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

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?

• hint: because of the spherical symmetry the electric field must be radial, what should the symmetry of the B field be and how can that be satisfied? May 16 at 13:19
• My instinct says that the B field should be zero given the symmetry of the problem, since if it pointed in any one direction you could rotate the system arbitrarily and it would point in a different direction. Is that a reasonable argument? May 16 at 13:48
• hint: yes it is reasonable! May 16 at 13:57

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
$$\mathbf{J}=\begin{cases} -\frac{q}{4\pi r^2\delta_t}\hat{r}\text{ if }rR \end{cases}$$
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.