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. This LC circuit will sustain an oscillation with frequency
`omega(2pi v=(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)C V^(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)L i^(2)`
`= (1)/(2) L q_(0)^(2) omega^(2)sin^(2)(omega t)`
`=(Q_(0)^(2))/(2C)sin^(2)(omega t) " " (because omega^(2)=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 `q_(0)` and C, both are time-independent.