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* Page found: Magnetische Multipole (eq math.2101.45)

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\begin{align}
& \bar{F}=\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times {{\nabla }_{r}}\left[ \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right] \\
& \bar{j}(\bar{r}\acute{\ })\times {{\nabla }_{r}}\left[ \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right]=-{{\nabla }_{r}}\times \left[ \left( \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right)\bar{j}(\bar{r}\acute{\ }) \right]+\left[ \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right]{{\nabla }_{r}}\times \bar{j}(\bar{r}\acute{\ }) \\
& {{\nabla }_{r}}\times \bar{j}(\bar{r}\acute{\ })=0 \\
& \Rightarrow \bar{F}=-\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r}}\times \left[ \left( \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right)\bar{j}(\bar{r}\acute{\ }) \right]=-{{\nabla }_{r}}\times \left( \bar{m}\times \bar{B}(\bar{r}) \right) \\
& \bar{F}=-{{\nabla }_{r}}\times \left( \bar{m}\times \bar{B}(\bar{r}) \right)=\left( \bar{m}\cdot {{\nabla }_{r}} \right)\bar{B}(\bar{r})=-{{\nabla }_{r}}\left( -\bar{m}\cdot \bar{B}(\bar{r}) \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&{\bar {F}}=\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\bar {j}}({\bar {r}}{\acute {\ }})\times {{\nabla }_{r}}\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]\\&{\bar {j}}({\bar {r}}{\acute {\ }})\times {{\nabla }_{r}}\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]=-{{\nabla }_{r}}\times \left[\left(\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right){\bar {j}}({\bar {r}}{\acute {\ }})\right]+\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]{{\nabla }_{r}}\times {\bar {j}}({\bar {r}}{\acute {\ }})\\&{{\nabla }_{r}}\times {\bar {j}}({\bar {r}}{\acute {\ }})=0\\&\Rightarrow {\bar {F}}=-\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\times \left[\left(\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right){\bar {j}}({\bar {r}}{\acute {\ }})\right]=-{{\nabla }_{r}}\times \left({\bar {m}}\times {\bar {B}}({\bar {r}})\right)\\&{\bar {F}}=-{{\nabla }_{r}}\times \left({\bar {m}}\times {\bar {B}}({\bar {r}})\right)=\left({\bar {m}}\cdot {{\nabla }_{r}}\right){\bar {B}}({\bar {r}})=-{{\nabla }_{r}}\left(-{\bar {m}}\cdot {\bar {B}}({\bar {r}})\right)\\\end{aligned}}

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MathML (91.416 KB / 8.446 KB) :

F ¯ = ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) × ∇ r ⁡ [ ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ] j ¯ ⁢ ( r ¯ ⁢ ´ ) × ∇ r ⁡ [ ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ] = - ∇ r × [ ( ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ) ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) ] + [ ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ] ⁢ ∇ r × j ¯ ⁢ ( r ¯ ⁢ ´ ) ∇ r × j ¯ ⁢ ( r ¯ ⁢ ´ ) = 0 ⇒ F ¯ = - ∫ d 3 ⁢ r ⁢ ´ ⁢ ∇ r × [ ( ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ) ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) ] = - ∇ r × ( m ¯ × B ¯ ⁢ ( r ¯ ) ) F ¯ = - ∇ r × ( m ¯ × B ¯ ⁢ ( r ¯ ) ) = ( m ¯ ⋅ ∇ r ) ⁢ B ¯ ⁢ ( r ¯ ) = - ∇ r ⁡ ( - m ¯ ⋅ B ¯ ⁢ ( r ¯ ) ) missing-subexpression ¯ 𝐹 superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 missing-subexpression ¯ 𝑗 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 subscript ∇ 𝑟 delimited-[] ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 ¯ 𝑗 ¯ 𝑟 ´ absent delimited-[] ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 subscript ∇ 𝑟 ¯ 𝑗 ¯ 𝑟 ´ absent missing-subexpression subscript ∇ 𝑟 ¯ 𝑗 ¯ 𝑟 ´ absent 0 missing-subexpression ⇒ absent ¯ 𝐹 superscript 𝑑 3 𝑟 ´ absent subscript ∇ 𝑟 delimited-[] ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 ¯ 𝑗 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ¯ 𝑚 ¯ 𝐵 ¯ 𝑟 missing-subexpression ¯ 𝐹 subscript ∇ 𝑟 ¯ 𝑚 ¯ 𝐵 ¯ 𝑟 ⋅ ¯ 𝑚 subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 subscript ∇ 𝑟 ⋅ ¯ 𝑚 ¯ 𝐵 ¯ 𝑟 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\bar{F}=\int{{}}{{d}% ^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times{{\nabla}_{r}}\left[\left(\bar% {r}\acute{\ }\right)\cdot\bar{B}(\bar{r})\right]\\ &\displaystyle\bar{j}(\bar{r}\acute{\ })\times{{\nabla}_{r}}\left[\left(\bar{r% }\acute{\ }\right)\cdot\bar{B}(\bar{r})\right]=-{{\nabla}_{r}}\times\left[% \left(\left(\bar{r}\acute{\ }\right)\cdot\bar{B}(\bar{r})\right)\bar{j}(\bar{r% }\acute{\ })\right]+\left[\left(\bar{r}\acute{\ }\right)\cdot\bar{B}(\bar{r})% \right]{{\nabla}_{r}}\times\bar{j}(\bar{r}\acute{\ })\\ &\displaystyle{{\nabla}_{r}}\times\bar{j}(\bar{r}\acute{\ })=0\\ &\displaystyle\Rightarrow\bar{F}=-\int{{}}{{d}^{3}}r\acute{\ }{{\nabla}_{r}}% \times\left[\left(\left(\bar{r}\acute{\ }\right)\cdot\bar{B}(\bar{r})\right)% \bar{j}(\bar{r}\acute{\ })\right]=-{{\nabla}_{r}}\times\left(\bar{m}\times\bar% {B}(\bar{r})\right)\\ &\displaystyle\bar{F}=-{{\nabla}_{r}}\times\left(\bar{m}\times\bar{B}(\bar{r})% \right)=\left(\bar{m}\cdot{{\nabla}_{r}}\right)\bar{B}(\bar{r})=-{{\nabla}_{r}% }\left(-\bar{m}\cdot\bar{B}(\bar{r})\right)\\ \end{aligned}}}

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F¯=∫d3r´j¯(r¯´)×∇r[(r¯´)⋅B¯(r¯)]j¯(r¯´)×∇r[(r¯´)⋅B¯(r¯)]=−∇r×[((r¯´)⋅B¯(r¯))j¯(r¯´)]+[(r¯´)⋅B¯(r¯)]∇r×j¯(r¯´)∇r×j¯(r¯´)=0⇒F¯=−∫d3r´∇r×[((r¯´)⋅B¯(r¯))j¯(r¯´)]=−∇r×(m¯×B¯(r¯))F¯=−∇r×(m¯×B¯(r¯))=(m¯⋅∇r)B¯(r¯)=−∇r(−m¯⋅B¯(r¯))

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