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When you say that "magnetic current" doesn't move - through a circuit - does this have any correlation to the flux lines around a conductor? These loops are shown with directional arrows, but this might indicate an impetus rather than rotation. I have a side project with a magnetic funnel antenna, and I'd like to be able to predict if the funnel is rotating.
Yes, there is a complete connection.
It is the name "flux" that is misleading. We think of the flux or flow of water when it is a purely mathematical concept, which also applies in statics, it is "the surface integral of the perpendicular component of a vector field over a surface".
In our case, this surface is the section of a magnetic circuit, therefore transverse. And this flux is conservative, we will always have the same flux whatever the surface which will cut the magnetic circuit, except of course if there are leaks, but then by widening the surface to "capture" the leaks, we will find again this flux constancy.
I think the best method is to reason with the image of the field lines. When an electron moves, it creates a magnetic field whose field lines (equipotentials) surround it in closed loops, in a plane perpendicular to its speed vector. If the electrons form a current, these loops encircle the conductor. If the velocity increases, we say that the number of lines increases. If the conductor passes near a ferromagnetic core, these loops will tend to concentrate inside, and very little in the air. It is the number of these closed lines that can be seen as the intensity of the flux. These lines are unbreakable, without beginning or end, never cross each other, and nothing moves along them.
To answer your question, the arrows only indicate the orientation of the field to know where north and south are, which tells us the direction of the velocity vector of the electrons that generate it. The only conventional force I know and that could rotate a funnel is the Lorentz force F=q.VxB.
In fact there is no mystery around all this when we reason by relativity. A charge seen in motion by an observer has its electric field compressed in the direction of the motion because of the contraction of the lengths. So its coulombic field is no longer seen isotropic around the charge. It is this anisotropy that causes the magnetic effects. The magnetic forces are only due to the electric forces of the coulombic field of the charge, deformed because of its motion.
The electric field, coulombic, is the primordial reality, and the relativity applied to the coulombic field of the moving charge is enough to explain all magnetism. Maxwell's equations + the Lorentz force, to which electromagnetism is generally summarized, are only a calculation facility, but which blurs the mental representation, paradoxically simpler, that one can make of the phenomenon thanks to relativity.