Magnet cores with high permeability but no ferromagnetism? I have a lab exercise I've been doing with my students for years, in which they suspend two steel balls at unequal heights from steel-core electromagnets, then break the circuit and drop the balls. By timing the sound of the hits with a microphone, we get a pretty high precision value of g. The apparatus is homemade.
Recently one of my lab groups cranked the power supply all the way up on their setup. The power supply got very hot, and I had them turn it down. Their results were not very good, and later when I got a chance to debug the setup a little more carefully I found that the problem seemed to be permanent magnetization of the steel cores. By reversing the current, I was able to demagnetize the cores pretty well (as measured by a Hall effect probe), and then the results became good again.
The workaround for now is to not use too much current, and put some masking tape over the knobs on the power supplies so that students won't be tempted to mess with them.
But I'm wondering if there is a better way to design this. I would have thought that the identical magnets would have been affected by the same amount by the permanent magnetization effect, so that the time difference between the balls would not be affected, but apparently the effects are unequal enough to matter.
Would it make sense to replace the steel-core magnets with magnets having cores made out of a material that has a high permeability but no ferromagnetism? Is there such a substance? If it was cheap and common, I would have thought I'd have seen it before.
 A: What did you use for the cores of the electromagnets?  Typical carbon steel gets magnetically hard and mechanically hard with carbon, so if you wound wire around bolts then you kinda did yourself in.
Silicon steel isn't as strong as carbon steel, but it stays much softer, magnetically.  It's what transformer and motor laminations are made out of.  It still exhibits some ferromagnatism, but not as much.  I'm not sure where you'd find it, though, except by taking transformers apart, or special-ordering a sheet of the stuff.
You could probably use ferrite rod, if you can find some that matches the diameter of your electromagnet cores.  This is the stuff that AM radio "loopstick" antennas are made out of.  You want to use a material that's made as a core material for inductors -- those are designed for high permeability and low remanance (because the remanance takes energy out of the circuit at each AC cycle, which is not a happy thing in an inductor).
If you make new magnets, take note of the fact that for a given size of coil, the field generated vs. the power dissipated in the coil is pretty constant -- but the voltage needed to generate that power goes up (and the current goes down) as the length of the wire.  So you can rewind your coils with more turns of finer wire and have them work with the power supply set on "high", and not have stifle your students' curiosity to save the apparatus.
