Bar magnet as an equivalent solenoid

A current loop is a magnetic dipole, and Ampere suggested that all magnetism comes from circulating currents. The field lines of a bar magnet and of a finite solenoid look alike, so a bar magnet can be pictured as a great many circulating currents, like the turns of a solenoid.

Cutting a magnet in half is like cutting a solenoid in half: two shorter solenoids, each weaker, with the lines still leaving one face and entering the other.

A small compass moved round a bar magnet and round a finite solenoid swings in the same way in both cases.

Far from a finite solenoid, at distance r along its axis, the field is B = (μ₀/4π)(2m/r³), where m is the solenoid's magnetic moment (NIA for N turns of area A carrying I). A bar magnet's far axial field, measured in the lab, has the same form.

So a bar magnet's magnetic moment equals that of the solenoid which would make the same field; its unit is A m², also written J/T.