Page:CunninghamExtension.djvu/10

Now pass to the consideration of the equilibrium radiation within a cavity in a body. The pressure is normal to the surface and the vector ${\frac {\sigma }{4\pi }}[EH]$ vanishes at all points.

Thus

$P_{X}=LP,\ P_{Y}=MP,\ P_{Z}=NP.$

Hence

${\begin{array}{lrl}p'_{x}ds=&LPdS&=Plds,\\\\p'_{y}ds=&{\frac {1}{\beta }}MPdS&=Pmds,\\\\p'_{z}ds=&{\frac {1}{\beta }}NPdS&=Pnds,\end{array}}$

Thus the pressure in the moving cavity is in the direction (l, m, n), and is equal to P.

Let the expression for the energy of the field be now similarly treated.

${\frac {1}{8\pi }}\left(e^{2}+h^{2}\right)={\frac {1}{8\pi }}\left\{E{}_{X}^{2}+H_{X}^{2}+\beta ^{2}\left(1+{\frac {v^{2}}{c^{2}}}\right)\left(E_{Y}^{2}+H_{Y}^{2}+E_{Z}^{2}+H_{z}^{2}\right)+{\frac {4\beta ^{2}v}{c}}\left(E_{Y}H_{z}-E_{z}H_{Y}\right)\right\}.$

In the cavity at rest when the radiation is in equilibrium, the mean values of $\left(E_{X}^{2}+H_{X}^{2}\right),\ \left(E_{Y}^{2}+H_{Y}^{2}\right),\ \left(E_{Z}^{2}+H_{Z}^{2}\right)$ are each equal to ${\frac {1}{3}}\left(E^{2}+H^{2}\right)$, and that of $\left(E_{Y}H_{Z}-E_{Z}H_{Y}\right)$ is zero, the average being taken over any interval of time very small, but large compared with the periods of the constituent radiation.

Hence, denoting mean values by a stroke,

${\frac {1}{8\pi }}{\overline {\left(e^{2}+h^{2}\right)}}={\frac {1}{8\pi }}{\overline {\left(E^{2}+H^{2}\right)}}\left\{{\frac {1}{3}}+{\frac {2\left(c^{2}+v^{2}\right)}{3\left(c^{2}-v^{2}\right)}}\right\},$

or

${\overline {\epsilon }}={\frac {3c^{2}+v^{2}}{3\left(c^{2}-v^{2}\right)}}{\overline {E}}.$

In identical manner, Planck's equation

${\overline {g}}={\frac {4v}{3\left(c^{2}-v^{2}\right)}}{\overline {E}}$

is obtained.

Finally,

${\overline {p}}={\overline {P}}={\frac {1}{3}}{\overline {E}}={\frac {c^{2}-v^{2}}{3c^{2}+v^{2}}}{\overline {\epsilon }},$ ${\overline {g}}={\frac {4v}{3c^{2}+v^{2}}}{\overline {\epsilon }}.$