GATE EC 2015 Set 2 — Question 42

The energy dissipated in the resistor is the difference between the initial total energy stored in the capacitors and the final total energy stored in the capacitors.Given:
$C_1 = 3 \text{ F}$, $V_{C1,initial} = 1 \text{ V}$
$C_2 = 1 \text{ F}$, $V_{C2,initial} = 3 \text{ V}$Initial energy in $C_1$: $E_{C1,initial} = \frac{1}{2}C_1V_{C1,initial}^2 = \frac{1}{2}(3 \text{ F})(1 \text{ V})^2 = 1.5 \text{ J}$
Initial energy in $C_2$: $E_{C2,initial} = \frac{1}{2}C_2V_{C2,initial}^2 = \frac{1}{2}(1 \text{ F})(3 \text{ V})^2 = 4.5 \text{ J}$
Total initial energy: $E_{initial} = E_{C1,initial} + E_{C2,initial} = 1.5 \text{ J} + 4.5 \text{ J} = 6 \text{ J}$When the switch closes, the capacitors are connected in parallel with the resistor. At steady state, the current through the resistor will be zero, and the capacitors will share a common voltage. Since there are no independent sources, the total charge in the isolated capacitor-resistor network is conserved.Initial charge on $C_1$: $Q_{C1,initial} = C_1V_{C1,initial} = (3 \text{ F})(1 \text{ V}) = 3 \text{ C}$
Initial charge on $C_2$: $Q_{C2,initial} = C_2V_{C2,initial} = (1 \text{ F})(3 \text{ V}) = 3 \text{ C}$Assuming the polarities are as shown (positive terminals connected together, negative terminals connected together), the total initial charge is:
$Q_{total,initial} = Q_{C1,initial} + Q_{C2,initial} = 3 \text{ C} + 3 \text{ C} = 6 \text{ C}$The total capacitance when connected in parallel is $C_{total} = C_1 + C_2 = 3 \text{ F} + 1 \text{ F} = 4 \text{ F}$.The final voltage across the parallel combination is $V_f = \frac{Q_{total,initial}}{C_{total}} = \frac{6 \text{ C}}{4 \text{ F}} = 1.5 \text{ V}$.$E_{final} = \frac{1}{2}C_{total}V_f^2 = \frac{1}{2}(4 \text{ F})(1.5 \text{ V})^2 = \frac{1}{2}(4)(2.25) = 4.5 \text{ J}$The energy dissipated in the resistor is the difference between the initial and final stored energy:
$E_{dissipated} = E_{initial} - E_{final} = 6 \text{ J} - 4.5 \text{ J} = 1.5 \text{ J}$The final answer is $\boxed{1.5}$