Answer :

Given:

Equation for voltage.

v=v_{m}sinωt

Formula used:

For an LCR circuit we have,

Here, L is Inductance , di/dt is the rate of change of current, I is the current, R is the resistance, q charges and C is the capacitance, V is the voltage.

We can substitute i in the form of dq/dt

This is the equation of motion for q(t).

For sinusoidal wave let

q =q_{m}sin(ωt+ϕ )= -q_{m}cos(ωt+ϕ )

i = dq/dt = q_{m}ωsin(ωt+ϕ)

Here, ω is the angular frequency and ϕ is the phase of the circuit.

i_{m} is the maximum current at maximum peak voltage v_{m}.

(ii)

At t=t_{0}, voltage source stops and R is short circuited.

Energy stored in inductor is

where L is the inductance and I is the A.C current flowing through the circuit.

We know that i=i_{m}sin(ωt+ϕ )= q_{m}ωsin(ωt+ϕ )….(1)

Here, X_{C} and X_{L} are reactive capacitance and reactive inductance respectively, v_{m} is the maximum peak voltage.

Substituting,

Similarly,

Energy stored in capacitor is

Here, q is the charge and C is the Capacitance.

From (1)

Substituting value of i_{m}

(iii)

Since it’s an LC oscillator, the capacitor will charge and discharge L. It will go to L and come back.

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