Question

In the electrical network, at $t<0$ (as given in the figure), key is placed on (1) till the capacitor got fully charged. Key is placed on (2) at $t=0$. Time when the energy in both the capacitor and the inductor will be same for the first time is

• A. $\frac{\pi }{4}\sqrt{LC}$
• B. $\frac{3\pi }{4}\sqrt{LC}$
• C. $\frac{\pi }{3}\sqrt{LC}$
• D. $\frac{2\pi }{3}\sqrt{LC}$

Answer 1

Answer: (A)

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

When energy on both is same, means energy on capacitor is half of its maximum energy.

$\frac{q^2}{2C}=\frac{1}{2}\frac{Q^2}{2C}\Rightarrow q=\frac{Q}{\sqrt{2}}$

$\Rightarrow Qcos\omega t=\frac{Q}{\sqrt{2}}\Rightarrow cos\omega t=\frac{1}{\sqrt{2}}$

$\Rightarrow \omega t=\frac{\pi }{4}\Rightarrow t=\frac{\pi }{4\omega }=\frac{\pi }{4}\sqrt{LC}$