AIPMT 2010 Physics and Chemistry Paper

A series combination of $n_1$ capacitors each of capacitance $C_1$ are connected to $4V$ source as shown in the figure.Total capacitance of the series combination of the capacitors is $\; \frac{1}{C_5} = \; \frac{1}{C_1} + \; \frac{1}{C_1} + \; \frac{1}{C_1} + \cdots$ upto $n_1$ terms $= \; \frac{n_1}{C_1}$ or $C_s = \; \frac{C_1}{n_1}$ Total energy stored in a series combination of the capacitors is $u_s = \; \frac{1}{2} C_s (4 V)^2$ $= \; \frac{1}{2} \left( \; \frac{C_1}{n_1} \right) (4 V)^2$   (using (i))...(ii) A parallel combination of $n_2$ capacitors each of capacitance $C_2$ are connected to $V$ source as shown in the figure.Total capacitance of the parallel combination of capacitors is $C_p = C_2 + C_2 + \cdots +$ upto $n_2$ terms $= n_2 C_2$ or $C_{c} = n_2 C_2$ .......(iii) Total energy stored in a parallel combination of capacitors is $u_p = \; \frac{1}{2} C_p V^2$ $= \; \frac{1}{2} (n_2 C_2) (V)^2$   (Using (iii))...(iv) According to the given problem, $U_s = U_p$ Substituting the values of $u_s$ and $u_p$ from equations (ii) and (iv), we get $\; \frac{1}{2} \; \frac{C_1}{n_1} (4 V)^2 = \; \frac{1}{2} (n_2 C_2) (V)^2$ or $\; \frac{C_1 16}{n_1} = n_2 C_2 \;$ or $C_2 = \; \frac{16 C_1}{n_1 n_2}$