Question

In the circuit shown in the figure, the ratio of the quality factor and the band width is__s

Answer 1

Correct Answer: 10

The relations for an LCR circuit are: \[ \Delta \omega = \frac{R}{L}, \quad Q = \frac{\omega_0}{\Delta \omega} = \omega_0 \cdot \frac{L}{R}. \] The resonant angular frequency \(\omega_0\) is given by: \[ \omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{3 \times 27 \times 10^{-6}}} = \frac{1}{9 \times 10^{-3}}. \] Simplify: \[ \omega_0 = 10^3 \, \text{rad/s}. \] The ratio \(\frac{Q}{\Delta \omega}\) is: \[ \frac{Q}{\Delta \omega} = \frac{\omega_0 \cdot L / R}{R / L} = \omega_0 \cdot \frac{L^2}{R^2}. \] Substituting the values: \[ \frac{Q}{\Delta \omega} = \sqrt{\frac{1}{LC}} \cdot \frac{L^2}{R^2}. \] Simplify further: \[ \frac{Q}{\Delta \omega} = \frac{1}{9 \times 10^{-3}} \cdot \frac{9}{100} = 10 \, \text{s}. \] Thus, the ratio of the quality factor to the bandwidth is: \[ \boxed{10 \, \text{s}}. \]