Alternating Current

JEE Main level, single correct, medium

59 of 190

Question 59 of 190

A capacitor CC is first charged fully with potential difference of V0V_{0} and disconnected from the battery. The charged capacitor is connected across an inductor having inductance LL. In t s 25%t \mathrm{~s} \ 25 \% of the initial energy in the capacitor is transferred to the inductor. The value of tt is ____\_\_\_\_ s.

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Answer: (B) πLC6\frac{\pi \sqrt{L C}}{6}

In an LC circuit, energy oscillates sinusoidally.
Initial energy in capacitor is E0=12CV02E_0 = \frac{1}{2}CV_0^2.
When 25% is transferred to the inductor, 75% remains in the capacitor.
Since energy in capacitor varies as EC(t)=E0cos⁡2(ωt)E_C(t) = E_0\cos^2(\omega t), we need cos⁡2(ωt)=0.75\cos^2(\omega t) = 0.75.
This gives sin⁡2(ωt)=0.25\sin^2(\omega t) = 0.25, so sin⁡(ωt)=0.5\sin(\omega t) = 0.5, yielding ωt=π6\omega t = \frac{\pi}{6}.
With ω=1LC\omega = \frac{1}{\sqrt{LC}}, we get t=πLC6t = \frac{\pi\sqrt{LC}}{6} seconds.

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