Two parallel sheets of charge

Prerequisites

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 electric field produced by the sheet was constant, both in magnitude and direction, with the direction of the field perpendicular to the sheet.

Motivation

Why would we care to calculate this? Do we ever have a single sheet that can be treated as infinite? Well, the answer is yes we do!  When our test charge is close to the sheet, and the edge of the sheet is far away from it in comparison to the distance of our test charge from the sheet, the result is the same as if the sheet were infinite. The far away parts of the sheet don't contribute very much to the field.  Even a 1 inch diameter sheet is large enough to treat as infinite if we consider only distances 1 mm or less away from it and don't get too close to the edge.  Indeed if you get close enough to the surface of any conductor, the electric field will look uniform. Even the membrane of a cell may be considered an infinite sheet when we consider its interaction with proteins that are tens of nanometers away from it. But actually a membrane represents an example of a slightly more complicated system: two parallel sheets of charges.

But a really useful case is when we have two equal and opposite (infinite) sheets parallel and very close to each other. While this seems like an unlikely toy model, it 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. The surfaces of both lipid bilayers are often charged, so membranes look a lot like two charged sheets that are about 6-10 nanometers apart.

Let's see how it works.

The fields from two sheets

We are going to take two sheets of equal and opposite amounts of charge that are large compared to how far away from them we will get. 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. (See our analysis of the single sheet at: A simple electric model: A sheet of charge.) 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 or away from the sheet as shown.

Now suppose we slide them towards each other. Here's what we get:

We've drawn the E fields from both plates everywhere: even on the other side of the complementary plate. Why? Remember that the E fields from individual charges are everywhere (and given by Coulomb's law)! They are not "blocked" by the presence of other charges. But we will see they can be cancelled by the E field from other charges.

How do the fields from the blue and the red sheet combine? When calculating electric fields, we simply add the field from every charge. So we can simply add the field of the blue set of charges and the red set of charges. The electric fields caused by any charges simply add!

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; and similarly in the region to the right of the red sheet. In that region we have equal and opposite arrows everywhere just as we did on the left. So everywhere OUTSIDE of the two sheets, their fields cancel each other.

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, the plates. the E fields add. The total field will look like this:

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

$$E = 4\pi k_C \sigma$$

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 as $1/4πε_0$, you may sometimes see this field written as $E = σ/ε_0$.)

Re-motivating

When we looked at a single sheet, we knew we really couldn't get away with the field being constant forever. No sheet is actually infinite. In the configuration shown above, with two equal and opposite sheets, we only really have to worry about the fields BETWEEN the sheets.

So as long as the distance between the sheets is small compared to the size of the sheets we can use the infinite sheet approximation! We know outside that the fields pretty much cancel. (There is some effect from the edges but it's small compared to what's happening between the sheets.)  Since we only relied on the distance between the sheets being small compared to the size of the sheets for our approximation, we can deform sheets on larger scales without worrying too much about the corrections to the infinite sheet model.

For example, we can model cell membranes that are rolled up into axons as if they were plane sheets since the axon is hundreds of nanometers or micrometers thick and so deforms the membrane on scales larger than the distance between the two sides of the membrane (~5-10 nm).

In the follow-on we'll apply this result to create a circuit device — the capacitor — that can be used to store electrical energy as a separation of charge.

Joe Redish 2/20/12 & Wolfgang Losert 2/22/13

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