Nature of the electromagnetic force between classical magnetic dipoles

The Lorentz force law of classical electrodynamics states that the force F exerted by the magnetic induction B on a particle of charge q moving with velocity V is given by F = qV × B. Since this force is orthogonal to the direction of motion, the magnetic field is said to be incapable of performing mechanical work. Yet there is no denying that a permanent magnet can readily perform mechanical work by pushing/pulling on another permanent magnet - or by attracting pieces of magnetizable material such as scrap iron or iron filings. We explain this apparent contradiction by examining the magnetic Lorentz force acting on an Amperian current loop, which is the model for a magnetic dipole. We then extend the discussion by analyzing the Einstein-Laub model of magnetic dipoles in the presence of external magnetic fields.