radiation is produced by a ponderable body, the values of the two constants must be determined by something in the constitution of this body, and these values can only have the universal meaning of which we have spoken, if all ponderable bodies have something in common. If we wish comple- tely to account for the form and dimensions of the curve, we shall have to discover these common features in the constitution of all ponderable matter.
59. I shall speak of three theories by which the problem has been at least partially solved, beginning with the one that goes farthest of all. This has been developed by Planck1), and leads to a definite formula for the function f(lT) in (129), viz. to
1) M. Planck, tiber irreversible Strahlungsvorgange, Ann. Phys. 1 (1900), p. 69; tiber das Gesetz cler Energieverteilung im Normalspektrum, Ann. Phys. 4 (1901), p. 563; tiber die Elementarquanta der Materie und der Elektrizitlit, ibid., p. 564; see also his book: Vorlesungen fiber die Theorie der Warme- strahlung, Leipzig, 1906.
THEORY OF PLANCK. 79
-JT-, (132)
«U —1
in which s is the basis of natural logarithms, whereas h and k are two universal physical constants.
Planck's theory is based on the assumption that every ponde- rable body contains an immense number of electromagnetic vibrators, or ,,resonators" as he calls them, each of which has its own period. If a body is enclosed within the perfectly reflecting walls we have so often mentioned, there will be a state of equilibrium, on the one hand between the resonators and the radiation in the ether, and on the other hand between the resonators and the ordinary heat motion of the molecules and atoms constituting the ponderable matter. The first of these equilibria can be examined by means of the electro- magnetic equations, and, in order to understand the second, one could try to trace the interchange of energy between the resonators and the ordinary particles. Planck, however, has not followed this course, which would lead us into very serious difficulties, but has found his formula by reasonings of a different kind.
In one of his papers he deduces it by examining what partition of the energy between the two sets of particles, the molecules and the resonators, is to be considered as the most probable one. Of course this is an expression, the precise meaning of which has to be fixed before we can make it the basis of the theory. I must abstain from explaining the sense in which it is understood by Planck. There is one point, however, in his theory to which I must refer for a moment. He is obliged to assume that the resonators can gain or lose energy, not quite gradually by infinitely small amounts, but only by certain portions of a definite finite magnitude. These por- tions are taken to be different for resonators of different frequencies. The portions of energy which we have to imagine when we speak of a resonator of the frequency n, have an amount that is given
by the expression
hn
It is in this way that the constant h is introduced into the equations.
As to the constant k, it has a very simple physical meaning. According to the kinetic theory of gases, the mean kinetic energy of the progressive motion of a molecule is equal for all gases, when compared at the same temperature. This mean energy is proportional to T, and if we represent it by ffcT, the quantity k will be the constant appearing in the formula (132).
Planck's law shows a most remarkable agreement with the experimental results of Lumrner and Pringsheim, and it is of
80 II. EMISSION AND ABSORPTION OF HEAT.
high value because it enables us to deduce from measurements on radiation the mean kinetic energy of a molecule, which, in its turn, leads us to the masses of the atoms in absolute measure. As the numbers obtained in this way1) are of the same order of magnitude as those that have been found by other means, there is undoubtedly much truth in the theory. Yet, we cannot say that the mechanism of the phenomena has been unveiled by it, and it must be admitted that it is difficult to see a reason for this partition of energy by finite portions, which are not even equal to each other, but vary from one resonator to the other.
60. I shall dwell somewhat longer on the second theory2), be- cause it is an application of the theory of electrons, and therefore properly belongs to my subject. In a certain sense, it may, I think, be considered as rather satisfactory, but it has the great defect of being confined to long waves. I may be permitted perhaps, by way of in- troduction, to tell you by what considerations I have been led to this theory. It is well known that, in general, the optical properties of ponderable bodies cannot be deduced quantitatively with any degree of accuracy from the electrical properties. For example, though Maxwell's theoretical inference, published long ago in his treatise, that good conductors for electricity must be but little transparent for light, is corroborated by the fact that metals are very opaque, yet, if we compare the optical constants of a metal, one of which is its coefficient of absorption, with the formulae of the electromagnetic theory of light, taking for the conductivity the ordinary value that is found by measurements on electric currents, there is a very wide disagreement. This shows, and so does the discrepancy between the refractive indices of dielectrics and the square root of their dielectric constants, that, in the case of the very rapid vibrations of light, cir- cumstances come into play with which we are not concerned in our experiments on steady or slowly alternating electric currents.
If this idea be right, we may hope to find a better agreement, if we examine the ,,optical" properties as we may continue to call them, not for rays of light, but for infra-red rays of the largest wave-lengths that are known to exist.
Now, in the case of the metals, this expectation has been verified in a splendid way by the measurements of the absorption that were made some years ago by Hagen and Rubens.3) These physicists have
1) Note 31.
2) Lorentz, On the emission and absorption by metals of rays of heat of great wave-lengths, Amsterdam Proc., 1902 — 03, p. GG6.
3) E. Hagen u. H. Rubens, tJber Beziehungen des Reflexions- und Emissionsvermogens der Metalle zu ihrem elektrischen Leitvermogen, Ann. Phys. 11 (1903), p. 873.
ABSORPTION BY A THIN METALLIC PLATE. 81
shown that rays whose wave-length is between 8 and 25 microns, are absorbed to a degree that may be calculated with considerable accuracy from the known conductivity.1) We can conclude from this that, in order to obtain a theory of absorption in the case of these long waves, we only have to understand the nature of a common current of con- duction. Moreover, if in this line of thought, we can form for our- selves a picture of the absorption, it must also be possible to get an insight into the way in which rays are emitted by a metal. Indeed, the universal validity of Kirchhoff's law clearly proves that the causes which produce the absorption by a body, and those which call forth its radiation, must be very closely related. Therefore, as soon as we have an adequate idea about a common current of conduction, we may hope to be able to explain the absorption and the emissivity of a metal, and to calculate the ratio between the two, i. e. our uni- versal function F(i, T~). However, we can only hope to succeed in this, if we confine ourselves to long waves.
Now, as we have already seen, a very satisfactory conception of the nature of a current of conduction has been worked out by Drude. We must therefore try to obtain a theory of the radiation and emission of metals that is based on his general principles, and in which we simply assume that the metal contains a large number of free electrons, moving with such speeds that their mean kinetic energy is equal to aT.
61. In doing so, we shall simplify as much as possible the cir- cumstances of the case. We shall consider a metal plate, whose thickness A is so small that the absorption may be considered as proportional to it, and that, in examining the emission, we need not consider the ab- sorption which the rays emitted by the back half of the plate undergo, while traversing the layers lying in front of it. We shall also confine ourselves to rays whose direction is perpendicular to the plate or makes an infinitely small angle with the normal. These assumptions will greatly facilitate our calculations without detracting from the generality of the final result. If we trust to Kirchhoff's law, the value which we shall find for the ratio between the emissivity and the coefficient of absorption may be expected to hold for all bodies and for all directions of the rays.
The calculation of the absorption is very easy. By the ordinary formulae of the electromagnetic field we find for the coefficient of absorption8)
1) Note 32. 2) Note 38.
Lorentz, Theory of electrons.
82 II- EMISSION AND ABSORPTION OF HEAT.
and here we have only to substitute the value of 0, given by Drude's theory. Using the formula (119), we find
62. The question now arises, in what manner a piece of metal in which free electrons are moving in all directions can be the source of a radiation. The answer is contained in what we have seen in a former lecture. We know that an electron can be the centre of an emission of energy only when its velocity changes. The cause of the emission must therefore be looked for in the impacts against the metallic atoms, by which the electron is made to rebound in a new direction, so that the radiation of heat, in the case we are now con- sidering, very much resembles the production of Rontgen rays, as it is explained in Wiechert's and J. J. Thomson's theory.
The mathematical operations required for the determination of the effects of the impacts are rather complicated, the more so because we must decompose the total radiation into the parts corresponding to different wave-lengths. I shall therefore give only a general outline of the calculations.
I must mention in advance that the decomposition of which I have spoken just now will be performed by means of Fourier's theorem, and that the duration of an impact will be taken to be ex- tremely small in comparison with the time of vibration of the rays considered. We shall even make the same assumption with regard to the time between two successive impacts of an electron. This is justified by the experiments of Hagen and Rubens. It is easily seen that the conductivity of a metal can be given by the formula (119), only if the electric force acts on the body either continually or at least for a time during which a large number of encounters of an electron take place. Therefore, the result found by flagen and Rubens, viz. that the absorption corresponds to the coefficient of conductivity, proves that the time during which the electric force acts in one and the same direction, i. e. half a period, contains very many times the interval between two successive encounters.
63. In § 51 we have considered the radiation from the body M through two infinitely small planes o and a'. We shall now suppose the first of these to be situated in the front surface of the thin metallic plate, and we shall fix our attention on the radiation issuing from the corresponding part cod of the plate, and directed towards the element to', parallel to o>, and situated at a point P of the line drawn normally to the plate from the centre 0 of the element to.
EMISSION BY A THIN PLATE. 83
We shall begin by taking into account only the component of the electric vibrations in a certain direction h perpendicular to OP.
Let us choose the point 0 as origin of coordinates, drawing the axis of e along OP, that of x in the direction //, and denoting the distance OP by r. According to what has been found in § 39, a single electron, moving with the velocity V in the part of the plate considered, will produce at P a dielectric displacement whose first component is given by
« dv,,. ~ ~dt '
if we take the value of the differential coefficient for the proper instant.
On account of our assumption as to the thickness of the plate, this instant may be represented for all the electrons in the portion ad
Dv ^ ~~ ~ » if ^ 's khe time f°r which we wish to determine the state
of things at the point P. We may therefore write for the first component of the dielectric displacement at P
The flow of energy through o' per unit of time will be
Since the motion of the electrons between the metallic atoms is highly irregular, we shall have, at rapidly succeeding instants, a large number of impacts in which the changes of the velocity are widely different. The state at P, which is due to all these impacts, will show the same irregularity. Nevertheless, we must try to deduce from the formulae relating to it, results concerning those quantities that can make themselves felt in actual experiments.
Results of this kind are obtained by considering the mean values of the variable quantities calculated for a sufficiently long lapse of time. We shall suppose this time to extend from t = 0 to t = &. If the mean value of d| is denoted by d|, we shall have for the flow of energy through 03' that is accessible to our means of observation
(135)
64. The introduction of this long time & is also very useful for the application of Fourier's theorem. Whatever be the way in which da changes from one instant to the next, we can always expand it in a series by the formula
84 H. EMISSION AND ABSORPTION OF HEAT.
where s is a positive whole number, each coefficient as being deter- mined by
o It appears from (136) that the frequency of one of the terms is
so that the corresponding wave-length is given by
The interval & being very large, the values of A belonging to small values of s will be so too; we shall not, however, have to speak of these very long waves, because they may be expected to represent no appreciable part of the total radiation. The rays with which we are concerned, will have wave-lengths below a certain upper limit i0; therefore, provided the time & (which we are free to choose as long as we like) be long enough, they will correspond to very high values of the number s. Now, if JLt and A4 + 1 are two successive wave- lengths, we shall have
_ _ __ __
I, s + i s + 1 '
which is a very small number. The wave-lengths corresponding to the successive terms in our series are thus seen to diminish by ex- ceedingly small steps. This means that, if we were to decompose the radiation represented by (136) into a spectrum, we should find a very large number of lines lying closely together. Their mutual distances may be indefinitely diminished by increasing the length of the time & and the values of s corresponding to the part of the spectrum we wish to consider. This is the way in which we can deduce from our formulae the existence of a continuous spectrum and the laws relating to it.
Let i and A + dk be two wave-lengths, which, from a physical point of view, may be said to lie infinitely near each other. If # is duly lengthened, the part of the spectrum corresponding to dk contains a large number of spectral lines, for which we find
EMISSION BY A THIN PLATE. H5
This is clear, if, after having written (138) in the form
_
i '
we observe that the number of lines is the same as the number of integers lying between the limits
i + m and i '
for which we may take the difference
because, in virtue of our supposition, this difference is much larger than 1.
We have now to substitute the value (136) in the equation (135). It is easily seen that the product of two terms of the series for dx will give 0, if integrated with respect to time between the limits 0 and &. Moreover
and (135) becomes
«=O9
(139)
This is the total flow of energy through to'. In order to find the part of it, corresponding to wave-lengths between I and /I + d)., we
have only to observe that the -rt -di spectral lines lying within that
interval, may be considered to have equal intensities.1)
In other terms, the value of at may be regarded as equal for each of them, so that they contribute to the sum in (139) an amount
Consequently, the part of the flow of energy, belonging to the inter- val of wave-lengths di, is given by
and our problem will be solved, if we succeed in calculating a*.
65. The following mathematical developments are somewhat more rigorous than those which I gave in my paper on the subject.
1) Note 34.
86 U. EMISSION AND ABSORPTION OF HEAT.
In fact, I shall now introduce Maxwell's law for the distribution of the velocities among the electrons, and take into account that the free paths are not all of the same length. At the same time I shall introduce a simplification for which I am indebted to Langevin1), and by which it will be possible to -give in a small space the essential part of the calculation.
By (134) and (137) we see that
a * — —
o
where the square brackets serve to indicate the value of vx at the
time t • --- c
The meaning of this equation is, that we must first, for one definite electron, calculate the integral, taking into account all the values of the acceleration occurring during the interval of time
between - - and & — — • This having been done, we have to take
the sum of the values that are found in this way for all the free electrons contained in the part ad of the plate.
Integrating by parts we find, because sin - - vanishes at the
limits,
2 1 /
T
for which, understanding by v^ the value at the time t, we may also write
By this artifice of partial integration, the problem is reduced to a much simpler one. If we had directly to calculate the integral in (141), we should have to attend to the intervals of time during which an electron is subjected to the force which makes it rebound from an atom against which it strikes; indeed, it is only during these intervals that there is an acceleration. On the other hand, the inte- gral in (142) is made up of parts, due, not only to the times of
1) See his translation of my paper in H. Abraham et P. Langevin, Les quantites elementaires d'e'lectricite, ions, electrons, corpuscules, Paris (1905), 1, p. 507.
EMISSION BY A THIN PLATE. 87
impact, but also to all intervening intervals. If we suppose the du- ration of an encounter to be very much smaller than the lapse of time between two successive collisions of an electron, we may even confine ourselves to the part that corresponds to the free paths be- tween these collisions.
While an electron travels over one of these free paths, its velo- city Vx is constant. We may also neglect the change in the factor
cos -.*- (t + — J , because the time between two encounters is supposed
to be very much smaller than the time of vibration corresponding to s. The part of a, which corresponds to one electron, and to the time during which it describes one of its free paths, is therefore given by
if we understand by r the time during which the path is travelled over. In the last factor we may take for t the value corresponding to the middle of the time r.
We shall now fix our attention on all the paths described by all the electrons during the time &. If we use the symbol S for denoting a sum relating to all these free paths, we shall have
66. We have to determine the square of the sum S. This may be done rather easily, because the products of two terms
TV,. COS -t-(t + -
whether they correspond to two different free paths of one and the same electron, or to two paths described by diS'erent electrons will give 0, if all taken together. Indeed, the velocities of two electrons are wholly independent of each other, and the same may be said of the velocities of one definite electron at two instants between which it has undergone one or more impacts.') Therefore, positive and negative values of V., being distributed quite indiscriminately between the terms of (143), positive and negative signs will be equally probable for the products of two terms.
It is seen in this way that we have only to calculate the sum of the squares of the several terms, so that we find
1) Note 35.
88 II. EMISSION AND ABSORPTION OF HEAT.
Now, since the irregular motion of the electrons takes place with the same intensity in all directions, we may replace v| by -J-V2. Therefore, writing I for the length r \ V \ of the free path, we find
a.a =
In the immense number of terms included in the sum, the length I is very different, and in order to effect the summation we may begin by considering only those terms for which it has a certain particular value. In these terms, which are still very numerous, the
angle - ^ (t + } has values that are distributed at random over an & \ c j
interval ranging from 0 to sn. The square of the cosine may there- fore be replaced by its mean value \, so that
67. The metallic atoms being considered as immovable, the velocity of an electron is not altered by a collision. We can there- fore fix our attention on a certain group of electrons which move along their zigzag-lines with a definite velocity u. During the time &, one of these particles describes a large number of free paths, this
number being given by
u&
IT1
if lm is the mean length of the paths. It can be shown1) that the number of paths whose length lies between I and I + dl, is
so that
i ±*L»«-fc<H
&
is the part of the sum S(Z2) contributed by these paths. Integrating with respect to I from 0 to oo, we find
2&ulm (145)
for the value of S(£2) in so far as it is due to one electron.
The total number of electrons in the part of the metallic plate under consideration is N(o4 and, by Maxwell's law, among these
(146)
1) Note 36.
KATIO BETWEEN EMISSIV1TY AND AHSOUPTION. 89
have velocities between u and u -\- du, the constant q being related to the velocity um whose square is equal to the mean value of u3, by the formula
3
In order to find the total value of S(P) we must multiply (145) by (146), and integrate the product between the limits u = 0 and u = oo. Supposing lm to be the same for all values of u1), we find
Finally, the equation (144) becomes
and the expression (140) for the radiation through the element w' takes the form
or, in virtue of (138), if, instead of lm, um, we simply write I, u,
2 e*lNu
3 r'i4
OHO
This is the energy radiated per unit of time, in so far as it belongs to wave-lengths between i and i. + di, and to the components of the vibrations in one direction h. Thus, the quantity we have cal- culated is exactly what was represented by (121), and on comparing the two expressions we find
for the euiissivity of the plate.
68. We have now to combine this with the value (133), which we have found for the coefficient of absorption. If Kirchhoff's
13