Okay, i can see your point. But what about direction dependent mu? We raise H in the big mu direction, then rotate it by 90 degrees, then lower H. On the way up, we get square curve, on the way down we get a line curve, so it must have gained energy. Thats assuming very large difference in mu depending on direction.

If we have a rotating uniform field,

and a permanent magnet lagging behind the angle,

the d(Flux)/dt through the area of the magnet will be constantly reduced,

That means that E.j_b is not zero i.e. the E field is doing work on the bound current. But the bound current doesn't get affected by the work therefore there is net energy balance, direction dependent on relative direction of field and magnet. If it were a wire the rotor current would reduce, conserving energy. But the bound current doesn't.

More information is needed to understand what you are describing.

Yes, it is the synchronous motor with much larger coils. But it seems my mind gets confused by the fact that the Poynting theorem makes the work equal to E.J, without including the work done by the magnetic field when it creates an EMF.

Since dB/dT passes through the poles rather than being at right angles, the idea was that work is being done on the bound current by the electric field.

And that the magnet will not demagnetize as a result.

The magnet and B -field rotate at the same rate.

Uh, maybe I really don't know what I was talking about.

I have made a logical error in the previous case. Now how about this:(see attachment)