Question

If 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 field and magnetic field is :

• A. $\pi \sqrt{LC}$
• B. $\frac{\pi }{4}\sqrt{LC}$
• C. $\frac{\pi }{2}\sqrt{LC}$
• D. $\frac{\pi }{6}\sqrt{LC}$

Answer 1

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

In $LC$ oscillation, energy is transferred $C$ to $L$ or $L$ to $C$, maximum energy in $L$ is $\frac{1}{2}{LI}_{max}^2$.
Maximum energy in $C$ is $\frac{q_{max}^2}{2C}$
Energy will be equal when,
$\frac{1}{2}{LI}^2=\frac{1}{2}\times \frac{1}{2}{LI}_{max}^2\Rightarrow I=\frac{I_{max}}{\sqrt{2}}$
$I=I_{max}\sin \omega t=\frac{1}{\sqrt{2}}{I}_{max}$
$\omega t=\frac{\pi }{4}\Rightarrow \frac{2\pi }{T}t=\frac{\pi }{4}\Rightarrow t=\frac{T}{8}$
$t=\frac{1}{8}2\pi \sqrt{LC}=\frac{\pi }{4}\sqrt{LC}$
$\Rightarrow t=\frac{\pi }{4}\sqrt{LC}$