Again with the rotor bars

I had a big thing going last year about the rotor bars in an induction motor, and I still can't figure it out. But once in a while someone comes along and posts a comment which purports to set everything straight, as one of my readers did yesterday on this post. John wants me to analyze the motor as magnet-on-magnet action instead of rotor bars in a magnetic field.

It's a good theory but I don't buy it. John doesn't deal with the essential problem: how do you calculate  the torque? The problem is that ILxB actually gives you pretty much the right answer, but fails to answer the question I asked with this picture:

how do you get torque if the field lines curve away from the rotor bars, avoiding the copper almost entirely?

I was tempted to answer this question by suggesting that the field lines exert a torque not on the copper bars, but on the atomic-level circulating currents in the magnetic domains of the iron. We know that iron can amplify a magnetic field up to a thousand times; this can only mean that the atomic-level circulating currents on the surface of the iron are a thousand times greater than the macroscopic copper-wound currents that we can see. Does this explain the torque of a motor?

I don't think so. My problem with this explanation is all of a sudden, there's way too much torque. Because the standard macroscopic analysis gets me in the right ballpark. I don't want a correction that makes the torque a thousand times bigger. The problem with the standard analysis isn't the result...it's that nagging question: don't the magnetic field lines curve away from the copper to stay inside the iron?

I want to show what I mean when I say the standard calculation gives the right answer (assuming the field passes through the copper bars, so I put some dimensions on my picture. With a 3-cm radius, the rotor shown here would look OK in an ordinary 1/2HP motor. It's not hard to see that there are about 10 square centimeters of rotor bar cross section, which would typically sustain a current of around 2000 Amps. A good strong magnetic field achievable with laminated iron would be around 1 Tesla (that's volt-seconds-per-meter-squared). Is this enough to calculate the torque? Just about...I still need to give you the length of the rotor bars. Let's call it 12 centimeters:

 (2000 Amps)(0.12 meters)(1 volt-sec/square meter) = 240 Newtons

To get the torque, we multiply by the radius (3 cm) which gives a torque of about 7 N-m. Now horsepower is torque times frequency (in radians per sec.) Remember that 3600 RPM is 377 radians per second...except the typical motor is four-pole, so you only get half of that...and also, the utilization factor of the copper bars is certainly under 50% with the sinusoidally varying currents and the phase shifts. Lumping it all together I get:

 (7.2 Newton-meters)(1/2 * 377 radians/sec)(1/2) = 650 Watts

which is one horsepower...just about where we want it.

Except for one problem: the magnetic field inside the copper bars is very low. Look at the picture again. The field goes into the iron. I could try and fix this by analyzing the atomic-level currents in the magnetic domanis (1000x greater than the copper bar current) that would really mess up my calculation. So I still really don't know what to do.

The irony is that I think I'm going to explain the great paradoxes in quantum mechanics but I still can't seem to figure out how a motor works.