An $L-C$ circuit contains a $196 \text{ pF}$ capacitor and a $441 \text{ } \mu\text{H}$ inductor. The frequency of electromagnetic radiation emitted by an antenna coupled to the $L-C$ circuit is:

For which two reasons is the discussion of $LC$ oscillations not realistic?

An $LCR$ circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having a damping constant $b$,the correct equivalence would be:

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 natural frequency of an $L-C$ circuit is $1,25,000 \text{ cycle/s}$. Then the capacitor $C$ is replaced by another capacitor with a dielectric medium of dielectric constant $K$. In this case,the frequency decreases by $25 \text{ kHz}$. The value of $K$ is

In an $LC$ oscillator,if values of inductance and capacitance become twice and eight times,respectively,then the resonant frequency of the oscillator becomes $x$ times its initial resonant frequency $\omega_0$. The value of $x$ is: