Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / J (Electromagnetism)

The integral Gauss's law isThe divergence theorem makes the left side ${\int }_V\nabla \cdot \mathbf{E}dV$. If the differential relation $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ holds, this proves the integral law. Conversely, if the integral identity holds for every suitable volume and the fields are sufficiently regular, a nonzero value of $\nabla \cdot \mathbf{E}-\rho /{ϵ}_0$ would persist in sign on a small neighborhood and contradict its zero integral. The differential law follows.

For a sphere of radius $r$, spherical symmetry makes the flux $4\pi r^{2}E(r)$ and the enclosed charge $4\pi {\int }_0^r{s}^2\rho (s)ds$. HenceThere is no extra point charge at the origin in the stipulated density distribution. Outside the charge support,