Vector potential and magnetic field of current-carrying finite arc segment in analytical form, Part I: Filament approximation

Analytical expressions for the components of the vector potential and magnetic field of a current-carrying circular arc filament of arbitrary length are derived. Components of the field gradient are also calculated. All the expressions developed consist of only known functions such as Jacobian elliptic functions, complete and incomplete elliptic integrals of the first and second kind, and thus permit a new time-saving efficient computation algorithm.