3-D Analytical Calculation of the Torque and Force Exerted Between Two Cuboidal Magnets

Abstract

Most of the systems working by magnet interactions can be calculated by superposition of the interactions between parallelepiped elementary magnets. Each elementary magnet is submitted to a force and a torque. By 3-D fully analytical calculation, up to now, only the force components problem has been solved. It was published for the first time in 1984. Until now, the torque components have never been analytically calculated because of the angular derivation. We have solved it, so that now all the interactions (energy, three forces components, and three torque components) can be expressed by analytical formulations. The only hypothesis is that the magnetizations J and J ' are supposed to be rigid and uniform in each magnet. The analytical calculation is made by replacing magnetizations by distributions of magnetic charges on the magnet poles. The torque is calculated for rotational movement of the second magnet around its center. The three components of the torque are written with functions based on logarithm and arc-tangent. The results have been verified and validated by comparison with finite-element calculation. Analytical calculation owns many advantages in comparison with other calculation methods. The 3-D analytical expressions are difficult to obtain, but the expressions of energy, force, and torque are very simple to use. For example, the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.