Question

An inductor and a resistor are connected in series to an AC source. If the power factor of the circuit is 0.5, the ratio of the resistance of the resistor and the reactance of the inductor is:

• A. 1:1
• B. 1:\(\sqrt{2}\)
• C. 1:\(\sqrt{3}\)
• D. 1:2

Hint:

When the power factor is 0.5 in a series R-L circuit, the ratio of resistance to reactance is \( \frac{1}{\sqrt{3}} \).

Answer 1

The Correct Option is C

The power factor (PF) of a series R-L circuit is given by: \[ \text{PF} = \cos \theta = \frac{R}{\sqrt{R^2 + X_L^2}} \] where \( R \) is the resistance and \( X_L \) is the reactance of the inductor. Given that the power factor is 0.5: \[ 0.5 = \frac{R}{\sqrt{R^2 + X_L^2}} \] Squaring both sides: \[ 0.25 = \frac{R^2}{R^2 + X_L^2} \] This simplifies to: \[ 0.25 (R^2 + X_L^2) = R^2 \] Expanding: \[ 0.25R^2 + 0.25X_L^2 = R^2 \] Rearranging: \[ 0.25X_L^2 = 0.75R^2 \] \[ X_L^2 = 3R^2 \] Taking the square root: \[ X_L = \sqrt{3}R \] Thus, the ratio of resistance \( R \) to reactance \( X_L \) is: \[ \frac{R}{X_L} = \frac{1}{\sqrt{3}} \] Therefore, the correct answer is option (3), 1:\(\sqrt{3}\).

Answer 2

Step 1: Understanding the power factor in an AC circuit
The power factor (PF) is defined as the cosine of the phase angle (\(\phi\)) between the voltage and the current in an AC circuit.
Mathematically, \(\text{PF} = \cos \phi\).

Step 2: Given data
Power factor, \(\cos \phi = 0.5\).
This implies \(\phi = \cos^{-1} 0.5 = 60^\circ\).

Step 3: Relation between resistance, reactance, and phase angle
For a series circuit with resistor \(R\) and inductive reactance \(X_L\), the phase angle is given by:
\(\tan \phi = \frac{X_L}{R}\).

Step 4: Calculate the ratio \(\frac{R}{X_L}\)
Since \(\phi = 60^\circ\), \(\tan 60^\circ = \sqrt{3} = \frac{X_L}{R}\).
Therefore, \(\frac{X_L}{R} = \sqrt{3}\) which gives \(\frac{R}{X_L} = \frac{1}{\sqrt{3}}\).

Step 5: Expressing ratio
The ratio of resistance to reactance is:
\(R : X_L = 1 : \sqrt{3}\).

Conclusion:
Hence, the ratio of the resistance of the resistor to the reactance of the inductor is \(1 : \sqrt{3}\).