A fully charged capacitor C with total initial charge `q_0` is connected to a coil of self-inductance L at t=0. Find the time at which the energy is stored equally between the electric and the magnetic fields. [Hint `q = q_0 cosomegat` ]

Explanation

Energy stored in capacitor, `E_C =(q^2)/(2C)`
`implies E_C= ((q_0cos omegat)^2)/(2C) implies E_C = (q_0^2cos^2omegat)/(2C)`
Energy stored in inductor,
`E_L =1/2 Ll^2 =1/2 L((dq)/(dt))^2 =1/2 Lq_0^2 omega^2 sin^2 omegat `
Now , `E_L = E_C`
`implies 1/2 Lq_0^2 omega^2 sin^2 omegat = (q_0^2cos^2omegat)/(2C)`
`implies1/2 Lq_0^2 1/(LC) sin^2omegat = (q_0^2cos^2omegat)/(2C)`
`implies tan^2 omegat=1 " " ( :. omega=1/(sqrtLC))`
`implies omegat = (4n +1)(pi)/4,(4n-1)pi/4`
`implies t = (4n+1)T/8 ,(4n -1)T/8 [ :. omega= 2 pi/T]`
`implies T/8,(3T)/8,(5T)/8.....`