Fields of Radiation in General

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

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

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

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

171. Let us now try to define for the general case the state of radiation in the thermodynamic-macroscopic sense as we did above in Sec. 107, et seq., for a stationary radiation. Every one of the three components of the electric field-strength, e.g., $E_z$ may, for the long time interval from $t=0$ to $t=T$, be represented at every point, e.g., at the origin of coordinates, by a Fourier’s integral, which in the present case is somewhat more convenient than the Fourier’s series (149):

where $C_{\nu }$ (positive) and ${\theta }_{\nu }$ denote certain functions of the positive variable of integration $\nu$. The values of these functions are not wholly determined by the behavior of $E_z$ in the time interval mentioned, but depend also on the manner in which $E_z$ varies as a function of the time beyond both ends of that interval. Hence the quantities $C_{\nu }$ and ${\theta }_{\nu }$ possess separately no definite physical significance, and it would be quite incorrect to think of the vibration $E_z$ as, say, a continuous spectrum of periodic vibrations with the constant amplitudes $C_{\nu }$. This may, by the way, be seen at once from the fact that the character of the vibration $E_z$ may vary with the time in any way whatever. How the spectral resolution of the vibration $E_z$ is to be performed and to what results it leads will be shown below (Sec. 174).

172. We shall, as heretofore (158), define $J$, the “intensity of the exciting vibration,”67 as a function of the time to be the mean value of $E_z^2$ in the time interval from $t$ to $t+\tau$, where $\tau$ is taken as large compared with the time $\frac{1}{\nu }$, which is the duration of one of the periodic partial vibrations contained in the radiation, but as small as possible compared with the time $T$. In this statement there is a certain indefiniteness, from which results the fact that $J$ will, in general, depend not only on $t$ but also on $\tau$. If this is the case one cannot speak of the intensity of the exciting vibration at all. For it is an essential feature of the conception of the intensity of a vibration that its value should change but unappreciably within the time required for a single vibration. (Compare above, Sec. 3.) Hence we shall consider in future only those processes for which, under the conditions mentioned, there exists a mean value of $E_z^2$ depending only on $t$. We are then obliged to assume that the quantities $C_{\nu }$ in (311) are negligible for all values of $\nu$ which are of the same order of magnitude as $\frac{1}{\tau }$ or smaller, i.e.,

In order to calculate $J$ we now form from (311) the value of $E_z^2$ and determine the mean value $\overline{E_z^2}$ of this quantity by integrating with respect to $t$ from $t$ to $t+\tau$, then dividing by $\tau$ and passing to the limit by decreasing $\tau$ sufficiently. Thus we get

If we now exchange the values of $\nu$ and $\nu \prime$, the function under the sign of integration does not change; hence we assume

and write:

or

And hence

If we now let $\tau$ become smaller and smaller, since $\nu \tau$ remains large, the denominator $(\nu \prime +\nu )\tau$ of the second fraction remains large under all circumstances, while that of the first fraction $(\nu \prime -\nu )\tau$ may decrease with decreasing value of $\tau$ to less than any finite value. Hence for sufficiently small values of $(\nu \prime -\nu )$ the integral reduces to

which is in fact independent of $\tau$. The remaining terms of the double integral, which correspond to larger values of $\nu \prime -\nu$, i.e., to more rapid changes with the time, depend in general on $\tau$ and therefore must vanish, if the intensity $J$ is not to depend on $\tau$. Hence in our case on introducing as a second variable of integration instead of $\nu$

we have

or

By this expression the intensity $J$ of the exciting vibration, if it exists at all, is expressed by a function of the time in the form of a Fourier’s integral.

173. The conception of the intensity of vibration $J$ necessarily contains the assumption that this quantity varies much more slowly with the time $t$ than the vibration $E_z$ itself. The same follows from the calculation of $J$ in the preceding paragraph. For there, according to (312), $\nu \tau$ and $\nu \prime \tau$ are large, but $(\nu \prime -\nu )\tau$ is small for all pairs of values $C_{\nu }$ and $C_{\nu \prime }$ that come into consideration; hence, a fortiori,

and accordingly the Fourier’s integrals $E_z$ in (311) and $J$ in (314) vary with the time in entirely different ways. Hence in the following we shall have to distinguish, as regards dependence on time, two kinds of quantities, which vary in different ways: Rapidly varying quantities, as $E_z$, and slowly varying quantities as $J$ and $I$ the spectral intensity of the exciting vibration, whose value we shall calculate in the next paragraph. Nevertheless this difference in the variability with respect to time of the quantities named is only relative, since the absolute value of the differential coefficient of $J$ with respect to time depends on the value of the unit of time and may, by a suitable choice of this unit, be made as large as we please. It is, therefore, not proper to speak of $J\left(t\right)$ simply as a slowly varying function of $t$. If, in the following, we nevertheless employ this mode of expression for the sake of brevity, it will always be in the relative sense, namely, with respect to the different behavior of the function $E_z\left(t\right)$.

On the other hand, as regards the dependence of the phase constant ${\theta }_{\nu }$ on its index $\nu$ it necessarily possesses the property of rapid variability in the absolute sense. For, although $\mu$ is small compared with $\nu$, nevertheless the difference ${\theta }_{\nu +\mu }-{\theta }_{\nu }$ is in general not small, for if it were, the quantities $A_{\mu }$ and $B_{\mu }$ in (314) would have too special values and hence it follows that $\frac{\partial {\theta }_{\nu }}{\partial \nu }\cdot \nu$ must be large. This is not essentially modified by changing the unit of time or by shifting the origin of time.

Hence the rapid variability of the quantities ${\theta }_{\nu }$ and also $C_{\nu }$ with $\nu$ is, in the absolute sense, a necessary condition for the existence of a definite intensity of vibration $J$, or, in other words, for the possibility of dividing the quantities depending on the time into those which vary rapidly and those which vary slowly—a distinction which is also made in other physical theories and upon which all the following investigations are based.

174. The distinction between rapidly variable and slowly variable quantities introduced in the preceding section has, at the present stage, an important physical aspect, because in the following we shall assume that only slow variability with time is capable of direct measurement. On this assumption we approach conditions as they actually exist in optics and heat radiation. Our problem will then be to establish relations between slowly variable quantities exclusively; for these only can be compared with the results of experience. Hence we shall now determine the most important one of the slowly variable quantities to be considered here, namely, the “spectral intensity” $I$ of the exciting vibration. This is effected as in (158) by means of the equation

By comparison with (313) we obtain:

By this expression the spectral intensity, $I$, of the exciting vibration at a point in the spectrum is expressed as a slowly variable function of the time $t$ in the form of a Fourier’s integral. The dashes over the expressions on the right side denote the mean values extended over a narrow spectral range for a given value of $\mu$. If such mean values do not exist, there is no definite spectral intensity.