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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 $t=0 \,min$, the magnetic field is abruptly switched $OFF$. Let $\omega_{0}=1 / \sqrt{L C}$ and ignore the resistance of the circuit.Then,
current in the circuit is $I(t)=(\phi / L) \cos \omega_{0} t$
magnitude of the charge on the capacitor is $|Q(t)|=2 C \omega_{0}\left|\sin \omega_{0} t\right|$
initial current in the circuit is infinite
initial charge on the capacitor is $C \omega_{0} \phi$
Solution
$(a)$ At $t=0$ capacitor is uncharged and flux of inductor is $\phi$.
Now, using $\phi=L I$, at $t=0$ current in circuit is
$I_{0}=\frac{\phi}{L}$
Instantaneous current in circuit is $\frac{d q}{d t}$. where, $q$ is solution of
$L \frac{d^{2} q}{d t^{2}}+\frac{q}{C}=0$
or $\quad \frac{d^{2} q}{d t^{2}}+\frac{q}{L C}=0$
At $t=0$, current in circuit is non-zero, so instantaneous current is given by
$=I_{0} \cos \omega t=\frac{\phi}{L} \cdot \cos \omega t$
