A new formula for calculating the magnetic force between two coaxial thick circular coils with rectangular cross-section

The magnetic force exerted by an array of two coaxial thick circular coils with rectangular cross-sections in air is important to both electrical and mechanical engineering applications. The magnetic force is typically calculated by taking the integral over the entire space defined by the array. This calculation even for this simple array is an intractable problem and numerical methods have been extensively used. In this work, the integration was subdivided into five regions, and in four of them, an analytical formula was found. The method proposed here is based on the Green’s function of the free space that leads to the elliptical integral of the first and second kind. The formula reveals new insights into how the geometry and relative positioning of the coils within the array determines the strength of the magnetic force. The thicker the coils are and the farther apart they are, the weaker the magnetic force is, and vice versa. This new formula is simpler and practically free of truncation errors, which are commonly encountered in numerical approximations. Several examples from the literature were used to corroborate the present formulation. The results show an excellent agreement with respect to the different numerical and semi-analytical approaches used by other authors.