In a series LCR resonant circuit, the quality factor is measured as 100 . If

(283)

Quality factor $=\frac{X_{\mathrm{L}}}{\mathrm{R}}=\frac{\omega \mathrm{L}}{\mathrm{R}}$

$\mathrm{Q}=\frac{1}{\sqrt{\mathrm{LC}}} \frac{\mathrm{L}}{\mathrm{R}}$

$\mathrm{Q}=\left(\frac{1}{\sqrt{\mathrm{C}}}\right) \frac{\sqrt{\mathrm{L}}}{\mathrm{R}}$

$\mathrm{Q}=\frac{\mathrm{XL}}{\mathrm{R}}=\frac{\omega \mathrm{L}}{\mathrm{R}}=\frac{1}{\sqrt{\mathrm{LC}}} \frac{\mathrm{L}}{\mathrm{R}}=\frac{1}{\mathrm{R}} \frac{\sqrt{\mathrm{L}}}{\sqrt{\mathrm{C}}}$

$\mathrm{Q}^{\prime}=\frac{\sqrt{2 \mathrm{~L}}}{\left(\frac{\mathrm{R}}{2}\right)^{\sqrt{\mathrm{C}}}}=2 \sqrt{2} \mathrm{Q}$

$\mathrm{Q}^{\prime}=282.84$