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

Explanation

Let `q_(0)` be the initial charge on a capcitor, Let the charged capacitor be connected to an inductor of inductance L as you have studied in Section 7.8, this LC cirucit will sustain an oscillation with frequency.
`omega(1=2piv=(1)/sqrt(LC)))`
At an instant t. charge q on the capacitor and the current t are given by:
`q(l)=q_(0)cos omegat`
`t(l)=-q_(0) omega sin omegat`
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)(omegat)`
Energy stored in the inductor at time t is
`U_(M)=(1)/(2)L t^(2)`
`=(1)/(2)L q_(0)^(2)omega^(2)sin^(2)(omegat)`
`=(q_(0)^(2))/(2C)sin^(2)(omegat)(because omega=1//sqrt(LC))`
Sum of energies
`U_(E)+U_(M)=(q_(0)^(2))/(2C)(cos^(2)omegat+sin^(2)omegat)`
`=(q_(0)^(2))/(2C)`
This sum is constant in time as `q_(0)`. and C , both are time-independent. Note that it is equal to the initial energy of the capacitor . why it is so? Think!