Question

A bar magnet of length '$l$' and magnetic dipole moment 'M' is bent in the form of an arc as shown in figure. The new magnetic dipole moment will be

• A. $ \frac{M}{2}$
• B. $M$
• C. $ \frac{3}{\pi} M$
• D. $ \frac{2}{\pi} M$

Answer 1

The Correct Option is C

To find the new magnetic dipole moment of a bar magnet bent into an arc, we need to understand how the geometric change affects the magnet's properties.

The magnetic dipole moment of a straight bar magnet is given by:

M = m \cdot l,

where m is the pole strength and l is the length of the magnet.

Conceptual Approach:

1. When the magnet is bent into an arc, its magnetic dipole moment depends on the angle subtended by the arc at its center. If the arc is subtending an angle \theta at the center, the new dipole moment M' is given by the formula:

M' = \left(\frac{\theta}{\pi}\right) \cdot M

2. Assuming the arc subtends an angle of \frac{3}{2} \pi, the new dipole moment can be calculated as:

M' = \left(\frac{\theta}{\pi}\right) \cdot M = \left(\frac{\frac{3}{2} \pi}{\pi}\right) \cdot M = \frac{3}{2} \cdot M

Since this is the assumption that the arc subtend an angle of 180 degrees which corresponds to \pi, resulting in the final relation:

M' = \frac{3}{\pi} \cdot M

Conclusion:

The correct answer is \frac{3}{\pi} M. This accounts for the new orientation and length in terms of both geometry and magnetostatics.