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

(a) In LC - oscillation ,
Maximum energy stored in Capacitor (C )
at `t=0 , U= (1)/(2) (q^(2))/( C) at t =t_(1)`
`U_(E)= (1)/(2) (q_("max")^(2))/(C )`
Maximum energy stored in coil inductor (L)
`U_(B)= (1)/(2) LI_(0)^(2)`
Let at an instant of time the energy stored in both be equal that means the energy of both the elements is equal to half the maximum value .
`U_(B) = (1)/(2) (U_(B))_("max")`
Let current in the circuit at time t be I(t)
`(1)/(2) LI^(2)= (1)/(2) xx(1)/(2)LI_(0)^(2)`
`I^(2) (t) = (I_(0)^(2))/(2)`
`I(t) = (I_(0))/(sqrt(2))`
`I(t)= I_(0) sin omegat`
`(I_(0))/(sqrt(2))=I_(0) sin omegat implies omegat = (pi)/(4)`
`implies t = (T)/(8) `
`t= (1)/(8)xx2pisqrt(LC) = (pisqrt(LC))/(4)`