The Theory of Heat Radiation (1914) — part 10 of 12

2 According to private information kindly furnished by the president, Mr. Warburg.

170 A SYSTEM OF OSCILLATORS

158. For small values of \T (i.e., small compared with the constant — ) equation (276) becomes

K j

an equation which expresses Wien's1 law of energy distribution. The specific intensity of radiation K then becomes, according to (274),

K = ~e "» (281)

c

and the space density of energy u is, from (275),

u-*^r£ (282)

C3

159. On the other hand, for large values of \T (276) becomes

rt-T

£x = yf (283)

a relation which was established first by Lord Rayleigh2 and which we may, therefore, call " Rayleigh' s law of radiation."

We then find for the specific intensity of radiation K from (274)

(284)

and from (275) for the space density of monochromatic radiation we get

8irkv2T

u=- (285)

c3

Rayleigh' s law of radiation is of very great theoretical interest, since it represents that distribution of energy which is obtained for radiation in statistical equilibrium with material molecules by means of the classical dynamics, and without introducing the hypothesis of quanta.3 This may also be seen from the fact that for a vanishingly small value of the quantity element of action, h, the general formula (276) degenerates into Rayleigh's formula (283). See also below, Sec. 168 et seq.

1 W. Wien, Wied. Ann. 58, p. 662, 1896.

2 Lord Rayleigh, Phil. Mag. 49, p. 539, 1900.

3 J. H. Jeans, Phil. Mag. Febr., 1909, p. 229, H. A. Lorentz, Nuovo Cimento V, vol. 16, 1908.

LAW OF NORMAL DISTRIBUTION OF ENERGY 171

160. For the total space density, u, of black radiation at any temperature T we obtain, from (275),

or

_ 3h kT

r OTT/i rv*av

Jo ° Jo ekT _ }

"v _Zhv v

'+e'kT+ . . . .)

e +e and, integrating term by term,

'h kT 4 : (286)

where a is an abbreviation for

=1.0823. (287)

This relation expresses the Stefan-Boltzmann law (75) and it also tells us that the constant of this law is given by

161. For that wave length Xm to which the maximum of the intensity of radiation corresponds in the spectrum of black radia tion, we find from (276)

On performing the differentiation and putting as an abbreviation

ch

k\mT we get

e-"+^_1=0.

5

The root of this transcendental equation is

0 = 4.9651, (289)

ch and accordingly \mT = —-) and this is a constant, as demanded

172 A SYSTEM OF OSCILLATORS

by Wien's displacement law. By comparison with (109) we find the meaning of the constant b, namely,

(290)

and, from (277),

6 = = = 0.289 cm- degree, (291)

p

while Lummer and Pringsheim found by measurements 0.294 and Paschen 0.292.

162. By means of the measured values1 of a and c% the universal constants h and A; may be readily calculated. For it follows from equations (277) and (288) that

(292)

487TQJ

Substituting the values of the constants a, c^, a, c, we get

0-27 erg sec., fc = 1.34-10-16-- (293)

degree

163. To ascertain the full physical significance of the quantity element of action, h, much further research work will be required. On the other hand, the value obtained for k enables us readily to state numerically in the C. G. S. system the general connection between the entropy S and the thermodynamic probability W as expressed by the universal equation (164). The general expression for the entropy of a physical system is

S = 1.34-10-16 log W -e™- (294)

degree

This equation may be regarded as the most general definition of entropy. Herein the thermodynamic probability W is an integral number, which is completely defined by the macroscopic state of the system. Applying the result expressed in (293) to the kinetic

1 Here as well as later on the value given above (79) has been replaced by a = 7.39- 10~15, obtained from <r = a c/4 = 5.54-10-B. This is the final result of the newest meas urements made by W. Westphal, according to information kindly furnished by him and Mr. H. Rubens. (Nov., 1912). [Compare p. 64, footnote. Tr.]

LAW OF NORMAL DISTRIBUTION OF ENERGY 173

theory of gases, we obtain from equation (194) for the ratio of the mass of a molecule to that of a mol,

(295)

that is to say, there are in one mol

--6.20X1023

CO

molecules, where the mol of oxygen, 02, is always assumed as 32 gr. Hence, for example, the absolute mass of a hydrogen atom ($#2 = 1-008) equals 1.62X1Q-24 gr. With these numer ical values the number of molecules contained in 1 cm.3 of an ideal gas at 0° C. and 1 atmosphere pressure becomes

The mean kinetic energy of translatory motion of a molecule at the absolute temperature T = l is, in the absolute C. G. S. system, according to (200),

-/c = 2.0MO-16 (297)

In general the mean kinetic energy of translatory motion of a molecule is expressed by the product of this number and the absolute temperature T.

The elementary quantity of electricity or the free charge of a monovalent ion pr electron is, in electrop tatic units,

4.67-10-10. (298)

Since absolute accuracy is claimed for the formulae here em ployed, the degree of approximation to which these numbers represent the corresponding physical constants depends only on the accuracy of the measurements of the two radiation constants a and c2.

164. Natural Units. — All the systems of units which have hitherto been employed, including the so-called absolute C. G. S. system, owe their origin to the coincidence of accidental circum-

174 A SYSTEM OF OSCILLATORS

stances, inasmuch as the choice of the units lying at the base of every system has been made, not according to general points of view which would necessarily retain their importance for all places and all times, but essentially with reference to the special needs of our terrestrial civilization.

Thus the units of length and time were derived from the pres ent dimensions and motion of our planet, and the units of mass and temperature from the density and the most important temperature points of water, as being the liquid which plays the most important part on the surface of the earth, under a pressure which corresponds to the mean properties of the atmosphere surrounding us. It would be no less arbitrary if, let us say, the invariable wave length of Na-light were taken as unit of length. For, again, the particular choice of Na from among the many chemical elements could be justified only, perhaps, by its com mon occurrence on the earth, or by its double line, which is in the range of our vision, but is by no means the only one of its kind. Hence it is quite conceivable that at some other time, under changed external conditions, every one of the systems of units which have so far been adopted for use might lose, in part or wholly, its original natural significance.

In contrast with this it might be of interest to note that, with the aid of the two constants h and k which appear in the universal law of radiation, we have the means of establishing units of length, mass, time, and temperature, which are independent of special bodies or substances, which necessarily retain their significance for all times and for all environments, terrestrial and human or otherwise, and which may, therefore, be described as " natural units."

The means of determining the four units of length, mass, time, and temperature, are given by the two constants h and k men tioned, together with the magnitude of the velocity of propaga tion of light in a vacuum, c, and that of the constant of gravita tion, /. Referred to centimeter, gram, second, and degrees Centigrade, the numerical values 'of these four constants are as follows:

sec

LAW OF NORMAL DISTRIBUTION OF ENERGY 175

sec

c = 3-10- ^ sec

cm

/ = 6.685-10-

If we now choose the natural units so that in the new system of measurement each of the four preceding constants assumes the value 1, we obtain, as unit of length, the quantity

Ifh

Y- = 3.99-10-33 cm, as unit of mass

Y- = 5.37-10-50, as unit of time

/^ = 1.33-10-43sec,

\C5

as unit of temperature

jr\V = 3.6O1032 degree.

These quantities retain their natural significance as long as the law of gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always the same, when measured by the most widely differing intelligences according to the most widely differing methods.

165. The relations between the intensity of radiation and the temperature expressed in Sec. 156 hold for radiation in a pure vacuum. If the radiation is in a medium of refractive index n, the way in which the intensity of radiation depends on the frequency and the temperature is given by the proposition of Sec. 39, namely, the product of the specific intensity of radiation K,, and the square of the velocity of propagation of the radiation

1 F. Richarz and 0. Krigar-Menzel, Wied. Ann. 66, p. 190, 1898.

176 A SYSTEM OF OSCILLATORS

has the same value for all substances. The form of this universal function (42) follows directly from (274)

hv*

Kq* = - q^ = hT (299)

a" ekT -I

Now, since the refractive index n is inversely proportional to the velocity of propagation, equation (274) is, in the case of a medium with the index of refraction n, replaced by the more general rela tion

hvV 1 K,--^- -*T- (300)

e kT — I

and, similarly, in place of (275) we have the more general relation

SirhvW 1

u= 3 ^r (301)

e kT — 1

These expressions hold, of course, also for the emission of a body which is black with respect to a medium with an index of refrac tion n.

166. We shall now use the laws of radiation we have obtained to calculate the temperature of a monochromatic unpolarized radiation of given intensity in the following case. Let the light pass normally through a small area (slit) and let it fall on an arbitrary system of diathermanous media separated by spherical surfaces, the centers of which lie on the same line, the axis of the system. Such radiation consists of homocentric pencils and hence forms behind every refracting surface a real or virtual image of the emitting surface, the image being likewise normal to the axis. To begin with, we assume the last as well as the first medium to be a pure vacuum. Then, for the determination of the temperature of the radiation according to equation (274), we need calculate only the specific intensity of radiation Kv in the last medium, and this is given by the total intensity of the monochromatic radiation /„, the size of the area of the image F, and the solid angle 12 of the cone of rays passing through a point of the image. For the specific intensity of radiation K, is, according to (13), determined by the fact that an amount

2K, da dQ dv dt

LAW OF NORMAL DISTRIBUTION OF ENERGY 177

of energy of unpolarized light corresponding to the interval of frequencies from vto v--dv is, in the time dt, radiated in a normal direction through an element of area do- within the conical element dti. If now da- denotes an element of the area of the surface image in the last medium, then the total monochromatic radia tion falling on the image has the intensity

lv is of the dimensions of energy, since the product dv dt is a mere number. The first integral is the whole area, F} of the image, the second is the solid angle, £2, of the cone of rays passing through a point of the surface of the image. Hence we get

OlX' 77I(-) /QPl^

v = *i\vrii) \6\jZi)

and, by making use of (274), for the temperature of the radiation hv 1

k 2hv*FQ , , (303)

If the diathermanous medium considered is not a vacuum but has an index of refraction nt (274) is replaced by the more general relation (300), and, instead of the last equation, we obtain

k 2hv*Fttnz \ (304)

log(-^/r-+1

or, on substituting the numerical values of c, h, and k,

0.479-10-10»

T — 7 — — r degree Centigrade.

/1.43-10-4WQn2 x logl- — -+1

V i v

In this formula, the natural logarithm is to be taken, and 7, is to be expressed in ergs, v in " reciprocal seconds," i.e., (seconds)"1, F in square centimeters. In the case of visible rays the second term, 1, in the denominator may usually be omitted.

The temperature thus calculated is retained by the radiation

considered, so long as it is propagated without any disturbing 12

178 A SYSTEM OF OSCILLATORS

influence in the diathermanous medium, however great the dis tance to which it is propagated or the space in which it spreads. For, while at larger distances an ever decreasing amount of energy is radiated through an element of area of given size, this is con tained in a cone of rays starting from the element, the angle of the cone continually decreasing in such a way that the value of K remains entirely unchanged. Hence the free expansion of radia tion is a perfectly reversible process. (Compare above, Sec. 144.) It may actually be reversed by the aid of a suitable concave mirror or a converging lens.

Let us next consider the temperature of the radiation in the other media, which lie between the separate refracting or reflect ing spherical surfaces. In every one of these media the radiation has a definite temperature, which is given by the last formula when referred to the real or virtual image formed by the radiation in that medium.

The frequency v of the monochromatic radiation is, of course, the same in all media; moreover, according to the laws of geomet rical optics, the product n2Ftt is the same for all media. Hence, if, in addition, the total intensity of radiation /„ remains constant on refraction (or reflection), T also remains constant, or in other words: The temperature of a homocentric pencil is not changed by regular refraction or reflection, unless a loss in energy of radiation occurs. Any weakening, however, of the total inten sity /„ by a subdivision of the radiation, whether into two or into many different directions, as in the case of diffuse reflection, leads to a lowering of the temperature of the pencil. In fact, a certain loss of energy by refraction or reflection does occur, in general, on a refraction or reflection, and hence also a lowering of the temperature takes place. In these cases a fundamental difference appears, depending on whether the radiation is weak ened merely by free expansion or by subdivision or absorption. In the first case the temperature remains constant, in the second it decreases.1

167. The laws of emission of a black body having been deter-

1 Nevertheless regular refraction and reflection are not irreversible processes; for the refracted and the reflected rays are coherent and the entropy of two coherent rays is not equal to the sum of the entropies of the separate rays. (Compare above, Sec. 104.) On the other hand, diffraction is an irreversible process. M. Laue, Ann. d. Phys. 31, p. 547, 1910.

LAW OF NORMAL DISTRIBUTION OF ENERGY 179

mined, it is possible to calculate, with the aid of Kirchhoff's law (48), the emissive power E of any body whatever, when its absorbing power A or its reflecting power 1 —A is known. In the case of metals this calculation becomes especially simple for long waves, since E. Hagen and H. Rubens1 have shown experimentally that the reflecting power and, in fact, the entire optical behavior of the metals in the spectral region mentioned is represented by the simple equations of Maxwell for an electromagnetic field with homogeneous conductors and hence depends only on the specific conductivity for steady electric currents. Accordingly, it is possible to express completely the emissive power of a metal for long waves by its electric conductivity combined with the for mulae for black radiation.2

168. There is, however, also a method, applicable to the case of long waves, for the direct theoretical determination of the elec tric conductivity and, with it, of the absorbing power, A, as well as the emissive power, E, of metals. This is based on the ideas of the electron theory, as they have been developed for the ther mal and electrical processes in metals by E. Rieckez and especially by P. Drude.* According to these, all such processes are based on the rapid irregular motions of the negative electrons, which fly back and forth between the positively charged molecules of mat ter (here of the metal) and rebound on impact with them as well as with one another, like gas molecules when they strike a rigid obstacle or one another. The velocity of the heat motions of the material molecules may be neglected compared with that of the electrons, since in the stationary state the mean kinetic energy of motion of a material molecule is equal to that of an electron, and since the mass of a material molecule is more than a thousand times as large as that of an electron. Now, if there is an electric field in the interior of the metal, the oppositely charged particles are driven in opposite directions with average velocities depend ing on the mean free path, among other factors, and this explains the conductivity of the metal for the electric current. On the other hand, the emissive power of the metal for the radiant heat follows from the calculation of the impacts of the electrons. For,

1 E. Hagen und H. Rubens, Ann. d. Phs.yll, p. 873, 1903. 1 E. Aschkinass, Ann. d. Phys. 17, p. 960, 1905.

3 E. Riecke, Wied. Ann. 66, p. 353, 1898.

4 P. Drude, Ann. d. Phys. 1, p. 566, 1900.

180 A SYSTEM OF OSCILLATORS

so long as an electron flies with constant speed in a constant direction, its kinetic energy remains constant and there is no radiation of energy; but, whenever it suffers by impact a change of its velocity components, a certain amount of energy, which may be calculated from electrodynamics and which may always be represented in the form of a Fourier's series, is radiated into the surrounding space, just as we think of Roentgen rays as being caused by the impact on the anticathode of the electrons ejected from the cathode. From the standpoint of the hypothesis of quanta this calculation cannot, for the present, be carried out without ambiguity except under the assumption that, during the time of a partial vibration of the Fourier series, a large number of impacts of electrons occurs, i.e., for comparatively long waves, for then the fundamental law of impact does not essentially matter.

Now this method may evidently be used to derive the laws of black radiation in a new way, entirely independent of that pre viously employed. For if the emissive power, E, of the metal, thus calculated, is divided by the absorbing power, A, of the same metal, determined by means of its electric conductivity, then, according to Kirchhoff's law (48), the result must be the emissive power of a black body, irrespective of the special substance used in the determination. In this manner H. A. Lorentz1 has, in a profound investigation, derived the law of radiation of a black body and has obtained a result the contents of which agree exactly with equation (283), and where also the constant k is related to the gas constant R by equation (193) . It is true that this method of establishing the laws of radiation is, as already said, restricted to the range of long waves, but it affords a deeper and very impor tant insight into the mechanism of the motions of the electrons and the radiation phenomena in metals caused by them. At the same time the point of view described above in Sec. Ill, 'according to which the normal spectrum may be regarded as consisting of a large number of quite irregular processes as elements, is expressly confirmed.

169. A further interesting confirmation of the law of radiation of black bodies for long waves and of the connection of the radiation constant k with the absolute mass of the material

i //. A. Lorentz, Proc. Kon. Akad. v. Wet. Amsterdam, 1903, p. 666.

LAW OF NORMAL DISTRIBUTION OF ENERGY 181

molecules was found by /. H. Jeans1 by a method previously used by Lord Rayleigh,2 which differs essentially from the one pursued here, in the fact that it entirely avoids making use of any special mutual action between matter (molecules, oscillators) and the ether and considers essentially only the processes in the vacuum through which the radiation passes. The starting point for this method of treatment is given by the following proposition of statistical mechanics. (Compare above, Sec. 140.) When irreversible processes take place in a system, which satisfies Hamilton's equations of motion, and whose state is determined by a large number of independent variables and whose total energy is found by addition of different parts depend ing on the squares of the variables of state, they do so, on the average, in such a sense that the partial energies corresponding to the separate independent variables of state tend to equality, so that finally, on reaching statistical equilibrium, their mean values have become equal. From this proposition the stationary distribution of energy in such a system may be found, when the independent variables which determine the state are known.

Let us now imagine a perfect vacuum, cubical in form, of edge I, and with metallically reflecting sides. If we take the origin of coordinates at one corner of the cube and let the axes of coordinates coincide with the adjoining edges, an electromagnetic process which may occur in this cavity is represented by the following system of equations:

, b-jry . Cirz x = cos — — sin — - sin — —(e\ cos 2irvt--e \ sin

mil

&TTX biry . CTTZ, Ey = sm — cos — — sin — (e2 cos 2rrf-fe'i sin 2ri»f),

III

_ 3LTTX , biry GTTZ.

E2 = sm — — sin — — cos ~T"(«J cos 2jrvt+e 3 sin 2rri),

Hx = sin— -— cos -- cos -— (^i sin Invt — li'i cos 2wvt),

ill

1 J. H. Jeans, Phil. Mag. 10, p. 91, 1905.

2 Lord Rayleigh, Nature 72, p. 54 and p. 243, 1905.

182 A SYSTEM OF OSCILLATORS

. C7T2:

sin — — cos — (h2 sin 2irvt — h 2 cos 2wvi),

III

a-irx iry , C7r£/7

•H2 =cos— - cos — — - sm — («i sin 2irvt — h 3 cos 2x^0, ill

where a, b, c represent any three positive integral numbers. The boundary conditions in these expressions are satisfied by the fact that for the six bounding surfaces z = 0, x = l, y = Q, y = l, 2 = 0, z = l the tangential components of the electric field-strength E vanish. Maxwell's equations of the field (52) are also satisfied, as may be seen on substitution, provided there exist certain condi tions between the constants which may be stated in a single proposition as follows : Let a be a certain positive constant, then there exist between the nine quantities written in the following square :

ac be cc

hi a

all the relations which are satisfied by the nine so-called " direc tion cosines" of two orthogonal right-handed coordinate systems, i.e., the cosines of the angles of any two axes of the systems.

Hence the sum of the squares of the terms of any horizontal or vertical row equals 1, for example,

(306)

Ai2+/*22+ h,2 = a2 = ei2+e22+e32.

Moreover the sum of the products of corresponding terms in any two parallel rows is equal to zero, for example,

LAW OF NORMAL DISTRIBUTION OF ENERGY 183

Moreover there are relations of the following form:

hi_ez _cc 63 be ^ a a 21 v a 2lv

and hence

hi = — (ce2— bea), etc. (308)

21 v

If the integral numbers a, b, c are given, then the frequency v is immediately determined by means of (306). Then among the six quantities d, ez, e3, hi, hz, h3, only two may be chosen arbi trarily, the others then being uniquely determined by them by linear homogeneous relations. If, for example, we assume e\ and e2 arbitrarily, e3 follows from (307) and the values of hi, hz, h3 are then found by relations of the form (308). Between the quantities with accent e\ , ez, esf, hi, hz, h3' there exist exactly the same relations as between those without accent, of which they are entirely independent. Hence two also of them, say hi and hz, may be chosen arbitrarily so that in the equations given above for given values of a, b, c four constants remain undetermined. If we now form, for all values of a b c whatever, expressions of the type (305) and add the corresponding field components, we again obtain a solution for Maxwell's equations of the field and the boundary conditions, which, however, is now so general that it is capable of representing any electromagnetic process possible in the hollow cube considered. For it is always possible to dispose of the constants ei, ez, hi, hz which have remained undetermined in the separate particular solutions in such a way that the process may be adapted to any initial state (£ = 0) whatever.

If now, as we have assumed so far, the cavity is entirely void of matter, the process of radiation with a given initial state is uniquely determined in all its details. It consists of a set of stationary vibrations, every one of which is represented by one of the particular solutions considered, and which take place entirely independent of one another. Hence in this case there can be no question of irreversibility and hence also none of any tendency to equality of the partial energies corresponding to the separate partial vibrations. As soon, however, as we assume the

184 A SYSTEM OF OSCILLATORS

presence in the cavity of only the slightest trace of matter which can influence the electrodynamic vibrations, e.g., a few gas molecules, which emit or absorb radiation, the process becomes chaotic and a passage from less to more probable states will take place, though perhaps slowly. Without considering any further details of the electromagnetic constitution of the molecules, we may from the law of statistical mechanics quoted above draw the conclusion that, among all possible processes, that one in which the energy is distributed uniformly among all the inde pendent variables of the state has the stationary character.

From this let us determine these independent variables. In the first place there are the velocity components of the gas mole cules. In the stationary state to every one of the three mutually independent velocity components of a molecule there corresponds on the average the energy \L where L represents the mean energy of a molecule and is given by (200). Hence the partial energy, which on the average corresponds to any one of the independent variables of the electromagnetic system, is just as large.

Now, according to the above discussion, the electro-magnetic state of the whole cavity for every stationary vibration corre sponding to any one system of values of the numbers a b c is determined, at any instant, by four mutually independent quan tities. Hence for the radiation processes the number of inde pendent variables of state is four times as large as the number of the possible systems of values of the positive integers a, b, c.

We shall now calculate the number of the possible systems of values a, b, c, which correspond to the vibrations within a certain small range of the spectrum, say between the frequencies v and v--dv. According to (306), these systems of values satisfy the inequalities

(309)

C

where not only — but also - - is to be thought of as a large c c

number. If we now represent every system of values of a, b, c graphically by a point, taking a, b, c as coordinates in an orthog onal coordinate system, the points thus obtained occupy one octant of the space of infinite extent, and condition (309) is

LAW OF NORMAL DISTRIBUTION OF ENERGY 185

equivalent to requiring that the distance of any one of these

21 v points from the origin of the coordinates shall lie. between

c

and - -- Hence the required number is equal to the c

number of points which lie between the two spherical surface-

.. 2lv , 2l(v+dv)

octants corresponding to the radii — and — — Now since

c c

to every point there corresponds a cube of volume 1 and vice versa, that number is simply equal to the space between the two spheres mentioned, and hence equal to

and the number of the independent variables of state is four times as large or

Since, moreover, the partial energy - corresponds on the aver-

3

age to every independent variable of state in the state of equilib rium, the total energy falling in the interval from v to v--dv becomes

IfrrZVdp - 3c«

Since the volume of the cavity is I3, this gives for the space density of the energy of frequency v

and, by substitution of the value of L = — from (200),

N

(310)

which is in perfect agreement with Rayleigh's formula (285). If the law of the equipartition of energy held true in all

186 A SYSTEM OF OSCILLATORS

cases, Rayleigh's law of radiation would, in consequence, hold for all wave lengths and temperatures. But since this possibility is excluded by the. measurements at hand, the only possible conclusion is that the law of the equipartition of energy and, with it, the system of Hamilton's equations of motion does not possess the general importance attributed to it in classical dynam ics. Therein lies the strongest proof of the necessity of a funda mental modification of the latter.

PART V IRREVERSIBLE RADIATION PROCESSES

CHAPTER I FIELDS OF RADIATION IN GENERAL

170. According to the theory developed in the preceding sec tion, the nature of heat radiation within an isotropic medium, when the state is one of stable thermodynamic equilibrium, may be regarded as known in every respect. The intensity of the radiation, uniform in all directions, depends for all wave lengths only on the temperature and the velocity of .propagation, accord ing to equation (300), which applies to black radiation in any medium whatever. But there remains another problem to be solved by the theory. It is still necessary to explain how and by what processes the radiation which is originally present in the medium and which may be assigned in any way whatever,- passes gradually, when the medium is bounded by walls imper meable to heat, into the stable state of black radiation, corre sponding to the maximum of entropy, just as a gas which is enclosed in a rigid vessel and in which there are originally cur rents and temperature differences assigned in any way whatever gradually passes into the state of rest and of uniform distribution of temperature.

To this much more difficult question only a partial answer can, at present, be given. In the first place, it is evident from the extensive discussion in the first chapter of the third part that, since irreversible processes are to be dealt with, the principles of pure electrodynamics alone will not suffice. For the second prin ciple of thermodynamics or the principle of increase of entropy is foreign to the contents of pure electrodynamics as well as of pure mechanics. This is most immediately shown by the fact that the fundamental equations of mechanics as well as those of electro dynamics allow the direct reversal of every process as regards time, which contradicts the principle of increase of entropy. Of course all kinds of friction and of electric conduction of cur-

189

190 IRREVERSIBLE RADIATION PROCESSES

rents must be assumed to be excluded; for these processes, since they are always connected with the production of heat, do not belong to mechanics or electrodynamics proper.

This assumption being made, the time t occurs in the funda mental equations of mechanics only in the components of acceleration; that is, in the form of the square of its differential. Hence, if instead of t the quantity — t is intrpduced as time variable in the equations of motion, they retain their form without change, and hence it follows that if in any motion of a system of material points whatever the velocity components of all points are sud denly reversed at any instant, the process must take place in the reverse direction. For the electrodynamic processes in a homogeneous non-conducting medium a similar statement holds. If in Maxwell's equations of the electrodynamic field — t is written everywhere instead of t, and if, moreover, the sign of the magnetic field-strength H is reversed, the equations remain unchanged, as can be readily seen, and hence it follows that if in any electrodynamic process whatever the magnetic field-strength is everywhere suddenly reversed at a certain instant, while the electric field-strength keeps its value, the whole process must take place in the opposite sense.

If we now consider any radiation processes whatever, taking place in a perfect vacuum enclosed by reflecting walls, it is found that, since they are completely determined by the principles of classical electrodynamics, there can be in their case no question of irreversibility of any kind. This is seen most clearly by con sidering the perfectly general formulae (305), which hold for a cubical cavity and which evidently have a periodic, i.e., reversible character. Accordingly we have frequently (Sec. 144 and 166) pointed out that the simple propagation of free radiation represents a reversible process. An irreversible element is introduced by the addition of emitting and absorbing sub stance.

171. Let us now try to define for the general case the state of radiation in the thermodynamic-macroscopic sense as we did above in Sec. 107, et seq., for a stationary radiation. Every one of the three components of the electric field-strength, e.g., E2 may, for the long time interval from t = 0 to t = T, be represented at every point, e.g., at the origin of coordinates, by a Fourier's

FIELDS OF RADIATION IN GENERAL 191

integral, which in the present case is somewhat more convenient than the Fourier's series (149) :

00

E2= fdi^cos (271-^-0,), (311)

o