>>16841335
Cont.
This is the fundamental 'superpotential' field of an irrotational vector potential, which has a singularity at the central axis of rotation that produces a nonzero B at the origin. Since B is zero everywhere else, χ is allowed to have a gradient everywhere else besides the origin.

What does this mean?
The χ field is a corkscrew of infinite width that winds around the z axis.

The infinite width is not a problem, it simply means that phenomena that depend on the path around the flux do not depend on distance from it.

One example is the Aharonov–Bohm effect, where an electron traveling around a long thin solenoid picks up a phase factor that depends on the magnetic flux inside the solenoid, but not distance from it. If this solenoid were bent into a closed toroid so that all flux were absolutely confined inside, the effect would still exist.

Another example is a loop of wire wound around a ferromagnetic rod in which there is a changing magnetic field. The electromotive force induced by the changing magnetic flux is independent of the diameter of the loop. If the flux were completely confined inside a toroidal core, it would still produce the same electromotive force.

That is because the electron isn’t actually experiencing the flux itself, but rather the corkscrew superpotential surrounding the flux lines.

A changing flux creates a changing gradient in the superpotential, and an electron in that path will be pumped along the gradient. Stated another way, a changing gradient generates an electric field, which places a force on the electron as expected.