Question

How do you find the current in the LC circuit?

Answer 1

According to Kirchhoff’s second law the sum of potential drop in a closed loop across every component is equal to zero. It is also known as Kirchhoff’s voltage law. This law also defines the conservation of charge across any closed loop.

Formula Used:
Potential drop across inductor:
${V_L} \Rightarrow L\dfrac{{dI}}{{dt}}$
Where, ${V_L}$ is the voltage across the inductor, $L$ is inductance, $I$ is current.
Potential drop across capacitor:
$\Rightarrow {V_C} = \dfrac{Q}{C}$
Where, ${V_C}$ is voltage across capacitor, $Q$ is charge, $C$ is capacitance

Complete step by step solution:
In the above circuit by applying Kirchhoff’s second law,
$\Rightarrow L\dfrac{{dI}}{{dt}} + \dfrac{Q}{C} = 0$ ............. (1)
By dividing equation (1) and differentiating with respect to $t$
$\Rightarrow \dfrac{{{d^2}I}}{{d{t^2}}} + \dfrac{1}{{LC}}\dfrac{{dQ}}{{dt}} = 0$.............. (2)
The flow of current across any circuit is equal to the rate at which charge accumulates across the positive plate of the capacitor.
$\Rightarrow I = \dfrac{{dQ}}{{dt}}$ ............. (3)
Where, resonant angular frequency,
$\Rightarrow \omega = \dfrac{1}{{\sqrt {LC} }}$ ........... (4)
Putting the values from equation (3) and (4) in equation (2)
$\Rightarrow \dfrac{{{d^2}I}}{{d{t^2}}} + {\omega ^2}I = 0$ ............ (5)
Solving equation (5) using differential equation,
We get,
$\Rightarrow I(t) = {I_0}\cos (\omega t - \phi )$
Where${I_0} > 0$ and $\phi$ are constant.
The value of current in LC circuit is,
$\therefore I(t) = {I_0}\cos (\omega t - \phi )$

Note:
Because of the absence of resistance in an ideal LC circuit. It consumes no energy which is completely different from the ideal form of RC, RL and RLC Circuit which consumes energy because of the presence of resistors.
The charge in LC circuit flows back and forth through the capacitor and inductor, the energy oscillates among capacitor and conductor till the point the internal resistance of components and connecting wire makes the oscillation disappear. This action of the circuit seems like a tuned action which is mathematically known as Harmonic Oscillation. That is the same as a pendulum moving back and forth or water flowing back and forth in a tank. Due to these reasons LC circuit also known as tuned circuit or tank circuit