For an $LCR$ series $AC$ circuit, $L = 25 \, mH$, $R = 3 \, \Omega$, and $C = 62.5 \, \mu F$. What is the frequency of the source at which resonance occurs (in $Hz$)?

$A$ 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 $\text{Watt}$	$(4)$ $2250$
	$(5)$ $500$

An $AC$ circuit consists of an inductor of inductance $0.5 \, H$ and a capacitor of capacitance $8 \, \mu F$ in series. The current in the circuit is maximum when the angular frequency of the $AC$ source is

In the circuit shown,the power factor of the circuit is $1$ and the power factor of the box is $\frac{3}{5}$. Find the reading of the ammeter in $A$.

In an $LCR$ circuit, at a particular angular frequency $\omega_0$, the capacitive reactance and inductive reactance are the same. If the angular frequency is doubled to $2\omega_0$, what will be the ratio of the reactance of the capacitor to that of the inductor?

For the series $LCR$ circuit shown in the figure,what is the resonance frequency and the amplitude of the current at the resonating frequency?