Determination of the optimal shape of permanent magnets of a given volume providing maximum strength of their magnetic coupling

The ponderomotive force of the adhesion of two permanent magnets depends on their shape. We present the results of determining the optimal shape of ellipsoidal magnets providing maximum magnetic adhesion between them. The interaction of two halves of a magnet, which is an ellipsoid of revolution, and a magnet in the form of a long rod with an elliptical cross section, is analyzed. Analytical formulas for the cohesion forces in these cases are obtained. For a fixed mass or volume of magnets, the problem of optimizing the adhesion force is solved and a geometric shape which provide the maximum adhesion force is determined. It is shown that in the case of a magnet in the form of an ellipsoid of revolution, the maximum adhesion force of its halves (ignoring the magnetic tension on the side surfaces) is achieved at an eccentricity of 0.625958. The magnitude of the maximum adhesion force exceeds the adhesion force of the halves of a uniformly magnetized spherical magnet of the same volume by 1.7%. In this case, the adhesion area of the ellipsoidal magnet will be less than the adhesion area of the spherical magnet by 28%. The optimal form of a bar magnet with an elliptical section with the maximum force of adhesion of its halves at a fixed volume of the magnet is determined. A formula is derived for the ponderomotive magnetostatic force of the interaction between the halves of a bar magnet with an elliptical section and the maximum force of interaction. Numerical estimates for a sintered NdFeB bar magnet showed that the ponderomotive force of interaction with a cross-sectional radius of 5 cm can reach 2 tons per 1 m of length. The results obtained can be used to improve the efficiency of devices based on permanent magnets.