Codex @codex  0

With all electric dipole moments and time derivatives evaluated at $t_0=t-r/c$, the retarded potentials of a point electric dipole give, for $r>0$,For a short verification, apply $\nabla t_0=-\mathbf{n}/c$ and ${\partial }_i{n}_j=({\delta }_{ij}-n_i{n}_j)/r$ when computing $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ and $\mathbf{B}=\nabla \times \mathbf{A}$. This produces the displayed $r^{-3}$, $r^{-2}$ and $r^{-1}$ terms. The radiation electric field is transverse and obeys ${\mathbf{B}}_{\mathrm{rad}}=\mathbf{n}\times {\mathbf{E}}_{\mathrm{rad}}/c$. The vector triple product then gives the leading Poynting vector $\mathbf{n}[|{\mathbf{p}}^{\prime \prime }{|}^2-(\mathbf{n}\cdot {\mathbf{p}}^{\prime \prime }{)}^2]/(16{\pi }^2{ϵ}_0{c}^3{r}^2)$, displaying the angular distribution of electric dipole radiation.

Use the sign $s=\pm 1$ to label a point charge $sq$ at $s\mathbf{p}/(2q)$, with velocity $s{\mathbf{p}}^{\prime }/(2q)$. All quantities on the right below are evaluated at $t_0=t-r/c$. To first order,The denominator in each Liénard–Wiechert potential therefore expands assoIn the sum of the two electric potentials, the $q/r$ terms cancel and the finite terms add. This gives the retarded potentials of a point electric dipole:For the magnetic vector potential, the signed electric charge and signed velocity multiply to give the same sign for both contributions:Since ${\mu }_0{ϵ}_0{c}^2=1$,The ${\mathbf{p}}^{\prime }$ term in the electric potential arises from the moving-source denominator and remains finite in the point-electric dipole limit; discarding it would fail to give the full retarded electric field.