The Theory of Heat Radiation (1914) — part 5 of 12

for any completely reflecting surface whatsoever, which we may assume to be at the bottom of the cylinder without in the least disturbing the stationary state of radiation. Hence we may also in all the foregoing considerations replace the reflecting metal by any completely reflecting or black body whatsoever, at the same temperature as the body forming the bottom, and it may be stated as a quite general law that the radiation pressure depends only on the properties of the radiation passing to and fro, not on the properties of the enclosing substance.

67. If, on raising the piston, the temperature of the black body forming the bottom is kept constant by a corresponding addition of heat from the heat reservoir, the process takes place 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 Sf = sV and the heat supplied from the heat reservoir is obtained by integrating (72) at constant T7,

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 %(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 some 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 U' = 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)

s* T'*V 4 IV'

• = — -VF>1. (82)

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 uv 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 uv and K,, 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 uv 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:

1 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 Reflector t + <5t „ , , f , . , f ,,

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

p - 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 —r 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

x+')

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+bt, 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, since

AB-A'B = and

c+v

v — -- v 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 6, 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+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 B AC

BA AC X " V

waves of the ray under consideration. We may further note that the angle of reflection 0' 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 B and 0' ', however, will be shown to be non-essential for our calculation. Moreover there are at th« time t+5t, 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 8t and v'bt waves leave the space through the plane CC' Hence we have

(BA' A

~VT+1

WIEN'S DISPLACEMENT LAW 75

but

BA-BA' = AA'= —

cos 6

AC-A'C' = AA' cos (d+Bf)

X =~f X = -•

v v

Hence

. c cos

c cos 0 — v cos (0+00

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 0 The difference between the two angles 0 and 6' is in any case of

the order of magnitude -; hence we may without appreciable c

error replace 6' by 0, thereby obtaining the following expression for the frequency of the reflected ray for oblique incidence

2v cos B\

•     -  83 
  

c I

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 < 0) . 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 0. 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 6, transmit the energy Idt 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 0

77'

and to the same degree of approximation the work done from the outside on the incident radiation in the time dt is

^°^IU. (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'dt. (85)

Hence we may summarize as follows: By the reflection of a monochromatic unpolarized pencil, incident at an angle 9 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'5t with the interval of frequency (/, i/'+d/), where /' is given by (85), / by (83), and accordingly di>'} the spectral breadth of the reflected pencil, by

(86) (87)

c A comparison of these values shows that

r=z/=^/

/ v dv

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 = 2K,, da- cos 6 dQ dv 5t, (88)

and hence the absolute value of the radiant energy which has been formed is, according to (85),

7'« = 2Mcr cos d dQ dvl+-—8t. (89)

\ c /

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 dff, while, in reality, the incident radiation is slightly increased by the motion of da 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 5t.

At the time t this radiant energy is, according to Sec. 23, V udv, at the time t--dt it is (Fu + 6 (Vu))dv, hence the change to be calculated is

d(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 5.

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 d v 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 (?,<£?) and leave this interval in the time 5t on account of reflection from the moving reflector, are simply those rays which strike the moving reflector in the time 5t. 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 do- of the

WIEN'S DISPLACEMENT LAW 79

reflecting surface at the angle of incidence 6, this energy is, according to (88) and (5),

l8t = 2Kvdo- cos 0 dtt dv dt = 2Kl, do- sin 0 cos 6 dd d^ dv dt.

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 0 from 0 to -, and with

2

respect to do- from 0 to F,

2ir F Kv dv dt. (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 (vydv) 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 Vj the frequency is increased by the reflection, the rays which must be considered have, at the time t, the frequency vi<v. Jf we now consider at the time t a monochromatic pencil of frequency (vi,dv), falling on the reflector at an angle of inci dence 0, a necessary and sufficient condition for its entrance, by reflection, into the interval (v,dv) is

/ 2t> cos 0\ / 2v cos 0

v—vi[l-\ — - I and dv — dvA l-\

c 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 (v,dv) in the time dt is obtained from (89), likewise by substituting v\ for v. It is

2K,i do- cos 8dQdvi[l + ^ C°S }dt = 2<lfldo- cos Bdtidvtt. N c /

Now we have

where we shall assume - to be finite.

80 DEDUCTIONS FROM ELECTRODYNAMICS

Hence, neglecting small quantities of higher order,

_ 2wcose dK

•Vi — *> " ^r~

c ov

Thus the energy required becomes

/ 2w cos 6 dK\

2dv( K, -- - — - ) sin 6 cos B d6 d$ dv 5t,

\ CO*/

and, integrating this expression as above, with respect to do-, $, and B, the total radiant energy which enters into the interval (i',dv) in the time dt becomes

•      <»> 
  

82. The difference of the two expressions (93) and (92) is equal to the whole change (90), hence

3 c di> or, according to (24),

1 du

-- pv v-~-

3 ov

or, finally, since Fvdt is equal to the decrease of the volume V, 1 du

u, (94)

3 ov

whence it follows that

• (#.-)?•

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 5£ 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 8V', 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

-Ffud,,

occurring on adiabatic compression, must be equal to the external work done against the radiation pressure

d.-. (96)

Now from (94) the change in the total energy is found to be

oo

sV f ou = — " '

or, by partial integration,

5V

ov /*

SU —-<[,»]. - iadr),

6 *J 0

and this expression is, in fact, identical with (96), since the prod uct i>u vanishes for v = 0 as well as f or v = <» . The latter might at first seem doubtful; but it is easily seen that, if v\ for v= °° had a value different from zero, the integral of u with respect to v taken from 0 to °° 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 :

bu^du 57 gdj-

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-|#(>19i (98)

where $ denotes an arbitrary function of the single argument vsV. Instead of this we may, on substituting v*V(j>(i>sV) for </>(V7), write

u = v4>(vV). (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 0 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' = ?"<£ (v'W). Therefore, if we put

v'V'=vV, (99a)

we shall also have

~=- andu'7' = u7, (99b)

/• y3

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 X/3~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 = \Jdv denotes the radiant energy lying between the frequencies pand v--dv, which is con tained in the volume V. Now since, according to (99a),

rf,or~=- (9*0

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

-W9

VW

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 </> 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 °° /T\

f* JL /» i J- \

u= I udv = — i v*F \ — )dv, (102)

J0 c3Jo V

T

or, on introducing — = x as the variable of integration instead v

of v,

CO

u = T^\F(^-dx. (103)

If we let the absolute constant

?s~\

a (104)

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

Ji =™ (105)