Equivalence of the Ampere and Biot-Savart force laws in magnetostatics

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Abstract
The equivalence of the Ampere and Biot-Savart force laws in magnetostatics is examined by considering the difference in the forces predicted by the two laws. The conditions under which this force difference vanishes are discussed, with special reference to the question of convergence of the integrals involved. The conclusions are that a complete current distribution exerts forces, which are the same according to both laws, when acting on a volume current element or a macroscopic part of a current distribution, irrespective of whether these are situated outside or inside the distribution exerting the force. These forces are shown to be normal to the current density vector at all points according to both force laws. Both the force distributions and stresses are predicted equal by both laws, thus proving the impossibility of devising an experiment which might differentiate between the two laws by measurements of forces exerted on a part of a circuit by the circuit itself. The equivalence of the two force laws in magnetostatistics is considered to be complete.