= Retarded potentials of a point electric dipole
{title2=$t_0=t-r/c$}
An ideal <electric dipole> at the origin with smooth <electric dipole moment> $\mathbf p(t)$ has <Lorenz gauge> <electric potential> and <magnetic vector potential>, for $r>0$,
$$
\phi=\frac1{4\pi\epsilon_0}\left(\frac{\mathbf n\cdot\mathbf p(t_0)}{r^2}+\frac{\mathbf n\cdot\mathbf p'(t_0)}{cr}\right),\qquad
\mathbf A=\frac{\mu_0}{4\pi r}\mathbf p'(t_0),\qquad \mathbf n=\frac{\mathbf r}r.
$$
For a short proof, place <electric charges> $sq$ at $s\mathbf p/(2q)$, $s=\pm1$, and let $q\to\infty$ with $\mathbf p$ fixed. The corresponding <retarded times> are $t_0+s\mathbf n\cdot\mathbf p(t_0)/(2qc)+O(q^{-2})$. Each <Liénard–Wiechert potential> has denominator $r-s(\mathbf n\cdot\mathbf p+r\mathbf n\cdot\mathbf p'/c)/(2q)+O(q^{-2})$. The opposite $q/r$ terms cancel, leaving the displayed <electric potential>. The products of signed <electric charge> and <velocity> add, giving the displayed <magnetic vector potential>. Bounded time <derivatives> on the relevant interval justify the <Taylor expansions>.
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