The 4-vector potential $A^{\mu}(t, \mathbf{x})$ (in the Lorenz gauge $\partial_{\mu} A^{\mu}=0$ ) due to a localised source with conserved 4-vector current $J^{\mu}$ is

$$A^{\mu}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{J^{\mu}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime}$$

where $t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c$. For a source that varies slowly in time, show that the spatial components of $A^{\mu}$ at a distance $r=|\mathbf{x}|$ that is large compared to the spatial extent of the source are

$$\left.\mathbf{A}(t, \mathbf{x}) \approx \frac{\mu_{0}}{4 \pi r} \frac{d \mathbf{P}}{d t}\right|_{t-r / c}$$

where $\mathbf{P}$ is the electric dipole moment of the source, which you should define. Explain what is meant by the far-field region, and calculate the leading-order part of the magnetic field there.

A point charge $q$ moves non-relativistically in a circle of radius $a$ in the $(x, y)$ plane with angular frequency $\omega$ (such that $a \omega \ll c$ ). Show that the magnetic field in the far-field at the point $\mathbf{x}$ with spherical polar coordinates $r, \theta$ and $\phi$ has components along the $\theta$ and $\phi$ directions given by

$$\begin{aligned} &B_{\theta} \approx-\frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \sin [\omega(t-r / c)-\phi] \\ &B_{\phi} \approx \frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \cos [\omega(t-r / c)-\phi] \cos \theta \end{aligned}$$

Calculate the total power radiated by the charge.