Potential Difference Across Capacitor in Steady State Circuit

To find the potential difference across the capacitor in the steady state, we analyze the circuit as follows:

1. Steady-State Behavior of the Capacitor

In a DC circuit under steady-state conditions, a capacitor acts as an open circuit. Therefore, no current flows through the branch containing the capacitor: $I_C = 0$

2. Potential at the Nodes

Let the common junction on the right side, where all three horizontal branches meet, be chosen as the reference node with potential: $V_{\text{right}} = 0\text{ V}$

Since no current flows through the top resistor ($R_{\text{top}} = 2\ \Omega$), there is no potential drop across it: $V_{\text{top-left}} - V_{\text{right}} = I_{\text{top}} \times R_{\text{top}} = 0 \times 2\ \Omega = 0\text{ V}$ $V_{\text{top-left}} = V_{\text{right}} = 0\text{ V}$

3. Current in the Active Loop

The remaining part of the circuit forms a single closed loop consisting of:

• The battery of emf $E = 2\text{ V}$
• The middle resistor $R_{\text{middle}} = 2\ \Omega$
• The bottom resistor $R_{\text{bottom}} = 6\ \Omega$

The total resistance of this loop is: $R_{\text{total}} = R_{\text{middle}} + R_{\text{bottom}} = 2\ \Omega + 6\ \Omega = 8\ \Omega$

The steady-state current $I$ flowing through this loop is given by Ohm's law: $I = \frac{E}{R_{\text{total}}} = \frac{2\text{ V}}{8\ \Omega} = 0.25\text{ A}$

4. Potential at the Middle-Left Node

The potential difference across the middle resistor is: $V_{\text{middle-left}} - V_{\text{right}} = I \times R_{\text{middle}} = 0.25\text{ A} \times 2\ \Omega = 0.5\text{ V}$

Since $V_{\text{right}} = 0\text{ V}$, we have: $V_{\text{middle-left}} = 0.5\text{ V}$

5. Potential Difference Across the Capacitor

The capacitor is connected between the top-left node and the middle-left node. Thus, the potential difference across the capacitor $V_C$ is: $V_C = |V_{\text{top-left}} - V_{\text{middle-left}}| = |0\text{ V} - 0.5\text{ V}| = 0.5\text{ V}$

Thus, the potential difference across the capacitor in steady state is 0.5 V.

Correct Option: A