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Display information for equation id:math.1432.92 on revision:1432

* Page found: Stationäre Ströme und Magnetfeld (eq math.1432.92)

Occurrences on the following pages:

Magnetische Multipole Eq: math.2101.33

Hash: 50033af4171a55e0b60029f733bd859f

TeX (original user input):

\begin{align}
& \bar{m}=\frac{1}{2}\oint\limits_{L}{{}}{{d}^{3}}r\acute{\ }\left( \bar{r}\acute{\ }\times \bar{j}(\bar{r}\acute{\ }) \right)=\frac{1}{2}\sum\limits_{i}{{}}{{q}_{i}}\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{r}\acute{\ }\times {{{\bar{v}}}_{i}}\delta \left( \bar{r}\acute{\ }-{{{\bar{r}}}_{i}} \right)=\frac{1}{2}\sum\limits_{i}{{}}{{q}_{i}}{{{\bar{r}}}_{i}}\times {{{\bar{v}}}_{i}}=\frac{1}{2}\sum\limits_{i}{{}}\frac{{{q}_{i}}}{{{m}_{i}}}{{m}_{i}}{{{\bar{r}}}_{i}}\times {{{\bar{v}}}_{i}} \\
& \frac{{{q}_{i}}}{{{m}_{i}}}=\frac{q}{m} \\
& \Rightarrow \bar{m}=\frac{q}{2m}\bar{L} \\
\end{align}

TeX (checked):

{\begin{aligned}&{\bar {m}}={\frac {1}{2}}\oint \limits _{L}{}{{d}^{3}}r{\acute {\ }}\left({\bar {r}}{\acute {\ }}\times {\bar {j}}({\bar {r}}{\acute {\ }})\right)={\frac {1}{2}}\sum \limits _{i}{}{{q}_{i}}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\bar {r}}{\acute {\ }}\times {{\bar {v}}_{i}}\delta \left({\bar {r}}{\acute {\ }}-{{\bar {r}}_{i}}\right)={\frac {1}{2}}\sum \limits _{i}{}{{q}_{i}}{{\bar {r}}_{i}}\times {{\bar {v}}_{i}}={\frac {1}{2}}\sum \limits _{i}{}{\frac {{q}_{i}}{{m}_{i}}}{{m}_{i}}{{\bar {r}}_{i}}\times {{\bar {v}}_{i}}\\&{\frac {{q}_{i}}{{m}_{i}}}={\frac {q}{m}}\\&\Rightarrow {\bar {m}}={\frac {q}{2m}}{\bar {L}}\\\end{aligned}}

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\begin{matrix} \bar{m} = \frac{1}{2} \oint_{L} d^{3} r \overset{´}{} (\bar{r} \overset{´}{} \times \bar{j} (\bar{r} \overset{´}{})) = \frac{1}{2} \sum_{i} q_{i} \int_{}^{} d^{3} r \overset{´}{} \bar{r} \overset{´}{} \times {\bar{v}}_{i} δ (\bar{r} \overset{´}{} - {\bar{r}}_{i}) = \frac{1}{2} \sum_{i} q_{i} {\bar{r}}_{i} \times {\bar{v}}_{i} = \frac{1}{2} \sum_{i} \frac{q_{i}}{m_{i}} m_{i} {\bar{r}}_{i} \times {\bar{v}}_{i} \\ \frac{q_{i}}{m_{i}} = \frac{q}{m} \\ \Rightarrow \bar{m} = \frac{q}{2 m} \bar{L} \end{matrix}

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Identifiers

$\bar{m}$

$L$

$d$

$r$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\bar{j}$

$\bar{r}$

$\overset{´}{}$

$i$

$q_{i}$

$r$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

${\bar{v}}_{i}$

$δ$

$\bar{r}$

$\overset{´}{}$

${\bar{r}}_{i}$

$i$

$q_{i}$

${\bar{r}}_{i}$

${\bar{v}}_{i}$

$i$

$q_{i}$

$m_{i}$

$m_{i}$

${\bar{r}}_{i}$

${\bar{v}}_{i}$

$q_{i}$

$m_{i}$

$q$

$m$

$\bar{m}$

$q$

$m$

$\bar{L}$

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