Question

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

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

Answer 1

Answer: (A)

The charge on the capacitor at any instant of time is
$Q=Q_{0}cos\left(\omega t\right)$
$\Rightarrow i=\frac{dQ}{dt}=-\omega Q_{0}sin\left(\omega t\right)$
$\frac{1}{2}L{i}^2=\frac{Q^2}{2C}$
${\left(\Rightarrow \frac{1}{2}L\left[\omega Q_{0}sin\left(\omega t\right)\right]\right)}^2=\frac{{\left(\left[Q_{0}cos\left(\omega t\right)\right]\right)}^2}{2C}$
$\omega =\frac{1}{\sqrt{LC}}$
$\Rightarrow tan\left(\omega t\right)=1$
$\omega t=ta{n}^{-1}\left(1\right)=\frac{\pi }{4}$
$t=\frac{\pi }{4\omega }=\frac{\pi }{4}\sqrt{LC}$