Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/14

Poynting^[1] to the emission of a moving surface. The mentioned authors confine themselves, however, to the case of perpendicular incidence or emission. Incidentally, Poynting^[1] has calculated by a different method the radiation pressure at perpendicular emission with the same result.

If we insert values (26) into (16), then we obtain the pressure upon the surfaces $A$ and $B$. For example, if we calculate the pressure acting upon surface $B$, and express this value for example by the density of the total incident radiation

${\frac {2\pi \ i\ \sin \psi \ d\psi }{c_{-}}}=d$,

then we obtain

(27)	$2d\cdot {\frac {c_{-}^{2}}{c^{2}}}{\frac {\cos ^{2}\psi }{1-\beta ^{2}}}=2d{\frac {(\cos \varphi -\beta )^{2}}{1-\beta ^{2}}},$

which of course is in agreement with the corresponding one of Abraham.^[2]

The pressure upon a moving mirror incidentally can be derived very easily from the mentioned hypothesis, and one also obtains the law of reflection with one stroke.^[3] It follows indeed immediately from it, that the intensities of the incident and reflected light behaves inversely as the wavelengths, and directly as the oscillation numbers.^[4]

§ 5.

Now we want to concern ourselves with the fraction of the energy in $R$, which is due to the apparent radiation and which is gained from mechanical work. If we set into (19) the values from (26) for $p_{1}$ and $p_{2}$, it becomes

$\epsilon '={\frac {2\pi wi_{0}}{c^{2}(1-\beta ^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi {\frac {\beta \cos ^{2}\psi +\beta (1-\beta ^{2}\sin ^{2}\psi )}{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}}.$

We put

(28)	$\epsilon '=\epsilon _{0}\tau ,$

1. J.H. Poynting, l. c. appendix.
2. M. Abraham, l. c. p. 257. eq. (18).
3. F. Hasenöhrl, Sitzungsber. d. k. Akad. d. Wissensch. zu Wien IIa. Meeting from 23. June 1904.
4. see M. Abraham, l. c. p. 252. eq. 11c).