Explain how time-dependent distributions of electric charge $\rho(\mathbf{x}, t)$ and current $\mathbf{j}(\mathbf{x}, t)$ can be combined into a four-vector $j^{a}(x)$ that obeys $\partial_{a} j^{a}=0$.

This current generates a four-vector potential $A^{a}(x)$. Explain how to find $A^{a}$ in the gauge $\partial_{a} A^{a}=0$.

A small circular loop of wire of radius $r$ is centred at the origin. The unit vector normal to the plane of the loop is $\mathbf{n}$. A current $I_{o} \sin \omega t$ flows in the loop. Find the three-vector potential $\mathbf{A}(\mathbf{x}, t)$ to leading order in $r /|\mathbf{x}|$.