Parallel Plate Capacitors, Electric Field, and the Energy Density

Numerical Calculation of the Electric Field Between Parallel Plates (created in Web VPython).

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…