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THE THEORY OF HEAT RADIATION
PLANCK and MASIUS

THE THEORY
OF
HEAT RADIATION
BY
DR. MAX PLANCK
PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN
AUTHORISED TRANSLATION BY
MORTON MASIUS, M. A., Ph. D. (Leipzig)
INSTRUCTOR IN PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE
WITH 7 ILLUSTRATIONS
PHILADELPHIA
P. BLAKISTON’S SON & CO.
1012 WALNUT STREET

Copyright, 1914, by P. Blakiston’s Son & Co.
THE · MAPLE · PRESS · YORK · PA

TRANSLATOR’S PREFACE
The present volume is a translation of the second edition of Professor Planck’s Waermestrahlung (1913). The profoundly original ideas introduced by Planck in the endeavor to reconcile the electromagnetic theory of radiation with experimental facts have proven to be of the greatest importance in many parts of physics. Probably no single book since the appearance of Clerk Maxwell’s Electricity and Magnetism has had a deeper inﬂuence on the development of physical theories. The great majority of English-speaking physicists are, of course, able to read the work in the language in which it was written, but I believe that many will welcome the opportunity oﬀered by a translation to study the ideas set forth by Planck without the diﬃculties that frequently arise in attempting to follow a new and somewhat diﬃcult line of reasoning in a foreign language.
Recent developments of physical theories have placed the quantum of action in the foreground of interest. Questions regarding the bearing of the quantum theory on the law of equipartition of energy, its application to the theory of speciﬁc heats and to photoelectric eﬀects, attempts to form some concrete idea of the physical signiﬁcance of the quantum, that is, to devise a “model” for it, have created within the last few years a large and ever increasing literature. Professor Planck has, however, in this book conﬁned himself exclusively to radiation phenomena and it has seemed to me probable that a brief r´esum´e of this literature might prove useful to the reader who wishes to pursue the subject further. I have, therefore, with Professor Planck’s permission, given in an appendix a list of the most important papers on the subjects treated of in this book and others closely related to them. I have also added a short note on one or two derivations of formulæ where the treatment in the book seemed too brief or to present some diﬃculties.

TRANSLATOR’S PREFACE

v

In preparing the translation I have been under obligation for advice and helpful suggestions to several friends and colleagues and especially to Professor A. W. Duﬀ who has read the manuscript and the galley proof.

Worcester, Mass., February, 1914.

Morton Masius.

PREFACE TO SECOND EDITION
Recent advances in physical research have, on the whole, been favorable to the special theory outlined in this book, in particular to the hypothesis of an elementary quantity of action. My radiation formula especially has so far stood all tests satisfactorily, including even the reﬁned systematic measurements which have been carried out in the Physikalisch-technische Reichsanstalt at Charlottenburg during the last year. Probably the most direct support for the fundamental idea of the hypothesis of quanta is supplied by the values of the elementary quanta of matter and electricity derived from it. When, twelve years ago, I made my ﬁrst calculation of the value of the elementary electric charge and found it to be 4.69 · 10−10 electrostatic units, the value of this quantity deduced by J. J. Thomson from his ingenious experiments on the condensation of water vapor on gas ions, namely 6.5 · 10−10 was quite generally regarded as the most reliable value. This value exceeds the one given by me by 38 per cent. Meanwhile the experimental methods, improved in an admirable way by the labors of E. Rutherford, E. Regener, J. Perrin, R. A. Millikan, The Svedberg and others, have without exception decided in favor of the value deduced from the theory of radiation which lies between the values of Perrin and Millikan.
To the two mutually independent conﬁrmations mentioned, there has been added, as a further strong support of the hypothesis of quanta, the heat theorem which has been in the meantime announced by W. Nernst, and which seems to point unmistakably to the fact that, not only the processes of radiation, but also the molecular processes take place in accordance with certain elementary quanta of a deﬁnite ﬁnite magnitude. For the hypothesis of quanta as well as the heat theorem of Nernst may be reduced to the simple proposition that the

PREFACE TO SECOND EDITION

vii

thermodynamic probability (Sec. 120) of a physical state is a deﬁnite integral number, or, what amounts to the same thing, that the entropy of a state has a quite deﬁnite, positive value, which, as a minimum, becomes zero, while in contrast therewith the entropy may, according to the classical thermodynamics, decrease without limit to minus inﬁnity. For the present, I would consider this proposition as the very quintessence of the hypothesis of quanta.
In spite of the satisfactory agreement of the results mentioned with one another as well as with experiment, the ideas from which they originated have met with wide interest but, so far as I am able to judge, with little general acceptance, the reason probably being that the hypothesis of quanta has not as yet been satisfactorily completed. While many physicists, through conservatism, reject the ideas developed by me, or, at any rate, maintain an expectant attitude, a few authors have attacked them for the opposite reason, namely, as being inadequate, and have felt compelled to supplement them by assumptions of a still more radical nature, for example, by the assumption that any radiant energy whatever, even though it travel freely in a vacuum, consists of indivisible quanta or cells. Since nothing probably is a greater drawback to the successful development of a new hypothesis than overstepping its boundaries, I have always stood for making as close a connection between the hypothesis of quanta and the classical dynamics as possible, and for not stepping outside of the boundaries of the latter until the experimental facts leave no other course open. I have attempted to keep to this standpoint in the revision of this treatise necessary for a new edition.
The main fault of the original treatment was that it began with the classical electrodynamical laws of emission and absorption, whereas later on it became evident that, in order to meet the demand of experimental measurements, the assumption of ﬁnite energy elements must

PREFACE TO SECOND EDITION

viii

be introduced, an assumption which is in direct contradiction to the fundamental ideas of classical electrodynamics. It is true that this inconsistency is greatly reduced by the fact that, in reality, only mean values of energy are taken from classical electrodynamics, while, for the statistical calculation, the real values are used; nevertheless the treatment must, on the whole, have left the reader with the unsatisfactory feeling that it was not clearly to be seen, which of the assumptions made in the beginning could, and which could not, be ﬁnally retained.
In contrast thereto I have now attempted to treat the subject from the very outset in such a way that none of the laws stated need, later on, be restricted or modiﬁed. This presents the advantage that the theory, so far as it is treated here, shows no contradiction in itself, though certainly I do not mean that it does not seem to call for improvements in many respects, as regards both its internal structure and its external form. To treat of the numerous applications, many of them very important, which the hypothesis of quanta has already found in other parts of physics, I have not regarded as part of my task, still less to discuss all diﬀering opinions.
Thus, while the new edition of this book may not claim to bring the theory of heat radiation to a conclusion that is satisfactory in all respects, this deﬁciency will not be of decisive importance in judging the theory. For any one who would make his attitude concerning the hypothesis of quanta depend on whether the signiﬁcance of the quantum of action for the elementary physical processes is made clear in every respect or may be demonstrated by some simple dynamical model, misunderstands, I believe, the character and the meaning of the hypothesis of quanta. It is impossible to express a really new principle in terms of a model following old laws. And, as regards the ﬁnal formulation of the hypothesis, we should not forget that, from the classical point of view, the physics of the atom really has always remained a very obscure, inac-

PREFACE TO SECOND EDITION

ix

cessible region, into which the introduction of the elementary quantum of action promises to throw some light.
Hence it follows from the nature of the case that it will require painstaking experimental and theoretical work for many years to come to make gradual advances in the new ﬁeld. Any one who, at present, devotes his eﬀorts to the hypothesis of quanta, must, for the time being, be content with the knowledge that the fruits of the labor spent will probably be gathered by a future generation.

Berlin, November, 1912.

The Author.

PREFACE TO FIRST EDITION
In this book the main contents of the lectures which I gave at the University of Berlin during the winter semester 1906–07 are presented. My original intention was merely to put together in a connected account the results of my own investigations, begun ten years ago, on the theory of heat radiation; it soon became evident, however, that it was desirable to include also the foundation of this theory in the treatment, starting with Kirchhoﬀ’s Law on emitting and absorbing power; and so I attempted to write a treatise which should also be capable of serving as an introduction to the study of the entire theory of radiant heat on a consistent thermodynamic basis. Accordingly the treatment starts from the simple known experimental laws of optics and advances, by gradual extension and by the addition of the results of electrodynamics and thermodynamics, to the problems of the spectral distribution of energy and of irreversibility. In doing this I have deviated frequently from the customary methods of treatment, wherever the matter presented or considerations regarding the form of presentation seemed to call for it, especially in deriving Kirchhoﬀ’s laws, in calculating Maxwell’s radiation pressure, in deriving Wien’s displacement law, and in generalizing it for radiations of any spectral distribution of energy whatever.
I have at the proper place introduced the results of my own investigations into the treatment. A list of these has been added at the end of the book to facilitate comparison and examination as regards special details.
I wish, however, to emphasize here what has been stated more fully in the last paragraph of this book, namely, that the theory thus developed does not by any means claim to be perfect or complete, although I believe that it points out a possible way of accounting for the processes of radiant energy from the same point of view as for the processes of molecular motion.

TABLE OF CONTENTS

PART I
FUNDAMENTAL FACTS AND DEFINITIONS

Chapter

Page

I. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II. Radiation at Thermodynamic Equilibrium. Kirchhoﬀ’s Law.

Black Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

PART II
DEDUCTIONS FROM ELECTRODYNAMICS AND THERMODYNAMICS
I. Maxwell’s Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 II. Stefan-Boltzmann Law of Radiation . . . . . . . . . . . . . . . . . . . . . . . . 69 III. Wien’s Displacement Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 IV. Radiation of Any Arbitrary Spectral Distribution of Energy.
Entropy and Temperature of Monochromatic Radiation . . . . 104 V. Electrodynamical Processes in a Stationary Field of Radiation 124

PART III
ENTROPY AND PROBABILITY
I. Fundamental Deﬁnitions and Laws. Hypothesis of Quanta . . 133 II. Ideal Monatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

TABLE OF CONTENTS

xii

Chapter

Page

III. Ideal Linear Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

IV. Direct Calculation of the Entropy in The Case of Thermody-

namic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

PART IV
A SYSTEM OF OSCILLATORS IN A STATIONARY FIELD OF RADIATION
I. The Elementary Dynamical Law for The Vibrations of an Ideal Oscillator. Hypothesis of Emission of Quanta . . . . . . . . . . . . . . 177
II. Absorbed Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 III. Emitted Energy. Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . . 189 IV. The Law of the Normal Distribution Of Energy. Elementary
Quanta Of Matter and Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . 197

PART V
IRREVERSIBLE RADIATION PROCESSES
I. Fields of Radiation in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 II. One Oscillator in the Field of Radiation . . . . . . . . . . . . . . . . . . . . 230 III. A System of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 IV. Conservation of Energy and Increase Of Entropy. Conclusion 241
List of Papers on Heat Radiation and the Hypothesis of Quanta by the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

PART I
FUNDAMENTAL FACTS AND DEFINITIONS

RADIATION OF HEAT
CHAPTER I
GENERAL INTRODUCTION
1. Heat may be propagated in a stationary medium in two entirely diﬀerent ways, namely, by conduction and by radiation. Conduction of heat depends on the temperature of the medium in which it takes place, or more strictly speaking, on the non-uniform distribution of the temperature in space, as measured by the temperature gradient. In a region where the temperature of the medium is the same at all points there is no trace of heat conduction.
Radiation of heat, however, is in itself entirely independent of the temperature of the medium through which it passes. It is possible, for example, to concentrate the solar rays at a focus by passing them through a converging lens of ice, the latter remaining at a constant temperature of 0◦, and so to ignite an inﬂammable body. Generally speaking, radiation is a far more complicated phenomenon than conduction of heat. The reason for this is that the state of the radiation at a given instant and at a given point of the medium cannot be represented, as can the ﬂow of heat by conduction, by a single vector (that is, a single directed quantity). All heat rays which at a given instant pass through the same point of the medium are perfectly independent of one another, and in order to specify completely the state of the radiation the intensity of radiation must be known in all the directions, inﬁnite in number, which pass through the point in question; for this
2

GENERAL INTRODUCTION

3

purpose two opposite directions must be considered as distinct, because the radiation in one of them is quite independent of the radiation in the other.
2. Putting aside for the present any special theory of heat radiation, we shall state for our further use a law supported by a large number of experimental facts. This law is that, so far as their physical properties are concerned, heat rays are identical with light rays of the same wave length. The term “heat radiation,” then, will be applied to all physical phenomena of the same nature as light rays. Every light ray is simultaneously a heat ray. We shall also, for the sake of brevity, occasionally speak of the “color” of a heat ray in order to denote its wave length or period. As a further consequence of this law we shall apply to the radiation of heat all the well-known laws of experimental optics, especially those of reﬂection and refraction, as well as those relating to the propagation of light. Only the phenomena of diﬀraction, so far at least as they take place in space of considerable dimensions, we shall exclude on account of their rather complicated nature. We are therefore obliged to introduce right at the start a certain restriction with respect to the size of the parts of space to be considered. Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wave lengths of the rays considered. With this assumption we may, without appreciable error, entirely neglect the inﬂuence of diﬀraction caused by the bounding surfaces, and everywhere apply the ordinary laws of reﬂection and refraction of light. To sum up: We distinguish once for all between two kinds of lengths of entirely diﬀerent orders of magnitude—dimensions of bodies and wave lengths. Moreover, even the diﬀerentials of the former, i.e., elements of length, area and volume, will be regarded as large compared with the corresponding powers of

FUNDAMENTAL FACTS AND DEFINITIONS

4

wave lengths. The greater, therefore, the wave length of the rays we wish to consider, the larger must be the parts of space considered. But, inasmuch as there is no other restriction on our choice of size of the parts of space to be considered, this assumption will not give rise to any particular diﬃculty.
3. Even more essential for the whole theory of heat radiation than the distinction between large and small lengths, is the distinction between long and short intervals of time. For the deﬁnition of intensity of a heat ray, as being the energy transmitted by the ray per unit time, implies the assumption that the unit of time chosen is large compared with the period of vibration corresponding to the color of the ray. If this were not so, obviously the value of the intensity of the radiation would, in general, depend upon the particular phase of vibration at which the measurement of the energy of the ray was begun, and the intensity of a ray of constant period and amplitude would not be independent of the initial phase, unless by chance the unit of time were an integral multiple of the period. To avoid this diﬃculty, we are obliged to postulate quite generally that the unit of time, or rather that element of time used in deﬁning the intensity, even if it appear in the form of a diﬀerential, must be large compared with the period of all colors contained in the ray in question.
The last statement leads to an important conclusion as to radiation of variable intensity. If, using an acoustic analogy, we speak of “beats” in the case of intensities undergoing periodic changes, the “unit” of time required for a deﬁnition of the instantaneous intensity of radiation must necessarily be small compared with the period of the beats. Now, since from the previous statement our unit must be large compared with a period of vibration, it follows that the period of the beats must be large compared with that of a vibration. Without this restriction it would be impossible to distinguish properly between “beats” and simple “vibra-

GENERAL INTRODUCTION

5

tions.” Similarly, in the general case of an arbitrarily variable intensity of radiation, the vibrations must take place very rapidly as compared with the relatively slower changes in intensity. These statements imply, of course, a certain far-reaching restriction as to the generality of the radiation phenomena to be considered.
It might be added that a very similar and equally essential restriction is made in the kinetic theory of gases by dividing the motions of a chemically simple gas into two classes: visible, coarse, or molar, and invisible, ﬁne, or molecular. For, since the velocity of a single molecule is a perfectly unambiguous quantity, this distinction cannot be drawn unless the assumption be made that the velocity-components of the molecules contained in suﬃciently small volumes have certain mean values, independent of the size of the volumes. This in general need not by any means be the case. If such a mean value, including the value zero, does not exist, the distinction between motion of the gas as a whole and random undirected heat motion cannot be made.
Turning now to the investigation of the laws in accordance with which the phenomena of radiation take place in a medium supposed to be at rest, the problem may be approached in two ways: We must either select a certain point in space and investigate the diﬀerent rays passing through this one point as time goes on, or we must select one distinct ray and inquire into its history, that is, into the way in which it was created, propagated, and ﬁnally destroyed. For the following discussion, it will be advisable to start with the second method of treatment and to consider ﬁrst the three processes just mentioned.
4. Emission.—The creation of a heat ray is generally denoted by the word emission. According to the principle of the conservation of energy, emission always takes place at the expense of other forms of

FUNDAMENTAL FACTS AND DEFINITIONS

6

energy (heat,1 chemical or electric energy, etc.) and hence it follows that only material particles, not geometrical volumes or surfaces, can emit heat rays. It is true that for the sake of brevity we frequently speak of the surface of a body as radiating heat to the surroundings, but this form of expression does not imply that the surface actually emits heat rays. Strictly speaking, the surface of a body never emits rays, but rather it allows part of the rays coming from the interior to pass through. The other part is reﬂected inward and according as the fraction transmitted is larger or smaller the surface seems to emit more or less intense radiations.
We shall now consider the interior of an emitting substance assumed to be physically homogeneous, and in it we shall select any volumeelement dτ of not too small size. Then the energy which is emitted by radiation in unit time by all particles in this volume-element will be proportional to dτ . Should we attempt a closer analysis of the process of emission and resolve it into its elements, we should undoubtedly meet very complicated conditions, for then it would be necessary to consider elements of space of such small size that it would no longer be admissible to think of the substance as homogeneous, and we would have to allow for the atomic constitution. Hence the ﬁnite quantity obtained by dividing the radiation emitted by a volume-element dτ by this element dτ is to be considered only as a certain mean value. Nevertheless, we shall as a rule be able to treat the phenomenon of emission as if all points of the volume-element dτ took part in the emission in a uniform manner, thereby greatly simplifying our calculation. Every point of dτ will then be the vertex of a pencil of rays diverging in all directions. Such a pencil coming from one single point of course does not represent
1Here as in the following the German “K¨orperw¨arme” will be rendered simply as “heat.” (Tr.)

GENERAL INTRODUCTION

7

a ﬁnite amount of energy, because a ﬁnite amount is emitted only by a ﬁnite though possibly small volume, not by a single point.
We shall next assume our substance to be isotropic. Hence the radiation of the volume-element dτ is emitted uniformly in all directions of space. Draw a cone in an arbitrary direction, having any point of the radiating element as vertex, and describe around the vertex as center a sphere of unit radius. This sphere intersects the cone in what is known as the solid angle of the cone, and from the isotropy of the medium it follows that the radiation in any such conical element will be proportional to its solid angle. This holds for cones of any size. If we take the solid angle as inﬁnitely small and of size dΩ we may speak of the radiation emitted in a certain direction, but always in the sense that for the emission of a ﬁnite amount of energy an inﬁnite number of directions are necessary and these form a ﬁnite solid angle.
5. The distribution of energy in the radiation is in general quite arbitrary; that is, the diﬀerent colors of a certain radiation may have quite diﬀerent intensities. The color of a ray in experimental physics is usually denoted by its wave length, because this quantity is measured directly. For the theoretical treatment, however, it is usually preferable to use the frequency ν instead, since the characteristic of color is not so much the wave length, which changes from one medium to another, as the frequency, which remains unchanged in a light or heat ray passing through stationary media. We shall, therefore, hereafter denote a certain color by the corresponding value of ν, and a certain interval of color by the limits of the interval ν and ν , where ν > ν. The radiation lying in a certain interval of color divided by the magnitude ν − ν of the interval, we shall call the mean radiation in the interval ν to ν . We shall then assume that if, keeping ν constant, we take the interval ν −ν suﬃciently small and denote it by dν the value of the mean radiation approaches a deﬁnite limiting value, independent of the size of dν, and

FUNDAMENTAL FACTS AND DEFINITIONS

8

this we shall brieﬂy call the “radiation of frequency ν.” To produce a ﬁnite intensity of radiation, the frequency interval, though perhaps small, must also be ﬁnite.
We have ﬁnally to allow for the polarization of the emitted radiation. Since the medium was assumed to be isotropic the emitted rays are unpolarized. Hence every ray has just twice the intensity of one of its plane polarized components, which could, e.g., be obtained by passing the ray through a Nicol’s prism.
6. Summing up everything said so far, we may equate the total energy in a range of frequency from ν to ν + dν emitted in the time dt in the direction of the conical element dΩ by a volume-element dτ to

dt · dτ · dΩ · dν · 2 ν.

(1)

The ﬁnite quantity ν is called the coeﬃcient of emission of the medium for the frequency ν. It is a positive function of ν and refers to a plane polarized ray of deﬁnite color and direction. The total emission of the volume-element dτ may be obtained from this by integrating over all directions and all frequencies. Since ν is independent of the direction, and since the integral over all conical elements dΩ is 4π, we get:

∞

dt · dτ · 8π

ν dν.

(2)

0

7. The coeﬃcient of emission depends, not only on the frequency ν, but also on the condition of the emitting substance contained in the volume-element dτ , and, generally speaking, in a very complicated way, according to the physical and chemical processes which take place in the elements of time and volume in question. But the empirical law that the emission of any volume-element depends entirely on what takes place inside of this element holds true in all cases (Prevost’s principle).

GENERAL INTRODUCTION

9

A body A at 100◦ C. emits toward a body B at 0◦ C. exactly the same amount of radiation as toward an equally large and similarly situated body B at 1000◦ C. The fact that the body A is cooled by B and heated by B is due entirely to the fact that B is a weaker, B a stronger emitter than A.
We shall now introduce the further simplifying assumption that the physical and chemical condition of the emitting substance depends on but a single variable, namely, on its absolute temperature T . A necessary consequence of this is that the coeﬃcient of emission depends, apart from the frequency ν and the nature of the medium, only on the temperature T . The last statement excludes from our consideration a number of radiation phenomena, such as ﬂuorescence, phosphorescence, electrical and chemical luminosity, to which E. Wiedemann has given the common name “phenomena of luminescence.” We shall deal with pure “temperature radiation” exclusively.
A special case of temperature radiation is the case of the chemical nature of the emitting substance being invariable. In this case the emission takes place entirely at the expense of the heat of the body. Nevertheless, it is possible, according to what has been said, to have temperature radiation while chemical changes are taking place, provided the chemical condition is completely determined by the temperature.
8. Propagation.—The propagation of the radiation in a medium assumed to be homogeneous, isotropic, and at rest takes place in straight lines and with the same velocity in all directions, diﬀraction phenomena being entirely excluded. Yet, in general, each ray suﬀers during its propagation a certain weakening, because a certain fraction of its energy is continuously deviated from its original direction and scattered in all directions. This phenomenon of “scattering,” which means neither a creation nor a destruction of radiant energy but sim-

FUNDAMENTAL FACTS AND DEFINITIONS

10

ply a change in distribution, takes place, generally speaking, in all media diﬀering from an absolute vacuum, even in substances which are perfectly pure chemically.1 The cause of this is that no substance is homogeneous in the absolute sense of the word. The smallest elements of space always exhibit some discontinuities on account of their atomic structure. Small impurities, as, for instance, particles of dust, increase the inﬂuence of scattering without, however, appreciably aﬀecting its general character. Hence, so-called “turbid” media, i.e., such as contain foreign particles, may be quite properly regarded as optically homogeneous,2 provided only that the linear dimensions of the foreign particles as well as the distances of neighboring particles are suﬃciently small compared with the wave lengths of the rays considered. As regards optical phenomena, then, there is no fundamental distinction between chemically pure substances and the turbid media just described. No space is optically void in the absolute sense except a vacuum. Hence a chemically pure substance may be spoken of as a vacuum made turbid by the presence of molecules.
A typical example of scattering is oﬀered by the behavior of sunlight in the atmosphere. When, with a clear sky, the sun stands in the zenith, only about two-thirds of the direct radiation of the sun reaches the surface of the earth. The remainder is intercepted by the atmosphere, being partly absorbed and changed into heat of the air, partly, however, scattered and changed into diﬀuse skylight. This phenomenon is produced probably not so much by the particles suspended in the atmosphere as by the air molecules themselves.
Whether the scattering depends on reﬂection, on diﬀraction, or on
1See, e.g., Lobry de Bruyn and L. K. Wolﬀ, Rec. des Trav. Chim. des PaysBas 23, p. 155, 1904.
2To restrict the word homogeneous to its absolute sense would mean that it could not be applied to any material substance.

GENERAL INTRODUCTION

11

a resonance eﬀect on the molecules or particles is a point that we may leave entirely aside. We only take account of the fact that every ray on its path through any medium loses a certain fraction of its intensity. For a very small distance, s, this fraction is proportional to s, say

βν s

(3)

where the positive quantity βν is independent of the intensity of radiation and is called the “coeﬃcient of scattering” of the medium. Inasmuch as the medium is assumed to be isotropic, βν is also independent of the direction of propagation and polarization of the ray. It depends, however, as indicated by the subscript ν, not only on the physical and chemical constitution of the body but also to a very marked degree on the frequency. For certain values of ν, βν may be so large that the straight-line propagation of the rays is virtually destroyed. For other values of ν, however, βν may become so small that the scattering can be entirely neglected. For generality we shall assume a mean value of βν. In the cases of most importance βν increases quite appreciably as ν increases, i.e., the scattering is noticeably larger for rays of shorter wave length;1 hence the blue color of diﬀuse skylight.
The scattered radiation energy is propagated from the place where the scattering occurs in a way similar to that in which the emitted energy is propagated from the place of emission, since it travels in all directions in space. It does not, however, have the same intensity in all directions, and moreover is polarized in some special directions, depending to a large extent on the direction of the original ray. We need not, however, enter into any further discussion of these questions.
9. While the phenomenon of scattering means a continuous modiﬁcation in the interior of the medium, a discontinuous change in both the
1Lord Rayleigh, Phil. Mag., 47, p. 379, 1899.

FUNDAMENTAL FACTS AND DEFINITIONS

12

direction and the intensity of a ray occurs when it reaches the boundary of a medium and meets the surface of a second medium. The latter, like the former, will be assumed to be homogeneous and isotropic. In this case, the ray is in general partly reﬂected and partly transmitted. The reﬂection and refraction may be “regular,” there being a single reﬂected ray according to the simple law of reﬂection and a single transmitted ray, according to Snell’s law of refraction, or, they may be “diﬀuse,” which means that from the point of incidence on the surface the radiation spreads out into the two media with intensities that are diﬀerent in diﬀerent directions. We accordingly describe the surface of the second medium as “smooth” or “rough” respectively. Diﬀuse reﬂection occurring at a rough surface should be carefully distinguished from reﬂection at a smooth surface of a turbid medium. In both cases part of the incident ray goes back to the ﬁrst medium as diﬀuse radiation. But in the ﬁrst case the scattering occurs on the surface, in the second in more or less thick layers entirely inside of the second medium.
10. When a smooth surface completely reﬂects all incident rays, as is approximately the case with many metallic surfaces, it is termed “reﬂecting.” When a rough surface reﬂects all incident rays completely and uniformly in all directions, it is called “white.” The other extreme, namely, complete transmission of all incident rays through the surface never occurs with smooth surfaces, at least if the two contiguous media are at all optically diﬀerent. A rough surface having the property of completely transmitting the incident radiation is described as “black.”
In addition to “black surfaces” the term “black body” is also used. According to G. Kirchhoﬀ 1 it denotes a body which has the property
1G. Kirchhoﬀ, Pogg. Ann., 109, p. 275, 1860. Gesammelte Abhandlungen, J. A. Barth, Leipzig, 1882, p. 573. In deﬁning a black body Kirchhoﬀ also assumes that the absorption of incident rays takes place in a layer “inﬁnitely thin.” We do not include this in our deﬁnition.

GENERAL INTRODUCTION

13

of allowing all incident rays to enter without surface reﬂection and not allowing them to leave again. Hence it is seen that a black body must satisfy three independent conditions. First, the body must have a black surface in order to allow the incident rays to enter without reﬂection. Since, in general, the properties of a surface depend on both of the bodies which are in contact, this condition shows that the property of blackness as applied to a body depends not only on the nature of the body but also on that of the contiguous medium. A body which is black relatively to air need not be so relatively to glass, and vice versa. Second, the black body must have a certain minimum thickness depending on its absorbing power, in order to insure that the rays after passing into the body shall not be able to leave it again at a diﬀerent point of the surface. The more absorbing a body is, the smaller the value of this minimum thickness, while in the case of bodies with vanishingly small absorbing power only a layer of inﬁnite thickness may be regarded as black. Third, the black body must have a vanishingly small coeﬃcient of scattering (Sec. 8). Otherwise the rays received by it would be partly scattered in the interior and might leave again through the surface.1
11. All the distinctions and deﬁnitions mentioned in the two preceding paragraphs refer to rays of one deﬁnite color only. It might very well happen that, e.g., a surface which is rough for a certain kind of rays must be regarded as smooth for a diﬀerent kind of rays. It is readily seen that, in general, a surface shows decreasing degrees of roughness for increasing wave lengths. Now, since smooth non-reﬂecting surfaces do not exist (Sec. 10), it follows that all approximately black surfaces
1For this point see especially A. Schuster, Astrophysical Journal, 21, p. 1, 1905, who has pointed out that an inﬁnite layer of gas with a black surface need by no means be a black body.

FUNDAMENTAL FACTS AND DEFINITIONS

14

which may be realized in practice (lamp black, platinum black) show appreciable reﬂection for rays of suﬃciently long wave lengths.
12. Absorption.—Heat rays are destroyed by “absorption.” According to the principle of the conservation of energy the energy of heat radiation is thereby changed into other forms of energy (heat, chemical energy). Thus only material particles can absorb heat rays, not elements of surfaces, although sometimes for the sake of brevity the expression absorbing surfaces is used.
Whenever absorption takes place, the heat ray passing through the medium under consideration is weakened by a certain fraction of its intensity for every element of path traversed. For a suﬃciently small distance s this fraction is proportional to s, and may be written

αν s.

(4)

Here αν is known as the “coeﬃcient of absorption” of the medium for a ray of frequency ν. We assume this coeﬃcient to be independent of the intensity; it will, however, depend in general in non-homogeneous and anisotropic media on the position of s and on the direction of propagation and polarization of the ray (example: tourmaline). We shall, however, consider only homogeneous isotropic substances, and shall therefore suppose that αν has the same value at all points and in all directions in the medium, and depends on nothing but the frequency ν, the temperature T , and the nature of the medium.
Whenever αν does not diﬀer from zero except for a limited range of the spectrum, the medium shows “selective” absorption. For those colors for which αν = 0 and also the coeﬃcient of scattering βν = 0 the medium is described as perfectly “transparent” or “diathermanous.” But the properties of selective absorption and of diathermancy may for a given medium vary widely with the temperature. In general we shall assume a mean value for αν. This implies that the absorption

GENERAL INTRODUCTION

15

in a distance equal to a single wave length is very small, because the distance s, while small, contains many wave lengths (Sec. 2).
13. The foregoing considerations regarding the emission, the propagation, and the absorption of heat rays suﬃce for a mathematical treatment of the radiation phenomena. The calculation requires a knowledge of the value of the constants and the initial and boundary conditions, and yields a full account of the changes the radiation undergoes in a given time in one or more contiguous media of the kind stated, including the temperature changes caused by it. The actual calculation is usually very complicated. We shall, however, before entering upon the treatment of special cases discuss the general radiation phenomena from a diﬀerent point of view, namely by ﬁxing our attention not on a deﬁnite ray, but on a deﬁnite position in space.
14. Let dσ be an arbitrarily chosen, inﬁnitely small element of area in the interior of a medium through which radiation passes. At a given instant rays are passing through this element in many diﬀerent directions. The energy radiated through it in an element of time dt in a deﬁnite direction is proportional to the area dσ, the length of time dt, and to the cosine of the angle θ made by the normal of dσ with the direction of the radiation. If we make dσ suﬃciently small, then, although this is only an approximation to the actual state of aﬀairs, we can think of all points in dσ as being aﬀected by the radiation in the same way. Then the energy radiated through dσ in a deﬁnite direction must be proportional to the solid angle in which dσ intercepts that radiation and this solid angle is measured by dσ cos θ. It is readily seen that, when the direction of the element is varied relatively to the direction of the radiation, the energy radiated through it vanishes when
π θ= .
2 Now in general a pencil of rays is propagated from every point of

FUNDAMENTAL FACTS AND DEFINITIONS

16

the element dσ in all directions, but with diﬀerent intensities in diﬀerent directions, and any two pencils emanating from two points of the element are identical save for diﬀerences of higher order. A single one of these pencils coming from a single point does not represent a ﬁnite quantity of energy, because a ﬁnite amount of energy is radiated only through a ﬁnite area. This holds also for the passage of rays through a so-called focus. For example, when sunlight passes through a converging lens and is concentrated in the focal plane of the lens, the solar rays do not converge to a single point, but each pencil of parallel rays forms a separate focus and all these foci together constitute a surface representing a small but ﬁnite image of the sun. A ﬁnite amount of energy does not pass through less than a ﬁnite portion of this surface.
15. Let us now consider quite generally the pencil, which is propagated from a point of the element dσ as vertex in all directions of space and on both sides of dσ. A certain direction may be speciﬁed by the angle θ (between 0 and π), as already used, and by an azimuth φ (between 0 and 2π). The intensity in this direction is the energy propagated in an inﬁnitely thin cone limited by θ and θ + dθ and φ and φ + dφ. The solid angle of this cone is

dΩ = sin θ · dθ · dφ.

(5)

Thus the energy radiated in time dt through the element of area dσ in the direction of the cone dΩ is:

dt dσ cos θ dΩ K = K sin θ cos θ dθ dφ dσ dt.

(6)

The ﬁnite quantity K we shall term the “speciﬁc intensity” or the “brightness,” dΩ the “solid angle” of the pencil emanating from a point of the element dσ in the direction (θ, φ). K is a positive function of position, time, and the angles θ and φ. In general the speciﬁc intensities of radiation in diﬀerent directions are entirely independent of one

GENERAL INTRODUCTION

17

another. For example, on substituting π − θ for θ and π + φ for φ in the function K, we obtain the speciﬁc intensity of radiation in the diametrically opposite direction, a quantity which in general is quite diﬀerent from the preceding one.
For the total radiation through the element of area dσ toward one side, say the one on which θ is an acute angle, we get, by integrating
π with respect to φ from 0 to 2π and with respect to θ from 0 to
2

2π

π 2

dφ dθ K sin θ cos θ dσ dt.

0

0

Should the radiation be uniform in all directions and hence K be a constant, the total radiation on one side will be

πK dσ dt.

(7)

16. In speaking of the radiation in a deﬁnite direction (θ, φ) one should always keep in mind that the energy radiated in a cone is not ﬁnite unless the angle of the cone is ﬁnite. No ﬁnite radiation of light or heat takes place in one deﬁnite direction only, or expressing it diﬀerently, in nature there is no such thing as absolutely parallel light or an absolutely plane wave front. From a pencil of rays called “parallel” a ﬁnite amount of energy of radiation can only be obtained if the rays or wave normals of the pencil diverge so as to form a ﬁnite though perhaps exceedingly narrow cone.
17. The speciﬁc intensity K of the whole energy radiated in a certain direction may be further divided into the intensities of the separate rays belonging to the diﬀerent regions of the spectrum which travel independently of one another. Hence we consider the intensity of radiation within a certain range of frequencies, say from ν to ν . If the

FUNDAMENTAL FACTS AND DEFINITIONS

18

interval ν − ν be taken suﬃciently small and be denoted by dν, the intensity of radiation within the interval is proportional to dν. Such radiation is called homogeneous or monochromatic.
A last characteristic property of a ray of deﬁnite direction, intensity, and color is its state of polarization. If we break up a ray, which is in any state of polarization whatsoever and which travels in a deﬁnite direction and has a deﬁnite frequency ν, into two plane polarized components, the sum of the intensities of the components will be just equal to the intensity of the ray as a whole, independently of the direction of the two planes, provided the two planes of polarization, which otherwise may be taken at random, are at right angles to each other. If their position be denoted by the azimuth ψ of one of the planes of vibration (plane of the electric vector), then the two components of the intensity may be written in the form

Kν cos2 ψ + Kν sin2 ψ

and

Kν sin2 ψ + Kν cos2 ψ.

(8)

Herein K is independent of ψ. These expressions we shall call the “components of the speciﬁc intensity of radiation of frequency ν.” The sum is independent of ψ and is always equal to the intensity of the whole ray Kν + Kν. At the same time Kν and Kν represent respectively the largest and smallest values which either of the components may have, namely,
π when ψ = 0 and ψ = . Hence we call these values the “principal
2 values of the intensities,” or the “principal intensities,” and the corresponding planes of vibration we call the “principal planes of vibration” of the ray. Of course both, in general, vary with the time. Thus we may write generally

∞

K = dν (Kν + Kν)

(9)

0

GENERAL INTRODUCTION

19

where the positive quantities Kν and Kν, the two principal values of the speciﬁc intensity of the radiation (brightness) of frequency ν, depend not only on ν but also on their position, the time, and on the angles θ and φ. By substitution in (6) the energy radiated in the time dt through the element of area dσ in the direction of the conical element dΩ assumes the value

∞

dt dσ cos θ dΩ dν (Kν + Kν)

(10)

0

and for monochromatic plane polarized radiation of brightness Kν:

dt dσ cos θ dΩ Kν dν = dt dσ sin θ cos θ dθ dφ Kν dν.

(11)

For unpolarized rays Kν = Kν, and hence

∞

K = 2 dν Kν,

(12)

0

and the energy of a monochromatic ray of frequency ν will be:

2 dt dσ cos θ dΩ Kν dν = 2 dt dσ sin θ cos θ dθ dφ Kν dν. (13)

When, moreover, the radiation is uniformly distributed in all directions,

the total radiation through dσ toward one side may be found from

(7) and (12); it is

∞

2π dσ dt Kν dν.

(14)

0

18. Since in nature Kν can never be inﬁnitely large, K will not have a ﬁnite value unless Kν diﬀers from zero over a ﬁnite range of frequencies. Hence there exists in nature no absolutely homogeneous or monochro-

matic radiation of light or heat. A ﬁnite amount of radiation contains

FUNDAMENTAL FACTS AND DEFINITIONS

20

always a ﬁnite although possibly very narrow range of the spectrum.

This implies a fundamental diﬀerence from the corresponding phenom-

ena of acoustics, where a ﬁnite intensity of sound may correspond to

a single deﬁnite frequency. This diﬀerence is, among other things, the

cause of the fact that the second law of thermodynamics has an impor-

tant bearing on light and heat rays, but not on sound waves. This will

be further discussed later on.

19. From equation (9) it is seen that the quantity Kν, the intensity of radiation of frequency ν, and the quantity K, the intensity of ra-

diation of the whole spectrum, are of diﬀerent dimensions. Further it

is to be noticed that, on subdividing the spectrum according to wave

lengths λ, instead of frequencies ν, the intensity of radiation Eλ of the wave lengths λ corresponding to the frequency ν is not obtained simply

by replacing ν in the expression for Kν by the corresponding value of λ

deduced from

q

ν=

(15)

λ

where q is the velocity of propagation. For if dλ and dν refer to the same

interval of the spectrum, we have, not Eλ = Kν, but Eλ dλ = Kν dν. By diﬀerentiating (15) and paying attention to the signs of corresponding

values of dλ and dν the equation

q dλ dν =
λ2 is obtained. Hence we get by substitution:

Eλ

=

qKν λ2

.

(16)

This relation shows among other things that in a certain spectrum the maxima of Eλ and Kν lie at diﬀerent points of the spectrum.

GENERAL INTRODUCTION

21

20. When the principal intensities Kν and Kν of all monochromatic rays are given at all points of the medium and for all directions, the state of radiation is known in all respects and all questions regarding it may be answered. We shall show this by one or two applications to special cases. Let us ﬁrst ﬁnd the amount of energy which is radiated through any element of area dσ toward any other element dσ . The distance r between the two elements may be thought of as large compared with the linear dimensions of the elements dσ and dσ but still so small that no appreciable amount of radiation is absorbed or scattered along it. This condition is, of course, superﬂuous for diathermanous media.
From any deﬁnite point of dσ rays pass to all points of dσ . These rays form a cone whose vertex lies in dσ and whose solid angle is

dσ cos(n , r)

dΩ =

r2

where n denotes the normal of dσ and the angle (n , r) is to be taken as an acute angle. This value of dΩ is, neglecting small quantities of higher order, independent of the particular position of the vertex of the cone on dσ.
If we further denote the normal to dσ by n the angle θ of (14) will be the angle (n, r) and hence from expression (6) the energy of radiation required is found to be:

dσ dσ cos(n, r) cos(n , r)

K·

r2

· dt.

(17)

For monochromatic plane polarized radiation of frequency ν the energy will be, according to equation (11),

dσ dσ cos(n, r) cos(n , r)

Kν dν ·

r2

· dt.

(18)

FUNDAMENTAL FACTS AND DEFINITIONS

22

The relative size of the two elements dσ and dσ may have any value whatever. They may be assumed to be of the same or of a diﬀerent order of magnitude, provided the condition remains satisﬁed that r is large compared with the linear dimensions of each of them. If we choose dσ small compared with dσ , the rays diverge from dσ to dσ , whereas they converge from dσ to dσ if we choose dσ large compared with dσ .
21. Since every point of dσ is the vertex of a cone spreading out toward dσ , the whole pencil of rays here considered, which is deﬁned by dσ and dσ , consists of a double inﬁnity of point pencils or of a fourfold inﬁnity of rays which must all be considered equally for the energy radiation. Similarly the pencil of rays may be thought of as consisting of the cones which, emanating from all points of dσ, converge in one point of dσ respectively as a vertex. If we now imagine the whole pencil of rays to be cut by a plane at any arbitrary distance from the elements dσ and dσ and lying either between them or outside, then the cross-sections of any two point pencils on this plane will not be identical, not even approximately. In general they will partly overlap and partly lie outside of each other, the amount of overlapping being diﬀerent for diﬀerent intersecting planes. Hence it follows that there is no deﬁnite cross-section of the pencil of rays so far as the uniformity of radiation is concerned. If, however, the intersecting plane coincides with either dσ or dσ , then the pencil has a deﬁnite cross-section. Thus these two planes show an exceptional property. We shall call them the two “focal planes” of the pencil.
In the special case already mentioned above, namely, when one of the two focal planes is inﬁnitely small compared with the other, the whole pencil of rays shows the character of a point pencil inasmuch as its form is approximately that of a cone having its vertex in that focal plane which is small compared with the other. In that case the “cross-section” of the whole pencil at a deﬁnite point has a deﬁnite meaning. Such a

GENERAL INTRODUCTION

23

pencil of rays, which is similar to a cone, we shall call an elementary pencil, and the small focal plane we shall call the ﬁrst focal plane of the elementary pencil. The radiation may be either converging toward the ﬁrst focal plane or diverging from the ﬁrst focal plane. All the pencils of rays passing through a medium may be considered as consisting of such elementary pencils, and hence we may base our future considerations on elementary pencils only, which is a great convenience, owing to their simple nature.
As quantities necessary to deﬁne an elementary pencil with a given ﬁrst focal plane dσ, we may choose not the second focal plane dσ but the magnitude of that solid angle dΩ under which dσ is seen from dσ. On the other hand, in the case of an arbitrary pencil, that is, when the two focal planes are of the same order of magnitude, the second focal plane in general cannot be replaced by the solid angle dΩ without the pencil changing markedly in character. For if, instead of dσ being given, the magnitude and direction of dΩ, to be taken as constant for all points of dσ, is given, then the rays emanating from dσ do not any longer form the original pencil, but rather an elementary pencil whose ﬁrst focal plane is dσ and whose second focal plane lies at an inﬁnite distance.
22. Since the energy radiation is propagated in the medium with a ﬁnite velocity q, there must be in a ﬁnite space a ﬁnite amount of energy. We shall therefore speak of the “space density of radiation,” meaning thereby the ratio of the total quantity of energy of radiation contained in a volume-element to the magnitude of the latter. Let us now calculate the space density of radiation u at any arbitrary point of the medium. When we consider an inﬁnitely small element of volume v at the point in question, having any shape whatsoever, we must allow for all rays passing through the volume-element v. For this purpose we shall construct about any point O of v as center a sphere of radius r,

FUNDAMENTAL FACTS AND DEFINITIONS

24

Fig. 1.
r being large compared with the linear dimensions of v but still so small that no appreciable absorption or scattering of the radiation takes place in the distance r (Fig. 1). Every ray which reaches v must then come from some point on the surface of the sphere. If, then, we at ﬁrst consider only all the rays that come from the points of an inﬁnitely small element of area dσ on the surface of the sphere, and reach v, and then sum up for all elements of the spherical surface, we shall have accounted for all rays and not taken any one more than once.
Let us then calculate ﬁrst the amount of energy which is contributed to the energy contained in v by the radiation sent from such an element dσ to v. We choose dσ so that its linear dimensions are small compared with those of v and consider the cone of rays which, starting at a point of dσ, meets the volume v. This cone consists of an inﬁnite number of conical elements with the common vertex at P , a point

GENERAL INTRODUCTION

25

of dσ, each cutting out of the volume v a certain element of length, f
say s. The solid angle of such a conical element is where f denotes r2
the area of cross-section normal to the axis of the cone at a distance r

from the vertex. The time required for the radiation to pass through

the distance s is:

s τ= .
q

From expression (6) we may ﬁnd the energy radiated through a certain

f element of area. In the present case dΩ = r2 and θ = 0; hence the energy is:

f

fs

τ dσ K = · K dσ.

(19)

r2

r2q

This energy enters the conical element in v and spreads out into the volume f s. Summing up over all conical elements that start from dσ and enter v we have

K dσ r2q

K dσ

fs =

v.

r2q

This represents the entire energy of radiation contained in the volume v, so far as it is caused by radiation through the element dσ. In order to obtain the total energy of radiation contained in v we must integrate over all elements dσ contained in the surface of the sphere. Denoting
dσ by dΩ the solid angle of a cone which has its center in O and
r2 intersects in dσ the surface of the sphere, we get for the whole energy:

v K dΩ.
q

FUNDAMENTAL FACTS AND DEFINITIONS

26

The volume density of radiation required is found from this by dividing

by v. It is

1

u = K dΩ.

(20)

q

Since in this expression r has disappeared, we can think of K as the

intensity of radiation at the point O itself. In integrating, it is to be

noted that K in general depends on the direction (θ, φ). For radiation

that is uniform in all directions K is a constant and on integration we

get:

4πK

u= .

(21)

q

23. A meaning similar to that of the volume density of the total radiation u is attached to the volume density of radiation of a deﬁnite frequency uν. Summing up for all parts of the spectrum we get:

∞

u = uν dν.

(22)

0

Further by combining equations (9) and (20) we have:

1

uν = q (Kν + Kν) dΩ,

(23)

and ﬁnally for unpolarized radiation uniformly distributed in all direc-

tions:

uν

=

8πKν . q

(24)

CHAPTER II
RADIATION AT THERMODYNAMIC EQUILIBRIUM. KIRCHHOFF’S LAW. BLACK RADIATION
24. We shall now apply the laws enunciated in the last chapter to the special case of thermodynamic equilibrium, and hence we begin our consideration by stating a certain consequence of the second principle of thermodynamics: A system of bodies of arbitrary nature, shape, and position which is at rest and is surrounded by a rigid cover impermeable to heat will, no matter what its initial state may be, pass in the course of time into a permanent state, in which the temperature of all bodies of the system is the same. This is the state of thermodynamic equilibrium, in which the entropy of the system has the maximum value compatible with the total energy of the system as ﬁxed by the initial conditions. This state being reached, no further increase in entropy is possible.
In certain cases it may happen that, under the given conditions, the entropy can assume not only one but several maxima, of which one is the absolute one, the others having only a relative signiﬁcance.1 In these cases every state corresponding to a maximum value of the entropy represents a state of thermodynamic equilibrium of the system. But only one of them, the one corresponding to the absolute maximum of entropy, represents the absolutely stable equilibrium. All the others are in a certain sense unstable, inasmuch as a suitable, however small, disturbance may produce in the system a permanent change in the equilibrium in the direction of the absolutely stable equilibrium. An example of this is oﬀered by supersaturated steam enclosed in a rigid
1See, e.g., M. Planck, Vorlesungen u¨ber Thermodynamik, Leipzig, Veit and Comp., 1911 (or English Translation, Longmans Green & Co.), Secs. 165 and 189, et seq.
27

FUNDAMENTAL FACTS AND DEFINITIONS

28

vessel or by any explosive substance. We shall also meet such unstable equilibria in the case of radiation phenomena (Sec. 52).
25. We shall now, as in the previous chapter, assume that we are dealing with homogeneous isotropic media whose condition depends only on the temperature, and we shall inquire what laws the radiation phenomena in them must obey in order to be consistent with the deduction from the second principle mentioned in the preceding section. The means of answering this inquiry is supplied by the investigation of the state of thermodynamic equilibrium of one or more of such media, this investigation to be conducted by applying the conceptions and laws established in the last chapter.
We shall begin with the simplest case, that of a single medium extending very far in all directions of space, and, like all systems we shall here consider, being surrounded by a rigid cover impermeable to heat. For the present we shall assume that the medium has ﬁnite coeﬃcients of absorption, emission, and scattering.
Let us consider, ﬁrst, points of the medium that are far away from the surface. At such points the inﬂuence of the surface is, of course, vanishingly small and from the homogeneity and the isotropy of the medium it will follow that in a state of thermodynamic equilibrium the radiation of heat has everywhere and in all directions the same properties. Then Kν, the speciﬁc intensity of radiation of a plane polarized ray of frequency ν (Sec. 17), must be independent of the azimuth of the plane of polarization as well as of position and direction of the ray. Hence to each pencil of rays starting at an element of area dσ and diverging within a conical element dΩ corresponds an exactly equal pencil of opposite direction converging within the same conical element toward the element of area.
Now the condition of thermodynamic equilibrium requires that the temperature shall be everywhere the same and shall not vary in time.

RADIATION AT THERMODYNAMIC EQUILIBRIUM

29

Therefore in any given arbitrary time just as much radiant heat must be absorbed as is emitted in each volume-element of the medium. For the heat of the body depends only on the heat radiation, since, on account of the uniformity in temperature, no conduction of heat takes place. This condition is not inﬂuenced by the phenomenon of scattering, because scattering refers only to a change in direction of the energy radiated, not to a creation or destruction of it. We shall, therefore, calculate the energy emitted and absorbed in the time dt by a volumeelement v.
According to equation (2) the energy emitted has the value

∞

dt v · 8π

ν dν

0

where ν, the coeﬃcient of emission of the medium, depends only on the frequency ν and on the temperature in addition to the chemical

nature of the medium.

26. For the calculation of the energy absorbed we shall employ the

same reasoning as was illustrated by Fig. 1 (Sec. 22) and shall retain

the notation there used. The radiant energy absorbed by the volume-

element v in the time dt is found by considering the intensities of all the

rays passing through the element v and taking that fraction of each of

these rays which is absorbed in v. Now, according to (19), the conical

element that starts from dσ and cuts out of the volume v a part equal

to f s has the intensity (energy radiated per unit time)

f dσ · r2 · K

or, according to (12), by considering the diﬀerent parts of the spectrum

separately:

f∞ 2 dσ r2 0 Kν dν.

FUNDAMENTAL FACTS AND DEFINITIONS

30

Hence the intensity of a monochromatic ray is:

f 2 dσ r2 Kν dν.
The amount of energy of this ray absorbed in the distance s in the time dt is, according to (4),

f dt ανs 2 dσ r2 Kν dν.

Hence the absorbed part of the energy of this small cone of rays, as found by integrating over all frequencies, is:

fs ∞

dt 2 dσ r2

0

ανKν dν.

When this expression is summed up over all the diﬀerent crosssections f of the conical elements starting at dσ and passing through v, it is evident that f s = v, and when we sum up over all elements dσ of the spherical surface of radius r we have

dσ r2 = 4π.

Thus for the total radiant energy absorbed in the time dt by the volumeelement v the following expression is found:

∞

dt v 8π ανKν dν.

(25)

0

By equating the emitted and absorbed energy we obtain:

∞

∞

ν dν = ανKν dν.

0

0

RADIATION AT THERMODYNAMIC EQUILIBRIUM

31

A similar relation may be obtained for the separate parts of the spectrum. For the energy emitted and the energy absorbed in the state of thermodynamic equilibrium are equal, not only for the entire radiation of the whole spectrum, but also for each monochromatic radiation. This is readily seen from the following. The magnitudes of ν, αν, and Kν are independent of position. Hence, if for any single color the absorbed were not equal to the emitted energy, there would be everywhere in the whole medium a continuous increase or decrease of the energy radiation of that particular color at the expense of the other colors. This would be contradictory to the condition that Kν for each separate frequency does not change with the time. We have therefore for each frequency the relation:

ν = ανKν, or

(26)

Kν

=

ν
αν

,

(27)

i.e.: in the interior of a medium in a state of thermodynamic equilibrium the speciﬁc intensity of radiation of a certain frequency is equal to the coeﬃcient of emission divided by the coeﬃcient of absorption of the medium for this frequency.
27. Since ν and αν depend only on the nature of the medium, the temperature, and the frequency ν, the intensity of radiation of a deﬁnite color in the state of thermodynamic equilibrium is completely deﬁned by the nature of the medium and the temperature. An exceptional case is when αν = 0, that is, when the medium does not at all absorb the color in question. Since Kν cannot become inﬁnitely large, a ﬁrst consequence of this is that in that case ν = 0 also, that is, a medium does not emit any color which it does not absorb. A second consequence is that if ν and αν both vanish, equation (26) is satisﬁed by every value of Kν. In a medium which is diathermanous for a certain color thermo-

FUNDAMENTAL FACTS AND DEFINITIONS

32

dynamic equilibrium can exist for any intensity of radiation whatever of that color.
This supplies an immediate illustration of the cases spoken of before (Sec. 24), where, for a given value of the total energy of a system enclosed by a rigid cover impermeable to heat, several states of equilibrium can exist, corresponding to several relative maxima of the entropy. That is to say, since the intensity of radiation of the particular color in the state of thermodynamic equilibrium is quite independent of the temperature of a medium which is diathermanous for this color, the given total energy may be arbitrarily distributed between radiation of that color and the heat of the body, without making thermodynamic equilibrium impossible. Among all these distributions there is one particular one, corresponding to the absolute maximum of entropy, which represents absolutely stable equilibrium. This one, unlike all the others, which are in a certain sense unstable, has the property of not being appreciably aﬀected by a small disturbance. Indeed we shall see later (Sec. 48) that among the inﬁnite number of values, which the quotient
ν can have, if numerator and denominator both vanish, there exists αν one particular one which depends in a deﬁnite way on the nature of the medium, the frequency ν, and the temperature. This distinct value of the fraction is accordingly called the stable intensity of radiation Kν, in the medium, which at the temperature in question is diathermanous for rays of the frequency ν.
Everything that has just been said of a medium which is diathermanous for a certain kind of rays holds true for an absolute vacuum, which is a medium diathermanous for rays of all kinds, the only diﬀerence being that one cannot speak of the heat and the temperature of an absolute vacuum in any deﬁnite sense.
For the present we again shall put the special case of diathermancy

RADIATION AT THERMODYNAMIC EQUILIBRIUM

33

aside and assume that all the media considered have a ﬁnite coeﬃcient of absorption.
28. Let us now consider brieﬂy the phenomenon of scattering at thermodynamic equilibrium. Every ray meeting the volume-element v suﬀers there, apart from absorption, a certain weakening of its intensity because a certain fraction of its energy is diverted in diﬀerent directions. The value of the total energy of scattered radiation received and diverted, in the time dt by the volume-element v in all directions, may be calculated from expression (3) in exactly the same way as the value of the absorbed energy was calculated in Sec. 26. Hence we get an expression similar to (25), namely,

∞

dt v 8π βνKν dν.

(28)

0

The question as to what becomes of this energy is readily answered. On account of the isotropy of the medium, the energy scattered in v and given by (28) is radiated uniformly in all directions just as in the case of the energy entering v. Hence that part of the scattered energy received in v which is radiated out in a cone of solid angle dΩ is obtained by
dΩ multiplying the last expression by . This gives
4π
∞
2 dt v dΩ βνKν dν,
0

and, for monochromatic plane polarized radiation,

dt v dΩ βνKν dν.

(29)

Here it must be carefully kept in mind that this uniformity of radiation in all directions holds only for all rays striking the element v

FUNDAMENTAL FACTS AND DEFINITIONS

34

taken together; a single ray, even in an isotropic medium, is scattered

in diﬀerent directions with diﬀerent intensities and diﬀerent directions

of polarization. (See end of Sec. 8.)

It is thus found that, when thermodynamic equilibrium of radiation

exists inside of the medium, the process of scattering produces, on

the whole, no eﬀect. The radiation falling on a volume-element from

all sides and scattered from it in all directions behaves exactly as if

it had passed directly through the volume-element without the least

modiﬁcation. Every ray loses by scattering just as much energy as it

regains by the scattering of other rays.

29. We shall now consider from a diﬀerent point of view

the radiation phenomena in the interior of a very extended

homogeneous isotropic medium which is in thermodynamic

equilibrium. That is to say, we shall conﬁne our attention,

not to a deﬁnite volume-element, but to a deﬁnite pencil,

and in fact to an elementary pencil (Sec. 21). Let this

pencil be speciﬁed by the inﬁnitely small focal plane dσ

at the point O (Fig. 2), perpendicular to the axis of the

pencil, and by the solid angle dΩ, and let the radiation

take place toward the focal plane in the direction of the

arrow. We shall consider exclusively rays which belong to

this pencil.

The energy of monochromatic plane polarized radiation

of the pencil considered passing in unit time through dσ

is represented, according to (11), since in this case dt = 1,

Fig. 2. θ = 0, by

dσ dΩ Kν dν.

(30)

The same value holds for any other cross-section of the pencil. For ﬁrst, Kν dν has everywhere the same magnitude (Sec. 25), and second,

RADIATION AT THERMODYNAMIC EQUILIBRIUM

35

the product of any right section of the pencil and the solid angle at which the focal plane dσ is seen from this section has the constant value dσ dΩ, since the magnitude of the cross-section increases with the distance from the vertex O of the pencil in the proportion in which the solid angle decreases. Hence the radiation inside of the pencil takes place just as if the medium were perfectly diathermanous.
On the other hand, the radiation is continuously modiﬁed along its path by the eﬀect of emission, absorption, and scattering. We shall consider the magnitude of these eﬀects separately.
30. Let a certain volume-element of the pencil be bounded by two cross-sections at distances equal to r0 (of arbitrary length) and r0 + dr0 respectively from the vertex O. The volume will be represented by dr0 · r02 dΩ. It emits in unit time toward the focal plane dσ at O a certain quantity E of energy of monochromatic plane polarized radiation. E may be obtained from (1) by putting

dt = 1,

dτ = dr0 r02 dΩ,

dσ dΩ = r02

and omitting the numerical factor 2. We thus get

E = dr0 · dΩ dσ ν dν.

(31)

Of the energy E, however, only a fraction E0 reaches O, since in every inﬁnitesimal element of distance s which it traverses before reaching O the fraction (αν + βν)s is lost by absorption and scattering. Let Er represent that part of E which reaches a cross-section at a distance r (< r0) from O. Then for a small distance s = dr we have

Er+dr − Er = Er(αν + βν) dr,

or,

dEr dr

=

Er (αν

+ βν),

FUNDAMENTAL FACTS AND DEFINITIONS

36

and, by integration,

Er = Ee(αν +βν )(r−r0)

since, for r = r0, Er = E is given by equation (31). From this, by putting r = 0, the energy emitted by the volume-element at r0 which reaches O is found to be

E0 = Ee−(αν+βν)r0 = dr0 dΩ dσ

e−(αν +βν )r0
ν

dν.

(32)

All volume-elements of the pencils combined produce by their emission an amount of energy reaching dσ equal to

∞

dΩ dσ dν ν
0

dr0 e−(αν +βν )r0

=

dΩ dσ

αν

ν
+

βν

dν.

(33)

31. If the scattering did not aﬀect the radiation, the total energy reaching dσ would necessarily consist of the quantities of energy emitted by the diﬀerent volume-elements of the pencil, allowance being made, however, for the losses due to absorption on the way. For βν = 0 expressions (33) and (30) are identical, as may be seen by comparison with (27). Generally, however, (30) is larger than (33) because the energy reaching dσ contains also some rays which were not at all emitted from elements inside of the pencil, but somewhere else, and have entered later on by scattering. In fact, the volume-elements of the pencil do not merely scatter outward the radiation which is being transmitted inside the pencil, but they also collect into the pencil rays coming from without. The radiation E thus collected by the volume-element at r0 is found, by putting in (29),

dt = 1,

ν = dr0 dΩ r02,

dσ dΩ = r02 ,

RADIATION AT THERMODYNAMIC EQUILIBRIUM

37

to be

E = dr0 dΩ dσ βνKν dν.

This energy is to be added to the energy E emitted by the volumeelement, which we have calculated in (31). Thus for the total energy contributed to the pencil in the volume-element at r0 we ﬁnd:

E + E = dr0 dΩ dσ ( ν + βνKν) dν.

The part of this reaching O is, similar to (32):

dr0 dΩ dσ ( ν + βν Kν ) dν e−r0(αν+βν).

Making due allowance for emission and collection of scattered rays entering on the way, as well as for losses by absorption and scattering, all volume-elements of the pencil combined give for the energy ultimately reaching dσ

dΩ dσ ( ν + βνKν) dν

∞
dr0 e−r0(αν+βν) = dΩ dσ
0

ν + βνKν dν, αν + βν

and this expression is really exactly equal to that given by (30), as may be seen by comparison with (26).
32. The laws just derived for the state of radiation of a homogeneous isotropic medium when it is in thermodynamic equilibrium hold, so far as we have seen, only for parts of the medium which lie very far away from the surface, because for such parts only may the radiation be considered, by symmetry, as independent of position and direction. A simple consideration, however, shows that the value of Kν, which was already calculated and given by (27), and which depends only on the temperature and the nature of the medium, gives the correct value of the intensity of radiation of the frequency considered for all directions up to points directly below the surface of the medium. For in

FUNDAMENTAL FACTS AND DEFINITIONS

38

the state of thermodynamic equilibrium every ray must have just the same intensity as the one travelling in an exactly opposite direction, since otherwise the radiation would cause a unidirectional transport of energy. Consider then any ray coming from the surface of the medium and directed inward; it must have the same intensity as the opposite ray, coming from the interior. A further immediate consequence of this is that the total state of radiation of the medium is the same on the surface as in the interior.
33. While the radiation that starts from a surface element and is directed toward the interior of the medium is in every respect equal to that emanating from an equally large parallel element of area in the interior, it nevertheless has a diﬀerent history. That is to say, since the surface of the medium was assumed to be impermeable to heat, it is produced only by reﬂection at the surface of radiation coming from the interior. So far as special details are concerned, this can happen in very diﬀerent ways, depending on whether the surface is assumed to be smooth, i.e., in this case reﬂecting, or rough, e.g., white (Sec. 10). In the ﬁrst case there corresponds to each pencil which strikes the surface another perfectly deﬁnite pencil, symmetrically situated and having the same intensity, while in the second case every incident pencil is broken up into an inﬁnite number of reﬂected pencils, each having a diﬀerent direction, intensity, and polarization. While this is the case, nevertheless the rays that strike a surface-element from all diﬀerent directions with the same intensity Kν also produce, all taken together, a uniform radiation of the same intensity Kν, directed toward the interior of the medium.
34. Hereafter there will not be the slightest diﬃculty in dispensing with the assumption made in Sec. 25 that the medium in question extends very far in all directions. For after thermodynamic equilibrium has been everywhere established in our medium, the equilibrium is,

RADIATION AT THERMODYNAMIC EQUILIBRIUM

39

according to the results of the last paragraph, in no way disturbed, if we assume any number of rigid surfaces impermeable to heat and rough or smooth to be inserted in the medium. By means of these the whole system is divided into an arbitrary number of perfectly closed separate systems, each of which may be chosen as small as the general restrictions stated in Sec. 2 permit. It follows from this that the value of the speciﬁc intensity of radiation Kν given in (27) remains valid for the thermodynamic equilibrium of a substance enclosed in a space as small as we please and of any shape whatever.
35. From the consideration of a system consisting of a single homogeneous isotropic substance we now pass on to the treatment of a system consisting of two diﬀerent homogeneous isotropic substances contiguous to each other, the system being, as before, enclosed by a rigid cover impermeable to heat. We consider the state of radiation when thermodynamic equilibrium exists, at ﬁrst, as before, with the assumption that the media are of considerable extent. Since the equilibrium is nowise disturbed, if we think of the surface separating the two media as being replaced for an instant by an area entirely impermeable to heat radiation, the laws of the last paragraphs must hold for each of the two substances separately. Let the speciﬁc intensity of radiation of frequency ν polarized in an arbitrary plane be Kν in the ﬁrst substance (the upper one in Fig. 3), and Kν in the second, and, in general, let all quantities referring to the second substance be indicated by the addition of an accent. Both of the quantities Kν and Kν depend, according to equation (27), only on the temperature, the frequency ν, and the nature of the two substances, and these values of the intensities of radiation hold up to very small distances from the bounding surface of the substances, and hence are entirely independent of the properties of this surface.
36. We shall now suppose, to begin with, that the bounding surface

FUNDAMENTAL FACTS AND DEFINITIONS

40

of the media is smooth (Sec. 9). Then every ray coming from the ﬁrst medium and falling on the bounding surface is divided into two rays, the reﬂected and the transmitted ray. The directions of these two rays vary with the angle of incidence and the color of the incident ray; the intensity also varies with its polarization. Let us denote by ρ (coeﬃcient of reﬂection) the fraction of the energy reﬂected, then the fraction transmitted is (1 − ρ), ρ depending on the angle of incidence, the frequency, and the polarization of the incident ray. Similar remarks apply to ρ the coeﬃcient of reﬂection of a ray coming from the second medium and falling on the bounding surface.
Now according to (11) we have for the monochromatic plane polarized radiation of frequency ν, emitted in time dt toward the ﬁrst medium (in the direction of the feathered arrow upper left hand in Fig. 3), from an element dσ of the bounding surface and contained in the conical element dΩ,

dt dσ cos θ dΩ Kν dν,

(34)

where

dΩ = sin θ dθ dφ.

(35)

This energy is supplied by the two rays which come from the ﬁrst and the second medium and are respectively reﬂected from or transmitted by the element dσ in the corresponding direction (the unfeathered arrows). (Of the element dσ only the one point O is indicated.) The ﬁrst ray, according to the law of reﬂection, continues in the symmetrically situated conical element dΩ, the second in the conical element

dΩ = sin θ dθ dφ

(36)

where, according to the law of refraction,

sin θ q

φ = φ and

=.

(37)

sin θ q

RADIATION AT THERMODYNAMIC EQUILIBRIUM

41

Fig. 3.

If we now assume the radiation (34) to be polarized either in the plane of incidence or at right angles thereto, the same will be true for the two radiations of which it consists, and the radiation coming from the ﬁrst medium and reﬂected from dσ contributes the part

ρ dt dσ cos θ dΩ Kν dν

(38)

while the radiation coming from the second medium and transmitted through dσ contributes the part

(1 − ρ ) dt dσ cos θ dΩ Kν dν.

(39)

The quantities dt, dσ, ν and dν are written without the accent, because they have the same values in both media.

FUNDAMENTAL FACTS AND DEFINITIONS

42

By adding (38) and (39) and equating their sum to the expression (34) we ﬁnd

ρ cos θ dΩ Kν + (1 − ρ ) cos θ dΩ Kν = cos θ dΩ Kν. Now from (37) we have

cos θ dθ cos θ dθ

=

q

q

and further by (35) and (36)

dΩ cos θ q 2

dΩ cos θ =

.

q2

Therefore we ﬁnd

q2 ρKν + (1 − ρ ) q2 Kν = K

or

Kν

·

q2

=

1−ρ .

Kν q 2 1 − ρ

37. In the last equation the quantity on the left side is independent

of the angle of incidence θ and of the particular kind of polarization;

hence the same must be true for the right side. Hence, whenever the

value of this quantity is known for a single angle of incidence and any

deﬁnite kind of polarization, this value will remain valid for all angles

of incidence and all kinds of polarization. Now in the special case when

the rays are polarized at right angles to the plane of incidence and strike

the bounding surface at the angle of polarization, ρ = 0, and ρ = 0.

The expression on the right side of the last equation then becomes 1;

hence it must always be 1 and we have the general relations:

ρ=ρ

(40)

RADIATION AT THERMODYNAMIC EQUILIBRIUM

43

and

q2Kν = q 2Kν.

(41)

38. The ﬁrst of these two relations, which states that the coeﬃcient

of reﬂection of the bounding surface is the same on both sides, is a

special case of a general law of reciprocity ﬁrst stated by Helmholtz.1

According to this law the loss of intensity which a ray of deﬁnite color

and polarization suﬀers on its way through any media by reﬂection,

refraction, absorption, and scattering is exactly equal to the loss suf-

fered by a ray of the same intensity, color, and polarization pursuing an

exactly opposite path. An immediate consequence of this law is that

the radiation striking the bounding surface of any two media is always

transmitted as well as reﬂected equally on both sides, for every color,

direction, and polarization.

39. The second formula (41) establishes a relation between the in-

tensities of radiation in the two media, for it states that, when ther-

modynamic equilibrium exists, the speciﬁc intensities of radiation of a

certain frequency in the two media are in the inverse ratio of the squares

of the velocities of propagation or in the direct ratio of the squares of

the indices of refraction.2

By substituting for Kν its value from (27) we obtain the following theorem: The quantity

q2Kν

=

q2 ν αν

(42)

1H. v. Helmholtz, Handbuch d. physiologischen Optik 1. Lieferung, Leipzig, Leop. Voss, 1856, p. 169. See also Helmholtz, Vorlesungen u¨ber die Theorie der W¨arme herausgegeben von F. Richarz, Leipzig, J. A. Barth, 1903, p. 161. The restrictions of the law of reciprocity made there do not bear on our problems, since we are concerned with temperature radiation only (Sec. 7).
2G. Kirchhoﬀ, Gesammelte Abhandlungen, Leipzig, J. A. Barth, 1882, p. 594. R. Clausius, Pogg. Ann. 121, p. 1, 1864.

FUNDAMENTAL FACTS AND DEFINITIONS

44

does not depend on the nature of the substance, and is, therefore, a universal function of the temperature T and the frequency ν alone.
The great importance of this law lies evidently in the fact that it states a property of radiation which is the same for all bodies in nature, and which need be known only for a single arbitrarily chosen body, in order to be stated quite generally for all bodies. We shall later on take advantage of the opportunity oﬀered by this statement in order actually to calculate this universal function (Sec. 165).
40. We now consider the other case, that in which the bounding surface of the two media is rough. This case is much more general than the one previously treated, inasmuch as the energy of a pencil directed from an element of the bounding surface into the ﬁrst medium is no longer supplied by two deﬁnite pencils, but by an arbitrary number, which come from both media and strike the surface. Here the actual conditions may be very complicated according to the peculiarities of the bounding surface, which moreover may vary in any way from one element to another. However, according to Sec. 35, the values of the speciﬁc intensities of radiation Kν and Kν remain always the same in all directions in both media, just as in the case of a smooth bounding surface. That this condition, necessary for thermodynamic equilibrium, is satisﬁed is readily seen from Helmholtz’s law of reciprocity, according to which, in the case of stationary radiation, for each ray striking the bounding surface and diﬀusely reﬂected from it on both sides, there is a corresponding ray at the same point, of the same intensity and opposite direction, produced by the inverse process at the same point on the bounding surface, namely by the gathering of diﬀusely incident rays into a deﬁnite direction, just as is the case in the interior of each of the two media.
41. We shall now further generalize the laws obtained. First, just as in Sec. 34, the assumption made above, namely, that the two media

RADIATION AT THERMODYNAMIC EQUILIBRIUM

45

extend to a great distance, may be abandoned since we may introduce

an arbitrary number of bounding surfaces without disturbing the ther-

modynamic equilibrium. Thereby we are placed in a position enabling

us to pass at once to the case of any number of substances of any size

and shape. For when a system consisting of an arbitrary number of

contiguous substances is in the state of thermodynamic equilibrium,

the equilibrium is in no way disturbed, if we assume one or more of

the surfaces of contact to be wholly or partly impermeable to heat.

Thereby we can always reduce the case of any number of substances

to that of two substances in an enclosure impermeable to heat, and,

therefore, the law may be stated quite generally, that, when any arbi-

trary system is in the state of thermodynamic equilibrium, the speciﬁc

intensity of radiation Kν is determined in each separate substance by the universal function (42).

42. We shall now consider a system in a state of thermodynamic

equilibrium, contained within an enclosure impermeable to heat and

consisting of n emitting and absorbing adjacent bodies of any size

and shape whatever. As in Sec. 36, we again conﬁne our attention

to a monochromatic plane polarized pencil which proceeds from an el-

ement dσ of the bounding surface of the two media in the direction

toward the ﬁrst medium (Fig. 3, feathered arrow) within the conical

element dΩ. Then, as in (34), the energy supplied by the pencil in unit

time is

dσ cos θ dΩ Kν dν = I.

(43)

This energy of radiation I consists of a part coming from the ﬁrst medium by regular or diﬀuse reﬂection at the bounding surface and of a second part coming through the bounding surface from the second medium. We shall, however, not stop at this mode of division, but shall further subdivide I according to that one of the n media from which

FUNDAMENTAL FACTS AND DEFINITIONS

46

the separate parts of the radiation I have been emitted. This point of view is distinctly diﬀerent from the preceding, since, e.g., the rays transmitted from the second medium through the bounding surface into the pencil considered have not necessarily been emitted in the second medium, but may, according to circumstances, have traversed a long and very complicated path through diﬀerent media and may have undergone therein the eﬀect of refraction, reﬂection, scattering, and partial absorption any number of times. Similarly the rays of the pencil, which coming from the ﬁrst medium are reﬂected at dσ, were not necessarily all emitted in the ﬁrst medium. It may even happen that a ray emitted from a certain medium, after passing on its way through other media, returns to the original one and is there either absorbed or emerges from this medium a second time.
We shall now, considering all these possibilities, denote that part of I which has been emitted by volume-elements of the ﬁrst medium by I1 no matter what paths the diﬀerent constituents have pursued, that which has been emitted by volume-elements of the second medium by I2, etc. Then since every part of I must have been emitted by an element of some body, the following equation must hold,

I = I1 + I2 + I3 + . . . In.

(44)

43. The most adequate method of acquiring more detailed information as to the origin and the paths of the diﬀerent rays of which the radiations I1, I2, I3, . . . In consist, is to pursue the opposite course and to inquire into the future fate of that pencil, which travels exactly in the opposite direction to the pencil I and which therefore comes from the ﬁrst medium in the cone dΩ and falls on the surface element dσ of the second medium. For since every optical path may also be traversed in the opposite direction, we may obtain by this consideration all paths along which rays can pass into the pencil I, however complicated they

RADIATION AT THERMODYNAMIC EQUILIBRIUM

47

may otherwise be. Let J represent the intensity of this inverse pencil, which is directed toward the bounding surface and is in the same state of polarization. Then, according to Sec. 40,

J = I.

(45)

At the bounding surface dσ the rays of the pencil J are partly reﬂected and partly transmitted regularly or diﬀusely, and thereafter, travelling in both media, are partly absorbed, partly scattered, partly again reﬂected or transmitted to diﬀerent media, etc., according to the conﬁguration of the system. But ﬁnally the whole pencil J after splitting into many separate rays will be completely absorbed in the n media. Let us denote that part of J which is ﬁnally absorbed in the ﬁrst medium by J1, that which is ﬁnally absorbed in the second medium by J2, etc., then we shall have

J = J1 + J2 + J3 + · · · + Jn.

Now the volume-elements of the n media, in which the absorption of the rays of the pencil J takes place, are precisely the same as those in which takes place the emission of the rays constituting the pencil I, the ﬁrst one considered above. For, according to Helmholtz’s law of reciprocity, no appreciable radiation of the pencil J can enter a volumeelement which contributes no appreciable radiation to the pencil I and vice versa.
Let us further keep in mind that the absorption of each volumeelement is, according to (42), proportional to its emission and that, according to Helmholtz’s law of reciprocity, the decrease which the energy of a ray suﬀers on any path is always equal to the decrease suﬀered by the energy of a ray pursuing the opposite path. It will then be clear that the volume-elements considered absorb the rays of the pencil J in

FUNDAMENTAL FACTS AND DEFINITIONS

48

just the same ratio as they contribute by their emission to the energy of the opposite pencil I. Since, moreover, the sum I of the energies given oﬀ by emission by all volume-elements is equal to the sum J of the energies absorbed by all elements, the quantity of energy absorbed by each separate volume-element from the pencil J must be equal to the quantity of energy emitted by the same element into the pencil I. In other words: the part of a pencil I which has been emitted from a certain volume of any medium is equal to the part of the pencil J (= I) oppositely directed, which is absorbed in the same volume.
Hence not only are the sums I and J equal, but their constituents are also separately equal or

J1 = I1, J2 = I2, . . . Jn = In.

(46)

44. Following G. Kirchhoﬀ 1 we call the quantity I2, i.e., the intensity of the pencil emitted from the second medium into the ﬁrst, the emissive power E of the second medium, while we call the ratio of J2 to J, i.e., that fraction of a pencil incident on the second medium which is absorbed in this medium, the absorbing power A of the second medium. Therefore

E = I2 (

I ),

A = J2 ( J

1).

(47)

The quantities E and A depend (a) on the nature of the two media, (b) on the temperature, the frequency ν, and the direction and the polarization of the radiation considered, (c) on the nature of the bounding surface and on the magnitude of the surface element dσ and that of the solid angle dΩ, (d) on the geometrical extent and the shape of the total surface of the two media, (e) on the nature and form of all

1G. Kirchhoﬀ, Gesammelte Abhandlungen, 1882, p. 574.

RADIATION AT THERMODYNAMIC EQUILIBRIUM

49

other bodies of the system. For a ray may pass from the ﬁrst into the second medium, be partly transmitted by the latter, and then, after reﬂection somewhere else, may return to the second medium and may be there entirely absorbed.
With these assumptions, according to equations (46), (45), and (43), Kirchhoﬀ ’s law holds,

E

A = I = dσ cos θ dΩ Kν dν,

(48)

i.e., the ratio of the emissive power to the absorbing power of any body is independent of the nature of the body. For this ratio is equal to the intensity of the pencil passing through the ﬁrst medium, which, according to equation (27), does not depend on the second medium at all. The value of this ratio does, however, depend on the nature of the ﬁrst medium, inasmuch as, according to (42), it is not the quantity Kν but the quantity q2Kν, which is a universal function of the temperature and frequency. The proof of this law given by G. Kirchhoﬀ l. c. was later greatly simpliﬁed by E. Pringsheim.1
45. When in particular the second medium is a black body (Sec. 10) it absorbs all the incident radiation. Hence in that case J2 = J, A = 1, and E = I, i.e., the emissive power of a black body is independent of its nature. Its emissive power is larger than that of any other body at the same temperature and, in fact, is just equal to the intensity of radiation in the contiguous medium.
46. We shall now add, without further proof, another general law of reciprocity, which is closely connected with that stated at the end of Sec. 43 and which may be stated thus: When any emitting and absorbing bodies are in the state of thermodynamic equilibrium, the part

1E. Pringsheim, Verhandlungen der Deutschen Physikalischen Gesellschaft, 3, p. 81, 1901.

FUNDAMENTAL FACTS AND DEFINITIONS

50

of the energy of deﬁnite color emitted by a body A, which is absorbed

by another body B, is equal to the part of the energy of the same color

emitted by B which is absorbed by A. Since a quantity of energy emitted

causes a decrease of the heat of the body, and a quantity of energy

absorbed an increase of the heat of the body, it is evident that, when

thermodynamic equilibrium exists, any two bodies or elements of bodies

selected at random exchange by radiation equal amounts of heat with

each other. Here, of course, care must be taken to distinguish between

the radiation emitted and the total radiation which reaches one body

from the other.

47. The law holding for the quantity (42) can be expressed in a

diﬀerent form, by introducing, by means of (24), the volume density uν of monochromatic radiation instead of the intensity of radiation Kν. We then obtain the law that, for radiation in a state of thermodynamic

equilibrium, the quantity

uν q3

(49)

is a function of the temperature T and the frequency ν, and is the same for all substances.1 This law becomes clearer if we consider that the

quantity

q3

uν dν ν3

(50)

also is a universal function of T , ν, and ν + dν, and that the product

uν dν

is,

according

to

(22),

the

volume

density

of

the

radiation q

whose

frequency lies between ν and ν +dν, while the quotient represents the

ν

wave length of a ray of frequency ν in the medium in question. The law

then takes the following simple form: When any bodies whatever are in

1In this law it is assumed that the quantity q in (24) is the same as in (37). This does not hold for strongly dispersing or absorbing substances. For the generalization applying to such cases see M. Laue, Annalen d. Physik 32, p. 1085, 1910.

RADIATION AT THERMODYNAMIC EQUILIBRIUM

51

thermodynamic equilibrium, the energy of monochromatic radiation of

a deﬁnite frequency, contained in a cubical element of side equal to the

wave length, is the same for all bodies.

48. We shall ﬁnally take up the case of diathermanous (Sec. 12)

media, which has so far not been considered. In Sec. 27 we saw that, in

a medium which is diathermanous for a given color and is surrounded

by an enclosure impermeable to heat, there can be thermodynamic

equilibrium for any intensity of radiation of this color. There must,

however, among all possible intensities of radiation be a deﬁnite one,

corresponding to the absolute maximum of the total entropy of the

system, which designates the absolutely stable equilibrium of radia-

tion. In fact, in equation (27) the intensity of radiation Kν for αν = 0

0

and

ν

=

0

assumes

the

value

, 0

and

hence

cannot

be

calculated

from

this equation. But we see also that this indeterminateness is removed

by equation (41), which states that in the case of thermodynamic equi-

librium the product q2Kν has the same value for all substances. From this we ﬁnd immediately a deﬁnite value of Kν which is thereby distinguished from all other values. Furthermore the physical signiﬁcance

of this value is immediately seen by considering the way in which that

equation was obtained. It is that intensity of radiation which exists in a

diathermanous medium, if it is in thermodynamic equilibrium when in

contact with an arbitrary absorbing and emitting medium. The volume

and the form of the second medium do not matter in the least, in partic-

ular the volume may be taken as small as we please. Hence we can for-

mulate the following law: Although generally speaking thermodynamic

equilibrium can exist in a diathermanous medium for any intensity of

radiation whatever, nevertheless there exists in every diathermanous

medium for a deﬁnite frequency at a deﬁnite temperature an intensity

of radiation deﬁned by the universal function (42). This may be called

FUNDAMENTAL FACTS AND DEFINITIONS

52

the stable intensity, inasmuch as it will always be established, when the medium is exchanging stationary radiation with an arbitrary emitting and absorbing substance.
49. According to the law stated in Sec. 45, the intensity of a pencil, when a state of stable heat radiation exists in a diathermanous medium, is equal to the emissive power E of a black body in contact with the medium. On this fact is based the possibility of measuring the emissive power of a black body, although absolutely black bodies do not exist in nature.1 A diathermanous cavity is enclosed by strongly emitting walls2 and the walls kept at a certain constant temperature T . Then the radiation in the cavity, when thermodynamic equilibrium is established for every frequency ν, assumes the intensity corresponding to the velocity of propagation q in the diathermanous medium, according to the universal function (42). Then any element of area of the walls radiates into the cavity just as if the wall were a black body of temperature T . The amount lacking in the intensity of the rays actually emitted by the walls as compared with the emission of a black body is supplied by rays which fall on the wall and are reﬂected there. Similarly every element of area of a wall receives the same radiation.
In fact, the radiation I starting from an element of area of a wall consists of the radiation E emitted by the element of area and of the radiation reﬂected from the element of area from the incident radiation I, i.e., the radiation which is not absorbed (1 − A)I. We have, therefore, in agreement with Kirchhoﬀ ’s law (48),
I = E + (1 − A)I.
1W. Wien and O. Lummer, Wied. Annalen, 56, p. 451, 1895. 2The strength of the emission inﬂuences only the time required to establish stationary radiation, but not its character. It is essential, however, that the walls transmit no radiation to the exterior.

RADIATION AT THERMODYNAMIC EQUILIBRIUM

53

If we now make a hole in one of the walls of a size dσ, so small that the intensity of the radiation directed toward the hole is not changed thereby, then radiation passes through the hole to the exterior where we shall suppose there is the same diathermanous medium as within. This radiation has exactly the same properties as if dσ were the surface of a black body, and this radiation may be measured for every color together with the temperature T .
50. Thus far all the laws derived in the preceding sections for diathermanous media hold for a deﬁnite frequency, and it is to be kept in mind that a substance may be diathermanous for one color and adiathermanous for another. Hence the radiation of a medium completely enclosed by absolutely reﬂecting walls is, when thermodynamic equilibrium has been established for all colors for which the medium has a ﬁnite coeﬃcient of absorption, always the stable radiation corresponding to the temperature of the medium such as is represented by the emission of a black body. Hence this is brieﬂy called “black” radiation.1 On the other hand, the intensity of colors for which the medium is diathermanous is not necessarily the stable black radiation, unless the medium is in a state of stationary exchange of radiation with an absorbing substance.
There is but one medium that is diathermanous for all kinds of rays, namely, the absolute vacuum, which to be sure cannot be produced in nature except approximately. However, most gases, e.g., the air of the atmosphere, have, at least if they are not too dense, to a sufﬁcient approximation the optical properties of a vacuum with respect to waves of not too short length. So far as this is the case the velocity
1M. Thiesen, Verhandlungen d. Deutschen Physikal. Gesellschaft, 2, p. 65, 1900.

FUNDAMENTAL FACTS AND DEFINITIONS

54

of propagation q may be taken as the same for all frequencies, namely,

c

=

3

×

1010

cm .

(51)

sec

51. Hence in a vacuum bounded by totally reﬂecting walls any state of radiation may persist. But as soon as an arbitrarily small quantity of matter is introduced into the vacuum, a stationary state of radiation is gradually established. In this the radiation of every color which is appreciably absorbed by the substance has the intensity Kν corresponding to the temperature of the substance and determined by the universal function (42) for q = c, the intensity of radiation of the other colors remaining indeterminate. If the substance introduced is not diathermanous for any color, e.g., a piece of carbon however small, there exists at the stationary state of radiation in the whole vacuum for all colors the intensity Kν of black radiation corresponding to the temperature of the substance. The magnitude of Kν regarded as a function of ν gives the spectral distribution of black radiation in a vacuum, or the so-called normal energy spectrum, which depends on nothing but the temperature. In the normal spectrum, since it is the spectrum of emission of a black body, the intensity of radiation of every color is the largest which a body can emit at that temperature at all.
52. It is therefore possible to change a perfectly arbitrary radiation, which exists at the start in the evacuated cavity with perfectly reﬂecting walls under consideration, into black radiation by the introduction of a minute particle of carbon. The characteristic feature of this process is that the heat of the carbon particle may be just as small as we please, compared with the energy of radiation contained in the cavity of arbitrary magnitude. Hence, according to the principle of the conservation of energy, the total energy of radiation remains essentially constant during the change that takes place, because the changes in

RADIATION AT THERMODYNAMIC EQUILIBRIUM

55

the heat of the carbon particle may be entirely neglected, even if its changes in temperature should be ﬁnite. Herein the carbon particle exerts only a releasing (auslo¨send) action. Thereafter the intensities of the pencils of diﬀerent frequencies originally present and having diﬀerent frequencies, directions, and diﬀerent states of polarization change at the expense of one another, corresponding to the passage of the system from a less to a more stable state of radiation or from a state of smaller to a state of larger entropy. From a thermodynamic point of view this process is perfectly analogous, since the time necessary for the process is not essential, to the change produced by a minute spark in a quantity of oxy-hydrogen gas or by a small drop of liquid in a quantity of supersaturated vapor. In all these cases the magnitude of the disturbance is exceedingly small and cannot be compared with the magnitude of the energies undergoing the resultant changes, so that in applying the two principles of thermodynamics the cause of the disturbance of equilibrium, viz. the carbon particle, the spark, or the drop, need not be considered. It is always a case of a system passing from a more or less unstable into a more stable state, wherein, according to the ﬁrst principle of thermodynamics, the energy of the system remains constant, and, according to the second principle, the entropy of the system increases.

PART II
DEDUCTIONS FROM ELECTRODYNAMICS AND THERMODYNAMICS

CHAPTER I

MAXWELL’S RADIATION PRESSURE
53. While in the preceding part the phenomena of radiation have been presented with the assumption of only well known elementary laws of optics summarized in Sec. 2, which are common to all optical theories, we shall hereafter make use of the electromagnetic theory of light and shall begin by deducing a consequence characteristic of that theory. We shall, namely, calculate the magnitude of the mechanical force, which is exerted by a light or heat ray passing through a vacuum on striking a reﬂecting (Sec. 10) surface assumed to be at rest.
For this purpose we begin by stating Maxwell’s general equations for an electromagnetic process in a vacuum. Let the vector E denote the electric ﬁeld-strength (intensity of the electric ﬁeld) in electric units and the vector H the magnetic ﬁeld-strength in magnetic units. Then the equations are, in the abbreviated notation of the vector calculus,

E˙ = c curl H

H˙ = −c curl E

(52)

div. E = 0

div. H = 0.

Should the reader be unfamiliar with the symbols of this notation, he may readily deduce their meaning by working backward from the subsequent equations (53).
54. In order to pass to the case of a plane wave in any direction we assume that all the quantities that ﬁx the state depend only on the time t and on one of the coordinates x , y , z , of an orthogonal right-handed system of coordinates, say on x . Then the equations (52)

57

DEDUCTIONS FROM ELECTRODYNAMICS

58

reduce to

∂Ex = 0 ∂t

∂Hx = 0 ∂t

∂Ey = −c ∂Hz

∂Hy = c ∂Ez

∂t

∂x

∂t

∂x

(53)

∂Ex = c ∂Hy

∂Hz = −c ∂Ey

∂t

∂x

∂t

∂x

∂Ex = 0 ∂x

∂Hx = 0. ∂x

Hence the most general expression for a plane wave passing through a

vacuum in the direction of the positive x -axis is

Ex = 0

Hx = 0

x

x

Ey = f

t− c

Hy = −g

t− c

(54)

x

Ez = g

t− c

x

Hz =

f

t− c

where f and g represent two arbitrary functions of the same argument. 55. Suppose now that this wave strikes a reﬂecting surface, e.g., the
surface of an absolute conductor (metal) of inﬁnitely large conductivity. In such a conductor even an inﬁnitely small electric ﬁeld-strength produces a ﬁnite conduction current; hence the electric ﬁeld-strength E in it must be always and everywhere inﬁnitely small. For simplicity we also suppose the conductor to be non-magnetizable, i.e., we assume the magnetic induction B in it to be equal to the magnetic ﬁeld-strength H, just as is the case in a vacuum.
If we place the x-axis of a right-handed coordinate system (xyz) along the normal of the surface directed toward the interior of the conductor, the x-axis is the normal of incidence. We place the (x y ) plane

MAXWELL’S RADIATION PRESSURE

59

Fig. 4.

in the plane of incidence and take this as the plane of the ﬁgure (Fig. 4). Moreover, we can also, without any restriction of generality, place the y-axis in the plane of the ﬁgure, so that the z-axis coincides with the z -axis (directed from the ﬁgure toward the observer). Let the common origin O of the two coordinate systems lie in the surface. If ﬁnally θ represents the angle of incidence, the coordinates with and without accent are related to each other by the following equations:

x = x cos θ − y sin θ y = x sin θ + y cos θ z=z

x = x cos θ + y sin θ y = −x sin θ + y cos θ z = z.

DEDUCTIONS FROM ELECTRODYNAMICS

60

By the same transformation we may pass from the components of the electric or magnetic ﬁeld-strength in the ﬁrst coordinate system to their components in the second system. Performing this transformation the following values are obtained from (54) for the components of the electric and magnetic ﬁeld-strengths of the incident wave in the coordinate system without accent,

Ex = − sin θ · f

Hx = sin θ · g

Ey = cos θ · f

Hy = − cos θ · g

(55)

Ez = g

Hz = f.

Herein the argument of the functions f and g is

x

x cos θ + y sin θ

t− =t−

.

(56)

c

c

56. In the surface of separation of the two media x = 0. According to the general electromagnetic boundary conditions the components of the ﬁeld-strengths in the surface of separation, i.e., the four quantities Ey, Ez, Hy, Hz must be equal to each other on the two sides of the surface of separation for this value of x. In the conductor the electric ﬁeld-strength E is inﬁnitely small in accordance with the assumption made above. Hence Ey and Ez must vanish also in the vacuum for x = 0. This condition cannot be satisﬁed unless we assume in the vacuum, besides the incident, also a reﬂected wave superposed on the former in such a way that the components of the electric ﬁeld of the two waves in the y and z direction just cancel at every instant and at every point in the surface of separation. By this assumption and the condition that the reﬂected wave is a plane wave returning into the interior of the vacuum, the other four components of the reﬂected wave are also completely determined. They are all functions of the single

MAXWELL’S RADIATION PRESSURE

61

argument

−x cos θ + y sin θ

t−

.

(57)

c

The actual calculation yields as components of the total electromagnetic

ﬁeld produced in the vacuum by the superposition of the two waves,

the following expressions valid for points of the surface of separation

x = 0,

Ex = − sin θ · f − sin θ · f = −2 sin θ · f

Ey = cos θ · f − cos θ · f = 0

Ez = g − g = 0

(58)

Hx = sin θ · g − sin θ · g = 0

Hy = − cos θ · g − cos θ · g = −2 cos θ · g

Hz = f + f = 2f.

In these equations the argument of the functions f and g is, according

to (56) and (57),

y sin θ

t−

.

c

From these values the electric and magnetic ﬁeld-strength within the

conductor in the immediate neighborhood of the separating surface

x = 0 is obtained:

Ex = 0

Hx = 0

Ey = 0

Hy = −2 cos θ · g

(59)

Ez = 0

Hz = 2f

y sin θ

where again the argument t−

is to be substituted in the functions

c

f and g. For the components of E all vanish in an absolute conductor

and the components Hx, Hy, Hz are all continuous at the separating

DEDUCTIONS FROM ELECTRODYNAMICS

62

surface, the two latter since they are tangential components of the ﬁeldstrength, the former since it is the normal component of the magnetic induction B (Sec. 55), which likewise remains continuous on passing through any surface of separation.
On the other hand, the normal component of the electric ﬁeldstrength Ex is seen to be discontinuous; the discontinuity shows the existence of an electric charge on the surface, the surface density of which is given in magnitude and sign as follows:

1

1

2 sin θ · f = sin θ · f.

(60)

4π

2π

In the interior of the conductor at a ﬁnite distance from the bounding surface, i.e., for x > 0, all six ﬁeld components are inﬁnitely small. Hence, on increasing x, the values of Hy and Hz, which are ﬁnite for x = 0, approach the value 0 at an inﬁnitely rapid rate.
57. A certain mechanical force is exerted on the substance of the conductor by the electromagnetic ﬁeld considered. We shall calculate the component of this force normal to the surface. It is partly of electric, partly of magnetic, origin. Let us ﬁrst consider the former, Fe. Since the electric charge existing on the surface of the conductor is in an electric ﬁeld, a mechanical force equal to the product of the charge and the ﬁeld-strength is exerted on it. Since, however, the ﬁeld-strength is discontinuous, having the value −2 sin θf on the side of the vacuum and 0 on the side of the conductor, from a well-known law of electrostatics the magnitude of the mechanical force Fe acting on an element of surface dσ of the conductor is obtained by multiplying the electric charge of the element of area calculated in (60) by the arithmetic mean of the electric ﬁeld-strength on the two sides. Hence

Fe

=

sin θ f
2π

dσ

(− sin θf )

=

− sin2 θ f 2 2π

dσ.

MAXWELL’S RADIATION PRESSURE

63

This force acts in the direction toward the vacuum and therefore exerts a tension.
58. We shall now calculate the mechanical force of magnetic origin Fm. In the interior of the conducting substance there are certain conduction currents, whose intensity and direction are determined by the vector I of the current density

c

I = curl H.

(61)

4π

A mechanical force acts on every element of space dτ of the conductor

through which a conduction current ﬂows, and is given by the vector

product

dτ

[I × H].

(62)

c

Hence the component of this force normal to the surface of the conduc-

tor x = 0 is equal to

dτ c (IyHz − IzHy).

On substituting the values of Iy and Iz from (61) we obtain

dτ 4π Hz

∂Hx − ∂Hz ∂z ∂x

− Hy

∂Hy − ∂Hx ∂x ∂y

.

In this expression the diﬀerential coeﬃcients with respect to y and z are negligibly small in comparison to those with respect to x, according to the remark at the end of Sec. 56; hence the expression reduces to

dτ −
4π

Hy

∂Hy ∂x

+

Hz

∂Hz ∂x

.

Let us now consider a cylinder cut out of the conductor perpendicular to the surface with the cross-section dσ, and extending from x = 0 to

DEDUCTIONS FROM ELECTRODYNAMICS

64

x = ∞. The entire mechanical force of magnetic origin acting on this cylinder in the direction of the x-axis, since dτ = dσ x, is given by

dσ ∞

Fm

=

− 4π

0

dx

Hy

∂Hy ∂x

+

Hz

∂Hz ∂x

.

On integration, since H vanishes for x = ∞, we obtain

or by equation (59)

dσ Fm = 8π

H2y + H2z x=0

Fm

=

dσ (cos2 2π

θ

·

g2

+

f 2).

By adding Fe and Fm the total mechanical force acting on the cylinder in question in the direction of the x-axis is found to be

F = dσ cos2 θ(f 2 + g2).

(63)

2π

This force exerts on the surface of the conductor a pressure, which acts

in a direction normal to the surface toward the interior and is called

“Maxwell’s radiation pressure.” The existence and the magnitude of

the radiation pressure as predicted by the theory was ﬁrst found by

delicate measurements with the radiometer by P. Lebedew.1

59. We shall now establish a relation between the radiation pressure

and the energy of radiation I dt falling on the surface element dσ of the

conductor in a time element dt. The latter from Poynting’s law of

energy ﬂow is

c I dt = 4π (EyHz − EzHy) dσ dt,

1P. Lebedew, Annalen d. Phys. 6, p. 433, 1901. See also E. F. Nichols and G. F. Hull, Annalen d. Phys. 12, p. 225, 1903.

MAXWELL’S RADIATION PRESSURE

65

hence from (55) I dt = c cos θ(f 2 + g2) dσ dt. 4π
By comparison with (63) we obtain

2 cos θ

F=

I.

(64)

c

From this we ﬁnally calculate the total pressure p, i.e., that mechanical force, which an arbitrary radiation proceeding from the vacuum and totally reﬂected upon incidence on the conductor exerts in a normal direction on a unit surface of the conductor. The energy radiated in the conical element
dΩ = sin θ dθ dφ

in the time dt on the element of area dσ is, according to (6),

I dt = K cos θ dΩ dσ dt,

where K represents the speciﬁc intensity of the radiation in the direction dΩ toward the reﬂector. On substituting this in (64) and integrating over dΩ we obtain for the total pressure of all pencils which fall on the surface and are reﬂected by it

2 p=

K cos2 θ dΩ,

(65)

c

the integration with respect to φ extending from 0 to 2π and with π
respect to θ from 0 to . 2
In case K is independent of direction as in the case of black radiation, we obtain for the radiation pressure

2K p=

2π
dφ

π
2 dθ cos2 θ sin θ = 4πK

c0

0

3c

DEDUCTIONS FROM ELECTRODYNAMICS

66

or, if we introduce instead of K the volume density of radiation u

from (21)

u

p= .

(66)

3

This value of the radiation pressure holds only when the reﬂection

of the radiation occurs at the surface of an absolute non-magnetizable

conductor. Therefore we shall in the thermodynamic deductions of

the next chapter make use of it only in such cases. Nevertheless it

will be shown later on (Sec. 66) that equation (66) gives the pressure

of uniform radiation against any totally reﬂecting surface, no matter

whether it reﬂects uniformly or diﬀusely.

60. In view of the extraordinarily simple and close relation between

the radiation pressure and the energy of radiation, the question might

be raised whether this relation is really a special consequence of the

electromagnetic theory, or whether it might not, perhaps, be founded

on more general energetic or thermodynamic considerations. To decide

this question we shall calculate the radiation pressure that would fol-

low by Newtonian mechanics from Newton’s (emission) theory of light,

a theory which, in itself, is quite consistent with the energy princi-

ple. According to it the energy radiated onto a surface by a light ray

passing through a vacuum is equal to the kinetic energy of the light

particles striking the surface, all moving with the constant velocity c.

The decrease in intensity of the energy radiation with the distance is

then explained simply by the decrease of the volume density of the light

particles.

Let us denote by n the number of the light particles contained in

a unit volume and by m the mass of a particle. Then for a beam of

parallel light the number of particles impinging in unit time on the

element dσ of a reﬂecting surface at the angle of incidence θ is

nc cos θ dσ.

(67)

MAXWELL’S RADIATION PRESSURE

67

Their kinetic energy is given according to Newtonian mechanics by

mc2

c3

I = nc cos θ dσ = nm cos θ dσ.

(68)

2

2

Now, in order to determine the normal pressure of these particles on the surface, we may note that the normal component of the velocity c cos θ of every particle is changed on reﬂection into a component of opposite direction. Hence the normal component of the momentum of every particle (impulse-coordinate) is changed through reﬂection by −2mc cos θ. Then the change in momentum for all particles considered will be, according to (67),

−2nm cos2 θ c2 dσ.

(69)

Should the reﬂecting body be free to move in the direction of the normal of the reﬂecting surface and should there be no force acting on it except the impact of the light particles, it would be set into motion by the impacts. According to the law of action and reaction the ensuing motion would be such that the momentum acquired in a certain interval of time would be equal and opposite to the change in momentum of all the light particles reﬂected from it in the same time interval. But if we allow a separate constant force to act from outside on the reﬂector, there is to be added to the change in momenta of the light particles the impulse of the external force, i.e., the product of the force and the time interval in question.
Therefore the reﬂector will remain continuously at rest, whenever the constant external force exerted on it is so chosen that its impulse for any time is just equal to the change in momentum of all the particles reﬂected from the reﬂector in the same time. Thus it follows that the force F itself which the particles exert by their impact on the surface

DEDUCTIONS FROM ELECTRODYNAMICS

68

element dσ is equal and opposite to the change of their momentum in unit time as expressed in (69)

F = 2nm cos2 θ c2 dσ

and by making use of (68),

4 cos θ

F=

I.

c

On comparing this relation with equation (64) in which all symbols have the same physical signiﬁcance, it is seen that Newton’s radiation pressure is twice as large as Maxwell’s for the same energy radiation. A necessary consequence of this is that the magnitude of Maxwell’s radiation pressure cannot be deduced from general energetic considerations, but is a special feature of the electromagnetic theory and hence all deductions from Maxwell’s radiation pressure are to be regarded as consequences of the electromagnetic theory of light and all conﬁrmations of them are conﬁrmations of this special theory.

CHAPTER II
STEFAN-BOLTZMANN LAW OF RADIATION
61. For the following we imagine a perfectly evacuated hollow cylinder with an absolutely tight-ﬁtting piston free to move in a vertical direction with no friction. A part of the walls of the cylinder, say the rigid bottom, should consist of a black body, whose temperature T may be regulated arbitrarily from the outside. The rest of the walls including the inner surface of the piston may be assumed as totally reﬂecting. Then, if the piston remains stationary and the temperature, T , constant, the radiation in the vacuum will, after a certain time, assume the character of black radiation (Sec. 50) uniform in all directions. The speciﬁc intensity, K, and the volume density, u, depend only on the temperature, T , and are independent of the volume, V , of the vacuum and hence of the position of the piston.
If now the piston is moved downward, the radiation is compressed into a smaller space; if it is moved upward the radiation expands into a larger space. At the same time the temperature of the black body forming the bottom may be arbitrarily changed by adding or removing heat from the outside. This always causes certain disturbances of the stationary state. If, however, the arbitrary changes in V and T are made suﬃciently slowly, the departure from the conditions of a stationary state may always be kept just as small as we please. Hence the state of radiation in the vacuum may, without appreciable error, be regarded as a state of thermodynamic equilibrium, just as is done in the thermodynamics of ordinary matter in the case of so-called inﬁnitely slow processes, where, at any instant, the divergence from the state of equilibrium may be neglected, compared with the changes which the total system considered undergoes as a result of the entire process.
69

DEDUCTIONS FROM ELECTRODYNAMICS

70

If, e.g., we keep the temperature T of the black body forming the bottom constant, as can be done by a suitable connection between it and a heat reservoir of large capacity, then, on raising the piston, the black body will emit more than it absorbs, until the newly made space is ﬁlled with the same density of radiation as was the original one. Vice versa, on lowering the piston the black body will absorb the superﬂuous radiation until the original radiation corresponding to the temperature T is again established. Similarly, on raising the temperature T of the black body, as can be done by heat conduction from a heat reservoir which is slightly warmer, the density of radiation in the vacuum will be correspondingly increased by a larger emission, etc. To accelerate the establishment of radiation equilibrium the reﬂecting mantle of the hollow cylinder may be assumed white (Sec. 10), since by diﬀuse reﬂection the predominant directions of radiation that may, perhaps, be produced by the direction of the motion of the piston, are more quickly neutralized. The reﬂecting surface of the piston, however, should be chosen for the present as a perfect metallic reﬂector, to make sure that the radiation pressure (66) on the piston is Maxwell’s. Then, in order to produce mechanical equilibrium, the piston must be loaded by a weight equal to the product of the radiation pressure p and the cross-section of the piston. An exceedingly small diﬀerence of the loading weight will then produce a correspondingly slow motion of the piston in one or the other direction.
Since the eﬀects produced from the outside on the system in question, the cavity through which the radiation travels, during the processes we are considering, are partly of a mechanical nature (displacement of the loaded piston), partly of a thermal nature (heat conduction away from and toward the reservoir), they show a certain similarity to the processes usually considered in thermodynamics, with the diﬀerence that the system here considered is not a material system, e.g., a

STEFAN-BOLTZMANN LAW OF RADIATION

71

gas, but a purely energetic one. If, however, the principles of thermodynamics hold quite generally in nature, as indeed we shall assume, then they must also hold for the system under consideration. That is to say, in the case of any change occurring in nature the energy of all systems taking part in the change must remain constant (ﬁrst principle), and, moreover, the entropy of all systems taking part in the change must increase, or in the limiting case of reversible processes must remain constant (second principle).
62. Let us ﬁrst establish the equation of the ﬁrst principle for an inﬁnitesimal change of the system in question. That the cavity enclosing the radiation has a certain energy we have already (Sec. 22) deduced from the fact that the energy radiation is propagated with a ﬁnite velocity. We shall denote the energy by U . Then we have

U = V u,

(70)

where u the volume density of radiation depends only on the temperature T of the black body at the bottom.
The work done by the system, when the volume V of the cavity increases by dV against the external forces of pressure (weight of the loaded piston), is p dV , where p represents Maxwell’s radiation pressure (66). This amount of mechanical energy is therefore gained by the surroundings of the system, since the weight is raised. The error made by using the radiation pressure on a stationary surface, whereas the reﬂecting surface moves during the volume change, is evidently negligible, since the motion may be thought of as taking place with an arbitrarily small velocity.
If, moreover, Q denotes the inﬁnitesimal quantity of heat in mechanical units, which, owing to increased emission, passes from the black body at the bottom to the cavity containing the radiation, the bottom or the heat reservoir connected to it loses this heat Q, and

DEDUCTIONS FROM ELECTRODYNAMICS

72

its internal energy is decreased by that amount. Hence, according to

the ﬁrst principle of thermodynamics, since the sum of the energy of

radiation and the energy of the material bodies remains constant, we

have

dU + p dV − Q = 0.

(71)

According to the second principle of thermodynamics the cavity containing the radiation also has a deﬁnite entropy. For when the heat Q passes from the heat reservoir into the cavity, the entropy of the reservoir decreases, the change being

Q −.
T

Therefore, since no changes occur in the other bodies—inasmuch

as the rigid absolutely reﬂecting piston with the weight on it does not

change its internal condition with the motion—there must somewhere Q
in nature occur a compensation of entropy having at least the value , T
by which the above diminution is compensated, and this can be nowhere

except in the entropy of the cavity containing the radiation. Let the

entropy of the latter be denoted by S.

Now, since the processes described consist entirely of states of equi-

librium, they are perfectly reversible and hence there is no increase in

entropy. Then we have

Q

dS − = 0,

(72)

T

or from (71)

dU + p dV

dS =

.

(73)

T

In this equation the quantities U , p, V , S represent certain proper-

ties of the heat radiation, which are completely deﬁned by the instanta-

neous state of the radiation. Therefore the quantity T is also a certain

STEFAN-BOLTZMANN LAW OF RADIATION

73

property of the state of the radiation, i.e., the black radiation in the cavity has a certain temperature T and this temperature is that of a body which is in heat equilibrium with the radiation.
63. We shall now deduce from the last equation a consequence which is based on the fact that the state of the system considered, and therefore also its entropy, is determined by the values of two independent variables. As the ﬁrst variable we shall take V , as the second either T , u, or p may be chosen. Of these three quantities any two are determined by the third together with V . We shall take the volume V and the temperature T as independent variables. Then by substituting from (66) and (70) in (73) we have

V du

4u

dS =

dT + dV.

(74)

T dT

3T

From this we obtain

∂S

V du

=

∂T V T dT

∂S

4u

=.

∂V T 3T

On partial diﬀerentiation of these equations, the ﬁrst with respect to V , the second with respect to T , we ﬁnd

∂2S 1 du 4 du 4u

∂T ∂V

= T

dT

= 3T

dT

− 3T 2

or du 4u = dT T

and on integration

u = aT 4

(75)

and from (21) for the speciﬁc intensity of black radiation

K = c · u = ac T 4.

(76)

4π

4π

DEDUCTIONS FROM ELECTRODYNAMICS

74

Moreover for the pressure of black radiation

p = aT4,

(77)

3

and for the total radiant energy

U = aT 4 · V.

(78)

This law, which states that the volume density and the speciﬁc intensity of black radiation are proportional to the fourth power of the absolute temperature, was ﬁrst established by J. Stefan1 on a basis of rather rough measurements. It was later deduced by L. Boltzmann2 on a thermodynamic basis from Maxwell’s radiation pressure and has been more recently conﬁrmed by O. Lummer and E. Pringsheim3 by exact measurements between 100◦ and 1300◦ C., the temperature being deﬁned by the gas thermometer. In ranges of temperature and for requirements of precision for which the readings of the diﬀerent gas thermometers no longer agree suﬃciently or cannot be obtained at all, the Stefan-Boltzmann law of radiation can be used for an absolute deﬁnition of temperature independent of all substances.
64. The numerical value of the constant a is obtained from measurements made by F. Kurlbaum.4 According to them, if we denote by St the total energy radiated in one second into air by a square centimeter of a black body at a temperature of t◦ C., the following equation holds

S100

− S0

=

0.0731

watt cm2

=

7.31

×

105

erg .
cm2 sec

(79)

1J. Stefan, Wien. Berichte, 79, p. 391, 1879. 2L. Boltzmann, Wied. Annalen, 22, p. 291, 1884. 3O. Lummer und E. Pringsheim, Wied. Annalen, 63, p. 395, 1897. Annalen d.
Physik 3, p. 159, 1900. 4F. Kurlbaum, Wied. Annalen, 65, p. 759, 1898.

STEFAN-BOLTZMANN LAW OF RADIATION

75

Now, since the radiation in air is approximately identical with the radiation into a vacuum, we may according to (7) and (76) put

St

=

πK

=

ac (273
4

+

t)4

and from this therefore

S100

− S0

=

ac (3734 4

−

2734),

a

=

3

×

4 × 7.31 × 105 1010 × (3734 − 2734)

=

7.061

×

10−15

cm3

erg degree4 .

Recently Kurlbaum has increased the value measured by him by 2.5 per cent.,1 on account of the bolometer used being not perfectly black, whence it follows that a = 7.24 · 10−15.
Meanwhile the radiation constant has been made the object of as accurate measurements as possible in various places. Thus it was measured by F´ery, Bauer and Moulin, Valentiner, F´ery and Drecq, Shakespear, Gerlach, with in some cases very divergent results, so that a mean value may hardly be formed.
For later computations we shall use the most recent determination made in the physical laboratory of the University of Berlin2

ac 4

=

σ

=

5.46

·

10−12

watt cm2 degree4 .

1F. Kurlbaum, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 580, 1912.
2According to private information kindly furnished by my colleague H. Rubens (July, 1912). (These results have since been published. See W. H. Westphal, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 987, 1912, Tr.)

DEDUCTIONS FROM ELECTRODYNAMICS

76

From this a is found to be

a

=

4 · 5.46 · 10−12 · 107 3 · 1010

=

7.28 · 10−15

erg cm3 degree4

which agrees rather closely with Kurlbaum’s corrected value. 65. The magnitude of the entropy S of black radiation found by
integration of the diﬀerential equation (73) is

S = 4 aT 3V.

(80)

3

In this equation the additive constant is determined by a choice that readily suggests itself, so that at the zero of the absolute scale of temperature, that is to say, when u vanishes, S shall become zero. From this the entropy of unit volume or the volume density of the entropy of black radiation is obtained,

S = s = 4 aT 3.

(81)

V

3

66. We shall now remove a restricting assumption made in order to enable us to apply the value of Maxwell’s radiation pressure, calculated in the preceding chapter. Up to now we have assumed the cylinder to be ﬁxed and only the piston to be free to move. We shall now think of the whole of the vessel, consisting of the cylinder, the black bottom, and the piston, the latter attached to the walls in a deﬁnite height above the bottom, as being free to move in space. Then, according to the principle of action and reaction, the vessel as a whole must remain constantly at rest, since no external force acts on it. This is the conclusion to which we must necessarily come, even without, in this case, admitting a priori the validity of the principle of action and reaction. For if the vessel

STEFAN-BOLTZMANN LAW OF RADIATION

77

should begin to move, the kinetic energy of this motion could originate only at the expense of the heat of the body forming the bottom or the energy of radiation, as there exists in the system enclosed in a rigid cover no other available energy; and together with the decrease of energy the entropy of the body or the radiation would also decrease, an event which would contradict the second principle, since no other changes of entropy occur in nature. Hence the vessel as a whole is in a state of mechanical equilibrium. An immediate consequence of this is that the pressure of the radiation on the black bottom is just as large as the oppositely directed pressure of the radiation on the reﬂecting piston. Hence the pressure of black radiation is the same on a black as on a reﬂecting body of the same temperature and the same may be readily proven for any completely reﬂecting surface whatsoever, which we may assume to be at the bottom of the cylinder without in the least disturbing the stationary state of radiation. Hence we may also in all the foregoing considerations replace the reﬂecting metal by any completely reﬂecting or black body whatsoever, at the same temperature as the body forming the bottom, and it may be stated as a quite general law that the radiation pressure depends only on the properties of the radiation passing to and fro, not on the properties of the enclosing substance.
67. If, on raising the piston, the temperature of the black body forming the bottom is kept constant by a corresponding addition of heat from the heat reservoir, the process takes place isothermally. Then, along with the temperature T of the black body, the energy density u, the radiation pressure p, and the density of the entropy s also remain constant; hence the total energy of radiation increases from U = uV to U = uV , the entropy from S = sV to S = sV and the heat supplied from the heat reservoir is obtained by integrating (72) at constant T ,
Q = T (S − S) = T s(V − V )

DEDUCTIONS FROM ELECTRODYNAMICS

78

or, according to (81) and (75),

Q = 4 aT 4(V

4 − V ) = (U

− U ).

3

3

Thus it is seen that the heat furnished from the outside exceeds the

increase in energy of radiation (U

− U ) by

1 3

(U

− U ).

This excess in

the added heat is necessary to do the external work accompanying the

increase in the volume of radiation.

68. Let us also consider a reversible adiabatic process. For this it

is necessary not merely that the piston and the mantle but also that

the bottom of the cylinder be assumed as completely reﬂecting, e.g., as

white. Then the heat furnished on compression or expansion of the

volume of radiation is Q = 0 and the energy of radiation changes only

by the value p dV of the external work. To insure, however, that in a

ﬁnite adiabatic process the radiation shall be perfectly stable at every

instant, i.e., shall have the character of black radiation, we may assume

that inside the evacuated cavity there is a carbon particle of minute size.

This particle, which may be assumed to possess an absorbing power

diﬀering from zero for all kinds of rays, serves merely to produce stable

equilibrium of the radiation in the cavity (Sec. 51 et seq.) and thereby

to insure the reversibility of the process, while its heat contents may

be taken as so small compared with the energy of radiation, U , that

the addition of heat required for an appreciable temperature change of

the particle is perfectly negligible. Then at every instant in the process

there exists absolutely stable equilibrium of radiation and the radiation

has the temperature of the particle in the cavity. The volume, energy,

and entropy of the particle may be entirely neglected.

On a reversible adiabatic change, according to (72), the entropy S

of the system remains constant. Hence from (80) we have as a condition

STEFAN-BOLTZMANN LAW OF RADIATION

79

for such a process

T 3V = const.,

or, according to (77),

4
pV 3 = const.,

i.e., on an adiabatic compression the temperature and the pressure of the radiation increase in a manner that may be deﬁnitely stated. The energy of the radiation, U , in such a case varies according to the law
U3 = S = const.,
T4 i.e., it increases in proportion to the absolute temperature, although the volume becomes smaller.
69. Let us ﬁnally, as a further example, consider a simple case of an irreversible process. Let the cavity of volume V , which is everywhere enclosed by absolutely reﬂecting walls, be uniformly ﬁlled with black radiation. Now let us make a small hole through any part of the walls, e.g., by opening a stopcock, so that the radiation may escape into another completely evacuated space, which may also be surrounded by rigid, absolutely reﬂecting walls. The radiation will at ﬁrst be of a very irregular character; after some time, however, it will assume a stationary condition and will ﬁll both communicating spaces uniformly, its total volume being, say, V . The presence of a carbon particle will cause all conditions of black radiation to be satisﬁed in the new state. Then, since there is neither external work nor addition of heat from the outside, the energy of the new state is, according to the ﬁrst principle, equal to that of the original one, or U = U and hence from (78)
T 4V = T 4V

T =4 V

T

V

DEDUCTIONS FROM ELECTRODYNAMICS

80

which deﬁnes completely the new state of equilibrium. Since V > V the temperature of the radiation has been lowered by the process.
According to the second principle of thermodynamics the entropy of the system must have increased, since no external changes have occurred; in fact we have from (80)

S

T 3V =

=4V

> 1.

(82)

S T 3V

V

70. If the process of irreversible adiabatic expansion of the radiation from the volume V to the volume V takes place as just described with the single diﬀerence that there is no carbon particle present in the vacuum, after the stationary state of radiation is established, as will be the case after a certain time on account of the diﬀuse reﬂection from the walls of the cavity, the radiation in the new volume V will not any longer have the character of black radiation, and hence no deﬁnite temperature. Nevertheless the radiation, like every system in a deﬁnite physical state, has a deﬁnite entropy, which, according to the second principle, is larger than the original S, but not as large as the S given in (82). The calculation cannot be performed without the use of laws to be taken up later (see Sec. 103). If a carbon particle is afterward introduced into the vacuum, absolutely stable equilibrium is established by a second irreversible process, and, the total energy as well as the total volume remaining constant, the radiation assumes the normal energy distribution of black radiation and the entropy increases to the maximum value S given by (82).

CHAPTER III
WIEN’S DISPLACEMENT LAW
71. Though the manner in which the volume density u and the speciﬁc intensity K of black radiation depend on the temperature is determined by the Stefan-Boltzmann law, this law is of comparatively little use in ﬁnding the volume density uν corresponding to a deﬁnite frequency ν, and the speciﬁc intensity of radiation Kν of monochromatic radiation, which are related to each other by equation (24) and to u and K by equations (22) and (12). There remains as one of the principal problems of the theory of heat radiation the problem of determining the quantities uν and Kν for black radiation in a vacuum and hence, according to (42), in any medium whatever, as functions of ν and T , or, in other words, to ﬁnd the distribution of energy in the normal spectrum for any arbitrary temperature. An essential step in the solution of this problem is contained in the so-called “displacement law” stated by W. Wien,1 the importance of which lies in the fact that it reduces the functions uν and Kν of the two arguments ν and T to a function of a single argument.
The starting point of Wien’s displacement law is the following theorem. If the black radiation contained in a perfectly evacuated cavity with absolutely reﬂecting walls is compressed or expanded adiabatically and inﬁnitely slowly, as described above in Sec. 68, the radiation always retains the character of black radiation, even without the presence of a
1W. Wien, Sitzungsberichte d. Akad. d. Wissensch. Berlin, Febr. 9, 1893, p. 55. Wiedemann’s Annal., 52, p. 132, 1894. See also among others M. Thiesen, Verhandl. d. Deutsch. phys. Gesellsch., 2, p. 65, 1900. H. A. Lorentz, Akad. d. Wissensch. Amsterdam, May 18, 1901, p. 607. M. Abraham, Annal. d. Physik 14, p. 236, 1904.
81

DEDUCTIONS FROM ELECTRODYNAMICS

82

carbon particle. Hence the process takes place in an absolute vacuum just as was calculated in Sec. 68 and the introduction, as a precaution, of a carbon particle is shown to be superﬂuous. But this is true only in this special case, not at all in the case described in Sec. 70.
The truth of the proposition stated may be shown as follows: Let the completely evacuated hollow cylinder, which is at the start ﬁlled with black radiation, be compressed adiabatically and inﬁnitely slowly to a ﬁnite fraction of the original volume. If, now, the compression being completed, the radiation were no longer black, there would be no stable thermodynamic equilibrium of the radiation (Sec. 51). It would then be possible to produce a ﬁnite change at constant volume and constant total energy of radiation, namely, the change to the absolutely stable state of radiation, which would cause a ﬁnite increase of entropy. This change could be brought about by the introduction of a carbon particle, containing a negligible amount of heat as compared with the energy of radiation. This change, of course, refers only to the spectral density of radiation uν, whereas the total density of energy u remains constant. After this has been accomplished, we could, leaving the carbon particle in the space, allow the hollow cylinder to return adiabatically and inﬁnitely slowly to its original volume and then remove the carbon particle. The system will then have passed through a cycle without any external changes remaining. For heat has been neither added nor removed, and the mechanical work done on compression has been regained on expansion, because the latter, like the radiation pressure, depends only on the total density u of the energy of radiation, not on its spectral distribution. Therefore, according to the ﬁrst principle of thermodynamics, the total energy of radiation is at the end just the same as at the beginning, and hence also the temperature of the black radiation is again the same. The carbon particle and its changes do not enter into the calculation, for its energy and entropy are vanish-

WIEN’S DISPLACEMENT LAW

83

ingly small compared with the corresponding quantities of the system. The process has therefore been reversed in all details; it may be repeated any number of times without any permanent change occurring in nature. This contradicts the assumption, made above, that a ﬁnite increase in entropy occurs; for such a ﬁnite increase, once having taken place, cannot in any way be completely reversed. Therefore no ﬁnite increase in entropy can have been produced by the introduction of the carbon particle in the space of radiation, but the radiation was, before the introduction and always, in the state of stable equilibrium.
72. In order to bring out more clearly the essential part of this important proof, let us point out an analogous and more or less obvious consideration. Let a cavity containing originally a vapor in a state of saturation be compressed adiabatically and inﬁnitely slowly.
“Then on an arbitrary adiabatic compression the vapor remains always just in the state of saturation. For let us suppose that it becomes supersaturated on compression. After the compression to an appreciable fraction of the original volume has taken place, condensation of a ﬁnite amount of vapor and thereby a change into a more stable state, and hence a ﬁnite increase of entropy of the system, would be produced at constant volume and constant total energy by the introduction of a minute drop of liquid, which has no appreciable mass or heat capacity. After this has been done, the volume could again be increased adiabatically and inﬁnitely slowly until again all liquid is evaporated and thereby the process completely reversed, which contradicts the assumed increase of entropy.”
Such a method of proof would be erroneous, because, by the process described, the change that originally took place is not at all completely reversed. For since the mechanical work expended on the compression of the supersaturated steam is not equal to the amount gained on expanding the saturated steam, there corresponds to a deﬁnite volume of

DEDUCTIONS FROM ELECTRODYNAMICS

84

the system when it is being compressed an amount of energy diﬀerent from the one during expansion and therefore the volume at which all liquid is just vaporized cannot be equal to the original volume. The supposed analogy therefore breaks down and the statement made above in quotation marks is incorrect.
73. We shall now again suppose the reversible adiabatic process described in Sec. 68 to be carried out with the black radiation contained in the evacuated cavity with white walls and white bottom, by allowing the piston, which consists of absolutely reﬂecting metal, to move downward inﬁnitely slowly, with the single diﬀerence that now there shall be no carbon particle in the cylinder. The process will, as we now know, take place exactly as there described, and, since no absorption or emission of radiation takes place, we can now give an account of the changes of color and intensity which the separate pencils of the system undergo. Such changes will of course occur only on reﬂection from the moving metallic reﬂector, not on reﬂection from the stationary walls and the stationary bottom of the cylinder.
If the reﬂecting piston moves down with the constant, inﬁnitely small, velocity v, the monochromatic pencils striking it during the motion will suﬀer on reﬂection a change of color, intensity, and direction. Let us consider these diﬀerent inﬂuences in order.1
74. To begin with, we consider the change of color which a monochromatic ray suﬀers by reﬂection from the reﬂector, which is moving with an inﬁnitely small velocity. For this purpose we consider ﬁrst the case of a ray which falls normally from below on the reﬂector
1The complete solution of the problem of reﬂection of a pencil from a moving absolutely reﬂecting surface including the case of an arbitrarily large velocity of the surface may be found in the paper by M. Abraham quoted in Sec. 71. See also the text-book by the same author. Electromagnetische Theorie der Strahlung, 1908 (Leipzig, B. G. Teubner).

WIEN’S DISPLACEMENT LAW

85

Fig. 5.

and hence is reﬂected normally downward. Let the plane A (Fig. 5)

represent the position of the reﬂector at the time t, the plane A the

position at the time t + δt, where the distance AA equals v δt, v de-

noting the velocity of the reﬂector. Let us now suppose a stationary

plane B to be placed parallel to A at a suitable distance and let us

denote by λ the wave length of the ray incident on the reﬂector and

by λ the wave length of the ray reﬂected from it. Then at a time t

there are in the interval AB in the vacuum containing the radiation

AB

AB

waves of the incident and waves of the reﬂected ray, as can

λ

λ

be seen, e.g., by thinking of the electric ﬁeld-strength as being drawn

at the diﬀerent points of each of the two rays at the time t in the form

of a sine curve. Reckoning both incident and reﬂected ray there are at

the time t

11 AB +
λλ

waves in the interval between A and B. Since this is a large number, it is immaterial whether the number is an integer or not.

DEDUCTIONS FROM ELECTRODYNAMICS

86

Similarly at the time t + δt, when the reﬂector is at A , there are

11 AB +
λλ

waves in the interval between A and B all told. The latter number will be smaller than the former, since in the
shorter distance A B there is room for fewer waves of both kinds than in the longer distance AB. The remaining waves must have been expelled in the time δt from the space between the stationary plane B and the moving reﬂector, and this must have taken place through the plane B downward; for in no other way could a wave disappear from the space considered.
Now ν δt waves pass in the time δt through the stationary plane B in an upward direction and ν δt waves in a downward direction; hence we have for the diﬀerence

11 (ν − ν) δt = (AB − A B) +
λλ

or, since

AB − A B = v δt,

and

c

c

λ= λ =

ν

ν

c+v

ν=

ν

c−v

or, since v is inﬁnitely small compared with c,

2v ν =ν 1+ .
c