Analytical Expressions for the Magnetic Field Generated by a Circular Arc Filament Carrying a Direct Current

This paper presents a simple, didactical and compact solution for the components of the magnetic vector potential and magnetic field generated by a circular arc filament carrying a direct current using the well-known incomplete elliptic integrals, their properties and formulas. Using a well-defined method, the expressions are obtained in Cartesian coordinates and then easily transformed to a compact expression in cylindrical coordinates. The paper provides numerical data for different study cases where arbitrary arc filaments are considered. Through Gauss' magnetic law, special cases, and comparisons with numerical methods, a degree of validation is provided for the derived expressions.These solutions represent a computational advantage for magnetic field calculations for applications like wireless power transfer. Finally, new integral formulas are derived using elliptic integrals.