Appendix 3 - Vortex rotations in a curl-free region

Summary

Figures 4.8 and 4.9a depict schematically the vortex rotations and idle-wheel translations associated with a uniform current density inside of a long, straight wire with uniform circular cross section. Inside the wire, the magnetic field grows linearly with distance from the axis; because of this, neighboring vortices rotate with different angular velocities; this engenders motion of the idle-wheel particles interposed between the vortices, constituting a nonzero current density J; and the inhomogeneity of the magnetic field H is associated with a nonzero value for curl H, which is equal to the nonzero current density J.

Outside of the wire, the H field falls off as 1/r, where r is the distance from the axis [E. R. Peck, Electricity and Magnetism (New York: McGraw-Hill, 1953), 214–17]. One might, at first thought, expect that because of this, neighboring vortices would rotate with different angular velocities, and this would engender motion of the idle-wheel particles, constituting a nonzero current density J. Even though the magnetic field H is inhomogeneous, however, curl H and hence curl ω* are zero outside the wire, and Maxwell's calculation leading to equation (3.7a) shows that in this situation there will be no net flux of the idle-wheel particles, and hence no current ι or J.