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Anderson Video - Metals in Electrostatic Equilibrium

Professor Anderson
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 Hello class, Professor Anderson here. Let's talk a little bit more about metals. What is sort of interesting about a metal? Well let's think about the following: if I have this big blob of metal and I put some charge on it, where is that charge gonna go? Well if I put a positive charge there and I put a positive charge right there, they are going to of course push apart because what signifies a metal is charge is free to move. Okay, that outermost valence electron is very loosely bound to each atom and so it can jump from one to the next, so if I take two positive charges and put them in there, this one is going to generate an electric field that is felt by the other one, that thing is going to feel a force, a force in that direction. Okay. This one is going to feel a force in that direction and so they're going to move and eventually what happens is they move all the way to the edge. Okay, they're no longer in the center, they have moved all the way to the edge and in fact if I put more charge on there, it's also going to move to the edge. So just a simple idea that charge is free to move makes you realize that any charge you put in that metal is going to push out to the edge, and so this is one of the rules of conductors: the E field is causing these charges to move and therefore charge resides on the surface. If all the charge is on the surface, then the E field inside is zero. Why is it zero? Because if it wasn't zero then charge would still be moving. Okay, the E field in here has to identically be zero everywhere inside the conductor, this is of course in the static equilibrium case. All right, so we've got two rules about metals conductors so far: charge resides on the surface, the E field is inside-- is inside, sorry, the E field inside is zero. If the E field is zero inside, what does that mean in terms of the potential? Well, when we were calculating potential, we said if I go from point A to point B and I want to look at the potential difference, that potential difference is going to be the integral of E over that path. But it's zero, the electric field in there is zero and so the potential difference is zero. And so this is rule number three about conductors, they are equipotentials. Okay, and they have the same potential everywhere. You kind of already knew this, right, because when you go to your wall plug and you see the wall plug has a voltage of 120 volts, here in the United States, it's 120 volts, and now I have this very long extension cord that's going to power my air conditioner, the voltage at the other end is also a hundred and twenty volts, why? Because it's metal, there's metal in that wire and so when you plug into 120 volts, that equipotential carries that voltage to the other side, it's still 120 volts. Alright, so we've got three rules about conductors, what else can we say about metals? Well, how about this, the electric field is pointing like that. Okay, but let's say it wasn't pointing like that. Let's say the electric field at the surface was pointing in some other direction, let's say it was pointing this way. If it was pointing that way, that means that the charge at the surface would feel a little bit of a force and would move, and when it moved down to this position, it would correct the electric field and make it perfectly outward again. So because we're in the static equilibrium case, the E field everywhere has to point at a right angle to the surface. Okay, all of these angles are perpendicular. Why? Because if it wasn't, charge would still be moving and then we're not in the static case anymore, so this is another rule for conductors: E field is perpendicular to the surface. All right, there's one more that we can get which is sort of obvious from charge from the number one which says the charge resides on the surface. If the charge resides on the surface, that means that there is zero net charge inside. Okay, in this region here, there is no charge, if there was charge it would migrate out to the surface. Now we know that a metal is made up of atoms which have protons and electrons, but what we're saying is they all balance out perfectly, evenly in the interior, to give a net charge of zero. All right, these are the five rules of conductors and it has some very interesting consequences for different geometrical shapes. All right, hopefully that's clear. Cheers!