# Explanation between the potential energy of a charge and the electric force experienced by a charge

Suppose there are two opposite charges and an infinite distance between the two. At this initial point, the electric potential energy of the two charges and the electric force experienced by each charge is zero. The two charges then move along the line connecting them and the distance between the two becomes small. At this final point, the electric potential energy of each of the charges is smaller compared to the initial point, but the electric force experienced by each of the charges is stronger.

Suppose we look at one of the charges. As that charge moves towards the other, the force that it experiences becomes stronger. Wouldn't that mean that the charge itself could do more "stuffs" (like it can push more because it is subjected to a greater magnitude of force) and hence have a higher "potential" (potential here means being able to and not the electric potential one)? If that charge has a higher "potential," why is it then that charge has lower electric potential energy?

This idea is obvious when I look at the equations, but intuitively, why is it that? Additionally, an explanation from the perspective of the charge moving along the electric field and how it affects the energy of the charge would be great. Furthermore, if I have a poor understanding of anything of what I've said, please correct me.

As that charge moves towards the other, the force that it experiences becomes stronger. Wouldn't that mean that the charge itself could do more "stuffs"

Your issue is that "potential energy" is not "the ability to do more 'stuffs'", or at least there isn't an objective way to understand this statement.

Potential energy has a precise definition. More specifically, by definition, if the conservative force has done work $$W$$ then the change in potential energy $$\Delta U$$ is equal to the negative of the work done: $$\Delta U=-W$$.

In this case, an increase in potential energy is associated with the "potential" for that energy to be converted into other forms. For example, if we just have the electrostatic force acting on the charges, there is "potential" for the charges to lose potential energy in order gain kinetic energy as the electrostatic force does work on the charges. The "potential" is not found in the larger force, "bigger" possible collisions if a collision happened now, etc.

Your intuition fails because the charges are opposite, and so they naturally attract each other. There is no potential for the charges to "push" anything as they are always pulling towards each other. This sort of thinking only works if the charges are like, so that they actually "push" each other.

Instead, a better intuition for potential is that the potential always decreases if you "un-pause" the system and let it evolve naturally. In this case, the potential of the initial state of two infinitely distant opposite charges is zero, and as the two charges naturally attract each other the potential will continuously decrease.

I should clarify that this intuition only applies if the interaction forces themselves are conservative. If nonconservative forces are involved (friction, magnetism ...) then no scalar potential can be sensibly defined.

You should start with defining the system you are dealing with which enables you to differentiate with internal and external forces.
In this case you seem to have defined the system as the two charges with no external forces acting on the system.

Now there are two ways of dealing with the system consisting of two charges.
The first is in terms of energy.
When the two charges of opposite sign get closer together the electric potential energy of the system decreases and the total kinetic energy of the two charges increases at the same rate.

The other way in terms of work done (force $$\times$$ displacement) by internal forces within the system and equating that to the increases in kinetic energy of the charges.

From this you will note that the change in potential energy is related to the work done by the internal forces as the resulting change in kinetic energy is the same for both approaches.
Thus for a given displacement when the forces (related to electric field strength) are large the change in potential energy is large.

However you then wrote, suppose we look at one of the charges, which implies that the system you have chosen was one charge.
In that case you cannot use the concept of potential energy for which a minimum of two charges is required.
The electric field produced by the other charge external to the system will exert an external force on the system which consists of one charge and do work on the system which will result in an increase in the kinetic energy of the system.

At infinity total energy is 0. 0 potential and 0 kinetic. Energy is conserved so total is always 0. With time potential energy goes below 0 and kinetic increasingly above 0. More kinetic energy gives you the impression that you can do more stuff, but you can also do stuff if your potential energy can get more negative. How much negative potential energy can get is what potential basicaly is.