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But increased electron flow WOULD be greater current.
Given a fixed current, the only way voltage can increase is if resistance increases (Ohms Law). So the same magnetic field results, because there are the same number of army men (electrons) moving through the jungle (wire), only now they are being pushed harder by a really mean general (higher voltage) through a nastier bog (higher resistance).
So, their speed doesn't increase because of the higher resistance.
By the way, Ben, that's a great idea! I'm going to try that.
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