Electrostatics
Let's go over some of the basic concepts. By the way, electrostatics refers to charge that isn't moving (i.e., static). Hopefully this is not the first time you are seeing the material, this should be a review.
Force
Between any two charges there exists a force between them, the strength and direction following Coulomb's Law.
Electric Field
Every charge creates an electric field, which transmits force to other charges. How this is done is still not known is actually a fundamental question for modern physics. My take on it.
Potential
The electric field is actually the gradient of the electric potential function. The potential of a point is the amount of work, per unit charge, it takes to move a positive test charge from infinity to that point (by convention). To deal with the nitpickers out there, the charge is small enough not to disturb the existing charges that are creating the field.
Thus, infinity is at zero potential (because you do not have to move the charge; therefore, no work (force times distance) is done). However, for all other points, a certain amount of work must be done to move the charge.
Potential can be positive or negative. Starting at infinity, if we move a positive test charge towards a fixed positive charge, it takes work on our part. Therefore it has positive potential. If we move a positive test charge towards a negative charge, it takes negative work (because the two attract, we get "free" work).
Important concept: Mother nature likes to reduce potential energy. Thus, given the option, a charge will move to the location of lowest potential. Two positive charges will repel (i.e, try to go to infinity, because 0 is less than a positive number), and a positive and negative charge will attract (0 is a higher potential than a negative number, so they will go to where they have negative potential). This is an important concept, and will help us when understanding electricity.
Charges in a field
As we said before, if you expose charges to a field they will move, namely to the point of lowest potential. But what if the charges are not floating in space, but are electrons in a metal? They can't leave the metal, so they will have to arrange themselves in some way while still remaining on the metal.
How do you think the charges will act when exposed to a field? Will they separate to create a larger or smaller field inside the metal? It's not hard to guess that they will seek to minimize the field where they can, namely inside the metal. Thus, the charges arrange themselves to cancel the field inside the metal. If the field was created by an external force, the charges in the metal cannot effect it, but the charges do cancel the field inside the metal. They reach some steady state, with the external field canceled in the metal, and all the charges motionless. If they didn't cancel the field inside the metal then the charges would be moving (due to the force of the field). Thus, we come to two important concepts:
Charges will try to cancel the field inside a metal, if possible.
The surface of the metal is an equipotential (if there is no field in the metal).
We just explained the first point, but let's look at it deeper. If the charges are fixed, then when they are exposed to a field they will try to cancel the field but cannot. If the charges can move, but do not cancel the field, they will keep moving, trying to cancel the field. Constantly moving charges -- are you thinking what I'm thinking? Yes, my friends, this is the beginning of electricity... but all in due time.
The second point means that the entire surface of the metal has the same value from the potential function. Because there is no field in the metal, it takes no work to move a charge on the surface of the metal. Thus, every point must have the same potential. Once you move a charge to any point on the equipotential, it takes no work to move it anywhere else on the equipotential.
That wacky inverse square law
The fact that electric field strength follows an inverse square law (decreases as r squared) leads to some interesting things. Let's get some geometric intuition for this. If you don't know what flux is, go find out.
Let us consider our surface as a cone (see diagram) [**expand this, use field of vision example**]. If we have a line of charge (one dimensional), as we move away from it our flux increases linearly (as r). However, our field strength decreases as r squared, so the net result is that our field decreases as r. Thus, for an infinite line of charge, the field at any point is proportional to 1/r.
If we have a plane of charge (two dimensional), as we move away our flux increases quadratically (as r squared). This is because our area is increasing in both dimensions. Our field strength is dropping as r squared, typical of an electric field. These two effects cancel, so we have a constant field strength at any location. At far locations, we have a weak field, but strong flux. At near locations, we have a small flux, but strong field. Amazingly, these two effects cancel at every point, so every point in space has the same electric field if there is an infinite plane of charge. What would happen if electric charges didn't follow an inverse square law. What if it decreased linearly (to the first power)? What if it decreased to the third power?
Now, we really can't make an infinite plane of charge. However, if we use the same principle of increasing area to combat decreasing field strength, we can create a region of constant field strength. Even better, if we can have two infinite planes, we might be able to have their fields cancel - we could have a bunch of charge and no field! Well, this idea was originally noted as the Iron Shell Theorem, and it worked for gravity (another inverse square field!).
Revisit Gauss's Theorem
[*** expand this out ***]
Iron Shell Time
The Iron Shell Theorem is as follows: if you are in the interior of a hollow sphere, then you feel no gravitational pull from the sphere. None. Now, this means you feel no gravity from the sphere; other objects still act on you, inside or outside the sphere. Sorry, you can't build a giant ball, jump in, and expect to float.
The reason this theorem holds is because of the reasoning before. Each wall of the sphere is like a plane to some extent. They both end up pulling evenly because the closer one has more strength but less flux, and the further one has more flux but less strength. But as we showed before, the two effects cancel. So, in a similar sphere made of electric charge, it creates no electric field inside the sphere.
Wow, that's pretty cool. What's even cooler is that for electric charges, the shape doesn't even have to be a sphere! That means you can have any closed shape, and, if charges are mobile, they will arrange themselves so there is no field in the metal and no field in the region the metal surrounds. And here's the kicker: even in the presence of external fields, there is still no field inside the region. If you aren't stunned, you should be. The charge, as Purcell puts it, has cleverly arranged itself to precisely cancel the field inside its region. That's quite a feat, I certainly couldn't look at an arbitrary shape and figure out how to arrange the charge so there was no field inside, for any external field. The reasoning for this isn't quite so intuitive, but can be explained. We must remember that the charges aren't thinking entities; the electric field is such that it moves the charges in such a way that they cancel the electric field inside. It is surprisingly elegant that our electric field, following a simple inverse square law, has this result. What about our other favorite inverse square law, gravity? Does this hold true as well?
Let's think about this. As we change the external field, the charges move around on the metal surface and end up canceling the field inside. No matter what we do outside, the field is still zero inside! We have shielded the region from outside electric fields. This is exactly what is done when an outside electric field needs to be blocked. For example, if we have a communications wire that we want to protect from outside noise, we surround it in metal. The metal does not even have to be solid; a wire mesh will work well, and only let a little bit of interference in. Can you think of some other applications of this? Suppose you are in a completely metal building. Do you think a radio or cell phone, which get information through fluctuating electric fields, will work properly? What about in an elevator? An armored car? Try this out sometime, it's pretty cool. Again, this is a remarkable result.
One thing to remember is that although there is no field inside the region, the charges on the surface of the metal will (and must) keep moving to cancel a changing field. Thus, a radio whose antenna is touching the metal region will get a reception because the charges on the entire metal region begin to move, giving the radio the signal it needs. So if you are in an elevator and your cell phone won't work, touch its antenna to the metal and see if the reception gets better. C'mon guys, be geeks with me, this stuff is cool!
Check your intuition
Ok, make sure you understand these points. This is why I made this site, to quickly write down points that took me a long time to intuitively understand:
Charges will move to minimize potential energy. Mother nature likes to minimize potential energy.
Charges will try to cancel a field inside a metal, if possible.
A metal's surface is an equipotential if #2 is met.
Electric field follows an inverse square law. Because of this:
We can use Gauss's theorem (flux = source)
Lines (one dimensional) of charge loose strength as 1/r
Planes (two dimensional) of charge have constant strength
8. Metal shields electric fields in the region it surrounds.
Got it? Good. Now you can understand electricity, which was one of the toughest things for me to intuitively understand. It's easy enough to memorize and use formulas, but I want to see the ins and outs of the formula, and to know why it works and what exactly it means.
