Question

Figure shows three concentric thin spherical shells $A,B$ and $C$ of radii $R,2R$ and $3R$. The shell $B$ is earthed and $A$ and $C$ are given charges $q$ and $2q$ respectively. Find the charges appearing on the outer surface of $C$.

• A. $-\frac{4}{3}q$
• B. $\frac{4}{3}q$
• C. $\frac{2}{3}q$
• D. $q$

Answer 1

The correct option is C $\frac{2}{3}q$


Since, there is no charge inside $A$. The whole charge $q$ given to the shell $A$ will appear on its outer surface. Charge on its inner surface will be zero. Moreover if a gaussian surface is drawn on the material of shell $B$, net charge enlcosed by it should be zero. Therefore, charge on its inner surface will be $-q$. Now , let $q^{\prime }$ be the charge on its outer surface, then charge on the inner surface of $C$ will be $-q^{\prime }$ and on its outer surface will be , $2q-(-q^{\prime })=2q+q^{\prime }$ as total charge on $C$ is $2q$.

Shell $B$ is earthed . Hence, its potential should be zero. $$V_B=0$$$\therefore \frac{1}{4\pi {\epsilon }_0}[\frac{q}{2R}-\frac{q}{2R}+\frac{q^{\prime }}{2R}-\frac{q^{\prime }}{3R}+\frac{2q+q^{\prime }}{3R}]=0$

Solving the equation we get, $q^{\prime }=-\frac{4}{3}q$

$\Rightarrow 2q+q^{\prime }=2q-\frac{4}{3}q=\frac{2}{3}q$

Therefore, charges on different surfaces in tabular form are given below:
$\text{A}$ $\text{B}$ $\text{C}$

Inner surface	$0$	$-q$	$\frac{4}{3}q$
Outer surface	$q$	$-\frac{4}{3}q$	$\frac{2}{3}q$

Hence, option (c) is the correct answer.