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 $q_1$ & $q$ be the charge on $1st$ & $2nd$ sphere

$\left(\sigma \right)$ surface charge density $=\frac{chargeonsphere}{surfacearea}$

${\sigma }_1=\frac{q_1}{4\pi r^2}$

${\sigma }_2=\frac{q_2}{4\pi R^2}$

${\sigma }_1={\sigma }_2=\sigma$

$\frac{q_1}{4\pi r^2}=\frac{q_2}{4\pi R^2}=\sigma$

$\frac{q_1}{r^2}=\frac{q_2}{R^2}$-----(2)

$q_1+q_2=Q$------(1) & (2)

we get

$q_1=\frac{Q{r}^2}{R^2+r^2}{q}_2=\frac{Q{R}^2}{R^2+r^2}$-----(3)

Potential at the centre will be due to both the spheres

$V=V_1+V_2$

$v=\frac{l{q}_1}{r}+\frac{k{q}_2}{R}$ put value from (3)

$v=\frac{kQ}{r^2+R^2}(r+R)$