= Retarded vector potential
For a localized <current density>, the retarded <magnetic vector potential> is
$$
\mathbf A(\mathbf r,t)=\frac{\mu_0}{4\pi}\int
\frac{\mathbf J(\mathbf r',t-|\mathbf r-\mathbf r'|/c)}{|\mathbf r-\mathbf r'|}\,d^3r'.
$$
At large $r$, replace the amplitude denominator by $r$. For an electrically short dipole, the delay difference across its length is negligible compared with an oscillation period, so evaluate all currents at $t_r=t-r/c$. The <continuity equation for electric charge> gives the identity
$$
\dot{\mathbf p}(t)=\frac{d}{dt}\int\mathbf r'\rho(\mathbf r',t)\,d^3r'
=-\int\mathbf r'\nabla'\cdot\mathbf J\,d^3r'
=\int\mathbf J\,d^3r',
$$
where localization makes the boundary term vanish. Hence
$$
\boxed{\mathbf A(\mathbf r,t)\simeq\frac{\mu_0}{4\pi r}\dot{\mathbf p}(t-r/c)
=-\frac{\mu_0\omega p_0}{4\pi r}\sin[\omega(t-r/c)]\widehat{\mathbf z}.}
$$
In this gauge its Cartesian direction is fixed and its magnitude depends only on $r$ and the retarded dipole motion, with $1/r$ spreading. Spherical components have the geometric factors $\cos\theta$ and $-\sin\theta$ above; the angular radiation pattern arises after differentiating the potentials, not from an isotropic radiated intensity.
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