Arrange the following series LCR circuits with parameters as given below in increasing order of their values of Q factor.
A. \( R=1\,\Omega, L=8\,\mathrm{mH}, C=2\,\mathrm{mF} \)
B. \( R=5\,\Omega, L=16\,\mathrm{mH}, C=4\,\mathrm{mF} \)
C. \( R=3\,\Omega, L=125\,\mathrm{mH}, C=5\,\mathrm{mF} \)
D. \( R=2\,\Omega, L=27\,\mathrm{mH}, C=3\,\mathrm{mF} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B, D, C, A
The quality factor of a series LCR circuit is
\( \displaystyle Q = \frac{\omega_0L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} \)
Only the ratio L/C matters, and mH/mF is the same for every circuit, so the values can be used directly.
\( \displaystyle \text{A: } Q = \frac{1}{1}\sqrt{\frac{8}{2}} = 2 \)
\( \displaystyle \text{B: } Q = \frac{1}{5}\sqrt{\frac{16}{4}} = 0.4 \)
\( \displaystyle \text{C: } Q = \frac{1}{3}\sqrt{\frac{125}{5}} = \frac{5}{3} \approx 1.67 \)
\( \displaystyle \text{D: } Q = \frac{1}{2}\sqrt{\frac{27}{3}} = 1.5 \)
Increasing order: B (0.4) < D (1.5) < C (1.67) < A (2), i.e. B, D, C, A.