Define Quality Factor (Q-factor) of a resonant series LCR circuit. State its physical significance and write down its mathematical expression.

Q: Define Quality Factor (Q-factor) of a resonant series LCR circuit. State its physical significance and write down its mathematical expression.

Definition of Quality Factor (Q-factor):\nThe Quality Factor (Q-factor) of a series resonant LCR circuit is defined as the ratio of the voltage drop across either the inductor or the capacitor at resonance to the applied voltage (which equals the voltage drop across the resistor).\nAlternatively, it is defined as $2\pi$ times the ratio of energy stored in the circuit to the energy dissipated per cycle. Mathematical Expression:\nAt resonance, $\omegar = \frac{1}{\sqrt{LC}}$. $$Q = \frac{VL}{VR} = \frac{I \omegar L}{I R} = \frac{\omegar L}{R}$$\nSubstituting $\omegar = \frac{1}{\sqrt{LC}}$: $$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$$ Physical Significance: Sharpness of Resonance: The Q-factor is a dimensionless quantity that measures the sharpness of the resonance peak. A higher Q-factor indicates a sharper response curve, meaning the circuit is more selective. Tuning Selectivity: In radio and communication receivers, a high Q-factor ensures that the circuit selectively tunes into a specific desired frequency while strongly attenuating neighboring unwanted frequencies.

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Definition of Quality Factor (Q-factor):
The Quality Factor (Q-factor) of a series resonant LCR circuit is defined as the ratio of the voltage drop across either the inductor or the capacitor at resonance to the applied voltage (which equals the voltage drop across the resistor).
Alternatively, it is defined as $2\pi$ times the ratio of energy stored in the circuit to the energy dissipated per cycle. Mathematical Expression:
At resonance, $\omegar = \frac{1}{\sqrt{LC}}$. $$Q = \frac{VL}{VR} = \frac{I \omegar L}{I R} = \frac{\omegar L}{R}$$
Substituting $\omegar = \frac{1}{\sqrt{LC}}$: $$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$$ Physical Significance: Sharpness of Resonance: The Q-factor is a dimensionless quantity that measures the sharpness of the resonance peak. A higher Q-factor indicates a sharper response curve, meaning the circuit is more selective. Tuning Selectivity: In radio and communication receivers, a high Q-factor ensures that the circuit selectively tunes into a specific desired frequency while strongly attenuating neighboring unwanted frequencies.

CBSE · Class 12 · Physics · Alternating CurrentDefine Quality Factor (Q-factor) of a resonant series LCR circuit. State its physical significance and write down its mathematical expression.