Torque on a Current Loop in a Magnetic Field

Learning Goal:

To understand the origin of the torque on a current loop due to the magnetic forces on the current-carrying wires.

This problem will show you how to calculate the torque on a magnetic dipole in a uniform magnetic field. We start with a rectangular current loop, the shape of which allows us to calculate the Lorentz forces explicitly. Then we generalize our result. Even if you already know the general formula to solve this problem, you might find it instructive to discover where it comes from.

Part A

A current I flows in a plane rectangular current loop with height w and horizontal sides b. The loop is placed into a uniform magnetic field B^→ in such a way that the sides of length w are perpendicular to B^→ (Figure 1) , and there is an angle θ between the sides of length b and B^→ (Figure 2).

Calculate τ, the magnitude of the torque about the vertical axis of the current loop due to the interaction of the current through the loop with the magnetic field.

Express the magnitude of the torque in terms of the given variables. You will need a trigonomeric function [e.g., sin(θ) or cos(θ)]. Use B for the magnitude of the magnetic field.

Part B

A current I flows around a plane circular loop of radius r, giving the loop a magnetic dipole moment of magnitude m. The loop is placed in a uniform magnetic field B^→, with an angle φ between the direction of the field lines and the magnetic dipole moment as shown in the figure.(Figure 3) Find an expression for the magnitude of the torque τ on the current loop.

Express the torque explicitly in terms of r, I, Π, φ, and B (where r and B are the magnitudes of the respective vector quantities). Do not use m. You will need a trigonomeric function [e.g.. sin(φ) or cos(φ)].