Consider a simple RC circuit as shown in Fig. 1.
Process 1: In the circuit the switch \( S \) is closed at \( t=0 \) and the capacitor is fully charged to voltage \( V_{0} \) (i.e., charging continues for time \( T \gg R C) \). In the process some dissipation \( \left(E_{\mathrm{D}}\right) \) occurs across the resistance \( R \). The amount of energy finally stored in the fully charged capacitor is \( E_{C} \).
Process 2: In a different process the voltage is first set to \( \frac{V_{0}}{3} \) and maintained for a charging time \( T \gg R C \). Then the voltage is raised to \( \frac{2 V_{0}}{3} \) without discharging the capacitor and again maintained for a time \( T \gg R C \). The process is repeated one more time by raising the voltage to \( V_{0} \) and the capacitor is charged to the same final voltage \( V_{0} \) as in process 1 .
These two processes are depicted in Fig. 2.
In process 2 , total energy dissipated across the resistance \( E_{D} \) is:
(1) \( E_{D}=\frac{1}{3}\left(\frac{1}{2} C V_{0}^{2}\right) \)
(2) \( E_{D}=3\left(\frac{1}{2} C V_{0}^{2}\right) \)
(3) \( E_{D}=\frac{1}{2} C V_{0}^{2} \)
(4) \( E_{D}=3 C V_{0}^{2} \)
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