Subtitles section Play video Print subtitles When positive charges move in this direction, then per definition, we say the current goes in this direction. When negative charges go in this direction, we also say the current goes in that direction, that's just our convention. If I apply a potential difference over a conductor, then I'm going to create an electric field in that conductor. And the electrons -- there are free electrons in a conductor -- they can move, but the ions cannot move, because they are frozen into the solid, into the crystal. And so when a current flows in a conductor, it's always the electrons that are responsible for the current. The electrons fuel the electric fields, and then the electrons try to make the electric field zero, but they can't succeed, because we keep the potential difference over the conductor. Often, there is a linear relationship between current and the potential, in which case, we talk about Ohm's Law. Now, I will try to derive Ohm's Law in a very crude way, a poor man's version, and not really one hundred percent kosher, it requires quantum mechanics, which is beyond the course -- beyond this course -- but I will do a job that still gives us some interesting insight into Ohm's Law. If I start off with a conductor, for instance, copper, at room temperature, three hundred degrees Kelvin, the free electrons in copper have a speed, an average speed of about a million meters per second. So this is the average speed of those free electrons, about a million meters per second. This in all directions. It's a chaotic motion. It's a thermal motion, it's due to the temperature. The time between collisions -- time between the collisions -- and this is a collision of the free electron with the atoms -- is approximately -- I call it tau -- is about three times ten to the minus fourteen seconds. No surprise, because the speed is enormously high. And the number of free electrons in copper per cubic meter, I call that number N, is about ten to the twenty-nine. There's about one free electron for every atom. So we get twen- ten to the twenty-nine free electrons per cubic meter. So now imagine that I apply a potential difference piece of copper -- or any conductor, for that matter -- then the electrons will experience a force which is the charge of the electron, that's my little E times the electric field that I'm creating, because I apply a potential difference. I realize that the force and the electric field are in opposite directions for electrons, but that's a detail, I'm interested in the magnitudes only. And so now these electrons will experience an acceleration, which is the force divided by the mass of the electron, and so they will pick up, eh, speed, between these colli- collisions, which we call the drift velocity, which is A times tau, it's just eight oh one. And so A equals F divided by M. E F is in the A, so we get E times E divided by the mass of the electrons, times tau. And that is the the drift velocity. When the electric field goes up, the drift velocity goes up, so the electrons move faster in the direction opposite to the current. If the time between collisions gets larger, they -- the acceleration lasts longer, so also, they pick up a larger speed, so that's intuitively pleasing. If we take a specific case, and I take, for instance, copper, and I apply over the -- over a wire -- let's say the wire has a length of 10 meters -- I apply a potential difference I call delta V, but I could have said just V -- I apply there a potential difference of ten volts, then the electric field -- inside the conductor, now -- is about one volt per meter. And so I can calculate, now, for that specific case, I can calculate what the drift velocity would be. So the drift velocity of those free electrons would be the charge of the electron, which is one point six times ten to the minus nineteen Coulombs. The E field is one, so I can forget about that. Tau is three times ten to the minus fourteen, as long as I'm room temperature, and the mass of the electron is about ten to the minus thirty kilograms. And so, if I didn't slip up, I found that this is five times ten to the minus three meters per second, which is half a centimeter per second. So imagine, due to the thermal motion, these free electrons move with a million meters per second. But due to this electric field, they only advance along the wire slowly, like a snail, with a speed on average of half a centimeter per second. And that goes very much against your and my own intuition, but this is the way it is. I mean, a turtle would go faster than these electrons. To go along a ten-meter wire would take half hour. Something that you never thought of. That it would take a half hour for these electrons to go along the wire if you apply potential difference of ten volts, copper ten meters long. Now, I want to massage this further, and see whether we can somehow squeeze out Ohm's Law, which is the linear relation between the potential and the current. So let me start off with a wire which has a cross-section A, and it has a length L, and I put a potential difference over the wire, plus here, and minus there, potential V, so I would get a current in this direction, that's our definition of current, going from plus to minus. The electrons, of course, are moving in this direction, with the drift velocity. And so the electric field in here, which is in this direction, that electric field is approximately V divided by L, potential difference divided by distance. In one second, these free electrons will move from left to right over a distance V D meters. So if I make any cross-section through this wire, anywhere, I can calculate how many electrons pass through that cross-section in one second. In one second, the volume that passes through here, the volume is V D times A but the number of free electrons per cubic meter is called N, so this is now the number of free electrons that passes, per second, through any cross-section. And each electron has a charge E, and so this is the current that will flow. The current, of course, is in this direction, but that's a detail. If I now substitute the drift velocity, which we have here, I substitute that in there, but then I find that the current -- I get a E squared, the charge squared, I get N, I get tau, I get downstairs, the mass of the electron, and then I get A times the electric field E. Because I have here, is electric field E. When you look at this here, that really depends only on the properties of by substance, for a given temperature. And we give that a name. We call this sigma, which is called conductivity. Conductivity. If I calculate, for copper, the conductivity, at room temperature, that's very easy, because I've given you what N is, on the blackboard there, ten to the twenty-nine, you know what tau is at room temperature, three times ten to the minus fourteen, so for copper, at room temperature, you will find about ten to the eighth. You will see more values fro sigma later on during this course. This is in SI units. I can massage this a little further, because E is V divided by L, and so I can write now that the current is that sigma times A times V divided by L. I can write it down a little bit differently, I can say V, therefore, equals L divided by sigma A, times I. And now, you're staring at Ohm's Law, whether you like it or not, because this is what we call the resistance, capital R. We often write down rho for one over sigma, and rho is called the resistivity. So either one will do. So you can also write down -- you can write down V equals I R, and this R, then, is either L divided by sigma A, or L times rho -- let me make it a nicer rho -- divided by A. That's the same thing. The units for resistance R is volts per ampere, but we call that ohm. And so the unit for R is ohm. And so if you want to know what the unit for rho and sigma is, that follows immediately from the equations. The unit for rho is then ohm-meters. So we have derived the resistance here in terms of the dimensions -- namely, the length and the cross-section -- but also in terms of the physics on an atomic scale, which, all by itself, is interesting. If you look at the resistance, you see it is proportional with the length of your wire through which you drive a current. Think of this as water trying to go through a pipe. If you make the pipe longer, the resistance goes up, so that's very intuitively pleasing.