# Parallel-plate capacitor

I'm trying to grasp parallel-plate capacitors in a class of electromagnetic field theory. My book leaves out a lot and I couldn't figure it out from what I found on Google.

1. The book's procedure is to place a charge $Q$ on the surface of the one conductor plate and $-Q$ on the other. Now would this charge only be distributed over the surfaces of the plates that face each other, or on all four surfaces?
2. Is it possible to prove that any kind of source of potential difference connected to a capacitor can be represented as two opposite charges like that?
3. The book treats one of the charged surfaces as a conductor surface (or this is my understanding) and this yields, with $\Delta S$ being area, $E_n=Q/\epsilon\Delta S=\rho/\epsilon$ when using Gauss' law (if you place say a thin box around the charged surface because the electric field within the conductor is zero when in equilibrium). I find this way of doing it very odd because first you place $Q$ on the one surface and $-Q$ on the other, and then disregard that you've put $-Q$ on the other surface. And I actually don't understand how come the $-Q$ does not matter. How can this work?
4. A different I think more intuitive way of doing it that I've found online is to treat each conductor as a sheet of charge. Then you have twice the area to spread out the charge over, so you get $E_n=\rho/2\epsilon$ directed inwards from each surface but with different signs, the sum which is $\rho/\epsilon$ in between the plates, just what we want. I like this because it uses the setup with $Q$ and $-Q$, plus it's easy to see how the fields cancel each other out. But why is it OK to consider a conductor to be a sheet of charge? Is it because the electric field will not be changed by passing through a conductor even though it will temporarily be zero within it?
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More on capacitors and factors of 2: physics.stackexchange.com/q/110480/2451 – Qmechanic May 12 '15 at 19:02

I won't answer the questions one after another but will try to share my form of understanding from which all the answers should be clear.

First of all, imagine what happens when you charge the capacitor. You take two parallel plates and connect each of them by a wire to one pole of a battery. Current from the source flows for a short time, not really going through the capacitor, but in a way that looks similar from the outside: electrons flow to plate A and plate B gives out other electrons which flow back to the other pole of the battery, keeping the chemical reaction in it going. (It wouldn't work well if more electrons left one pole than came back into the other.)

Since the capacitor plates are split by an insulator, the charge of the electrons accumulates on plate A. Opposite charge of the same absolute amount arises on the other plate, since we pulled electrons out of it. Both the plates contribute to an electric field which is strongest and almost homogeneous in the are between the two plates. This creates a potential difference, which happens to be in the same direction as the battery's EMF, thus there's less reason for more current to charge the capacitor.

The process stops when the voltage across the capacitor matches the battery's EMF. (This would strictly speaking take an infinite amount of time but nothing's perfect anyway.) Now you have two charged plates of opposite signs. The plates are not the only charged thing, the rest of the circuit, which is also conductive, is still connected to them. However, there's a slight difference, as the electrons on the surface of plate A facing the plate B are attracted by the Coulomb force of the positive charge of plate B. The electrons covering the rest of the negatively charged surface are repelled from each other and from the aforementioned area, too. We can effectively neglect them and say that only the side of the plate A facing plate B contains negative charge and vice versa.

Now we come to the numbers. Some negative charge -$Q$ is distributed over an area $S$ on plate A and a positive charge $+Q$ is distributed over an area $S$ on plate B. This immediately gives that the area charge density is $-Q/S$ or $+Q/S$, respectively. The next step to get us closer to the relation between charge $Q$ and the corresponding voltage $U$ is Gauss' law. It states that the area integral of $E$ over any surface equals $1/\epsilon$ of the total charge of the volume enclosed the surface, i.e. inside. This really says that any charge outside the surface has zero net effect on the integral (answering your 3.)! How can this be the case? It's because the flow lines created by outside charges enter the surface but leave it later. This contributes to the surface integral by both signs. And the particular form of Coulomb's law is mathematically so very clever that these exactly cancel out. So only charges located inside work, because they don't share this behaviour.

So, let us take some reasonable surface. If we encompassed the whole capacitor in it, the net charge inside is zero, which would render Gauss somewhat useless. If we only cut parts of the plates or took parts of the surface not parallel to them, this would just mean extra work. So we naturally take only one of the places, whole, and a bit of free space around it (so that it's really clear what is inside and what is not). Let us say we take the positive plate. The charge inside is $+Q$. The total surface of the box is somewhat bigger than two times the area $S$. However, we remember that almost all of the electric field is acting between the places, perpendicular to them, and is homogeneous in space. So there will only be some electric field crossing our surface on an area of size $S$ which really lies "between" the plates, and the $E$ will be more or less constant and with at a right angle over this area. Thus, Gauss says that $$E.S = \frac{Q}{\epsilon}.$$

Finally, the voltage, dealing with a homogeneous field, is just $E.d$, giving our desired relation defining capacity, $$U = E.d = \frac{d}{S}\cdot\frac{Q}{\epsilon} \quad \Rightarrow \quad Q = \frac{\epsilon S}{d} U =: C.U.$$

You might wonder why I assumed above that the field behaves the way it does. This was an important point in using the law. Well, we can try to reach the answer without this assumption, using only what we know about sheets of charge, thus deciding not to use Gauss' law. For this purpose, let the plate A be a sheet of negative charge and plate B a sheet of positive charge. We know that either of them generates the $E$ field of strength $\frac\rho{2\epsilon}$ on its either side. This seems to be half of what I said above, and moreover also out of the capacitor. But there's actually no contradiction, just a different way of reaching the same answer.

For an easier description, let the plates are vertical, plate A is on the left and plate B on the right. Plate A is negative, thus creates a field of strength $\frac\rho{2\epsilon}$ pointing to the right on the left of the capacitor, to the left inside the capacitor and to the left to the right of it, too. Plate B is positive, so its field points away from it in all the three regions. It turns out that the two $\frac\rho{2\epsilon}$ are in different directions and cancel out when outside of the capacitor, but they help each other and sum up to $\frac\rho\epsilon$ inside. This is the same result Gauss' law gave us.

Remember that in this way, there is no selection of volume and its enclosing surface and all that stuff, we must consider everything. Gauss' law is another way, equally correct. It gives the same answer usually more quickly at the cost of the need of more assumptions or better said, intuition. Feel free to choose what suits you better, but be prepared to need both, eventually.

Final remark on the charges being exactly opposite: well, apart from the fact that there are fluctuations and such which would make the balance inexact anyway, there actually does not need to be any. I used a capacitor involved in an electrically neutral circuit. However, you may very well take it out, charge one plate using, say, a van der Waals generator, and not let the charge carriers charge leave the other plate. In this case, the charges on the two plates are not of the same size. However, Coulomb forces act which repel the charge of the same sign as on plate A away from plate B, concentrating it on other parts of the conductor's surface and inducing an opposite charge on the facing side. The charge on A also redistributes. As a result, the capacitor would behave as if it was neutral with charges $\frac Q2$ and $-\frac Q2$ on its plates (in terms of both potential difference and electric field), with an ambient electric field of a discharged capacitor with equal charges on both sides superimposed to it. You could measure this part of the field outside of the capacitor. However, when a capacitor is used in circuits, which is the most common use one may need to deal with, no net charge of it as a whole changes its properties in any way, so we always just safely and silently ignore this possibility.

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1. The surface charge densities on the two outside-facing surfaces are necessarily zero. The easiest way to see this is by constructing a pill-box Gaussian surface that goes through all four surfaces so that the total enclosed charge cancels to zero. This implies that normal components of the electric field near the two outermost surfaces are zero as well. Finally, the charge density on the surface of a conductor is directly proportional the magnitude of the normal component of the electric field outside the conductor. It's worthwhile to note that this kind of argument generalizes to more complicated surfaces pretty effortlessly.

2. This question is putting the buggy before the horse. A "capacitor" in the context of electrostatics defined simply as a pair of conductors with non-overlapping spatial boundaries. The source of a potential difference between the two plates is nothing other than the existence of charges on the conducting surfaces. Those charges are only guaranteed to be +Q and -Q when the source of potential difference is a single battery connected to the capacitor leads in a simple closed circuit (charge conservation).

3. You aren't disregarding it per se. The laws of physics are. Gauss's Law in integral form is completely blind to any circumstance outside the boundary of integration. I'm struggling to come up with better response than "the math deems it so", but that really is a large part of the case here. Would say you complain that Newton's laws "disregard" a particle's previous history when it's state at a certain time determines all future motion? Hopefully someone else can offer a more physically intuitive explanation.

4. My response to (1.) shows why you can't do what you described. We're allowed to choose how much charge to put on some conductor, but we don't choose how that charge arranged. The charges would immediately reconfigure themselves, unless you were able to guess in advance the proper equilibrium distribution. Also, bear in mind that objects like infinite surface of zero width and the like are physically meaningful only as results of limiting procedures applied to corresponding finite-valued cases.

As a final random note, terms like "electric potential difference" and "capacitor" unfortunately have very different definitions depending on whether the context is elementary circuit theory or electrodynamics, despite referring to supposedly the same physical attributes/objects. Wikipedia did a better job setting me straight on this issue than any of the standard textbooks ever did.