Analytical Lorentz Force Model Between 1-D Linear Currents in Arbitrary Relative Positions and Directions

We derive an analytical formula for the Lorentz force and torque between 1-D linear current elements in arbitrary relative positions explicitly without remaining integral. A circuit is represented as the links of linear elements, and the force and torque on an element are represented as equivalent nodal forces on either side of it. We then treat the circuit as a link mechanism. Furthermore, we consider the singularity that appears in connected elements in a single circuit due to zero distance and avoid it using an approximation with a small radius. This singularity treatment makes the formula applicable even for calculation of the partial force from the entire circuit. Since the formula does not require numerical integration, it can reduce computation time. The accuracy of the approximation is also evaluated by comparing our calculations with the existing experimental results in the literature and numerical integrations.