An inductor of $0.5 \,mH$,a capacitor of $200 \,\mu F$ and a resistor of $2 \,\Omega$ are connected in series with a $220 \,V$ ac source. If the current is in phase with the emf,the frequency of ac source will be ................ $\times 10^{2} \,Hz$

$A$ series $LCR$ circuit is connected to a $45 \sin (\omega t) \text{ V}$ source. The resonant angular frequency of the circuit is $10^5 \text{ rad s}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8 \times 10^4 \text{ rad s}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L = 50 \text{ mH}$, match each entry in List-$I$ with an appropriate value from List-$II$ and choose the correct option.

List-$I$	List-$II$
$(P)$ $I_0$ in $\text{mA}$	$(1)$ $44.4$
$(Q)$ The quality factor of the circuit	$(2)$ $18$
$(R)$ The bandwidth of the circuit in $\text{rad s}^{-1}$	$(3)$ $400$
$(S)$ The peak power dissipated at resonance in $\text{Watt}$	$(4)$ $2250$
	$(5)$ $500$

The inductance $L$,capacitance $C$,and resistance $R$ are the values of the components connected in series to an $AC$ source of angular frequency $\omega$. The inductive and capacitive reactances are $X_L$ and $X_C$ respectively. If the circuit is purely resistive,then

$A$ 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?

In an $L-C-R$ series circuit,the value of only capacitance $C$ is varied. The resulting variation of resonance frequency $f_0$ as a function of $C$ can be represented as

In an $LCR$ circuit,the resonance frequency of the circuit increases to two times its initial value by changing the capacitance from $C$ to $C^{\prime}$ and the resistance from $100 \ \Omega$ to $400 \ \Omega$,while the inductance $L$ is kept constant. The ratio $C / C^{\prime}$ is: