Question

Prove that the force acting on a current-carrying wire, joining two fixed points a and b in a uniform magnetic field, is independent of the shape of the wire.

Answer 1

Explanation:

Given:

In the wire’s region, the uniform magnetic field = B

Let’s consider that the electric current via the wire be i.

Wire’s length between two points b and a = l

Magnetic force is shown as

$F \rightarrow=i ! \rightarrow \times B \rightarrow F \rightarrow=i | B \sin \theta$

Two wires of length l can be considered, one circular and other straight.

The radius of the circular wire is 2πa = l

• Assume that along the z direction, the magnetic field is showing. It can be observed that in the xy plane, both the wires are lying, so that the angle between the magnetic field and the area vector is 90°.
• The length of the straight wire is l and this lies in the magnetic field that is uniform and with strength B:  Force, F = ilB
• The length of the circular wire: l = 2πa
• The angle between the magnetic field and area vector will again be 90°.
• On the circular wire, the force is  F = ilB
• In magnitude, both the forces are identical. This suggests that the magnetic force is not dependent on the wire’s shape and rests on the orientation and length of the wire.
• Consequently, the magnetic force is not dependent on the wire’s shape.