# How to find the charge at time = t (at any instant)

Homework Statement:
A circuit consists of a coil with inductance ##L## and an uncharged capacitor of capacitance ##C##. The coil is in a constant uniform magnetic field such that the flux through the coil is ##\phi##. At time ##t = 0##, the magnetic field was abruptly switched off. Let ##\omega_0= \frac{1}{\sqrt(LC)}## and ignore the resistance of the circuit. Then,

1) Magnitude of charge on the capacitor is ##|Q(t)|= 2C \omega_0\phi\sin(\omega_o t)##

2) Magnitude of charge on the capacitor is ##|Q(t)|= C \omega_0\phi\sin(\omega_o t)##

3) Initial current in the circuit is infinite.

4) The cyclotron frequencies of all the particles are same.

It is a more than one correct type, with the answers being 2 and 4.
Relevant Equations:
Charge on Capacitor ## Q = CV ##
Emf induced in inductor ##V (or E) = -\frac{d\phi}{dt}##
I was not able to derive the charge on the capacitor. But then, I arbitrarily assumed ##\phi=B.A## (Dot product of Magnetic field and Area)

Then, proceeding as follows,

##\phi=BA\cos(\omega_0 t)##
##\frac{d\phi}{dt}=−BA\omega_0\sin(\omega_0 t)##

Now at ##t=0, \phi=BA\cos(0)=BA##
Therefore,
##\frac{d\phi}{dt}=−\phi\omega_0\sin(\omega_0 t)##

Now,
##V(t)=−\frac{d\phi}{dt}##
##V(t)=\phi\omega_0\sin(\omega_0 t)##

And finally,
##Q(t)=CV(t)##
##Q(t)=C\phi\omega_0\sin(\omega_0 t)##
Which corresponds to option 2.

Now, since I arbitrarily assumed the value of ##\phi##, I don't know if it is correct.
Also, I don't understand the 4th option. Can you help in that as well?

Thank You.

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