For a series LCR circuit, values of R, L and C are given in the statements below. Arrange them in order of increasing Q factor.
(A) R = 10 Ω, L = 2.5 H, C = 1.6 μF
(B) R = 20 Ω, L = 1.0 H, C = 2.5 μF
(C) R = 10 Ω, L = 3.6 H, C = 1.6 μF
(D) R = 20 Ω, L = 2.5 H, C = 2.5 μF
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B), (D), (A), (C)
The quality factor of a series LCR circuit measures the sharpness of resonance:
\( \displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \)
Compute \( \sqrt{L/C} \) and divide by R for each circuit:
(A) R = 10 Ω, L = 2.5 H, C = 1.6 μF — \( \displaystyle Q = \frac{1}{10}\sqrt{\frac{2.5}{1.6\times10^{-6}}} = \frac{1250}{10} = 125 \)
(B) R = 20 Ω, L = 1.0 H, C = 2.5 μF — \( \displaystyle Q = \frac{1}{20}\sqrt{\frac{1.0}{2.5\times10^{-6}}} = \frac{632}{20} \approx 31.6 \)
(C) R = 10 Ω, L = 3.6 H, C = 1.6 μF — \( \displaystyle Q = \frac{1}{10}\sqrt{\frac{3.6}{1.6\times10^{-6}}} = \frac{1500}{10} = 150 \)
(D) R = 20 Ω, L = 2.5 H, C = 2.5 μF — \( \displaystyle Q = \frac{1}{20}\sqrt{\frac{2.5}{2.5\times10^{-6}}} = \frac{1000}{20} = 50 \)
Increasing order: 31.6, 50, 125, 150, that is (B), (D), (A), (C).