Equivalent magnetic charge in helicoidal magnets

In this paper we show that magnetostatic models of permanent magnets, based on distributions of magnetic charge and shaped by a helicoidal geometry of cylindrical type, have the volume charge density $ρM$ equal to zero. This result is valid when (i) the modulus of the vector physical magnetization density $M$ has a constant value in all the points of the helicoidal magnet and (ii) $M$ has the same direction of an oriented straight line having a constant angle with respect to the normal line of a cylindrical helix. Another case study concerns the permanent magnets with conchospiral geometry and magnetization $M$⁠. For this kind of magnet we show that, in general, $ρM≠0$⁠. Furthermore, in relation to the nonobvious result $ρM=0$ obtained for the cylindrical helicoidal permanent magnets, some geometrical physical considerations are illustrated. With reference to these observations, in order to understand if $ρM$ is equal to zero or not without considering divergence computation, the possibility of defining a criterion based only on the geometry and magnetization of the magnet is discussed. Finally, an application of the results obtained from the previous analysis is shown. After drawing an analytical formulation of the surface charge density $σM$ relative to a cylindrical helicoidal magnet, a complete evaluation of the field distribution around this magnet is performed. The results are presented by using cylindrical polar graphics of the magnetic flux $B$⁠.