$A$ series $LCR$ circuit driven by $300 \, V$ at a frequency of $50 \, Hz$ contains a resistance $R = 3 \, k\Omega$, an inductor of inductive reactance $X_{L} = 250 \pi \, \Omega$ and an unknown capacitor. The value of capacitance to maximize the average power should be: (Take $\pi^{2} = 10$) (in $\mu F$)

$A$ tuned amplifier circuit is used to generate a carrier frequency of $ 2 \text{ MHz} $ for amplitude modulation. The value of $ \sqrt{LC} $ is

In the circuit shown,the ratio of the quality factor to the bandwidth is: (in $\text{ s}$)

The capacitor of an oscillatory circuit is enclosed in a container. When the container is evacuated,the resonance frequency of the circuit is $10\, kHz$. When the container is filled with a gas,the resonance frequency changes by $50\, Hz$. The dielectric constant of the gas is

$A$ series $LCR$ circuit with $L=0.12\, H$,$C=480\, nF$,$R=23\, \Omega$ is connected to a $230\, V$ variable frequency supply.
$(a)$ What is the source frequency for which current amplitude is maximum? Obtain this maximum value.
$(b)$ What is the source frequency for which average power absorbed by the circuit is maximum? Obtain the value of this maximum power.
$(c)$ For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
$(d)$ What is the $Q$-factor of the given circuit?

$A$ resistor $R$,an inductor $L$,and a capacitor $C$ are connected in series to an oscillator of frequency $n$. If the resonant frequency is $n_r$,then the current lags behind the voltage when: