Question

In a series $LCR$ resonant circuit, the quality factor is measured as $100.$ If the inductance is increased by two fold and resistance is decreased by two fold, then the quality factor after this change will be___ (Approximate the answer to the nearest integer)

Answer 1

Answer: 283

Quality factor $=\frac{X_L}{R}=\frac{\omega L}{R}$
$Q=\frac{1}{\sqrt{LC}}\frac{L}{R}$
$Q=\left(\frac{1}{\sqrt{C}}\right)\frac{\sqrt{L}}{R}$
$Q=\frac{XL}{R}=\frac{\omega L}{R}=\frac{1}{\sqrt{LC}}\frac{L}{R}=\frac{1}{R}\frac{\sqrt{L}}{\sqrt{C}}$
$Q^{{}^{\prime }}=\frac{\sqrt{2L}}{\left(\frac{R}{2}\right)\sqrt{C}}=2\sqrt{2}Q$
$Q^{{}^{\prime }}=282.84$