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 three-dimensional 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. We speak of the field contribution of the yoke as an ''image field'' although it is associated with a distinctly diffuse distribution of image currents or magnetic moments. At every point on the boundary of the yoke the total scalar potential is constant, so Vi = -Vd where i=image and d=direct contribution from the conductors. If we describe both potentials as a series of ''harmonic components'' with respect to azimuthal dependence, then the nature of the boundary condition is such that a de-coupling of one harmonic component from another is preserved and therefore it is also true that Vi(n)=-Vd(n) at the iron interface, where n is a harmonic number. If we solve the appropriate individual differential equations for the scalar potential functions Vi(n) throughout the iron-free region, with the proper applied boundary condition for the scalar potential of each harmonic number, we shall achieve upon summation the appropriate potential function to describe the field contribution of the surrounding high-permeability iron. 2 refs., 3 figs