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:

In the $LC$ circuit shown below,the current is in the direction shown. At this time :-

$A$ $2\,\mu F$ capacitor is charged to $100\,V$. After disconnecting the battery,a $20\,mH$ coil is connected across the charged capacitor. The capacitor discharges through the inductor and the current in the circuit is oscillatory. Assuming the resistance of the coil is zero,the maximum current in the circuit is.....$A$

Show that in the free oscillations of an $LC$ circuit,the sum of energies stored in the capacitor and the inductor is constant in time.

An oscillator circuit consists of an inductance of $0.5 \ mH$ and a capacitor of $20 \ \mu F$. The resonant frequency of the circuit is nearly......$Hz$

Initially,the key was placed on $(1)$ until the capacitor got fully charged. Now,the key is placed on $(2)$ at $t = 0$. Find the minimum time when the energy in both the capacitor and the inductor will be the same.