The Dipole in a Uniform Magnetic Field

Torque on a Magnetic Dipole in a Uniform Field

When a magnetic dipole (like a compass needle) of magnetic moment $\mathbf{m}$ is placed in a uniform magnetic field $\mathbf{B}$, it experiences a torque. This torque tends to rotate the dipole to align it with the field.

The torque $\boldsymbol{\tau}$ is given by the cross product:

In magnitude, this is:

where:

• $m$ is the magnitude of the magnetic moment,
• $B$ is the magnitude of the uniform magnetic field,
• $\theta$ is the angle between $\mathbf{m}$ and $\mathbf{B}$.

This torque is a restoring torque — it always acts to bring the dipole towards $\theta = 0^\circ$ (alignment with the field).

Magnetic Potential Energy

The magnetic potential energy $U_m$ of the dipole in the field is defined as the work done by an external agent to rotate the dipole from a reference orientation to a given angle $\theta$.

Starting from the torque expression, the work done (and hence the change in potential energy) for an infinitesimal angular displacement $d\theta$ is:

Integrating from a reference angle $\theta_0$ to $\theta$:

Choosing the zero of potential energy at $\theta_0 = 90^\circ$ (when the dipole is perpendicular to the field), we set $U_m(90^\circ) = 0$. Then $\cos\theta_0 = \cos 90^\circ = 0$, giving:

This can be written compactly as the dot product:

Interpretation of Potential Energy …