A magnetized wire of moment M is bent into an arc of a...

A magnetized wire of moment $M$ is bent into an arc of a circle subtending an angle of $60^{\circ}$ at the centre, the new magnetic moment is
(A) $\frac{2 M}{\pi}$
(B) $\frac{M}{\pi}$
(C) $\frac{3 \sqrt{3} M}{\pi}$
(D) $\frac{3 M}{\pi}$

Key Concepts

Chord of a Circular Arc

For a uniformly magnetized arc of a circle, the process of summing the magnetic moment contributions is equivalent to finding the chord of the arc. The chord represents the resultant vector obtained from the integration of the tangent directions over the arc and has a magnitude proportional to the sine of half the subtended angle relative to the total angular span. This concept ties the geometric properties of the arc—such as its radius and subtended angle—to the resultant magnetic moment.

Magnetic Moment

The magnetic moment is a vector quantity that characterizes the strength and direction of a magnetic source. In the context of a uniformly magnetized material, it is obtained by integrating the contributions of the magnetic dipole density over the length or volume of the object. This concept is central to understanding how the overall magnetization arises from microscopic sources, such as individual atomic moments or current elements.

Vector Addition in Curved Geometries

When a magnetized object, such as a wire with a uniform magnetic moment per unit length, is bent into a curve, the individual magnetic moment contributions along the object no longer point in the same direction. Instead, one must add these vector contributions along the path of the curve. The net magnetic moment is given by the vector sum of these segments, and its magnitude depends on how the directions of the contributions add or cancel out. This approach involves integrating the tangent unit vectors along the curved path.

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