Question

A non-conducting spherical ball of radius $R$ contains a spherically symmetric charge with volume charge density $\rho =k{r}^n$, where $r$ is the distance from the center of the ball and $n$ is a constant. The value of $n$ such that the electric field inside the ball is directly proportional to square of distance from the centre is

Answer 1

By Gauss law, we know that
$\oint \overrightarrow{E}.d\overrightarrow{A}=\frac{q_{in}}{{ϵ}_o}=\frac{\int \rho dV}{{ϵ}_o}$
$dV=4\pi r^{2}dr$ for the shell element. The element is shown in the figure at a distance $r$ from the centre $O$
$\Rightarrow E\times 4\pi r^2=\frac{k\int r^n\times 4\pi r^{2}dr}{{ϵ}_o}=\frac{4\pi k}{{ϵ}_o}\frac{r^{n+3}}{(n+3)}$
$\Rightarrow E=\frac{k}{(n+3){ϵ}_o}\left(r^{(n+1)}\right)$
For the electric field to be directly proportional to the square of distance from the centre,
$n+1=2\Rightarrow n=1$