77.

Secondly, let us calculate the change in energy, which the moving reflector produces in the incident radiation, and let us consider from the outset the general case of oblique incidence. Let a monochromatic, infinitely thin, unpolarized pencil of rays, which falls on a surface element of the reflector at the angle of incidence $\theta$, transmit the energy $I\delta t$ to the reflector in the time $\delta t$. Then, ignoring vanishingly small quantities, the mechanical pressure of the pencil of rays normally to the reflector is, according to equation [eqn:(64)] (64), $𝖥=\frac{2\cos \theta }{c}I,$ and to the same degree of approximation the work done from the outside on the incident radiation in the time $\delta t$ is $𝖥v\delta t=\frac{2v\cos \theta }{c}I\delta t.\mathrm{\backslash Label}[eqn](84)\text{upshape (84)}$ According to the principle of the conservation of energy this amount of work must reappear in the energy of the reflected radiation. Hence the reflected pencil has a larger intensity than the incident one. It produces, namely, in the time $\delta t$ the energyIt is clear that the change in intensity of the reflected radiation caused by the motion of the reflector can also be derived from purely electrodynamical considerations, since electrodynamics are consistent with the energy principle. This method is somewhat lengthy, but it affords a deeper insight into the details of the phenomenon of reflection. $I\delta t+𝖥v\delta t=I(1+\frac{2v\cos \theta }{c})\delta t=I^{\prime }\delta t.\mathrm{\backslash Label}[eqn](85)\text{upshape (85)}$ Hence we may summarize as follows: By the reflection of a monochromatic unpolarized pencil, incident at an angle $\theta$ on a reflector moving toward the radiation with the infinitely small velocity $v$, the radiant energy $I\delta t$, whose frequencies extend from $\nu$ to $\nu +d\nu$, is in the time $\delta t$ changed into the radiant energy $I^{\prime }\delta t$ with the interval of frequency $({\nu }^{\prime },{\nu }^{\prime }+d{\nu }^{\prime })$, where $I^{\prime }$ is given by [eqn:(85)] (85), ${\nu }^{\prime }$ by [eqn:(83)] (83), and accordingly $d{\nu }^{\prime }$, the spectral breadth of the reflected pencil, by $d{\nu }^{\prime }=d\nu (1+\frac{2v\cos \theta }{c}).\mathrm{\backslash Label}[eqn](86)\text{upshape (86)}$ A comparison of these values shows that $\frac{I^{\prime }}{I}=\frac{{\nu }^{\prime }}{\nu }=\frac{d{\nu }^{\prime }}{d\nu }.\mathrm{\backslash Label}[eqn](87)\text{upshape (87)}$

The absolute value of the radiant energy which has disappeared in this change is, from equation [eqn:(13)] (13), $I\delta t=2{𝖪}_{\nu }d\sigma \cos \theta d\Omega d\nu \delta t,\mathrm{\backslash Label}[eqn](88)\text{upshape (88)}$ and hence the absolute value of the radiant energy which has been formed is, according to [eqn:(85)] (85), $I^{\prime }\delta t=2{𝖪}_{\nu }d\sigma \cos \theta d\Omega d\nu (1+\frac{2v\cos \theta }{c})\delta t.\mathrm{\backslash Label}[eqn](89)\text{upshape (89)}$

Strictly speaking these last two expressions would require an infinitely small correction, since the quantity $I$ from equation [eqn:(88)] (88) represents the energy radiation on a stationary element of area $d\sigma$, while, in reality, the incident radiation is slightly increased by the motion of $d\sigma$ toward the incident pencil.

The corresponding additional terms may, however, be omitted here without appreciable error, since the correction caused by them would consist merely of the addition to the energy change here calculated of a comparatively infinitesimal energy change of the same kind with an external work that is infinitesimal of the second order.