Forces in interacting ferromagnetic conductors subjected to electrical currents and magnetic fields

We revisit the classical problem of Lorentz forces exerted on conductors subjected simultaneously to electrical currents and external magnetic fields within the framework of magnetostatics, i.e. when all field quantities are time-independent. In contrast to the well-known results pertaining to non-magnetic materials, we consider here ferromagnetic materials and study the influence of the magnetic constitutive law on the forces exerted on these conductors. Following the general setting for the coupled magnetoelastic boundary value problem in three dimensions (Lagrangian and Eulerian descriptions), we restrict attention to the two-dimensional problem of a single, two or many interacting parallel conductors of infinite extent and circular sections. Both analytical and numerical (FEM) results are presented. For a single conductor, where the magnetic properties do not influence the force exerted, we calculate the magnetization and magnetic stress fields; analytically for the linear magnetic response and numerically for the general nonlinear case with saturation. For two parallel conductors, the magnetic properties affect significantly the Lorentz forces when the conductors are placed close to each other, as the magnetic fields outside them are strongly influenced by the conductors' magnetic response. For the case of an infinite array of parallel conductors, there is no influence of their magnetic properties on the Lorentz forces when same direction currents are applied, while only a small magnetic effect is found for currents applied in alternating directions, even for closely spaced conductors.