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

Let `q_(0)` be the initial charge on a capacitor. Let the charged capacitor be connected to an inductor of inductance L. As you have studied in Section 7.8, this LC circuit will sustain an oscillation with frquency
`omega(=2piv=(1)/(sqrt(LC)))`
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) Lt^(2)`
`=(1)/(2) Lq_(0)^(2)omega^(2) sin^(2)(omega t)`
`=(q_(0)^(2))/(2C)sin^(2)(omega t) (because omegao=1//sqrt(LC))`
Sum of energies
`U_(E )+U_(M)=(q_(0)^(2))/(2C)(cos^(2)omega t+sin^(2)omega t)`
`=(q_(0)^(2))/(2C)`
This sum is constant in time as qo and C, both are time-independent. Note that it is equal to the initial energy of the capacitor. Why it is so? Think!