Question

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 $\varphi$. At time $t=0min$, the magnetic field is abruptly switched OFF. Let ${\omega }_0=1/\sqrt{LC}$ and ignore the resistance of the circuit. Then,

• A. current in the circuit is $I(t)=(\varphi /L)\cos {\omega }_{0}t$
• B. magnitude of the charge on the capacitor is $|Q(t)|=2C{\omega }_0|\sin {\omega }_{0}t|$
• C. initial current in the circuit is infinite
• D. initial charge on the capacitor is $C{\omega }_0\varphi$

Answer 1

Answer: (A)

At $t=0$ capacitor is uncharged and flux of inductor is $\varphi .$

Now, using $\varphi =LI,$ at $t=0$ current in circuit is
$I_0=\frac{\varphi }{L}$
Instantaneous current in circuit is $\frac{dq}{dt}.$
where, $q$ is solution of
$L\frac{d^{2}q}{d{t}^2}+\frac{q}{C}=o$
or $\frac{d^{2}q}{d{t}^2}+\frac{q}{LC}=o$
At $t=0,$ current in circuit is non-zero,
so instantaneous current is given by
$=I_0\cos \omega t=\frac{\varphi }{L}.\cos \omega t$