Complementary uses of the magnetic vector potential for the understanding and teaching of electromagnetics

This work reports on the uses of the magnetic vector potential A for understanding, visualizing and teaching electromagnetic phenomena. The teaching benefits of this work include examples of numerical modeling that facilitate understanding, and complemenatary visualization approaches. To facilitate discussion of A we first include a brief section looking at ϕ in a system comprised of two hollow, positively charged spheres, demonstrating that the work required to bring the two spheres close together Q ϕ is equal to change in the total integrated electric field energy density. We then introduce a system comprised of a hollow, positively charged sphere interacting with two parallel, current-carrying wires, for which A wires is determined as a function of position. Two calculations of the field momentum of the system are compared and shown to be equal: the first is Q A wires , the second is the integration over all space of the interaction momentum density ϵ 0 E × B . We conclude by analyzing the trajectory of Q near the wires, establishing a visual connection between particle motion and the magnetic vector potentials sourced by the wires, including a discussion of relativistic implications.