$A$ capacitor is fully charged with a battery. Then the battery is removed and a coil is connected with the capacitor in parallel. How does the current vary?

In the $LCR$ circuit shown in the figure,the ac driving voltage is $V = V_m \sin \omega t$.
$(a)$ Write down the equation of motion for $q(t)$.
$(b)$ At $t = t_0$,the voltage source is removed and $R$ is short-circuited. Write down the energy stored in each of $L$ and $C$ at this instant.
$(c)$ Describe the subsequent motion of charges.

The switch is in position $A$ for a long time. At time $t=0$,it is shifted to position $B$. Find the maximum charge that will accumulate on the capacitor.

$A$ $LC$ circuit is in the state of resonance. If $C = 0.1 \mu F$ and $L = 0.25 \ H$. Neglecting the ohmic resistance of the circuit,what is the frequency of oscillations in $Hz$?

$A$ charged $30\; \mu F$ capacitor is connected to a $27\; mH$ inductor. What is the angular frequency of free oscillations of the circuit?

The natural frequency of an $LC$ circuit is $120 \ kHz$. When the capacitor in the circuit is totally filled with a dielectric material,the natural frequency of the circuit decreases by $20 \ kHz$. The dielectric constant of the material is: