Question

An ac circuit has an inductor and a resistor of resistance $R$ in series, such that $X_L=3R.$ Now, a capacitor is added in series such that $X_C=2R.$ The ratio of new power factor with the old power factor of the circuit is $\sqrt{5}:x.$ The value of $9x$ is ______.

• A. 9.0
• B. 9
• C. 9.00

Answer 1

Answer: (A), (B), (C)

In an RLC circuit,
Power factor is defined as $\frac{R}{Z}$

When only inductor and resistance are present in the circuit, power factor
$P_1=\frac{R}{Z_1}=\frac{R}{\sqrt{X_L^2+R^2}}=\frac{R}{\sqrt{9R^2+R^2}}=\frac{1}{\sqrt{10}}$

When capacitor is also added to circuit, power factor
$P_2=\frac{R}{Z_2}=\frac{R}{\sqrt{(X_L-X_C{)}^2+R^2}}=\frac{R}{\sqrt{(3R-2R{)}^2+R^2}}=\frac{1}{\sqrt{2}}$

Ratio of new powerfactor with old power factor is $\frac{P_2}{P_1}=\frac{\sqrt{10}}{\sqrt{2}}=\frac{\sqrt{5}}{1}=\frac{\sqrt{5}}{x}$
Here, $x=1$

Hence, value of $9x$ is $9,9.0,9.00$