An a.c. source is applied to a series LR circuit with X_L=3 R and power factor is X₁ . Now a capacitor with X_C=R is added in series to LR circuit and the power factor is X₂ . The ratio X₁ to X₂ is

Step-by-step solution

The ratio of power factors X₁ to X₂ is determined by comparing the impedance characteristics of two circuits. For the series LR circuit with X_L = 3R , the impedance is Z₁ = R^2 + (3R)^2 = R 10 . The power factor is then X₁ = R Z₁ = 1 10 . For the series LCR circuit with X_L = 3R and X_C = R , the impedance becomes Z₂ = R^2 + (3R - R)^2 = R 5 . This yields a power factor of X₂ = R Z₂ = 1 5 . The desired ratio is X₁ X₂ = 1/ 10 1/ 5 = 5 10 = 1 2 = 1 2 , corresponding to a ratio of 1 : 2 .