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

Inductive Reactance and Inductance Relationship:
- Inductive reactance (\( X_L \)) is calculated using the formula:\[X_L = \omega L\]- The provided graph illustrates the relationship between \( X_L \) and \( \omega \) for two inductors, \( L_1 \) and \( L_2 \). The angles shown in the graph signify:\[\tan 30^\circ = \frac{X_{L1}}{X_{L2}}\]- Using trigonometric principles:\[\tan 30^\circ = \frac{1}{\sqrt{3}}\]- As \( X_L \) is directly proportional to \( L \), we can establish:\[\frac{L_1}{L_2} = \frac{X_{L1}}{X_{L2}} = \frac{1}{3}\]Therefore, the correct ratio is \( \frac{1}{3} \).