Q. 31

# In the LCR circui

Given:
Equation for voltage.
v=vmsinωt
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 =qmsin(ωt+
ϕ )= -qmcos(ωt+ϕ )
i = dq/dt = qmωsin(ωt+
ϕ)
Here, ω is the angular frequency and
ϕ is the phase of the circuit.
im is the maximum current at maximum peak voltage vm.
(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=imsin(ωt+
ϕ )= qmωsin(ωt+ϕ )….(1)

Here, XC and XL are reactive capacitance and reactive inductance respectively, vm 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 im

(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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