Capacitors A and B are identical. Capacitor A is charged so it stores \(4\text{J}\), of energy and capacitor B is uncharged. The capacitor are then connected in parallel. The total stored energy in the capacitors is now:
1. \(16\text{J}\)
2. \(8\text{J}\)
3. \(4\text{J}\)
4. \(2\text{J}\)
View Answer
Let \(C\) be the capacitance of each capacitor. For capacitor A, initial energy \(U_A = \frac{Q_A^2}{2C} = 4\text{J}\), so \(Q_A = \sqrt{8C}\). Capacitor B is uncharged, so \(Q_B = 0\). When connected in parallel, total charge is conserved: \(Q_{total} = Q_A + Q_B = \sqrt{8C}\). The equivalent capacitance is \(C_{eq} = C + C = 2C\). The final total energy is \(U_{total} = \frac{Q_{total}^2}{2C_{eq}} = \frac{(\sqrt{8C})^2}{2(2C)} = \frac{8C}{4C} = 2\text{J}\).
Assertion (A): When two capacitors of capacitance \(300 \text{ pF}\) and \(600 \text{ pF}\) which can work upto maximum potential of \(4 \text{ kV}\) and \(3 \text{ kV}\) respectively, are connected in series, their combination can work upto maximum potential of \(7 \text{ kV}\).
Reason (R): In series combination, maximum working potential will be sum of maximum working potential of individual capacitors.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For capacitors in series, the maximum charge \(Q_{\text{max}}\) the combination can hold is the minimum of individual \(C V_{\text{max}}\). Here, \(Q_{1, \text{max}}\ = 300 \text{ pF} \cdot 4 \text{ kV} = 1200 \text{ pC}\) and \(Q_{2, \text{max}}\ = 600 \text{ pF} \cdot 3 \text{ kV} = 1800 \text{ pC}\). So, \(Q_{\text{max}}\ = 1200 \text{ pC}\). Equivalent capacitance \(C_{\text{eq}}\ = (300 \cdot 600) / (300 + 600) = 200 \text{ pF}\). The maximum potential for the combination is \(V_{\text{max}}\ = Q_{\text{max}} / C_{\text{eq}}\ = 1200 \text{ pC} / 200 \text{ pF} = 6 \text{ kV}\). Thus, both Assertion (A) and Reason (R) are false.
Assertion (A): Two parallel plates having unequal charges have same capacitance as that of equal and opposite charges on same plates and same configuration.
Reason (R): Capacitance of system/ configuration is independent of charge on plates.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. The capacitance of a parallel plate capacitor, \(C = \frac{\epsilon_0 A}{d}\), is a geometric property and does not depend on the specific charge values on the plates, only their configuration.\nR: True. Capacitance is an intrinsic property dependent on geometry and dielectric, not on charge or potential.\n(R) correctly explains (A).
Assertion (A): Electrolytic capacitors have larger capacities.
Reason (R): Electrolytic capacitors have a positive and a negative terminal.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true; electrolytic capacitors offer high capacitance due to their very thin dielectric layer and large effective plate area. Reason (R) is also true, as all capacitors have two terminals. However, the presence of terminals doesn't explain why they have *larger* capacities, so (R) is not the correct explanation for (A).