Class Content

Prerequisites:

• The Electric field
• The electric potential
• A simple electric model: a sheet of charge

In a previous reading (A simple electric model: a sheet of charge) we studied the simple model of what the field would look like from a very large (treated as infinitely large) sheet of charge. As an analytic exercise, this was mildly interesting. It showed that if we assumed that the edges of the sheet were very far away and we ignored the discrete nature of charge, then the field produced by the sheet was constant, independent of the distance from the sheet, and perpendicular to it.

Motivation

This seems like a pretty bizarre and uninteresting result. Who cares? Do we ever have a single sheet like that? Well, the answer is yes, we do. One case is the surface of a conductor. The field near to the surface of a conductor looks sort of like this. But the really important case is when we have two equal and opposite sheets parallel and very close to each other. This is the basis for an important electrical device: the capacitor. This is a standard piece of electrical equipment, found in essentially every electrical instrument. It allows the storage of electrostatic energy. But besides being important to electrical engineers, it has relevance to us as well.

1. It allows us to define a fundamental electrical property, capacitance, that allows us to quantify information about the separation of charge in any physical system.
2. It provides a model for many useful biological systems, in particular, the cell membrane. 

Let's see how it works.

The fields from the sheets

We are going to take two large (at least compared to how far away from them we will get) sheet of equal and opposite amounts of charge. The field from a sheet of positive charge (blue) is shown at the left below. If the charge density on the sheet is σ (C/m²), the E field will have a magnitude E = 2πk_cσ on either side, pointing away from the sheet as shown. The field from a sheet of negative charge (red) is shown at the right below. If the charge density on the sheet is -σ (C/m²), the E field will have a magnitude E = 2πk_cσ on either side, pointing towards from the sheet as shown.

Now suppose we slide them towards each other. Remember: the fields simply add. They don't know anything about any other charges. So when we slide them close to each other here's what we get:

On the left side, there are arrows pointing to the left that come from the blue sheet of positive charges and arrows pointing to the right that come from the red sheet of negative charges. Since the fields are independent of distance, if the sheets have that same but opposite charge densities, the fields from each sheet will cancel in the region to the left of the blue sheet. Similarly in the region to the right of the red sheet. We see in that region equal and opposite arrows everywhere just as we did on the left.

But in between the two sheets the arrows are in the SAME direction. The ones from the positive (blue) sheet point away from it -- to the right. The ones from the negative (red) sheet point towards it -- again to the right. So in between, they add. The total field will look like this:

The field outside the sheets will be 0. The field inside the sheets will point from positive to negative and have a value of

E = 4πk_cσ

where +σ is the charge density on the positive sheet and -σ is the charge density on the negative sheet. (Since k_c is sometimes written, 1/4πε₀, you may sometimes see this field written as E = σ/ε₀.)

Re-motivating

When we looked at a single sheet, it was troubling since we knew we really couldn't get away with the field being constant forever. No sheet is actually infinite. But in the configuration shown above, with two equal and opposite sheets, we only really have to worry about the fields BETWEEN the sheets -- which don't go very far away at all. We know outside the pretty much cancel (there is some effect from the edges) but it's small compared to what's happening between the sheets. We can even roll them up into an axon, cell membrane, or capacitor without worrying too much about the corrections to the infinite sheet model (as long as the sheets are not curved too sharply).

In the follow-on we'll look at the capacitor and see how we can quantify what's going on.

Follow-on:

• The capacitor

Joe Redish 2/20/12