Analytical computation of the magnetic field of a conical permanent magnet with arbitrarily uniform magnetization

Abstract

This study presents a fast-computed analytical model to compute the magnetic field generated by a cone with arbitrarily uniform magnetization. Based on the charge model, algebraic and geometrical analyses, the analytical expressions of the axial, azimuthal, and radial components of the magnetic field produced by the cone at any given point in space are derived. The model was in excellent agreement with the Finite Element Model (FEM), as is proved in this research. It is demonstrated that it took an average of less than 1 ms to execute each expression at a single point on the modern personal computer, whereas the FEM simulation consumed 779 s. Moreover, using the developed model, the magnetic field distribution of a cone used in magnetic resonance imaging with varying magnetization is analyzed. As a result, the distributions of the axial and radial components are a cosine-like wave with opposite directions, except for the case where the slope angle of the magnetization is equal to π/2; or, in other words, the magnetization is axial. On the other hand, the distribution of the azimuthal component is a sine-like wave, except for the noted case where the magnetization is axial.

Analytical Models and Magnetic Position Systems

“…), and expressions such as elliptic integrals that are easily evaluated. Solutions that can be found in the literature include cuboid [5], cylinder [6,7], cone [8], triangle [9], and other basic shapes with homogeneous magnetizations, of which most are implemented in Magpylib. Finding new solutions is an on-going effort.…”