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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

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  • $\begingroup$ There should be only single dot above $p$ in that expression for $A$. $\endgroup$ – Ján Lalinský Jan 5 at 3:07
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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, $\mathbf{J} =0$.

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  • $\begingroup$ Don’t we have a current through the dipole though? Or is the j in Maxwell only referring to ‘free currents’? What even constitutes a ‘free current’ $\endgroup$ – PhysicsMathsLove Jan 4 at 23:10
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    $\begingroup$ Yes, that's fine. But your $I_{enc}$ in the region you're looking for the $E$-field is zero. You can see it that way, that's fine but then you could say that "well we shouldn't get rid of it yet then. That's fine. However, another way of thinking about it is what you are writing down is maxwells equations **in a given region ** (i.e. a domain). And since you are solving maxwells equations in the region where $\mathbf{J} =0$ we may just go ahead and set it to zero. $\endgroup$ – InertialObserver Jan 4 at 23:12

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