Page:EB1911 - Volume 22.djvu/804

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787
RADIATION


derived from a theory, namely, that of the ordinary electrodynamic equations, which considers the velocity of the matter, or rather of the electrons associated with it, to be so small compared with that of radiation that the square of the ratio of these velocities can be neglected. The formula above obtained is of general application, and shows that for high values of v the pressure must fall off. It has been urged as an objection to the thermodynamic reversibility of a ray (§ 8) that the work of the radiant pressure exerted at its front is lost, as there is no obstacle to sustain it; but on an obstacle moving with the velocity of the wave-front the pressure would vanish, so that this objection does not now hold.

In every such case of an advancing perfect reflector the aggregate amplitude of the superposed incident and reflected wave-trains, of different wave-lengths and periods, will be represented by

mv C2 mc

E-{-5 -2a s1nc v(x - vt) sin6 U(x-ul),

thus the appearance presented will be that of a train of waves each of length (1-u/c)21r/m, and progressing with the velocity u of the reflector, which travels at one of the nodes of the train. This slowly travelling wave-train corresponds to the stationary train which would be produced by a stationary perfect reflector; but the amplitude is now a varying quantity which, once uniform vibration has been fully established along any path, may itself be described as running on after the manner of a superposed wave-train of very great wave-length (0/u-r)21r/m and of very great velocity 6”/v. A somewhat similar state of things arises when a wave-train is incident on a stationary reflector very nearly normally, as may sometimes be seen with incoming rollers along a shelving beach; the visible disturbance at a reflecting ridge, arising from each single wave-crest, then rushes along the ridge at a speed which is at first sight surprising, as it is enormously in excess of the speed possible for any simple train of waves travelling into quiescent aether. 3. Wien's Law.-Let us consider a spherical enclosure filled with radiation, and having walls of ideal perfectly reflecting quality so that none of the radiation can escape. If there is no material body inside it, any arbitrarily assigned constitution of this radiation will be permanent. Let us suppose that the radius a of the enclosure is shrinking with extremely small velocity u. A ray inside it, incident at angle L, will always be incident on the walls in its successive reflexions at the same angle, except as regards a negligible change due to the motion of the reflector (§ 2); and the length of its path between successive reflexions is 20 cos L. Each undulation on this ray will thus undergo reflexion at intervals of time equal to za cos L/0, where c is the velocity of light, and it is easily verified that on each reflexion it, is shortened by the fraction 2U cos L/c of itself: thus in the very long time T required to complete the shrinkage it is shortened by the fraction uTa, which is 5a/a where 5a is the total shrinkage in radius, and is independent of the value of L. The wave-length of each undulation in the radiation inside the enclosure is therefore reduced in the same ratio as the radius. Now suppose that the constitution of the enclosed radiation corresponded initially to a definite temperature. During the shrinkage thermal equilibrium must be maintained among its constituents; otherwise there would be a running down of their energies towards uniformity'of temperature, if material radiating bodies are present, which would be superposed on the mechanical operations belonging to the shrinkage, and the process could not be reversible. Such a state of affairs is not possible, for it would land us in processes of the following type. Expand the enclosure, gaining the mechanical work of the radiant pressure against its walls, whatever that may be. Then equalize the intensities of the constituent radiations to those corresponding to a common temperature, by taking advantage of the absorptions of material bodies at the actual temperatures of these radiations; when this is done, as it may actually be to some extent by aid of the sifting produced by partitions which transmit some kinds of radiation more rapidly than others, a further gain of work can be obtained at the expense of the radiant energy. Then contract the remaining radiant energy to its previous volume, which requires an expenditure of less work on the walls of the enclosure than the expansion of the greater amount of radiation originally afiorded; and, finally, gain still more work by again equalizing the temperatures of its constituents, The energy now remaining, being of smaller amount and under similar conditions, must have a temperature lower than the initial one. This process might be repeated indefinitely, and would constitute an engine without an extraneous refrigerator, violating Carnot's principle by deriving an unlimited supply of. mechanical work from thermal sources at a uniform temperature.

Thus, independently of any knowledge of the intensity of the mechanical pressure of radiation, or indeed of whether such a pressure exists at all, it is established that the shrinkage of the enclosure must directly transform the contained radiation to the constitution which corresponds to some definite new temperature. Now we have seen that the wave-lengths of its constituents are all reduced in the same ratio by this process. If, then, we can prove that the intensities of these constituents are also all changed in a common ratio by the reflexions at the shrinking envelope, it will follow that the distributions of the radiation among the various wave-lengths are, at these two temperatures, and therefore at any two temperatures, homologous, in the sense that the intensity curves, after the Wave-lengths in one of them have been reduced in a ratio depending definitely on the two temperatures, differ only in the absolute scale of magnitude of the ordinates.

This procedure modifies Wienfs argument by employing a uniformly shrinking spherical enclosure (cf. Brit. Assoc. Report, tooo). If the enclosure is not spherical, the angles of incidence at successive reflexions of the same ray will differ by finite amounts; we must then estimate the average effect of the shrinkage. In the form of enclosure here employed all rays are affected alike, and no averaging is required; while by the principle of Stewart and Kirchhoff what is established for any one form is of general validity.

4. Pressure of N afural Radiation.-The question reserved above has now to be settled. At first sight it might have appeared that the reflexion is simply total; but, as has been seen in § 2, the advancing perfect reflector does work against the pressure of the radiation, and this work must be changed into radiant energy and thus go to increase the intensity of the reflected ray. Considering electric radiation incident at angle L, the tangential electric force is annulled at the reflector; hence the amplitude of the electric vibration is conserved on reflexion, though its phase is reversed. As already seen, the wave-length is shortened approximately by the fraction 2U cos L/c in each reflexion; thus, just as in § 2, the energy transmitted per unit time per unit area is increased in the same ratio; andrallowing for the factor cos L of foreshortening, there is therefore a radiant pressure equal to 'the total density of radiant energy in front of the reflector multiplied by COS2L. This argument, being independent of the wave-length, applies to each constituent of the radiation in this direction separately; thus their energies are all increased in the same ratio by the reflexion, as was to be proved. When we are dealing with the natural radiation in an enclosure, which is distributed equally in all directions, this factor .COSZL must be averaged; and we thus attain Boltzmann's result that the radiant pressure is then one-third of the density of radiant energy in front of the reflector, this statement holding good as regards each constituent of the natural radiation taken separately.

5. Adiabaf-ic Relations.-Consider the enclosure filled with radiation of energy-density E at volume V, of any given constitution but devoid of special direction, and let it be shrunk to volume V - 5V against its own pressure; if the density thereby become E 1- 61-E, the conservation of the energy requires EV-I-§ E5V=(E-5E)(V-6V),

so, that § E<§ V+V5E=o, or E varies as V'ii.

Again-but now with a restriction to radiation with its energy