Generalization of Helmholtz coil problem

The primary intent of this work is to propose a simple analytical method for designing coil systems for homogeneous and gradient magnetostatic field generation. Coil system consists of two identical coaxial (regular) polygonal current loops. In the space between the loops, there is nearly homogeneous or nearly linear distribution of the magnetic field along the axes depending on the currents' direction. First, we derived a suitable, simple and general expression for the magnetic field along the axes due to a polygonal current loop. We emphasize the importance of the role of this expression for further analysis. The total on-axes magnetic field is the result of superposition of the magnetic fields that each loop generates separately. The proper distance between the loops and the current orientation make the system to become either Helmholtz coil or anti-Helmholtz coil. In this paper we give exact, analytical and general expression for this optimal distance that provides the magnetic field to be homogeneous (linear) as much as possible. We based our study on Taylor series expansion of the total magnetic field, demanding that the first contaminating term must be canceled, in both, symmetric and asymmetric case.