The warning here is that I don't really know what I'm talking about - I'm doing this here in hopes that Will or Sam will be able to correct me.

Electromagnets are limited by the amount of copper and iron available in the space that you want to fill with electromagnet. There is an ideal balance between the two: you want to fill all of the space, while balancing the amount of coil with the amount of iron core such that the coil produces just enough magnetic field to ['saturate'](https://en.wikipedia.org/wiki/Saturation_(magnetic)) the core.

Iron cores can be ballpark-saturated at about 2 Teslas - this is a measure of 'flux density' - for reference, most Neodymium Magnets have a flux density of 1.2 - 1.4 T. 

The magnetic field produced through the center of a coil is proportional to the number of turns (N) in the coil and the current (I) running through them. The tradeoff there is that to get more turns, we need thinner wire, and then longer wire, such that the voltage we need to drive the same current becomes large. This is something that is easy to spec in a spreadsheet - and you'll see that for the same total power flowing through the conductor, we generate the same field strength no matter whether we choose high voltage / high turns / low amperage ~ or ~ low voltage / less turns / more amperage (in the same amount of coil space). 

[Wikipedia is great for this](https://en.wikipedia.org/wiki/Electromagnet) 

The field strength $`H = \frac{IN}{L}`$ where H is the magnetic field strength, I is amperage, N is the number of turns, and L is the length of the coil.

The strength of the field is related to the Flux Density (T) in the coil by some mystic powers that I have yet to understand, so here's kind of where I have to stop. Intuitively, from my time in COMSOL, about a 1/3 relation is good: 2 parts iron for 1 part coil, given the space. Then you can just tune the diameter of your wire to determine the number of loops you'll have, and adjust given the driving voltage you can generate. 12V is friendly and available in PC power supplies, etc.

As for motor design and electromagnet design, the best way to think of generating force is that 'force is exerted in whichever direction decreases the energy in the system' - so if you imagine magnetic field lines coming out of your coil, they want to find a low-stress path to completing their loops. We can use permanent magnets to create complimentary field lines that will 'want' (pardon Aristotelian definition) to align with the field you're generating with the electromagnet. Critical to this is that a 'return path' is included on the back of the magnet ring: some iron that lets the magnetic field lines return through the other side.

... tbd