A charge $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R(R>r)$ such that the…

A charge $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R(R>r)$ such that the surface densities are equal. The potential at the common centre is $\frac{1}{4 \pi \varepsilon_{0}}$ times-
  1. $Q\left[\frac{r+R}{r^{2}+R^{2}}\right]$
  2. $\frac{Q}{2}\left(\frac{r+R}{r^{2}+R^{2}}\right)$
  3. $2 Q\left(\frac{r+R}{r^{2}+R^{2}}\right)$
  4. zero

Solution

$q_{r}+q_{R}=Q$ ...(1)
$\frac{q_{r}}{q_{R}}=\frac{4 \pi r^{2} \sigma}{4 \pi R^{2} \sigma}=\frac{r^{2}}{R^{2}} \Rightarrow \frac{q_{r}}{q_{R}}=\frac{r^{2}}{R^{2}} \ldots(2)$
From (1) and (2) $q_{r}=\frac{Q r^{2}}{R^{2}+r^{2}}$ and $q_{R}=\frac{Q R^{2}}{R^{2}+r^{2}}$
So, $V=\frac{q_{r}}{4 \pi \varepsilon_{0} r}+\frac{q_{R}}{4 \pi \varepsilon_{0} R}$
$\Rightarrow V=\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{r+R}{r^{2}+R^{2}}\right)$

Asked in: JEE Mains - Electrostatics - Test 4

Practice more Electrostatics questions on Aicharya