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 20 Today, I will quantify the ability of a circuit to fight a magnetic flux that is produced by the circuits themselves. If you have a circuit and you run a current through the circuit, then you create some magnetic fields, and if the currents are changing then the magnetic fields are changing. And so there will be an induced EMF in that circuit that fights the change, and we express that in terms of a self-inductance: L, self-inductance, and the word self speaks for itself. It's doing it to itself. Magnetic flux that is produced by a circuit is always proportional to the current. You double the current, the magnetic flux doubles. And so it is the proportionality constant that we call L, that is the self- inductance, and so therefore the induced EMF equals minus d phi/dt, that is Faraday's Law. And so that becomes minus L dI/dt. L is only a matter of geometry. L is not a function of the current itself. I will calculate for you a very simple case of the self-inductance of a solenoid. Let this be a solenoid and this is a closed circuit, and we run a current I through the solenoid, and the radius of these windings is little r. Let's say there're N windings and the length of the solenoid is little l.Perhaps you'll remember that we earlier derived, using Ampere's Law, that the magnetic field inside the solenoid is mu 0 times I times capital N divided by l. This is the number of windings per meter. If we attach an open surface to this closed loop, very difficult to imagine what that open surface looks like -- we discussed it many times -- inside this solenoid you have sort of a staircase-like of surface. That magnetic field penetrates that surface N times because you have N loops. And so the magnetic flux, phi of B, is simply the area by little r squared, which is the surface area of one loop, because I assume that the magnetic field is constant inside the solenoid, and I assume that it is 0 outside, which is a very good approximation. So we get pi little r squared surface area of one loop, but we have N loops and then we have to multiply that by that constant magnetic field. So we get an N squared because we have an N here, mu 0 I divided by L. And this we call L times I. That's our definition for self-inductance. And so the self-inductance L is purely geometry. It's pi little R squared, capital N squared divided by L times mu 0. Let me check this. Pi little r squared, I have a capital N squared, mu 0, that's correct, divided by little l. And so we can calculate, for instance, what this self-inductance is for a solenoid that we have used in class several times. We had one whereby we had 2800 windings.R I think was something like 5 centimeters -- you have to work SI of course, be careful -- and we had a length, was 0.6 meters. We had it several times out here, and if you substitute those numbers in there, you will find that the self-inductance of that solenoid is 0.1 in SI units and we call the SI units Henry, capital H. It would be the same as volt-seconds per ampere, but no one would ever use that. We call that Henry. Every circuit has a finite value for the self-inductance, however small that may be. Sometimes it's so small that we ignore it, but if you take a simple loop, a simple current, just one wire that goes around -- whether it is a rectangle or whether it is a circle it doesn't make any difference -- it always produces a magnetic field. It always produces a magnetic flux through the surface, and so it always has a finite self-inductance. Maybe only 9- nano Henrys, maybe only micro Henrys, but it's never 0. And so now what I want to do is to show you the remarkable consequences of the presence of a self-inductor in a circuit, and I start very simple. I have here a battery which has EMF V. I have here a switch, and here are the self-inductor. We always draw a self-inductor in a circuit with these coils, and we also have in series a resistor, which we always indicate with this, these teeth. And I close this switch when there is no current running. In other words, at time T equals 0 when I close the switch, there is no current.When I close this switch the current wants to increase, but the self-inductance says uh-huh, uh-huh, take it easy, Lenz law, I don't like the change of such a current. So the self-inductance is fighting the current that wants to go through it. There comes a time that the self-inductance loses the fight, if you wait long enough, and then of course the current has reached a maximum volume, which you can find with Ohm's Law, because the self-inductance itself has no resistor. Think of the self-inductance as made of super-conducting material. There's no resistance. And so without knowing much about physics, you can make a plot about the current that is going to flow as a function of time. You start out with 0 and then ultimately, if you wait long enough, you reach a maximum current which is given by Ohm's Law, which is simply V divided by R. And you slowly approach that value. And how slowly depends on the value of the self-inductance. If the self-inductance is very high, it might climb up like this, so this is a high value for L. If the self-inductance is very low, that is a low value for L. If the self-inductance were 0, it would come up instantaneously, but I just convinced you that there I no such thing as 0 self-inductance. There's always something finite, no matter how small. And so this is qualitatively what you would expect if you use your stomach and if you don't use your brains yet. There's nothing wrong with using your stomach occasionally, but now I want to do this in a move civilized way, and I want to use my brains, and when I use my brains I have to set up an equation for this circuit.And if you read your book, you will find that Mr. Giancoli tells you to use Kirchhoff’s Loop Rule. But Mr. Giancoli doesn't