Three-Dimensional Analytical Calculation of Permanent Magnet Interactions by “Magnetic Node” Representation

The interaction force and torque between two parallelepiped magnets can be analytically calculated by “magnetic nodes.” The obtained expressions only depend on the corner positions. It leads F. Bancel to propose the notion of magnetic nodes situated at the corner positions. The force and torque components can be directly calculated with the magnetic nodes representation. The only hypothesis is that the magnets are supposed to be perfect (polarizations J and J
^'
constant, rigid and uniform). The analytical calculation is made by replacing polarizations by distributions of magnetic charges on the poles. The torque is calculated for rotational movement of the second magnet around its center. The three components of the force and the torque can be written with functions based on logarithm and arc-tangent. The representation by magnetic nodes is correct from a mathematical point of view. But attention must be paid to the fact that the magnetic nodes cannot be isolated from the others. The whole force is correct even when the value on the magnetic nodes is not. Meanwhile, the magnetic node representation is very suitable for global interaction understanding and for force and torque calculation.