= Fields of a time-dependent electric dipole
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 $r>0$,
$$
\mathbf E=\frac1{4\pi\epsilon_0}\left[
\frac{3\mathbf n(\mathbf n\cdot\mathbf p)-\mathbf p}{r^3}
+\frac{3\mathbf n(\mathbf n\cdot\mathbf p')-\mathbf p'}{cr^2}
+\frac{\mathbf n(\mathbf n\cdot\mathbf p'')-\mathbf p''}{c^2r}\right],\qquad
\mathbf B=\frac{\mu_0}{4\pi}\left(\frac{\mathbf p'}{r^2}+\frac{\mathbf p''}{cr}\right)\times\mathbf n.
$$
For a short verification, apply $\nabla t_0=-\mathbf n/c$ and $\partial_i n_j=(\delta_{ij}-n_in_j)/r$ when computing $\mathbf E=-\nabla\phi-\partial_t\mathbf A$ 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 $\mathbf B_{\rm rad}=\mathbf n\times\mathbf E_{\rm rad}/c$. The <vector triple product> then gives the leading <Poynting vector> $\mathbf n[|\mathbf p''|^2-(\mathbf n\cdot\mathbf p'')^2]/(16\pi^2\epsilon_0c^3r^2)$, displaying the <angular distribution of electric dipole radiation>.
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