MIT OpenCourseWare http://ocw.mit.edu 8.02 Electricity and Magnetism, Spring 2002 Please use the following citation format: Lewin, Walter, 8.02 Electricity and Magnetism, Spring 2002 (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMIT OpenCourseWare http://ocw.mit.edu 8.02 Electricity and Magnetism, Spring 2002 Transcript – Lecture 9 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 0, 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 100 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, 300 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 3 times 10 to the -14 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 10 to the 29. There's about one free electron for every atom. So we get twen- 10 to the 29 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, a speed, between these collisions, which we call the drift velocity, which is A times tau, it's just 8.01. And so A equals F divided by Me. F is e E, 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 10 volts, then the electric field -- inside the conductor, now -- is about 1 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 1.6 times 10 to the -19 Coulombs. The E field is 1, so I can forget about that. Tau is 3 times 10 to the -14, as long as I'm room temperature, and the mass of the electron is about 10 to the -30 kilograms. And so, if I didn't slip up, I found that this is 5 times 10 to the -3 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 10-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 10 volts, copper 10 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, andminus 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 1 second, these free electrons will move from left to right over a distance Vd meters. So if I make any cross-section through this wire, anywhere, I can calculate how many electrons pass through that cross-section in 1 second. In 1 second, the volume that passes through here, the volume is Vd 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
View Full Document
Unlocking...