The use of a relaxation method to calculate the 3D magnetic field contribution of an iron yoke

A computational procedure has been developed for calculating the 3-D field produced by an axisymmetric iron yoke of high permeability in the presence of a system of conductors. The procedure is particularly applicable to the end regions of multipole magnets of the sort used in particle accelerators. The field produced by the conductors is calculated using the Biot-Savart law. If one solves the appropriate individual differential equations for specified scalar potential functions throughout the iron-free region, with the proper applied boundary condition for the scalar potential of each harmonic number, one will achieve upon summation the appropriate potential function to describe the field contribution of the surrounding high-permeability iron