Past exam of the optics and electrodynamics course of the University of Cambridge / 2018 / Section B / B3 / Vanishing field components

For a dipole along $z$, axial symmetry removes dependence on azimuth $\varphi$. In the Lorenz gauge, its magnetic vector potential points along $z$, so it has spherical components $A_r=A_z\cos \theta$, $A_{\theta }=-A_z\sin \theta$ and $A_{\varphi }=0$. The scalar potential is also independent of $\varphi$. Thus $E_{\varphi }=0$, while taking $\nabla \times \mathbf{A}$ leaves only an azimuthal magnetic field:The generally nonzero components are $E_r,E_{\theta },B_{\varphi }$. This statement includes the full dipole near field. In the leading $1/r$ radiation zone term, $E_r$ also vanishes; its nonzero full-field contributions fall faster than $1/r$. Confusing those two levels of approximation would incorrectly suggest four exact zeros everywhere.