Member-only story
Parallel Plate Capacitors, Electric Field, and the Energy Density
My goal is to derive the energy density for the electric field. However, I don’t want to leave out any steps so I’m basically going to start from almost the very beginning.
Here’s my plan:
- Determine the electric field due to parallel charged plates.
- Use this to find the work needed to pull plates apart.
- From that, and the volume of the field inside the plates, determine the energy density.
Let’s get started. Oh, and before you say it — I’m not going to use Gauss’s law (except at the end). Yes, it would be much easier that way but only if you assume the electric field has both a constant magnitude and direction (which isn’t fair to just start off with that).
Electric field and superposition
It’s not completely necessary, but I’m going to use a circular plate. If I want to find the electric field due to this plate, it’s important to remember that it’s charged with individual point charges. It’s fine to assume a continuous charge distribution (even though at the atomic level this is clearly not true).
Recall that the electric field at any point is the vector sum of the field due to multiple charges. We call this the super position principle. But this means that we need a nice way to write the electric field due to an individual charge at some generic location. Here’s a diagram.
Here I have two vector locations. The charge is at r_q and the place where I want to find the electric field is at r_o (observation location). Remember that r-hat is a unit vector in the direction of r. If you don’t have this in your expression, you won’t have a vector value for the electric field (which we totally need).
For the electric field due to a plate, I could do a surface area integral to find the the net field. However, I’m going to sort of a trick instead. What if I cut the disk into concentric rings? If I want to do that, I’m going to need an expression for the electric field due to a ring.
