A capacitor C is first charged fully with potential difference of V₀ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s 25% of the initial energy in the capacitor is transferred to the inductor. The value of t i $s_______s.$
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Question

A capacitor C is first charged fully with potential difference of V₀ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s 25% of the initial energy in the capacitor is transferred to the inductor. The value of t i $s_______s.$

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G1 Physics · Accepted Answer

Initially, all the energy is stored in the capacitor. After time t, 25% of the initial energy has been transferred to the inductor. Therefore 75% of the initial energy remains in the capacitor. For an ideal LC oscillator, Q(t) = Q₀ cos(ωt), where ω = 1/√(LC). Capacitor energy is proportional to Q², so U(C,t)/U(C,0) = Q(t)²/Q₀² = cos²(ωt). Given, cos²(ωt) = 3 4 \frac{3}{4} 4 3 ​ . For the first positive time, cos(ωt) = √3/2, so ωt = π/6. Therefore, t = π/(6ω) = π√(LC)/6. Correct option: B.

A capacitor C is first charged fully with potential difference of V₀ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s 25% of the initial energy in the capacitor is transferred to the inductor. The value of t i s_______s.