On the interaction between a current density and a vector potential: Amp\`{e}re force, Helmholtz tension and Larmor torque

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 Amp\`{e}re'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 Amp\`{e}re'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 Amp\`{e}re's hairpin demonstration and the more modern ones of the Graneaus and of Saumont.