Seite:De Induction in rotirenden Kugeln (Hertz) 078.png

	Heinrich Hertz: Ueber die Induction in rotirenden Kugeln

77

Dann gelten für ${\mathfrak {x}}\ {\mathfrak {y}}\ {\mathfrak {z}}\,$ die Gleichungen:

${\begin{aligned}{\mathfrak {x}}&=-\varepsilon {\frac {\partial \varphi }{\partial x}}+\varepsilon {\mathfrak {X}}\\{\mathfrak {y}}&=-\varepsilon {\frac {\partial \varphi }{\partial y}}+\varepsilon {\mathfrak {Y}}\\{\mathfrak {z}}&=-\varepsilon {\frac {\partial \varphi }{\partial z}}+\varepsilon {\mathfrak {Z}}\end{aligned}}\,$

${\frac {\partial {\mathfrak {x}}}{\partial x}}+{\frac {\partial {\mathfrak {y}}}{\partial y}}+{\frac {\partial {\mathfrak {z}}}{\partial z}}={\frac {1}{4\pi }}{\mathit {\Delta }}\varphi ;\,$

für $\varrho =R\,$

${\mathfrak {x}}x+{\mathfrak {y}}y+{\mathfrak {z}}z={\frac {\varrho }{4\pi }}\left[{\frac {\partial \varphi _{i}}{\partial \varrho }}-{\frac {\partial \varphi _{a}}{\partial \varrho }}\right].\,$

Daraus folgt, für $\varphi :\,$

${\mathit {\Delta }}\varphi ={\frac {4\pi \varepsilon }{1+4\pi \varepsilon }}\left({\frac {\partial {\mathfrak {X}}}{\partial x}}+{\frac {\partial {\mathfrak {Y}}}{\partial y}}+{\frac {\partial {\mathfrak {Z}}}{\partial z}}\right)\,$

und für $\varrho =R:\,$

$(1+4\pi \varepsilon ){\frac {\partial \varphi _{i}}{\partial \varrho }}-{\frac {\partial \varphi _{a}}{\partial \varrho }}={\frac {4\pi \varepsilon }{\varrho }}(x{\mathfrak {X}}+y{\mathfrak {Y}}+z{\mathfrak {Z}}).\,$

Im äussern Raum muss sein ${\mathit {\Delta }}\varphi =0.\,$

Ist wieder $\chi _{n}\,$ das $n\,$te Glied des äussern Potentiales, so wie oben (Seite 10):

${\frac {\partial {\mathfrak {X}}}{\partial x}}+{\frac {\partial {\mathfrak {Y}}}{\partial y}}+{\frac {\partial {\mathfrak {Z}}}{\partial z}}=2\omega {\frac {\partial \chi _{n}}{\partial z}}\,$

$x{\mathfrak {X}}+y{\mathfrak {Y}}+z{\mathfrak {Z}}=\omega \left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right).\,$

Um die Bedingungsgleichungen zu erfüllen, setzen wir:

$\varphi =\varphi ^{0}+\varphi '\,$

${\begin{aligned}{\varphi ^{0}}_{i}&={\frac {4\pi \varepsilon }{1+4\pi \varepsilon }}{\frac {\omega }{n+1}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)\\{\varphi ^{0}}_{a}&={\frac {4\pi \varepsilon }{1+4\pi \varepsilon }}{\frac {\omega }{n+1}}\left[\left({\frac {R}{\varrho }}\right)^{n+2}{\frac {n}{n+1}}{\overline {\left({\frac {\varrho ^{2}}{2n+1}}{\frac {\partial \chi _{n}}{\partial z}}-z\chi _{n}\right)}}+\left({\frac {R}{\varrho }}\right)^{n}{\frac {R^{2}}{2n+1}}{\overline {\frac {\partial \chi _{n}}{\partial z}}}\right]\end{aligned}}\,$

Empfohlene Zitierweise:
Heinrich Hertz: Ueber die Induction in rotirenden Kugeln, Berlin 1880, Seite 77. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:De_Induction_in_rotirenden_Kugeln_(Hertz)_078.png&oldid=- (Version vom 31.7.2018)