A magnetised thin rod of length L and magnetic moment is bent in the form of an arc , which makes an angle 90° at the centre.What is the new magnetic moment?

To determine the new magnetic moment of a magnetized thin rod that has been bent into an arc, we need to consider how the bending affects the distribution of the magnetic moment. Initially, the rod has a magnetic moment \( \vec{m} \) and a length \( L \). When bent into a 90° arc, the geometry changes, but the total magnetic moment will remain the same in magnitude, though its direction will change.

Understanding Magnetic Moment

The magnetic moment \( \vec{m} \) of a magnetized object is a vector quantity that represents the strength and direction of its magnetic field. For a straight rod, the magnetic moment can be calculated using the formula:

• Magnetic Moment (m) = m * L

where \( m \) is the magnetization (magnetic moment per unit volume) and \( L \) is the length of the rod.

Effect of Bending the Rod

When the rod is bent into a 90° arc, the length of the arc becomes \( L' = \frac{L}{4} \) for each segment of the arc, but the total magnetic moment remains unchanged. The magnetic moment of the entire arc can be thought of as being concentrated at the center of the arc.

Calculating the New Magnetic Moment

For a rod bent into an arc, the magnetic moment can be expressed as:

• New Magnetic Moment (m') = m * L

Since the bending does not change the total magnetic moment, we can say:

• m' = m

However, the direction of the magnetic moment vector will now point perpendicular to the plane of the arc due to the geometry of the bend. This means that while the magnitude remains the same, the orientation of the magnetic moment has changed.

Visualizing the Change

Imagine holding a straight magnet and then bending it into a quarter-circle. The strength of the magnet remains the same, but the way it interacts with external magnetic fields will differ due to its new orientation. The magnetic moment vector, which initially pointed along the length of the rod, now points out of the plane formed by the arc.

Conclusion

In summary, when a magnetized thin rod of length \( L \) and magnetic moment \( \vec{m} \) is bent into a 90° arc, the new magnetic moment remains equal in magnitude to the original magnetic moment, but its direction changes to be perpendicular to the plane of the arc. Thus, the new magnetic moment can be expressed as:

• New Magnetic Moment = m

This understanding is crucial in applications involving magnetic fields, such as in electric motors or magnetic sensors, where the orientation of the magnetic moment can significantly influence performance.