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Multipolstrahlung Eq: math.2133.23

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TeX (original user input):

\begin{align}
& \dot{\Phi }\left( \bar{r},t \right)+{{c}^{2}}\nabla \cdot \bar{A}\left( \bar{r},t \right)=0 \\
& \Rightarrow \frac{\partial }{\partial t}\Phi \left( \bar{r},t \right)=-\frac{1}{{{\varepsilon }_{0}}{{\mu }_{0}}}\nabla \cdot \bar{A}\left( \bar{r},t \right)=-\frac{1}{4\pi {{\varepsilon }_{0}}}\nabla \left[ \frac{1}{r}\dot{\bar{p}}\left( t-\frac{r}{c} \right) \right] \\
& \Rightarrow \Phi \left( \bar{r},t \right)=-\frac{1}{4\pi {{\varepsilon }_{0}}}\nabla \left[ \frac{1}{r}\bar{p}\left( t-\frac{r}{c} \right) \right]+{{\Phi }_{stat.}}\left( {\bar{r}} \right) \\
& {{\Phi }_{stat.}}\left( {\bar{r}} \right)=0(obda) \\
& \Rightarrow \Phi \left( \bar{r},t \right)=-\frac{1}{4\pi {{\varepsilon }_{0}}}\nabla \left[ \frac{1}{r}\bar{p}\left( t-\frac{r}{c} \right) \right]=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{1}{c{{r}^{2}}}\bar{r}\dot{\bar{p}}\left( t-\frac{r}{c} \right)+\frac{1}{{{r}^{3}}}\bar{r}\bar{p}\left( t-\frac{r}{c} \right) \right] \\
& \frac{1}{c{{r}^{2}}}\bar{r}\dot{\bar{p}}\left( t-\frac{r}{c} \right)\tilde{\ }\frac{1}{r} \\
& \frac{1}{{{r}^{3}}}\bar{r}\bar{p}\left( t-\frac{r}{c} \right)\tilde{\ }\frac{1}{{{r}^{2}}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{\dot {\Phi }}\left({\bar {r}},t\right)+{{c}^{2}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)=0\\&\Rightarrow {\frac {\partial }{\partial t}}\Phi \left({\bar {r}},t\right)=-{\frac {1}{{{\varepsilon }_{0}}{{\mu }_{0}}}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\nabla \left[{\frac {1}{r}}{\dot {\bar {p}}}\left(t-{\frac {r}{c}}\right)\right]\\&\Rightarrow \Phi \left({\bar {r}},t\right)=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\nabla \left[{\frac {1}{r}}{\bar {p}}\left(t-{\frac {r}{c}}\right)\right]+{{\Phi }_{stat.}}\left({\bar {r}}\right)\\&{{\Phi }_{stat.}}\left({\bar {r}}\right)=0(obda)\\&\Rightarrow \Phi \left({\bar {r}},t\right)=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\nabla \left[{\frac {1}{r}}{\bar {p}}\left(t-{\frac {r}{c}}\right)\right]={\frac {1}{4\pi {{\varepsilon }_{0}}}}\left[{\frac {1}{c{{r}^{2}}}}{\bar {r}}{\dot {\bar {p}}}\left(t-{\frac {r}{c}}\right)+{\frac {1}{{r}^{3}}}{\bar {r}}{\bar {p}}\left(t-{\frac {r}{c}}\right)\right]\\&{\frac {1}{c{{r}^{2}}}}{\bar {r}}{\dot {\bar {p}}}\left(t-{\frac {r}{c}}\right){\tilde {\ }}{\frac {1}{r}}\\&{\frac {1}{{r}^{3}}}{\bar {r}}{\bar {p}}\left(t-{\frac {r}{c}}\right){\tilde {\ }}{\frac {1}{{r}^{2}}}\\\end{aligned}}

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\begin{aligned} \dot{Φ} (\bar{r}, t) + c^{2} \nabla \cdot \bar{A} (\bar{r}, t) = 0 \\ \Rightarrow \frac{\partial}{\partial t} Φ (\bar{r}, t) = - \frac{1}{ε_{0} μ_{0}} \nabla \cdot \bar{A} (\bar{r}, t) = - \frac{1}{4 π ε_{0}} \nabla [\frac{1}{r} \dot{\bar{p}} (t - \frac{r}{c})] \\ \Rightarrow Φ (\bar{r}, t) = - \frac{1}{4 π ε_{0}} \nabla [\frac{1}{r} \bar{p} (t - \frac{r}{c})] + Φ_{s t a t .} (\bar{r}) \\ Φ_{s t a t .} (\bar{r}) = 0 (o b d a) \\ \Rightarrow Φ (\bar{r}, t) = - \frac{1}{4 π ε_{0}} \nabla [\frac{1}{r} \bar{p} (t - \frac{r}{c})] = \frac{1}{4 π ε_{0}} [\frac{1}{c r^{2}} \bar{r} \dot{\bar{p}} (t - \frac{r}{c}) + \frac{1}{r^{3}} \bar{r} \bar{p} (t - \frac{r}{c})] \\ \frac{1}{c r^{2}} \bar{r} \dot{\bar{p}} (t - \frac{r}{c}) \tilde{} \frac{1}{r} \\ \frac{1}{r^{3}} \bar{r} \bar{p} (t - \frac{r}{c}) \tilde{} \frac{1}{r^{2}} \end{aligned}

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Identifiers

$\dot{Φ}$

$\bar{r}$

$t$

$c$

$\bar{A}$

$\bar{r}$

$t$

$t$

$Φ$

$\bar{r}$

$t$

$ε_{0}$

$μ_{0}$

$\bar{A}$

$\bar{r}$

$t$

$π$

$ε_{0}$

$r$

$\dot{\bar{p}}$

$t$

$r$

$c$

$Φ$

$\bar{r}$

$t$

$π$

$ε_{0}$

$r$

$\bar{p}$

$t$

$r$

$c$

$Φ$

$s$

$t$

$a$

$t$

$\bar{r}$

$Φ$

$s$

$t$

$a$

$t$

$\bar{r}$

$o$

$b$

$d$

$a$

$Φ$

$\bar{r}$

$t$

$π$

$ε_{0}$

$r$

$\bar{p}$

$t$

$r$

$c$

$π$

$ε_{0}$

$c$

$r$

$\bar{r}$

$\dot{\bar{p}}$

$t$

$r$

$c$

$r$

$\bar{r}$

$\bar{p}$

$t$

$r$

$c$

$c$

$r$

$\bar{r}$

$\dot{\bar{p}}$

$t$

$r$

$c$

$\tilde{}$

$r$

$r$

$\bar{r}$

$\bar{p}$

$t$

$r$

$c$

$\tilde{}$

$r$

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