An AC circuit has an inductor and resistor R in series with Xₗ=3R. A capacitor is added in series so that X꜀=2R. The ratio of new power factor to old power factor is √5:x. Find x.
Physics · AC Circuits
JEE Mains 2021

Question

An AC circuit has an inductor and resistor R in series with Xₗ=3R. A capacitor is added in series so that X꜀=2R. The ratio of new power factor to old power factor is √5:x. Find x.

G1 Physics · Accepted Answer

Old power factor = R/√(R²+9R²)=1/√10. After adding the capacitor, net reactance is R, so new power factor = R/√(R²+R²)=1/√2. Therefore new/old=√5=√5/1, so x=1.