I'm quite new to physics and I don't totally understand the concepts of potential energy and work. I've been watching MIT courses and I got pretty confused, considering the following situation:

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We have 2 charges: +4 and -1. At the top there is a point, where the eletric field is 0, which means there will be no force on a test charge that we place here, consequently it will gain no velocity and it will stay in place. But the electric potential at this point is not zero, moreover it becomes smaller, when we go above that point (in this picture). Charges tend to move from higher potential to lower potential, so according to this statement, the charge should be willing to go up. But there is no force that could push it. I'm pretty sure, I'm wrong at some point, so I would like you to explain this situation in details.


Think about a ball sitting right at the apex of a hill. Is the gravitational potential energy at the top of the hill larger than the gravitational potential energy at the bottom of the hill? Yes, it is. Will the ball suddenly start rolling down the hill because of this point? No, not unless something gives it an initial push.

This is the point in your studies where you need to replace these adages such as "charges move from high to low potential energy" with some mathematical reasoning. Indeed, just as how the force $\mathbf F$ relates to the potential energy $U$ by $$\mathbf F=-\nabla U$$ so the electric field $\mathbf E$ is related to the potential $V$ by $$\mathbf E=-\nabla V$$

These equations give us the "movement from higher to lower potential energy". Forces point in the direction of decreasing potential energy. However, since you have found a point in space where $\mathbf E=0$, you have found a point in space where $\nabla V=0$. The charge does not move from this point, and it doesn't matter if there exist points of lower potential energy at other places. This is exactly what we reasoned through with the ball on the hill.


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