Power Factor of Series LCR Circuit Driven by AC Source

To find the power factor of a series $LCR$ circuit, we use the relation: $\text{Power factor } (\cos\phi) = \frac{R}{Z}$

where $R$ is the resistance and $Z$ is the total impedance of the circuit.

The impedance $Z$ of a series $LCR$ circuit is given by: $Z = \sqrt{R^2 + (X_C - X_L)^2}$

Given:

• Resistance, $R = 80\ \Omega$
• Inductive reactance, $X_L = 20.0\ \Omega$
• Capacitive reactance, $X_C = 80.0\ \Omega$

Substituting these values into the expression for impedance $Z$: $Z = \sqrt{80^2 + (80.0 - 20.0)^2}$ $Z = \sqrt{80^2 + 60^2}$ $Z = \sqrt{6400 + 3600}$ $Z = \sqrt{10000} = 100\ \Omega$

Now, substituting $R$ and $Z$ into the power factor formula: $\cos\phi = \frac{80}{100} = 0.8$

Hence, the power factor of the circuit is $0.8$, which corresponds to Option A.