V Electrodynamical Processes in a Stationary Field of Radiation

Chapter V
Electrodynamical Processes in a Stationary Field of Radiation

107. We shall now consider from the standpoint of pure electrodynamics the processes that take place in a vacuum, which is bounded on all sides by reﬂecting walls and through which heat radiation passes uniformly in all directions, and shall then inquire into the relations between the electrodynamical and the thermodynamic quantities.

The electrodynamical state of the ﬁeld of radiation is determined at every instant by the values of the electric ﬁeld-strength $E$ and the magnetic ﬁeld-strength $H$ at every point in the ﬁeld, and the changes in time of these two vectors are completely determined by Maxwell’s ﬁeld equations (52), which we have already used in Sec. 53, together with the boundary conditions, which hold at the reﬂecting walls. In the present case, however, we have to deal with a solution of these equations of much greater complexity than that expressed by (54), which corresponds to a plane wave. For a plane wave, even though it be periodic with a wave length lying within the optical or thermal spectrum, can never be interpreted as heat radiation. For, according to Sec. 16, a ﬁnite intensity $K$ of heat radiation requires a ﬁnite solid angle of the rays and, according to Sec. 18, a spectral interval of ﬁnite width. But an absolutely plane, absolutely periodic wave has a zero solid angle and a zero spectral width. Hence in the case of a plane periodic wave there can be no question of either entropy or temperature of the radiation.

108. Let us proceed in a perfectly general way to consider the components of the ﬁeld-strengths $E$ and $H$ as functions of the time at a deﬁnite point, which we may think of as the origin of the coordinate system. Of these components, which are produced by all rays passing through the origin, there are six; we select one of them, say $E_{z}$ , for closer consideration. However complicated it may be, it may under all circumstances be written as a Fourier’s series for a limited time interval, say from $t = 0$ to $t = T$ ; thus

E_{z} = \sum_{n = 1}^{n = \infty} C_{n} cos (\frac{2 π n t}{T} - θ_{n})

(149)

where the summation is to extend over all positive integers $n$ , while the constants $C_{n}$ (positive) and $θ_{n}$ may vary arbitrarily from term to term. The time interval $T$ , the fundamental period of the Fourier’s series, we shall choose so large that all times $t$ which we shall consider hereafter are included in this time interval, so that $0 < t < T$ . Then we may regard $E_{z}$ as identical in all respects with the Fourier’s series, i.e., we may regard $E_{z}$ as consisting of “partial vibrations,” which are strictly periodic and of frequencies given by

ν = \frac{n}{T} .

Since, according to Sec. 3, the time diﬀerential $d t$ required for the deﬁnition of the intensity of a heat ray is necessarily large compared with the periods of vibration of all colors contained in the ray, a single time diﬀerential $d t$ contains a large number of vibrations, i.e., the product $ν d t$ is a large number. Then it follows a fortiori that $ν t$ and, still more,

ν T = n is enormously large

(150)

for all values of $ν$ entering into consideration. From this we must conclude that all amplitudes $C_{n}$ with a moderately large value for the ordinal number $n$ do not appear at all in the Fourier’s series, that is to say, they are negligibly small.

109. Though we have no detailed special information about the function $E_{z}$ , nevertheless its relation to the radiation of heat aﬀords some important information as to a few of its general properties. Firstly, for the space density of radiation in a vacuum we have, according to Maxwell’s theory,

u = \frac{1}{8 π} (\bar{E_{x}^{2}} + \bar{E_{y}^{2}} + \bar{E_{z}^{2}} + \bar{H_{x}^{2}} + \bar{H_{y}^{2}} + \bar{H_{z}^{2}}) .

Now the radiation is uniform in all directions and in the stationary state, hence the six mean values named are all equal to one another, and it follows that

u = \frac{3}{4 π} \bar{E_{z}^{2}} .

(151)

Let us substitute in this equation the value of $E_{z}$ as given by (149). Squaring the latter and integrating term by term through a time interval, from $0$ to $t$ , assumed large in comparison with all periods of vibration $\frac{1}{ν}$ but otherwise arbitrary, and then dividing by $t$ , we obtain, since the radiation is perfectly stationary,

u = \frac{3}{8 π} \sum C_{n}^{2} .

(152)

From this relation we may at once draw an important conclusion as to the nature of $E_{z}$ as a function of time. Namely, since the Fourier’s series (149) consists, as we have seen, of a great many terms, the squares, $C_{n}^{2}$ , of the separate amplitudes of vibration the sum of which gives the space density of radiation, must have exceedingly small values. Moreover in the integral of the square of the Fourier’s series the terms which depend on the time $t$ and contain the products of any two diﬀerent amplitudes all cancel; hence the amplitudes $C_{n}$ and the phase-constants $θ_{n}$ must vary from one ordinal number to another in a quite irregular manner. We may express this fact by saying that the separate partial vibrations of the series are very small and in a “chaotic”³³ state.

For the speciﬁc intensity of the radiation travelling in any direction whatever we obtain from (21)

K = \frac{c u}{4 π} = \frac{3 c}{32 π^{2}} \sum C_{n}^{2} .

(153)

110. Let us now perform the spectral resolution of the last two equations. To begin with we have from (22):

u = \int_{0}^{\infty} u_{ν} d ν = \frac{3}{8 π} \sum_{1}^{\infty} C_{n}^{2} .

(154)

On the right side of the equation the sum $\sum$ consists of separate terms, every one of which corresponds to a separate ordinal number $n$ and to a simple periodic partial vibration. Strictly speaking this sum does not represent a continuous sequence of frequencies $ν$ , since $n$ is an integral number. But $n$ is, according to (150), so enormously large for all frequencies which need be considered that the frequencies $ν$ corresponding to the successive values of $n$ lie very close together. Hence the interval $d ν$ , though inﬁnitesimal compared with $ν$ , still contains a large number of partial vibrations, say $n'$ , where

d ν = \frac{n'}{T} .

(155)

If now in (154) we equate, instead of the total energy densities, the energy densities corresponding to the interval $d ν$ only, which are independent of those of the other spectral regions, we obtain

u_{ν} d ν = \frac{3}{8 π} \sum_{n}^{n + n'} C_{n}^{2},

or, according to (155),

u_{ν} = \frac{3 T}{8 π} \cdot \frac{1}{n'} \sum_{n}^{n + n'} C_{n}^{2} = \frac{3 T}{8 π} \cdot \bar{C_{n}^{2}},

(156)

where we denote by $\bar{C_{n}^{2}}$ the average value of $C_{n}^{2}$ in the interval from $n$ to $n + n'$ . The existence of such an average value, the magnitude of which is independent of $n$ , provided $n'$ be taken small compared with $n$ , is, of course, not self-evident at the outset, but is due to a special property of the function $E_{z}$ which is peculiar to stationary heat radiation. On the other hand, since many terms contribute to the mean value, nothing can be said either about the magnitude of a separate term $C_{n}^{2}$ , or about the connection of two consecutive terms, but they are to be regarded as perfectly independent of each other.

In a very similar manner, by making use of (24), we ﬁnd for the speciﬁc intensity of a monochromatic plane polarized ray, travelling in any direction whatever,

K_{ν} = \frac{3 c T}{64 π^{2}} \bar{C_{n}^{2}} .

(157)

From this it is apparent, among other things, that, according to the electromagnetic theory of radiation, a monochromatic light or heat ray is represented, not by a simple periodic wave, but by a superposition of a large number of simple periodic waves, the mean value of which constitutes the intensity of the ray. In accord with this is the fact, known from optics, that two rays of the same color and intensity but of diﬀerent origin never interfere with each other, as they would, of necessity, if every ray were a simple periodic one.

Finally we shall also perform the spectral resolution of the mean value of $E_{z}^{2}$ , by writing

E_{z}^{2} = J = \int_{0}^{\infty} J_{ν} d ν .

(158)

Then by comparison with (151), (154), and (156) we ﬁnd

J_{ν} = \frac{4 π}{3} u_{ν} = \frac{T}{2} \bar{C_{n}^{2}} .

(159)

According to (157), $J_{ν}$ is related to $K_{ν}$ , the speciﬁc intensity of radiation of a plane polarized ray, as follows:

K_{ν} = \frac{3 c}{32 π^{2}} J_{ν} .

(160)

111. Black radiation is frequently said to consist of a large number of regular periodic vibrations. This method of expression is perfectly justiﬁed, inasmuch as it refers to the resolution of the total vibration in a Fourier’s series, according to equation (149), and often is exceedingly well adapted for convenience and clearness of discussion. It should, however, not mislead us into believing that such a “regularity” is caused by a special physical property of the elementary processes of vibration. For the resolvability into a Fourier’s series is mathematically self-evident and hence, in a physical sense, tells us nothing new. In fact, it is even always possible to regard a vibration which is damped to an arbitrary extent as consisting of a sum of regular periodic partial vibrations with constant amplitudes and constant phases. On the contrary, it may just as correctly be said that in all nature there is no process more complicated than the vibrations of black radiation. In particular, these vibrations do not depend in any characteristic manner on the special processes that take place in the centers of emission of the rays, say on the period or the damping of the emitting particles; for the normal spectrum is distinguished from all other spectra by the very fact that all individual diﬀerences caused by the special nature of the emitting substances are perfectly equalized and eﬀaced. Therefore to attempt to draw conclusions concerning the special properties of the particles emitting the rays from the elementary vibrations in the rays of the normal spectrum would be a hopeless undertaking.

In fact, black radiation may just as well be regarded as consisting, not of regular periodic vibrations, but of absolutely irregular separate impulses. The special regularities, which we observe in monochromatic light resolved spectrally, are caused merely by the special properties of the spectral apparatus used, e.g., the dispersing prism (natural periods of the molecules), or the diﬀraction grating (width of the slits). Hence it is also incorrect to ﬁnd a characteristic diﬀerence between light rays and Roentgen rays (the latter assumed as an electromagnetic process in a vacuum) in the circumstance that in the former the vibrations take place with greater regularity. Roentgen rays may, under certain conditions, possess more selective properties than light rays. The resolvability into a Fourier’s series of partial vibrations with constant amplitudes and constant phases exists for both kinds of rays in precisely the same manner. What especially distinguishes light vibrations from Roentgen vibrations is the much smaller frequency of the partial vibrations of the former. To this is due the possibility of their spectral resolution, and probably also the far greater regularity of the changes of the radiation intensity in every region of the spectrum in the course of time, which, however, is not caused by a special property of the elementary processes of vibration, but merely by the constancy of the mean values.

112. The elementary processes of radiation exhibit regularities only when the vibrations are restricted to a narrow spectral region, that is to say in the case of spectroscopically resolved light, and especially in the case of the natural spectral lines. If, e.g., the amplitudes $C_{n}$ of the Fourier’s series (149) diﬀer from zero only between the ordinal numbers $n = n_{0}$ and $n = n_{1}$ , where $\frac{n_{1} - n_{0}}{n_{0}}$ is small, we may write

E_{z} = C_{0} cos (\frac{2 π n_{0} t}{T} - θ_{0}),

(161)

where

\begin{array}{l} C_{0} cos θ_{0} & = \sum_{n_{0}}^{n_{1}} C_{n} cos (\frac{2 π (n - n_{0}) t}{T} - θ_{n}) \\ C_{0} sin θ_{0} & = - \sum_{n_{0}}^{n_{1}} C_{n} sin (\frac{2 π (n - n_{0}) t}{T} - θ_{n}) \end{array}

and $E_{z}$ may be regarded as a single approximately periodic vibration of frequency $ν_{0} = \frac{n_{0}}{T}$ with an amplitude $C_{0}$ and a phase-constant $θ_{0}$ which vary slowly and irregularly.

The smaller the spectral region, and accordingly the smaller $\frac{n_{1} - n_{0}}{n_{0}}$ , the slower are the ﬂuctuations (“Schwankungen”) of $C_{0}$ and $θ_{0}$ , and the more regular is the resulting vibration and also the larger is the diﬀerence of path for which radiation can interfere with itself. If a spectral line were absolutely sharp, the radiation would have the property of being capable of interfering with itself for diﬀerences of path of any size whatever. This case, however, according to Sec. 18, is an ideal abstraction, never occurring in reality.

ptail

top

Chapter VElectrodynamical Processes in a Stationary Field of Radiation

Chapter V
Electrodynamical Processes in a Stationary Field of Radiation