Quality Factor And Resonance

According to NCERT Class 12, Chapter 7 (Alternating Current), the quality factor of a series resonant circuit measures the sharpness of its resonance and is given by $Q=\frac{X_L}{R}=\frac{{\omega }_{0}L}{R}$ evaluated at the resonant frequency, where a higher $Q$ means a narrower, sharper resonance peak. Substituting the reactance and resistance: $Q=\frac{500}{25}=20$. The option 0.05 inverts the ratio, computing resistance over reactance. The option 525 wrongly adds the two quantities instead of dividing. The option 10 halves the correct ratio, perhaps by using twice the reactance. A plausibility check confirms that the quality factor is dimensionless, since it is a ratio of two quantities measured in ohms, and a value of 20 indicates a fairly selective circuit useful in tuning applications. A high quality factor implies the circuit strongly favours the resonant frequency while sharply rejecting nearby frequencies, which is exactly why such circuits are used in radio receivers. The quality factor also equals the ratio of the resonant frequency to the bandwidth of the resonance curve, so a $Q$ of 20 means the band of frequencies effectively passed is only one-twentieth of the resonant frequency. Physically, a high quality factor corresponds to low resistance, because the resistance is the only element that dissipates energy and thereby damps the oscillations. In a radio tuner, choosing components that give a large quality factor allows the receiver to isolate a single station while suppressing neighbouring channels, which is why designers carefully minimise resistive losses in resonant tuning stages to keep the quality factor as high as possible.