Question

In an ac circuit, if the rms value of current is 2 A and the wattless current is \(\sqrt{3}\) A, then the power factor of the circuit is

• A. \(\frac{\sqrt3}{2}\)
• B. \(0.3\)
• C. \(0.5\)
• D. \(\frac1{\sqrt3}\) \bigskip

Hint:

For AC circuits, \[ I^2=I_a^2+I_w^2 \] which is analogous to the Pythagoras theorem for current components.

Answer 1

The Correct Option is C

Concept: In an AC circuit, the total current can be resolved into two mutually perpendicular components: \[ I= \text{Active Current} + \text{Wattless Current} \] The active current contributes to real power consumption. The wattless current contributes only to reactive power. If \[ I=\text{Total current} \] and \[ I_w=\text{Wattless current} \] then \[ I_a=\sqrt{I^2-I_w^2} \] The power factor is \[ \cos\phi=\frac{I_a}{I} \]

Step 1: Write the given values.
\[ I=2\,A \] \[ I_w=\sqrt3\,A \]

Step 2: Calculate the active current component.
Using \[ I_a=\sqrt{I^2-I_w^2} \] \[ =\sqrt{(2)^2-(\sqrt3)^2} \] \[ =\sqrt{4-3} \] \[ =1\,A \]

Step 3: Calculate the power factor.
\[ \cos\phi=\frac{I_a}{I} \] \[ =\frac{1}{2} \] \[ =0.5 \]

Step 4: State the answer.
Hence the power factor of the circuit is \[ \boxed{0.5} \]