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)

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

Answer 2

: 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

$\frac{1}{2}\frac{Q^{2}}{C} = \frac{1}{2}\left\lbrack \frac{1}{2}\frac{Q_{0}^{2}}{C} \right\rbrack$

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 $t = \frac{\pi}{4\omega} = \frac{\pi}{4 \times (1/\sqrt{LC)}} = \frac{\pi\sqrt{LC}}{4}$