A series LC circuit has an inductor L and a capacitor C. At time t = 0 , the capacitor has charge Q 0 and the current in the circuit is zero. Match the following quantities at different times with t

Question

A series LC circuit has an inductor L and a capacitor C. At time $t = 0$ , the capacitor has charge $Q_{0}$ and the current in the circuit is zero. Match the following quantities at different times with their values.

List - I	List - II
(P) Current at $t = \frac{π}{2 ω _{0}}$	(1) $0$
(Q) Charge on capacitor at $t = \frac{π}{ω _{0}}$	(2) $Q_{0}$
(R) Energy stored in inductor at $t = \frac{π}{2 ω _{0}}$	(3) $\frac{Q _{0}^{2}}{2 C}$
(S) Total energy in the circuit at any time $t$	(4) $Q_{0} ω_{0}$
	(5) $\frac{Q _{0}^{2}}{2 C}$

TestHub · Accepted Answer

The charge on the capacitor is q ( t ) = Q 0 ​ cos ( ω 0 ​ t ) and current is i ( t ) = − Q 0 ​ ω 0 ​ sin ( ω 0 ​ t ) , where ω 0 ​ = 1/ L C ​ . At t = 2 ω 0 ​ π ​ , i = − Q 0 ​ ω 0 ​ sin ( π /2 ) = − Q 0 ​ ω 0 ​ . So magnitude is Q 0 ​ ω 0 ​ (P-4). At t = ω 0 ​ π ​ , q = Q 0 ​ cos ( π ) = − Q 0 ​ (Q-1, considering magnitude or value). At t = 2 ω 0 ​ π ​ , q = 0 , so all energy is in the inductor. Maximum energy in inductor is equal to initial energy in capacitor, 2 C Q 0 2 ​ ​ (R-3). Total energy remains constant at 2 C Q 0 2 ​ ​ (S-3).

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