New analytical formulas for calculating magnetic field

Simple analytical formulas have been derived for calculating the magnetic field generated by a very general and versatile uniform current-density finite element consisting of a rectangular-cross-section prism terminated at each end by an arbitrary oblique end-plane. These formulas enable the user to calculate efficiently the exact magnetic field generated by any system built of straight, rectangular-cross-section, uniform-current-density conductors. The formulas may also be used to calculate the magnetic field generated by any system containing curved elements such as circular coils, helixes, toruses, torsatrons, saddle coils, and yin-yang coils by approximating the curved elements by concatenated short, straight, trapezoidal segments. The accuracy and simplicity of the formulas have been verified using MACSYMA, the symbolic manipulator developed at the Massachusetts Institute of Technology. Using MACSYMA to successively differentiate and simplify the formulas, the authors have rederived all of the published simpler special cases, often in more compact or more symmetric equivalent forms. The accuracy of the new formulas for curved geometries has been verified by comparison with existing codes for circular coils.