Question

The percentage increase in the reactance of an ac circuit, when its power factor changes form, 1/2 to 1/4 is
(A) 200%
(B) 124%
(C) 50%
(D) 426%

Answer 1

Hint
The power factor for AC circuit is given by $\cos \theta = \dfrac{R}{Z}$, $Z = \sqrt {{R^2} + {X^2}} $here, X is reactance and Z is the impedance. For the two cases resistance R is constant so we can find the values of reactance after that we find the percentage increase in reactance by using the formula i.e. $percentage = \dfrac{{{X_2} - {X_1}}}{{{X_1}}} \times 100$

Complete step by step answer
The power factor of an AC circuit is $\cos \theta = \dfrac{R}{Z}$
Here, R is resistance and Z is the impedance
And we also know that, $Z = \sqrt {{R^2} + {X^2}} $
Where, X is the reactance.
let $X_1$ is the reactance corresponding to the power factor 1/2, then we can write it as
$ \Rightarrow \dfrac{1}{2} = \dfrac{R}{{\sqrt {{R^2} + X_1^2} }}$
Squaring and rearranging both sides, we get
$ \Rightarrow {R^2} + X_1^2 = 4{R^2}$
$ \Rightarrow {X_1} = \sqrt 3 R = 1.73R$ ………………….. (1)
Let $X_2$ is the reactance corresponding to the power factor 1/4, then we write it as
$ \Rightarrow \dfrac{1}{4} = \dfrac{R}{{\sqrt {{R^2} + X_2^2} }}$
Again, squaring and rearranging above equation, we get
$ \Rightarrow {R^2} + X_2^2 = 16{R^2}$
$ \Rightarrow {X_2} = \sqrt {15} R = 3.87R$ ……………………….. (2)
Now, we have to find out the value of percentage increase in the reactance, this can be found as.
$percentage = \dfrac{{{X_2} - {X_1}}}{{{X_1}}} \times 100$
Now, substitute the values X2 and X1 in above equation, we get
$ \Rightarrow percentage = \dfrac{{3.87 - 1.72}}{{1.72}} \times 100$
$ \Rightarrow percentage = 1.24 \times 100 = 124\% $
Hence, the percentage increase in the reactance is 124%.
Thus, option (C) is correct.

Note
It must be noticed that there is the difference between the impedance and the reactance. Impedance is the measure of overall opposition to the current produced in the circuit. While reactance is the opposition involved only due to the presence of the inductor or capacitor. The relation between both is $Z = \sqrt {{R^2} + {X^2}} $, Z is the impedance and X is the reactance, R is the resistance.