A fully charged capacitor C with initial charge `q_(0)` is connected to a coil of self inductance L at t=0. The time at which the energy is stored equally between the electric and the magnetic fields is

Options

A. `(pi)/(4) sqrt(LC)`
B. `2 pi sqrt(LC)`
C. `sqrt(LC)`
D. `pi sqrt(LC)`

Answer

A

Explanation

During discharging of capacitor `C` through inductance `L`, let at any instant, charge in capacitor be `q` and current in inductance be changing at the rate of `dI//dt`
The emf equation of the circuit is
`(Q)/(C ) + L (dI)/(dt) = 0` or `(Q)/(C ) + L (d^(2) Q)/(dt^(2)) = 0`
or `(d^(2) Q)/(dt^(2)) + (Q)/(LC) = 0`
It is a differntial equation of second order with shows that the charge is oscillaroty in `L -C` circuit, i.e., `Q = Q_(0) sin omega t`.
Comparing (i) with the relation
`(d^(2) Q)/(dt^(2) + omega^(2) Q = 0`
We have `omega^(2) = (1)/(LC)` or `omega = (1)/(sqrt(LV))`.
Max. energy stored in capacitor `= (1)/(2) (Q_(0)^(2))/(C )`
Let at instant `t`, the energy be stored eqally between electric and magnetic field. Then energy stored in electric field at instant `t`is
`(1)/(2) (Q^(2))/(C ) = (1)/(2) [(1)/(2) (Q_(0)^(2))/(C )]` or `Q^(2) = (Q_(0)^(2))/(2)` or `Q = (Q_(0))/(sqrt2)`
`Q_(0) sin omega t = (Q_(0))/(sqrt2)` or `omega t = (pi)/(4)`
or `t = (pi)/(4 omega) = (pi)/(4 xx (1//sqrt(LC))) = (pi sqrt(LC))/(4)`