= Vanishing field components
For a dipole along $z$, axial symmetry removes dependence on azimuth $\phi$. 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 $A_\phi=0$. The scalar potential is also independent of $\phi$. Thus $E_\phi=0$, while taking $\nabla\times\mathbf A$ leaves only an azimuthal <magnetic field>:
$$
\boxed{E_\phi=B_r=B_\theta=0.}
$$
The generally nonzero components are $E_r,E_\theta,B_\phi$. 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.
Back to article page
