Question

A solid nonconducting sphere of radius $R$ has a uniform charge distribution of volume charge density, $\rho ={\rho }_0\frac{r}{R}$, where ${\rho }_0$ is a constant and r is the distance from the centre of the sphere. Show that:
(a) the total charge on the sphere is $Q=\pi {\rho }_0{R}^3$
(b) the electric field inside the sphere has a magnitude given by,
$E=\frac{KQ{r}^2}{R^4}$

Answer 1

a) : Total charge , $Q={\int }_0^R\rho dV={\int }_0^R\frac{{\rho }_{0}r}{R}4\pi r^{2}dr=\frac{4\pi {\rho }_0}{R}\left[R^4/4\right]=\pi {\rho }_0{R}^3$

b) By Gauss's law, the field inside sphere, $E.4\pi r^2=\frac{Q_{en}}{{ϵ}_0}$

Here, $Q_{en}={\int }_0^r\rho dV={\int }_0^r\frac{{\rho }_{0}r}{R}4\pi r^{2}dr=\frac{4\pi {\rho }_0}{R}\left[r^4/4\right]=\pi {\rho }_0{r}^4/R=Q{r}^4/R^4$

So, $E.4\pi r^2=\frac{Q{r}^4}{R^4{ϵ}_0}$

or $E=\frac{KQ{r}^2}{R^4}$ where $K=\frac{1}{4\pi {ϵ}_0}$