Question

When a capacitor discharges through a resistance R, the time constant is $\tau$ and the maximum current in the circuit is $i_0$

• A. the initial charge on the capacitor was $i_0\tau$
• B. the initial charge on the capacitor was $\frac{1}{2}{i}_0\tau$
• C. the initial energy stored in the capacitor was $i_0^{2}R\tau$
• D. the initial energy stored in the capacitor was $\frac{1}{2}{i}_0^{2}R\tau$

Answer 1

Answer: (A), (D)

The correct options are
A the initial charge on the capacitor was $i_0\tau$
D the initial energy stored in the capacitor was $\frac{1}{2}{i}_0^{2}R\tau$

The current at instant t in the circuit is $I=\frac{V_0}{R}{e}^{-t/CR}$
thus, $i_0=\frac{V_0}{R}$ and time constant , $\tau =CR$
Initial charge, $Q_0=C{V}_0=\left(\frac{\tau }{R}\right)\left(i_{0}R\right)=i_0\tau$.
Initial energy stored in the capacitor is $U_0=\frac{1}{2}C{V}_0^2=\frac{1}{2}C{V}_0.V_0=\frac{1}{2}{i}_0\tau .\left(i_{0}R\right)=\frac{1}{2}{i}_0^{2}R\tau$