Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/4

9. Application to cavity radiation.

We base our calculation on the relative ray path. We consider radiation that encloses angles between $\phi$ and $\phi +d\phi$ with the direction of motion; it carries – in unit volume through the unit surface of a perpendicular (co-moving) plane – the energy amount:

$2\pi J\ \sin \phi \ \cos \phi \ d\phi .$

We call $J$ the intensity of the total (relative) radiation. If this radiation is incident upon an absorbing surface, it performs the pressure work:^[1]

$q\cdot {\frac {2\pi J\sin \phi \cos \phi \ d\phi }{c}}\cdot \cos \varphi =2\pi J\sin \phi \cos \phi \ d\phi \ \beta \cos \varphi ,$

where $\varphi$ is the angle between the absolute radiation direction and the direction of motion. The difference:

$2\pi J\ \sin \phi \ \cos \phi \ d\phi (1-\beta \ \cos \varphi )=2\pi i\ \sin \phi \ \cos \phi \ d\phi$

we call the true (relative) radiation. The true radiation intensity

$i=J(1-\beta \cos \varphi )$

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is crucial for the heat transport between bodies of equal velocity.^[2]

We employ the standpoint of Lorentz's contraction hypothesis and introduce the angle $\phi '$ by the equation

${\begin{array}{rl}\operatorname {tg} \ \phi '=&\varkappa \ \operatorname {tg} \ \phi \\\varkappa {}^{2}=&1-\beta ^{2}\end{array}}$

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1. M. Abraham, Boltzmann-Festschrift, p. 90, 1904. Compare for instance F. Hasenöhrl, Jahrb. d. Radioaktivität, 2, p. 281 (1905).
2. This terminology agrees with the one used in an earlier paper (Ann. d. Phys., 15 [1904]). There, $i$ and $i_{0}$ was written instead of $J$ and $i$. See also Jahrb. d. Radioaktivität und Elektronik, 2, p. 283 (1905).