Electrodynamics confusion - Hertzian dipole

Question

I am studying a course in Electrodynamics and we are just covering retarded potentials and the Hertzian dipole.

In my lecture notes, we have calculated the magnetic vector potential $A$ in the Lorenz gauge as $$A = \frac{\mu_0}{4 \pi r} [\ddot{p}]$$ where the square brackets indicated evaluated at the retarded time.

Now the confusion comes in once we start to compute the fields $B$ and $E$.

Calculation of $B$ is easy enough using $B = \nabla \times A$, but for $E$ we are using $$\frac{1}{c^2} \frac{\partial E}{\partial t} = \nabla \times B$$ but where has the $\mu_0 j$ gone from this equation (Ampere's law??)?? I don't know if it has anything to do with the fact we are neglected terms in $\frac{1}{r^2}$ and higher, but I am quite confused about where this equation has come from.

Any insight would be appreciated

Answer 1

You are calculating the $E$ field far away due to a radiating dipole. That is, your entire universe is a radiating dipole and you want to know what the $E$ field looks like some distance away in the absence of anything else, as if you added a current in the mix then that would be the $E$ field for a radiating dipole and due to a free current. Since our desired analysis is to study a dipole only, we simply impose that there are no free currents in our region of interest. That is, $\mathbf{J} =0$.