An alternating e.m.f. of frequency $v (= \frac{1}{2 π \sqrt{LC}})$ is applied to a series LCR circuit. For this frequency of the applied e.m.f.,

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The quality factor of the circuit is $ωL / R or 1 / ωCR$ and this is a measure of the voltage magnification (produced by the circuit at resonance) as well as the sharpness of resonance of the circuit.

The current in the circuit is in phase with the applied e.m.f. and the voltage across R equals this applied emf.

The sum of the p.d’s across the inductance and capacitance equals the applied e.m.f. which is 180⁰ ahead of phase of the current in the circuit.

The circuit is at resonance and its impedance is made up only of a reactive part.

answer is B, D.

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Detailed Solution

LCR series resonance circuit:

The given frequency is resonant frequency at which current in the circuit is in phase with applied emf and the voltage across R equals this applied emf.

The quality factor of LCR series circuit is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the voltage across the inductor or capacitor at resonance to the voltage across the resistor at resonance .

It is given by:

$\begin{aligned} Q-factor = \frac{ω_{r} L}{R} \\ Q-factor = \frac{L}{R \sqrt{L C}} \\ Q-factor = \frac{1}{R} \sqrt{\frac{L}{C}} \end{aligned}$