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

[asadahmad_7869]

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.
Please help me with any other alternative to this question?
Also, I don't understand the 4th option. Can you help in that as well?

Thank You.

 

[PeroK]

Please sort out your Latex tags!

 

[asadahmad_7869]

Please sort out your Latex tags!

Sorry for that, I just corrected them!