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Display information for equation id:math.2095.13 on revision:2095

* Page found: Magnetostatische Feldgleichungen (eq math.2095.13)

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\begin{align}
& \Rightarrow \nabla \cdot \bar{A}(\bar{r})=-\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\cdot \left( \frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \right)-\frac{\partial }{\partial t}\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{\rho (\bar{r}\acute{\ },t)}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \\
& \frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{\rho (\bar{r}\acute{\ },t)}{\left| \bar{r}-\bar{r}\acute{\ } \right|}={{\mu }_{0}}{{\varepsilon }_{0}}\Phi (\bar{r},t) \\
& \Rightarrow \nabla \cdot \bar{A}(\bar{r})=-\frac{{{\mu }_{0}}}{4\pi }\oint\limits_{S\infty }{{}}{{d}^{3}}\bar{f}\acute{\ }\left( \frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \right)-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial }{\partial t}\Phi (\bar{r},t) \\
\end{align}

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⇒ ∇ ⋅ A ¯ ⁢ ( r ¯ ) = - μ 0 4 ⁢ π ⁢ ∫ R 3 d 3 ⁢ r ⁢ ´ ⁢ ∇ r ⁢ ´ ⋅ ( j ¯ ⁢ ( r ¯ ⁢ ´ ) | r ¯ - r ¯ ⁢ ´ | ) - ∂ ∂ ⁡ t ⁢ μ 0 4 ⁢ π ⁢ ∫ R 3 d 3 ⁢ r ⁢ ´ ⁢ ρ ⁢ ( r ¯ ⁢ ´ , t ) | r ¯ - r ¯ ⁢ ´ | μ 0 4 ⁢ π ⁢ ∫ R 3 d 3 ⁢ r ⁢ ´ ⁢ ρ ⁢ ( r ¯ ⁢ ´ , t ) | r ¯ - r ¯ ⁢ ´ | = μ 0 ⁢ ε 0 ⁢ Φ ⁢ ( r ¯ , t ) ⇒ ∇ ⋅ A ¯ ⁢ ( r ¯ ) = - μ 0 4 ⁢ π ⁢ ∮ S ⁢ ∞ d 3 ⁢ f ¯ ⁢ ´ ⁢ ( j ¯ ⁢ ( r ¯ ⁢ ´ ) | r ¯ - r ¯ ⁢ ´ | ) - μ 0 ⁢ ε 0 ⁢ ∂ ∂ ⁡ t ⁢ Φ ⁢ ( r ¯ , t ) missing-subexpression ⇒ absent ⋅ ∇ ¯ 𝐴 ¯ 𝑟 subscript 𝜇 0 4 𝜋 subscript superscript 𝑅 3 ⋅ superscript 𝑑 3 𝑟 ´ absent subscript ∇ 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent ¯ 𝑟 ¯ 𝑟 ´ absent 𝑡 subscript 𝜇 0 4 𝜋 subscript superscript 𝑅 3 superscript 𝑑 3 𝑟 ´ absent 𝜌 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent missing-subexpression subscript 𝜇 0 4 𝜋 subscript superscript 𝑅 3 superscript 𝑑 3 𝑟 ´ absent 𝜌 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent subscript 𝜇 0 subscript 𝜀 0 Φ ¯ 𝑟 𝑡 missing-subexpression ⇒ absent ⋅ ∇ ¯ 𝐴 ¯ 𝑟 subscript 𝜇 0 4 𝜋 subscript contour-integral 𝑆 superscript 𝑑 3 ¯ 𝑓 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent ¯ 𝑟 ¯ 𝑟 ´ absent subscript 𝜇 0 subscript 𝜀 0 𝑡 Φ ¯ 𝑟 𝑡 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\Rightarrow\nabla% \cdot\bar{A}(\bar{r})=-\frac{{{\mu}_{0}}}{4\pi}\int_{{{R}^{3}}}{{}}{{d}^{3}}r% \acute{\ }{{\nabla}_{r\acute{\ }}}\cdot\left(\frac{\bar{j}(\bar{r}\acute{\ })}% {\left|\bar{r}-\bar{r}\acute{\ }\right|}\right)-\frac{\partial}{\partial t}% \frac{{{\mu}_{0}}}{4\pi}\int_{{{R}^{3}}}{{}}{{d}^{3}}r\acute{\ }\frac{\rho(% \bar{r}\acute{\ },t)}{\left|\bar{r}-\bar{r}\acute{\ }\right|}\\ &\displaystyle\frac{{{\mu}_{0}}}{4\pi}\int_{{{R}^{3}}}{{}}{{d}^{3}}r\acute{\ }% \frac{\rho(\bar{r}\acute{\ },t)}{\left|\bar{r}-\bar{r}\acute{\ }\right|}={{\mu% }_{0}}{{\varepsilon}_{0}}\Phi(\bar{r},t)\\ &\displaystyle\Rightarrow\nabla\cdot\bar{A}(\bar{r})=-\frac{{{\mu}_{0}}}{4\pi}% \oint\limits_{S\infty}{{}}{{d}^{3}}\bar{f}\acute{\ }\left(\frac{\bar{j}(\bar{r% }\acute{\ })}{\left|\bar{r}-\bar{r}\acute{\ }\right|}\right)-{{\mu}_{0}}{{% \varepsilon}_{0}}\frac{\partial}{\partial t}\Phi(\bar{r},t)\\ \end{aligned}}}

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⇒∇⋅A¯(r¯)=−μ04π∫R3d3r´∇r´⋅(j¯(r¯´)|r¯−r¯´|)−∂∂tμ04π∫R3d3r´ρ(r¯´,t)|r¯−r¯´|μ04π∫R3d3r´ρ(r¯´,t)|r¯−r¯´|=μ0ε0Φ(r¯,t)⇒∇⋅A¯(r¯)=−μ04π∮S∞d3f¯´(j¯(r¯´)|r¯−r¯´|)−μ0ε0∂∂tΦ(r¯,t)

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