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Display information for equation id:math.2139.22 on revision:2139

* Page found: Polarisation (eq math.2139.22)

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Polarisation Eq: math.2141.22

Hash: 7bb3910342d5f550e2e338b72db94f25

TeX (original user input):

\begin{align}
& \Phi \left( \bar{r},t \right)=\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s{{\Phi }_{m}}\left( \bar{r}+\bar{s},t \right)=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{{{\rho }_{m}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}+\bar{s}-\bar{r}\acute{\ } \right|}{c} \right)}{\left| \bar{r}+\bar{s}-\bar{r}\acute{\ } \right|} \\
& \bar{r}\acute{\ }\acute{\ }:=\bar{r}\acute{\ }-\bar{s} \\
& \Phi \left( \bar{r},t \right)=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\acute{\ }\frac{{{\rho }_{m}}\left( \bar{r}\acute{\ }\acute{\ }+\bar{s},t-\frac{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}{c} \right)}{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|} \\
& =\frac{1}{4\pi {{\varepsilon }_{0}}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\acute{\ }\frac{1}{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s{{\rho }_{m}}\left( \bar{r}\acute{\ }\acute{\ }+\bar{s},t-\frac{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}{c} \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&\Phi \left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\Phi }_{m}}\left({\bar {r}}+{\bar {s}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {{{\rho }_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)}{\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}}\\&{\bar {r}}{\acute {\ }}{\acute {\ }}:={\bar {r}}{\acute {\ }}-{\bar {s}}\\&\Phi \left({\bar {r}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}\\&={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\\\end{aligned}}

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\begin{matrix} Φ (\bar{r}, t) = \frac{1}{Δ V} \int_{Δ V}^{} d^{3} s Φ_{m} (\bar{r} + \bar{s}, t) = \frac{1}{4 π ε_{0}} \frac{1}{Δ V} \int_{Δ V}^{} d^{3} s \int_{R^{3}}^{} d^{3} r \overset{´}{} \frac{ρ_{m} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} + \bar{s} - \bar{r} \overset{´}{} |}{c})}{| \bar{r} + \bar{s} - \bar{r} \overset{´}{} |} \\ \bar{r} \overset{´}{} \overset{´}{} := \bar{r} \overset{´}{} - \bar{s} \\ Φ (\bar{r}, t) = \frac{1}{4 π ε_{0}} \frac{1}{Δ V} \int_{Δ V}^{} d^{3} s \int_{R^{3}}^{} d^{3} r \overset{´}{} \overset{´}{} \frac{ρ_{m} (\bar{r} \overset{´}{} \overset{´}{} + \bar{s}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} \overset{´}{} |}{c})}{| \bar{r} - \bar{r} \overset{´}{} \overset{´}{} |} \\ = \frac{1}{4 π ε_{0}} \int_{R^{3}}^{} d^{3} r \overset{´}{} \overset{´}{} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} \overset{´}{} |} \frac{1}{Δ V} \int_{Δ V}^{} d^{3} s ρ_{m} (\bar{r} \overset{´}{} \overset{´}{} + \bar{s}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} \overset{´}{} |}{c}) \end{matrix}

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Identifiers

$Φ$

$\bar{r}$

$t$

$Δ$

$V$

$Δ$

$V$

$s$

$Φ_{m}$

$\bar{r}$

$\bar{s}$

$t$

$π$

$ε_{0}$

$Δ$

$V$

$Δ$

$V$

$s$

$R$

$r$

$\overset{´}{}$

$ρ_{m}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{s}$

$\bar{r}$

$\overset{´}{}$

$c$

$\bar{r}$

$\bar{s}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\bar{s}$

$Φ$

$\bar{r}$

$t$

$π$

$ε_{0}$

$Δ$

$V$

$Δ$

$V$

$s$

$R$

$r$

$\overset{´}{}$

$\overset{´}{}$

$ρ_{m}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$\bar{s}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$c$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$π$

$ε_{0}$

$R$

$r$

$\overset{´}{}$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$Δ$

$V$

$Δ$

$V$

$s$

$ρ_{m}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$\bar{s}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\overset{´}{}$

$c$

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