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17.4: CGS Magnetic Moment, and Lip Service to SI

Equation 17.3.1 is the equation (written in the convention of quantity calculus, in which symbols stand for physical quantities rather than for their numerical values in some particular system of units) for the magnetic field at a large distance on the equator of a magnet. The equation is valid in any coherent system of units whatever, and its validity is not restricted to any particular system of units. Example of systems of units in which Equation 17.3.1 are valid include SI, CGS EMU, and British Imperial Units.

If CGS EMU are used, the quantity \(\mu_0/(4 \pi)\) has the numerical value 1. Consequently, when working exclusively in CGS EMU, Equation 17.3.1 is often written as

\[B = \frac{p}{r^3}. \label{17.4.1}\]

This equation appears not to balance dimensionally. However, the equation is not written according to the conventions of quantity calculus, and the symbols do not stand for physical quantities. Rather, they stand for their numerical values in a particular system of units. Thus \(r\) is the distance in cm, \(B\) is the field in gauss, and \(p\) is the magnetic moment in dyne cm per gauss. However, because of the deceptive appearance of the equation, a common practice, for example, in calculating the magnetic moment of a planet is to measure its surface equatorial field, multiply it by the cube of the planet’s radius, and then quote the magnetic moment in “\(\text{G cm}^3\)”. While the numerical result is correct for the magnetic moment in CGS EMU, the units quoted are not.

While some may consider objections to incorrect units to be mere pedantry (and who would presumably therefore see nothing wrong with quoting a length in grams, as long as the actual number is correct), the situation becomes more difficult when a writer, wishing to pay lip service to SI, attempts to use Equation \(\ref{17.4.1}\) using SI units, by multiplying the surface equatorial field in \(\text{T}\) by the cube of the planet’s radius, and then giving the magnetic moment in “\(\text{T m}^3\)”, a clearly disastrous recipe!

Of course, some may use Equation \(\ref{17.4.1}\) as a definition of magnetic moment. If that is so, then the quantity so defined is clearly not the same quantity, physically, conceptually, dimensionally or numerically, as the quantity defined as magnetic moment in Section 17.2.

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