17.5: Possible Alternative Definitions of Magnetic Moment

Although the standard SI definition of magnetic moment is described in Section 17.2, and there is little reason for anyone who wishes to be understood by others to use any other, the previous paragraph suggested that there might be more than one choice as to how one wishes to define magnetic moment. Do we use equation 17.2.1 or equation 17.4.1 as the definition? (They are clearly different concepts.) Additional degrees of freedom as to how one might choose to define magnetic moment depend on whether we elect to use magnetic field $H$ or magnetic field $B$ in the definition, or whether the permeability is or is not to include the factor $4 \pi$ in its definition – that is, whether we elect to use a “rationalized” or “unrationalized” definition of permeability.

If one chooses to define the magnetic moment as the maximum torque experienced in unit external magnetic field, there is still a choice as to whether by magnetic field we choose $H$ or $B$. Thus we could define magnetic moment by either of the following two equations:

$$\tau = p_1 H$$

or

$$\tau = p_2 B.$$

Alternatively, we could choose to define the magnetic moment is terms of the field on the equator. In that case we have a choice of four. We can choose to use $B$ or $H$ for the magnetic field, and we can choose to exclude or include $4 \pi$ in the denominator:

$$B = \frac{p_3}{r^3},$$

$$H = \frac{p_4}{r^3},$$

$$B = \frac{p_5}{4 \pi r^3},$$

$$H = \frac{p_6}{4 \pi r^3}.$$

These six possible definitions of magnetic moment are clearly different quantities, and one may well wonder why to list them all. The reason is that all of them are to be found in current scientific literature. To give some hint as to the unnecessary complications introduced when authors depart from the simple SI definition, I list in Table $\text{XVII.1}$ the dimensions of each version of magnetic moment, the CGS EM unit, the SI unit, and the conversion factor between CGS and SI. The conversion factors cannot be obtained simply by referring to the dimensions, because this does not take into account the inclusion or exclusion of $4 \pi$ in the permeability. The correct factors can be obtained from the units, for example by noting that $1 \ \text{Oe} = 10^{-3}/(4 \pi) \text{A m}^{-1}$ and $1 \ \text{G} = 10^{-4} \text{T}$.

$\text{TABLE XVII.1}$: DIMENSIONS, CGS AND SI UNITS, AND CONVERSION FACTORS FOR MAGNETIC MOMENTS

$$\begin{array}{l |c c c c c} \nonumber & \text{Dimensions} & 1 \text{CGS EMU} & = & \text{Conversion Factor} & \text{ SI unit} \\ \hline p_1 & \text{ML}^{3}\text{T}^{-1}\text{Q}^{-1} & 1 \ \text{dyn cm Oe}^{-1} & = & 4 \pi \times 10^{-10} & \text{N m (A/m)}^{-1} \\ p_2 & \text{L}^2\text{T}^{-1}\text{Q} & 1 \ \text{dyn cm G}^{-1} & = & 10^{-3} & \text{N m T}^{-1}\\ p_3 & \text{ML}^{3}\text{T}^{-1}\text{Q}^{-1} & 1 \ \text{G cm}^3 & = & 10^{-10} & \text{T m}^3\\ p_4 & \text{L}^2\text{T}^{-1}\text{Q} & 1 \ \text{Oe cm}^3 & = & 10^{-3}/4\pi & \text{A m}^2 \\ p_5 & \text{ML}^3\text{T}^{-1}\text{Q}^{-1} & 1 \ \text{G cm}^3 & = & 10^{-10} & \text{T m}^3 \\ p_6 & \text{L}^2\text{T}^{-1}\text{Q} & 1 \ \text{Oe cm}^3 & = & 10^{-3}/4 \pi & \text{A m}^2 \\ \end{array}$$