Past exam of the optics and electrodynamics course of the University of Cambridge / 2018 / Section B / B3 / Retarded vector potential

For a localized current density, the retarded magnetic vector potential isAt 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 identitywhere localization makes the boundary term vanish. HenceIn 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.