Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/7

${\begin{array}{rl}H&={\frac {2\pi vi'}{c\varkappa ^{2}}}\int _{0}^{\pi }\sin \phi 'd\phi '(1+\beta \ \cos \ \phi ')\\\\&={\frac {4\pi v}{\varkappa ^{2}}}\cdot {\frac {i'}{c}}.\end{array}}$

However, according to (16) it is:

$H={\sqrt {1-\beta ^{2}}}\cdot U_{0}=\varkappa \cdot {\frac {4\pi v_{0}}{c}}i_{0},$

where $i_{0}$ is the radiation intensity in the resting cavity. Thus it must be

$\varkappa v_{0}i_{0}={\frac {1}{\varkappa ^{2}}}vi'$;

or, since $v=\varkappa v_{0}$:

$i'=\varkappa ^{2}i_{0}.$

This is in agreement with the generally valid theorems of the theory of H. A. Lorentz.^[1]

If we set in accordance with the Stefan-Boltmann law

$i_{0}=\sigma T_{0}^{4},$

then it follows (see. (14)):

$i'=\varkappa ^{2}\sigma T_{0}^{4}={\frac {\sigma }{\varkappa ^{2}}}T^{4}.$

The constant of the Stefan-Boltzmann law is thus to be divided by $\varkappa ^{2}$.

The energy density of the true radiation is:

${\frac {H}{v}}={\frac {4\pi }{c}}\cdot {\frac {i'}{\varkappa ^{2}}}={\frac {4\pi }{c}}i_{0};$

thus it has the same value as in the resting cavity.

The total energy follows from (17): it has the value:

$U={\frac {1}{\sqrt {1-\beta ^{2}}}}\left(1+{\frac {1}{3}}\beta ^{2}\right)U_{0}={\frac {4\pi v_{0}i_{0}}{c}}\cdot {\frac {1+{\frac {1}{3}}\beta ^{2}}{\sqrt {1-\beta ^{2}}}}.$

1. See for instance, M. Abraham, Theorie der Elektrizität, II, p. 282 (1905)