JEE Main 2023 · 24 January · morning · Numerical answer · +4 / −0
In the circuit shown in the figure, the ratio of the quality factor and the band width is ___________ s.
Answer and solution
Answer 10
Solution
Physics sir · 3 steps
Write Q and the bandwidth in terms of R, L and C, and the ratio needs only two short calculations.
- Resonant frequency. For the series LCR circuit, ω₀ = 1/√LC = 1/√(3 × 27 × 10⁻⁶) = 1/(9 × 10⁻³) = 1000/9 rad s⁻¹. The 230 V source value is not needed.
- Q and bandwidth. The quality factor is Q = (ω₀ L)/R = ((1000/9) × 3)/10 = 100/3. The bandwidth is Δω = R/L = 10/3 rad s⁻¹.
Q/Δω = (ω₀ L / R)/(R / L) = (100/3)/(10/3) s
- So. The ratio of the quality factor to the bandwidth is 10 s.
The common slip: 62.8. Taking the bandwidth in hertz, Δ f = R/(2π L), instead of the angular bandwidth R/L makes the ratio 2π times larger, about 62.8 instead of 10.
The idea to keep. In a series LCR circuit, the bandwidth is Δω = R/L and Q = ω₀/Δω = ω₀ L/R. Use angular bandwidth unless the question asks for it in hertz.
Answer from the NTA final answer key. Solution by PrepWiser.