In the LCR circuit shown in Fig 7.7, the ac driving voltage is

(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.

Given:
Equation for voltage.
v=v_msinωt
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_msin(ωt+ϕ )= -q_mcos(ωt+ϕ )
i = dq/dt = q_mωsin(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=t₀, 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_msin(ωt+ϕ )= q_mωsin(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.