OF THE UNIVERSITY OF
Rtf
ASTRONOMY DEPT ,
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 JjAASUJS, 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
SEP 29
ASTRONOMY DEFT;
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 quan- tum 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 resume 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 formula) where the treatment in the book seemed too brief or to present some difficulties.
vi TRANSLATOR'S PREFACE
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 manu- script and the galley proof.
MORTON MASIUS. WORCESTER, MASS., February, 1914.
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 radi- ation 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 hypothe- sis 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 ele- mentary quanta of a definite finite magnitude. For the hypoth- esis of quanta as well as the heat theorem of Nernst may be re- duced to the simple proposition that the thermodynamic proba- bility (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
vii
viii PREFACE TO SECOND EDITION
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 satis- factorily completed. While many physicists, through conserva- tism, 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 com- pelled 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 be introduced, an assumption which is in direct contradiction to the fundamental ideas of classical electro- dynamics. 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 calcula- tion, 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
PREFACE TO SECOND EDITION ix
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 satis- factory 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 ele- mentary physical processes is made clear in every respect or may be demonstrated by some simple dynamical model, misunder- stands, I believe, the character and the meaning of the hy- pothesis of quanta. It is impossible to express a really new principle in terms of a model following old laws. And, as re- gards the final formulation of the hypothesis, we should not forget that, from the classical point of view, the physics of the atom really has alwrays remained a very obscure, inacces- sible 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.
THE AUTHOR. BERLIN,
November, 1912.
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 be- came evident, however, that it was desirable to include also the foundation of this theory in the treatment, starting with Kirch- hoff'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 KirchhofFs 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.
XI
TABLE OF CONTENTS
PART I FUNDAMENTAL FACTS AND DEFINITIONS
CHAPTER PAGE
I. General Introduction 1
II. Radiation at Thermodynamic Equilibrium. Kirchhoff's Law.
Black Radiation 22
PART II
DEDUCTIONS FROM ELECTRODYNAMICS AND THERMODYNAMICS
r I. Maxwell's Radiation Pressure 49
II. Stefan-Boltzmann Law of Radiation 59
III. Wien' s Displacement Law 69
IV. Radiation of any Arbitrary Spectral Distribution of Energy. Entropy and Temperature of Monochromatic Radiation. . . 87
V. Electrodynamical Processes in a Stationary Field of Radiation . . 103
PART III ENTROPY AND PROBABILITY
I. Fundamental Definitions and Laws. Hypothesis of Quanta . . 113 II. Ideal Monatomic Gases 127
III. Ideal Linear Oscillators 135
IV. Direct Calculation of the Entropy in the Case of Thermodynamic Equilibrium .144
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 151
II. Absorbed Energy 155
III. Emitted Energy. Stationary State .161
IV. The Law of the Normal Distribution of Energy. Elementary Quanta of Matter and of Electricity 167
xiii
xiv TABLE OF CONTENTS
PART V IRREVERSIBLE RADIATION PROCESSES
r I. Fields of Radiation in General 189
II. One Oscillator in the Field of Radiation 196
III. A System of Oscillators 200
IV. Conservation of Energy and Increase of Entropy. Conclusion . . 205 List of Papers on Heat Radiation and the Hypothesis of Quanta
by the Author 216
Appendices 218
Errata . . 225
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 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.
1
2 FUNDAMENTAL FACTS AND DEFINITIONS
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 radia- tion," 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 experi- mental 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 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 trans-
GENERAL INTRODUCTION 3
mitted 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, obvi- ously 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 inten- sity of a ray of constant period and amplitude would not be inde- pendent 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 peri- odic 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 "vibrations." 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 suffi- ciently 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
4 FUNDAMENTAL FACTS AND DEFINITIONS
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 sup- posed 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 conserva- tion of energy, emission always takes place at the expense of other forms of 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 volume-element dr 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 dr. Should we attempt a closer analysis of the process of emission and resolve it into its elements, we should undoubtedly meet very complicated con- ditions, 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
1 Here as in the following the German "Korperwarme" will be rendered simply as "heat." (Tr.)
GENERAL INTRODUCTION 5
obtained by dividing the radiation emitted by a volume-element dr by this element dr is to be considered only as a certain mean value. Nevertheless, we shall as a rule be able to treat the phe- nomenon of emission as if all points of the volume-element dr took part in the emission in a uniform manner, thereby greatly simplifying our calculation. Every point of dr 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 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 dr 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 in- finitely small and of size dti we maj^ speak of the radiation emitted in a certain direction, but always in the sense that for the emis- sion 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 experi- mental 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 v 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 cer- tain color by the corresponding value of v, and a certain interval of color by the limits of the interval v and /, where /> v. The radiation lying in a certain interval of color divided by the magni- tude v'-v of the interval, we shall call the mean radiation in the interval v to /. We shall then assume that if, keeping v constant,
6 FUNDAMENTAL FACTS AND DEFINITIONS
we take the interval v'-v sufficiently small and denote it by dv the value of the mean radiation approaches a definite limiting value, independent of the size of dv, and this we shall briefly call the "radiation of frequency v." 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 NicoVs prism.
6. Summing up everything said so far, we may equate the total energy in a range of frequency from v to v-\-dv emitted in the time dt in the direction of the conical element cZl2 by a volume element dr to
The finite quantity e, is called the coefficient of emission of the medium for the frequency v. It is a positive function of v and refers to a plane polarized ray of definite color and direction. The total emission of the volume-element dr may be obtained from this by integrating over all directions and all frequencies. Since €„ is independent of the direction, and since the integral over all conical elements dti is 4rr, we get:
00
dt-dr.S* j tvdv. ^ (2)
7. The coefficient of emission e depends, not only on the fre- quency v, but also on the condition of the emitting substance contained in the volume-element dr, 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 ele- ment holds true in all cases (Prevost's principle). 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
GENERAL INTRODUCTION 7
by B and heated by Br 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 sub- stance depends on but a single variable, namely, on its absolute temperature T. A necessary consequence of this is that the coefficient of emission e depends, apart from the frequency v 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 com^ pletely 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, diffrac- tion 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 phenome- non of " scattering," which means neither a creation nor a destruction of radiant energy but simply 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 homogene- ous 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, appre- ciably affecting its general character. Hence, so-called ''turbid"
i See, e.g., Lobry de Bruyn and L. K. Wolff, Rec. des Trav. China, des Paya-Bas 23, p. 155, 1904.
8 FUNDAMENTAL FACTS AND DEFINITIONS
media, i.e., such as contain foreign particles, may be quite prop- erly regarded as optically homogeneous,1 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 phenom- ena, 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 atmos- phere as by the air molecules themselves.
Whether the scattering depends on reflection, on diffraction, or on 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 frac- tion is proportional to s, say
As (3)
where the positive quantity $v is independent of the intensity of radiation and is called the "coefficient of scattering" of the me- dium. Inasmuch as the medium is assumed to be isotropic, fa is also independent of the direction of propagation and polariza- tion of the ray. It depends, however, as indicated by the subscript v, 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 v, &v may be so large that the straight-line propagation of the rays is virtually destroyed. For other values of v, however, 0, may become so small that the scattering can
1 To restrict the word homogeneous to its absolute sense would mean that it could not be applied to any material substance.
GENERAL INTRODUCTION 9
be entirely neglected. For generality we shall assume a mean value of ft,. In the cases of most importance ft increases quite appreciably as v 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 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 trans- mitted 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 intensi- ties 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
i Lord Rayleigh, Phil. Mag., 47, p. 379, 1899.
10 FUNDAMENTAL FACTS AND DEFINITIONS
called " white." The other extreme, namely, complete trans- mission 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. Kirchhoff1 it denotes a body which has the property 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 ab- sorbing a body is, the smaller the value of this minimum thick- ness, 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.2
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
1 G. Kirchhoff, Pogg. Ann., 109, p. 275, 1860. Gesammelte Abhandlungen, J. A. Earth, 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.
2 For 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.
GENERAL INTRODUCTION 11
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 which may be real- ized 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 some- times 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 frac- tion 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
«,* (4)
Here av is known as the " coefficient of absorption" of the me- dium for a ray of frequency v. 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 ho- mogeneous isotropic substances, and shall therefore suppose that av has the same value at all points and in all directions in the medium, and depends on nothing but the frequency v, the tem- perature T, and the nature of the medium.
Whenever av does not differ from zero except for a limited range of the spectrum, the medium shows "selective" absorption. For those colors for which av = 0 and also the coefficient of scattering ^ = 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 «„. This implies that the absorption in a distance equal to a single wave length is very small, because the distance s, while small, contains many wave lengths (Sec. 2).
12 FUNDAMENTAL FACTS AND DEFINITIONS
13. The foregoing considerations regarding the emission, the propagation, and the absorption of heat rays suffice for a mathe- matical 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 da 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 da, the length of time dt and to the cosine of the angle 6 made by the normal of do- with the direction of the radiation. If we make da sufficiently small, then, although this is only an approximation to the actual state of affairs, we can think of all points in da as being affected by the radiation in the same way. Then the energy radiated through da in a definite direction must be pro- portional to the solid angle in which da intercepts that radiation and this solid angle is measured by da cos 6. 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
.
Now in general a pencil of rays is propagated from every point of the element da 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
GENERAL INTRODUCTION 13
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 da as vertex in all direc-' tions of space and on both sides of do-. A certain direction may be specified by the angle 9 (between 0 and TT), as already used, and by an azimuth <£ (between 0 and 2ii) . The intensity in this direction is the energy propagated in an infinitely thin cone lim- ited by 6 and B+dB and </> and 0+d0. The solid angle of this cone is
dfi = sin B'dB'dQ. (5)
Thus the energy radiated in time dt through the element of area da in the direction of the cone d£l is:
dt da cos ddttK = K sin B cos B dd d<f> da dt. (6)
The finite quantity K we shall term the "specific intensity" or the " brightness," d®, the "solid angle" of the pencil emanating from a point of the element da in the direction (0, <£). K is a positive function of position, time, and the angles B and 4>. In general the specific intensities of radiation in different directions are entirely independent of one another. For example, on sub- stituting TT — B for B and TT+ <£ for <£ 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 da toward one side, say the one on which B is an acute angle, we get, by integrating with respect to 0 from 0 to 2?r and with respect to
B from 0 to
id* r
t/ o t/ o
dBK sin B cos B da dt.
14 FUNDAMENTAL FACTS AND DEFINITIONS
Should the radiation be uniform in all directions and hence K be a constant, the total radiation on one side will be
TT K da dt. (7)
16. In speaking of the radiation in a definite direction (6, 0) 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 spec- trum which travel independently of one another. Hence we consider the intensity of radiation within a certain range of fre- quencies, say from v to /. If the interval v'—v be taken suffi- ciently small and be denoted by dv, the intensity of radiation within the interval is proportional to dv. 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 v, 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 posi- tion be denoted by the azimuth \l/ 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
and K,sinV+K/cosV (8)
Herein K is independent of \f/. These expressions we shall call
GENERAL INTRODUCTION 15
the " components of the specific intensity of radiation of frequency v." The sum is independent of \f/ and is always equal to the intensity of the whole ray K,, + K/. At the same time Kv and K/ represent respectively the largest and smallest values which
either of the components may have, namely, when \f/ = 0 and \f/ = ~ •
Hence we call these values the " principal values of the intensi- ties/' 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
•S:
(9)
where the positive quantities K,, and K/, the two principal values of the specific intensity of the radiation (brightness) of fre- quency v, depend not only on v but also on their position, the time, and on the angles 6 and <£. By substitution in (6) the energy radiated in the time dt through the element of area da in the direc- tion of the conical element d& assumes the value
00
dt da cos 6 dtt [dv (K,+ K/) (10)
I
and for monochromatic plane polarized radiation of brightness K,:
dt da cos B dtt K,, dv = dt da- sin 6 cos 0 dd d$ K,, dv. (11) For unpolarized rays K,, = K/, and hence
oo
K = 2 (dv K,, (12)
I \dv K
and the energy of a monochromatic ray of frequency v will be: 2dt da- cos e dQ K, dv = 2dt da- sin 6 cos 6 dd d<f> K, dv.(l3) When, moreover, the radiation is uniformly distributed in all directions, the total radiation through dcr toward one side may be found from (7) and (12) ; it is
I
K>. (14)
16 FUNDAMENTAL FACTS AND DEFINITIONS
18. Since in nature K,, can never be infinitely large, K will not have a finite value unless K,, differs from zero over a finite range of frequencies. Hence there exists in nature no absolutely homogeneous or monochromatic radiation of light or heat. A finite amount of radiation contains always a finite although possi- bly very narrow range of the spectrum. This implies a funda- mental difference from the corresponding phenomena 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 important 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 K,,, the intensity of radiation of frequency v, and the quantity K, the intensity of radiation of the whole spectrum, are of different dimensions. Further it is to be noticed that, on subdividing the spectrum according to wave lengths X, instead of frequencies v, the intensity of radiation #\.of the wave lengths X correspond- ing to the frequency v is not obtained simply by replacing v in the expression for K,, by the corresponding value of X deduced from
V = I (15)
A
where q is the velocity of propagation. For if d\ and dv refer to the same interval of the spectrum, we have, not Ex = K,,, but Ex d\ = K, dv. By differentiating (15) and paying attention to the signs of corresponding values of d\ and dv the equation
is obtained. Hence we get by substitution:
This relation shows among other things that in a certain spectrum the maxima of Ex and K,, lie at different points of the spectrum. 20. When the principal intensities K, and K/ of all mono- chromatic rays are given at all points of the medium and for all directions, the state of radiation is known in all respects and all
GENERAL INTRODUCTION 17
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 do- toward any other element dcr'. The distance r between the two elements may be thought of as large compared with the linear dimensions of the elements da- and da' but still so small that no appreciable amount of radiation is absorbed or scattered along it. This condition is, of course, superfluous for diather- manous media.
From any definite point of da rays pass to all points of da' . These rays form a cone whose vertex lies in da and whose solid angle is
dQ = da' cos (/, r) r2
where v denotes the normal of da' and the angle (V, r) is to be taken as an acute angle. This value of dtt is, neglecting small quantities of higher order, independent of the particular position of the vertex of the cone on da.
If we further denote the normal to da by v the angle 6 of (14) will be the angle (v, r) and hence from expression (6) the energy of radiation required is found to be :
dada' cos(v,r)-cos(v',r)
K- - — — - at. (17)
r2
For monochromatic plane polarized radiation of frequency v the energy will be, according to equation (11),
The relative size of the two elements da and da' 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 da small compared with da', the rays diverge from da to daf, whereas they converge from da to da', if we choose da large compared with da'.
21. Since every point of da is the vertex of a cone spreading out toward da', the whole pencil of rays here considered, which is
18 FUNDAMENTAL FACTS AND DEFINITIONS
defined by da and dcr', 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 da, converge in one point of da' 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 da and da' 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 over- lapping 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, how- ever, the intersecting plane coincides with either da or da' , 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 inas- much 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 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 elemen- tary 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 da} we may choose not the second focal plane da' but the magnitude of that solid angle dti under which da' is seen from da. On the other hand, in the case of an arbi- trary pencil, that is, when the two focal planes are of the same order of magnitude, the second focal plane in general cannot be
GENERAL INTRODUCTION 19
replaced by the solid angle d!2 without the pencil changing markedly in character. For if, instead of da' being given, the magnitude and direction of dft, to be taken as constant for all points of do-, is given, then the rays emanating from do- do not any longer form the original pencil, but rather an elementary pencil whose first focal plane is da 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 magni- tude 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 0 of v as center a sphere of radius r, r being large compared with the linear dimensions of v but still so small that no appreciable absorption or scattering of the radia- tion 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 do- FlG
on the surface of the sphere, and
reach v, and then sum up for all elements of the spherical sur- face, 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 con- tributed to the energy contained in v by the radiation sent from such an element do- to v. We choose do- so that its linear dimen- sions are small compared with those of v and consider the cone of rays which, starting at a point of do-, meets the volume v. This cone consists of an infinite number of conical elements with the
20 FUNDAMENTAL FACTS AND DEFINITIONS
common vertex at P, a point of da, each cutting out of the volume v a certain element of length, say s. The solid angle of such a
conical element is — 2 where / denotes 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
T = ~
q
From expression (6) we may find the energy radiated through a certain element of a hence the energy is :
certain element of area. In the present case dft = — and 0 = 0;
--Kda. (19)
r2q
This energy enters the conical element in v and spreads out into the volume fs. Summing up over all conical elements that start from da and enter v we have
Kda Kda
This represents the entire energy of radiation contained in the volume v, so far as it is caused by radiation through the element da. In order to obtain the total energy of radiation contained in v we must integrate over all elements da contained in the sur-
da face of the sphere. Denoting by d!2 the solid angle — of a
cone which has its center in 0 and intersects in da the surface of the sphere, we get for the whole energy:
The volume density of radiation required is found from this by dividing by v. It is
u=- \KdQ. (20)
GENERAL INTRODUCTION 21
Since in this expression r has disappeared, we can think of K as the intensity of radiation at the point 0 itself. In integrating, it is to be noted that K in general depends on the direction (6, <£). For radiation that is uniform in all directions K is a constant and on integration we get:
4irK u = -q- (21)
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 u,,. Summing up for all parts of the spec- trum we get:
u= I u,,dv. (22)
Further by combining equations (9) and (20) we have:
(23)
and finally for unpolarized radiation uniformly distributed in all directions :
u, = ^ (24)
CHAPTER II
RADIATION AT THERMODYNAMIC EQUILIBRIUM. KIRCHHOFF'S LAW. BLACK RADIATION
24. We shall now apply the laws enunciated in the last chap- ter 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 sur- rounded 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 equilib- rium, 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 max- imum value of the entropy represents a state of thermodynamic equilibrium of the system. But only one of them, the one cor- responding 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, distur- bance 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 vessel or by any explosive substance. We shall also meet such unstable equilibria in the case of radiation phenomena (Sec. 52).
1 See, e.g., M. Planck, Vorlesungen viber Thermodynamik, Leipzig, Veit and Comp., 1911 (or English Translation, Longmans Green & Co.), Sees. 165 and 189, et seq.
22
RADIATION AT THERMODYNAMIC EQUILIBRIUM 23
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 con- sistent 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 imper- meable 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 thermody- namic equilibrium the radiation of heat has everywhere and in all directions the same properties. Then K,,, the specific intensity of radiation of a plane polarized ray of frequency v (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 da and diverging within a conical element d$l corresponds an exactly equal pencil of oppo- site 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. Therefore in any given arbitrary time just as much radiant heat must be absorbed as is emitted in each vol- ume-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 scat- tering refers only to a change in direction of the energy radiated, not to a creation or destruction of it. We shall, therefore, cal-
24 FUNDAMENTAL FACTS AND DEFINITIONS
culate the energy emitted and absorbed in the time dt by a volume-element v.
According to equation (2) the energy emitted has the value
CO
dt v8w I e, dv
vSir I
Jo
where €„, the coefficient of emission of the medium, depends only on the frequency v 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 con- sidering 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 da- and cuts out of the volume v a part equal to fs has the intensity (energy radiated per unit time)
rf./r
d°- ~*'K
or, according to (12), by considering the different parts of the spectrum separately:
Hence the intensity of a monochromatic ray is:
2 da ~ K, dv.
The amount of energy of this ray absorbed in the distance s in the time dt is, according to (4),
dtavs2d(r- K, dv. r2
Hence the absorbed part of the energy of this small cone of rays, as found by integrating over all frequencies, is :
RADIATION AT THERMODYNAMIC EQUILIBRIUM 25
When this expression is summed up over all the different cross- sections / of the conical elements starting at da and passing through v, it is evident that 2/s = v, and when we sum up over all elements da of the spherical surface of radius r we have
J
d*
— =47T.
Thus for the total radiant energy absorbed in the time dt by the volume-element v the following expression is found:
oo
f «,K,
dt V STT I av K, dv. (25)
By equating the emitted and absorbed energy we obtain:
dp.
I e, dv = I a, K,
•Jo tJ o
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 monochro- matic radiation. This is readily seen from the following. The magnitudes of e,, «„, and Ky 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 con- tinuous 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 K,, for each separate frequency does not change with the time. We have therefore for each frequency the relation:
e, = av K,, or (26)
K,= — ' (27)
OLV
i.e. : in the interior of a medium in a state of thermodynamic equi- librium the specific intensity of radiation of a certain frequency is equal to the coefficient of emission divided by the coefficient of absorp- tion of the medium for this frequency.
26 FUNDAMENTAL FACTS AND DEFINITIONS
27. Since ev and «„ depend only on the nature of the medium, the temperature, and the frequency v, 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 tempera- ture. An exceptional case is when a,, = 0, that is, when the medium does not at all absorb the color in question. Since K, cannot become infinitely large, a first consequence of this is that in that case ev = 0 also, that is, a medium does not emit any color which it does not absorb. A second consequence is that if ev and av both vanish, equation (26) is satisfied by every value of Ky. In a medium which is diathermanous for a certain color thermodynamic 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 — can have, if numerator and denom-
Civ
inator both vanish, 'there exists one particular one which depends in a definite way on the nature of the medium, the frequency v, and the temperature. This distinct value of the fraction is accordingly called the stable intensity of radiation K,, in the me- dium, which at the temperature in question is diathermanous for rays of the frequency v.
Everything that has just been said of a medium which is dia- thermanous for a certain kind of rays holds true for an absolute
RADIATION AT THERMODYNAMIC EQUILIBRIUM 27
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 diather- mancy 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 weaken- ing 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 Sir I ft K, dp. (28)
I
The question as to what becomes of this energy is readily an- swered. On account of the isotropy of the medium, the energy scattered in v and given by (28) is radiated uniformly in all direc- tions 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 dti is obtained by multiplying the last expres- sion by -r-. This gives
00
2 dt v dtt ( ft K, dv,
and, for monochromatic plane polarized radiation,
dt v dQ ft K, dv. (29)
Here it must be carefully kept in mind that this uniformity of radiation in all directions holds only for all rays striking the ele- ment v 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.)
28 FUNDAMENTAL FACTS AND DEFINITIONS
It is thus found that, when thermodynamic equilibrium of ra- diation exists inside of the medium, the process of scattering pro- duces, on the whole, no effect. The radiation falling on a volume- element from all sides and scattered from it in all directions be- haves 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 homogene- ous 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 da at the point 0 (Fig. 2), perpen- dicular to the axis of the pencil, and by the solid angle dft, 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 radi- FIG. 2. ation of the pencil considered passing in unit time through da is represented, according to (11), since in this case dt = l, 6 = 0, by
da da K, dv. (30)
The same value holds for any other cross-section of the pencil. For first, K,, dv has everywhere the same magnitude (Sec. 25), and second, the product of any right section of the pencil and the solid angle at which the focal plane da is seen from this sec- tion has the constant value da dfi, since the magnitude of the cross-section increases with the distance from the vertex 0 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
RADIATION AT THERMODYNAMIC EQUILIBRIUM 29
two cross-sections at distances equal to r0 (of arbitrary length) and r0-\-dr0 respectively from the vertex 0. The volume will be represented by dr0-r02 dtt. It emits in unit time toward the focal plane da- at 0 a certain quantity E of energy of monochro- matic plane polarized radiation. E may be obtained from (1) by putting
dt = l, dr = dr0 r02 dQ, dQ = -g
TO
and omitting the numerical factor 2. We thus get
E = dr0-dttd<r e, dp. (31)
Of the energy E, however, only a fraction E0 reaches 0, since in every infinitesimal element of distance s which it traverses before reaching 0 the fraction (o:,,+^)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 0. Then for a small distance s = dr we have
Er+dr-Er = Er(ar+&)dr, or,
^-fl,(*+&),
dr
and, by integration,
Er= Ee(a»+V(r-r*}
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 0 is found to be
E0 = Ee -(a"+^ro = dr0 dQ do- e, e~(o"+p"^dv. (32)
All volume-elements of the pencils combined produce by their emission an amount of energy reaching da- equal to
I
da dv e, \ dr0 e~ <°>+Wro = dQd<r -- dv. (33)
31. If the scattering did not affect the radiation, the total energy reaching do- 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
30 FUNDAMENTAL FACTS AND DEFINITIONS
on the way. For ft = 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 do- 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 Er thus collected by the volume- element at r0 is found, by putting in (29),
do- dt =1, v = dr0 da r02, da = — >
/)•> 2 'o
to be
E' = dr0 da da ft K, dv.
This energy is to be added to the energy E emitted by the vol- ume-element, 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 da do- fo+ft K,) dv. The part of this reaching 0 is, similar to (32) : dr0 da do- (e, + ft K,) dv e~ro(<x >+ft>>
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 do-
CO
r* \ R K
da do- fe+ft K,) dv dr0 e -r0(a"+ft^ = da daP"-dv,
C
and this expression is real]y 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 homo- geneous isotropic medium when it is in thermodynamic equilib- rium 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 independ- ent of position and direction. A simple consideration, however,
RADIATION AT THERMODYNAMIC EQUILIBRIUM 31
shows that the value of K,,, 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 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 trans- port 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 sur- face 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 K,, also produce, all taken together, a uniform radiation of the same intensity K,,, 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 thermo- dynamic equilibrium has been everywhere established in our me- dium, the equilibrium is, 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
32 FUNDAMENTAL FACTS AND DEFINITIONS
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 K,, 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 what- ever.
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 consid- erable 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 v polarized in an arbitrary plane be K,, in the first sub- stance (the upper one in Fig. 3), and K/ 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 K,, and K/ depend, according to equation (27), only on the tem- perature, the frequency v, 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 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 p (coefficient of reflection) the fraction of the energy reflected, then the fraction transmitted is (1-p), p depending on the angle of incidence, the frequency, and the polarization of the incident ray. Similar remarks apply to
RADIATION AT THERMODYNAMIC EQUILIBRIUM 33
p 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 v, emitted in time dt toward the first medium (in the direction of the feathered arrow upper left
Bounding Surface
FIG. 3.
hand in Fig. 3), from an element do- of the bounding surface and contained in the conical element d!2
where
dt da cos 6 d!2 K, dv, dO d<fr.
(34) (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 da in the corresponding direction (the unfeathered arrows). (Of the element da only the one point 0 is indicated.) The first ray, according to the law of reflection, continues in the symmetrically situated conical element d!2, the second in the conical element
(36)
(37)
^sin 0' dtf dcj>' where, according to the law of refraction,
sin0 q —; =^ sm0' qf
34 FUNDAMENTAL FACTS AND DEFINITIONS
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 do- contributes the part
p dt do- cos 0 da K, dv (38)
while the radiation coming from the second medium and trans- mitted through do- contributes the part
(1-p') dt da cos 0' dtf K/ dv. (39)
The quantities dt, do-, v and dv are written without the accent, because they have the same values in both media.
By adding (38) and (39) and equating their sum to the expres- sion (34) we find
p cos 0 dfi K,,+(l-p') cos 0' dQ' K/ = cos 0 dO K,. Now from (37) we have
cos 0 d0_cos tf d0' q q'
and further by (35) and (36)
d 12 cos 0 q'2 dO' cos 0'=— — - — — •
Therefore we find
or
K/ g'2 1-p
37. In the last equation the quantity on the left side is inde- pendent of the angle of incidence 0 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
RADIATION AT THERMODYNAMIC EQUILIBRIUM 35
bounding surface at the angle of polarization, p = 0, and p' = 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 :
P-P' (40)
and
?2 K,=9" K; (4i)
38. The"jirst 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. l 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 suffered 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 intensities of radiation in the two media, for it states that, when thermodynamic equilibrium exists, the specific intensities of radia- tion 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 K,, its value from (27) we obtain the fol- lowing theorem: The quantity
q*K, = q2 — (42)
OLV
does not depend on the nature of the substance, and is, therefore,
a universal function of the temperature T and the frequency v 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
1 H. v. Helmholtz, Handbuch d. physiologischen Optik 1. Lieferung, Leipzig, Leop. Voss, 1856, p. 169. See also Helmholtz, Vorlesungen fiber die Theorie der Warme herausgegeben von F. Richarz, Leipzig, J. A. Earth, 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. Earth, 1882, p. 594. R. Claurius, Pogg. Ann. 121, p. 1, 1864.
36 FUNDAMENTAL FACTS AND DEFINITIONS
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 func- tion (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 sur- face 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 K,, and K/ 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 thermo- dynamic equilibrium, is satisfied is readily seen from Helm- holtz's law of reciprocity, according to which, in the case of sta- tionary 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 direc- tion,, 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 extend to a great distance, may be abandoned since we may introduce an arbitrary number of bounding surfaces without disturbing the thermodynamic 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
RADIATION AT THERMODYNAMIC EQUILIBRIUM 37
that of two substances in an enclosure impermeable to heat, and, therefore, the law may be stated quite generally, that, when any arbitrary system is in the state of thermodynamic equilibrium, the specific intensity of radiation K,, is determined in each separate substance by the universal function (42).
42. We shall now consider a system in a state of thermody- namic equilibrium, contained within an enclosure impermeable to heat and consisting of n emitting and absorbing adjacent bod- ies of any size and shape whatever. As in Sec. 36, we again con- fine our attention to a monochromatic plane polarized pencil which proceeds from an element d<r of the bounding surface of the two media in the direction toward the first medium (Fig. 3, feathered arrow) within the conical element dfi. Then, as in (34) , the energy supplied by the pencil in unit time is
dff cos 6 dtt K, dv = 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 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, scat- tering, and partial absorption any number of times. Similarly the rays of the pencil, which coming from the first medium are reflected at d<r, 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 I\ no matter what paths the different constituents
38 FUNDAMENTAL FACTS AND DEFINITIONS
have pursued, that which has been emitted by volume-elements of the second medium by 72, etc. Then since every part of I must have been emitted by an element of some body/ the follow- ing equation must hold,
I = /1 + /2 + /8+ 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 7i, 72, Is, 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 dti and falls on the surface element da- of the second me- dium. 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 7, however complicated they 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 dcr 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 /2, etc., then we shall have
J = /1+j-24-j3+ +Jn.
Now the volume-elements of the n media, in which the absorp- tion 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 7, the first one considered above. For, according to Helmholtz's law of reciprocity, no appreciable radiation of the pen- cil J can enter a volume-element which contributes no appreci- able radiation to the pencil 7 and vice versa.
RADIATION AT THERMODYNAMIC EQUILIBRIUM 39
Let us further keep in mind that the absorption of each volume- element 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 de- « crease 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 / in just the same ratio as they contribute by their emission to the energy of the opposite pencil /. 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 / must be equal to the quantity of energy emitted by the same element into the pencil 7. 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 JT( = 7) oppositely directed, which is absorbed in the same volume.
Hence not only are the sums 7 and J equal, but their constitu- ents are also separately equal or
«7i = 7!, /2 = 72, ...... /n = 7n. (46)
44. Following G. Kirchhoff1 we call the quantity 72, 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 J% to /, 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
A=(^l). (47)
The quantities E and A depend (a) on the nature of the two media, (b) on the temperature, the frequency v, 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 da and that of the solid angle dfi, (d) on the geometrical extent and the shape of the total surface of the two media, (e) on the nature and form of all other bodies of the system. For a ray may pass from the first into the second medium, be partly trans- mitted by the latter, and then, after reflection somewhere else,
i(?. Kirchhoff, Gesammelte Abhandlungcn, 1882, p. 574.
40 FUNDAMENTAL FACTS AND DEFINITIONS
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,
Tjl
- = I = do- cos e dtt K, dv, (48)
A.
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 Ky but the quantity g2 K,,, which is a univer- sal function of the temperature and frequency. The proof of this law given by G. Kirchhoff I.e. 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 Jz = J, A = l, and E = A^.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 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.
1 E. Pringsheim, Verhandlungen der Deutschen Physikalischen Gesellschaft, 3, p. 81, 1901.
RADIATION AT THERMODYNAMIC EQUILIBRIUM 41
47. The law holding for the quantity (42) can be expressed in a different form, by introducing, by means of (24), the volume density uv of monochromatic radiation instead of the intensity of radiation K,,. We then obtain the law that, for radiation in a state of thermodynamic equilibrium, the quantity
u, <Z3 (49)
is a function of the temperature T and the frequency v, and is the same for all substances.1 This law becomes clearer if we consider that the quantity
u, d,-^ (50)
vz
also is a universal function of T, v, and v+dv, and that the product uv dv is, according to (22), the volume density of the radiation whose frequency lies between v and v-\-dv, while the
quotient — represents the wave length of a ray of frequency v in v
the medium in question. The law then takes the following sim- ple form : When any bodies whatever are in thermodynamic equilib- rium, 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 radiation. In fact, in equa- tion (27) the intensity of radiation K,, for «„ = () and e,, = 0
assumes the value—-' 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
1 In 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.
42 FUNDAMENTAL FACTS AND DEFINITIONS
.equilibrium the product g2 Ky has the same value for all sub- stances. From this we find immediately a definite value of Kv which is thereby distinguished from all other values. Further- more 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 particular the volume may be taken as small as we please. Hence we can formulate 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 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 diather- manous 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 *>, assumes the intensity corre- sponding 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
1 W. Wien and 0. Lummer, Wied. Annalen, 56, p. 451, 1895.
2 The 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 radia- tion to the exterior.
RADIATION AT THERMODYNAMIC EQUILIBRIUM 43
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 7 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),
If we now make a hole in one of the walls of a size do-, 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 diather- manous medium as within. This radiation has exactly the same properties as if da 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 dia- thermanous 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
1 M. Thiesen, Verhandlungen d. Deutschen Physikal. Gesellschaft, 2, p. 65, 1900.
44 FUNDAMENTAL FACTS AND DEFINITIONS
this is the case the velocity of propagation q may be taken as the same for all frequencies, namely,
PTYl
(51)
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 sta- tionary state of radiation is gradually established. In this the radiation of every color which is appreciably absorbed by the substance has the intensity K,, 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 diatherma- nous 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 K, of black radiation corresponding to the temperature of the substance. The magnitude of Kv regarded as a function of v 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 charac- teristic feature of this process is that the heat of the carbon par- ticle 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 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 (auslosend) action. Thereafter the intensities of the pencils of different frequencies originally present and having different frequencies, directions, and different states of polari-
RADIATION AT THERMODYNAMIC EQUILIBRIUM 45
zation 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 super- saturated vapor. In all these cases the magnitude of the dis- turbance 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 com- mon 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 equa- tions 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 ( .
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 xf, y', z', of an ortho- gonal right-handed system of coordinates, say on x1 '. Then the equations (52) reduce to
.
d E,, dH^
~
49
50
DEDUCTIONS FROM ELECTRODYNAMICS
= c
da;'
— c
(53)
= 0
= 0
Hence the most general expression for a plane wave passing through a vacuum in the direction of the positive z'-axis is
0
= 0
(54)
H.,
Vacuum x< 0
where / and gr 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 pro- duces a finite conduction cur- rent; 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-mag- netizable, 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 z-axis of a right-handed coordinate system (xyz) along the normal of the sur- face directed toward the interior of the conductor, the x-axis is the normal of incidence. We place the (x'yr) plane in the plane of incidence and take this as the plane of the figure (Fig. 4) . Moreover, we can also, without
FIG. 4.
MAXWELL'S RADIATION PRESSURE 51
any restriction of generality, place the ?/-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 0 of the two coordinate systems lie in the surface. If finally 6 represents the angle of incidence, the coordinates with and with- out accent are related to each other by the following equations:
x = x' cos 0 — y' sin 0 xf = x cos 6+y sin 6
y = x' sin B+y' cos 0 y' = — x sin d+y cos 0
z' = z
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,
/K .
Ex = — smd-f Hx = sin0-g(
Ey = cos0-/ Hy = — cos0-0
Ez = g H. = /
Herein the argument of the functions / and g is
. z' . s cos 0+y sin 0 i -- = t ---
c c
56. In the surface of separation of the two media x = 0. Ac- cording to the general electromagnetic boundary conditions the components of the field-strengths in the surface of separation, i.e., the four quantities Ey, EZ} \-\y, 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 Ev and Ez must vanish also in the vacuum for x = 0. This con- dition cannot be satisfied unless we assume in the vacuum, besides the incident, also a reflected wave superposed on the for- mer 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 assump- tion and the condition that the reflected wave is a plane wave returning into the interior of the vacuum, the other four compo-
52 DEDUCTIONS FROM ELECTRODYNAMICS
nents of the reflected wave are also completely determined. They are all functions of the single argument
-x cos 0+y sin 0
t — (ol)
c
The actual calculation yields as components of the total electro- magnetic 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 = -sin0-/- sin0-/= - 2 sin0-/
Ey = cos0-/ - cos0-/ = 0
E* = g -g = 0 (58)
Hx = sin 0-gr — sin0-0 = 0
Hj, = — COS0-0 — cosd-g = —2 cosd-g
H. =/+/=2/.
In these equations the argument of the functions / and g is, ac- cording to (56) and (57),
From these values the electric and magnetic field-strength within the conductor in the immediate neighborhood of the separating surface x = Q is obtained:
* * (59)
Ey = 0 \-\v = -2 cosd-g
Ez = 0 H2 = 2f
where again the argument t --- — is to be substituted in the
C
functions / and g. For the components of E all vanish in an abso- lute conductor and the components H^, Hj/, H2 are all continuous at the separating surface, the two latter since they are tangential components of the field-strength, 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 field- strength Ex is seen to be discontinuous; the discontinuity shows
MAXWELL'S RADIATION PRESSURE 53
the existence of an electric charge on the surface, the surface density of which is given in magnitude and sign as follows:
- 2 sin0./=— sin0./. (60)
TtTT £W
In the interior of the conductor at a finite distance from the bounding surface, i.e., for x>0, all six field components" are infi- nitely small. Hence, on increasing x, the values of Hy and H2, which are finite for x = Q, 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 con- sider 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 9f on the side of the vacuum and 0 on the side of the conductor, from a well-known law of electrostatics the mag- nitude of the mechanical force Fe acting on an element of surface da 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
sin 6 f sin20 „,
e=~2^ f da(-~sm Bfi = ~^~f d«
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
l=— curlH. (61)
4rr
A mechanical force acts on every element of space dr of the con- ductor through which a conduction current flows, and is given by the vector product
-[IH] (62)
c
54 DEDUCTIONS FROM ELECTRODYNAMICS
Hence the component of this force normal to the surface of the conductor x = 0 is equal to
— (I.H.-I.H,).
c
On substituting the values of \y and I2 from (61) we obtain
In this expression the differential coefficients with respect to y and 0 are negligibly small in comparison to those with respect to x, according to the remark at the end of Sec. 56; hence the expres- sion reduces to
&H, &H.\
~
drl
~\ H
Let us now consider a cylinder cut out of the conductor perpen- dicular to the surface with the cross-section da, and extending from x = 0 to z=oo. The entire mechanical force of magnetic origin acting on this cylinder in the direction of the z-axis, since dr = da x, is given by
4rr
e/ o
On integration, since H vanishes f or x — oo , we obtain
or by equation (59)
By adding Fe and Fm the total mechanical force acting on the cylinder in question in the direction of the z-axis is found to be
do-
F = — cos20 (f+02). (63)
ZTT
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
MAXWELL'S RADIATION PRESSURE 55
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 Idt falling on the surface element da of the conductor in a time element dt. The latter from Poynting's law of energy flow is
c
Idt = — (Ey)r\z — EzHy) da dt, 4rr
hence from (55)
Idt = — cos 0 (/2+02) da dt. 4?r
By comparison with (63) we obtain
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 con- ductor exerts in a normal direction on a unit surface of the con- ductor. The energy radiated in the conical element
da = sin 0 d0 d(j>
in the time dt on the element of area da is, according to (6), Idt=K cos 0 da da dt,
where K represents the specific intensity of the radiation in the direction d a toward the reflector. On substituting this in (64) and integrating over da we obtain for the total pressure of all pencils which fall on the surface and are reflected by it
cos2 e da, (65)
the integration with respect to $ extending from 0 to 2?r and with
respect to 0 from 0 to — • 2
» P. Lebedew, Annalen d. Phys., 6, p. 433, 1901. See also E. F. Nichols and O. F. Hull, Annalen d. Phys., 12, p. 225, 1903.
56 DEDUCTIONS FROM ELECTRODYNAMICS
In case K is independent of direction as in the case of black radiation, we obtain for the radiation pressure
IT 2K Cl f 7 ^K
p =— I d4> I dd cos2 0 sin 0
Jf d<p I ( t/ °
3c
or, if we introduce instead of K the volume density of radiation u from (21)
P = y. (66)
This value of the radiation pressure holds only when the reflec- tion of the radiation occurs at the surface of an absolute non- magnetizable conductor. Therefore we shall in the thermody- namic 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 thermo- dynamic considerations. To decide this question we shall cal- culate the radiation pressure that would follow by Newtonian mechanics from Newton's (emission) theory of light, a theory which, in itself, is quite consistent with the energy principle. 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 da- of a reflecting surface at the angle of incidence 0 is
nc cos 0 da. (67)
MAXWELL'S RADIATION PRESSURE 57
Their kinetic energy is given according to Newtonian mechanics by
9?? C (*
I = nc cos 0 do- — - — = nm cos 6—-d<r. (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 6 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 6. Then the change in momentum for all particles considered will be, according to (67),
-2nm cos2 0 c2 dor. (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 momen- tum 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, when- ever 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 element da is equal and opposite to the change of their momentum in unit time as expressed in (69)
F = 2 nm cos2 0 c2 do- and by making use of (68),
4 cos 0
r — 1 .
c
On comparing this relation with equation (64) in which all symbols have the same physical significance, it is seen that
58 DEDUCTIONS FROM ELECTRODYNAMICS
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 Max- well'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 out- side. The rest of the walls including the inner surface of the pis- ton may be assumed as totally reflecting. Then, if the piston remains stationary and the temperature, T, constant, the radia- tion in the vacuum will, after a certain time, assume the charac- ter 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 com- pressed 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 ther- modynamics 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.
If, e.g., we keep the temperature T of the black body forming the bottom constant, as can be done by a suitable connection
59
60 DEDUCTIONS FROM ELECTRODYNAMICS
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 accel- erate 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 pres- sure (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 gas, but a purely ener- getic 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 revers- ible processes must remain constant (second principle).
STEFAN-BOLTZMANN LAW OF RADIATION 61
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=Vu, (70)
where u the volume density of radiation depends only on the temperature of T 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 pdV, 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 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+pdV-Q = 0. (71)
According to the second principle of thermodynamics the cav- ity 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 — inas- much as the rigid absolutely reflecting piston with the weight on it does not change its internal condition with the motion — there
62 DEDUCTIONS FROM ELECTRODYNAMICS
must somewhere in nature occur a compensation of entropy hav- ing at least the value — > by which the above diminution is com- pensated, 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 equilibrium, they are perfectly reversible and hence there is no increase in entropy. Then we have
dS-| = 0, (72)
or from (71)
dS = — (73)
In this equation the quantities U, p, V, S represent certain properties of the heat radiation, which are completely defined by the instantaneous state of the radiation. Therefore the quantity T is also a certain 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 quan- tities any two are determined by the third together with V. We shall take the volume V and the temperature T as indepen- dent variables. Then by substituting from (66) and (70) in (73) we have
- dV. (74)
j. a± ol
From this we obtain
/cXS\ _VduL \t>T/v~T dT
STEFAN-BOLTZMANN LAW OF RADIATION 63
On partial differentiation of these equations, the first with respect to V, the second with respect to T, we find
1 du 4 du 4u
or
du _4:U
~dT^~T and on integration
u = aT* (75)
and from (21) for the specific intensity of black radiation
X = --W = fCr*. (76)
4r 4?r
Moreover for the pressure of black radiation
P=lT<, (77)
o
and for the total radiant energy
. (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 /. 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 0. Lummer and E. Pringsheim* 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 thermome- ters 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
1 J. Stefan, Wien. Berichte, 79, p. 391, 1879.
2 L. Boltzmann, Wied. Annalen, 22, p. 291, 1884.
8 0. Lummer und E. Pringsheim, Wied. Annalen, 63, p. 395, 1897. Annalen d. Physik, 3, p. 159, 1900.
4 F. Kurlbaum, Wied. Annalen, 65, p. 759, 1898.
64 DEDUCTIONS FROM ELECTRODYNAMICS
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
Sioo-£o = 0.0731
cm2 cm2 sec
Now, since the radiation in air is approximately identical with the radiation into a vacuum, we may according to (7) and (76) put
and from this
= irK = -- (273+Z)4 4
= — (3734-2734), 4
therefore
a-. .. . =7.061X10-*-
3 X 1010 X (3734 - 2734) cm3 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 Fery, Bauer and Moulin, Valentiner, Fery 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 detertnina- tion made in the physical laboratory of the University of Berlin2
— = o- = 5.46-10-12 — 0Wf
4 cm2 degree4
From this a is found to be
erg
3-1010 cm3 degree4
which agrees rather closely with Kurlbaum's corrected value.
1F. Kurlbaum, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 580, 1912.
2 According 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.)
STEFAN-BOLTZMANN LAW OF RADIATION 65
65. The magnitude of the entropy S of black radiation found by integration of the differential equation (73) is
S = ^-aT*V. (80)
o
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,
|=«=|or». (si)
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 con- stantly 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 should begin to move, the kinetic energy of this motion could originate only at the ex- pense 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 de- crease, 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
66 DEDUCTIONS FROM ELECTRODYNAMICS
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 corresppnding addition of heat from the heat reservoir, the process takes place isother- mally. 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,
or, according to (81) and (75),
Thus it is seen that the heat furnished from the outside exceeds the increase in energy of radiation (U'—U) by J ( 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 pdV 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 char- acter 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
STEFAN-BOLTZMANN LAW OF RADIATION 67
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 for such a process
T3F = const., or, according to (77),
4
= 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
-=-S = const.,
i.e., it increases in proportion to the absolute temperature, al- though 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 reflect- ing walls. The radiation will at first be of a very irregular char- acter; after spme time, however, it will assume a stationary con- dition 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
68 DEDUCTIONS FROM ELECTRODYNAMICS
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 Uf — U and hence from (78)
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)
_ ~VV'
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 radia- tion 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 prin- ciple, 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 compara- tively little use in finding the volume density u,, corresponding to a definite frequency v, and the specific intensity of radiation K,, of monochromatic radiation, which are related to each other by equation (24) and ton 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 u,, and Kv for black radiation in a vacuum and hence, according to (42), in any medium whatever, as functions of v 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 solu- tion 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 u,, and K, of the two arguments v 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 evac- uated 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 radia- tion, even without the presence of a 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:
i W. Wien, Sitzungsberichte d. Akad. d. Wissensch. Berlin, Febr. 9, 1893, p. 55. Wiede- mann'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.
69
70 DEDUCTIONS FROM ELECTRODYNAMICS
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 equilib- rium 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 com- pared with the energy of radiation. This change, of course, refers only to the spectral density of radiation uv, 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 radia- tion, not on its spectral distribution. Therefore, according to the first principle of thermodynamics, the total energy of radia- tion 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 calcu- lation, for its energy and entropy are vanishingly small com- pared 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 com- pletely 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
WIEN'S DISPLACEMENT LAW 71
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 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 radia- tion 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
72 DEDUCTIONS FROM ELECTRODYNAMICS
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. l 74. To begin with, we consider the change of color which a mono- chromatic ray suffers by reflection from the reflector, which is A moving with an infinitely small veloc-
Reflector t ., -^ , •, . . -,
/ ity. For this purpose we consider
X Reflectort + $t ,. , . , . , . ..
first the case of a ray which falls
normally from below on the reflector and hence is reflected normally down- ward. Let the plane A (Fig. 5) repre- sent the position of the reflector at the
B — ""stationary" time t, the plane A' the position at
F - the time t-\-dt, where the distance
A A' equals vdt, v denoting 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 X the wave length of the ray incident on the reflector and by X' 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 radia- tion — — waves of the incident and — - waves of the reflected ray, X X
as can 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
waves in the interval between A and B. Since this is a large num- ber, it is immaterial whether the number is an integer or not.
1 The 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 73
Similarly at the time t+dt, when the reflector is at A', there are
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 dt 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 vbt waves pass in the time dt through the stationary plane B in an upward direction and v'bt waves in a downward direction; hence we have for the difference
or, snce
AB-A'B = vdt, and
v v
c+v If' = -- If
c — v or, since v is infinitely small compared with c,
75. When the radiation does not fall on the reflector normally but at an acute angle of incidence 0, it is possible to pursue a very similar line of reasoning, with the difference that then A, the point of intersection of a definite ray BA with the reflector at the time t, has not the same position on the reflector as the point of intersection, A', of the same ray with the reflector at the time t-i-dt (Fig. 6). The number of waves which lie in the interval
BA at the time t is -- Similarly, at the time t the number of
A
waves in the interval AC representing the distance of the point
74
DEDUCTIONS FROM ELECTRODYNAMICS
A from a wave plane CC', belonging to the reflected ray and
AC stationary in the vacuum, is — -•
A
Hence there are, all told, at the time t in the interval BAC
BA AC X "" V
waves of the ray under consideration. We may further note that the angle of reflection 6' is not exactly equal to the angle
Reflector t Reflector t + 5 1
Stationary
FIG. 6.
of incidence, but is a little smaller as can be shown by a simple geometric consideration based on Huyghens' principle. The difference of 6 and B' ', however, will be shown to be non-essential for our calculation. Moreover there are at the time t+8t, when the reflector passes through A',
BA' A'C' ~ ~
waves in the distance BA'C'. The latter number is smaller than the former and the difference must equal the total number of waves which are expelled in the time dt from the space which is bounded by the stationary planes BB' and CC'.
Now vdt waves enter into the space through the plane BB' in the time dt and v'U waves leave the space through the plane CC' Hence we have
WIEN'S DISPLACEMENT LAW 75
but
BA-BA'-AA'~'*
COS 0
AC-A'C' = AAr w$ (0+0')
v v'
Hence
, c cos 0+y
c cos B — v cos (0+0')
This relation holds for any velocity v of the moving reflector. Now, since in our case v is infinitely small compared with c, we have the simpler expression
c cos 6 The difference between the two angles 6 and 0' is in any case of
the order of magnitude -; hence we may without appreciable c
error replace 6' by 6, thereby obtaining the following expression for the frequency of the reflected ray for oblique incidence
/, , 2v cos 0\
v'=v I H — -I (83)
\ c /
76. From the foregoing it is seen that the frequency of all rays which strike the moving reflector are increased on reflection, when the reflector moves toward the radiation, and decreased, when the reflector moves in the direction of the incident rays (v<Q). However, the total radiation of a definite frequency v striking the moving reflector is by no means reflected as monochromatic radia- tion but the change in color on reflection depends also essentially on the angle of incidence 6. Hence we may not speak of a cer- tain spectral " displacement " of color except in the case of a sin- gle pencil of rays of definite direction, whereas in the case of the entire monochromatic radiation we must refer to a spectral " dispersion." The change in color is the largest for normal inci- dence and vanishes entirely for grazing incidence.
77. Secondly, let us calculate the change in energy, which the
76 DEDUCTIONS FROM ELECTRODYNAMICS
moving reflector produces in the incident radiation, and let us consider from the outset the general case of oblique incidence. Let a monochromatic, infinitely thin, unpolarized pencil of rays. which falls on a surface element of the reflector at the angle of incidence 0, transmit the energy I8t to the reflector in the time 5t. Then, ignoring vanishingly small quantities, the mechanical pressure of the pencil of rays normally to the reflector is, accord- ing to equation (64),
2 cos e
c
and to the same degree of approximation the work done from the outside on the incident radiation in the time 5t is
^0!_V (84)
According to the principle of the conservation of energy this amount of work must reappear in the energy of the reflected radia- tion. Hence the reflected pencil has a larger intensity than the incident one. It produces, namely, in the time dt the energy1
= I( \
(85)
Hence we may summarize as follows: By the reflection of a monochromatic unpolarized pencil, incident at an angle 0 on a reflector moving toward the radiation with the infinitely small velocity v, the radiant energy Idt, whose frequencies extend from v to v+dv, is in the time dt changed into the radiant energy I'bt with the interval of frequency (/, v'-\-dv'), where /' is given by (85), v' by (83), and accordingly dv', the spectral breadth of the reflected pencil, by
(86)
c A comparison of these values shows that
>-'=?-' ' '• 1 (87)
I v dp
1 It is clear that the change in intensity of the reflected radiation caused by the motion of the reflector can also be derived from purely electrodynamical considerations, since elec- trodynamics are consistent with the energy principle. This method is somewhat lengthy, but it affords a deeper insight into the details of the phenomenon of reflection.
WIEN'S DISPLACEMENT LAW 77
The absolute value of the radiant energy which has disappeared in this change is, from equation (13),
l5t = 2Kv da cos 0 dtt dv dt, (88)
and hence the absolute value of the radiant energy which has been formed is, according to (85),
7'« = 2K,d<r cos B dtt dvl+tt. (89)
\ c I
Strictly speaking these last two expressions would require an infinitely small correction, since the quantity / from equation (88) represents the energy radiation on a stationary element of area d<r, while, in reality, the incident radiation is slightly increased by the motion of do- toward the incident pencil. The additional terms resulting therefrom may, however, be omitted here without appreciable error.
78. As regards finally the changes in direction, which are im- parted to the incident ray by reflection from the moving reflector, we need not calculate them at all at this stage. For if the motion of the reflector takes place sufficiently slowly, all irregularities in the direction of the radiation are at once equalized by further reflection from the walls of the vessel. We may, indeed, think of the whole process as being accomplished in a very large number of short intervals, in such a way that the piston, after it has moved a very small distance with very small velocity, is kept at rest for a while, namely, until all irregularities produced in the directions of the radiation have disappeared as the result of the reflection from the white walls of the hollow cylinder. If this procedure be carried on sufficiently long, the compression of the radiation may be continued to an arbitrarily small fraction of the original volume, and while this is being done, the radiation may be always regarded as uniform in all directions. This continuous process of equalization refers, of course, only to difference in the direction of the radiation; for changes in the color or intensity of the radiation of however small size, having once occurred, can evidently never be equalized by reflection from totally reflecting stationary walls but continue to exist forever.
79. With the aid of the theorems established we are now in a position to calculate the change of the density of radiation for
78 DEDUCTIONS FROM ELECTRODYNAMICS
every frequency for the case of infinitely slow adiabatic compres- sion of the perfectly evacuated hollow cylinder, which is filled with uniform radiation. For this purpose we consider the radia- tion at the time t in a definite infinitely small interval of fre- quencies, from v to v+dv, and inquire into the change which the total energy of radiation contained in this definite constant interval suffers in the time dt.
At the time t this radiant energy is, according to Sec. 23, V udv, at the time t-\-dt it is (Vu + d (Vu))dv, hence the change to be calculated is
S(Vu)dv. (90)
In this the density of monochromatic radiation u is to be regarded as a function of the mutually independent variables v and t, the differentials of which are distinguished by the symbols d and d.
The change of the energy of monochromatic radiation is pro- duced only by the reflection from the moving reflector, that is to say, firstly by certain rays, which at the time t belong to the interval (v,dv), leaving this interval on account of the change in color suffered by reflection, and secondly by certain rays, which at the time t do not belong to the interval (v,dv), coming into this interval on account of the change in color suffered on reflection. Let us calculate these influences in order. The calculation is greatly simplified by taking the width of this interval dv so small that
dv is small compared with -v, . (91)
c
a condition which can always be satisfied, since dv and v are mutually independent.
80. The rays which at the time t belong to the interval (v,dv) and leave this interval in the time 8t on account of reflection from the moving reflector, are simply those rays which strike the moving reflector in the time dt. For the change in color which such a ray undergoes is, from (83) and (91), large compared with dv, the width of the whole interval. Hence we need only cal- culate the energy, which in the time dt is transmitted to the re- flector by the rays in the interval (v,dv).
For an elementary pencil, which falls on the element da- of the
WIEN'S DISPLACEMENT LAW 79
reflecting surface at the angle of incidence 0, this energy is, according to (88) and (5),
!8t = 2Kvda cos 6 dtt dp dt = 2Kv do- sin 0 cos 0 dd d<f> dv 5t.
Hence we obtain for the total monochromatic radiation, which falls on the whole surface F of the reflector, by integration with
respect to <£ from 0 to 2?r, with respect to 6 from 0 to -, and with
2i
respect to da- from 0 to F,
2ir F Kp dv 5t. (92)
Thus this radiant energy leaves, in the time dt, the interval of frequencies (v,dv) considered.
81. In calculating the radiant energy which enters the interval (v,dv) in the time dt on account of reflection from the moving reflector, the rays falling on the reflector at different angles of incidence must be considered separately. Since in the case of a positive v, the frequency is increased by the reflection, the rays which must be considered have, at the time t, the frequency PI<P. If we now consider at the time t a monochromatic pencil of frequency (vi,dvi), falling on the reflector at an angle of inci- dence 6, a necessary and sufficient condition for its entrance, by reflection, into the interval (v,dv) is
/ 2v cos 0\ / 2v cos 0\
p=pi\l-\ — -) and dv = dvA H —
\ c I \ c /
These relations are obtained by substituting v\ and v respectively in the equations (83) and (86) in place of the frequencies before and after reflection v and v' .
The energy which this pencil carries into the interval (pi,dp) in the time dt is obtained from (89), likewise by substituting PI for v. It is
2K,i do- cos 6d$ldpi(l + — -)dt = 2Kvi da- cos BdttdvU.
\ c /
Now we have
where we shall assume - - to be finite. OP
80 DEDUCTIONS FROM ELECTRODYNAMICS
Hence, neglecting small quantities of higher order,
2?t;cos B dK
•V, — K* -- " ^T~
c dv Thus the energy required becomes
_ /.. 2w cos B dK\
2cM K,— - - I sin 8 cos 0 dB d<f> dv 5t,
\ c dv I
and, integrating this expression as above, with respect to do-, <f>, and 0, the total radiant energy which enters into the interval vdv in the time dt becomes
dv U. (93)
3 c ov I
82. The difference of the two expressions (93) and (92) is equal to the whole change (90), hence
3 c Ov or, according to (24),
1 du
-- Fv v—Bt = 3 OP
or, finally, since Fvdt is equal to the decrease of the volume V, 1 du
, (94)
3 dv
whence it follows that
/?bu \SV
•u-(is-u)r (95)
This equation gives the change of the energy density of any definite frequency v, which occurs on an infinitely slow adiabatic compression of the radiation. It holds, moreover, not only for black radiation, but also for radiation originally of a perfectly arbitrary distribution of energy, as is shown by the method of derivation.
Since the changes taking place in the state of the radiation in the time dt are proportional to the infinitely small velocity v and are reversed on changing the sign of the latter, this equation holds for any sign of 5F; hence the process is reversible.
WIEN'S DISPLACEMENT LAW 81
83. Before passing on to the general integration of equation (95) let us examine it in the manner which most easily suggests itself. According to the energy principle, the change in the radiant energy
'S.
udv,
occurring on adiabatic compression, must be equal to the external work done against the radiation pressure
udv. (96)
Now from (94) the change in the total energy is found to be
C
J
or, by partial integration,
00
8V., "
3\L - JO _
and this expression is, in fact, identical with (96) , since the prod- uct vu vanishes for v = 0 as well as f or v — <» . The latter might at first seem doubtful; but it is easily seen that, if vu for v— <» had a value different from zero, the integral of u with respect to v taken from 0 to oo could not have a finite value, which, however, certainly is the case.
84. We have already emphasized (Sec. 79) that u must be regarded as a function of two independent variables, of which we have taken as the first the frequency v and as the second the time t. Since, now, in equation (95) the time t does not explicitly appear, it is more appropriate to introduce the volume V, which depends only on t, as the second variable instead of t itself. Then equation (95) may be written as a partial differential equation as follows:
From this equation, if, for a definite value of V, u is known as a function of v, it may be calculated for all other values of V as a
82 DEDUCTIONS FROM ELECTRODYNAMICS
function of v. The general integral of this differential equation, as may be readily seen by substitution, is
U=10((,3F)) (Q8)
where 0 denotes an arbitrary function of the single argument i>3F. Instead of this we may, on substituting v*V<j>(i>*V) for 0<V7), write
u = v*<f>(v*V). (99)
Either of the last two equations is the general expression of Wien's displacement law.
If for a definitely given volume V the spectral distribution of energy is known (i.e., u as a function of v), it is possible to deduce therefrom the dependence of the function (f> on its argument, and thence the distribution of energy for any other volume V, into which the radiation filling the hollow cylinder may be brought by a reversible adiabatic process.
84a. The characteristic feature of this new distribution of energy may be stated as follows : If we denote . all quantities referring to the new state by the addition of an accent, we have the following equation in addition to (99)
u' = /34> (v'*V). Therefore, if we put
V>*V'=V*V, (99a)
we shall also have
^7=- -andu'y' = uF, (99b)
/* VA
i.e., if we coordinate with every frequency v in the original state that frequency v' which is to v in the inverse ratio of the cube roots of the respective volumes, the corresponding energy densities u' and u will be in the inverse ratio of the volumes.
The meaning of these relations will be more clearly seen, if we write
V^__V V3~X3
This is the number of the cubes of the wave lengths, which correspond to the frequency v and are contained in the volume
WIEN'S DISPLACEMENT LAW 83
of the radiation. Moreover udvV = \Jdi> denotes the radiant energy lying between the frequencies vand v-\-dv, which is con- tained in the volume V. Now since, according to (99a),
<jFOr — =- (99C)
V V
we have, taking account of (99b),
These results may be summarized thus: On an infinitely slow reversible adiabatic change in volume of radiation contained in a cavity and uniform in all directions, the frequencies change in such a way that the number of cubes of wave lengths of every frequency contained in the total volume remains unchanged, and the radiant energy of every infinitely small spectral interval changes in proportion to the frequency.
85. Returning now to the discussion of Sec. 73 we introduce the assumption that at first the spectral distribution of energy is the normal one, corresponding to black radiation. Then, accord- ing to the law there proven, the radiation retains this property without change during a reversible adiabatic change of volume and the laws derived in Sec. 68 hold for the process. The radia- tion then possesses in every state a definite temperature T, which depends on the volume V according to the equation derived in that paragraph,
TW = const. =T'W. (100)
Hence we may now write equation (99) as follows:.
or
Therefore, if for a single temperature the spectral distribution of black radiation, i.e., u as a function of v, is known, the depen- dence of the function <f> on its argument, and hence the spec- tral distribution for .any other temperature, may be deduced therefrom.
84 DEDUCTIONS FROM ELECTRODYNAMICS
If we also take into account the law proved in Sec. 47, that, for the black radiation of a definite temperature, the product ug3 has for all media the same value, we may also write
where now the function F no longer contains the velocity of propagation.
86. For the total radiation density in space of the black radia- tion in the vacuum we find
1 = C"di> = -
(102)
T
or, on introducing — = re as the variable of integration instead v
of v,
00
J. I F (X) j /-t f\o\
u— — i— Lax. UUD;
c3 I x5
t/o
If we let the absolute constant
=a (104)
^x »
the equation reduces to the form of the Stefan-Boltzmann law of radiation expressed in equation (75).
87. If we combine equation (100) with equation (99a) we obtain
Hence the laws derived at the end of Sec. 84a assume the fol- lowing form: On infinitely slow reversible adiabatic change in volume of black radiation contained in a cavity, the temperature T varies in the inverse ratio of the cube root of the volume V, the frequencies v vary in proportion to the temperature, and the radiant energy \Jdv of an infinitely small spectral interval varies in the same ratio. Hence the total radiant energy U as the sum of the energies of all spectral intervals varies also in proportion to the temperature, a statement which agrees with the
WIEN'S DISPLACEMENT LAW 85
conclusion arrived at already at the end of Sec. 68, while the space density of radiation, u = — > varies in proportion to the
fourth power of the temperature, in agreement with the Stefan- Boltzmann law.
88. Wien's displacement law may also in the case of black radiation be stated for the specific intensity of radiation K,, of a plane polarized monochromatic ray. In this form it reads according to (24)
(106)
If, as is usually done in experimental physics, the radiation inten- sity is referred to wave lengths X instead of frequencies v, accord- ing to (16), namely
P eK, Ex = ^
equation (106) takes the following form:
(107)
This form of Wien's displacement law has usually been the start- ing-point for an experimental test, the result of which has in all cases been a fairly accurate verification of the law.1
89. Since Ex vanishes for X = 0 as well as for X = °° , Ex must have a maximum with respect to X, which is found from the equation
dE 5j\T\ . 1 T
where F denotes the differential coefficient of F with respect to its argument. Or
(108) c c i c
KT
This equation furnishes a definite value for the argument — , so
1 E.g., F. Paschen, Sitzungsber. d. Akad. d. Wissensch. Berlin, pp. 405 and 959, 1899. 0. Lummer und E. Pringsheim, Verhandlungen d. Deutschen physikalischen Gesellschaft 1, pp. 23 and 215, 1899. Annal. d. Physik 6, p. 192, 1901.
86 DEDUCTIONS FROM ELECTRODYNAMICS
that for the wave length Xm corresponding to the maximum of the radiation intensity E^ the relation holds
6. (109)
With increasing temperature the maximum of radiation is therefore displaced in the direction of the shorter wave lengths. The numerical value of the constant b as determined by Lummer and Pringsheim1 is
6 = 0.294 cm. degree. (110)
* Paschen2 has found a slightly smaller value, about 0.292.
We may emphasize again at this point that, according to Sec. 19, the maximum of E^ does not by any means occur at the same point in the spectrum as the maximum of K,, and that hence the significance of the constant b is essentially dependent on the fact that the intensity of monochromatic radiation is referred to wave lengths, not to frequencies.
90. The value also of the maximum of Ex is found from (107) by putting X=Xm. Allowing for (109) we obtain
Ema* = const. T5, (111)
i.e., the value of the maximum of radiation in the spectrum of the black radiation is proportional to the fifth power of the absolute temperature.
Should we measure the intensity of monochromatic radiation not by Ex but by K,,, we would obtain for the value of the radia- tion maximum a quite different law, namely,
Kmax = const. T\ (112)
1 0. Lummer und E. Pringsheim, 1. c.
2 F. Paschen, Annal. d. Physik, 6, p. 657, 1901.
CHAPTER IV
RADIATION OF ANY ARBITRARY SPECTRAL DISTRI- BUTION .OF ENERGY. ENTROPY AND TEMPERA- TURE OF MONOCHROMATIC RADIATION
91. We have so far applied Wien's displacement law only to the case of black radiation; it has, however, a much more general importance. For equation (95) , as has already been stated, gives, for any original spectral distribution of the energy radiation con- tained in the evacuated cavity and radiated uniformly in all direc- tions, the change of this energy distribution accompanying a reversible adiabatic change of the total volume. Every state of radiation brought about by such a process is perfectly stationary and can continue infinitely long, subject, however, to the con- dition that no trace of an emitting or absorbing substance exists in the radiation space. For otherwise, according to Sec. 51, the distribution of energy would, in the course of time, change through the releasing action of the substance irreversibly, i.e., with an increase of the total entropy, into the stable distribution correponding to black radiation.
The difference of this general case from the special one dealt with in the preceding chapter is that we can no longer, as in the case of black radiation, speak of a definite temperature of the radiation. Nevertheless, since the second principle of thermo- dynamics is supposed to hold quite generally, the radiation, like every physical system which is in a definite state, has a definite entropy, S = Vs. This entropy consists of the entropies of the monochromatic radiations, and, since the separate kinds of rays are independent of one another, may be obtained by addition. Hence
00 00
s= (sdv, S = V fsdv, (113)
Jo J o
where sdv denotes the entropy of the radiation of frequencies between v and v+dv contained in unit volume. S is a definite
87
88 DEDUCTIONS FROM ELECTRODYNAMICS
function of the two independent variables v and u and in the following will always be treated as such.
92. If the analytical expression of the function s were known, the law of energy distribution in the normal spectrum could immediately be deduced from it; for the normal spectral distri- bution of energy or that of black radiation is distinguished from all others by the fact that it has the maximum of the entropy of radiation S.
Suppose then we take s to be a known function of v and u. Then as a condition for black radiation we have
dS = Q, (114)
for any variations of energy distribution, which are possible with a constant total volume V and constant total energy of radiation U. Let the variation of energy distribution be char- acterized by making an infinitely small change 5u in the energy u of every separate definite frequency v. Then we have as fixed conditions
CO
67 = 0 and (*8udv = 0. (115)
The changes d and 6 are of course quite independent of each other.
Now since dV = 0, we have from (114) and (113)
or, since v remains unvaried
I
'bs — du
and, by allowing for (115), the validity of this equation for all values of 5u whatever requires that
ds
-= const. (116)
oti
for all different frequencies. This equation states the law of energy distribution in the case of black radiation.
93. The constant of equation (116) bears a simple relation to the temperature of black radiation. For if the black radiation,
SPECTRAL DISTRIBUTION OF ENERGY 89
by conduction into it of a certain amount of heat at constant vol- ume V, undergoes an infinitely small change in energy dU, then, according to (73), its change in entropy is
*«? su -¥'
However, from (113) and (116),
5S=V \ ^ 5u dv = ~V \ Su dv = -
hence
and the above quantity, which was found to be the same for all frequencies in the case of black radiation, is shown to be the recip- rocal of the temperature of black radiation.
Through this law the concept of temperature gains sig- nificance also for radiation of a quite arbitrary distribution of energy. For since s depends only on u and v, monochromatic radiation, which is uniform in all directions and has a definite energy density u, has also a definite temperature given by (117), and, among all conceivable distributions of energy, the normal one is characterized by the fact that the radiations of all frequencies have the same temperature.
Any change in the energy distribution consists of a passage of energy from one monochromatic radiation into another, and, if the temperature of the first radiation is higher, the energy transformation causes an increase of the total entropy and is hence possible in nature without compensation; on the other hand, if the temperature of the second radiation is higher, the total entropy decreases and therefore the change is impossible in nature, unless compensation occurs simultaneously, just as is the case with the transfer of heat between two bodies of different tem- peratures.
94. Let us now investigate Wien's displacement law with regard to the dependence of the quantity s on the variables u and v.
90 DEDUCTIONS FROM ELECTRODYNAMICS
From equation (101) it follows, on solving for T and substituting the value given in (117), that
//^3ii\ ?Vo
(118)
where again F represents a function of a single argument and the constants do not contain the velocity of propagation c. On integration with respect to the argument we obtain
C3U
the notation remaining the same. In this form Wien's displace- ment law has a significance for every separate monochromatic radiation and hence also for radiations of any arbitrary energy distribution.
95. According to the second principle of thermodynamics, the total entropy of radiation of quite arbitrary distribution of energy must remain constant on adiabatic reversible compression. We are now able to give a direct proof of this proposition on the basis of equation (119). For such a process, according to equation (113), the relation holds:
CXI
J.
t/ o
Ids dv(V— SU+S57)- (120)
Here, as everywhere, s should be regarded as a function of u and v, and dv = Q.
Now for a reversible adiabatic change of state the relation (95) holds. Let us take from the latter the value of 6u and substitute. Then we have
_,y ("<,,{*
Jo [du
-u+s
In this equation the differential coefficient of u with respect to v refers to the spectral distribution of energy originally assigned arbitrarily and is therefore, in contrast to the partial differential coefficients, denoted by the letter d.
SPECTRAL DISTRIBUTION OF ENERGY 91
Now the complete differential is:
cte_ds du bs dv du dv dj>
Hence by substitution:
But from equation (119) we obtain by differentiation
— -=-pl— j and — =-^W— J- _jpf— -J (122)
Hence
~ = 2s-3u-^ (123)
oi> du
On substituting this in (121), we obtain
or,
as it should be. That the product vs vanishes also for v = oo may be shown just as was done in Sec. 83 for the product i>u.
96. By means of equations (118) and (119) it is possible to give to the laws of reversible adiabatic compression a form in which their meaning is more clearly seen and which is the generalization of the laws stated in Sec. 87 for black radiation and a supplement to them. It is, namely, possible to derive (105) again from (118) and (99b). Hence the laws deduced in Sec. 87 for the change of frequency and temperature of the monochromatic radiation energy remain valid for a radiation of an originally quite arbitrary distribution of energy. The only difference as compared with the black radiation consists in the fact that now every frequency has its own distinct temperature.
Moreover it follows from (119) and (99b) that
92 DEDUCTIONS FROM ELECTRODYNAMICS
Now sdvV = Sdv denotes the radiation entropy between the frequencies v and v+dv contained in the volume V. Hence on account of (125), (99a), and (99c)
S'dv' = Sdv, (126)
i.e., the radiation entropy of an infinitely small spectral interval remains constant. This is another statement of the fact that the total entropy of radiation, taken as the sum of the entropies of all monochromatic radiations contained therein, remains constant.
97. We may go one step further, and, from the entropy s and the temperature T of an unpolarized monochromatic radia- tion which is uniform in all directions, draw a certain conclusion regarding the entropy and temperature of a single, plane polar- ized, monochromatic pencil. That every separate pencil also has a certain entropy follows by the second principle of thermo- dynamics from the phenomenon of emission. For since, by the act of emission, heat is changed into radiant heat, the entropy of the emitting body decreases during emission, and, along with this decrease, there must be, according to the principle of increase of the total entropy, an increase in a different form of entropy as a compensation. This can only be due to the energy of the emitted radiation. Hence every separate, plane polarized, mono- chromatic pencil has its definite entropy, which can depend only on its energy and frequency and which is propagated and spreads into space with it. We thus gain the idea of entropy radiation, which is measured, as in the analogous case of energy radiation, by the amount of entropy which passes in unit time through unit area in a definite direction. Hence statements, exactly similar to those made in Sec. 14 regarding energy radia- tion, will hold for the radiation of entropy, inasmuch as every pencil possesses and conveys, not only its energy, but also its entropy. Referring the reader to the discussions of Sec. 14, we shall, for the present, merely enumerate the most important laws for future use.
98. In a space filled with any radiation whatever the entropy radiated in the time dt through an element of area do- in the direction of the conical element dtt is given by an expression of the form
dt d<r cos 6dttL=L sin 0 cos 6 dB d$ dv dt. (127)
SPECTRAL DISTRIBUTION OF ENERGY 93
The positive quantity L we shall call the " specific intensity of entropy radiation" at the position of the element of area do- in the direction of the solid angle dtt. L is, in general, a function of position, time, and direction.
The total radiation of entropy through the element of area da toward one side, say the one where 6 is an acute angle, is ob- tained by integration with respect to $ from 0 to 2?r and with
respect to 6 from 0 to -. It is
2x x
da- dt I d<f> I dd L sin 6 cos 6.
J<> J o
When the radiation is uniform in all directions, and hence L constant, the entropy radiation through da- toward one side is
*Ld<r dt. (128)
The specific intensity L of the entropy radiation in every direc- tion consists further of the intensities of the separate rays belong- ing to the different regions of the spectrum, which are propagated independently of one another. Finally for a ray of definite color and intensity the nature of its polarization is characteristic. When a monochromatic ray of frequency v consists of two mutually independent1 components, polarized at right angles to each other, with the principal intensities of energy radiation (Sec. 17) K,, and K/, the specific intensity of entropy radiation is of the form
(129)
The positive quantities Lv and L'v in this expression, the principal intensities of entropy radiation of frequency v, are determined by the values of K, and K/. By substitution in (127), this gives for the entropy which is radiated in the time
1 " Independent" in the sense of " noncoherent." If, e.g., a ray with the principal intensities K and K' is elliptically polarized, its entropy is not equal to l_+L_', but equal to the entropy of a plane polarized ray of intensity K + K'. For an elliptically polarized ray may be transformed at once into a plane polarized one, e.g., by total reflection. For the en- tropy of a ray with coherent components see below Sec. 104, et seq.\
94 DEDUCTIONS FROM ELECTRODYNAMICS
dt through the element of area dcr in the direction of the conical element dti the expression
00
dt do- cos 6 dtt
and, for monochromatic plane polarized radiation,
dt da- cos e dtt Lv dv = Lv dv sin 6 cos 6 dd d(j> do- dt. (130) For unpolarized rays LV = \JV and (129) becomes.
For radiation which is uniform in all directions the total entropy radiation toward one side is, according to (128),
2?r da dt
99. From the intensity of the propagated entropy radiation the expression for the space density of the radiant entropy may also be obtained, just as the space density of the radiant energy follows from the intensity of the propagated radiant energy. (Compare Sec. 22.) In fact, in analogy with equation (20), the space density, s, of the entropy of radiation at any point in a vacuum is
!, (131)
where the integration is to be extended over the conical elements which spread out from the point in question in all directions. L is constant for uniform radiation and we obtain
(132) c
By spectral resolution of the quantity L, according to equation (129), we obtain from (131) also the space density of the mono- chromatic radiation entropy:
8--f(L+L')da,
C"
and for unpolarized radiation, which is uniform in all directions
s = — (133)
SPECTRAL DISTRIBUTION OF ENERGY 95
100. As to how the entropy radiation L depends on the energy radiation K Wien's displacement law in the form of (119) affords immediate information. It follows, namely, from it, considering (133) and (24), that
L=-/
2
K\
r)
* /
and, moreover, on taking into account (118),
&L_<te_l_ bK~du~T Hence also
V>KN
(136)
\ va /
or
T\
-) (137)
v I
It is true that these relations, like the equations (118) and (119), were originally derived for radiation which is unpolarized and uniform in all directions. They hold, however, generally in the case of any radiation whatever for each separate monochro- matic plane polarized ray. For, since the separate rays behave and are propagated quite independently of one another, the inten- sity, L, of the entropy radiation of a ray can depend only on the intensity of the energy radiation, K, of the same ray. Hence every separate monochromatic ray has not only its energy but also its entropy defined by (134) and its temperature defined by (136).
101. The extension of the conception of temperature to a single monochromatic ray, just discussed, implies that at the same point in a medium, through which any rays whatever pass, there exist in general an infinite number of temperatures, since every ray passing through the point has its separate temperature, and, moreover, even the rays of different color traveling in the same direction show temperatures that differ according to the spectral distribution of energy. In addition to all these tempera- tures there is finally the temperature of the medium itself, which at the outset is entirely independent of the temperature of the radiation. This complicated method of consideration lies in the
96 DEDUCTIONS FROM ELECTRODYNAMICS
nature of the case and corresponds to the complexity of the physical processes in a medium through which radiation travels in such a way. It is only in the case of stable thermodynamic equilibrium that there is but one temperature, which then is common to the medium itself and to all rays of whatever color crossing it in different directions.
In practical physics also the necessity of separating the concep- tion of radiation temperature from that of body temperature has made itself felt to a continually increasing degree. Thus it has for some time past been found advantageous to speak, not only of the real temperature of the sun, but also of an " apparent" or " effective" temperature of the sun, i.e., that temperature which the sun would need to have in order to send to the earth the heat radiation actually observed, if it radiated like a black body. Now the apparent temperature of the sun is obviously nothing but the actual temperature of the solar rays,1 depending entirely on the nature of the rays, and hence a property of the rays and not a property of the sun itself. Therefore it would be, not only more convenient, but also more correct, to apply this notation directly, instead of speaking of a fictitious temperature of the sun, which can be made to have a meaning only by the introduction of an assumption that does not hold in reality.
Measurements of the brightness of monochromatic light have recently led L. Holborn and F. KuHbaum2 to the introduction of the concept of " black" temperature of a radiating surface. The black temperature of a radiating surface is measured by the brightness of the rays which it emits. It is in general a separate one for each ray of definite color, direction, and polarization, which the surface emits, and, in fact, merely represents the temperature of such a ray. It is, according to equation (136), determined by its brightness (specific intensity), K, and its frequency, i>, without any reference to its origin and previous states. The definite numerical form of this equation will be given below in Sec. 166. Since a black body has the maximum emissive power, the temperature of an emitted ray can never be higher than that of the emitting body.
1 On the average, since the solar rays of different color do not have exactly the same temperature.
2 L. Holborn und F. Kurlbaum, Annal. d. Physik., 10, p. 229, 1903.
SPECTRAL DISTRIBUTION OF ENERGY 97
102. Let us make one more simple application of the laws just found to the special case of black radiation. For this, according to (81), the total space density of entropy is
s = -a*T. (138)
o
Hence, according to (132), the specific intensity of the total entropy radiation in any direction is
L = ~aT°, (139)
07T
and the total entropy radiation through an element of area da- toward one side is, according to (128),
. (140)
3
As a special example we shall now apply the two principles of thermodynamics to the case in which the surface of a black body of temperature T and of infinitely large heat capacity is struck by black radiation of temperature Tf coming from all directions. Then, according to (7) and (76), the black body emits per unit area and unit time the energy
and, according to (140), the entropy
f-
On the other hand, it absorbs the energy
It and the entropy
1$
Hence, according to the first principle, the total heat added to the body, positive or negative according as T' is larger or smaller than T, is
Q = — T 4- — T4 =— (T 4-T4),
98 DEDUCTIONS FROM ELECTRODYNAMICS
and, according to the second principle, the change of the entire entropy is positive or zero. Now the entropy of the body changes
by — , the entropy of the radiation in the vacuum by
Hence the change per unit time and unit area of the entire entropy of the system considered is
In fact this relation is satisfied for all values of T and T". The minimum value of the expression on the left side is zero ; this value is reached when T=T'. In that case the process is reversible. If, however, T differs from T', we have an appreciable increase of entropy; hence the process is irreversible. In particular we find that if T = 0 the increase in entropy is » t i.e., the absorption of heat radiation by a black body of vanishingly small tempera- ture is accompanied by an infinite increase in entropy and cannot therefore be reversed by any finite compensation. On the other hand for T' = 0, the increase in entropy is only equal to
a c
— T3, i.e., the emission of a black body of temperature T without
\Z
simultaneous absorption of heat radiation is irreversible without compensation, but can be reversed by a compensation of at least the stated finite amount. For example, if we let the rays emitted by the body fall back on it, say by suitable reflection, the body, while again absorbing these rays, will necessarily be at the same time emitting new rays, and this is the compensation required by the second principle.
Generally we may say : Emission without simultaneous absorp- tion is irreversible, while the opposite process, absorption without emission, is impossible in nature.
103. A further example of the application of the two principles of thermodynamics is afforded by the irreversible expansion of originally black radiation of volume V and temperature T to the larger volume V as considered above in Sec. 70, but in the absence of any absorbing or emitting substance whatever. Then
SPECTRAL DISTRIBUTION OF ENERGY 99
not only the total energy but also the energy of every separate frequency v remains constant; hence, when on account of diffuse reflection from the walls the radiation has again become uniform in all directions, UVV = ufvVtm, moreover by this relation, according to (118), the temperature T'v of the monochromatic radiation of frequency v in the final state is determined. The actual calcula- tion, however, can be performed only with the help of equation (275) (see below). The total entropy of radiation, i.e., the sum of the entropies of the radiations of all frequencies,
-f,
Jo
dv,
must, according to the second principle, be larger in the final state than in the original state. Since T'v has different values for the different frequencies v, the final radiation is no longer black. Hence, on subsequent introduction of a carbon particle into the cavity, a finite change of the distribution of energy is obtained, and simultaneously the entropy increases further to the value S' calculated in (82).
104. In Sec. 98 we have found the intensity of entropy radia- tion of a definite frequency in a definite direction by adding the entropy radiations of the two independent components K and K', polarized at right angles to each other, or
L(K)+L(K'), (141)
where L denotes the function of K given in equation (134). This method of procedure is based on the general law that the entropy of two mutually independent physical systems is equal to the sum of the entropies of the separate systems.
If, however, the two components of a ray, polarized at right angles to each other, are not independent of each other, this method of procedure no longer remains correct. This may be seen, e.g., on resolving the radiation intensity, not with reference to the two principal planes of polarization with the principal intensities K and K', but with reference to any other two planes at right angles to each other, where, according to equation (8), the intensities of the two components assume the following values
K cos2 1//+ K' sin2 ^ = K" (142)
100 DEDUCTIONS FROM ELECTRODYNAMICS
In that case, of course, the entropy radiation is not equal to L(K") + L(K'").
Thus, while the energy radiation is always obtained by the summation of any two components which are polarized at right angles to each other, no matter according to which azimuth the resolution is performed, since always
(143)
a corresponding equation does not hold in general for the entropy radiation. The cause of this is that the two components, the intensities of which we have denoted by K" and K'", are, unlike K and K', not independent or noncoherent in the optic sense. In such a case
L(K'0 + L(K''0>L(K) + L(K'), (144)
as is shown by the following consideration.
Since in the state of ther mo dynamic equilibrium all rays of the same frequency have the same intensity of radiation, the intensities of radiation of any two plane polarized rays will tend to become equal, i.e., the passage of energy between them will be accompanied by an increase of entropy, when it takes place in the direction from the ray of greater intensity toward that of smaller intensity. Now the left side of the inequality (144) represents the entropy radiation of two noncoherent plane polar- ized rays with the intensities K" and K'", and the right side the entropy radiation of two noncoherent plane polarized rays with the intensities K and K'. But, according to (142), the values of K" and K'" lie between K and K'; therefore the inequality (144) holds.
At the same time it is apparent that the error committed, when the entropy of two coherent rays is calculated as if they were noncoherent, is always in such a sense that the entropy found is too large. The radiations K" and K'" are called " partially coherent," since they have some terms in common. In the special case when one of the two principal intensities K and K' vanishes entirely, the radiations K" and K'" are said to be " completely coherent," since in that case the expression for one radiation may be completely reduced to that for the other. The entropy of two completely coherent plane polarized rays is equal
SPECTRAL DISTRIBUTION OF ENERGY 101
to the entropy of a single plane polarized ray, the energy of which is equal to the sum of the two separate energies.
105. Let us for future use solve also the more general problem of calculating the entropy radiation of a ray consisting of an arbitrary number of plane polarized noncoherent components Ki, K2, K3, ..... , the planes of vibration (planes of the electric vector) of which are given by the azimuths i/% T/% ^3? ..... This problem amounts to finding the principal intensities K0 and K0' of the whole ray; for the ray behaves in every physical respect as if it consisted of the noncoherent com- ponents Ko and K</. For this purpose we begin by establishing the value K^, of the component of the ray for an azimuth \f/ taken arbitrarily. Denoting by / the electric vector of the ray in the direction \l/, we obtain this value K^, from the equation f=fi cos (^i-iM+/2 cos (fo-iM+/3 cos (^3--iW+ ..... » where the terms on the right side denote the projections of the vectors of the separate components in the direction ^, by squaring and averaging and taking into account the fact that /i, /2, /a, . . are noncoherent
or ^ = cos sn sn cos
where A = K! cos2 i£i+K2 cos2 ^2+ ...... (145)
C = 2(Ki sin i/'i cos ^1+ K2 sin i^2 cos
The principal intensities K0 and Ko7 of the ray follow from this expression as the maximum and the minimum value of K^, according to the equation
— = 0 or. tan 2\1/ = — d\p A—B
Hence it follows that the principal intensities are
± V(A-£)2 + C2), (146)
or, by taking (145) into account,
102 DEDUCTIONS FROM ELECTRODYNAMICS
Then the entropy radiation required becomes:
L(Ko) + L(K0'). (148)
106. When two ray components K and K', polarized at right angles to each other, are noncoherent, K and K' are also the prin- cipal intensities, and the entropy radiation is given by (141). The converse proposition, however, does not hold in general, that is to say, the two components of a ray polarized at right angles to each other, which correspond to the principal intensities K and K', are not necessarily noncoherent, and hence the entropy radia- tion is not always given by (141).
This is true, e.g., in the case of elliptically polarized light. There the radiations K and K' are completely coherent and their entropy is equal to L(K+K'). This is caused by the fact that it is possible to give the two ray components an arbitrary dis- placement of phase in a reversible manner, say by total reflection. Thereby it is possible to change elliptically polarized light to plane polarized light and vice versa.
The entropy of completely or partially coherent rays has been investigated most thoroughly by M. Laue.1 For the significance of optical coherence for thermodynamic probability see the next part, Sec. 119.
i M. Laue, Annalen d. Phys., 23, p. 1, 1907.
CHAPTER V
ELECTRODYNAMICAL PROCESSES IN A STATIONARY FIELD OF RADIATION
107. We shall DOW consider from the standpoint of pure elec- trodynamics the processes that take place in a vacuum, which is bounded on all sides by reflecting walls and through which heat radiation passes uniformly in all directions, and shall then inquire into the relations between the electrodynamical and the thermodynamic quantities.
The electrodynamical state of the field of radiation is deter- mined at every instant by the values of the electric field-strength E and the magnetic field-strength H at every point in the field, and the changes in time of these two vectors are completely determined by Maxwell's field equations (52), which we have already used in Sec. 53, together with the boundary conditions, which hold at the reflecting walls. In the present case, however, we have to deal with a solution of these equations of much greater complexity than that expressed by (54), which corresponds to a plane wave. For a plane wave, even though it be periodic with a wave length lying within the optical or thermal spectrum, can never be interpreted as heat radiation. For, according to Sec. 16, a finite intensity K of heat radiation requires a finite solid angle of the rays and, according to Sec. 18, a spectral interval of finite width. But an absolutely plane, absolutely periodic wave has a zero solid angle and a zero spectral width. Hence in the case of a plane periodic wave there can be no question of either entropy or temperature of the radiation.
108. Let us proceed in a perfectly general way to consider the components of the field-strengths E and H as functions of the time at a definite point, which we may think of as the origin of the coordinate system. Of these component's, which are pro- duced by all rays passing through the origin, there are six; we select one of them, say E*, for closer consideration. However
103
104 DEDUCTIONS FROM ELECTRODYNAMICS
complicated it may be, it may under all circumstances be written as a Fourier's series for a limited time interval, say from £ = 0 to t = T; thus
(149)
where the summation is to extend over all positive integers n, while the constants Cn (positive) and 8n may vary arbitrarily from term to term. The time interval T, the fundamental period of the Fourier's series, we shall choose so large that all times t which we shall consider hereafter are included in this time interval, so that 0<£<T. Then we may regard Ez as identical in all respects with the Fourier's series, i.e., we may regard Ez as consisting of " partial vibrations," which are strictly periodic and of frequencies given by
n
"=f
Since, according to Sec. 3, the time differential dt required for the definition of the intensity of a heat ray is necessarily large compared with the periods of vibration of all colors contained in the ray, a single time differential dt contains a large number of vibrations, i.e., the product vdt is a large number. Then it follows a fortiori that vt and, still more,
vT = n is enormously large (150)
for, all values of v entering into consideration. From this we must conclude that all amplitudes Cn with a moderately large value for the ordinal number n do not appear at all in the Fourier's series, that is to say, they are negligibly small.
109. Though we have no detailed special information about the function Ez, nevertheless its relation to the radiation of heat affords some important information as to a few of its general properties. Firstly, for the space density of radiation in a vacuum we^have, according to Maxwell's theory,
u = ~ (^+17+E?+H72+IH72+Hl72). Now the radiation is uniform in all directions and in the stationary
STATIONARY FIELD OF RADIATION 105
state, hence the six mean values named are all equal to one another, and it follows that
u = l^*' (151)
Let us substitute in this equation the value of Ez as given by (149) . Squaring the latter and integrating term by term through a time interval, from 0 to t, assumed large in comparison with all
periods of vibration - but otherwise arbitrary, and then divid- v
ing by t, we obtain, since the radiation is perfectly stationary,
«=-
From this relation we may at once draw an important conclu- sion as to the nature of Ez as a function of time. Namely, since the Fourier's series (149) consists, as we have seen, of a great many terms, the squares, Cn2, of the separate amplitudes of vibration the sum of which gives the space density of radiation, must have exceedingly small values. Moreover in the integral of the square of the Fourier's series the terms which depend on the time t and contain the products of any two different amplitudes all cancel; hence the amplitudes Cn and the phase-constants 6n must vary from one ordinal number to another in a quite irregular manner. We may express this fact by saying that the separate partial vibrations of the series are very small and in a " chaotic"1 state.
For the specific intensity of the radiation travelling in any direction whatever we obtain from (21)
110. Let us now perform the spectral resolution of the last two equations. To begin with we have from (22) :
On the right side of the equation the sum ^ consists of separate
1 Compare footnote to page 116 (Tr.).
106 DEDUCTIONS FROM ELECTRODYNAMICS
terms, every one of which corresponds to a separate ordinal number n and to a simple periodic partial vibration. Strictly speaking this sum does not represent a continuous sequence of frequencies v, since n is an integral number. But n is, according to (150), so enormously large for all frequencies which need be considered that the frequencies v corresponding to the successive values of n lie very close together. Hence the interval dv, though infinitesimal compared with v, still contains a large number of partial vibrations, say nf, where
dv = ^ (155)
If now in (154) we equate, instead of the total energy densities, the energy