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 centre of the ball and $n$ is a constant. What should be $n$ such that the electric field inside the ball is directly proportional to the square of distance from the centre?

Answer 1

By Gauss's law,

$\oint \overrightarrow{E}.d\overrightarrow{A}=\frac{q_{\text{enclosed}}}{{ϵ}_0}=\frac{\int \rho dV}{{ϵ}_0}\{\because \rho =\frac{\text{charge}}{\text{volume}}\}$

$\Rightarrow E\times 4\pi r^2=\frac{{\int }_0^{r}k{r}^n\times 4\pi r^{2}dr}{{ϵ}_0}$

$\Rightarrow E\times 4\pi r^2=\frac{4\pi k}{{ϵ}_0}\frac{r^{(n+3)}}{(n+3)}$

$\Rightarrow E=\frac{k}{{ϵ}_0}\frac{r^{(n+1)}}{(n+3)}$

Given that, $E\propto r^2$

On comparing,

$n+1=2$

$\Rightarrow n=1$