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Display information for equation id:math.2155.28 on revision:2155

* Page found: Mikroskopisches Modell der Polarisierbarkeit (eq math.2155.28)

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Mikroskopisches Modell der Polarisierbarkeit Eq: math.2156.28

Hash: 1b26d5682289eafdbfadd81477baf121

TeX (original user input):

\begin{align}
& {{\Phi }_{0}}\left( {\bar{r}} \right)={{\Phi }_{0}}\left( \bar{r}-\frac{1}{2}{{{\bar{r}}}_{0}} \right)-{{\Phi }_{0}}\left( \bar{r}+\frac{1}{2}{{{\bar{r}}}_{0}} \right) \\
& \approx -{{{\bar{r}}}_{0}}\nabla {{\Phi }_{0}}\left( {\bar{r}} \right) \\
& \nabla {{\Phi }_{0}}\left( {\bar{r}} \right)=-{{{\bar{E}}}_{0}} \\
& \Rightarrow {{\Phi }_{0}}\left( {\bar{r}} \right)\approx {{{\bar{r}}}_{0}}{{{\bar{E}}}_{0}}=\frac{Q}{4\pi {{\varepsilon }_{0}}}\left\{ \begin{matrix}
\frac{{{{\bar{r}}}_{0}}\bar{r}}{{{a}^{3}}}r\le a  \\
\frac{{{{\bar{r}}}_{0}}\bar{r}}{{{r}^{3}}}r\ge a  \\
\end{matrix} \right.=\frac{1}{4\pi {{\varepsilon }_{0}}}\left\{ \begin{matrix}
\frac{\bar{p}\bar{r}}{{{a}^{3}}}r\le a  \\
\frac{\bar{p}\bar{r}}{{{r}^{3}}}r\ge a  \\
\end{matrix} \right. \\
& \bar{p}:=Q{{{\bar{r}}}_{0}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\Phi }_{0}}\left({\bar {r}}\right)={{\Phi }_{0}}\left({\bar {r}}-{\frac {1}{2}}{{\bar {r}}_{0}}\right)-{{\Phi }_{0}}\left({\bar {r}}+{\frac {1}{2}}{{\bar {r}}_{0}}\right)\\&\approx -{{\bar {r}}_{0}}\nabla {{\Phi }_{0}}\left({\bar {r}}\right)\\&\nabla {{\Phi }_{0}}\left({\bar {r}}\right)=-{{\bar {E}}_{0}}\\&\Rightarrow {{\Phi }_{0}}\left({\bar {r}}\right)\approx {{\bar {r}}_{0}}{{\bar {E}}_{0}}={\frac {Q}{4\pi {{\varepsilon }_{0}}}}\left\{{\begin{matrix}{\frac {{{\bar {r}}_{0}}{\bar {r}}}{{a}^{3}}}r\leq a\\{\frac {{{\bar {r}}_{0}}{\bar {r}}}{{r}^{3}}}r\geq a\\\end{matrix}}\right.={\frac {1}{4\pi {{\varepsilon }_{0}}}}\left\{{\begin{matrix}{\frac {{\bar {p}}{\bar {r}}}{{a}^{3}}}r\leq a\\{\frac {{\bar {p}}{\bar {r}}}{{r}^{3}}}r\geq a\\\end{matrix}}\right.\\&{\bar {p}}:=Q{{\bar {r}}_{0}}\\\end{aligned}}

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\begin{matrix} Φ_{0} (\bar{r}) = Φ_{0} (\bar{r} - \frac{1}{2} {\bar{r}}_{0}) - Φ_{0} (\bar{r} + \frac{1}{2} {\bar{r}}_{0}) \\ \approx - {\bar{r}}_{0} \nabla Φ_{0} (\bar{r}) \\ \nabla Φ_{0} (\bar{r}) = - {\bar{E}}_{0} \\ \Rightarrow Φ_{0} (\bar{r}) \approx {\bar{r}}_{0} {\bar{E}}_{0} = \frac{Q}{4 π ε_{0}} {\begin{matrix} \frac{{\bar{r}}_{0} \bar{r}}{a^{3}} r \leq a \\ \frac{{\bar{r}}_{0} \bar{r}}{r^{3}} r \geq a \end{matrix} = \frac{1}{4 π ε_{0}} {\begin{matrix} \frac{\bar{p} \bar{r}}{a^{3}} r \leq a \\ \frac{\bar{p} \bar{r}}{r^{3}} r \geq a \end{matrix} \\ \bar{p} := Q {\bar{r}}_{0} \end{matrix}

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Identifiers

$Φ_{0}$

$\bar{r}$

$Φ_{0}$

$\bar{r}$

${\bar{r}}_{0}$

$Φ_{0}$

$\bar{r}$

${\bar{r}}_{0}$

${\bar{r}}_{0}$

$Φ_{0}$

$\bar{r}$

$Φ_{0}$

$\bar{r}$

${\bar{E}}_{0}$

$Φ_{0}$

$\bar{r}$

${\bar{r}}_{0}$

${\bar{E}}_{0}$

$Q$

$π$

$ε_{0}$

${\bar{r}}_{0}$

$\bar{r}$

$a$

$r$

$a$

${\bar{r}}_{0}$

$\bar{r}$

$r$

$r$

$a$

$π$

$ε_{0}$

$\bar{p}$

$\bar{r}$

$a$

$r$

$a$

$\bar{p}$

$\bar{r}$

$r$

$r$

$a$

$\bar{p}$

$Q$

${\bar{r}}_{0}$

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