Question

In the circuit diagram shown, initially there is no energy in the inductor and the capacitor. The switch is closed at $t=0$. Find the current $I$ as a function of time if $R=\sqrt{L/C}$.

Answer 1

${\tau }_L=\frac{L}{R}$
and ${\tau }_C=CR,\frac{{\tau }_C}{{\tau }_L}=\frac{C{R}^2}{L}=\frac{C}{L}\cdot \frac{L}{C}=1$
$\therefore {\tau }_L={\tau }_C$
$\therefore$ for the given condition ${\tau }_L={\tau }_C=\tau$
Now in $L-R$ circuit
$I_1=\frac{V}{R}(1-e^{-t/{\tau }_L})$
In $CR$ circuit, $I_2=\frac{V}{R}{e}^{-t/{\tau }_L}$
$\therefore I=I_1+I_2=\frac{V}{R}=$ constant