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:

An $LC$ circuit contains a $20 \; mH$ inductor and a $50 \; \mu F$ capacitor with an initial charge of $10 \; mC$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$.
$(a)$ What is the total energy stored initially? Is it conserved during $LC$ oscillations?
$(b)$ What is the natural frequency of the circuit?
$(c)$ At what time is the energy stored $(i)$ completely electrical (i.e.,stored in the capacitor)? $(ii)$ completely magnetic (i.e.,stored in the inductor)?
$(d)$ At what times is the total energy shared equally between the inductor and the capacitor?
$(e)$ If a resistor is inserted in the circuit,how much energy is eventually dissipated as heat?

If maximum energy is stored in a capacitor at $t=0$,then the time after which the current in the circuit will be maximum is:

In an $LC$ circuit,the capacitor has a maximum charge $q_0$. The value of $\left| \frac{di}{dt} \right|_{\max }$ is

Compare the oscillations in an $LC$ circuit with the oscillations of a block attached to a spring.

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.