Show that the charges oscillate with a frequency given by ` omega = (1)/( sqrt(LC))`when a charged capacitor of capacitance C is connected to an inductor of inductance L.

Explanation

Consider a capacitor of capacitance C. Let it be charged `q_0` and connected to an ideal inductor of inductance L as shown in fig.
The capacitor begins to discharge. As it does so an emf is induced in the inductor. At any instant p.d. across the capacitor = p.d. across the inductor.
`(q)/(C ) =-L (dI)/( dt)`
`(q)/(C ) +l (dl)/(dt)=0`
Where, q and I are the charge and current at any instant
But ` I=(dI)/(dt) to (2)`
`therefore (q)/(C ) +L ((d)/(d t)) ((dq)/( dt)) =0`
` ((d^2 q)/(dt^2))+(q)/(LC) =0 to (3)`
The above relation is in the form of
` (d^2 x)/( dt^2) + omega ^2x =0 to (4)`
On comparing (3) and (4) we get
`omega^2 = (1)/(LC)`
` omega=(1)/(sqrt(LC))`