Question

In an AC circuit, the reactance is equal to the resistance. The power factor of the circuit will be:
A) 1
B) $\dfrac{1}{2}$
C) $\dfrac{1}{\sqrt 2}$
D) None of the above

Answer 1

Hint: In the question we have to use the concept of the Impedance of an AC circuit. We have to use the concept of the power factor of an Ac circuit and use the fact that the reactance is equal to the impedance to calculate power factor.

Complete step by step answer:
In an AC circuit the power factor is the ratio of the original power that is used to do work on the load and the apparent power given to the circuit.
It is a trigonometric function that gives the variation of the power in the ac circuit with the phase difference.
If we write the expression of the power of an AC circuit it is as follows:
$P = {V_{rms}} \times {I_{rms}} \times \cos \phi $
In the above expression P is the power of a circuit.
${V_{rms}}$ is the root mean square potential, and
${I_{rms}}$ is the root mean square current
And $\cos \phi $ is the Power factor.
We have to calculate this power factor in the above question.
The relation between the impedance, resistance is given as:
$\tan \phi = \dfrac{Z}{R}$
Where $\phi $ is the phase difference between the resistance and the impedance. R is the resistance of the circuit and Z is the impedance of the circuit. This expression is derived from the power triangle diagram.
Since we have been given that the resistance and impedance are equal to each other hence, the ratio of them will become one:
$
\tan \phi = \dfrac{R}{Z}\\
\implies \tan \phi = \dfrac{Z}{Z}\\
\implies \tan \phi = 1\\
\implies \phi = {\tan ^{ - 1}}1\\
\implies \phi = \dfrac{\pi }{4}
$
Hence the phase difference has become known to use therefore the power factor of the circuit will be:
$
{\rm{power\,factor = cos}}\phi \\
\implies {\rm{power \, factor = cos}}\dfrac{\pi }{4}\\
\therefore {\rm{power \, factor = }}\dfrac{1}{{\sqrt 2 }}
$

Hence the power factor of the above circuit will be equal to $\dfrac{1}{{\sqrt 2 }}$, and the correct option is (C).

Note:
In these types of questions the most important thing to consider and calculate is the phase difference, it holds a very high importance in the problems of AC circuits. The phase difference expression is derived from the impedance triangle or the power triangle, it shows the phases of the individual components connected in the circuit and the resultant phase difference.