# 17.3: The Magnetic Field on the Equator of a Magnet

By the “equator” of a magnet I mean a plane normal to its magnetic moment vector, passing through the mid-point of the magnet.

The magnetic field at a point at a distance *r* on the equator of a magnet may be expressed as a series of terms of successively higher powers of \(1/r\) (the first term in the series being a term in \(r^{-3}\)), and the higher powers decrease rapidly with increasing distance. At large distances, the higher powers become negligible, so that, at a large distance from a small magnet, the magnitude of the magnetic field produced by the magnet is given approximately by

\[B = \frac{\mu_0}{4 \pi} \frac{p}{r^3}.\]

For example, if the surface magnetic field on the equator of a planet has been measured, and the magnetic properties of the planet are being modelled in terms of a small magnet at the centre of the planet, the dipole moment can be calculated by multiplying the surface equatorial magnetic field by \(\mu_0/(4 \pi)\) times the cube of the radius of the planet. If \(\text{B}\), \(\mu_0\)_{ }and \(r\) are expressed respectively in \(\text{T, H m}^{-1}\) and \(\text{m}\), the magnetic moment will be in \(\text{N m T}^{-1}\).