# eli5: why does a negative charge accelerate towards the region of high electric potential?

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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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A positive point charge moves towards lower potential. A negative point charge is the opposite of a positive charge, so it moves away from lower potential. The opposite of lower potential is higher potential, so a negative charge moves towards higher potential.

It’s not that things “want to move towards lower potential”, it’s that things want to move in a certain direction, and we define “potential” based on that direction. Electric charges move in two different directions, so we just had to choose one direction to call low and one to call high.

Things can have positive or negative electric charge, but the “high/low” phrasing brings to mind gravity, and things can only have positive mass (which you might think of as “gravitational charge”).

But you can think of something like a helium balloon, which is lighter than surrounding air, as kind-of-sort-of behaving like it has negative “gravitational charge,” and it rises, that is, moves towards higher gravitational potential.

I suspect that the word “potential” is messing you up. It’s really not important. What IS important is that positively charged things tend to move in the direction of negatively charged things, and vice versa. If particle A is less positively charged than particle B, then they will be attracted to each other, even though both are “positive”. It’s all relative.

When we talk about electric field potential, it’s just describing the spatial distribution of charges. More positive (less negative) thisaway, more negative (less positive) thataway.

This question is philosophy rather than physics.

There are no objective ways to determine whether the potential is high or low, or whether the charge is positive or negative. These are what philosophers call “useful fiction”. What you can see, at the end of the day, is that certain charge move in a certain way under the influences of other charges. If you assign certain charge a value of positive, some other charge negative values, and write out an equation for the electric potential based on them, you can find the correct way these things moves. That’s why it’s useful.

But the choice of assigning positive or negative charge is arbitrary. It’s a human invention. Assigning values at all to charge allows us to write simpler theory than not doing it, but there are multiple valid way of doing the assignments. Hence it’s “fiction”. For example, you can rewrite the entire Newtonian physics without even talking about force at all; but it’s quite inconvenient to study it without using the concept of force. If someone asked on ELI5 “why greater force causes greater acceleration” the answer would be similar “force is an useful fiction; that’s how we define the concept of force”.

>the source charge doesn’t matter

Here is something I can say that’s part of physics. The source charge do not have instant effect on the test charge: the limit is the speed of light. You can confirm this without even looking at Einstein’s special relativity: even from Maxwell’s equation, the effect of the source charge only move at the speed of light (in fact, this issue is what lead people to realize that either Maxwell is wrong or Newton is wrong, and special relativity confirmed that it’s Newton). Therefore, from modern perspective, we do not consider the source charge to act directly on the test charge. What happened is that there is an electromagnetic (EM) field that permeate the entire universe, and other charge particles interact with this EM field: the EM field cause acceleration on the charged particle, but the charged particle also cause ripple in the EM field. In this formulation, we were able to explain the delay in the effect of the source charge on test charge.

(note that the test charge also distort the EM field; that’s why people define test charge to be an ideal small charge: the charge is too small to cause any changes to the EM field, but big enough for it to be affected by the EM field; this is just a non-mathematical way of talking about order of approximation)

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.