59.

We shall now establish a relation between the radiation pressure and the energy of radiation $Idt$ falling on the surface element $d\sigma$ of the conductor in a time element $dt$. The latter from Poynting's law of energy flow is $Idt=\frac{c}{4\pi }({⋿}_y{𝖧}_z-{⋿}_z{𝖧}_y)d\sigma dt,$ hence from [eqn:(55)] (55) $Idt=\frac{c}{4\pi }\cos \theta (f^2+g^2)d\sigma dt.$ By comparison with [eqn:(63)] (63) we obtain $𝖥=\frac{2\cos \theta }{c}I.\mathrm{\backslash Label}[eqn](64)\text{upshape (64)}$

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 conductor exerts in a normal direction on a unit surface of the conductor. The energy radiated in the conical element $d\Omega =\sin \theta d\theta d\varphi$ in the time $dt$ on the element of area $d\sigma$ is, according to [eqn:(6)] (6), $Idt=K\cos \theta d\Omega d\sigma dt,$ where $K$ represents the specific intensity of the radiation in the direction $d\Omega$ toward the reflector. On substituting this in [eqn:(64)] (64) and integrating over $d\Omega$ we obtain for the total pressure of all pencils which fall on the surface and are reflected by it $p=\frac{2}{c}\int K{\cos }^2\theta d\Omega ,\mathrm{\backslash Label}[eqn](65)\text{upshape (65)}$ the integration with respect to $\varphi$ extending from $0$ to $2\pi$ and with respect to $\theta$ from $0$ to ${\displaystyle \frac{\pi }{2}}$.

In case $K$ is independent of direction as in the case of black radiation, we obtain for the radiation pressure $p=\frac{2K}{c}{\int }_0^{2\pi }d\varphi {\int }_0^{\frac{\pi }{2}}d\theta {\cos }^2\theta \sin \theta =\frac{4\pi K}{3c}$ or, if we introduce instead of $K$ the volume density of radiation $u$ from [eqn:(21)] (21) $p=\frac{u}{3}.\mathrm{\backslash Label}[eqn](66)\text{upshape (66)}$

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