Seite:De Induction in rotirenden Kugeln (Hertz) 013.png

	Heinrich Hertz: Ueber die Induction in rotirenden Kugeln

12

entsprechend für $V\,$ und $W;\,$ ausserdem die gewöhnlichen Stetigkeitsbedingungen. Allen diesen Bedingungen ist genügt, wenn ^[1] man setzt:

${\begin{aligned}U_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial x}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+x{\frac {\partial \chi _{n}}{\partial z}}\right\rbrace \\V_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial y}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+y{\frac {\partial \chi _{n}}{\partial z}}\right\rbrace \\W_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial z}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+z{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right\rbrace \end{aligned}}\,$

${\begin{aligned}U_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}U_{i}\left({\frac {1}{\varrho }}\right)\\V_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}V_{i}\left({\frac {1}{\varrho }}\right)\\W_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}W_{i}\left({\frac {1}{\varrho }}\right).\end{aligned}}\,$

Aus diesen $U\ V\ W\,$ wollen wir die magnetisirenden Kräfte im Innern, nämlich

${\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}},\,$ etc.

berechnen, und dieselben

$=-{\frac {n+1}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial z}},\,$ etc.

setzen, wir erhalten so die Funktion ${\mathit {\Psi }}\,$ (§ 1,5). Wir finden nämlich:

${\begin{aligned}-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial z}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial z}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial z}}{\mathit {\Psi _{i}}}\\-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial x}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial x}}{\mathit {\Psi _{i}}}\\-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial y}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial y}}{\mathit {\Psi _{i}}}.\end{aligned}}\,$

Also ergiebt sich

${\begin{aligned}{\mathit {\Psi _{i}}}&={\frac {4\pi R^{2}}{(2n+1)(n+1)}}{\frac {\omega }{k}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)\\&={\frac {4\pi R^{2}}{(2n+1)(n+1)}}{\frac {\omega }{k}}\chi _{n}',\end{aligned}}\,$

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1. Aufsuchung der Funktion ${\mathit {\Psi }}.\,$ WS: Die Randnotiz wurde als Fußnote übertragen.

Empfohlene Zitierweise:
Heinrich Hertz: Ueber die Induction in rotirenden Kugeln, Berlin 1880, Seite 12. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:De_Induction_in_rotirenden_Kugeln_(Hertz)_013.png&oldid=- (Version vom 31.7.2018)