Why does the rotor need slots?

I wrote the other day about the importance of the air gap in an induction motor. It seems after all the rest of the world doesn't share my concern about this parameter. I looked up and down the internet and the unanimous opinion is that the air gap just has to be big enough to provide mechanical clearance for the rotor.

I found some good pictures of typical rotor stampings on this website, so we can see what we are talking about. Here is a good one:

You can see that the linear profile is about fifty percent copper and fifty percent iron. You want iron to carry the magnetic field, and you want lots of copper to provide a path for current to flow. (The fifty-fifty compromise is not that uncommon in these engineering situations, and not only because people are too lazy to work out the true optimum. But that's another story.)

But I'm still not convinced that this is entirely right. As I explained in my last article, if you want torque then the rotor current has to flow in a region where there is a strong magnetic field. What good is the magnetic field if it's inside the iron? It has to cut across the copper bars, and I can't for the life of me see why that should happen in this typical rotor configuration.

You have to understand something about what we call the magnetic permeability of iron. As compared to all ordinary materials, including in this case such things as air and copper, the relative permeability of magnetic iron is on the order of 1000:1. That means, in anthropomorphic terms, that a line of magnetic flux would just as rather pass through 1000 millimeters of iron rather than jump a gap of 1 millimeter through air or copper. If you'll forgive my shaky graphics, we can see what this means for this particular rotor in a typical magnetic field situation:

The field lines will do just about anything to stay inside the iron. So what good is all that copper if it's not in the magnetic field? I just don't get it.

Obviously people have been building motors that work for 100 years. So maybe I'm missing something here. I just can't see what's wrong with my analysis.

According to my theory, I wouldn't even have rotor slots. My rotor would be a solid iron cylinder, and my "squirrel cage" would be a smooth copper coating about 3 millimeters thick all around the suface of the rotor cylinder. That way the magnetic field lines would have to  cut through the copper to complete their circuit. Yes, it would mean the motor has lower inductance, which means it would need a higher magnetising current. But it does no good to design for low magnetising current if the tradeoff is that your magnetic field manages to avoid the rotor bars.

So that's my theory. You tell me what's wrong with it.