In a series LR circuit with X_L = R . Power factor is P₁ . If a capacitor of capacitance C with X_c = X_L is added to the circuit the power factor becomes P₂ . The ratio of P₁ to P₂ will be :

Step-by-step solution

For the initial series LR circuit, the impedance is Z₁ = R^2 + X_L^2 . Given X_L = R , we have Z₁ = R^2 + R^2 = 2 R . The power factor P₁ = R Z₁ = R 2 R = 1 2 . When a capacitor with X_C = X_L is added in series, the new impedance is Z₂ = R^2 + (X_L - X_C)^2 . Since X_C = X_L , the circuit is in resonance and Z₂ = R . The new power factor P₂ = R Z₂ = R R = 1 . The ratio of P₁ to P₂ is P₁ P₂ = 1/ 2 1 = 1 2 . Thus, P₁ : P₂ = 1 : 2 .