Question

Two short bar magnets 'A' and 'B' (having magnetic moments \( M_1 \) and \( M_2 \) respectively) are kept one above the other with their magnetic axes perpendicular to each other. If their resultant at a point on the axis of magnet 'A' is inclined at 45° with the axis of magnet 'A', then the ratio of magnetic moments \( \frac{M_2}{M_1} \) is

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

Hint:

When two magnets are perpendicular to each other, the resultant magnetic moment can be calculated using the Pythagorean theorem, and the angle can help find the ratio of their magnetic moments.

Answer 1

The Correct Option is A

Step 1: Resultant magnetic moment.
The magnetic moment of a bar magnet is given by \( M \), and the field produced by each magnet at a point is proportional to its magnetic moment. Since the axes of the two magnets are perpendicular, the resultant magnetic moment is given by: \[ M_{\text{resultant}} = \sqrt{M_1^2 + M_2^2} \]
Step 2: Using the given angle.
The resultant magnetic moment is also inclined at an angle \( \theta = 45^\circ \) with the axis of magnet 'A'. Thus, \[ \tan \theta = \frac{M_2}{M_1} = 1 \] This implies that: \[ \frac{M_2}{M_1} = 2 \]
Step 3: Conclusion.
Thus, the ratio of the magnetic moments is \( 2 : 1 \), which corresponds to option (A).