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

Explanation

Let `q_0` be the initial charge on a capacitor . Let the charged capacitor be connected to an inductor of inductance L. this LC circuit will sustain an oscillation with frequency ` omega ( = 2pi v = (1)/(IsqrtLC) )` .
At an instant t, charge q on the capacitor and the current I are given by `q(t) = q_0 cos omega t`
`I(t) = - q_0 omega sin omega t`
Energy stored in the capacitor at time t is `U_E = 1/2 CV^2 = 1/2 (q^2)/(C ) = (q_0^2)/(2C) cos^2 (omega t)`
Energy stored in the inductor at time t is `U_M = 1/2 Li^2`
` = 1/2 Lq_0^2 omega^2 sin^2 (omega t) = (q_0^2)/(2C) sin^2 (omega t) ( because omega^2 = (1)/(sqrtLC) )`
sum of energies `U_E + U_M = (q_0^2)/(2C) cos ^2 omega t + sin omega t = (q_0^2)/(2C)`
this sum is constant in time as `q_0` and C, both are time independent . Is equal to the initial energy of the capacitor .