# Electrostatics: Potential

There are two very important concepts that sound very similar: electric potential and electric potential energy. They are not the same thing! They are related to each other, but it's very important to distinguish between the two.

Let's begin with electric potential energy. Recall from the chapter on Work-Energy that we defined potential energy $U$ associated with a particular, conservative, force as the negative of the work done by that force. $$ U \equiv - W = - \int \vec{F}\cdot d\vec r $$. That same definition still holds true.

## For Uniform Electric Fields:

In the case where the field is uniform (constant in magnitude and direction), the electric potential energy (-work) is easy to calculate. $$ U = -W = \int \vec F \cdot d\vec r = \int q\vec E \cdot d\vec r$$ But since $\vec E$ is constant, we can pull it out of the integral. $$ U = q\vec E \cdot \int d\vec r = q\vec E\cdot\vec r $$

(Recall the the dot product can be calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.)

Example: A region of space has an electric field $\vec E = 500$N/C$\hat x$. Calculate the change in potential energy in moving a charge $q=2\mu$C from point A at the origin to point B at $x=3$m.

Solution: Since both the electric field vector and the vector $r$ (from A to B) are in the same direction ($\hat x$), you can calculate the potential energy by: $$U = -q\vec \cdot \vec r = -(2\times 10^{-6}C)(500 N/C)(3m)\cos 0 = 3000 J $$

## For Point Charges

The force between two point charges is given by Coulomb's Law $$F = k\frac{Qq}{r^2}\hat r$$. Substituting this expression for force into our definition of potential energy allows us to solve for the electric potential energy for point charges.