Show that in the free oscillations of an LC circuit, the sum of energies stored in the capacitor and the inductor is constant in time.

Explanation

Suppose at time `t=0`, a capacitor with charge `q_(0)` is connected to an inductor ( without any magnetic field inside it) and the circuit is made closed. Now, capacitor starts getting discharged through an inductor. If charge on capacitor is q at time t then,
`q = _(0 ) cos ( omega t ) ` ...(1)
( Where` omega =` angular frequency of LC oscillations `= ( 1)/( sqrt( LC ))` ) ..(2)
Energy in the electric field between the plates of a capacitor at time t will be.
`U_(E ) = ( 1)/(2) (q^(2))/( C )`
`:. U_(E ) = ( 1)/(2) (q_(0)^(2)cos^(2) ( omegat))/( C ) ` ....(3)
Now, current through an inductor at time t is,
`i = ( dq)/( dt)`
`= ( d)/( dt) { q_(0) cos ( omega t )}`
`= q_(0) { - sin ( omega t )} omegat`
`:. i = - q_(0) omega sin ( omega t )` ....(4)
Energy in the magnetic field inside an indcutor at time t will be,
`U _(B) = ( 1)/(2) Li^(2)`
`:. U_(B ) = ( 1)/(2) Lq_(0)^(2) omega^(2) ( omega t )` [ From equation (4) ]
`:. U_(B) = (1)/(2) Lq_(0)^(2) xx (1)/( LC) sin^(2) ( omegat )`
`:. U_(B ) = (1)/(2) ( q_(0)^(2))/( C ) sin^(2) ( omega t ) ` ...(5)
Total energy stored in capacitor and in an inductor at time t will be ,
`U_(E ) + U_(B ) = (1)/(2) ( q_(0)^(2))/( C ) { cos ^(2) ( omega t ) |+ sin^(2) ( omega t) }`
`:. U_(E ) + U_(B ) = ( 1)/(2) (q_(0)^(2))/( C )` ...(6)
`:. U_(E ) + U_(B )` = constant `( :. q_(0)` and C are constants )