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Nun nehmen wir wieder
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{\displaystyle \epsilon =1}
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{\displaystyle \mu =1}
Nr. 13
(13)
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{\displaystyle \left\{{\begin{aligned}{\mathfrak {K}}_{1}&={\mathfrak {\varrho E_{1}+\left[J_{1}B\right]=\varrho E+\left[JB\right]}}=\\&={\frac {n}{4\pi }}{\mathfrak {E}}\ \mathrm {div} {\mathfrak {E}}-{\frac {n}{4\pi }}[{\mathfrak {E}}\ \mathrm {rot} {\mathfrak {E}}]-{\frac {[{\mathfrak {H}}\ \mathrm {rot} {\mathfrak {H}}]}{4\pi }}-{\frac {n}{4\pi }}{\frac {\partial [{\mathfrak {EH]}}}{\partial t}}\end{aligned}}\right.}
Unter Berücksichtigung von (12) und
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{\displaystyle \mathrm {div} {\mathfrak {H}}=0}
(14)
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{\displaystyle \int \limits _{V}{\mathfrak {K}}_{1}dv=\int \limits _{F}{\mathfrak {T}}df-\int {\frac {n}{4\pi }}{\frac {\partial [{\mathfrak {EH]}}}{\partial t}}dv}
wo
(15)
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{\displaystyle {\mathfrak {T}}={\frac {n}{4\pi }}\left({\mathfrak {E\cdot En-n\cdot {\frac {E^{2}}{2}}}}\right)+{\frac {1}{4\pi }}\left({\mathfrak {H\cdot Hn-n\cdot {\frac {H^{2}}{2}}}}\right)}
ist. Aus (8) und (5) folgt dann weiter
(16)
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{\displaystyle {\mathfrak {R}}=-{\frac {n}{4\pi }}\left({\mathfrak {Ev\cdot E-v\cdot {\frac {E^{2}}{2}}}}\right)-{\frac {1}{4\pi }}\left({\mathfrak {Hv\cdot H-v\cdot {\frac {H^{2}}{2}}}}\right)}
Aus (14) und (9) ergibt sich dann
(17)
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{\displaystyle \int \limits _{\infty }{\mathfrak {K}}_{1}dv+\int \limits _{f}{\mathfrak {M}}df=-{\frac {d}{dt}}\int {\frac {n[{\mathfrak {EH}}]}{4\pi }}dv}
und demnach ist die Impulsdichte
(18)
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{\displaystyle {\mathfrak {g}}={\frac {n}{4\pi }}[{\mathfrak {EH}}]}
also genau dieselbe wie bei den Lorentz schen Gleichungen.
Weiter erhalten wir unter Heranziehung der beiden Hauptgleichungen (1) und (2) Nr. 13 und der Beziehung
(19)
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{\displaystyle \mathrm {div} [{\mathfrak {EH}}]={\mathfrak {H}}\ \mathrm {rot} {\mathfrak {E}}-{\mathfrak {E}}\ \mathrm {rot} {\mathfrak {H}}}
(20)
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{\displaystyle {\mathfrak {K_{1}v}}+Q_{1}={\mathfrak {J_{1}E_{1}+\varrho E_{1}v+\left[J_{1}B\right]v=JE}}=}
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{\displaystyle =-{\frac {\mathrm {div} [{\mathfrak {EH}}]}{4\pi }}-{\frac {1}{8\pi }}{\frac {\partial }{\partial t}}\left\{{\mathfrak {E}}^{2}n+{\mathfrak {H}}^{2}\right\}}
Wir setzen nun
(21)
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{\displaystyle \psi ={\frac {1}{8\pi }}\left({\mathfrak {E}}^{2}n+{\mathfrak {H}}^{2}\right)}
Dann folgt aus (20) und (7)
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{\displaystyle \mathrm {div} \left({\mathfrak {S-{\frac {[EH]}{4\pi }}+\psi v+R}}\right)=0}
weshalb wir schreiben
(22)
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{\displaystyle {\mathfrak {S={\frac {[EH]}{4\pi }}-\psi v-R}}}
![{\displaystyle \left\{{\begin{aligned}{\mathfrak {K}}_{1}&={\mathfrak {\varrho E_{1}+\left[J_{1}B\right]=\varrho E+\left[JB\right]}}=\\&={\frac {n}{4\pi }}{\mathfrak {E}}\ \mathrm {div} {\mathfrak {E}}-{\frac {n}{4\pi }}[{\mathfrak {E}}\ \mathrm {rot} {\mathfrak {E}}]-{\frac {[{\mathfrak {H}}\ \mathrm {rot} {\mathfrak {H}}]}{4\pi }}-{\frac {n}{4\pi }}{\frac {\partial [{\mathfrak {EH]}}}{\partial t}}\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7189f320cc96c0e28a6cfb4c91fcddd4263f434)
![{\displaystyle \int \limits _{V}{\mathfrak {K}}_{1}dv=\int \limits _{F}{\mathfrak {T}}df-\int {\frac {n}{4\pi }}{\frac {\partial [{\mathfrak {EH]}}}{\partial t}}dv}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1129a4f6f9ce4d9f4e2b785ab1b3856ff5e3bcf5)
![{\displaystyle \int \limits _{\infty }{\mathfrak {K}}_{1}dv+\int \limits _{f}{\mathfrak {M}}df=-{\frac {d}{dt}}\int {\frac {n[{\mathfrak {EH}}]}{4\pi }}dv}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f40e6749032970ef1d5f76bbd3a39e044a8a22)
![{\displaystyle {\mathfrak {g}}={\frac {n}{4\pi }}[{\mathfrak {EH}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916c99325781a046fe6e60932c74ed708d0af89c)
![{\displaystyle \mathrm {div} [{\mathfrak {EH}}]={\mathfrak {H}}\ \mathrm {rot} {\mathfrak {E}}-{\mathfrak {E}}\ \mathrm {rot} {\mathfrak {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ea4a8cf1f4afba308b8503184c03ecb32652b3)
![{\displaystyle {\mathfrak {K_{1}v}}+Q_{1}={\mathfrak {J_{1}E_{1}+\varrho E_{1}v+\left[J_{1}B\right]v=JE}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b86031561fe4da7ef4fd933d170deb79a573352)
![{\displaystyle =-{\frac {\mathrm {div} [{\mathfrak {EH}}]}{4\pi }}-{\frac {1}{8\pi }}{\frac {\partial }{\partial t}}\left\{{\mathfrak {E}}^{2}n+{\mathfrak {H}}^{2}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3395fdc0b890340199dc58e529d8852f1a083303)
![{\displaystyle \mathrm {div} \left({\mathfrak {S-{\frac {[EH]}{4\pi }}+\psi v+R}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd37e8e1d43e6fbb485cb854e7c7337cb995344)
![{\displaystyle {\mathfrak {S={\frac {[EH]}{4\pi }}-\psi v-R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee54f037a073a4a00b1b168de8e1ef4670b47f41)
