Physics Bootcamp

Section 38.5 Force and Torque on Magnetic Dipole

Fundamentally, magnets consist of microscopic magnets. For instance, an electron itself is a tiny magnet. Each microscopic magnet, even a single electron as a magnet, always consists of both North and South poles. That is, every magnet is made up of magnetic dipoles.

In the case of electric dipoles, we had defined a quantity called dipole moment, which was equal to the product of charge and distance between two charges. Although, we do not have individual positive and negative magnetic charges, we can still define a magnetic dipole moment by the way tiny magnets behave when placed in an external magnetic field.

We use vector \(\vec \mu\) to denote magnetic dipole moment. In an external magnetic field \(\vec B\text{,}\) we find that tiny magnets tends to align with the external field.

If the external magnetic field is uniform, then there is no force on the dipole. But there is a torque on the dipole if dipole is not aligned with the field. The torque vector \(\vec \tau\) is given by

\begin{equation} \vec \tau = \vec \mu \times \vec B,\label{eq-torque-on-a-magnetic-dipole}\tag{38.5.1} \end{equation}

from which we ascertain that the magnitude of torque

\begin{equation} \tau = \mu B |\sin\,\theta|,\label{eq-torque-on-a-magnetic-dipole-1}\tag{38.5.2} \end{equation}

where \(\theta\) is the angle between the dipole and the field.

From (38.5.2) we can deduce unit of magnetic dipole moment.

We will find that this unit can aalso be expressed as \(\text{Am}^2\text{.}\) In these units, the intrinsic magnetic dipole moment of an electron is approximately \(9.28\times 10^{−24} \text{J/T}\text{.}\)

Due to this torque, it costs energy to change the direction of a magnetic dipole moment in an external field. Similar to the way we learned for electric dipoles. Suppose that a magnetic dipole is oriented at an angle \(\theta_i\) with respect to an external magnetic field and you want to change its orientation to an angle \(\theta_f\text{.}\) You will need to do the following work.

This will be equal to the change in magnetic potential energy of the dipole between the two orientations. Assuming zero of potential energy when the orientation angle is \(90^\circ\) with respect to the direction of the magnetic field, we get the following for potential energy formula when angle is \(\theta\text{.}\)

This says that magnetic potential energy of a magnetic dipole varies form \(-\mu B\text{,}\) when aligned to \(+\mu B\) when anti-aligned with the external field.