Page:Zur Theorie der Strahlung in bewegten Körpern. Berichtigung.djvu/2

electromagnetic momentum which coincides with the direction of the system and which is contained in the cavity, we have to calculate expression (1) with $cos\varphi$, then to integrate with respect to $\varphi$ from O to $\pi$, and then to multiply the result with the volume of cavity $h$. If we additionally substitute for $c'$ its value^[1], then the momentum becomes

${\begin{array}{rl}G&={\frac {2\pi i_{0}}{c^{2}}}h{\overset {\pi }{\underset {0}{\int }}}{\frac {\sin \varphi \cos \varphi \ d\varphi }{(1+\beta ^{2}-2\beta \cos \varphi )^{2}}}\\\\&={\frac {\epsilon _{0}h}{c}}\left({\frac {1}{2\beta }}{\frac {1+\beta ^{2}}{(1-\beta ^{2})^{2}}}-{\frac {1}{4\beta ^{2}}}\log {\frac {1+\beta }{1-\beta }}\right).\end{array}}.$

Now, the longitudinal electromagnetic mass is given by ${\tfrac {1}{c}}{\tfrac {dG}{d\beta }}$;^[2] thus it becomes equal to (when higher terms are neglected):

${\frac {4}{3}}{\frac {h\epsilon _{0}}{c^{2}}}.$

This is half of the value given by me.

After it was sought in vain after a principal difference, I found that this difference stems from a calculation error, unfortunately committed by me in my paper. At p. 362, line 6 from above,

not ${\frac {2\beta _{1}}{c^{2}(1-\beta _{1}^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\dots$ shall be stated, but ${\frac {4\beta _{1}}{c^{2}(1-\beta _{1}^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\dots$,

therefore the heat absorbed by the walls of the cavity when the system is accelerated, is:

$Q=h\epsilon _{0}\left(\varkappa _{1}-2\delta \beta {\frac {\partial \varkappa _{1}}{\partial \beta }}\right);$

however, since furthermore the walls have given off the heat $h\epsilon _{0}\varkappa _{1}$, we can say that the walls of the cavity (when accelerated by $\delta _{\tau }$) have altogether given off the heat

$2h\epsilon _{0}\delta \beta {\frac {\partial \varkappa }{\partial \beta }}=2h\epsilon _{0}\delta \varkappa$

1. F. Hasenöhrl, l. c. p. 347, Equation (1).
2. M. Abraham, Ann. d. Phys. 10. p. 150. (Gl. 16a) 1903.