Symmetry of Magnetostatic Fields Generated by Toroidal Helicoidal Magnets

In this paper, the axial symmetry of the magnetic field generated by a permanent magnet of helicoidal toroidal kind is shown. In the first part of this paper, we illustrate the shape of the magnet and the number of areas where the field is calculated to demonstrate the symmetry. We define quantitatively the size of the toroidal helical magnet and the regions where the magnetostatic field is evaluated. The field is carried out for each angular sector that represents the regions where the magnetic flux density is computed. This calculation is performed with reference to a matrix of points belonging to each sector. Two sets of evaluations are performed. The first one is referred to a less dense matrix of points relative to all the regions. The aim of this computation is to demonstrate the axial symmetry of the field. The second set of calculations concerns the field evaluation using a much higher dense matrix of points. Using these data, we are able to interpolate the same field with a high precision. This second evaluation of the field is carried out with reference to only the flat region facing the first coil of the helical toroidal magnet. The use of an interpolation surface through the final points of the magnetic induction vectors previously computed allows a very fast evaluation of the field virtually in all the infinite points of the angular sector. The symmetry enables us to drastically reduce the time computation of the magnetostatic field in the points of interest.