I think your reasoning is wrong here. Looking at your Figure 2 I agree that you can use the equivalent current concept for a permanent magnet to deduce forces on the magnet. With the positive charge moving towards the rim you depict the force on the charge correctly, but the force on the magnet is not the induction force (acting on the charge carriers of the current) that you predict, at that point the force is zero. If you meant that to represent a rotational force on the magnet then that also is wrong. This is one of the paradoxes for charge moving towards (or away from) a line current. If your line current came from minus infinity and went to plus infinity then Newton’s Law of action and reaction would be broken. However for closed currents as you depict Newton’s Law holds, and in your case the forces on the current loop are maximum towards the top and the bottom in the counter direction that you show. The magnetic field ahead of the moving charge forms concentric circles around the velocity axis, and you can deduce the force on each current element from their intersection with the current using Fleming’s LH rule. They integrate to a net force on the whole circular current, i.e. a linear force on the magnet, not a rotational one. If you consider the field circles to be moving at the charge velocity, then the induction into each current element by Fleming’s RH rule is zero, the force there is radial. Note tor the charge moving away from the current loop in Figure 2 you have the force arrow on the moving charge in the wrong direction.

Smudge

I'm really glad someone is willing to have an intellectual discussion about this. And even more glad it's you Smudge, seeing your experience and knowledge on this topic.

I totally agree that this idea is riddled with paradoxes and controversies one of which you mentioned is the reaction force paradox. This is also related to the question of where the recoil of a railgun is located. And as research has shown this is at the back of the the railgun where the return path connection is made, not so in the rails. As the next thesis on that also concludes:

An Experimental Study of Electromagnetic Lorentz Force and Rail RecoilThe relevant conclusion:

combined with the results from the split rail Lorentz-recoil measurements (Figure 21), leads to the conclusion that there are not any internal stresses within the rails.

Or the "Railgun recoil, ampere tension, and the laws of electrodynamics" paper attached below which indeed shows no longitudinal force EVEN using the classical and forgotten Ampere force law (which imo is a good thing):

Again I quote:

The net longitudinal force on any element of the armature, and onthe armature as a while, is zero to with the accuracy of the calculation.

Another more recent paper (which cites the previous) also shows with Finite element modeling that this force is is mainly on the transverse part of the circuit rather longitudinal parts.

Comparative study on longitudinal momentum characteristics of L-type and I-type breech in railgunThe simulation results show that the longitudinal force on the breech is concentrated on the current injector plates. The longitudinal force on the power connector of L-type breech is much bigger than that of the I-type breech, up to nine times.

To me this makes total sense as well, even from an intuitive point of view without any equations. If you would replace the free flowing current with water in a U-shaped tube and apply pressure on both ends, the resulting pressure will emerge perpendicular to the surface causing a reaction force at the returning U path of the tube at the bottom. Moving the whole container down, it would be silly to think that the force would be longitudinally on the straight tube pieces cause them to crunch as I illustrate in

**Fig 3**.

So now that we have established that I can comment on your points. The thing is that the force equation aka the Lorentz force goes hand in hand with the Biot-Savart Law as you know, to be of any kind of use. This implies that you indeed can calculate the force acting on a free moving charge due to a magnetic field. But it's always assumed that this magnet field source comes from a closed current loop. The Biot-Savart Law breaks down the other way around, as a free moving charge is not a current carrying loop.

This is why, in my brain, I always go back to classical formulations such as the Ampere Force law (not the one about parallel current wires) or Weber's electrodynamics. That can handle charge on charge interactions in a Newtonian way i.e. along the connecting radial vector. This allows you to analyze interactions of isolated charged particles in an n-body simulation. I show this in

**fig. 4**. Here a free charge moving towards a "wire" will act on the current elements (depicted as red circles) in such a way that "opposes" the current, when it passes the wire the direction is reversed, and when combined we indeed see that the total force would be perpendicular to the "wire".

This is also the basis of this proposed idea/experiment. I have yet to come across any such experiment, and a magnet in a homopolar motor is not a current carrying wire either so this proposed current "inducing" force can only manifest as a mechanical force on a perpetually spinning electron I believe.

The key here is that the charge never crosses the edge/wire where everything is reversed including the force that just induced an EMF. In fact I intentionally left something wrong in fig 2 that you mentioned, but it's not the force direction as this should be in accordance with the left hand rule no? What I left out is the fact that when the charge moves back no matter the direction of the current loop, this proposed EMF will always be in the opposite direction. This is why the the current loop must have NO current when the charge needs to move back. It can only have current when the charge moves towards it. I see this as a square wave sigbak, where the signal is off when the charge moves back again.

I hope this will interest someone enough to do some basic experiments, all of this is deduced from first principles thinking. There is no magic Russian special sauce or UFO alien technology involved. This experiment can be performed with reasonable voltages, obviously the higher the better but regardless the induced voltage may be tiny.