In a series $LCR$ circuit,a resistor of $300 \ \Omega$,a capacitor of $25 \ \text{nF}$ and an inductor of $100 \ \text{mH}$ are used. For maximum current in the circuit,the angular frequency of the ac source is $. . . . \times 10^4 \ \text{rad s}^{-1}$.

At resonance,the value of current in a series $L-C-R$ circuit is: (Symbols have their usual meanings.)

The frequencies at which the current amplitude in an $LCR$ series circuit becomes $\frac{1}{\sqrt{2}}$ times its maximum value are $212\,rad\,s^{-1}$ and $232\,rad\,s^{-1}$. The value of resistance in the circuit is $R = 5\,\Omega$. The self-inductance in the circuit is $.........\,mH$.

In a series $LCR$ circuit,the capacitance is changed from $C$ to $4C$. To keep the resonance frequency unchanged,the new inductance should be:

$A$ 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 inductor coil is connected to a capacitor and an $AC$ source of rms voltage $8 \, V$ in series. The rms current in the circuit is $16 \, A$ and is in phase with the emf. If this inductor coil is connected to a $6 \, V$ $DC$ battery, the magnitude of the steady current is (in $A$)