At t=0, let Q = Q 0 , and I = 0 in an LC circuit. (a) At the first moment when t

Question

At t=0, let Q = Q0, and I = 0 in an LC circuit. (a) At the first moment when the energy is shared equally bythe inductor and the capacitor, what is the charge on thecapacitor? I'm usingQ(t)=Q0cos(t), but I don't know what to do fromthere. (b) How much time has elapsed (in terms of the period T)?I know the answer is T/8, but I don't knowhow to prove this. (a) At the first moment when the energy is shared equally bythe inductor and the capacitor, what is the charge on thecapacitor? I'm usingQ(t)=Q0cos(t), but I don't know what to do fromthere. (b) How much time has elapsed (in terms of the period T)?I know the answer is T/8, but I don't knowhow to prove this.

Explanation / Answer

Q = CV, so at t =0, v = Qo/C and energy = 1/2 C(Qo/C)2 which is Qo/(2C). If the phase of theoscillator is at 0 when the i is 0, then the equal energy point isat 45 degrees. At 90 degrees, the energy will all be in theinductor. So, since the period is one full circle, 45 degreesis 1/8 of this. You can think of an axis of V vs I, and theoscillator goes in a circle around the origin.
The I and C will both be .707 of the peak values of v and i(sin 45 = cos 45 = .707). Since Q = CV, when v goes down by.707, so will Q. So, charge is /707 Qo

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