Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Charge distribution on shells is as shown in the figure below



Since, there is no charge present inside shell $A$. 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 shell $\text{B}$, net charge enclosed 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 $\text{C}$ will be $-q^{\prime }$ and on its outer surface will be, $2q-(-q^{\prime })=2q+q^{\prime }$ as total charge on $\text{C}$ is $2q$.

Given that Shell $\text{B}$ is earthed. Hence, its potential should be zero. i.e,

$V_B=0$

$\Rightarrow \frac{1}{4\pi {\epsilon }_0}[\frac{q}{2R}-\frac{q}{2R}+\frac{q^{\prime }}{2R}-\frac{q^{\prime }}{3R}+\frac{2q+q^{\prime }}{3R}]=0$

Solving this equation, we get

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

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

Therefore, charges on different surfaces in tabular form are given below:



Hence, option (d) is the correct answer.