In the LCR circuit the ac driving voltage is v = vm sin ωt.

Question:

In the LCR circuit the ac driving voltage is v = vm sin ωt.

(i) Write down the equation of motion for q (t).

(ii) At t = t0, the voltage source stops and R is short-circuited. Now write down how much energy is stored in each of L and C.

(iii) Describe subsequent motion of charges.

Solution:

(i) The equation for variation of motion of charge with respect to time is given as

L d2q(t)/dt + R dq(t)/dt + q(t)/C = Vm sin ωt

(ii) The energy stored in each of L and C is given as

Uc = 1/2Cω2[Vm2/R2+(XC – Xl)2] cos2 (ωto + ϕ)

(iii) For the circuit to become LC oscillator, R needs to be short-circuited. By doing so, the capacitor will continue to discharge and all the energy will be transferred to L and back and forth. This way there will be an oscillation of energy from electrostatic to magnetic.

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