Current density in the presence of a stationary point charge of varying magnitude

Question

I have been trying to solve the following problem out of Modern Electrodynamics by Andrew Zangwill (Problem 20.8):

A charge density $\rho(\vec{r},t) = q(t)\delta(r) $ where $ q(t) = 0$ for $t < 0$ and $q(t) = q_0$ for $t > \tau$. Calculate E and B using symmetry and elementary methods

I tried to determine the current density of this setup using the continuity equation: $\frac{\partial \rho}{\partial t} = -\nabla \cdot \vec J$, so $-\dot{q}(t)\delta(r) = \nabla \cdot \vec J$. This is satisfied by a function J of the form $\frac{-\dot{q}(t)\hat{r}}{4\pi r^2}$ plus a term which has zero divergence. But this answer seems completely nonsensical - how could it be that there is current density everywhere in space for that time? Am I possibly missing an extra delta term that would localise the density? If not, how do I make sense of this answer?

Answer 1

Your result for the current density actually looks right to me. Your charge suddenly appears at $r=0$, but it doesn't come... from anywhere. Or in other words it has to come from infinitely far away and instantly appear at $r=0$. There will need to be current density throughout all of space to facilitate this. If I set up a current density that will transport charge from a uniformly charged spherical surface of radius $R$ to $r=0$ in a time $\delta_T$, it will be

$$ \mathbf{J}=\begin{cases} -\frac{q}{4\pi r^2\delta_t}\hat{r}\text{ if }r<R\\ 0\text{ if }r>R \end{cases} $$

through each spherical surface of radius $r$, charge is flowing at a rate $q/\delta_t$. The charge is leaving the surface at radius $R$ and appearing at the center of the coordinate system. So we set $R\rightarrow\infty$, and $\delta_t\rightarrow 0$ (replacing $1/\delta_t$ with $\delta(t)$). Then we get your solution for the charge density.