Energy Stored and Dissipated in Single Step RC Charging Process

In Process 1, a capacitor of capacitance $C$ is charged by a battery of constant voltage $V_0$ from an initial uncharged state ($q = 0$) to a fully charged state ($q = C V_0$).

1. Work Done by the Battery ($W_{\text{battery}}$): The total charge that flows through the battery during the charging process is: $Q = C V_0$ The work done by the battery maintained at constant potential $V_0$ is given by: $W_{\text{battery}} = Q V_0 = (C V_0) V_0 = C V_0^2$

2. Energy Stored in the Capacitor ($E_C$): The electrostatic potential energy stored in the capacitor when fully charged to potential $V_0$ is: $E_C = \frac{1}{2} C V_0^2$

3. Energy Dissipated across the Resistor ($E_D$): By the law of conservation of energy, the difference between the total energy supplied by the battery and the energy stored in the capacitor is dissipated as heat across the resistor: $E_D = W_{\text{battery}} - E_C$ $E_D = C V_0^2 - \frac{1}{2} C V_0^2 = \frac{1}{2} C V_0^2$

4. Relationship between $E_C$ and $E_D$: Comparing the expressions for $E_C$ and $E_D$: $E_C = \frac{1}{2} C V_0^2 \quad \text{and} \quad E_D = \frac{1}{2} C V_0^2$ $E_C = E_D$

This corresponds to Option A.