If the power factor in an AC circuit changes | vedclass

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(A) The power factor is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X^2}}$.Initially,$\frac{R}{\sqrt{R^2 + X^2}} = \frac{1}{3}$. Squaring

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Increase by $200 \%$

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(A) The power factor is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X^2}}$.Initially,$\frac{R}{\sqrt{R^2 + X^2}} = \frac{1}{3}$. Squaring both sides: $\frac{R^2}{R^2 + X^2} = \frac{1}{9} \implies 9R^2 = R^2 + X^2 \implies X^2 = 8R^2 \implies X = R\sqrt{8}$.Finally,$\frac{R}{\sqrt{R^2 + (X')^2}} = \frac{1}{9}$. Squaring both sides: $\frac{R^2}{R^2 + (X')^2} = \frac{1}{81} \implies 81R^2 = R^2 + (X')^2 \implies (X')^2 = 80R^2 \implies X' = R\sqrt{80}$.The ratio of reactances is $\frac{X'}{X} = \frac{R\sqrt{80}}{R\sqrt{8}} = \sqrt{10} \approx 3.16$.The percentage change in reactance is $\frac{X' - X}{X} 	imes 100 = (\sqrt{10} - 1) 	imes 100 \approx (3.16 - 1) 	imes 100 = 216 \%$.Rounding to the nearest given option,the reactance increases by approximately $200 \%$.

If the power factor in an $AC$ circuit changes from $\frac{1}{3}$ to $\frac{1}{9}$,then by what percent will the reactance change (approximately),if the resistance remains constant?

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