The Theory of Heat Radiation (1914) — part 4 of 12

50. Thus far all the laws derived in the preceding sections for diathermanous media hold for a definite frequency, and it is to be kept in mind that a substance may be diathermanous for one color and adiathermanous for another. Hence the radiation of a medium completely enclosed by absolutely reflecting walls is, when thermodynamic equilibrium has been established for all colors for which the medium has a finite coefficient of absorption, always the stable radiation corresponding to the temperature of the medium such as is represented by the emission of a black body. Hence this is briefly called " black" radiation.1 On the other hand, the intensity of colors for which the medium is dia thermanous is not necessarily the stable black radiation, unless the medium is in a state of stationary exchange of radiation with an absorbing substance.

There is but one medium that is diathermanous for all kinds of rays, namely, the absolute vacuum, which to be sure cannot be produced in nature except approximately. However, most gases, e.g., the air of the atmosphere, have, at least if they are not too dense, to a sufficient approximation the optical properties of a vacuum with respect to waves of not too short length. So far as

i M . Thiesen, Verhandlungen d. Deutschen Physikal. Gesellschaft, 2, p. 65, 1900.

44 FUNDAMENTAL FACTS AND DEFINITIONS

this is the case the velocity of propagation q may be taken as the same for all frequencies, namely,

PTT1

c = 3X!010- (51)

sec

51. Hence in a vacuum bounded by totally reflecting walls any state of radiation may persist. But as soon as an arbitrarily small quantity of matter is introduced into the vacuum, a sta tionary state of radiation is gradually established. In this the radiation of every color which is appreciably absorbed by the substance has the intensity K,, corresponding to the temperature of the substance and determined by the universal function (42) for q = c, the intensity of radiation of the other colors remaining indeterminate. If the substance introduced is not diatherma- nous for any color, e.g., a piece of carbon however small, there exists at the stationary state of radiation in the whole vacuum for all colors the intensity K, of black radiation corresponding to the temperature of the substance. The magnitude of K,, regarded as a function of v gives the spectral distribution of black radiation in a vacuum, or the so-called normal energy spectrum, which depends on nothing but the temperature. In the normal spectrum, since it is the spectrum of emission of a black body, the intensity of radiation of every color is the largest which a body can emit at that temperature at all.

52. It is therefore possible to change a perfectly arbitrary radiation, which exists at the start in the evacuated cavity with perfectly reflecting walls under consideration, into black radiation by the introduction of a minute particle of carbon. The charac teristic feature of this process is that the heat of the carbon par ticle may be just as small as we please, compared with the energy of radiation contained in the cavity of arbitrary magnitude. Hence, according to the principle of the conservation of energy, the total energy of radiation remains essentially constant during the change that takes place, because the changes in the heat of the carbon particle may be entirely neglected, even if its changes in temperature should be finite. Herein the carbon particle exerts only a releasing (auslosend) action. Thereafter the intensities of the pencils of different frequencies originally present and having different frequencies, directions, and different states of polari-

RADIATION AT THERMODYNAMIC EQUILIBRIUM 45

zation change at the expense of one another, corresponding to the passage of the system from a less to a more stable state of radiation or from a state of smaller to a state of larger entropy. From a thermodynamic point of view this process is perfectly analogous, since the time necessary for the process is not essential, to the change produced by a minute spark in a quantity of oxy- hydrogen gas or by a small drop of liquid in a quantity of super saturated vapor. In all these cases the magnitude of the dis turbance is exceedingly small and cannot be compared with the magnitude of the energies undergoing the resultant changes, so that in applying the two principles of thermodynamics the cause of the disturbance of equilibrium, viz., the carbon particle, the spark, or the drop, need not be considered. It is always a case of a system passing from a more or less unstable into a more stable state, wherein, according to the first principle of thermodynamics, the energy of the system remains constant, and, according to the second principle, the entropy of the system increases.

PART II

DEDUCTIONS FROM ELECTRODYNAMICS AND THERMODYNAMICS

CHAPTER I MAXWELL'S RADIATION PRESSURE

53. While in the preceding part the phenomena of radiation have been presented with the assumption of only well known elementary laws of optics summarized in Sec. 2, which are com mon to all optical theories, we shall hereafter make use of the electromagnetic theory of light and shall begin by deducing a consequence characteristic of that theory. We shall, namely, calculate the magnitude of the mechanical force, which is exerted by a light or heat ray passing through a vacuum on striking a reflecting (Sec. 10) surface assumed to be at rest.

For this purpose we begin by stating Maxwell's general equa tions for an electromagnetic process in a vacuum. Let the vector E denote the electric field-strength (intensity of the electric field) in electric units and the vector H the magnetic field-strength in magnetic units. Then the equations are, in the abbreviated notation of the vector calculus,

E = c curl H H = — c curl E . .

div. E = 0 div. H = 0

Should the reader be unfamiliar with the symbols of this notation, he may readily deduce their meaning by working backward from the subsequent equations (53).

54. In order to pass to the case of a plane wave in any direction we assume that all the quantities that fix the state depend only on the time t and on one of the coordinates x', yf, z', of an ortho gonal right-handed system of coordinates, say on x'. Then the equations (52) reduce to

to dx' to

49

~

50

DEDUCTIONS FROM ELECTRODYNAMICS €./ bhV bH3> bf

= c

= 0

-.-§ (53)

= 0

Hence the most general expression for a plane wave passing through a vacuum in the direction of the positive o/-axis is

0

'(<-•'

,-!

c

= 0

(54)

Vacuum CC< 0

Conductor

where / and g represent two arbitrary functions of the same argument.

55. Suppose now that this wave strikes a reflecting surface, e.g., the surface of an absolute conductor (metal) of infinitely

large conductivity. In such a conductor even an infinitely small electric field-strength pro duces a finite conduction cur rent; hence the electric field- strength E in it must be always and everywhere infinitely small. For simplicity we also suppose the conductor to be non-mag- netizable, i.e., we assume the magnetic induction B in it to be equal to the magnetic field- strength H, just as is the case in a vacuum.

If we place the z-axis of a right-handed coordinate system (xyz) along the normal of the sur face directed toward the interior

of the conductor, the x-axis is the normal of incidence. We place the (x'yf) plane in the plane of incidence and take this as the plane of the figure (Fig. 4). Moreover, we can also, without

FIG. 4.

MAXWELL'S RADIATION PRESSURE 51

any restriction of generality, place the ?/-axis in the plane of the figure, so that the 2-axis coincides with the z'-axis (directed from the figure toward the observer). Let the common origin 0 of the two coordinate systems lie in the surface. If finally 6 represents the angle of incidence, the coordinates with and with out accent are related to each other by the following equations :

x = xr cos 0 — yf sin 6 x' = x cos 0+y sin 6

y = x' sin B--y' cos 6 y' — — x sin 6--y cos 0

z = zf zf = z

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

Ez = — smd-f -\x = s

Ey = cos0-/ Hy = - cos0-0 . ,

E. = g H. - /

Herein the argument of the functions / and g is

x' x cos 6+y sin 6

t -- = t ---- ww

c c

56. In the surface of separation of the two media x = 0. Ac cording to the general electromagnetic boundary conditions the components of the field-strengths in the surface of separation, i.e., the four quantities Etf, E3, Hj,, H* must be equal to each other on the two sides of the surface of separation for this value of x. In the conductor the electric field-strength E is infinitely small in accordance with the assumption made above. Hence Ey and Ez must vanish also in the vacuum for x = 0. This con dition cannot be satisfied unless we assume in the vacuum, besides the incident, also a reflected wave superposed on the for mer in such a way that the components of the electric field of the two waves in the y and z direction just cancel at every instant and at every point in the surface of separation. By this assump tion and the condition that the reflected wave is a plane wave returning into the interior of the vacuum, the other four compo-

52 DEDUCTIONS FROM ELECTRODYNAMICS

nents of the reflected wave are also completely determined. They are all functions of the single argument

— x cos 0+y sin 0 c~

The actual calculation yields as components of the total electro magnetic field produced in the vacuum by the superposition of the two waves, the following expressions valid for points of the surface of separation x = 0,

Ex = -sin0-/- sin0-/ = - 2 sin0-/

Ev = cos0-/ — cos0-/ = 0

E2 = g -g = 0 (58)

Hz = sin0-g — smO-g = 0

Hy = — COS0-0 — cosd-g = —2 cosd-g

H, =/+/ = 2/.

In these equations the argument of the functions / and g is, ac cording to (56) and (57),

y sin 0

I,

c

From these values the electric and magnetic field-strength within the conductor in the immediate neighborhood of the separating surface x = 0 is obtained :

Ey = 0 Hy = -2 cosd-g

E, = 0 Hz = 2f

where again the argument t ----- — is to be substituted in the

c

functions / and g. For the components of E all vanish in an abso lute conductor and the components Hx, Hj,, Hz are all continuous at the separating surface, the two latter since they are tangential components of the field-strength, the former since it is the normal component of the magnetic induction B (Sec. 55), which likewise remains continuous on passing through any surface of separation. On the other hand, the normal component of the electric field- strength Ex is seen to be discontinuous; the discontinuity shows

MAXWELL'S RADIATION PRESSURE 53

the existence of an electric charge on the surface, the surface density of which is given in magnitude and sign as follows:

— 2 sin0-/= — smO-f. (60)

4rr 2ir

In the interior of the conductor at a finite distance from the bounding surface, i.e., for x>0, all six field components" are infi nitely small. Hence, on increasing x, the values of Hv and Hz, which are finite for x = Q, approach the value 0 at an infinitely rapid rate.

57. A certain mechanical force is exerted on the substance of the conductor by the electromagnetic field considered. We shall calculate the component of this force normal to the surface. It is partly of electric, partly of magnetic, origin. Let us first con sider the former, Fe. Since the electric charge existing on the surface of the conductor is in an electric field, a mechanical force equal to the product of the charge and the field-strength is exerted on it. Since, however, the field-strength is discontinuous, having the value —2 sin Of on the side of the vacuum and 0 on the side of the conductor, from a well-known law of electrostatics the mag nitude of the mechanical force Fe acting on an element of surface da of the conductor is obtained by multiplying the electric charge of the element of area calculated in (60) by the arithmetic mean of the electric field-strength on the two sides. Hence

sin S sin20

F' = ^T~ / d*(-m Of) = — — / da

ATT AIT

This force acts in the direction toward the vacuum and therefore exerts a tension.

58. We shall now calculate the mechanical force of magnetic origin Fm. In the interior of the conducting substance there are certain conduction currents, whose intensity and direction are determined by the vector I of the current density

l=— curl H. (61)

4?r

A mechanical force acts on every element of space dr of the con ductor through which a conduction current flows, and is given by the vector product

-[I'H] (62)

54 DEDUCTIONS FROM ELECTRODYNAMICS

Hence the component of this force normal to the surface of the conductor x = 0 is equal to

— (I.H.-I.H,).

c

On substituting the values of \y and I2 from (61) we obtain

5H,\1

" a* /I

,

2te~H" d* a*

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

Let us now consider a cylinder cut out of the conductor perpen dicular to the surface with the cross-section do-, and extending from x = 0 to x = oo . The entire mechanical force of magnetic origin acting on this cylinder in the direction of the z-axis, since dr = da x, is given by

m ,

4?T

On integration, since H vanishes f or x = oo , we obtain

or by equation (59)

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

F = ^cos20(.P+<72). (63)

This force exerts on the surface of the conductor a pressure, which acts in a direction normal to the surface toward the interior and is

MAXWELL'S RADIATION PRESSURE 55

called "Maxwell's radiation pressure." The existence and the magnitude of the radiation pressure as predicted by the theory was first found by delicate measurements with the radiometer by P . Lebedew.1

59. We shall now establish a relation between the radiation pressure and the energy of radiation Idt falling on the surface element da- of the conductor in a time element dt. The latter from Poynting's law of energy flow is

hence from (55)

= — (EtfH,-E,Hy) d<rdt,

47T

Idt = —cos 0 (/2+02) dadt. 4?r

By comparison with (63) we obtain

F JMS^. (64)

c

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

dtt = sin 0 dd d<f>

in the time dt on the element of area da- is, according to (6), Idt=K cos 9 dtt do- dt,

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

If*

p = - I K cos2 d dQ, (65)

c

the integration with respect to <£ extending from 0 to 2-rr and with respect to 0 from 0 to —

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

56 DEDUCTIONS FROM ELECTRODYNAMICS

In case K is independent of direction as in the case of black radiation, we obtain for the radiation pressure

IT

p = \ d<f> I dB cos2 0 sin 0 =- "

Jd<f> I ( «y °

3c

or, if we introduce instead of K the volume density of radiation u from (21)

P =y. (66)

This value of the radiation pressure holds only when the reflec tion of the radiation occurs at the surface of an absolute non- magnetizable conductor. Therefore we shall in the thermody- namic deductions of the next chapter make use of it only in such cases. Nevertheless it will be shown later on (Sec. 66) that equation (66) gives the pressure of uniform radiation against any totally reflecting surface, no matter whether it reflects uniformly or diffusely.

60. In view of the extraordinarily simple and close relation between the radiation pressure and the energy of radiation, the question might be raised whether this relation is really a special consequence of the electromagnetic theory, or whether it might not, perhaps, be founded on more general energetic or thermo- dynamic considerations. To decide this question we shall cal culate the radiation pressure that would follow by Newtonian mechanics from Newton's (emission) theory of light, a theory which, in itself, is quite consistent with the energy principle. According to it the energy radiated onto a surface by a light ray passing through a vacuum is equal to the kinetic energy of the light particles striking the surface, all moving with the constant velocity c. The decrease in intensity of the energy radiation with the distance is then explained simply by the decrease of the volume density of the light particles.

Let us denote by n the number of the light particles contained in a unit volume and by m the mass of a particle. Then for a. beam of parallel light the number of particles impinging in unit time on the element do- of a reflecting surface at the angle of incidence 6 is

nc cos 6 da. (67)

MAXWELL'S RADIATION PRESSURE 57

Their kinetic energy is given according to Newtonian mechanics

by

/YY) /"»2 y>3

I = nc cos 0 da = nm cos 6 — da. (68)

2 2

Now, in order to determine the normal pressure of these particles on the surface, we may note that the normal component of the velocity c cos 6 of every particle is changed on reflection into a component of opposite direction. Hence the normal component of the momentum of every particle (impulse-coordinate) is changed through reflection by — 2mc cos 6. Then the change in momentum for all particles considered will be, according to (67),

-2nm cos2 6 c2 d<r. (69)

Should the reflecting body be free to move in the direction of the normal of the reflecting surface and should there be no force acting on it except the impact of the light particles, it would be set into motion by the impacts. According to the law of action and reaction the ensuing motion would be such that the momen tum acquired in a certain interval of time would be equal and opposite to the change in momentum of all the light particles reflected from it in the same time interval. But if we allow a separate constant force to act from outside on the reflector, there is to be added to the change in momenta of the light particles the impulse of the external force, i.e., the product of the force and the time interval in question.

Therefore the reflector will remain continuously at rest, when ever the constant external force exerted on it is so chosen that its impulse for any time is just equal to the change in momentum of all the particles reflected from the reflector in the same time. Thus it follows that the force F itself which the particles exert by their impact on the surface element da is equal and opposite to the change of their momentum in unit time as expressed in (69)

F = 2 nm cos2 6 c2 da and by making use of (68),

_ 4 cos 0 c

On comparing this relation with equation (64) in which all symbols have the same physical significance, it is seen that

58 DEDUCTIONS FROM ELECTRODYNAMICS

Newton's radiation pressure is twice as large as Maxwell's for the same energy radiation. A necessary consequence of this is that the magnitude oi^Maxwell's radiation pressure cannot be deduced from general energetic considerations, but is a special feature of the electromagnetic theory and hence all deductions from Max well's radiation pressure are to be regarded as consequences of the electromagnetic theory of light and all confirmations of them are confirmations of this special theory.

CHAPTER II STEFAN-BOLTZMANN LAW OF RADIATION

61. For the following we imagine a perfectly evacuated hollow cylinder with an absolutely tight-fitting piston free to move in a vertical direction with no friction. A part of the walls of the cylinder, say the rigid bottom, should consist of a black body, whose temperature T may be regulated arbitrarily from the out side. The rest of the walls including the inner surface of the pis ton may be assumed as totally reflecting. Then, if the piston remains stationary and the temperature, T, constant, the radia tion in the vacuum will, after a certain time, assume the charac ter of black radiation (Sec. 50) uniform in all directions. The specific intensity, K, and the volume density, u, depend only on the temperature, T, and are independent of the volume, V, of the vacuum and hence of the- position of the piston.

If now the piston is moved downward, the radiation is com pressed into a smaller space; if it is moved upward the radiation expands into a larger space. At the same time the temperature of the black body forming the bottom may be arbitrarily changed by adding or removing heat from the outside. This always causes certain disturbances of the stationary state. If, however, the arbitrary changes in V and T are made sufficiently slowly, the departure from the conditions of a stationary state may always be kept just as small as we please. Hence the state of radiation in the vacuum may, without appreciable error, be regarded as a state of thermodynamic equilibrium, just as is done in the ther modynamics of ordinary matter in the case of so-called infinitely slow processes, where, at any instant, the divergence from the state of equilibrium may be neglected, compared with the changes which the total system considered undergoes as a result of the entire process.

If, e.g., we keep the temperature T of the black body forming the bottom constant, as can be done by a suitable connection

59

60 DEDUCTIONS FROM ELECTRODYNAMICS

between it and a heat reservoir of large capacity, then, on raising the piston, the black body will emit more than it absorbs, until the newly made space is filled with the same density of radiation as was the original one. Vice versa, on lowering the piston the black body will absorb the superfluous radiation until the original radiation corresponding to the temperature T is again established. Similarly, on raising the temperature T of the black body, as can be done by heat conduction from a heat reservoir which is slightly warmer, the density of radiation in the vacuum will be correspondingly increased by a larger emission, etc. To accel erate the establishment of radiation equilibrium the reflecting mantle of the hollow cylinder may be assumed white (Sec. 10), since by diffuse reflection the predominant directions of radiation that may, perhaps, be produced by the direction of the motion of the piston, are more quickly neutralized. The reflecting surface of the piston, however, should be chosen for the present as a perfect metallic reflector, to make sure that the radiation pres sure (66) on the piston is Maxwell's. Then, in order to produce mechanical equilibrium, the piston must be loaded by a weight equal to the product of the radiation pressure p and the cross- section of the piston. An exceedingly small difference of the loading weight will then produce a correspondingly slow motion of the piston in one or the other direction.

Since the effects produced from the outside on the system in question, the cavity through which the radiation travels, during the processes we are considering, are partly of a mechanical nature (displacement of the loaded piston), partly of a thermal nature (heat conduction away from and toward the reservoir), they show a certain similarity to the processes usually considered in thermodynamics, with the difference that the system here considered is not a material system, e.g., a gas, but a purely ener getic one. If, however, the principles of thermodynamics hold quite generally in nature, as indeed we shall assume, then they must also hold for the system under consideration. That is to say, in the case of any change occurring in nature the energy of all systems taking part in the change must remain constant (first principle), and, moreover, the entropy of all systems taking part in the change must increase, or in the limiting case of revers ible processes must remain constant (second principle).

STEFAN-BOLTZMANN LAW OF RADIATION 61

62. Let us first establish the equation of the first principle for an infinitesimal change of the system in question. That the cavity enclosing the radiation has a certain energy we have already (Sec. 22) deduced from the fact that the energy radiation is propagated with a finite velocity. We shall denote the energy by U. Then we have

U=Vu, (70)

where u the volume density of radiation depends only on the temperature of T the black body at the bottom.

The work done by the system, when the volume V of the cavity increases by dV against the external forces of pressure (weight of the loaded piston), is pdV, where p represents Maxwell's radiation pressure (66). This amount of mechanical energy is therefore gained by the surroundings of the system, since the weight is raised. The error made by using the radiation pressure on a stationary surface, whereas the reflecting surface moves during the volume change, is evidently negligible, since the motion may be thought of as taking place with an arbitrarily small velocity.

If, moreover, Q denotes the infinitesimal quantity of heat in mechanical units, which, owing to increased emission, passes from the black body at the bottom to the cavity containing the radiation, the bottom or the heat reservoir connected to it loses this heat Q, and its internal energy is decreased by that amount. Hence, according to the first principle of thermodynamics, since the sum of the energy of radiation and the energy of the material bodies remains constant, we have

dU+pdV-Q = Q. (71)

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

_Q

T

Therefore, since no changes occur in the other bodies — inas much as the rigid absolutely reflecting piston with the weight on it does not change its internal condition with the motion — there

62 DEDUCTIONS FROM ELECTRODYNAMICS

must somewhere in nature occur a compensation of entropy hav ing at least the value — > by which the above diminution is com pensated, and this can be nowhere except in the entropy of the cavity containing the radiation. Let the entropy of the latter be denoted by S.

Now, since the processes described consist entirely of states of equilibrium, they are perfectly reversible and hence there is no increase in entropy. Then we have

dS-- = 0, (72)

rji * \ /

or from (71)

ds = ^~ (73)

In this equation the quantities U, p} V, S represent certain properties of the heat radiation, which are completely defined by the instantaneous state of the radiation. Therefore the quantity T is also a certain property of the state of the radiation, i.e., the black radiation in the cavity has a certain temperature T and this temperature is that of a body which is in heat equilibrium with the radiation.

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

dS = -~dT+—dV (74)

From this we obtain

3>T/v=T dT

STEFAN-BOLTZMANN LAW OF RADIATION 63

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

52>S 1 du 4 du 4u

or

du 4-u

and on integration

u = aT* (75)

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

tf = £ . u = ~ Z". (76)

4?T 47T

Moreover for the pressure of black radiation

P=fr4, (77)

and for the total radiant energy

U = aT*-V. (78)

This law, which states that the volume density and the specific intensity of black radiation are proportional to the fourth power of the absolute temperature, was first established by /. Stefan1 on a basis of rather rough measurements. It was later deduced by L. Boltzmann2 on a thermodynamic basis from Maxwell's radiation pressure and has been more recently confirmed by 0. Lummer and E. Pringsheim* by exact measurements between 100° and 1300° C., the temperature being defined by the gas thermometer. In ranges of temperature and for requirements of precision for which the readings of the different gas thermome ters no longer agree sufficiently or cannot be obtained at all, the Stefan-Boltzmann law of radiation can be used for an absolute definition of temperature independent of all substances.

64. The numerical value of the constant a is obtained from measurements made by F. Kurlbaum.* According to them, if

1 /. Stefan, Wien. Berichte, 79, p. 391, 1879.

2 L. Boltzrnann, Wied. Annalen, 22, p. 291, 1884.

3 0. Lummer und E. Pringsheim, Wied. Annalen, 63, p. 395, 1897. Annalen d. Physik, 3, p. 159, 1900.

4 F. Kurlbaum, Wied. Annalen, 65, p. 759, 1898.

64 DEDUCTIONS FROM ELECTRODYNAMICS

we denote by St the total energy radiated in one second into air by a square centimeter of a black body at a temperature of t° C., the following equation holds

Sioo-& = 0.0731 -~ = 7.31X105 **g cm2 cm2 sec

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

and from this

therefore

= (273+04 4

dC

-S0 = — (3734-2734), 4

4X7.31X105 __ erg

=7.061 X10~15-

3 X 1010 X (3734 - 2734) cm3 degree4

Recently Kurlbaum has increased the value measured by him by 2.5 per cent.,1 on account of the bolometer used being not perfectly black, whence it follows that a = 7.24-10~15.

Meanwhile the radiation constant has been made the object of as accurate measurements as possible in various places. Thus it was measured by Fery, Bauer and Moulin, Valentiner, Fery and Drecq, Shakespear, Gerlach, with in some cases very divergent results, so that a mean value may hardly be formed.

For later computations we shall use the most recent determina tion made in the physical laboratory of the University of Berlin2

^C = cr = 5.46-10-12--^t^- 4 cm* degree*

From this a is found to be

4-5.46-10-12-107 erg

a =

• = 7.28-10-15-

3-1010 cm3 degree4

which agrees rather closely with Kurlbaum' 8 corrected value.

1 F. Kurlbaum, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 580, 1912.

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

STEFAN-BOLTZMANN LAW OF RADIATION 65

65. The magnitude of the entropy S of black radiation found by integration of the differential equation (73) is

S-$aT*V. (80)

3

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

66. We shall now remove a restricting assumption made in order to enable us to apply the value of Maxwell's radiation pressure, calculated in the preceding chapter. Up to now we have assumed the cylinder to be fixed and only the piston to be free to move. We shall now think of the whole of the vessel, consisting of the cylinder, the black bottom, and the piston, the latter attached to the walls in a definite height above the bottom, as being free to move in space. Then, according to the principle of action and reaction, the vessel as a whole must remain con stantly at rest, since no external force acts on it. This is the conclusion to which we must necessarily come, even without, in this case, admitting a priori the validity of the principle of action and reaction. For if the vessel should begin to move, the kinetic energy of this motion could originate only at the ex pense of the heat of the body forming the bottom or the energy of radiation, as there exists in the system enclosed in a rigid cover no other available energy; and together with the decrease of energy the entropy of the body or the radiation would also de crease, an event which would contradict the second principle, since no other changes of entropy occur in nature. Hence the vessel as a whole is in a state of mechanical equilibrium. An immediate consequence of this is that the pressure of the radiation on the black bottom is just as large as the oppositely directed pressure of the radiation on the reflecting piston. Hence the pressure of black radiation is the same on a black as on a reflecting body of the same temperature and the same may be readily proven

66 DEDUCTIONS FROM ELECTRODYNAMICS