I was told this idea works for any sign of the source charge: in any potential, a positive test charge will always naturally move towards a lower potential and a negative test charge towards a higher potential.
I can kind of understand why a positive charge would move towards the region of lower potential (because don’t things in general want to do that? but if the source charge doesn’t matter, I’m just confused).
Can someone explain like I’m stupid why a negative charge would move towards higher potential?
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Potential isn’t a physical/measurable quantity. It’s a mathematical tool, so we can define it the way we want.
The way we WANT to define potential is that for lets say gravity the things have a force pointing from higher to lower potential, in other words the steepest decrease of potential.
For electrostatics we got two types of charge and we define potential for the positive case and the negative charge works the opposite. I show the math to see how the sign of the charge changes nothing:
So the field in electrostatics is the E=-grad(V) where V is the potential. Now grad is just a derivitive, we don’t need to worry about it its an operation just like + or ×. Now froce on a charge q is F=qE. So if you have V, then F=q(-grad(V)) so F = -q grad(V). So grad(V) is a vector that points in the steepest increase of potential, and the minus will turn it around to point towards the steepest decrease in potential and if q is q=-|q| so we got negative charge we have another minus which flips the force vector again to point in the steepest increase in potential.
Alternatively you can think like this: if you are a negative charge instead imagine yourself as a positive charge and add a minus sign to the potentials. So if potential is at point A 25 and at point B its 10, a positive charge would move from A to B but a negative change “sees” -25 and -10 and -25 is lower so it moves from B to A. But the math doesn’t require us to think too hard on the definition and how “motion happens in the direction of potential decrease.” Not necessarily the case, it’s not more than a definition.
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