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 $=\frac{1}{2}\frac{Q_0^2}{C}$
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
$\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}\left[\frac{1}{2}\frac{Q_0^2}{C}\right]$
or $Q^2=\frac{Q_0^2}{2}$ 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 $t=\frac{\pi }{4\omega }$
$=\frac{\pi }{4\times \left(1/\sqrt{LC}\right)}$
$=\frac{\pi \sqrt{LC}}{4}$