Question

The value of $R$ for which current through battery doesn't change with time $t$ is

• A. R = $\sqrt{L/C}$
• B. R = $L/C$
• C. R = $C/L$
• D. R = $L C$

Answer 1

Answer: (A)

R = $\sqrt{L/C}$

Answer 2

For the current through the battery to be constant with time, the circuit must be in a steady state. The current through the RL branch is given by $I_{RL}(t) = \frac{\epsilon}{R}(1 - e^{-Rt/L})$. The current through the RC branch is given by $I_{RC}(t) = \frac{\epsilon}{R}e^{-t/RC}$. The total current through the battery is $I(t) = I_{RL}(t) + I_{RC}(t) = \frac{\epsilon}{R}(1 - e^{-Rt/L}) + \frac{\epsilon}{R}e^{-t/RC}$. For $I(t)$ to be constant, its derivative with respect to time must be zero: $\frac{dI}{dt} = \frac{\epsilon}{L}e^{-Rt/L} - \frac{\epsilon}{RC}e^{-t/RC} = 0$. This implies $\frac{1}{L}e^{-Rt/L} = \frac{1}{RC}e^{-t/RC}$. For this equality to hold for all $t$, the time constants must be equal: $\frac{R}{L} = \frac{1}{RC}$. Solving for $R$, we get $R^2 = \frac{L}{C}$, which gives $R = \sqrt{\frac{L}{C}}$.