A magnetic field based on Ampère's force law

Ampère'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 Ampère's force law is a function of a double-differential of this field. We call this field the Ampère–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 Ampère–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 Ampère–Weber field is based on Ampère'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 Ampère–Weber field B, like Ampère's force law, is more of a curiosity than a viable alternative to the usual magnetic field B. If the Ampère–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.