The "inverse problem" to the evaluation of magnetic fields

In the design of superconducting magnet elements, such as may be required to guide and focus ions in a particle accelerator, one frequently premises some particular current distribution and then proceeds to compute the consequent magnetic field through use of the laws of Biot and Savart or of Ampere. When working in this manner one of course may need to revise frequently the postulated current distribution before arriving at a resulting magnetic field of acceptable field quality. It therefore is of interest to consider an alternative ("inverse") procedure in which one specifies a desired character for the field required in the region interior to the winding and undertakes then to evaluate the current distribution on the specified winding surface that would provide this desired field. By evaluating the specified potential in the region interior to the winding along the interface, we have determined that a relaxation solution to the potential in the region outside the winding can be converged and used to calculate wire location. We have demonstrated this method by applying a slightly modified version of the program POISSON to a periodic alternating sinusoidal quadrupole field.