3D analytical calculation of the forces exerted between two cuboidal magnets

The interaction forces exerted between permanent magnets are used in many magneto-mechanical devices (magnetic bearings, couplings, etc...). By analytical calculation, 2D problems can be solved easily, when simple shaped magnets are used. Usually, the 3D problems are numerically computed, by using a finite element method for example. This paper presents the 3D calculation of the interaction forces exerted between two cuboidal magnets, by analytical means only. The obtained expressions are rather complicated, but a pocket programmable calculator is sufficient to the force calculation. By derivation, the analytical expressions of the stiffnesses can be easily obtained. In addition, the 3D analytical calculation allows a simple optimization of the magnet dimensions.