(a) For sphere with radius a, Potential $= V$ (since, both the charged spheres are connected by a wire) $\text{ Charge } = Q_a$ $\text{ Capacitance } = C_a$ $\text{ Electric field, } = E_a = \frac{Q_a}{4 \pi \varepsilon_0 a^2}$ For sphere with radius $b$, Potential $= V$ (since both the charged spheres are connected by a wire) $\text{ Charge } = Q_b$ $\text{ Capacitance } = C_b$ $\text{ Electric field } = E_b$ $= \frac{Q_b}{4 \pi \varepsilon_0 b^2}$ $\text{ Now, } \frac{E_a}{E_b} = \frac{Q_a}{Q_b} \times \frac{b^2}{a^2}$ ......(1) $\frac{Q_a}{Q_b} = \frac{C_a V}{C_b V}$ $\text{ Or, } \frac{Q_a}{Q_b} = \frac{C_a}{C_b}$ $\text{ Or, } \frac{Q_a}{Q_b} = \frac{a}{b}$ .......(2) (since, $\frac{C_d}{C_b} = \frac{a}{b}$ ) Putting in equation (1) $\frac{E_a}{E_b} = \left( \frac{a}{b} \right) \times \left( \frac{b^2}{a^2} \right)$ $\therefore \frac{E_a}{E_b} = \frac{b}{a}$ OR (b) (i) Initially, Charge on A capacitor $= CV$ Charge on $B$ capacitor $= 0$ After connecting A with B, $\text{ Final potential } = V' = \frac{\text{ Total Charge }}{\text{ total capacitance }}$ $= \frac{Q+0}{C+2C} = \frac{Q}{3C}$ Final charge on A capacitor after redistribution $= C V' = \frac{C Q}{3 C} = \frac{Q}{3}$ Final charge on B capacitor after redistribution $= 2 C V' = \frac{2 C Q}{3 C} = \frac{2 Q}{3}$So, the ratio of charges $= \frac{Q/3}{2Q/3} = 1:2$(ii) Initially energy stored in $A = \frac{1}{2} (CV)^2$Finally energy stored in $A = \frac{1}{2} C V^2$$= \frac{1}{2} \times C \times \left( \frac{Q}{3C} \right)^2$$= \frac{1}{2} \times C \times \left( \frac{V}{3} \right)^2$ $= \frac{C V^2}{18}$Finally energy stored in $B = \frac{1}{2} 2 (CV)^2$$= \frac{1}{2} \times 2 C \times \left( \frac{Q}{3C} \right)^2$$= \frac{1}{2} \times 2 C \times \left( \frac{V}{3} \right)^2$$= \frac{C V^2}{9}$Total energy stored in A and B$= \frac{C V^2}{18} + \frac{C V^2}{9}$$= \frac{C V^2}{6}$So, the required ratio $= \frac{\text{ Find energy stored in } A \text{ and } B}{\text{ Initial energy stored in } A}$ $= \frac{ \frac{C V^2}{6} }{ \frac{C V^2}{2} }$$= \frac{1}{3}$