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A capacitor is first fully charged by placing the key on position (1). At t = 0 the key is switched to position (2), connecting the charged capacitor across an inductor to form an LC circuit. Find the earliest time at which the energy stored in the capacitor equals the energy stored in the inductor.

1. pi*sqrt(L*C)/4
2. pi*sqrt(L*C)/2
3. 5*pi*sqrt(L*C)/4
4. 5*pi*sqrt(L*C)/2

Correct answer: pi*sqrt(L*C)/4

Solution

In an LC circuit q = q0*cos(wt), w = 1/sqrt(LC). Capacitor energy U_C = q²/(2C) ~ cos²(wt); inductor energy U_L = (1/2)L*i² ~ sin²(wt). They are equal when cos²(wt) = sin²(wt), i.e. tan²(wt) = 1 => wt = pi/4 (first instance). Thus t = (pi/4)/w = (pi/4)*sqrt(LC) = pi*sqrt(LC)/4.

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