Question

In a circuit, a resistor \( R \) and an inductor \( L \) are connected in series with an AC voltage source \( V = V_0 \sin(\omega t) \). The current \( I(t) \) through the circuit lags the applied voltage \( V(t) \) by \( \frac{\pi}{3} \) radians. If the impedance of the circuit is \( Z = \sqrt{R^2 + (\omega L)^2} \), determine the power factor of the circuit.

A. \( \cos \left( \frac{\pi}{3} \right) \)

B. \( \cos \left( \frac{\pi}{6} \right) \)

C. \( \cos \left( \frac{2\pi}{3} \right) \)

D. \( \cos \left( \frac{\pi}{4} \right) \)

Answer 1

The power factor of a circuit, which is the cosine of the phase angle \(\phi\) between the voltage and current, can be calculated using the given phase lag. Here, the current lags the voltage by \(\frac{\pi}{3}\) radians.

The power factor \(\text{pf}\) is given by:

\[
\text{pf} = \cos(\phi)
\]

Since \(\phi = \frac{\pi}{3}\):

\[
\text{pf} = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\]

So, the power factor of the circuit is \(\frac{1}{2}\). The correct answer is:

A. \( \cos \left( \frac{\pi}{3} \right) \)