A coil of inductive reactance $31 \, \Omega$ has a resistance of $8 \, \Omega$. It is placed in series with a

Question

A coil of inductive reactance $31 \, \Omega$ has a resistance of $8 \, \Omega$. It is placed in series with a condenser of capacitive reactance $25 \, \Omega$. The combination is connected to an a.c. source of $110$ V. The power factor of the circuit is:

Accepted Answer

The power factor ($\cos \phi$) of a series LCR circuit is defined as the ratio of resistance ($R$) to impedance ($Z$), given by $\cos \phi = \frac{R}{Z}$ . 
First, calculate the impedance $Z$. The formula for impedance in a series LCR circuit is $Z = \sqrt{R^2 + (X_L - X_C)^2}$ . 
Given: 
Resistance $R = 8 \, \Omega$ 
Inductive Reactance $X_L = 31 \, \Omega$ 
Capacitive Reactance $X_C = 25 \, \Omega$ 
Calculate the net reactance: $X = X_L - X_C = 31 - 25 = 6 \, \Omega$. 
Calculate the impedance: $Z = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \, \Omega$. 
Calculate the power factor: $\cos \phi = \frac{R}{Z} = \frac{8}{10} = 0.8$.

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