Question

A fully charged capacitor $C$ with initial charge $Q_0$ is connected to a coil of self inductance $L$ at $t = 0$. The time at which the energy is stored equally between the electric and the magnetic field is

• A. $\frac{\pi}{4}\sqrt{LC}$
• B. $2\pi\sqrt{LC}$
• C. $\sqrt{LC}$
• D. $\pi\sqrt{LC}$

Answer 1

Answer: (A)

As $\omega^{2}=\frac{1}{LC}$ or $\omega=\frac{1}{\sqrt{LC}}$
Maximum energy stored in capacitor $=\frac{1}{2} \frac{Q^{2}_{0}}{C}$
Let at any instant $t$, the energy be stored equally between electric and magnetic field. Then energy stored in electric field at instant $t$ is
$\frac{1}{2} \frac{Q^{2}}{C}=\frac{1}{2} \left[\frac{1}{2} \frac{Q^{2}_{0}}{C}\right]$
or $Q^{2}=\frac{Q^{2}_{0}}{2}$ or $Q=\frac{Q_{0}}{\sqrt{2}} \Rightarrow Q_{0}\,cos\,\omega t=\frac{Q_{0}}{\sqrt{2}}$
or $\omega t =\frac{\pi}{4}$ or $t=\frac{\pi}{4\omega}$
$=\frac{\pi}{4\times\left(1/ \sqrt{LC} \right)}$
$=\frac{\pi\sqrt{LC}}{4}$