Inherent electric potential energy at one location

Question

My question is this: Why can't electric potential energy be defined at one point. Surely a point charge has some inherent electric potential energy cause by force applied by the source field. The applied force would cause the charge to accelerate. The acceleration approaches zero as you get farther from the field, so the velocity could also approach a known limit (depending on the function). That velocity can be used to calculate the maximum kinetic energy, which would equal the original electric potential energy imparted on the charge by the field. I know this is a classical treatment of the situation, but what's wrong with this line of thinking?

Answer 1

It is customary to set the potential to zero at infinity, but that's not a necessary choice. We can set it to anything we like without any changes to the physics of the problem since the absolute value of a potential doesn't matter. The only thing that matters is the gradient of the potential, which is a derivative. The derivative of a constant is zero. Is that your question?

Answer 2

Electrical potential energy (measured in joule) cannot be defined just by defining a point, because it depends on the size of the charge that is placed at that point. Electrical potential, which is the electrical potential energy per unit charge, is defined in very much the way you outlined, but it is measured in joule per coulomb, usually abbreviated to volt. If you have a spot with a potential of 200 V and place a 4 $\mu$C charge there, it will have electrical potential energy of 800 $\mu$J.

In most situations it is the change in potential between two points that is the useful quantity. This is the potential difference, often called the voltage.

Answer 3

That velocity can be used to calculate the maximum kinetic energy, which would equal the original electric potential energy imparted on the charge by the field.

You could do this, and you would get a result equivalent to the potential distribution.

However it's much simpler to simply take path integral for a convenient path through the electric field to determine the potential than to work out the actual trajectory of a particle in the field.

And once you've determined the potential distribution, it's much easier to work out the final velocity from the potential difference between two points than it was to work out the potential distribution from the electric field.