Question

A charge Q is distributed over two concentric hollow spheres of radii r and R (> r) such that the surface densities are equal. Find the potential at the common centre.

Answer 1

Let $q_1$ and $q_2$ be the charges on them.
${\sigma }_1={\sigma }_2$
$\therefore \frac{q_1}{4\pi r^2}=\frac{q_2}{4\pi R^2}$
$\therefore \frac{q_1}{q_2}=\frac{r^2}{R^2}$
i.e.charge on them is distributed in above ratio
or $q_1=\frac{r^2}{r^2+R^2}Q$ and $q_2=\frac{R^2}{r^2+R^2}Q$
$\therefore$Potential at centre
V= Potential due to $q_1$+ Potential due to $q_2$
$\therefore V=\frac{1}{4\pi {ϵ}_0}\frac{q_1}{r}+\frac{1}{4\pi {ϵ}_0}.\frac{q_2}{R}=\frac{Q(R+r)}{4\pi {ϵ}_0(r^2+R^2)}$