A capacitor $C$ is first charged fully with potential difference of $V_0$ and disconnected from the

Question

A capacitor $C$ is first charged fully with potential difference of $V_0$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance $L$. In $t \mathrm{~s}$ $25 \%$ of the initial energy in the capacitor is transferred to the inductor. The value of $t$ is $\_\_\_\_$ s.

Professor PC Thomas Classes · Answer

Let total energy be $U_0 = \frac{1}{2}\frac{Q_0^2}{C}$. When $25\%$ energy transfers to the inductor, $75\%$ energy remains in the capacitor. So, $U_c = 0.75 U_0 = \frac{3}{4} U_0 \Rightarrow \frac{1}{2}\frac{q^2}{C} = \frac{3}{4} (\frac{1}{2}\frac{Q_0^2}{C}) \Rightarrow q = \frac{\sqrt{3}}{2} Q_0$. In an LC circuit, charge oscillates as $q = Q_0 \cos(\omega t)$. Thus, $\cos(\omega t) = \frac{\sqrt{3}}{2} \Rightarrow \omega t = \frac{\pi}{6}$. Given $\omega = \frac{1}{\sqrt{LC}}$, we get $t = \frac{\pi \sqrt{LC}}{6}$.