Question 88: AC fundamentals

Question

A series combination of resistor of resistance $$100 ~\Omega$$, inductor of inductance $$1 ~\mathrm{H}$$ and capacitor of capacitance $$6.25 ~\mu \mathrm{F}$$ is connected to an ac source. The quality factor of the circuit will be __________

Accepted Answer

The Q factor (quality factor) in a series RLC circuit can be given by the formula:
$ Q = \frac{X_L}{R} = \frac{\omega L}{R} $
where ($X_L$) is the inductive reactance, (R) is the resistance, (L) is the inductance, and ($\omega$) is the angular frequency.
In a series RLC circuit at resonance, the resonant frequency (f) is given by
$ f = \frac{1}{2\pi\sqrt{LC}} $
or equivalently, the angular frequency ($\omega$) at resonance is
$ \omega = \frac{1}{\sqrt{LC}} $
Substituting (L = 1H) and ($C = 6.25 \mu F = 6.25 	imes 10^{-6} F$) into the equation for (\omega) gives
$\omega = \frac{1}{\sqrt{1H 	imes 6.25 	imes 10^{-6}F}}$ = 400 rad/s
Substituting ($\omega = 400 rad/s$), (L = 1H), and ($R = 100\Omega$) into the equation for the Q factor gives
$ Q = \frac{\omega L}{R} = \frac{400 rad/s 	imes 1H}{100\Omega} = 4 $
So, the Q factor of the circuit is 4.

AC fundamentals

A series combination of resistor of resistance $100 ~\Omega, inductor of inductance 1 ~\mathrm{H} and capacitor of capacitance 6.25 ~\mu \mathrm{F}$ is connected to an ac source. The quality factor of the circuit will be __________

Choose the correct answer: