Unified Analytical and Numerical Evaluation of Axial Magnetic Force in Coaxial Air-Core Coils

: Accurate prediction of magnetic interaction forces is important for electromagnetic actuators, inductive coupling systems, and calibration devices. This paper presents a unified analytical and numerical framework for evaluating the axial magnetic force between two finite-dimensional, perfectly coaxial, air-core cylindrical coils under steady currents. The model assumes uniform purely azimuthal current density and neglects radial and axial current components, winding-pitch effects, magnetic materials, misalignment, and transient phenomena. Starting from the Biot-Savart law and Lorentz force formulation, the coil-coil interaction integralisderived and reduced using cylindrical symmetry, leaving only the axial resultant force. Three complementary methods are developed in MATLAB: a semi-analytical elliptic-integral formulation, a direct trapezoidal numerical-integration method, and a filament-based mutual-inductance method. The methods are computationally benchmarked for representative thin-wall, moderate finite-radius, mixed-radius, and large finite-radius coil geometries. The results show consistent force predictions, with relative half-spread values below approximately 4% for the cases considered. Discretization sensitivity and error-source analysis are included to clarify numerical accuracy and convergence. The proposed framework provides a transparent benchmark for axial force evaluation in idealized coaxial air-core coil systems.