My physics knowledge is quite poor, but I've always wondered if it was possible to use the rotation of the earth as some kind of alternator in conjunction with an orbital station. Is that idea fundamentally ignorant? Or is it something that has been toyed with before?
It's not an answer to your question but there are indeed ways to utilize interaction of Earth's magnetic field with spacecrafts.
Power production (or thrust if reversing the process by feeding a current into the wire) is described in [0].
Use for stabilization and attitude control is described in [1]. These might be combined with [2], which have more immediate effect but from time to time need to be 'desaturated' which is where [1] comes in again.
Well, you need a rotor and a stator. Alternators work by transferring energy to the stator, which is extracted as an electrical current thanks to it being held in place.
You cannot really "hold" something in place (barring an external propulsion system that costs energy), so while you could extract some energy at first due to inertia, you will slow the orbiter down and barely collect more (if any) than you needed to put it there in the first place.
An alternative but close system was described in Stephen Baxter's Sapce (Manifold series), with (IIRC) superconducting loops around Io, to collect energy from Jupiter's magnetic field, technically extracting energy from Io's momentum.
