What good are the rotor slots?

Last week I argued that the normal design of rotor slots in an induction motor seems to guarantee that the magnetic field will avoid passing through the current-carrying rotor bars. Here is the picture I drew:

If the field bypasses the rotor bars, then where is the force to turn the motor? And yet they've been building motors this way for a hundred years. So I've been racking my brains trying to figure out what's wrong with my analysis. Here are some of the thoughts I've had:

1. Maybe I'm not taking into account the total magnetic field. The lines I've drawn are intended to show the field due to the magnetising current in the stator. What about the load current in the stator, and the induced currents in the rotor? Don't they generate their own fields, therefore nullifying the significance of the simple drawing I've shown here?

I don't think so. First of all, we don't like to draw the fields from the rotor currents, because a conductor feels a force mainly due to all the other fields, not its own field. But then, what about the fields due to the load currents flowing in the stator? That's a little trickier to answer, but we can get a pretty good idea by looking at the ideal case where the air gap goes to zero and there is no leakage flux. In that case, we have an ideal transformer, and the magnetic fields created by the secondary currents are exactly nullified by the additional currents that flow on the primary side. Once again, we return to the situation where the only magnetic field we need to consider is the original field due to the magnetising current in the stator. Just as I've drawn.

2. Maybe you want to argue that the case of no leakague flux becomes degenerate, and the motor doesn't work. I don't buy that. Analyzed as a transformer, the case of no leakage corresponds to the perfect transformer, in which case the rotor currents are purely resistive. I already showed that the case of pure resistive current corresponds to the ideal magnet configuration, with the stator field at right angles to the rotor field. If that doesn't generate torque, what does?

3. What about the saturation of the magnetic iron? Doesn't this change the field path?

I don't think so. I hate to go out on a limb, but I don't think we need to worry about that. After all, we can run a motor on half voltage, keeping it well within the magnetic limits of the iron, and I don't think the performance is all that different, qualitatively speaking, once you account for the lower power levels.

4. What about analyzing the forces between the magnetised rotor vs the magnetised stator, instead of trying to analyze the forces between the stator field and the rotor currents? Maybe we need to look on it as magnet-on-magnet?

I don't think so, but I'm on shaky ground here. I think the magnet-on-magnet analysis is just a different version of the same physics. I really don't think it's a whole nother set of forces in addition to the forces on the conductors. It's just really hard for me to believe that the total force can't be analyzed just by looking at the force on the rotor bars. Maybe you can analyze it by looking at the magnets instead,  but I'll be really surprised if it turns out you have to look at both sets of forces. In any case, it still seems really wrong that there are no forces to speak of on the rotor bars.

5. Doesn't all the flux have to cross through the rotor bars anyways? This is a scary argument that keeps pushing its way into my mind. The stator field is moving a little faster than the rotor, so those lines of force do in fact sweep through the rotor bars as the rotor orientations shifts backwards relative to the stator field. Yes, the field lines crowd into the iron salients, but then they snap across the gaps very quickly. Is there some additional component of magnetic force related to the velocity of the field lines in addition to their density?

It's a tempting argument but I just can't buy it. The force on a conductor has got to be based on the current and the field, and nothing else. There's no additional term in the equations which brings in the velocity of the field lines. Not that I can see.

And that's just about it. You can see I've been turning it over and over in my mind, and I just can't see how those rotor slots are designed to acheive any kind of torque in the motor. As I said the other day, my idea would have been to have a solid cylindrical rotor coated with a thin layer of copper conductor. So the magnetic field lines would have to cut through the copper to complete their circuit from pole to pole through the magnetic iron. I still can't see what's wrong with my theory.

We have to ask the question: if I can't figure out how a motor works, how am I going to figure out the quantum structure of the helium atom? That's a tough question, and I'm the first to admit it.