Figure shows a system of three concentric metal shells $$A,B$$  and $$C$$ with radii $$a,2a$$  and $$3a$$ respectively. Shell $$B$$ is earthed and shell $$C$$ is given a charge $$Q.$$ Now if shell $$C$$ is connected to shell $$A,$$ then the final charge on the shell $$B,$$ is equal to

Solution :
From given conditions, $${V_A} = {V_C}$$   and $${V_B} = 0$$
$$\eqalign{ & \Rightarrow {V_B} = \frac{{k\left( {Q - {q_1}} \right)}}{{3a}} + \frac{{k{q_2}}}{{2a}} + \frac{{k{q_1}}}{{2a}} = 0 \cr & \Rightarrow 2Q + {q_1} + 3{q_2} = 0\,......\left( {\text{i}} \right) \cr} $$

$$\eqalign{ & {\text{Using}}\,\,{V_A} = {V_C} \cr & \frac{{k\left( {Q - {q_1}} \right)}}{{3a}} + \frac{{k{q_2}}}{{3a}} + \frac{{k{q_1}}}{{3a}} \cr & = \frac{{k{q_1}}}{a} + \frac{{k\left( {Q - {q_1}} \right)}}{{3a}} + \frac{{k{q_2}}}{{2a}} \cr & \Rightarrow {q_1} = - \frac{{{q_2}}}{4}\,......\left( {{\text{ii}}} \right) \cr & {\text{Using it in }}\left( 1 \right),{\text{ }}{q_2} = - \frac{8}{{11}}Q \cr} $$