The Bar Magnet

A bar magnet is modelled as carrying a north pole of pole strength $+q_m$ and a south pole of pole strength $-q_m$, with the two poles separated inside the magnet by a distance $2l$. Because the magnet carries two poles of equal magnitude but opposite sign, it is called a magnetic dipole -- directly analogous to an electric dipole (two equal and opposite charges separated by a fixed distance). Its magnetic dipole moment is defined as $\vec m=q_m(2\vec l)$, where the vector $2\vec l$ points from the south pole to the north pole, in exact analogy with how the electric dipole moment is defined pointing from the negative to the positive charge. The SI unit of pole strength $q_m$ is the ampere-metre (A m), and the SI unit of magnetic dipole moment $m$ is ampere-metre squared (A m$^2$).

Two more terms are needed before any field formula can be written down. The axis of a bar magnet is the (unique) straight line passing through both of its poles -- every bar magnet has exactly one axis. The equator is a line passing through the centre of the magnet, perpendicular to its axis; the plane containing all such lines is called the equatorial plane, and the locus of points on that plane which are equidistant from the magnet's centre is called the equatorial circle (echoing, though not identical to, the everyday geographical use of the word "equator" -- any circle of any diameter drawn on the equatorial plane, centred on the magnet, is technically "an equator"). …