Question

A series \( L, R \) circuit connected with an AC source \( E = (25 \sin 1000 \, t) \, \text{V} \) has a power factor of \( \frac{1}{\sqrt{2}} \). If the source of emf is changed to \( E = (20 \sin 2000 \, t) \, \text{V} \), the new power factor of the circuit will be:

• A. \( \frac{1}{\sqrt{2}} \)
• B. \( \frac{1}{\sqrt{3}} \)
• C. \( \frac{1}{\sqrt{5}} \)
• D. \( \frac{1}{\sqrt{7}} \)

Answer 1

The Correct Option is C

This analysis focuses on how the power factor of an \( L, R \) circuit is affected by changes in AC source frequency. The process will be examined systematically.

The initial AC source is described by \( E = 25 \sin 1000 \, t \, \text{V} \), with an associated power factor of \( \frac{1}{\sqrt{2}} \). The following fundamental concepts are prerequisite:

1. The power factor, \( \cos \phi \), in an \( L, R \) circuit is mathematically defined as \( \cos \phi = \frac{R}{Z} \), where \( Z \) represents the total circuit impedance, computed by \( Z = \sqrt{R^2 + (X_L)^2} \).
2. Inductive reactance \( X_L \) is determined by the formula \( X_L = \omega L \), with \( \omega = 2\pi f \) denoting the angular frequency.

The initial angular frequency of the circuit is:

\(\omega_1 = 1000 \, \text{rad/s}\)

When the source is modified to \( E = 20 \sin 2000 \, t \, \text{V} \), the resultant angular frequency is:

\(\omega_2 = 2000 \, \text{rad/s}\)

The subsequent examination addresses the transformation of the power factor:

1. At the initial frequency \( \omega_1 = 1000 \, \text{rad/s} \), the inductive reactance is calculated as \( X_{L1} = 1000L \). The corresponding power factor is:
2. With the revised frequency \( \omega_2 = 2000 \, \text{rad/s} \), the inductive reactance becomes \( X_{L2} = 2000L \). Consequently, the new power factor requires calculation:
3. From the initial power factor relationship, \( \frac{1}{\sqrt{2}} = \frac{R}{\sqrt{R^2 + (1000L)^2}} \), the following deduction is made:
4. By substituting the derived condition \( R^2 = (1000L)^2 \) into the expression for \( \cos \phi_2 \):

Therefore, the updated power factor for the circuit, when the frequency is elevated to 2000 rad/s, is \( \frac{1}{\sqrt{5}} \).