Mathematical Derivation of the Continuum Limit of the Magnetic Force between Two Parts of a Rigid Crystalline Material

The topic of this paper is a mathematically rigorous derivation of the continuum limit of the magnetic force between two parts of a rigid magnetized body. For this we start from a discrete setting of magnetic dipoles fixed to a scaled Bravais lattice, The limit as l corresponds to the passage to the continuum. The magnetic dipole moments are scaled in such a way that we obtain a finite total magnetic moment per unit volume. Under certain regularity assumptions on the magnetization and the boundaries we derive a force formula in the passage from the discrete setting to the continuum. Compared with a corresponding magnetic-force formula which has been previously discussed in the literature, the limiting force consists of an additional explicit local surface term, which is due to short-range effects and which reflects the lattice approximation of the underlying hypersingular integral.