Three concentric, conducting spherical shells $A, B$ and $C$ have radii $a = 10 cm, b = 20 cm$ and $c = 30 cm$ respectively. The innermost shell A is earthed and charges $q_2 = 4\mu C$ and $q_3 = 3\mu C$ are given to shell $B$ and $C$ respectively. Calculate charges $q_1$ induced on shell $A$ and energy stored in the system.

System of three concentric shells is as shown in Fig.(a).

Since, shell $A$ is earthed, therefore its potential is zero. But its potential is

$$\begin{gathered} V=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\left[\frac{q_1}{a}+\frac{q_2}{b}+\frac{q_3}{c}\right] \\ \therefore \left(\frac{q_1}{a}+\frac{q_2}{b}+\frac{q_3}{c}\right)=0 \\ or q1=-3\mu C \mathit{A}\mathit{n}\mathit{s}\mathbf{.} \end{gathered}$$

Now, charges on different surfaces will be as shown in Fig.(b).

To calculate energy stored in the system, it can be as considered in the parts:

(i)  a spherical capacitor having radii $a$ and $b$ and having charges

                         $\left|q_1\right| = 3\mu C.$

     Its capacitance, $C_1=\frac{4\pi {\epsilon }_{0}ab}{(b-a)}$

(ii)  a spherical capacitor having radii b and c and having charge $(q_2 + q_1) = 1\mu C.$

      Its capacitance, $C_2=\frac{4\pi {\epsilon }_{0}bc}{(c-b)}$

(iii)  an isolated sphere of radius $c$ and having charge $(q_3 + q_2 + q_1) = 4\mu C.$

      Its capacitance, $C_3=4{\mathrm{\pi \epsilon }}_0\mathrm{c}$

$$\begin{gathered} \therefore Energy stored in the system \\ =\frac{{q_1}^2}{2C_1}+\frac{{(q_1+q_2)}^2}{2C_2}+\frac{{(q_1+q_2+q_3)}^2}{2C_3} \\ =0.45 joule \mathbf{ }\mathit{A}\mathit{n}\mathit{s}. \end{gathered}$$