A magnetic field based on Ampere's force law

Ampere's force law for steady currents was not historically associated with a magnetic field, but it could have been. A magnetic field, inspired by work of Helmholtz in 1870, can be defined such that the double-differential form of Ampere's force law is a function of a double-differential of this field. We call this field the Ampere-Weber field, B, and show that its divergence is zero everywhere, as is that of the usual, but different, magnetic field B of Maxwellian electrodynamics. The curl of the Ampere-Weber field is nonzero everywhere in static examples, in contrast to that of the usual magnetic field B. We illustrate the field B for three examples, which exhibit patterns of field lines quite different from those of the usual magnetic field. As the Ampere-Weber field is based on Ampere's force law for steady currents, it does not extrapolate well to the Lorentz force on a moving charge in a magnetic field. That is, the Ampere-Weber field B, like Ampere's force law, is more of a curiosity than a viable alternative to the usual magnetic field B. If the Ampere-Weber field had been invented in the mid-1800s, it would have been a distraction more than a step toward a generally valid electromagnetic field theory. (C) 2023 Published under an exclusive license by American Association of Physics Teachers.