Bulk Forces and Interface Forces in Assemblies of Magnetized Pieces of Matter

When two magnets are stuck together, where do magnetic forces operate and which formulas should one apply to compute them? Such frequently asked questions do not find immediate answers in the literature on forces, mainly because the force field is obtained, by the Virtual Power Principle (VPP), as a (mathematical, vector-valued) distribution, not as a plain vector field, which would be more convenient for practical computation. We intend to show how to represent this single distribution by two vector fields, one of them borne by the bulk of the matter, the other one localized at material interfaces where discontinuities of permeability, of magnetization, etc., do occur. Suitable extensions of the classical Maxwell tensor play an important role in the computation (by the so-called “pillbox trick”) of this interface vector field.