Question

In the circuit shown, $R=\sqrt{\frac{L}{C}}$. Switch S is closed at t = 0. The current through C and L would be equal after a time t equal to

• A. RC
• B. RC $\ln \left(2\right)$
• C. RL $\ln \left(2\right)$
• D. LC

Answer 1

The correct option is B RC $\ln \left(2\right)$

$i_L=\frac{\epsilon }{R}(1-e^{-Rt/L})$
$i_C=\frac{\epsilon }{R}{e}^{-t/RC}$
$i_L=i_C\Rightarrow \frac{\epsilon }{R}{e}^{-t/RC}=\frac{\epsilon }{R}(1-e^{-Rt/L})$
Substituting $R^{2}C$ in place of $L$
$\frac{\epsilon }{R}{e}^{-t/RC}=\frac{\epsilon }{R}(1-e^{-t/RC})$
$\Rightarrow e^{\frac{-t}{RC}}=\frac{1}{2}$
$\Rightarrow t=RC\ln 2$