An approach for calculation of the magnetic force arising between two electric current-carrying filaments having a circular and closed-curve of arbitrary shape is developed. The developed approach is based on the recently formulated segmentation method applied for the calculation of the mutual induction for a similar filament system. Employing the fact that any curve can be interpolated by a set of line segments with the desired accuracy and deriving the set of formulas for calculating of the magnetic force between a circular filament and line segment, the developed approach was also successfully applied for the estimation of the distribution of magnetic force along the closed-curve in addition to the resulting one. As illustrative examples, the calculation of the magnetic force and its distribution between the circular filament and the following closed-curves such as polygons, circles and a 3D curve was efficiently performed by using the developed approach. Also, the developed method was applied for the calculation of the resultant magnetic force between the rigid bodies including permanent magnets and current-carrying coils. The results of calculation were validated successfully by using FEM method and the analytical formulas available in the literature.

The sets of expressions for calculation of sixteen components of magnetic stiffness over torque between two current-carrying circular filaments arbitrarily positioning in the space are derived by using mutual inductance and Grover's method. The expressions are presented in the analytical form via integral equations, whose kernels include the elliptic functions of first and second kinds. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.