Abstract.
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.
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Schlömerkemper, A. Mathematical Derivation of the Continuum Limit of the Magnetic Force between Two Parts of a Rigid Crystalline Material. Arch. Rational Mech. Anal. 176, 227–269 (2005). https://doi.org/10.1007/s00205-004-0354-1
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DOI: https://doi.org/10.1007/s00205-004-0354-1