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Question: If a fully charged capacitor C with initial charge \({q_0}\) is connected to a coil of self inductan...

If a fully charged capacitor C with initial charge q0{q_0} is connected to a coil of self inductance L at t=0t = 0 . The time at which the energy is stored equally between the electric field and magnetic field is:
(A) πLC\pi \sqrt {LC}
(B) π4LC\dfrac{\pi }{4}\sqrt {LC}
(C) π2LC\dfrac{\pi }{2}\sqrt {LC}
(D) π6LC\dfrac{\pi }{6}\sqrt {LC}

Explanation

Solution

Hint We must know a few basic concepts to solve this question. The energy stored in a magnetic field is equal to the work needed to produce a current through the inductor. The capacitance of the capacitor is the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors forming the capacitor.

Complete step by step solution:
As we know the energy in the electric field is equal to the energy present in the magnetic field.
  Uelectric=Umagnetic\;{U_{electric}} = {U_{magnetic}}
Uelectric=UTotal2{U_{electric}} = \dfrac{{{U_{Total}}}}{2}​​
q22C=q022(2C)\dfrac{{{q^2}}}{{2C}} = \dfrac{{{q_0}^2}}{{2(2C)}}
Thus,q=q02q = \dfrac{{{q_0}}}{{\sqrt 2 }}
Charge on the capacitor varies sinusoidally .
Henceω=1LC\omega = \dfrac{1}{{\sqrt {LC} }}
As initial charge is maximum
 q=q0cos⁡ωt cos⁡ωt=12 ωt=π4 t=π4ω t=πLC4  \ q = {q_0}\cos \omega t \\\ \cos \omega t = \dfrac{1}{{\sqrt 2 }} \\\ \omega t = \dfrac{\pi }{4} \\\ t = \dfrac{\pi }{{4\omega }} \\\ t = \dfrac{{\pi \sqrt {LC} }}{4} \\\ \

Hence the correct option is B.

Additional information:
The capacitance of the capacitor depends on both the shape and the size of the conductor, separation between the conductors and the dielectric medium between the conductors. The current generated by a changing electric field in an inductor is proportional to the rate of change of the magnetic field. This effect is called inductance.

Note:
When a conductor carries a current, a magnetic field is produced around the conductor. The resulting magnetic flux is directly proportional to the current. If the current changes, the change in magnetic flux is proportional to the time rate of change in current by a factor called inductance. Due to energy conservation, the energy needed to drive the original current must have an outlet. For an inductor, that outlet is the magnetic field. The energy stored by an inductor is equal to the work needed to produce a current through the inductor.