Page:Zur Theorie der Strahlung bewegter Koerper.djvu/9

and

$p_{2}={\frac {i'}{B}}\left(-\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\right)$

(8)

or

$p_{2}=i_{0}{\frac {-\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}_{+}{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}=$ $=i_{0}{\frac {\cos \phi -\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}(1-\sigma ^{2}){\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}.$

(9)

If we insert this into (2), then:

$L=2\pi cDi_{0}\cdot \int _{0}^{\pi /2}{\frac {\sin \phi \ d\phi }{{\mathfrak {B}}(1-\sigma ^{2}){\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}$ $\left[\cos \ \phi \left({\frac {1}{{\mathfrak {B}}_{-}}}-{\frac {1}{{\mathfrak {B}}_{+}}}\right)+\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\left({\frac {1}{{\mathfrak {B}}_{-}}}+{\frac {1}{{\mathfrak {B}}_{+}}}\right)\right]=$ $={\frac {4\pi cDi_{0}\sigma }{{\mathfrak {B}}^{2}(1-\sigma ^{2})^{2}}}\int _{0}^{\pi /2}\sin \phi \ d\phi {\frac {1+\cos ^{2}\phi -\sigma ^{2}\sin ^{2}\phi }{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}$

If we additionally put the energy content of the resting cavity $R$:

${\frac {4\pi i_{0}D}{\mathfrak {B}}}=E_{0},$

then after execution of the integration

$L=E_{0}{\frac {c^{2}}{{\mathfrak {B}}^{2}}}{\frac {1}{(1-\sigma ^{2})^{2}}}\left[{\frac {1}{2}}+{\frac {1}{2\sigma ^{2}}}-{\frac {(1-\sigma ^{2})^{2}}{4\sigma ^{3}}}\log {\frac {1+\sigma }{1-\sigma }}\right].$

If we neglect herein magnitudes of order $\sigma ^{3}$, then it becomes

$L=E_{0}{\frac {c^{2}}{B^{2}}}\cdot {\frac {4}{3}}.$

This expression has now the form and the dimension of a kinetic energy. Thus one can say, that the kinetic energy of our system was apparently increased by $L$,