Explain how LC oscillations takes place in the circuit.

Explanation

Figure shows a capacitor with initial charge `q_(m)` connected to ideal inductor.
The electrical energy stored in the charged capacitor is `U = (1)/(2) (q^(2)m)/(C )`
Since circuit is open there is no current in the circuit. The energy in the inductor is zero.
`:.` Total energy of LC circuit, `U = U_(E ) = ( 1)/(2) ( q^(2) m )/( C )`

At t=0, the switch is closed and the capacitor starts to discharge. This shown in figure .
As the current increases it sets up a magnetic field in the inductor and there by magnetic energy stored will be `U_(B) = ( 1)/(2) LI^(2)`.
As the current reaches its maximum value `I_(m)` and `t = ( T )/( 4)` , all the energy is stored in the magnetic field `U_(B) = ( 1)/(2) LI_(m)^(2)`. This is shown in figure.
Since maximum electrical energy of capacitor equals the maximum magnetic energy from inductor. The capacitor now has no charge and hence no energy.
The current now starts charging the capacitor as shown in figure.
This process continues till the capacitor is fully charged at time `t = (T)/( 2)` as shown in figure . But it is charged with a polarity opposite to its initial state as in figure.
The whole process just described will now repeat itselt till the system reverts to its original state . Thus the energy in the system oscillates between the capacitor and the inductor.
Hence, charge between two plates oscillates by inductor. This phenomenon is known as LCR circuit.