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 magnetic fields is

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

Answer 1

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

charge on the capacitor at any time $t$,

$q=q_{0}cos\omega t\dots \left(i\right)$

At a time when energy stored equally in electric and magnetic field,
Energy of a capacitor = $\frac{1}{2}$( Total energy initially stored in the capacitor)

$\frac{1}{2}\frac{q^2}{C}=\frac{1}{2}\left(\frac{1}{2}\frac{q_0^2}{C}\right)\Rightarrow q=\frac{q_0}{\sqrt{2}}$

From Equation (i),

$\frac{q_0}{\sqrt{2}}=q_{0}cos\omega t$

$cos\omega t=\frac{1}{\sqrt{2}}\Rightarrow \omega t=co{s}^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi }{4}$

So, $t=\frac{\pi }{4\omega }=\frac{\pi }{4}\sqrt{LC}(\because \omega =\frac{1}{\sqrt{LC}})$