Question

The power factor of an A.C. circuit having resistance (R) and inductance (L) connected in series and an angular velocity is $\omega $ is
A) $\dfrac{R}{{\omega L}}$
B) $\dfrac{R}{{{{({R^2} + {\omega ^2}{L^2})}^{1/2}}}}$
C) $\dfrac{{\omega L}}{R}$
D) $\dfrac{R}{{{{({R^2} - {\omega ^2}{L^2})}^{1/2}}}}$

Answer 1

Hint: Power factor is defined as the ratio of the real power used to do work and the apparent power is supplied to the circuit. Power factor can get values in the range of 0 to 1.


Complete step by step answer:
Step I:
Reactance as the name suggests measures the opposition offered to the flow of current in the circuit. But it is different from resistance. Because in reactance the energy is stored in the reactance and returns later to the circuit and energy stored is continuously lost in the circuit.
Step II:
When potential difference across a conductor is developed and the value of current changes, it is known as inductive reactance.
Inductive Reactance due to inductor is written as, ${X_L} = 2\pi fL$
Where $2\pi f = \omega $ is the angular frequency.
Therefore, ${X_L} = \omega L$ ---(i)
${X_L}$ is the inductive reactance
$L$ is the reactance of the conductor.
Step III:
In series circuit, the impedance is given by:
$Z = \sqrt {{R^2} + X_L^2} $
Substituting value of ${X_L}$ from equation (i),
$Z = \sqrt {{R^2} + {{(\omega L)}^2}} $ ---(ii)
Step IV:
The total phase angle of an a.c. circuit is given by
$\tan \theta = \dfrac{{\omega L}}{R}$ ---(iii)
Step V:
The power factor is given by $\cos \theta $. Therefore,
$\cos \theta = \dfrac{1}{{\sqrt {1 + {{\tan }^2}\theta } }}$
Substituting values of $\tan \theta $ from equation (iii),
$\cos \theta = \dfrac{1}{{\sqrt {1 + \dfrac{{{\omega ^2}{L^2}}}{{{R^2}}}} }}$
$\cos \theta = \dfrac{1}{{\sqrt {\dfrac{{{R^2} + {\omega ^2}{L^2}}}{{{R^2}}}} }}$
\[\cos \theta = \dfrac{R}{{\sqrt {{R^2} + {\omega ^2}{L^2}} }}\]

$\therefore$ Option B is the correct answer.

Note: Sometimes impedance can be mixed with reactance. But in actual they are both different. Where reactance is the resistance to the flow of current and stores energy, impedance includes both resistance and reactance. The resistance of impedance occurs due to the collision between the particles of the conductor with the electrons. But reactance arises when there are changing electric and magnetic fields in the circuit with alternating currents.