Analytical Calculation of Cuboïdal Magnet Interactions in 3D Jean-Paul YONNET and Hicham ALLAG Laboratoire de Génie Electrique de Grenoble, G2E Lab (UMR 5269 CNRS/INPG-UJF), Institut Polytechnique de Grenoble, ENSE3, BP46, 38402 St Martin d’Hères cedex, France Tel : 33 (0) 476 82 62 97 – Fax : 33 (0) 476 82 63 00 E-mail : <Jean-Paul.Yonnet@g2elab.grenoble-inp.fr> Topics : Analyses of EM field and force field – Material: Permanent Magnets Abstract : A synthesis of all the analytical expressions of the interaction energy, force components and torque components is presented. It allows the analytical calculation of all the interactions when the magnetizations are in any direction. The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. 1. Introduction The analytical expressions are a very powerful and a very fast method to calculate magnetic interactions. It is why the analytical expressions of all the interactions, energy, forces and torques between two cuboïdal magnets are very important results [1] – [7]. Many problems can be solved by the addition of element interactions. The simpler shape of elementary volume is the parallelepiped, with its cuboïdal volume. It is why many 3D calculations can be made by the way of 3D interactions between two elementary magnets of cuboïdal shape. Until a recent time, only the force components between two magnets with their magnetization direction parallel to one edge of the parallelepipeds have been analytically solved [1]. Two new results have been recently obtained: - the torque between two magnets [8], - the force components and torque when the directions of the magnetization are perpendicular [9] – [12]. Figure 1: basic geometrical disposition Consequently, by combining parallel and perpendicular magnetization directions, the interaction energy and all the components of force and torque can be calculated by fully analytical expressions, for any magnetization direction, and for any relative position between the two magnets. The only two hypotheses are that the magnets own a cuboïdal shape (Figure 1), and they are uniformly magnetized. To complete the results, the field and induction analytical calculation are given in Annex by using the same parameters. 2. The basic mathematical model The interactions between two parallelepiped magnets are studied. Their edges are respectively parallel (Fig. 1). The magnetizations J and J’ are supposed to be rigid and uniform in each magnet. The dimensions of the first magnet are 2a x 2b x 2c, and its polarization is J. Its center is O, the origin of the axes Oxyz. For the second magnet, the dimensions are 2A x 2B x 2C, its polarization is J’, and the coordinates of its centre O’ are (α, β, γ). The side 2a is parallel to the side 2A, and so on. The magnet dimensions are given on Table 1. Table 1 : Magnet dimensions and position Axis Ox Oy Oz First Magnet (J) 2a 2b 2c Second Magnet (J’) 2A 2B 2C Second Magnet Position O’ α β γ The magnetization directions shown on Figure 2 correspond to the case when the polarizations J and J’ have the same direction, parallel to the side 2c. Note that the calculation stays valid when they are in opposite direction; only the expression sign is reversed. The polarizations J and J’ are supposed to be rigid and uniform. They can be replaced by distributions of magnetic charges on the poles. It is the coulombian representation of the magnetization. Their density σ is defined by n J r r ⋅ = σ . On the example of Figure 2, since J is perpendicular to the surfaces 2a x 2b and oriented to the top, these faces wear the density σ = +J on the upper face (North Pole), and σ = -J on the lower face (South Pole). All the analytical calculations have been made by successive integrals. We have determined the scalar potential created by one charged surface. From this scalar potential, the induction components can be obtained by derivation. For the whole magnets, we have calculated the interaction energy. The forces and the torques can be deduced by linear and angular derivation. The most difficult analytical calculation is the interaction energy in 3D. It is made by four successive integrations. The first one gives a logarithm function. In the second one, you have two logarithm and two arc-tangent functions. The last one owns many complex functions based on logarithm and arc-tangent functions.