**Pre Reqs:** Electric Field, Work-Energy, Potential Energy

If you are already familiar with the topics listed in the pre-reqs above, this will be uber-simple.

## Potential energy – short version

The work-energy principle basically says:

In this most basic form, the energy is just kinetic energy (if you are not going near the speed of light). BUT…if you have a force that is conservative (meaning the work done does not depend on the path you take), then you can make it a potential energy and move it to the other side.

**Warning:** you can not have a force and have that force do both work AND be a potential energy.

If you make the work done by a force a potential energy then that change in potential energy will be:

There is one small problem. What if the force is not constant? In that case, this really doesn’t work, but you get the idea (you would need some calculus to fix this for non-constant forces). But that is potential energy in a nutshell.

## Electric Potential

As an example, let me start with the case of a constant electric field. This is not so crazy of an idea. It is a pretty good assumption to say the electric field inside of a parallel plate capacitor makes a constant electric field (where one side of the capacitor has a positive charge and one side has a negative charge of the same magnitude).

Now suppose that I take a positive charge from the negative side of the capacitor and move it to the positive side. During this time there is an electric force on the charge in the opposite direction it moves. The work done by the electric force would be:

If I now consider this work as a potential energy, it would have to be negative. So, moving from the negative side to the positive side the change in electric potential energy would be:

I will leave it as a homework question, but it is not too difficult to show that this change in potential energy does not depend on the path the charge takes from one side to the other.

What if I change the charge that I am moving? Well, then the change in electric potential energy would be different. How about I just find the change in electric potential energy per unit charge? I could write that as:

And this is the electric potential – though it is often called just “potential”. Really it is the change in electric potential and it has units of Joules per Coulombs or Volts. Some people even call this the voltage (but I like to call it potential difference – to emphasize that I am dealing with a change).

A couple of other points:

- The relationship between change in potential energy and change in electric potential is true even if the field is not constant.
- In the above, I did the change in potential going from the negative plate to the positive plate. This gave a positive change in potential. If I had gone the other way (in the same direction as the electric field) the change in potential would have been negative.
- Please be careful with the equation ΔV = -Ed. I have seen many student use this as the general expression for potential. However, this is
**only true for a constant electric field**