WEBVTT 00:01.570 --> 00:04.460 Prof: All right, I'm going to begin by 00:04.462 --> 00:06.962 summarizing stuff done near the end. 00:06.960 --> 00:10.120 Usually stuff done near the end is maybe hurried, 00:10.122 --> 00:13.022 or also you may have forgotten where we were, 00:13.022 --> 00:14.342 so it's helpful. 00:14.340 --> 00:20.250 The first of what I did last time is called Lenz and Faraday 00:20.248 --> 00:23.938 law, and it says the following - in 00:23.941 --> 00:28.031 any circuit made with a real conductor, 00:28.030 --> 00:37.630 if you integrate the electric and magnetic force on a charge, 00:37.630 --> 00:41.600 unit charge around a closed loop--this is called the 00:41.599 --> 00:48.949 electromotive force-- that = - the rate of change of 00:48.951 --> 00:53.401 flux through that loop. 00:53.400 --> 00:57.310 And the flux is defined as follows. 00:57.310 --> 01:03.450 The flux is just the integral of the magnetic field over an 01:03.445 --> 01:07.145 area whose boundary is this loop. 01:07.150 --> 01:09.510 So this loop is a boundary of that, 01:09.510 --> 01:13.720 so if you have a loop, and there is some flux coming 01:13.716 --> 01:16.276 out of it, you integrate that, 01:16.280 --> 01:18.870 and you take the rate of change. 01:18.870 --> 01:21.030 That covers a variety of different phenomena. 01:21.030 --> 01:24.320 It's what I was trying to explain last time. 01:24.319 --> 01:27.199 First of all, if you ask, "Why does this 01:27.197 --> 01:28.177 thing change? 01:28.180 --> 01:29.810 Why does the flux change?" 01:29.810 --> 01:32.680 it can change for two reasons. 01:32.680 --> 01:35.990 It can change because B itself is changing 01:35.992 --> 01:36.932 with time. 01:36.930 --> 01:39.430 Or even if B is not changing with time, 01:39.431 --> 01:42.491 but changing with space, if a loop happens to be moving, 01:42.488 --> 01:44.988 that can also change the flux through it. 01:44.989 --> 01:46.309 For whatever reason, it will change. 01:46.310 --> 01:50.340 So this total derivative means the total rate of change of flux 01:50.337 --> 01:53.127 for the two reasons I mentioned, or both. 01:53.129 --> 01:56.349 You can have a time dependent, space dependent field in which 01:56.346 --> 01:57.736 a loop is dragged along. 01:57.739 --> 02:00.399 Then at any instant, electromotive force, 02:00.400 --> 02:04.130 namely the total force on the charge, on the unit charge, 02:04.125 --> 02:06.715 is given by rate of change of flux. 02:06.718 --> 02:11.408 Then I said we can write this integral, this rate of change is 02:11.411 --> 02:13.181 made up of two parts. 02:13.180 --> 02:20.320 One was the rate of change due to explicit time dependence of 02:20.324 --> 02:26.874 the field on the old surface another one that looks like 02:26.872 --> 02:32.352 v x B⋅dl. 02:32.348 --> 02:36.318 This was what I did the last time. 02:36.318 --> 02:38.588 And we got that result--I'll remind you briefly, 02:38.590 --> 02:40.580 but I certainly don't want to do it again, 02:40.580 --> 02:46.860 is that if you had a loop that was doing this initially, 02:46.860 --> 02:51.290 it was doing that a little later, the loop is moving, 02:51.288 --> 02:54.928 then if you want to find the change in flux between that one 02:54.931 --> 02:57.661 and this one, the natural thing to do is to 02:57.660 --> 03:00.060 integrate the flux on that loop later, 03:00.060 --> 03:03.910 subtract from it the original loop earlier. 03:03.908 --> 03:06.348 But calculationally, it's easier to use the fact 03:06.346 --> 03:09.456 that if you want to find the flux through a surfacing bounded 03:09.455 --> 03:11.935 by this loop, you don't have to simply take 03:11.944 --> 03:13.344 the easiest loop you see. 03:13.340 --> 03:16.080 You can take any loop with that as a boundary. 03:16.080 --> 03:21.380 And we cleverly chose the surface to be the one containing 03:21.382 --> 03:25.292 the original surface and the two walls, 03:25.288 --> 03:27.358 and the circular walls, if you like, 03:27.360 --> 03:31.660 that add onto it and produce a new surface. 03:31.658 --> 03:34.788 So if you take one flat sheet and glue the sides of the 03:34.792 --> 03:38.332 cylinder, you get a rim here, and that's the final surface. 03:38.330 --> 03:41.920 The advantage of doing it is that when you find the flux at a 03:41.917 --> 03:44.247 later time using this joint surface, 03:44.250 --> 03:48.050 one contribution will come from exactly the old surface, 03:48.050 --> 03:50.800 and the difference there is only because of explicit time 03:50.800 --> 03:51.440 dependence. 03:51.440 --> 03:54.490 The other will come from the sides, and the sides, 03:54.491 --> 03:56.541 you can see, are spanned by little 03:56.544 --> 03:57.484 rectangles. 03:57.479 --> 04:00.149 This side of the rectangle is v times dt. 04:00.150 --> 04:01.590 This side is dl. 04:01.590 --> 04:05.010 And the dot product with the flux will be that term. 04:05.008 --> 04:09.188 And you rearrange the product you get this. 04:09.188 --> 04:15.838 Therefore now we can balance that with that, 04:15.840 --> 04:21.420 or if you take a problem where nothing is changing with time, 04:21.420 --> 04:22.480 then you can just forget this term, 04:22.480 --> 04:27.060 and this term will match that term. 04:27.060 --> 04:31.010 Then we are left with the following result - the line 04:31.005 --> 04:35.175 integral of the electric field on any loop = the surface 04:35.178 --> 04:39.498 integral of the rate of change of B over that fixed 04:39.504 --> 04:40.344 loop. 04:40.339 --> 04:45.389 So here is the fixed surface, and here is the boundary of the 04:45.391 --> 04:46.741 fixed surface. 04:46.740 --> 04:47.990 You have to understand the difference. 04:47.990 --> 04:52.270 We started out with a statement about a real physical loop. 04:52.269 --> 04:56.549 The physical loop was moving and there is an emf driving 04:56.553 --> 04:58.583 charges around the loop. 04:58.579 --> 05:00.019 It's got two parts. 05:00.019 --> 05:02.009 One had to do with the motion of the loop. 05:02.009 --> 05:03.969 The other had to do with the changing of the field. 05:03.970 --> 05:08.570 This relation, you see, has no need of a real 05:08.565 --> 05:10.755 loop to be present. 05:10.759 --> 05:13.459 This one definitely needs a real loop to be present, 05:13.461 --> 05:16.481 a real conducting loop, because whose velocity is this? 05:16.480 --> 05:19.960 It's a velocity of each portion of the loop that's moving. 05:19.959 --> 05:23.639 But this one is about a fixed contour in space and has to do 05:23.639 --> 05:26.759 with the line integral of E around the fixed 05:26.757 --> 05:28.377 contour with this one. 05:28.379 --> 05:30.929 So it is true even if you remove that loop. 05:30.930 --> 05:33.750 It's a basic statement about electric and magnetic fields 05:33.754 --> 05:36.784 that tell you that if you have a changing magnetic field, 05:36.779 --> 05:39.889 a time dependent magnetic field, it will generate an 05:39.894 --> 05:43.524 electric field which, unlike the electrostatic field, 05:43.521 --> 05:46.451 will have a line integral which is not 0. 05:46.449 --> 05:47.389 That's the main point. 05:47.389 --> 05:50.159 It's a non conservative electric field, 05:50.160 --> 05:54.240 whose origin is not electric charges, but really changing 05:54.242 --> 05:55.702 magnetic fields. 05:55.699 --> 06:00.779 So this is called Faraday's law. 06:00.778 --> 06:03.398 Faraday's law is a very profound statement about 06:03.396 --> 06:06.846 electric and magnetic fields and that's what's going to be more 06:06.850 --> 06:07.630 important. 06:07.629 --> 06:12.789 But the original formula is able to cover all situations 06:12.788 --> 06:15.508 where loops move in fields. 06:15.509 --> 06:19.979 This - sign is due to Lenz and the - sign is going to save your 06:19.975 --> 06:20.475 life. 06:20.480 --> 06:23.330 The - sign tells you, if you want to know when you 06:23.334 --> 06:26.484 drag a loop or when you change the flux through it, 06:26.480 --> 06:29.800 what is it going to do, the - sign tells you it's going 06:29.802 --> 06:31.652 to fight the change in flux. 06:31.649 --> 06:34.749 Namely, if an emf generated, which way will it point? 06:34.750 --> 06:37.130 Will it move charges this way or that way? 06:37.129 --> 06:40.829 Answer is, it will move charges in such a way that they produce 06:40.827 --> 06:43.447 their own current, which will produce its own 06:43.452 --> 06:44.052 field. 06:44.050 --> 06:47.720 That field will fight the change that you're trying to 06:47.716 --> 06:48.406 produce. 06:48.410 --> 06:51.500 In other words, if you take a loop here and 06:51.499 --> 06:55.989 you've got a bar magnet sending some lines of flux through it, 06:55.985 --> 06:59.145 you move the magnet closer to the loop. 06:59.149 --> 07:02.309 That's an example of rate of change of flux through the loop. 07:02.310 --> 07:03.940 Which way will the current flow? 07:03.939 --> 07:05.419 The answer is very simple. 07:05.420 --> 07:08.730 You're trying to increase the flux through the loop. 07:08.730 --> 07:11.390 It will fight it by decreasing the flux going upwards, 07:11.389 --> 07:13.949 so it will try to produce a flux going downwards. 07:13.949 --> 07:18.919 Therefore it will have a current that looks like this. 07:18.920 --> 07:21.040 If you have a current that looks like this, 07:21.040 --> 07:24.870 let me see, you do that and the thumb points down, 07:24.870 --> 07:28.870 and therefore the magnetic field it produces will oppose 07:28.867 --> 07:29.737 the field. 07:29.740 --> 07:34.020 So this rule is very important. 07:34.019 --> 07:36.719 That's what distinguishes human being from primates. 07:36.720 --> 07:40.350 The primates cannot do this rule. 07:40.350 --> 07:43.470 In fact, there are lots of cave drawings of apes trying to 07:43.473 --> 07:45.973 invent solenoids, and they always kept saying 07:45.973 --> 07:48.893 this, and they thought the magnetic field was parallel to 07:48.886 --> 07:49.506 the coil. 07:49.509 --> 07:51.919 So one day, we figured this out. 07:51.920 --> 07:55.320 That's the beginning of--it's even better than fire, 07:55.322 --> 07:57.662 finding out that right hand rule. 07:57.660 --> 08:00.630 Okay, so this is the rule you should use. 08:00.629 --> 08:02.539 So whenever I do any calculation of emf, 08:02.535 --> 08:04.485 I'm not going to worry about the sign. 08:04.490 --> 08:06.880 In the end, we'll fix the sign so it makes sense. 08:06.879 --> 08:10.359 That's going to be our formula. 08:10.360 --> 08:13.930 All right, so now let's continue with this one, 08:13.925 --> 08:18.185 and I remind you how we'd explained everything regarding 08:18.190 --> 08:20.130 the other experiment. 08:20.129 --> 08:26.009 Remember we had a magnetic field going into the board, 08:26.014 --> 08:28.684 only up to some point. 08:28.680 --> 08:33.600 To the right of this point, it's everywhere uniform. 08:33.600 --> 08:40.100 Then you had a loop which we were dragging to the right. 08:40.100 --> 08:44.590 And I put a light bulb here, and I said if you drag it, 08:44.592 --> 08:46.842 the light bulb will glow. 08:46.840 --> 08:48.900 Now we can understand because there's an emf, 08:48.899 --> 08:51.859 and the emf is coming because in this example, 08:51.860 --> 08:55.870 the term that really matters for the emf is the v x 08:55.865 --> 08:56.985 B term. 08:56.990 --> 08:58.750 There is no dB/dt. 08:58.750 --> 09:00.260 It's fixed. 09:00.259 --> 09:02.649 Somebody's holding a magnet, shoving flux into the 09:02.645 --> 09:03.275 blackboard. 09:03.279 --> 09:04.189 You're carrying the loop. 09:04.190 --> 09:06.870 So it's the v x B that gives you the emf. 09:06.870 --> 09:07.860 That's fine. 09:07.860 --> 09:10.950 That's the part we're not very impressed with in this law. 09:10.950 --> 09:13.050 And let's look at the - sign. 09:13.048 --> 09:17.748 The - sign will be telling you which way the current will flow. 09:17.750 --> 09:21.080 You are increasing the flux going into the blackboard, 09:21.082 --> 09:24.792 therefore the current that is generated will try to put flux 09:24.793 --> 09:26.873 coming out of the blackboard. 09:26.870 --> 09:31.270 And that means the current will flow that way. 09:31.269 --> 09:33.779 And that agrees with what you expect from fundamental 09:33.783 --> 09:35.913 principles, because if the wire is moving 09:35.908 --> 09:38.378 to the right, B is into the board, 09:38.380 --> 09:41.570 v x B will produce an emf this way. 09:41.570 --> 09:43.620 So we understand everything here. 09:43.620 --> 09:47.880 I also explained to you that the energy balance is that the 09:47.881 --> 09:50.861 work done on the wire, on the light bulb, 09:50.860 --> 09:53.780 is paid for by the person pulling this loop, 09:53.779 --> 09:55.359 because the minute you have a current, 09:55.360 --> 09:57.850 that current doesn't like to be dragged across a field. 09:57.850 --> 10:01.310 There is some BlI force and when you overcome the force 10:01.307 --> 10:04.477 with your mechanical force, you convert mechanical energy 10:04.481 --> 10:05.901 to electrical energy. 10:05.899 --> 10:10.239 There's one subtlety that is usually overlooked, 10:10.243 --> 10:14.593 which is the following - if you go to this wire, 10:14.586 --> 10:17.356 here is the piece of wire. 10:17.360 --> 10:20.720 There are some charges in here the wire has moving at a 10:20.724 --> 10:21.974 velocity v. 10:21.970 --> 10:24.220 And there is a B somewhere, maybe like that. 10:24.220 --> 10:28.340 And we said v x B is a certain force. 10:28.340 --> 10:32.490 But if there's a current flowing in the wire, 10:32.490 --> 10:36.170 it also has a speed u along the wire. 10:36.168 --> 10:38.698 This is something we saw even in the loop problem. 10:38.700 --> 10:45.730 So the real force of a charge on the wire is not simply 10:45.727 --> 10:51.317 v x B, but v u x 10:51.323 --> 10:55.883 B⋅dl. 10:55.879 --> 10:58.309 Because it's not just the motion of the wire that we take 10:58.313 --> 11:00.313 for the velocity, because charges are moving in 11:00.313 --> 11:00.883 the wire. 11:00.879 --> 11:05.349 The net velocity is v u, to some number like 11:05.350 --> 11:05.900 that. 11:05.899 --> 11:09.879 But when you do the emf, you don't have to worry about 11:09.875 --> 11:13.095 this part of the velocity along the wire. 11:13.100 --> 11:15.850 I'll give you a second to figure out why. 11:15.850 --> 11:17.240 But just take that part. 11:17.240 --> 11:18.790 Why am I allowed to ignore it? 11:18.788 --> 11:22.618 Because u x B is perpendicular to u, 11:22.615 --> 11:25.345 and dl is parallel to u. 11:25.350 --> 11:28.190 u is the direction of motion in the wire and dl 11:28.192 --> 11:29.142 is along the wire. 11:29.139 --> 11:31.569 So if you take a vector, cross it with something else 11:31.565 --> 11:34.125 and take the dot product with another vector parallel to 11:34.129 --> 11:35.389 itself, you will get 0. 11:35.389 --> 11:39.559 That's why you don't worry about the extra term in 11:39.557 --> 11:41.257 computing the emf. 11:41.259 --> 11:43.729 This is a subtle point for those of you who wake up with a 11:43.725 --> 11:45.235 sweat in the middle of the night. 11:45.240 --> 11:47.840 I'm just trying to tell you, calm down, it's okay. 11:47.840 --> 11:53.640 It's really present as a force, but not as an emf. 11:53.639 --> 11:57.839 In fact, it's going to push the charge perpendicular to the wire 11:57.837 --> 12:00.767 and not do any work going around the loop. 12:00.769 --> 12:04.489 So this is one topic. 12:04.490 --> 12:10.810 I'm going to illustrate the reality of this result by 12:10.812 --> 12:18.352 describing to you the operation of what's called a betatron. 12:18.350 --> 12:21.610 You remember the cyclotron, what it does, 12:21.607 --> 12:22.257 right? 12:22.259 --> 12:27.209 You had these Ds and the charge was jumping around from one D to 12:27.210 --> 12:32.240 the other when you put a certain voltage, so this is positive and 12:32.238 --> 12:33.808 that's negative. 12:33.808 --> 12:37.288 It bends in a magnetic field which is going in the plane into 12:37.293 --> 12:38.283 the blackboard. 12:38.279 --> 12:40.689 Then it gets another kick and when it comes here, 12:40.692 --> 12:43.412 you reverse the polarity, so that this becomes and this 12:43.405 --> 12:44.105 becomes -. 12:44.110 --> 12:49.520 It picks up another kick in velocity and it keeps doing 12:49.518 --> 12:50.218 that. 12:50.220 --> 12:54.040 And the remarkable thing that was noticed is if you write the 12:54.043 --> 12:57.653 equation mv^(2)/R, which is the force you need to 12:57.653 --> 13:03.813 bend into a circle, that's going to be qvB. 13:03.808 --> 13:12.308 So if you cancel the velocity here, you find v/R to be 13:12.307 --> 13:14.287 qB/m. 13:14.288 --> 13:18.448 v/R is the angular frequency with which it rotates 13:18.445 --> 13:21.335 and that's independent of the radius. 13:21.340 --> 13:24.820 Therefore you have to reverse the polarity with the same 13:24.815 --> 13:28.345 regularity, with the same frequency, even as the particle 13:28.354 --> 13:30.444 picks up more and more speed. 13:30.440 --> 13:32.630 It's not that you have to keep changing the rate at which you 13:32.634 --> 13:33.334 flip the voltage. 13:33.330 --> 13:35.680 Then you don't have any means of doing it. 13:35.678 --> 13:37.488 But if it's flipping at a definite rate, 13:37.494 --> 13:40.284 then of course you can get a generator that generates voltage 13:40.284 --> 13:42.244 at that rate and you can make it work. 13:42.240 --> 13:46.820 The key to all of that was this remarkable fact that omega does 13:46.822 --> 13:48.822 not depend on the radius. 13:48.820 --> 13:53.450 But that has a weakness that as the particle picks up speed, 13:53.450 --> 13:57.200 eventually you will have to use the fact that the real momentum 13:57.202 --> 14:01.912 of a particle is not mv, but mv divided by this 14:01.908 --> 14:06.718 factor, which comes from relativistic 14:06.721 --> 14:07.961 effects. 14:07.960 --> 14:11.120 Then you can show the centrifugal force you need. 14:11.120 --> 14:13.870 The centripetal force you need is not given by 14:13.873 --> 14:19.673 mv/R, but rather it's given by this 14:19.674 --> 14:25.794 momentum times omega, where momentum is given by this 14:25.794 --> 14:27.884 formula and not by this. 14:27.879 --> 14:30.399 Now that's a homework problem where you'll get enough time to 14:30.395 --> 14:31.105 think about it. 14:31.110 --> 14:34.360 That will show you that once the momentum formula deviates 14:34.359 --> 14:37.779 appreciably from p = mv to p equal to this, 14:37.779 --> 14:40.569 then condition for the cyclotron orbit, 14:40.570 --> 14:44.960 the frequency being independent of radius will fail. 14:44.960 --> 14:47.060 Essentially, as the particle gets more 14:47.061 --> 14:49.051 velocity, it's harder to push it. 14:49.048 --> 14:52.338 Your ability to accelerate it changes and therefore this 14:52.341 --> 14:53.421 condition fails. 14:53.418 --> 14:58.078 So the cyclotron can accelerate particles only up to velocities 14:58.077 --> 15:02.057 where the relativistic corrections are unimportant. 15:02.058 --> 15:05.298 But the betatron, which I'm going to describe to 15:05.297 --> 15:07.667 you, is a device that actually 15:07.673 --> 15:12.143 manages to accelerate particles even as their motion becomes 15:12.142 --> 15:16.192 relativistic, even as the momentum is given 15:16.191 --> 15:17.811 by that formula. 15:17.808 --> 15:20.338 So I will tell you how that works. 15:20.340 --> 15:26.890 You've got these poles of some magnet, the side view, 15:26.888 --> 15:30.538 producing a magnetic field. 15:30.539 --> 15:32.149 Let's look at the top view. 15:32.149 --> 15:35.799 In the top view, you've got some magnetic field. 15:35.798 --> 15:43.558 Let's say it's coming out of the blackboard. 15:43.558 --> 15:48.328 So if you put a particle here, some velocity v, 15:48.333 --> 15:53.293 v x B is to the right and it will bend. 15:53.289 --> 15:55.329 But here's what we want to do. 15:55.330 --> 16:02.600 We're going to have it go in a circle of definite radius and 16:02.604 --> 16:05.074 yet pick up speed. 16:05.070 --> 16:07.610 I'll show you how that happens. 16:07.610 --> 16:12.530 So what you do is, this is not a fixed bar magnet. 16:12.528 --> 16:16.298 It's an electromagnet, the current in which you're 16:16.303 --> 16:21.083 changing, and therefore the field is in fact time dependent. 16:21.080 --> 16:25.500 Furthermore, the field will be strong near 16:25.504 --> 16:29.504 the middle and weak near the edges. 16:29.500 --> 16:34.290 Now if that field is changing, then if you draw any loop of 16:34.288 --> 16:38.008 radius r, the flux through that loop is 16:38.005 --> 16:39.075 changing. 16:39.080 --> 16:42.610 Therefore the electric field will obey this condition, 16:42.613 --> 16:44.683 2Πr times E. 16:44.678 --> 16:51.618 The electric field will then also be going around in circles. 16:51.620 --> 16:53.440 It will encircle the magnetic field. 16:53.440 --> 16:56.410 If the magnetic field is coming from ceiling to the floor, 16:56.410 --> 16:59.330 electric field will be horizontal, horizontal circles. 16:59.330 --> 17:02.130 And by symmetry, they'll be circles centered at 17:02.126 --> 17:03.766 the center of the magnet. 17:03.769 --> 17:06.159 That's the direction of the electric field. 17:06.160 --> 17:08.920 This field coming from top to bottom, if it changes in 17:08.915 --> 17:11.975 strength, will generate a field in a horizontal plane that's 17:11.984 --> 17:13.394 going round in circles. 17:13.390 --> 17:17.010 So if you put a charge there, you'll push it and it will 17:17.006 --> 17:17.726 speed up. 17:17.730 --> 17:19.010 How big is the field? 17:19.009 --> 17:21.469 Well, the field is constant on the circle of radius r. 17:21.470 --> 17:23.300 It's pointing azimuthally. 17:23.298 --> 17:26.948 So its line integral is just 2Πr times E at 17:26.949 --> 17:27.629 r. 17:27.630 --> 17:34.360 That's going to be = (forget the - sign) d/dt of 17:34.364 --> 17:40.854 B⋅dA inside that circle. 17:40.848 --> 17:43.748 Now that B⋅dA 17:43.752 --> 17:47.352 I'm going to write as Πr^(2) ties some 17:47.345 --> 17:48.945 average B. 17:48.950 --> 17:50.350 That's how we define the average. 17:50.348 --> 17:53.708 The total flux through the radius when the field is varying 17:53.707 --> 17:56.947 with distance can always be written as the area times the 17:56.950 --> 18:00.250 average B and the time derivative of that is, 18:00.250 --> 18:07.230 if you like, d/dt of 18:07.233 --> 18:13.953 B_bar. 18:13.950 --> 18:22.480 So I'm going to write now the equation I get. 18:22.480 --> 18:26.510 E of r, if you solve for it, 18:26.506 --> 18:32.096 is going to be r/2 times d/dt of the average 18:32.103 --> 18:33.383 B. 18:33.380 --> 18:40.390 Average is inside the circle of radius r. 18:40.390 --> 18:44.350 Now if that's the electric field you multiply both sides by 18:44.349 --> 18:48.719 q, which is the charge of the object, that's the force. 18:48.720 --> 18:52.430 That force is the rate of change of momentum. 18:52.430 --> 19:00.230 So rate of change of momentum is qr/2 times rate of 19:00.226 --> 19:04.326 change of the average field. 19:04.328 --> 19:06.728 Now if this is the rate of change and that's equal to the 19:06.731 --> 19:11.301 rate of change of B, then p of t = 19:11.295 --> 19:18.205 qr/2 times the average B at time t. 19:18.210 --> 19:19.580 Actually, I'm making one assumption here. 19:19.579 --> 19:26.959 Do you know what that is? 19:26.960 --> 19:28.910 I'm assuming the initial momentum was 0, 19:28.910 --> 19:31.060 because if you know only the rate of change, 19:31.058 --> 19:32.708 you can add a constant to it. 19:32.710 --> 19:35.550 So I'm assuming that t = 0, p was 0. 19:35.548 --> 19:38.898 So that's the momentum of this guy after time t. 19:38.900 --> 19:42.320 So as the average B is growing, the momentum is 19:42.323 --> 19:46.203 growing, it has this value at this instant at that radius. 19:46.200 --> 19:49.730 Now what else do we need to make sure that solution makes 19:49.733 --> 19:50.243 sense. 19:50.240 --> 19:56.290 What do you need? 19:56.288 --> 19:57.718 Is there anything else to this story? 19:57.720 --> 19:58.590 Yes? 19:58.589 --> 20:01.209 Would you like to guess? 20:01.210 --> 20:05.770 If a particle picks up speed, why would it continue to be in 20:05.770 --> 20:08.400 that orbit that I've shown here. 20:08.400 --> 20:14.090 What does it take to keep it in that orbit. 20:14.089 --> 20:14.819 Pardon me? 20:14.818 --> 20:16.818 Student: The force of that entity. 20:16.818 --> 20:18.468 Prof: Which force, I'm sorry? 20:18.470 --> 20:20.940 Student: If it's in an orbit of constant radius, 20:20.942 --> 20:23.412 then the velocity's increasing the force, and that's the 20:23.414 --> 20:24.004 increase. 20:24.000 --> 20:26.460 Prof: Right, but at a given velocity. 20:26.460 --> 20:28.600 Even at a given velocity, you need a force to bend 20:28.596 --> 20:29.726 something into a circle. 20:29.730 --> 20:31.780 You guys know that. 20:31.778 --> 20:34.708 Remember that force from last term? 20:34.710 --> 20:40.670 That force is mv^(2) over r. 20:40.670 --> 20:43.920 That comes from saying that the particle has a velocity momentum 20:43.923 --> 20:46.043 like that now, momentum like that a little 20:46.042 --> 20:46.562 later. 20:46.558 --> 20:49.518 There's a change in momentum pointing towards the center and 20:49.515 --> 20:52.315 you can calculate that as mv^(2) over r. 20:52.318 --> 20:55.278 So who's going to provide that force? 20:55.279 --> 20:57.669 In other words, things don't move in a circle 20:57.670 --> 20:59.790 unless you pull them into the center. 20:59.788 --> 21:05.378 My question is, who's going to do that for this 21:05.380 --> 21:06.110 guy? 21:06.108 --> 21:08.688 Who's going to provide that force? 21:08.690 --> 21:18.330 Any idea? 21:18.328 --> 21:20.228 Don't worry about the sign or anything. 21:20.230 --> 21:27.360 Is there anything that can push this guy towards the center? 21:27.359 --> 21:28.509 Want to make a guess? 21:28.509 --> 21:29.289 Student: The magnetic field? 21:29.288 --> 21:30.688 Prof: It's the magnetic force, 21:30.690 --> 21:33.950 because that's a charged particle, moving in a magnetic 21:33.946 --> 21:36.376 field, and the force of that is 21:36.380 --> 21:39.530 q times v times B. 21:39.529 --> 21:43.579 But this B is the B at the orbital radius. 21:43.579 --> 21:44.759 It's not the average B. 21:44.759 --> 21:47.579 This is the B where the particle really is. 21:47.578 --> 21:49.478 So we're going to keep the particle at a fixed radius. 21:49.480 --> 21:54.120 It's the B at that radius that matters. 21:54.118 --> 22:01.538 Therefore you cancel the velocity, you find that mv = 22:01.536 --> 22:07.736 qrB_0 and that's the momentum. 22:07.740 --> 22:16.090 But we also saw the momentum at time t = this. 22:16.088 --> 22:18.598 They are both expressions for the momentum at time t. 22:18.598 --> 22:21.438 This just says the magnetic field hopefully is strong enough 22:21.442 --> 22:22.072 to bend it. 22:22.068 --> 22:24.848 That's the field that you need, but I'm just going to equate 22:24.853 --> 22:26.413 the momentum computed two ways. 22:26.410 --> 22:31.430 That tells me this interesting result, that 22:38.848 --> 22:41.018 So let's summarize what I'm saying. 22:41.019 --> 22:42.859 I want you to visualize this. 22:42.858 --> 22:45.288 You've got a magnetic field from ceiling to floor. 22:45.289 --> 22:47.139 It's pointing down, let's say. 22:47.140 --> 22:50.800 You put a little charge there, it won't do anything. 22:50.798 --> 22:54.088 But if you change that field, because of this law, 22:54.088 --> 22:56.088 Faraday's law, the change in field will 22:56.086 --> 22:58.186 produce a circulating electric field, 22:58.190 --> 23:00.830 that circles the changing magnetic flux and going around 23:00.827 --> 23:03.067 in circles, and it will accelerate the 23:03.065 --> 23:07.035 particle along a circle, because it is bending around a 23:07.037 --> 23:07.677 circle. 23:07.680 --> 23:11.100 But at the same time, you need the right magnetic 23:11.097 --> 23:13.517 field to keep it in that circle. 23:13.519 --> 23:16.369 But the particle is speeding up. 23:16.368 --> 23:19.478 Therefore the force needed is also increasing. 23:19.480 --> 23:22.120 So you're in addition to increasing the average field 23:22.122 --> 23:25.132 over the loop over the circle, to produce the emf, 23:25.134 --> 23:29.214 you need to have the right field at the boundary to provide 23:29.212 --> 23:33.432 exactly the right force needed to keep it on an orbit of that 23:33.432 --> 23:35.192 radius at that time. 23:35.190 --> 23:37.630 Therefore the story is, as time passes, 23:37.631 --> 23:40.591 the field increases in strength, the particle's 23:40.585 --> 23:42.765 tangential momentum increases. 23:42.769 --> 23:45.749 And the radial force to keep it in orbit at the tangential 23:45.748 --> 23:48.788 momentum also increases, and the demand when you look at 23:48.786 --> 23:50.986 it says that the field at the periphery, 23:53.965 --> 23:55.615 the average field. 23:55.618 --> 23:58.118 So you'll have to build a magnet very carefully. 23:58.118 --> 24:01.188 You can always build a magnet whose field varies as you go off 24:01.185 --> 24:03.745 center that gets weak when you go off the center. 24:03.750 --> 24:07.850 It should go off the center in such a way that by the time you 24:07.854 --> 24:11.024 come to this radius, the field strength there is 24:11.016 --> 24:12.426 half the average. 24:12.430 --> 24:15.020 Once you design a magnet that way you just crank up the 24:15.015 --> 24:18.075 current and let a particle go at the radius, it will pick up more 24:18.078 --> 24:19.418 and more and more speed. 24:19.420 --> 24:22.760 Even though it's going faster and faster, the field at the 24:22.763 --> 24:26.403 orbit will be just right to push it towards the center with the 24:26.402 --> 24:27.402 right amount. 24:27.400 --> 24:31.000 One of the homework problems was to show something I made a 24:31.001 --> 24:33.301 big deal about, namely, here I've used 24:33.299 --> 24:35.039 relativistic kinematics. 24:35.038 --> 24:37.828 Momentum was mv, the force to the center is 24:37.833 --> 24:38.863 mv^(2)/r. 24:38.858 --> 24:42.088 But the homework problem shows you that even in the 24:42.087 --> 24:44.867 relativistic case, the momentum is mv 24:44.865 --> 24:47.895 divided by all of that, this is still true. 24:47.900 --> 24:51.550 So the betatron, in spite of the simple example 24:51.548 --> 24:54.008 I've given here, actually works, 24:54.007 --> 24:57.257 even if the particle is relativistic. 24:57.259 --> 24:59.829 In fact, the only reason the betatron eventually fails is 24:59.832 --> 25:02.362 that when particles start going in a circle at very, 25:02.358 --> 25:04.428 very high speed and they are charged, 25:04.430 --> 25:06.940 they start radiating energy. 25:06.940 --> 25:09.320 That's another aspect that we have not discussed in our course 25:09.317 --> 25:11.307 yet, but accelerating charges 25:11.305 --> 25:13.155 radiate energy, and eventually, 25:13.163 --> 25:14.583 you cannot put in enough energy. 25:14.578 --> 25:17.198 It radiates more than you can give it. 25:17.200 --> 25:19.030 So then you need other things called synchrotrons. 25:19.028 --> 25:22.208 So that's a constant struggle for people building 25:22.207 --> 25:23.197 accelerators. 25:23.200 --> 25:27.070 It's easy to get more and more velocity, but if you bend them 25:27.068 --> 25:29.328 into a circle, they start radiating, 25:29.326 --> 25:31.386 emit gamma rays, emit light. 25:31.390 --> 25:35.920 And that energy eventually is so big that you cannot 25:35.923 --> 25:39.483 compensate it with your pushing force. 25:39.480 --> 25:45.390 So the next topic I want to discuss is in a totally 25:45.390 --> 25:50.470 different vein, and that has to do with more 25:50.471 --> 25:54.611 practical issues like this one. 25:54.608 --> 26:03.418 This is going to be a power generator. 26:03.420 --> 26:07.330 Remember, I told you one way to make electricity in your house 26:07.325 --> 26:11.225 is to take this coil and tell somebody to carry it and run. 26:11.230 --> 26:14.080 Then the light bulb will glow here. 26:14.078 --> 26:16.748 Or you can have somebody carry the magnet the other way and the 26:16.747 --> 26:18.337 light bulb will glow in your house. 26:18.338 --> 26:21.998 But there's a more clever way to make the light bulb glow, 26:22.000 --> 26:24.250 which I think you guys have seen. 26:24.250 --> 26:27.250 By the way, I think these are all things you have seen in one 26:27.253 --> 26:29.413 form or another, so I'm not going to put too 26:29.406 --> 26:31.106 much energy into these things. 26:31.108 --> 26:34.508 I just want to tell you things you may have missed. 26:34.509 --> 26:36.499 So here's some magnet. 26:36.500 --> 26:41.090 Here's some field lines going from here to there, 26:41.093 --> 26:42.723 north to south. 26:42.720 --> 26:52.280 In that magnet you put a coil, which looks like this. 26:52.279 --> 26:55.629 The plane of the coil that's pointing in that direction of 26:55.627 --> 26:57.697 the magnetic moment, or if you want, 26:57.699 --> 26:59.329 the area vector points that way, 26:59.329 --> 27:03.949 the B field is horizontal. 27:03.950 --> 27:09.060 Now if you spin that coil, I think you know what will 27:09.056 --> 27:09.936 happen. 27:09.940 --> 27:13.580 The flux through the coil is going to change and the rate of 27:13.583 --> 27:15.933 change of flux will give you an emf. 27:15.930 --> 27:17.380 So what is the emf? 27:17.380 --> 27:20.040 First you've got to find the flux through the coil. 27:20.038 --> 27:22.998 The flux through the coil is the area of the coil, 27:23.001 --> 27:25.661 the magnetic field, times cosine of the angle 27:25.662 --> 27:26.692 between them. 27:26.690 --> 27:31.760 This would be angle theta in the figure. 27:31.759 --> 27:36.569 But you have attached this to an axle and you're going to just 27:36.567 --> 27:39.087 keep spinning it mechanically. 27:39.088 --> 27:42.768 And you agree to spin it at a uniform rate, 27:42.769 --> 27:43.469 omega. 27:43.470 --> 27:47.760 So theta is omega t. 27:47.759 --> 27:52.389 Then you can see that the emf, which is 27:52.392 --> 27:59.712 -dΦ/dt = ABsinωt 27:59.707 --> 28:03.727 times another ω. 28:03.730 --> 28:06.450 That's the emf. 28:06.450 --> 28:08.080 Now what about the - sign. 28:08.079 --> 28:08.859 Forget the - sign. 28:08.858 --> 28:12.958 We can figure out what the emf--which way the current will 28:12.961 --> 28:13.971 try to flow. 28:13.970 --> 28:16.360 Look at this coil here. 28:16.358 --> 28:19.368 Let's decide how we want to turn this guy. 28:19.368 --> 28:22.978 If you turn it in the manner I've shown here, 28:22.979 --> 28:27.819 it's going from some angle like that, eventually to an angle 28:27.817 --> 28:31.097 perpendicular to the magnetic field. 28:31.098 --> 28:32.778 What's happening to the flux through it? 28:32.779 --> 28:37.249 Is it increasing or decreasing? 28:37.250 --> 28:38.490 Increasing. 28:38.490 --> 28:41.760 Increasing in this direction. 28:41.759 --> 28:45.249 So it will fight it by trying to produce a flux going in the 28:45.248 --> 28:46.488 opposite direction. 28:46.490 --> 28:49.600 Therefore it will try to produce a current that looks 28:49.601 --> 28:50.321 like this. 28:50.318 --> 28:52.468 That's the direction of your emf. 28:52.470 --> 28:56.510 Emf, given a chance, will drive a current as shown 28:56.509 --> 29:01.369 here, because that one by right hand rule will have the flux 29:01.374 --> 29:03.604 going the opposite way. 29:03.598 --> 29:05.408 So that's why I don't care about the - sign. 29:05.410 --> 29:09.480 I know which way the current will flow if I let it flow. 29:09.480 --> 29:13.460 But now if this is an open circuit like this, 29:13.461 --> 29:18.531 the two wires sit here, what do you think will happen? 29:18.528 --> 29:22.418 What do you think will happen in that case? 29:22.420 --> 29:27.400 If you're an electric charge in that wire, what will you do 29:27.396 --> 29:28.766 under the emf? 29:28.769 --> 29:33.199 You will follow the emf and you will run from this terminal to 29:33.198 --> 29:34.358 that terminal. 29:34.358 --> 29:39.028 So let me blow up that picture for you. 29:39.029 --> 29:44.849 If the current is trying to go that way, then charges will 29:44.846 --> 29:48.416 leave like this and pile up here. 29:48.420 --> 29:52.450 They cannot go very far because it's an open circuit. 29:52.450 --> 29:56.450 At some point, these guys piled up here--so 29:56.453 --> 30:02.083 this is the induced electric field obeying that equation. 30:02.078 --> 30:03.928 In fact, it's not induced electric field. 30:03.930 --> 30:08.210 This is just the v x B force on the magnet, 30:08.210 --> 30:09.510 on the charges. 30:09.509 --> 30:12.879 But then these charges will produce and electrostatic force 30:12.882 --> 30:15.952 called the Coulomb field, which is the electric field due 30:15.946 --> 30:18.816 to charges, which will oppose it, 30:18.816 --> 30:22.806 till the two cancel inside the wire. 30:22.808 --> 30:25.058 In other words, this wire is assumed to be a 30:25.061 --> 30:26.111 perfect conductor. 30:26.108 --> 30:28.528 You all know that you cannot have an electric field in a 30:28.528 --> 30:29.408 perfect conductor. 30:29.410 --> 30:31.370 You can say, "Hey, why do you have an 30:31.368 --> 30:32.658 electric field now?" 30:32.660 --> 30:34.840 The real statement is, in a perfect conductor, 30:34.839 --> 30:37.509 you cannot have any net charge on the charged particles, 30:37.506 --> 30:39.926 because then they will pick up infinite speed. 30:39.930 --> 30:42.050 So if there's a v x B force, 30:42.053 --> 30:44.853 that is actually canceled by an electrostatic force. 30:44.848 --> 30:49.938 The two of them cancel inside the wire. 30:49.940 --> 30:53.350 And the line integral of the v x B, 30:53.348 --> 30:55.908 which is the emf, will numerically equal the 30:55.910 --> 30:59.240 integral of the electric field from this terminal to that 30:59.244 --> 31:00.024 terminal. 31:00.019 --> 31:02.459 But now if you put the whole thing in a box and you don't 31:02.455 --> 31:04.105 know anything and you come outside, 31:04.108 --> 31:06.368 what you will find is this will be positively charged, 31:06.369 --> 31:07.579 this will be negatively charged. 31:07.578 --> 31:11.828 There'll be voltage between those two which will numerically 31:11.826 --> 31:13.046 equal your emf. 31:13.049 --> 31:16.689 This will be the polarity. 31:16.690 --> 31:18.130 So I'm giving you an option. 31:18.130 --> 31:20.630 If you don't want to look under the hood, you simply say 31:20.632 --> 31:22.682 whenever you rotate a coil, you get an emf. 31:22.680 --> 31:24.350 That emf is like a voltage. 31:24.349 --> 31:26.049 Case closed. 31:26.048 --> 31:28.488 But if you really want to know what's happening inside the wire 31:28.493 --> 31:31.843 that made up that coil, inside the wire the net force 31:31.835 --> 31:36.275 on the charges is actually 0, or an infinitesimal amount left 31:36.279 --> 31:38.059 to make them move this way. 31:38.058 --> 31:41.418 That's because the v x B force is countered by 31:41.419 --> 31:44.889 an internal electrostatic force due to pile up of charges that 31:44.892 --> 31:47.172 opposes it so that inside it's free. 31:47.170 --> 31:50.140 It's just like the ski experiment I told you, 31:50.140 --> 31:52.030 where you have a ski lift. 31:52.029 --> 31:54.989 You come down the ski lift, you come here, 31:54.987 --> 31:58.087 and here is the lift that carries you up. 31:58.088 --> 32:01.978 But during the time gravity's acting down and the force of the 32:01.980 --> 32:04.660 lift is exactly equal to mg and gravity. 32:04.660 --> 32:08.000 And once you're outside the lift, it's gravity that brings 32:07.999 --> 32:09.229 you down back here. 32:09.230 --> 32:12.410 Similarly, inside the thing the two guys are opposed to each 32:12.407 --> 32:12.837 other. 32:12.838 --> 32:15.878 Outside this region there is no v x B force, 32:15.875 --> 32:18.695 but the electrostatic force has a line integral that's 32:18.698 --> 32:20.188 independent of the path. 32:20.190 --> 32:22.970 So integral that way the same as the integral that way. 32:22.970 --> 32:26.560 So if you put a circuit here, it will produce a voltage 32:26.563 --> 32:30.493 difference with this being positive, that being negative. 32:30.490 --> 32:35.730 So the bottom line is that if you have open circuit, 32:35.730 --> 32:38.440 and if you put a volt meter that measures voltage, 32:38.440 --> 32:41.810 you will get exactly that time dependent voltage. 32:41.809 --> 32:44.199 So this voltage is not constant. 32:44.200 --> 32:51.020 It will look like this, t versus V. 32:51.019 --> 32:54.079 So in the linear loop, when you drag it and run, 32:54.080 --> 32:55.970 you get a constant voltage. 32:55.970 --> 33:00.290 Here you get a time dependent one, and most of the supplies in 33:00.285 --> 33:03.605 the world are either 50 cycles or 60 cycles, 33:03.608 --> 33:09.508 and they come from rotating a coil in this magnetic field. 33:09.509 --> 33:19.609 Now how much is the work done by the person rotating this coil 33:19.612 --> 33:21.602 right now? 33:21.598 --> 33:24.808 I'm telling you the coil is made of massless, 33:24.814 --> 33:26.864 perfectly conducting wire. 33:26.858 --> 33:32.078 What work do you have to do to spin it? 33:32.079 --> 33:33.269 Any ideas? 33:33.269 --> 33:37.329 Student: > 33:37.328 --> 33:38.228 Prof: Which one, dE? 33:38.230 --> 33:39.920 Student: Edq. 33:39.920 --> 33:42.370 Prof: Yeah, but I'm saying the total 33:42.366 --> 33:46.036 force--you don't have to do any work because there is no current 33:46.038 --> 33:47.668 flowing in the coil yet. 33:47.670 --> 33:48.480 There's no current. 33:48.480 --> 33:56.670 If there's no current, it's BlI force. 33:56.670 --> 34:01.360 So what we want to do then is do something more interesting 34:01.362 --> 34:05.572 where you bring it here, connect it to a resistor. 34:05.569 --> 34:07.949 Then the current will flow. 34:07.950 --> 34:09.050 Now we're getting something. 34:09.050 --> 34:11.240 Till now, we were getting nothing from the generator. 34:11.239 --> 34:14.579 In other words, look, where's a socket here? 34:14.579 --> 34:16.529 That's a socket. 34:16.530 --> 34:19.740 There's voltage there waiting for you to use. 34:19.739 --> 34:22.289 But you don't pay, because you're not drawing any 34:22.288 --> 34:22.818 current. 34:22.820 --> 34:25.790 You don't pay just because someone gave you the voltage. 34:25.789 --> 34:29.179 You take a battery in your hand, you don't run a current, 34:29.184 --> 34:30.704 you don't pay anything. 34:30.699 --> 34:32.639 That's what the situation is. 34:32.639 --> 34:35.569 But if I stick my fingers into that socket, then there is 34:35.567 --> 34:36.087 current. 34:36.090 --> 34:38.100 Then we'll all pay. 34:38.099 --> 34:40.549 The loss is I^(2)R, right? 34:40.550 --> 34:43.570 That's when you've got to explain to yourself, 34:43.565 --> 34:45.235 who is paying for this? 34:45.239 --> 34:50.489 Because we can see that the power in the resistor will be 34:50.489 --> 34:51.989 I^(2)R. 34:51.989 --> 34:55.129 I, if you want, is E^(2)/R, 34:55.128 --> 34:58.838 emf or voltage squared over R. 34:58.840 --> 35:03.270 That gives me ω^(2)A^(2)B^(2) 35:03.268 --> 35:06.978 sin^(2)ωt/R. 35:06.980 --> 35:09.550 That's the rate at which this power is consumed by the 35:09.547 --> 35:10.077 resistor. 35:10.079 --> 35:15.489 Someone's got to pay for that. 35:15.489 --> 35:17.649 The someone, I think you can imagine now, 35:17.650 --> 35:20.200 now that I've closed the circuit and a current is 35:20.197 --> 35:24.417 actually flowing, there is a torque on that loop 35:24.423 --> 35:26.643 in a magnetic field. 35:26.639 --> 35:28.919 The torque on any loop, you'll remember, 35:28.920 --> 35:30.560 is μ x B. 35:30.559 --> 35:33.609 μ is the magnetic moment. 35:33.610 --> 35:37.530 That happens to be area of the loop, current in the loop, 35:37.534 --> 35:42.094 B times the sine of the angle between area and B. 35:42.090 --> 35:45.350 That is torque. 35:45.349 --> 35:48.549 So that's the torque that you've got to fight. 35:48.550 --> 35:51.270 So you will have to apply mechanical force to turn that. 35:51.268 --> 35:54.678 If you're turning the crank by hand, the minute you put a load, 35:54.675 --> 35:56.595 you'll find it's hard to turn it. 35:56.599 --> 35:59.399 And the power, just like power is force times 35:59.404 --> 36:03.104 velocity, for rotational motion it is torque times angle of 36:03.101 --> 36:03.931 velocity. 36:03.929 --> 36:11.409 That gives me ABωI sinθ. 36:11.409 --> 36:14.339 That's the power, the mechanical power, 36:14.336 --> 36:16.106 P_m. 36:16.110 --> 36:20.120 But then that = ABω 36:20.115 --> 36:24.635 sinθ times I. 36:24.639 --> 36:26.399 What is I? 36:26.400 --> 36:30.980 The current in the loop is the emf divided by R. 36:30.980 --> 36:34.740 If you put the emf I got for you somewhere, 36:34.739 --> 36:42.299 then you'll find it's equal to A^(2)B^(2)ω^(2)/R 36:42.300 --> 36:46.080 sin^(2)θ. 36:46.079 --> 36:47.519 So this is how you have to balance. 36:47.518 --> 36:50.808 We did a similar calculation here. 36:50.809 --> 36:54.149 This is also a generator and if you find out the minute there's 36:54.148 --> 36:57.358 a current flowing here, you have I^(2)R or 36:57.360 --> 36:59.750 E^(2)/R energy loss here. 36:59.750 --> 37:01.870 But the minute the current flows through the loop, 37:01.871 --> 37:04.561 you will not be able to pull it to the right without paying the 37:04.556 --> 37:04.986 force. 37:04.989 --> 37:07.559 That force times the velocity will be the power and the two 37:07.556 --> 37:08.216 will balance. 37:08.219 --> 37:12.389 Similarly, here is a rotational problem, and the torque times 37:12.393 --> 37:16.083 angular velocity will exactly equal the power here. 37:16.079 --> 37:18.489 So the generators, you've got hydroelectric 37:18.487 --> 37:20.777 generators, or you've got steam, 37:20.782 --> 37:24.342 turbines that are spinning, initially, because the turbines 37:24.335 --> 37:25.985 in the real world have a real mass, 37:25.989 --> 37:27.159 it's not easy to spin them. 37:27.159 --> 37:29.269 You spend some power, but if you ignore that, 37:29.268 --> 37:32.768 the minute you put a device in your house into the socket, 37:32.768 --> 37:34.908 your toaster, you start drawing current, 37:34.909 --> 37:37.869 that current's got to flow right through the generator. 37:37.869 --> 37:40.409 It's going to make it that much harder to rotate the generator. 37:40.409 --> 37:43.049 That's when the steam turbines do their work. 37:43.050 --> 37:48.030 That's when they pay for it. 37:48.030 --> 37:54.480 Okay, so this is the end of the story about how to get power out 37:54.478 --> 37:57.548 of this and how it balances. 37:57.550 --> 38:11.090 I'm going to a fairly different notion called inductors and 38:11.085 --> 38:14.115 inductance. 38:14.119 --> 38:16.189 So here is the following phenomenon. 38:16.190 --> 38:23.400 I take some cardboard tube and around the cardboard tube I wrap 38:23.403 --> 38:30.273 some wire and I connect this to some alternating voltage. 38:30.268 --> 38:36.068 That means there is going to be some magnetic flux going through 38:36.065 --> 38:37.165 that coil. 38:37.170 --> 38:41.590 Then I bring a second wire, maybe I wrap it round a couple 38:41.594 --> 38:45.324 of times or a few times, and I leave it there. 38:45.320 --> 38:48.110 That's it. 38:48.110 --> 38:52.300 The question is, what will happen if I look at 38:52.300 --> 38:54.910 the two ends of this wire? 38:54.909 --> 38:57.019 You can see what will happen. 38:57.018 --> 39:02.788 If this current changes, the flux through this solenoid 39:02.791 --> 39:03.861 changes. 39:03.860 --> 39:06.790 That means the flux through this little guy alsochanges. 39:06.789 --> 39:09.579 Every loop has flux going through it and that also 39:09.579 --> 39:10.149 changes. 39:10.150 --> 39:11.380 So there's going to be an emf. 39:11.380 --> 39:14.670 For every turn of wire here, they'll be an emf equal to the 39:14.670 --> 39:16.090 rate of change of flux. 39:16.090 --> 39:20.260 And therefore that emf will try to draw a current, 39:20.262 --> 39:24.092 just like before, and the charges will pile up 39:24.094 --> 39:27.164 maybe like this at some instant. 39:27.159 --> 39:33.379 Now if you are outside all of this, you will just think there 39:33.380 --> 39:36.700 is a voltage available to you. 39:36.699 --> 39:40.379 But I want you to understand the origin of that voltage. 39:40.380 --> 39:42.110 This is like the ski lift here. 39:42.110 --> 39:46.310 Charges, once they come into this region, are pushed up the 39:46.306 --> 39:49.776 wire and later on, if you're connected to a load, 39:49.780 --> 39:51.880 they can drive a current. 39:51.880 --> 39:53.740 This is just like the generator I had there, 39:53.739 --> 39:57.199 except the flux is changing not due to any mechanical motion of 39:57.195 --> 40:01.345 a coil in a fixed field, but it's fixed coils in a 40:01.349 --> 40:03.999 changing magnetic field. 40:04.000 --> 40:06.290 So you understand, when I change the current here, 40:06.288 --> 40:07.688 you will get a voltage here. 40:07.690 --> 40:10.660 This is the first thing Faraday noticed, is that he thought 40:10.664 --> 40:13.234 first that magnetic field may produce a current. 40:13.230 --> 40:14.370 It didn't. 40:14.369 --> 40:17.459 Then he found a changing magnetic field is able to 40:17.460 --> 40:20.550 produce a current and this is why this happens. 40:20.550 --> 40:23.500 Once again, I want you to think about the following question - 40:23.496 --> 40:26.296 we are going to take such things and put them in electrical 40:26.297 --> 40:27.407 circuits and so on. 40:27.409 --> 40:29.839 We'll draw all kinds of circuits later on. 40:29.840 --> 40:32.400 And what I will do in all those calculations, 40:32.396 --> 40:35.476 I will say I start here, I go around a loop and I come 40:35.478 --> 40:38.208 back and the change in voltage should be 0. 40:38.210 --> 40:41.200 I'm going to use that principle. 40:41.199 --> 40:46.049 But there is one flaw if you don't think about it that says 40:46.054 --> 40:48.404 maybe I shouldn't do that. 40:48.400 --> 40:52.050 You remember that you can define a voltage only for a 40:52.047 --> 40:55.437 conservative force, whereas you definitely have non 40:55.438 --> 40:58.238 conservative forces that work in this problem. 40:58.239 --> 41:00.649 That's why they have a line integral not equal to 0. 41:00.650 --> 41:03.900 And yet we apply the laws of conservative forces, 41:03.900 --> 41:06.610 or a notion of a voltage in a circuit. 41:06.610 --> 41:08.010 I want to explain why that's allowed. 41:08.010 --> 41:10.910 If it never bothered you, you can ignore this part. 41:10.909 --> 41:13.379 But it's important to understand how we can have a 41:13.384 --> 41:16.014 notion of a voltage in the presence of time dependent 41:16.010 --> 41:17.020 magnetic fields. 41:17.018 --> 41:19.888 The line integral of E is not 0. 41:19.889 --> 41:25.029 Once again, what happens is, if you look at this coil, 41:25.028 --> 41:30.848 there may be an electric field that is induced now due to the 41:30.847 --> 41:33.367 changing flux of that. 41:33.369 --> 41:37.129 That will try to build up charges here and take them out 41:37.132 --> 41:38.092 of this end. 41:38.090 --> 41:39.340 After a while, these guys will say 41:39.344 --> 41:41.364 "Enough" and they'll start fighting you, 41:41.360 --> 41:44.210 till they set up an electrostatic field, 41:44.210 --> 41:48.720 a Coulomb field, inside the wire that cancels 41:48.722 --> 41:49.442 this. 41:49.440 --> 41:52.000 You understand, the integral of this electric 41:52.003 --> 41:55.443 field from top to bottom will numerically equal the integral 41:55.442 --> 41:58.592 of the induced electric field from bottom to top, 41:58.590 --> 42:01.070 because at every point they're equal in magnitude. 42:01.070 --> 42:07.320 Because you cannot have a non-zero field inside a wire. 42:07.320 --> 42:09.990 So the two fields have to be canceled. 42:09.989 --> 42:13.609 But this electrostatic field, built by these charges, 42:13.610 --> 42:16.400 is a conservative field, so if it does any work going 42:16.396 --> 42:19.976 this way, it will also do work going that 42:19.976 --> 42:20.446 way. 42:20.449 --> 42:22.909 Therefore if you don't open the black box, 42:22.909 --> 42:26.689 you don't know what's inside, if you just took the two wires 42:26.688 --> 42:29.728 coming out of it, the static field coming from 42:29.726 --> 42:33.176 this will be able to do work going from to - terminal. 42:33.179 --> 42:37.099 And if you put a resistor on its path, it will deliver some 42:37.097 --> 42:38.177 energy to you. 42:38.179 --> 42:40.319 The trick is, yes, there are changing 42:40.324 --> 42:43.004 magnetic fields, but they're hidden inside the 42:43.003 --> 42:43.543 coil. 42:43.539 --> 42:46.169 In the region outside the coil, we don't have to worry about 42:46.168 --> 42:49.538 the changing magnetic field, where an electric potential can 42:49.543 --> 42:52.763 be defined as the integral of the electric field. 42:52.760 --> 42:56.100 The electric field outside is entirely electrostatic. 42:56.099 --> 43:01.049 Electric field inside is a combination of electrostatic and 43:01.050 --> 43:03.440 induced one, which cancel. 43:03.440 --> 43:05.870 Once current begins to flow, you may worry that these 43:05.873 --> 43:06.953 charges will go away. 43:06.949 --> 43:11.269 There'll always be some electric charges making sure 43:11.268 --> 43:15.588 that the field inside the coil continues to be 0. 43:15.590 --> 43:18.810 Again, there's one abuse of terminology here, 43:18.809 --> 43:20.749 because I'm saying electrostatic, 43:20.751 --> 43:24.461 and yet this is a problem where the electric field you need is 43:24.456 --> 43:27.626 actually changing with time, because the rate of change of 43:27.625 --> 43:30.105 flux is changing with time, induced electric field is 43:30.106 --> 43:31.156 changing with time. 43:31.159 --> 43:34.699 The compensating electrostatic field is also changing with 43:34.704 --> 43:35.144 time. 43:35.139 --> 43:38.229 So you really should not use the laws of Coulomb for this 43:38.231 --> 43:39.841 problem, but it turns out, 43:39.844 --> 43:43.224 even though Coulomb's law is valid only strictly for fixed 43:43.217 --> 43:46.317 electric charges, as long as the velocities 43:46.317 --> 43:50.017 required is not too big, you can continue to use 43:50.021 --> 43:53.411 Coulomb's notion of an electrostatic force. 43:53.409 --> 43:56.169 Basic question is, you're telling the charges to 43:56.173 --> 43:58.413 constantly rearrange back and forth. 43:58.409 --> 44:00.269 First you want this terminal to be . 44:00.268 --> 44:02.028 A short time later, you may want that to be . 44:02.030 --> 44:05.260 How quickly can they rearrange? 44:05.260 --> 44:07.370 So that's connected to something called plasma 44:07.373 --> 44:08.833 frequency of these materials. 44:08.829 --> 44:10.899 And unless the frequencies are like a trillion, 44:10.896 --> 44:12.466 you don't have to worry about it. 44:12.469 --> 44:15.649 But any normal problem, like 50 cycles per second, 44:15.650 --> 44:19.090 the charges will be able to keep up with this changing 44:19.090 --> 44:19.740 field. 44:19.739 --> 44:22.669 In other words, can the charges go to the edge 44:22.670 --> 44:26.580 of a conductor and screen any internal field you're trying to 44:26.577 --> 44:27.357 produce? 44:27.360 --> 44:29.260 If it's a static field, they will kill it, 44:29.260 --> 44:31.810 because you're giving them all the time in the world. 44:31.809 --> 44:34.629 But if you keep changing your mind, one minute the external 44:34.630 --> 44:37.060 field goes this way, so the charges in the metal go 44:37.061 --> 44:38.231 that way to kill it. 44:38.230 --> 44:40.250 Suddenly you reverse it,they've got to go this way. 44:40.250 --> 44:42.120 So how quickly can they do this dance? 44:42.119 --> 44:44.209 There's a limit to how quickly they can do it. 44:44.210 --> 44:46.360 But that's a very, very high frequency, 44:46.362 --> 44:48.632 so you don't have to worry about that. 44:48.630 --> 44:58.390 Now we saw that the emf in the second coil = the flux in the 44:58.389 --> 45:04.509 second coil divided by d/dt. 45:04.510 --> 45:07.350 Now you've got to be a little careful about flux. 45:07.349 --> 45:13.359 Normally, the flux due to the magnetic field is defined as the 45:13.356 --> 45:17.786 integral of B⋅dA. 45:17.789 --> 45:19.739 But if you've got a coil made up of two loops, 45:19.739 --> 45:21.649 for example, and there's a magnetic flux 45:21.650 --> 45:24.300 going through them, then the emf is not simply the 45:24.304 --> 45:26.884 rate of change of the flux through one of them, 45:26.880 --> 45:30.330 but you've got to double it, because it's like two batteries 45:30.327 --> 45:31.027 in series. 45:31.030 --> 45:35.020 An emf is running round and round and round, 45:35.023 --> 45:39.483 so really, this is N_2 times the 45:39.481 --> 45:43.941 rate of change of the magnetic flux to that. 45:43.940 --> 45:46.950 So this Φ is not simply the magnetic 45:46.949 --> 45:47.489 flux. 45:47.489 --> 45:50.839 It's the magnetic flux times the number of turns the second 45:50.844 --> 45:51.254 coil. 45:51.250 --> 45:53.490 That's what the real emf is. 45:53.489 --> 45:57.959 A real emf is really rate of change of the flux linking with 45:57.956 --> 46:00.376 your whole circuit, dt. 46:00.380 --> 46:06.100 That equals number of turns times the actual magnetic flux 46:06.096 --> 46:07.296 dt. 46:07.300 --> 46:09.150 Do you follow that? 46:09.150 --> 46:11.940 Emf around a single loop is the rate of change of flux, 46:11.936 --> 46:14.926 but if your coil loops around twice, the end-to-end voltage 46:14.931 --> 46:16.171 will be double that. 46:16.170 --> 46:17.310 If I loops around three times, 46:17.309 --> 46:19.569 it will be triple that, assuming the same magnetic flux 46:19.570 --> 46:25.180 is going through all of them, which is the assumption I made 46:25.184 --> 46:26.044 here. 46:26.039 --> 46:31.549 So now you can see here that the flux in the second coil is 46:31.545 --> 46:37.045 due to the current in the first coil and the coefficient of 46:37.052 --> 46:41.992 proportionality is called the mutual inductance. 46:41.989 --> 46:47.179 So mutual inductance is how much flux you can get in the 46:47.182 --> 46:51.812 second coil per unit current in the first coil. 46:51.809 --> 46:59.419 Then you can write the emf in the second coil as 46:59.422 --> 47:07.362 −M_12 dI_1/dt. 47:07.360 --> 47:10.650 M_12 is called the mutual inductance of the 47:10.652 --> 47:13.112 first coil with respect to the second one. 47:13.110 --> 47:15.640 It's true, but very hard to prove that M_12 47:15.643 --> 47:17.323 is the same as M_21. 47:17.320 --> 47:22.100 Not at all obvious that if you drove a current in the first 47:22.099 --> 47:24.279 coil, it's going to have a magnetic 47:24.284 --> 47:27.124 field that's threading the second coil and you can show 47:27.121 --> 47:30.491 that the flux per current in the first coil due to the current in 47:30.485 --> 47:34.765 the second, it's given by the same number. 47:34.769 --> 47:36.249 M_12--yes? 47:36.250 --> 47:38.660 Student: How did the number of turns go away? 47:38.659 --> 47:40.509 Prof: Where did it go away? 47:40.510 --> 47:42.280 In this one you mean? 47:42.280 --> 47:45.580 That's all included in the definition of M. 47:45.579 --> 47:48.209 I'm going to now calculate M so you will see the 47:48.206 --> 47:49.566 number of turns coming in. 47:49.570 --> 47:54.450 So let's do the calculation of M for a simple problem. 47:54.449 --> 47:57.739 Here's one solenoid. 47:57.739 --> 48:00.479 It's got N_1 turns. 48:00.480 --> 48:01.560 And there's another guy. 48:01.559 --> 48:04.319 I'm just going to show you one turn of it, but it can have 48:04.317 --> 48:05.717 N_2 turns. 48:05.719 --> 48:09.809 And the flux is going through this. 48:09.809 --> 48:12.649 Now you remember that the magnetic field due to any 48:12.650 --> 48:14.920 solenoid is μ_0 times 48:14.923 --> 48:20.933 n, where n is the number of 48:20.931 --> 48:27.321 turns per unit length times I. 48:27.320 --> 48:35.430 Therefore the magnetic field in this solenoid = 48:35.427 --> 48:41.067 μ_0 n_2 48:41.068 --> 48:49.528 I_2--I'm sorry about that one. 48:49.530 --> 48:52.120 This is the first wire of the first coil. 48:52.119 --> 49:00.139 It's n_1I _1. 49:00.139 --> 49:05.989 And the flux of the magnetic field is μ_0n 49:05.994 --> 49:12.064 _1I_1 times the cross sectional area 49:12.056 --> 49:13.696 of the coil. 49:13.699 --> 49:16.819 Now the flux linking with the second one = 49:16.815 --> 49:20.915 μ_0n _1I_1-- 49:20.920 --> 49:24.950 sorry, n_1AN _2 times 49:24.954 --> 49:26.934 I_1. 49:26.929 --> 49:30.859 You see that? 49:30.860 --> 49:34.600 So the first coil has some wrapping density of wires, 49:34.596 --> 49:37.826 n_1 turns per unit length. 49:37.829 --> 49:40.269 I've shown you long back from Ampere's law, 49:40.266 --> 49:43.226 the magnetic field that produced this has that flux, 49:43.226 --> 49:44.616 that B value. 49:44.619 --> 49:48.089 The integral of the field, which is the magnetic flux, 49:48.088 --> 49:49.788 is just area times that. 49:49.789 --> 49:54.259 But the linking with the second coil is that flux times the 49:54.264 --> 49:57.124 number of turns in the second coil. 49:57.119 --> 50:00.509 That by definition = the mutual inductance 50:00.512 --> 50:05.642 M_21I_1 (I'm not going to call it 21 50:05.641 --> 50:06.471 or 12. 50:06.469 --> 50:13.159 It's independent of the order) = μ_0n 50:13.155 --> 50:17.395 _1N_2A. 50:17.400 --> 50:21.510 That's the mutual inductance. 50:21.510 --> 50:23.540 So if I give you two closed loops and I say, 50:23.543 --> 50:25.343 "Find the mutual inductance," 50:25.342 --> 50:27.142 here's what you're supposed to do. 50:27.139 --> 50:31.109 Drive a current in the first one and produce some flux lines, 50:31.106 --> 50:32.226 magnetic lines. 50:32.230 --> 50:34.760 Some of them will penetrate the second one. 50:34.760 --> 50:37.060 You count how many penetrate the second one, 50:37.061 --> 50:39.951 multiplied by the number of turns, if they exist in the 50:39.949 --> 50:42.679 second one, divide by the current producing it. 50:42.679 --> 50:46.269 That's the mutual inductance. 50:46.268 --> 50:53.248 So inductance is measured in henries, another thing for you. 50:53.250 --> 51:02.920 And usually you may find millihenries or microhenries for 51:02.923 --> 51:05.173 common use. 51:05.170 --> 51:08.620 Notice a very interesting result. 51:08.619 --> 51:11.649 The flux is going through both coils, you understand that? 51:11.650 --> 51:13.180 The same flux is going through both coils. 51:13.179 --> 51:16.009 Maybe you're happier if I drew the coil like this. 51:16.010 --> 51:18.020 Here's a doughnut coil, right? 51:18.019 --> 51:18.959 You bring the wire here. 51:18.960 --> 51:21.530 I told you how you can wrap it around many times. 51:21.530 --> 51:24.430 Then the other coil, the secondary coil, 51:24.434 --> 51:27.644 is also wrapped around the same doughnut. 51:27.639 --> 51:31.459 It must be clear to you that the magnetic field is going 51:31.456 --> 51:33.256 through all these coils. 51:33.260 --> 51:37.000 The emf on the first one is proportional to the number of 51:36.996 --> 51:41.126 turns on the first one times the rate of change of the magnetic 51:41.132 --> 51:41.802 field. 51:41.800 --> 51:44.930 The emf on the second one is N_2 times 51:44.927 --> 51:47.997 dΦ _B/dt. 51:48.000 --> 51:52.610 Therefore E_1/E_2 51:52.614 --> 51:57.824 = N_1/N _2. 51:57.820 --> 52:00.930 The same flux is going around the doughnut. 52:00.929 --> 52:02.539 One guy has N_1 turns around it, 52:02.538 --> 52:04.178 other has N_2 turns around it. 52:04.179 --> 52:06.459 Emf, you remember, is not simply rate of change of 52:06.458 --> 52:07.248 magnetic flux.. 52:07.250 --> 52:09.640 It's that multiplied by the number of turns. 52:09.639 --> 52:11.209 They both have the same flux going through them, 52:11.210 --> 52:12.510 but this has N_1 turns, 52:12.512 --> 52:13.852 this has N_2 turns. 52:13.849 --> 52:16.579 You can see this ratio. 52:16.579 --> 52:18.669 That's a very powerful result. 52:18.670 --> 52:22.600 You know what this gadget is called? 52:22.599 --> 52:25.999 It's a transformer, because you put in one voltage 52:25.998 --> 52:28.078 and you get another voltage. 52:28.079 --> 52:31.009 You can have a step up transformer or a step down 52:31.014 --> 52:31.874 transformer. 52:31.869 --> 52:34.799 If you drive a current from here, and you pick it up here, 52:34.800 --> 52:37.840 that's a step down transformer, because you're going down in 52:37.835 --> 52:39.065 the number of turns. 52:39.070 --> 52:42.680 If you connect your power supply to this one and you pull 52:42.675 --> 52:45.825 it out of that one, it's a step up transformer. 52:45.829 --> 52:49.049 So you can step up or step down, but you can only do it for 52:49.052 --> 52:49.332 AC. 52:49.329 --> 52:52.559 You cannot do it for DC, at least, not in any simple 52:52.559 --> 52:52.939 way. 52:52.940 --> 52:58.410 You need the changing thing to do a transformer. 52:58.409 --> 53:02.809 But I don't have the time or techniques to convince you, 53:02.809 --> 53:07.529 in spite of the ratio of emfs, you don't gain or lose energy 53:07.530 --> 53:09.130 by transformers. 53:09.130 --> 53:11.490 In other words, this is not a device for 53:11.487 --> 53:12.877 manufacturing energy. 53:12.880 --> 53:17.250 You will find out that if you connect a load here and it's 53:17.246 --> 53:20.536 drawing some power, the same power has to be 53:20.541 --> 53:22.611 provided by the source. 53:22.610 --> 53:24.470 So it's not a way to manufacture energy. 53:24.469 --> 53:25.579 It's like a lever. 53:25.579 --> 53:29.929 You have a long thing and you're trying to lift some 53:29.929 --> 53:34.359 object here, and you're trying to life with--I got it 53:34.364 --> 53:35.564 backwards. 53:35.559 --> 53:38.869 So this is a huge object, a tiny object, 53:38.867 --> 53:40.817 you can balance them. 53:40.820 --> 53:44.570 By varying the distance, you can have a tiny guy lifting 53:44.565 --> 53:47.965 a big one in the inverse ratio of the distances. 53:47.969 --> 53:49.989 But you don't get any free mileage out of this, 53:49.989 --> 53:52.539 because if you look at the work done by similar triangles, 53:52.539 --> 53:56.129 this will have to move a lot that will have to move a little. 53:56.130 --> 53:59.210 So the work done is the same, but doesn't mean it's useless, 53:59.213 --> 54:02.093 because this is the only way you can lift something very 54:02.086 --> 54:02.606 heavy. 54:02.610 --> 54:06.340 Likewise in a transformer, you may not have the ability to 54:06.335 --> 54:09.005 give 5,000 volts, starting with 200 volts, 54:09.014 --> 54:11.894 but you can if you use this transformer. 54:11.889 --> 54:13.789 Quite often you want to step it down. 54:13.789 --> 54:17.809 In all the gadgets you use in your house, you start with 110 54:17.809 --> 54:20.939 volts, you want to step it down to some number, 54:20.943 --> 54:23.603 so you use a step down transformer. 54:23.599 --> 54:31.869 All right, so that's the stuff on inductance. 54:31.869 --> 54:35.849 Now we are going to come--this part is really a curiosity. 54:35.849 --> 54:39.219 I'm not going to use it very much, the notion of mutual 54:39.217 --> 54:40.027 inductance. 54:40.030 --> 54:44.010 Mutual inductance is one coil trying to generate a voltage in 54:44.007 --> 54:46.857 a second coil, because they share a flux. 54:46.860 --> 54:49.620 And when it's changing one of them, it's also changing the 54:49.619 --> 54:50.199 other one. 54:50.199 --> 54:52.369 They don't have to be really coaxial. 54:52.369 --> 54:55.909 You can put a second loop way over here, and maybe some other 54:55.905 --> 54:58.495 extra flux coming out is penetrating this. 54:58.500 --> 55:01.640 That's the mutual inductance between this guy and that guy 55:01.635 --> 55:02.015 also. 55:02.018 --> 55:06.358 Any time the flux of one coil can go through another one, 55:06.360 --> 55:08.770 you have a mutual inductance, because if you change the 55:08.773 --> 55:11.023 current in the first coil, you're going to generate a 55:11.023 --> 55:12.053 voltage in the second coil. 55:12.050 --> 55:16.640 That's why you need to know that proportionality. 55:16.639 --> 55:20.569 All right, so now we are going to do the most important circuit 55:20.574 --> 55:24.134 element, which is an inductor, and it looks like this. 55:24.130 --> 55:26.410 That's a coil of wire. 55:26.409 --> 55:28.029 Say some current is coming in like this. 55:28.030 --> 55:33.490 It's wrapped around some solenoid, in the form of a 55:33.489 --> 55:34.689 solenoid. 55:34.690 --> 55:38.820 This wire is a perfect conductor, therefore it takes no 55:38.817 --> 55:42.407 voltage at all to drive a current through it. 55:42.409 --> 55:46.589 You put a battery there and it just burns immediately. 55:46.590 --> 55:52.360 But if you put a time dependent current you will need a voltage, 55:52.360 --> 55:55.530 because a time dependent current will produce a time 55:55.532 --> 55:57.962 dependent magnetic flux through this. 55:57.960 --> 56:00.550 So let us say the current was originally 0. 56:00.550 --> 56:04.410 You're trying to increase it and produce a magnetic flux 56:04.407 --> 56:04.897 here. 56:04.900 --> 56:10.030 Then an emf would be generated that opposes it. 56:10.030 --> 56:11.790 And we can ask, how much is the emf? 56:11.789 --> 56:17.689 The emf is the rate of change of flux to that coil, 56:17.690 --> 56:22.170 and I'm going to assume that the flux through that coil = the 56:22.172 --> 56:26.212 current in the very same coil time a number called self 56:26.208 --> 56:27.328 inductance. 56:27.329 --> 56:31.759 The self inductance is how much flux do you produce by a current 56:31.760 --> 56:33.520 going through yourself? 56:33.518 --> 56:36.128 Not on another coil, on yourself. 56:36.130 --> 56:39.680 Every coil, when it carries current, will have some flux 56:39.684 --> 56:41.434 threading through itself. 56:41.429 --> 56:44.039 So that ratio is called the self inductance, 56:44.039 --> 56:45.679 also measured in henries. 56:45.679 --> 56:51.339 So this becomes −LdI/dt. 56:51.340 --> 56:53.290 You will calculate L in a moment, 56:53.289 --> 56:57.009 but I'm just telling you that as long as dI/dt is not 56:57.005 --> 56:59.505 0, you will have to oppose that 56:59.512 --> 57:03.272 back emf with a voltage from some other supply. 57:03.268 --> 57:06.528 So in this example, the current is going like this. 57:06.530 --> 57:09.460 Let us say it's trying to increase. 57:09.460 --> 57:13.800 If it is trying to increase, then the back emf will be set 57:13.802 --> 57:17.462 up, an electro emf will be set up to fight it. 57:17.460 --> 57:20.950 It will try to push charges that way, from this terminal to 57:20.954 --> 57:21.924 that terminal. 57:21.920 --> 57:23.640 It will pile up charges there. 57:23.639 --> 57:27.489 But if you don't look under the hood and just went outside, 57:27.489 --> 57:30.479 you'll have charges and - charges and an electrostatic 57:30.483 --> 57:33.313 field here will be able to push them like this, 57:33.309 --> 57:36.109 maybe through a circuit. 57:36.110 --> 57:37.660 So it's the same story again and again. 57:37.659 --> 57:39.889 There is no net field inside the coil. 57:39.889 --> 57:43.179 The electromotive force is canceled by a Coulomb force. 57:43.179 --> 57:45.599 But the Coulomb force, if it has a line integral here, 57:45.601 --> 57:47.341 will have the same line integral there, 57:47.336 --> 57:49.206 because it's independent of the path. 57:49.210 --> 57:51.980 So if you don't look inside, you will just find there is 57:51.976 --> 57:54.336 some field which has lines from here to here. 57:54.340 --> 57:56.530 That's the voltage. 57:56.530 --> 57:59.620 But it will be a voltage drop, so that if you want another 57:59.619 --> 58:02.059 convention for the voltage, it's like this. 58:02.059 --> 58:05.889 If the current is going this way and it's increasing in this 58:05.891 --> 58:09.791 direction, this will be and that will be - at the instant. 58:09.789 --> 58:12.579 Just like a resistor, it flows downhill, 58:12.579 --> 58:14.939 this will also flow downhill, meaning this is higher than 58:14.942 --> 58:18.452 this, provided the current is trying 58:18.447 --> 58:23.197 to increase in the direction of the current. 58:23.199 --> 58:27.009 So we can have the following very simple circuit. 58:27.010 --> 58:28.770 We have a battery here. 58:28.768 --> 58:34.478 You've got a resistor there and we have an inductor there. 58:34.480 --> 58:39.800 This is some voltage V, this is R, 58:39.804 --> 58:43.434 this is some L henries. 58:43.429 --> 58:46.869 So let us write a circuit equation and here is where I 58:46.869 --> 58:50.699 spend an inordinate amount of time justifying what I'm about 58:50.699 --> 58:51.349 to do. 58:51.349 --> 58:53.489 I say start here. 58:53.489 --> 58:57.879 There's going to be a voltage defined everywhere except inside 58:57.878 --> 58:59.028 the black box. 58:59.030 --> 58:59.690 You understand? 58:59.690 --> 59:02.130 You cannot define a voltage inside the black box where 59:02.134 --> 59:03.894 there's an inductor, because there is a 59:03.885 --> 59:05.865 non-conservative electric field inside. 59:05.869 --> 59:07.429 But we promise not to go there. 59:07.429 --> 59:09.339 Then from here to here, you go up by 59:09.342 --> 59:10.602 V_0. 59:10.599 --> 59:13.469 Then here, current flows downhill, 59:13.469 --> 59:16.909 so you drop RI and here, if this is the sense of the 59:16.909 --> 59:20.939 current and it's increasing, your loop will go from there to 59:20.938 --> 59:24.468 there and it will jump this and come to this end. 59:24.469 --> 59:28.539 The drop from here to here is LdI/dt, 59:28.539 --> 59:32.419 and the whole thing should add up to 0. 59:32.420 --> 59:36.140 So here's a one word summary for those of you who have heard 59:36.143 --> 59:39.933 enough - we learned the notion of voltage can be defined, 59:39.929 --> 59:44.309 or a potential, only in a conservative problem. 59:44.309 --> 59:47.699 But a changing magnetic field inductor definitely is producing 59:47.704 --> 59:49.714 a field which is not conservative. 59:49.710 --> 59:52.340 So if you go deep into the coil, you will have problems 59:52.335 --> 59:53.255 defining voltage. 59:53.260 --> 59:55.990 But if you come outside the coil, I've tried to show you 59:55.994 --> 59:58.704 over and over again, all it looks like is there is a 59:58.702 --> 1:00:00.922 voltage difference between the two plates, 1:00:00.920 --> 1:00:04.170 the two terminals, and that value is 1:00:04.172 --> 1:00:05.662 LdI/dt. 1:00:05.659 --> 1:00:07.189 So when you do your circuit equation, 1:00:07.190 --> 1:00:08.770 you go from there, all the way back here, 1:00:08.768 --> 1:00:15.198 you skip these funny elements, and around them you still have 1:00:15.202 --> 1:00:17.992 the notion of a voltage. 1:00:17.989 --> 1:00:25.419 So this is the equation to solve at any time for an 1:00:25.418 --> 1:00:28.388 LR circuit. 1:00:28.389 --> 1:00:31.379 Now prior to this, let me give you a very 1:00:31.376 --> 1:00:34.806 interesting result, which we will use a lot. 1:00:34.809 --> 1:00:37.949 If you take an inductor which had current I = 0 and you 1:00:37.954 --> 1:00:40.844 managed to drive a current through it and slowly built up 1:00:40.842 --> 1:00:43.922 the current, it's got some other value I at 1:00:43.916 --> 1:00:48.776 the end, what was the work done to do 1:00:48.775 --> 1:00:49.655 that? 1:00:49.659 --> 1:00:52.489 I will now show you that when you drive a current through an 1:00:52.492 --> 1:00:54.642 inductor, you are doing some work, 1:00:54.641 --> 1:00:57.591 because when you start driving the current, 1:00:57.590 --> 1:01:00.670 it's opposing you with a voltage LdI/dt, 1:01:00.670 --> 1:01:03.570 and you're ramming it down that, up in spite of that 1:01:03.572 --> 1:01:06.162 opposition, so that the work you do, 1:01:06.163 --> 1:01:08.993 the power, is I times 1:01:08.985 --> 1:01:11.925 LdI/dt, which is, 1:01:11.929 --> 1:01:17.589 if you like the rate of change of energy. 1:01:24.342 --> 1:01:26.992 IL squared. 1:01:26.989 --> 1:01:36.259 I'm sorry, LI squared. 1:01:36.260 --> 1:01:39.870 You see that from the rules of calculus, the derivative of this 1:01:39.873 --> 1:01:42.093 guy is LI times dI/dt. 1:01:42.090 --> 1:01:47.630 Therefore the integral of the power is simply 1:01:53.543 --> 1:01:58.203 current, started at I = 0. 1:01:58.199 --> 1:02:01.929 So it takes some energy to build up a current in the 1:02:01.925 --> 1:02:02.725 inductor. 1:02:02.730 --> 1:02:04.640 That's the point. 1:02:04.639 --> 1:02:08.009 Just like it takes some energy to charge up a capacitor. 1:02:08.010 --> 1:02:12.390 I showed you when you charge up a capacitor, here's a capacitor. 1:02:12.389 --> 1:02:15.179 It's got some charges, , -, you want to charge it even 1:02:15.181 --> 1:02:18.081 more, you're going to ram more positive charges and more 1:02:18.079 --> 1:02:19.449 negative charges here. 1:02:19.449 --> 1:02:22.299 You fight it harder and the total work done is 1:02:22.297 --> 1:02:23.877 q^(2)/2C. 1:02:23.880 --> 1:02:26.250 Similarly, when you have an inductor and you're trying to 1:02:26.248 --> 1:02:28.868 increase the current through it from 0 to some final value, 1:02:28.869 --> 1:02:31.839 this is the amount of work done by you. 1:02:31.840 --> 1:02:36.170 That energy is stored in the inductor. 1:02:36.170 --> 1:02:39.180 I want to look a little bit about inductors, 1:02:39.184 --> 1:02:41.784 but first let's calculate L. 1:02:41.780 --> 1:02:45.790 So let's calculate L for a simple solenoid. 1:02:45.789 --> 1:02:48.709 Here is my solenoid. 1:02:48.710 --> 1:02:54.160 Remember, L is defined as the flux linkage divided by 1:02:54.164 --> 1:02:55.464 the current. 1:02:55.460 --> 1:02:58.670 It's going to be a one line calculation so it's going to be 1:02:58.666 --> 1:02:59.326 very easy. 1:02:59.329 --> 1:03:03.439 The magnetic field = μ_0 little 1:03:03.442 --> 1:03:04.592 nI. 1:03:04.590 --> 1:03:08.110 Little n you remember, always is the number of turns 1:03:08.114 --> 1:03:09.364 per unit length. 1:03:09.360 --> 1:03:14.680 The flux of the magnetic field is μ_0 nI 1:03:14.675 --> 1:03:17.735 times the area of cross section. 1:03:17.739 --> 1:03:23.079 That's the total magnetic flux. 1:03:23.079 --> 1:03:26.689 But the flux linkage is μ_0 little n 1:03:26.688 --> 1:03:31.118 IA times big N, because every loop of the coil 1:03:31.123 --> 1:03:33.233 links with its own flux. 1:03:33.230 --> 1:03:42.150 But that is by definition LI. So you can see that 1:03:42.146 --> 1:03:50.736 L = μ_0 little n big NA. 1:03:50.739 --> 1:03:54.599 That's the self inductance of the coil. 1:03:54.599 --> 1:03:56.359 What does this mean? 1:03:56.360 --> 1:04:01.450 If this = 5 henries, it means that if you shove 1 1:04:01.451 --> 1:04:07.281 ampere through this guy, a flux equal to 5 tesla squared 1:04:07.284 --> 1:04:11.744 meters will be linked to that circuit. 1:04:11.739 --> 1:04:16.529 So the thing I want to do now is to equate this energy, 1:04:21.056 --> 1:04:22.206 the coil. 1:04:22.210 --> 1:04:25.000 So you've got 1 over 2. 1:04:25.000 --> 1:04:32.960 L is μ_0 little n, 1:04:32.960 --> 1:04:36.760 big NAI^(2). 1:04:40.744 --> 1:04:47.874 μ_0n^(2) times lA times 1:04:47.869 --> 1:04:49.559 I^(2). 1:04:49.559 --> 1:04:50.949 In other words, I've written, 1:04:50.945 --> 1:04:53.815 using this formula to write little n as big N 1:04:53.818 --> 1:04:54.708 over l. 1:04:54.710 --> 1:04:58.200 times A, is the volume in which there's 1:04:58.204 --> 1:04:59.684 a magnetic field. 1:04:59.679 --> 1:05:04.899 So this looks like 1 over 2μ_0 times 1:05:04.902 --> 1:05:11.112 μ_0nI whole squared, times l 1:05:11.106 --> 1:05:13.606 times A. 1:05:13.610 --> 1:05:15.640 But who is μ_0nI. 1:05:15.639 --> 1:05:18.989 μ_0nI is the magnetic field. 1:05:18.989 --> 1:05:25.379 So this looks like B^(2)/2μ_0 1:05:25.380 --> 1:05:30.040 times the volume of the solenoid. 1:05:30.039 --> 1:05:33.919 From that we learn that when you have a magnetic field, 1:05:33.918 --> 1:05:37.508 there's an energy of B^(2)/2μ_0 1:05:37.512 --> 1:05:38.592 per volume. 1:05:38.590 --> 1:05:42.980 So the energy density of the magnetic field = 1:05:42.983 --> 1:05:46.283 B^(2)/2μ_0. 1:05:46.280 --> 1:05:49.410 Let me remind you the electric field energy, 1:05:49.407 --> 1:05:52.027 energy density for electric field, is 1:05:52.027 --> 1:05:55.007 ε_0/2 E^(2). 1:05:55.010 --> 1:05:56.930 You might remember that formula. 1:05:56.929 --> 1:05:58.589 So they are very similar formulas, 1:05:58.590 --> 1:06:02.030 except μ_0, which is normally upstairs in 1:06:02.034 --> 1:06:04.094 every formula, comes downstairs here, 1:06:04.086 --> 1:06:06.666 and ε_0 which is always downstairs in every 1:06:06.672 --> 1:06:16.082 formula, comes upstairs here. 1:06:16.079 --> 1:06:20.779 So let me summarize what you should remember from all of 1:06:20.780 --> 1:06:21.380 this. 1:06:21.380 --> 1:06:27.850 When you have a circuit element called an inductor, 1:06:27.853 --> 1:06:36.013 it's just a coil of wire that's wrapped around some solenoid. 1:06:36.010 --> 1:06:38.720 And when you change the current through the inductor, 1:06:38.717 --> 1:06:40.017 it's going to fight it. 1:06:40.019 --> 1:06:41.359 It's not like a resistor. 1:06:41.360 --> 1:06:43.340 A resistor fights any current. 1:06:43.340 --> 1:06:47.570 An inductor fights only a change in current, 1:06:47.574 --> 1:06:51.614 so that's all summarized in this equation, 1:06:51.612 --> 1:06:55.752 the voltage = LdI/dt RI. 1:06:55.750 --> 1:06:59.360 This is the circuit we're going to look at. 1:06:59.360 --> 1:07:02.440 Maybe a switch is open like that. 1:07:02.440 --> 1:07:05.320 There's R, that's L, 1:07:05.317 --> 1:07:08.627 that's a switch, that's the voltage. 1:07:08.630 --> 1:07:13.450 When you close the switch, you've got to ask yourself, 1:07:13.449 --> 1:07:16.449 what's the current going to be? 1:07:16.449 --> 1:07:22.329 What will be the current infinitesimally after the switch 1:07:22.329 --> 1:07:23.589 is closed? 1:07:23.590 --> 1:07:25.140 Yes? 1:07:25.139 --> 1:07:29.169 Suppose it was not 0, but .2 amps, 1:07:29.172 --> 1:07:31.742 what's the problem? 1:07:31.739 --> 1:07:35.179 After all, you closed the circuit. 1:07:35.179 --> 1:07:35.999 Yes? 1:07:36.000 --> 1:07:40.100 Student: You have energy stored in the inductor 1:07:40.099 --> 1:07:41.769 without ___________. 1:07:41.768 --> 1:07:43.158 Prof: Yeah, first of all, 1:07:45.317 --> 1:07:47.467 and you can ask, "Who had the time to do 1:07:47.469 --> 1:07:47.909 that?" 1:07:47.909 --> 1:07:49.459 Nobody. 1:07:49.460 --> 1:07:53.050 More importantly, if LdI/dt is the voltage 1:07:53.045 --> 1:07:57.075 across this, it will become infinite if dI/dt is 1:07:57.077 --> 1:07:58.047 infinite. 1:07:58.050 --> 1:08:01.660 A current that jumps from 0 to something in no time, 1:08:01.655 --> 1:08:03.985 that's got infinite derivative. 1:08:03.989 --> 1:08:06.849 So any quantity whose derivative is bounded cannot 1:08:06.846 --> 1:08:09.116 jump in its value just from calculus. 1:08:09.119 --> 1:08:11.789 So the current in the inductor will never jump. 1:08:11.789 --> 1:08:16.089 Likewise if you have a capacitor, with some charge on 1:08:16.092 --> 1:08:17.862 it, and you close the circuit, 1:08:17.860 --> 1:08:19.820 the charge on the capacitor initially was 1:08:19.817 --> 1:08:21.737 Q_0, will remain 1:08:21.738 --> 1:08:25.488 Q_0 one femto-second after you close it, 1:08:25.488 --> 1:08:28.048 because charge on it, the rate of change of the 1:08:28.051 --> 1:08:29.751 charge, is a current. 1:08:29.750 --> 1:08:33.080 The current is finite in any real problem. 1:08:33.078 --> 1:08:37.028 So capacitors cannot abruptly change the charge they have, 1:08:37.025 --> 1:08:41.315 and inductors cannot abruptly change the current they carry. 1:08:41.319 --> 1:08:43.469 If you want, they are connected to energy. 1:08:43.470 --> 1:08:46.090 The energy in the capacitor is Q^(2)/2C, 1:08:46.090 --> 1:08:47.760 therefore if q changes abruptly, 1:08:47.760 --> 1:08:51.530 the energy changes by a finite amount in infinitesimal time. 1:08:51.529 --> 1:08:54.669 Nobody can deliver that energy or take out energy at that rate. 1:08:54.670 --> 1:08:57.390 Similarly for the inductor. 1:08:57.390 --> 1:09:01.360 So I'll tell you what's in store on Wednesday. 1:09:01.359 --> 1:09:05.359 We're going to come back and look at this LR circuits and 1:09:05.355 --> 1:09:08.705 look at LC circuits and look at LCR circuits. 1:09:08.710 --> 1:09:11.520 That's the kind of stuff I think you've all done before in 1:09:11.520 --> 1:09:13.600 high school, but I think still have to do 1:09:13.600 --> 1:09:16.520 that, because that's the kind of stuff that may be more useful 1:09:16.519 --> 1:09:18.959 than many of the other things I'm talking about. 1:09:18.960 --> 1:09:21.190 But it won't be in the greatest depth. 1:09:21.189 --> 1:09:23.199 I just want to hit the high points. 1:09:23.199 --> 1:09:27.999