Home exercise 3.3.7: Charge from the potential

For $V(r)=k{r}^3$ the electric field is

Gauss's law in differential form, ${\rho }_v=\nabla \cdot D={\epsilon }_0\nabla \cdot E$, with the divergence in spherical coordinates:

The same follows in one line from Poisson's equation: $\Delta V=\frac{1}{r^2}\frac{d}{dr}\left(r^2\cdot 3k{r}^2\right)=12kr=-{\rho }_v/{\epsilon }_0$.

The total charge in the sphere is

Units: $k$ is in $V/{\mathrm{m}}^3$, so ${\epsilon }_{0}kr$ is in $(F/m)(V/{\mathrm{m}}^3)\mathrm{m}=C/{\mathrm{m}}^3$, and ${\epsilon }_{0}k{b}^4$ in C. Gauss's law in integral form confirms $Q$ (remark below).