2021/04/22 10:44 更新
点電荷や点電流の運動により生じる電磁場
76 いいね ブックマーク
目次
$$\gdef\ele{\epsilon_{0}} \gdef\mag{\mu_{0}} \gdef\lig{c} \gdef\tret{\tau_{\mathrm{r}}} \gdef\F#1{\bm{F}\left(#1\right)} \gdef\pPhi#1#2{\varphi\!\left(#1,#2\right)} \gdef\pA#1#2{\bm{A}\!\left(#1,#2\right)} \gdef\E#1#2{\bm{E}\left(#1,#2\right)} \gdef\B#1#2{\bm{B}\left(#1,#2\right)} \gdef\density#1#2{\rho\!\left(#1,#2\right)} \gdef\current#1#2{\bm{j}\!\left(#1,#2\right)} \gdef\rsource#1{\bm{r}_{\mathrm{s}}(#1)} \gdef\dotrsource#1{\dot{\bm{r}}_{\mathrm{s}}(#1)} \gdef\ddotrsource#1{\ddot{\bm{r}}_{\mathrm{s}}(#1)} \gdef\dddotrsource#1{\dot{\ddot{\bm{r}}}_{\mathrm{s}}(#1)} \gdef\n#1{\bm{n}(#1)}$$

ローレンツ力

$$\F{t} = \int_{V} d\bm{r} \left[ \density{t}{\bm{r}} \E{t}{\bm{r}} + \current{t}{\bm{r}} \times \B{t}{\bm{r}} \right] .$$

遅延ポテンシャル

$$\begin{aligned} \frac{\pPhi{t}{\bm{r}}}{\lig} & = \frac{\mag}{4\pi} \int_{-\infty}^{\infty} d\lig\tau \int_{V} d\bm{p} \frac{\lig\density{\tau}{\bm{p}}}{|\bm{r}-\bm{p}|} \delta\left(\lig\tau-\lig t+|\bm{r}-\bm{p}|\right) , \\ \pA{t}{\bm{r}} & = \frac{\mag}{4\pi} \int_{-\infty}^{\infty} d\lig\tau \int_{V} d\bm{p} \frac{\current{\tau}{\bm{p}}}{|\bm{r}-\bm{p}|} \delta\left(\lig\tau-\lig t+|\bm{r}-\bm{p}|\right) , \end{aligned} \qquad \lig^{2} = \frac{1}{\ele\mag} .$$

点電荷

$$\begin{aligned} \lig \density{\tau}{\bm{p}} &= q \lig \delta( \bm{p} - \rsource{\tau} ) , \\ \current{\tau}{\bm{p}} &= q \lig \delta( \bm{p} - \rsource{\tau} ) \frac{\dotrsource{\tau}}{\lig} . \end{aligned}$$

$$\F{t} = q \left[ \E{t}{\rsource{t}} + \dotrsource{t} \times \B{t}{\rsource{t}} \right] .$$

ポテンシャル

$$\begin{aligned} 4\pi\ele \frac{\pPhi{t}{\bm{r}}}{\lig} & = \frac{q}{\lig} \frac{1}{|\bm{r}-\rsource{\tret}|} \frac{1}{\left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|} , \\ \pA{t}{\bm{r}} & = \frac{\pPhi{t}{\bm{r}}}{\lig} \frac{\dotrsource{\tret}}{\lig} . \end{aligned}$$
$$\lig\tret - \lig t + |\bm{r}-\rsource{\tret}| = 0 , \qquad \n{\tau} \equiv \frac{\bm{r}-\rsource{\tau}}{|\bm{r}-\rsource{\tau}|} . $$

Wikipedia「リエナール・ヴィーヘルト・ポテンシャル」(2021-01-23)

電磁場

$$\begin{aligned} 4\pi\ele \E{t}{\bm{r}} = &\hphantom{\,+\;} q \frac{ \left( 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right) \left[ \n{\tret} - \frac{\dotrsource{\tret}}{\lig} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{3}} \\ & + q \frac{ \n{\tret} \times \left\{ \left[ \n{\tret} - \frac{\dotrsource{\tret}}{\lig} \right] \times \frac{\ddotrsource{\tret}}{\lig^{2}} \right\} }{ |\bm{r}-\rsource{\tret}| \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{3}} , \\ 4\pi\ele \B{t}{\bm{r}} = &\; \frac{\n{\tret}}{\lig} \times 4\pi\ele \E{t}{\bm{r}} . \end{aligned}$$

点電流

$$\begin{aligned} \lig \density{\tau}{\bm{p}} &= 0 , \\ \current{\tau}{\bm{p}} &= - \bm{m} \times \nabla \delta( \bm{p} - \rsource{\tau} ) . \end{aligned}$$

$$\F{t} = \nabla \left[ \bm{m} \cdot \B{t}{\rsource{t}} \right] .$$

ポテンシャル

$$\begin{aligned} \frac{\pPhi{t}{\bm{r}}}{\lig} = &\; 0, \\ \frac{4\pi}{\mag} \pA{t}{\bm{r}} = &\hphantom{\,+\;} \frac{ \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{3}} \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \\ & - \frac{ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{3}} \left[ \n{\tret} \times \bm{m} \right] \\ & - \frac{ \left[ \n{\tret} \cdot \frac{ \ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}| \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{3}} \left[ \n{\tret} \times \bm{m} \right] . \end{aligned}$$
$$\lig\tret - \lig t + |\bm{r}-\rsource{\tret}| = 0 , \qquad \n{\tau} \equiv \frac{\bm{r}-\rsource{\tau}}{|\bm{r}-\rsource{\tau}|} . $$

arXiv:1806.07089「The vector potential of a point magnetic dipole」(2018-06-20)

電磁場

$$\begin{aligned} \frac{1}{\lig} \frac{4\pi}{\mag} \E{t}{\bm{r}} = &\hphantom{\,+\;} \frac{ 2 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \\ & - \frac{ 3 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right] \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \left\{ \left[ \n{\tret} \times \bm{m} \right] + \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \right\} \\ &+ \frac{ 3 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \left[ \n{\tret} \times \bm{m} \right] \\ & - \frac{ 3 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \left\{ \left[ \n{\tret} \times \bm{m} \right] + \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \right\} \\ &- \frac{ \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \left[ \frac{\ddotrsource{\tret}}{\lig^{2}} \times \bm{m} \right] \\ &+ \frac{ 6 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right] \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] - 3 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \frac{\dotrsource{\tret}}{\lig} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \left[ \n{\tret} \times \bm{m} \right] \\ &+ \frac{ 3 \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right]^{2} + \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \n{\tret} \cdot \frac{\dddotrsource{\tret}}{\lig^{3}} \right] }{ |\bm{r}-\rsource{\tret}| \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \left[ \n{\tret} \times \bm{m} \right] , \\ \frac{4\pi}{\mag} \B{t}{\bm{r}} = &\hphantom{\,+\;} \frac{ 2 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \frac{\dotrsource{\tret}}{\lig} \times \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \\ &- \frac{ 3 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right] \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \left\{ \frac{\dotrsource{\tret}}{\lig} \times \left[ \n{\tret} \times \bm{m} \right] + \n{\tret} \times \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \right\} \\ &+ \frac{ 3 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \n{\tret} \times \left[ \n{\tret} \times \bm{m} \right] \\ &+ \frac{ 2 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right] \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{3} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \bm{m} \\ &- \frac{ 3 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \left\{ \frac{\dotrsource{\tret}}{\lig} \times \left[ \n{\tret} \times \bm{m} \right] + \n{\tret} \times \left[ \frac{\dotrsource{\tret}}{\lig} \times \bm{m} \right] \right\} \\ & - \frac{ \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \left\{ \frac{\ddotrsource{\tret}}{\lig^{2}} \times \left[ \n{\tret} \times \bm{m} \right] + \n{\tret} \times \left[ \frac{\ddotrsource{\tret}}{\lig^{2}} \times \bm{m} \right] \right\} \\ &+ \frac{ 6 \left[ 1 - \left| \frac{\dotrsource{\tret}}{\lig} \right|^{2} \right] \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] - 3 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \frac{\dotrsource{\tret}}{\lig} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5}} \n{\tret} \times \left[ \n{\tret} \times \bm{m} \right] \\ &+ \frac{ 2 \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right]^{2} \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right] }{ |\bm{r}-\rsource{\tret}|^{2} \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \bm{m} \\ &+ \frac{ 3 \left[ \n{\tret} \cdot \frac{\ddotrsource{\tret}}{\lig^{2}} \right]^{2} + \left[ 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right] \left[ \n{\tret} \cdot \frac{\dddotrsource{\tret}}{\lig^{3}} \right] }{ |\bm{r}-\rsource{\tret}| \left| 1 - \n{\tret} \cdot \frac{\dotrsource{\tret}}{\lig} \right|^{5} } \n{\tret} \times \left[ \n{\tret} \times \bm{m} \right] . \end{aligned}$$