A simple LC circuit oscillates between current flowing one way, charging the cap

Question

A simple LC circuit oscillates between current flowing one way, charging the capacitor, then current flowing the other way and charging the capacitor in the other orientation. Click here for an animation from Wikipedia. Energy in the circuit remains constant. When the current is flowing, the energy stored is all stored in the inductor. When the current stops flowing, it is because all the energy is stored in the capacitor. (a) When the charge on the capacitor is half of its maximum value, what fraction of the total energy is stored in the inductor? What fraction of the total energy is stored in capacitor? (b) When the charge on the capacitor is half of its maximum value, what is the current in the inductor? Express your answer as a fraction of the maximum current, when all the energy is stored in the inductor.

Explanation / Answer

it is known that the energy stored is constant .

so when capacitor is fully charged there is no current through the inductor.

let the maximum charge on capacitor be Q

so max energy = 0.5Q2/C

now when charge on capacitor in Q/2

energy in capacitor = 0.5* ( Q/2) 2 / C=0.125 Q2/C

hence fraction of energy on capacitor = 1/4 of total energy.

so inductor has stored 3/4 of the energy.

b) the the maximum current in inductor be I . ( at this point capacitor will have 0 charge)

now, maximum energy is constant = C = 0.5L * I 2 = 0.5Q2/C

now when charge on capacitor is Q/2

energy on capacitor = 0.125 Q2/C

as discussed earlier , energy on inductor is 3/4 of max energy

so current at that instant be i

we have,

0.5*L*i2 = 3/4 * 0.5L * I 2

=> i = sqrt ( 3/4) * I

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