リエナール・ヴィーヘルト・ポテンシャル
A ( r , t ) = μ 0 4 π ∫∫ q v s ( t ′ ) δ 3 ( r ′ − r s ( t ′ ) ) | r − r ′ | δ ( t ′ − t r ′ ) d t ′ d 3 r ′ . {\displaystyle {\boldsymbol {A}}({\boldsymbol {r}},t)={\frac {\mu _{0}}{4\pi }}\iint {\frac {q{\boldsymbol {v}}_{\mathrm {s} }(t')\delta ^{3}({\boldsymbol {r'}}-{\boldsymbol {r}}_{\mathrm {s} }(t'))}{|{\boldsymbol {r}}-{\boldsymbol {r}}'|}}\delta (t'-t_{\mathrm {r} }')\,dt'\,d^{3}{\boldsymbol {r}}'\,.}
また、積分の順序を入れ替えれば以下の式を得る。
φ ( r , t ) = 1 4 π ε 0 ∫∫ δ ( t ′ − t r ′ ) | r − r ′ | q δ 3 ( r ′ − r s ( t ′ ) ) d 3 r ′ d t ′ , {\displaystyle \varphi ({\boldsymbol {r}},t)={\frac {1}{4\pi \epsilon _{0}}}\iint {\frac {\delta (t'-t_{\mathrm {r} }')}{|{\boldsymbol {r}}-{\boldsymbol {r}}'|}}q\delta ^{3}({\boldsymbol {r'}}-{\boldsymbol {r}}_{\mathrm {s} }(t'))\,d^{3}{\boldsymbol {r}}'dt'\,,}
φ ( r , t ) = 1 4 π ε 0 ( q ( 1 − n ⋅ β s ) | r − r s | ) t r , {\displaystyle \varphi ({\boldsymbol {r}},t)={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{(1-{\boldsymbol {n}}\cdot {\boldsymbol {\beta }}_{\mathrm {s} })|{\boldsymbol {r}}-{\boldsymbol {r}}_{\mathrm {s} }|}}\right)_{t_{\mathrm {r} }}\,,}
A ( r , t ) = μ 0 c 4 π ( q β s ( 1 − n ⋅ β s ) | r − r s | ) t r = β s ( t r ) c φ ( r , t ) . {\displaystyle {\boldsymbol {A}}({\boldsymbol {r}},t)={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{\mathrm {s} }}{(1-{\boldsymbol {n}}\cdot {\boldsymbol {\beta }}_{\mathrm {s} })|{\boldsymbol {r}}-{\boldsymbol {r}}_{\mathrm {s} }|}}\right)_{t_{\mathrm {r} }}={\frac {{\boldsymbol {\beta }}_{\mathrm {s} }(t_{\mathrm {r} })}{c}}\varphi ({\boldsymbol {r}},t)\,.}