On the interaction between a current density and a vector potential: Ampère force, Helmholtz tension and Larmor torque

Abstract

Several mathematical formulae are used nowadays in order to compute a magnetic torque. We demonstrate that its more general expression is the vectorial product of the current density with the vector potential. We associate this Larmor's torque with Ampère's force and more specifically with Helmholtz mechanical tension, which is at the origin of the longitudinal stresses in "open" circuits carrying current. We show that Ampère's force enters into the realm of Newtonian Electrodynamics and we explain the absence of contradiction with special relativity. Hence, we provide for the first time a theoretical basis for the numerous experiments, which claimed to have demonstrated the existence of the longitudinal mechanical tension starting with the historical Ampère's hairpin demonstration and the more modern ones of the Graneaus and of Saumont.