First slide

Electrical Resonance series circuit

Question

Which of the following combinations should be selected for better tuning of an L-C-R circuit used for communication?

Moderate

A

$R = 20 Ω, L = 1.5 H, C = 35 μ F$
B

$R = 25 Ω, L = 2.5 H, C = 45 μ F$
C

$R = 15 Ω, L = 3.5 H, C = 30 μ F$
D

$R = 25 Ω, L = 1.5 H, C = 45 μ F$

Solution

Quality factor of an L-C-R circuit is given by,
$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$
$\begin{array}{l} Q_{1} = \frac{1}{20} \sqrt{\frac{1.5}{35 \times 10^{- 6}}} = 50 \times \sqrt{\frac{3}{70}} = 10.35 \\ Q_{2} = \frac{1}{25} \times \sqrt{\frac{2.5}{45 \times 10^{- 6}}} = 40 \times \sqrt{\frac{5}{90}} = 9.43 \\ Q_{3} = \frac{1}{15} \sqrt{\frac{3.5}{30 \times 10^{- 6}}} = \frac{100}{15} \sqrt{\frac{35}{3}} = 22.77 \\ Q_{4} = \frac{1}{25} \times \sqrt{\frac{1.5}{45 \times 10^{- 6}}} = \frac{40}{\sqrt{30}} = 7.30 \end{array}$
Clearly $Q_{3}$ is maximum of $Q_{1} Q_{2} Q_{3}$ and $Q_{4}$ . Hence, option
(c) should be selected for better tuning of an L-C-R circuit.

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