Question

Figure shows the variation of inductive reactance \( X_L \) of two ideal inductors of inductance \( L_1 \) and \( L_2 \) with angular frequency \( \omega \). The value of \( \frac{L_1}{L_2} \) is: 

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

Hint:

Inductive reactance \( X_L = \omega L \) increases linearly with angular frequency \( \omega \). The slope of the graph provides a direct ratio of the inductances.

Answer 1

The Correct Option is D

Relationship Between Inductance and Reactance:
- The inductive reactance is given by the formula: \[ X_L = \omega L \] - The given graph shows the variation of \( X_L \) with \( \omega \) for two inductors \( L_1 \) and \( L_2 \). The angles in the graph indicate that: \[ \tan 30^\circ = \frac{X_{L1}}{X_{L2}} \] - From trigonometry, \[ \tan 30^\circ = \frac{1}{\sqrt{3}} \] - Since \( X_L \) is proportional to \( L \), we write: \[ \frac{L_1}{L_2} = \frac{X_{L1}}{X_{L2}} = \frac{1}{3} \] Thus, the correct answer is \( \frac{1}{3} \).