The Theory of Heat Radiation (1914) — part 3 of 12

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 d£l is obtained by multiplying the last expres sion by -r- . This gives

00

f

Jo

K, dv,

and, for monochromatic plane polarized radiation,

dtvdSlhK,, 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 do- at the point 0 (Fig. 2), perpen dicular to the axis of the pencil, and by the solid angle dfi, 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 dfi 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 dtt, 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 do- at 0 a certain quantity E of energy of monochro matic plane polarized radiation. E may be obtained from (1) by putting

do- dt = l, dr = dr0 r02 dtt, dtt = -^

TO

and omitting the numerical factor 2. We thus get

E = dr0'dttda e, dv. (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 (<*„+ fi^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

or,

— dr

and, by integration,

since, for r = r0, Er = E is given by equation (31). Fromthis,-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 dfi do- c, -•*+**• dv. (32)

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

00

d!2 da dv e, f dr0 e-(a>+ft>)r° = dttda —^— dv. (33)

f 1

31. If the scattering did not affect the radiation, the total energy reaching dcr 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 Ef thus collected by the volume- element at r0 is found, by putting in (29),

dt =1, v = dr0 dQ, r

do-

to be

E' = dr0 dQ 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 dtt do- (e,+ft K,) dv. The part of this reaching 0 is, similar to (32) : dr0 dti da (e, + ft K.) dv e~ro(a -"W

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-

dti da fe+ft K,) d

and this expression is realty 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 Kv, 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 da of the bounding surface and contained in the conical element dtt,

where

dt do- cos 6 d$l K,, dv, d!2 = sin0 d6 d<f>.

(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$l, the second in the conical element

dn' = sin 0' dtf dct>' where, according to the law of refraction,

sin0 q

^-=^ sm0 q

(36) (37)

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 da 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 do- cos 0' da' 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 da K, + (l-p/) cos 0' daf K/ = cos 0 da K,. Now from (37) we have

cos 0 dd cos 0' dd'

q q'

and further by (35) and (36)

; , da cos 0 q'2

Therefore we find

;* ?"„,_„

or

K,

K/ q'* 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

<? K,-g" K/ (41) 38. The first of these two relations, which states that the coefficient of reflection of the bounding surface is the same on both sides, is a special case of a general law of reciprocity first stated by Helmholtz.1 According to this law the loss of intensity which a ray of definite color and polarization suffers on its way through any media by reflection, refraction, absorption, and scattering is exactly equal to the loss 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 <?2K, = <?*-- (42) «„ 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 iiber 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. Clausius, 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 Ky 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- holt^ 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 Kv 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 da of the bounding surface of the two media in the direction toward the first medium (Fig. 3, feathered arrow) within the conical element d£l. Then, as in (34), the energy supplied by the pencil in unit time is da cos 0 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 da, 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, / = /1 + /2 + /3+ /,. (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 /i, J2, 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 dQ, 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 da 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 Ji} that which is finally absorbed in the second medium by J2, etc., then we shall have J = Jl + J* + J*+ +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 I, 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 I 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 J in just the same ratio as they contribute by their emission to the energy of the opposite pencil 7. Since, moreover, the sum I of the energies given off by emission by all volume-elements is equal to the sum J of the energies absorbed by all elements, the quantity of energy absorbed by each separate volume-element from the pencil J must be equal to the quantity of energy emitted by the same element into the pencil I. In other words : the part of a pencil I which has been emitted from a certain volume of any medium is equal to the part of the pencil J( = I) oppositely directed, which is absorbed in the same volume. Hence not only are the sums / and J equal, but their constitu ents are also separately equal or Jl = 7i, J2=/2, Jn=In> (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 Jz to J, i.e., that fraction of a pencil incident on the second medium which is absorbed in this medium, the absorbing power A of the second medium. Therefore E = h(<I), A^(<\). (47) J 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 do- and that of the solid angle dtt, (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, iG. Kirchhoff, Gcsammelte Abhandlungen, 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, E — = I = da cos 0 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 K, but the quantity g2 Ky, 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 = Ar.i.e., the emissive power of a black body is independent of its nature. Its emissive power is larger than that of any other body at the same temperature and, in fact, is just equal to the intensity of radiation in the contiguous medium. 46. We shall now add, without further proof, another general law of reciprocity, which is closely connected with that stated at the end of Sec. 43 and which may be stated thus: When any emitting and absorbing bodies are in the state of thermodynamic equilibrium, the part 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 u, 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, dv-£ (50) V3 also is a universal function of T, i>, 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 €„ = () 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 q2 K,, has the same value for all sub stances. From this we find immediately a definite value of K,, 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 v, 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 O. 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 da, 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.