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 fields is

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

Answer 1

The correct option is B $\frac{\pi }{4}\sqrt{LC}$

Given $U_{electric}=U_{magnetic}$
$U_{electric}=\frac{U_{Total}}{2}$
$\frac{q^2}{2C}=\frac{{q_0}^2}{2\left(2C\right)}$
Thus $q=\frac{q_0}{\sqrt{2}}$
Charge on capacitor varies sinusoidally .

$\omega =\frac{1}{\sqrt{LC}}$
$q=q_0\cos \omega t$ ( As initial charge is maximum )
$\cos \omega t=\frac{1}{\sqrt{2}}$
$\omega t=\frac{\pi }{4}$

$⟹t=\frac{\pi }{4\omega }$

$=\frac{\pi \sqrt{LC}}{4}$