$$\gdef\ele{\epsilon_{0}}
\gdef\mag{\mu_{0}}
\gdef\lig{c}
\gdef\tret{\tau_{\mathrm{r}}}
\gdef\F#1{\bm{F}\left(#1\right)}
\gdef\pPhi#1#2{\varphi\!\left(#1,#2\right)}
\gdef\pA#1#2{\bm{A}\!\left(#1,#2\right)}
\gdef\E#1#2{\bm{E}\left(#1,#2\right)}
\gdef\B#1#2{\bm{B}\left(#1,#2\right)}
\gdef\density#1#2{\rho\!\left(#1,#2\right)}
\gdef\current#1#2{\bm{j}\!\left(#1,#2\right)}
\gdef\rsource#1{\bm{r}_{\mathrm{s}}(#1)}
\gdef\dotrsource#1{\dot{\bm{r}}_{\mathrm{s}}(#1)}
\gdef\ddotrsource#1{\ddot{\bm{r}}_{\mathrm{s}}(#1)}
\gdef\dddotrsource#1{\dot{\ddot{\bm{r}}}_{\mathrm{s}}(#1)}
\gdef\n#1{\bm{n}(#1)}$$
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
$$\begin{aligned}
\lig
\density{\tau}{\bm{p}}
&=
q
\lig
\delta(
\bm{p}
-
\rsource{\tau}
)
,
\\
\current{\tau}{\bm{p}}
&=
q
\lig
\delta(
\bm{p}
-
\rsource{\tau}
)
\frac{\dotrsource{\tau}}{\lig}
.
\end{aligned}$$
力
$$\F{t}
=
q
\left[
\E{t}{\rsource{t}}
+
\dotrsource{t}
\times
\B{t}{\rsource{t}}
\right]
.$$
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
$$\begin{aligned}
\lig
\density{\tau}{\bm{p}}
&=
0
,
\\
\current{\tau}{\bm{p}}
&=
-
\bm{m}
\times
\nabla
\delta(
\bm{p}
-
\rsource{\tau}
)
.
\end{aligned}$$
力
$$\F{t}
=
\nabla
\left[
\bm{m}
\cdot
\B{t}{\rsource{t}}
\right]
.$$
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}$$
@norkron
