Question

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

• A. $\frac{\sqrt{2}}{\pi {\epsilon }_0}\frac{Q(R+r)}{(R^2+r^2)}$
• B. $\frac{1}{2\pi {\epsilon }_0}\frac{Q(R+r)}{(R^2+r^2)}$
• C. $\frac{1}{4\pi {\epsilon }_0}\frac{Q(R+r)}{(R^2+r^2)}$
• D. $\frac{1}{\pi {\epsilon }_0}\frac{Q(R-r)}{(R^2+r^2)}$

Answer 1

The correct option is C $\frac{1}{4\pi {\epsilon }_0}\frac{Q(R+r)}{(R^2+r^2)}$

Let charge on surface of sphere of radius R be $q_A$ and charge on surface of sphere of radius r be $q_B$
Since there surface charge densities is equal
$\frac{q_A}{R^2}=\frac{q_B}{r^2}$
and
$q_A+q_B=Q$
on solving both the equations we get
$q_A=\frac{Q{R}^2}{R^2+r^2}$

$q_B=\frac{Q{r}^2}{R^2+r^2}$
potential at the common center will be
$V=\frac{k{q}_A}{R}+\frac{k{q}_B}{r}$

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

$V=\frac{Q(R+r)}{4\pi {ϵ}_o(R^2+r^2)}$