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* Page found: Magnetisierung (eq math.2144.18)

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\begin{align}
& \bar{A}\left( \bar{r},t \right)=\frac{{{\mu }_{0}}}{4\pi }\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\left[ \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\dot{\bar{P}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right)+{{\nabla }_{r}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right) \right] \\
& \int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)= \\
& =-\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)+\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}{{\nabla }_{r\acute{\ }}}\times \bar{M}\left( \bar{r}\acute{\ },t\acute{\ } \right) \\
& t\acute{\ }=t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \\
& \int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)=0 \\
& \Rightarrow \bar{A}\left( \bar{r},t \right)=\frac{{{\mu }_{0}}}{4\pi }\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\left[ \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\dot{\bar{P}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right)+\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}{{\nabla }_{r\acute{\ }}}\times \bar{M}\left( \bar{r}\acute{\ },t\acute{\ } \right) \right] \\
\end{align}

TeX (checked): 

{\begin{aligned}&{\bar {A}}\left({\bar {r}},t\right)={\frac {{\mu }_{0}}{4\pi }}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}\left[{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\dot {\bar {P}}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{{\nabla }_{r}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)\right]\\&\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)=\\&=-\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r{\acute {\ }}}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)+\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}\times {\bar {M}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\\&t{\acute {\ }}=t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\\&\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r{\acute {\ }}}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)=0\\&\Rightarrow {\bar {A}}\left({\bar {r}},t\right)={\frac {{\mu }_{0}}{4\pi }}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}\left[{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\dot {\bar {P}}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}\times {\bar {M}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\right]\\\end{aligned}}

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MathML (160.426 KB / 13.655 KB) :

A ¯ ⁢ ( r ¯ , t ) = μ 0 4 ⁢ π ⁢ ∫ d 3 ⁢ r ⁢ ´ ⁢ [ 1 | r ¯ - r ¯ ⁢ ´ | ⁢ P ¯ ˙ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) + ∇ r × ( 1 | r ¯ - r ¯ ⁢ ´ | ⁢ M ¯ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) ) ] ∫ d 3 ⁢ r ⁢ ´ ⁢ ∇ r × ( 1 | r ¯ - r ¯ ⁢ ´ | ⁢ M ¯ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) ) = = - ∫ d 3 ⁢ r ⁢ ´ ⁢ ∇ r ⁢ ´ × ( 1 | r ¯ - r ¯ ⁢ ´ | ⁢ M ¯ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) ) + ∫ d 3 ⁢ r ⁢ ´ ⁢ 1 | r ¯ - r ¯ ⁢ ´ | ⁢ ∇ r ⁢ ´ × M ¯ ⁢ ( r ¯ ⁢ ´ , t ⁢ ´ ) t ⁢ ´ = t - | r ¯ - r ¯ ⁢ ´ | c ∫ d 3 ⁢ r ⁢ ´ ⁢ ∇ r ⁢ ´ × ( 1 | r ¯ - r ¯ ⁢ ´ | ⁢ M ¯ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) ) = 0 ⇒ A ¯ ⁢ ( r ¯ , t ) = μ 0 4 ⁢ π ⁢ ∫ d 3 ⁢ r ⁢ ´ ⁢ [ 1 | r ¯ - r ¯ ⁢ ´ | ⁢ P ¯ ˙ ⁢ ( r ¯ ⁢ ´ , t - | r ¯ - r ¯ ⁢ ´ | c ) + 1 | r ¯ - r ¯ ⁢ ´ | ⁢ ∇ r ⁢ ´ × M ¯ ⁢ ( r ¯ ⁢ ´ , t ⁢ ´ ) ] missing-subexpression ¯ 𝐴 ¯ 𝑟 𝑡 subscript 𝜇 0 4 𝜋 superscript 𝑑 3 𝑟 ´ absent delimited-[] 1 ¯ 𝑟 ¯ 𝑟 ´ absent ˙ ¯ 𝑃 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 subscript ∇ 𝑟 1 ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 missing-subexpression superscript 𝑑 3 𝑟 ´ absent subscript ∇ 𝑟 1 ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 absent missing-subexpression absent superscript 𝑑 3 𝑟 ´ absent subscript ∇ 𝑟 ´ absent 1 ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 superscript 𝑑 3 𝑟 ´ absent 1 ¯ 𝑟 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ´ absent missing-subexpression 𝑡 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 missing-subexpression superscript 𝑑 3 𝑟 ´ absent subscript ∇ 𝑟 ´ absent 1 ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 0 missing-subexpression ⇒ absent ¯ 𝐴 ¯ 𝑟 𝑡 subscript 𝜇 0 4 𝜋 superscript 𝑑 3 𝑟 ´ absent delimited-[] 1 ¯ 𝑟 ¯ 𝑟 ´ absent ˙ ¯ 𝑃 ¯ 𝑟 ´ absent 𝑡 ¯ 𝑟 ¯ 𝑟 ´ absent 𝑐 1 ¯ 𝑟 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ´ absent ¯ 𝑀 ¯ 𝑟 ´ absent 𝑡 ´ absent {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\bar{A}\left(\bar{r}% ,t\right)=\frac{{{\mu}_{0}}}{4\pi}\int{{}}{{d}^{3}}r\acute{\ }\left[\frac{1}{% \left|\bar{r}-\bar{r}\acute{\ }\right|}\dot{\bar{P}}\left(\bar{r}\acute{\ },t-% \frac{\left|\bar{r}-\bar{r}\acute{\ }\right|}{c}\right)+{{\nabla}_{r}}\times% \left(\frac{1}{\left|\bar{r}-\bar{r}\acute{\ }\right|}\bar{M}\left(\bar{r}% \acute{\ },t-\frac{\left|\bar{r}-\bar{r}\acute{\ }\right|}{c}\right)\right)% \right]\\ &\displaystyle\int{{}}{{d}^{3}}r\acute{\ }{{\nabla}_{r}}\times\left(\frac{1}{% \left|\bar{r}-\bar{r}\acute{\ }\right|}\bar{M}\left(\bar{r}\acute{\ },t-\frac{% \left|\bar{r}-\bar{r}\acute{\ }\right|}{c}\right)\right)=\\ &\displaystyle=-\int{{}}{{d}^{3}}r\acute{\ }{{\nabla}_{r\acute{\ }}}\times% \left(\frac{1}{\left|\bar{r}-\bar{r}\acute{\ }\right|}\bar{M}\left(\bar{r}% \acute{\ },t-\frac{\left|\bar{r}-\bar{r}\acute{\ }\right|}{c}\right)\right)+% \int{{}}{{d}^{3}}r\acute{\ }\frac{1}{\left|\bar{r}-\bar{r}\acute{\ }\right|}{{% \nabla}_{r\acute{\ }}}\times\bar{M}\left(\bar{r}\acute{\ },t\acute{\ }\right)% \\ &\displaystyle t\acute{\ }=t-\frac{\left|\bar{r}-\bar{r}\acute{\ }\right|}{c}% \\ &\displaystyle\int{{}}{{d}^{3}}r\acute{\ }{{\nabla}_{r\acute{\ }}}\times\left(% \frac{1}{\left|\bar{r}-\bar{r}\acute{\ }\right|}\bar{M}\left(\bar{r}\acute{\ }% ,t-\frac{\left|\bar{r}-\bar{r}\acute{\ }\right|}{c}\right)\right)=0\\ &\displaystyle\Rightarrow\bar{A}\left(\bar{r},t\right)=\frac{{{\mu}_{0}}}{4\pi% }\int{{}}{{d}^{3}}r\acute{\ }\left[\frac{1}{\left|\bar{r}-\bar{r}\acute{\ }% \right|}\dot{\bar{P}}\left(\bar{r}\acute{\ },t-\frac{\left|\bar{r}-\bar{r}% \acute{\ }\right|}{c}\right)+\frac{1}{\left|\bar{r}-\bar{r}\acute{\ }\right|}{% {\nabla}_{r\acute{\ }}}\times\bar{M}\left(\bar{r}\acute{\ },t\acute{\ }\right)% \right]\\ \end{aligned}}}

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A¯(r¯,t)=μ04π∫d3r´[1|r¯−r¯´|P¯˙(r¯´,t−|r¯−r¯´|c)+∇r×(1|r¯−r¯´|M¯(r¯´,t−|r¯−r¯´|c))]∫d3r´∇r×(1|r¯−r¯´|M¯(r¯´,t−|r¯−r¯´|c))==−∫d3r´∇r´×(1|r¯−r¯´|M¯(r¯´,t−|r¯−r¯´|c))+∫d3r´1|r¯−r¯´|∇r´×M¯(r¯´,t´)t´=t−|r¯−r¯´|c∫d3r´∇r´×(1|r¯−r¯´|M¯(r¯´,t−|r¯−r¯´|c))=0⇒A¯(r¯,t)=μ04π∫d3r´[1|r¯−r¯´|P¯˙(r¯´,t−|r¯−r¯´|c)+1|r¯−r¯´|∇r´×M¯(r¯´,t´)]

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