The frequency of an $L-C$ circuit is $f_{1}$. If a resistance $R$ is also added to it,the frequency becomes $f_{2}$. The ratio $\frac{f_{2}}{f_{1}}$ will be:

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?

$A$ resonant $AC$ circuit contains a capacitor of capacitance $10^{-6} \ F$ and an inductor of $10^{-4} \ H$. The frequency of electrical oscillations will be:

In the given circuit, the resonant frequency is (in $ Hz$)

If a $LC$ circuit is considered analogous to a harmonically oscillating spring-block system,which energy of the $LC$ circuit would be analogous to potential energy and which one analogous to kinetic energy?

The magnetic energy in an inductor changes from maximum value to minimum value in $5 \ ms$. When connected to an $A.C.$ source,the frequency of the source is (in $Hz$)