# 17.2: The SI Definition of Magnetic Moment

- Page ID
- 5523

If a magnet is placed in an external magnetic field \(\textbf{B}\), it will experience a torque. The magnitude of the torque depends on the orientation of the magnet with respect to the magnetic field. There are two oppositely-directed orientations in which the magnet will experience the greatest torque, and the magnitude of the magnetic moment is **defined** as the *maximum torque experienced by the magnet when placed in unit external magnetic field.* The magnitude and direction of the torque is given by the equation

\[\boldsymbol{\tau} = \textbf{p} \times \textbf{B}. \label{1}\]

The SI unit for magnetic moment is clearly \(\text{N m T}^{-1}\).

If an electric current \(I\) flows in a plane coil of area \(\textbf{A}\)** **(recall that area is a vector quantity – hence the boldface), the torque it will experience in a magnetic field is given by

\[\boldsymbol{\tau} = I \textbf{A} \times \textbf{B}. \label{2}\]

This means that the magnetic moment of the coil is given by

\[\textbf{p} = I \textbf{A}. \label{3}\]

Thus the unit \(\text{A m}^2\) is also a correct SI unit for magnetic moment, though, unless the concept of “current in a coil” needs to be emphasized in a particular context, it is perhaps better to stick to \(\text{N m T}^{-1}\).

While “\(\text{J T}^{-1}\)” is also formally dimensionally correct, it is perhaps better to restrict the unit “joule” to work or energy, and to use \(\text{N m}\) for torque. Although these are dimensionally similar, they are conceptually rather different. For this reason, the occasional practice seen in atomic physics of expressing magnetic moments in \(\text{MeV T}^{-1}\) is not entirely appropriate, however convenient it may sometimes seem to be in a field in which masses and momenta are often conveniently expressed in \(\text{MeV}/c^2\) and \(\text{MeV}/c\).

It is clear that the unit “\(\text{T M}^3\)”, often seen for “magnetic moment” is not dimensionally correct for magnetic moment as defined above, so that, whatever quantity is being expressed by the often-seen “\(\text{T M}^3\)”, it is not the conventionally defined concept of magnetic moment.

The *magnetization* \(\textbf{M}\)** **of a material is defined by the equation

\[\textbf{B} = \mu_0 (\textbf{H} + \textbf{M}) \label{4}\]

Equations \(\ref{2}\) and \(\ref{4}\) for the definitions of magnetic moment and magnetization are consistent with the alternative concept of magnetization as “magnetic moment per unit volume”.