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17.2: The SI Definition of Magnetic Moment

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  • 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”.