Question

A resistor of resistance \( R \) and an inductor of inductive reactance \( R \) are connected in series to an ac source. A capacitor of capacitive reactance \( 2R \) is then connected in series with \( L \) and \( R \). The ratio of the power factors of LR and LCR circuits is

• A. 1:1
• B. 1:2
• C. 1:3
• D. 2:3

Hint:

The power factor for circuits with inductive and capacitive reactances is determined by the impedance, which depends on both the resistance and reactances.

Answer 1

The Correct Option is A

The power factor \( p.f. \) in an ac circuit is defined as the ratio of the resistance \( R \) to the impedance \( Z \), that is, \[ p.f. = \frac{R}{Z} \] For the LR circuit: - The impedance is given by \( Z_{\text{LR}} = \sqrt{R^2 + X_L^2} = \sqrt{R^2 + R^2} = \sqrt{2} R \), - The power factor is \( p.f. = \frac{R}{\sqrt{2} R} = \frac{1}{\sqrt{2}} \). For the LCR circuit: - The impedance is given by \( Z_{\text{LCR}} = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (R - 2R)^2} = \sqrt{R^2 + R^2} = \sqrt{2} R \), - The power factor is \( p.f. = \frac{R}{\sqrt{2} R} = \frac{1}{\sqrt{2}} \). Thus, the ratio of the power factors is \( 1:1 \). Hence, the correct answer is \( \boxed{1:1} \).