1 00:00:00 --> 00:00:00 I'd like to thank you for your evaluations. 2 00:00:00 --> 00:00:03,883 I'd like to thank you for your evaluations. 3 00:00:03,883 --> 00:00:09,246 They were very useful to me. I already sent e-mail to about 4 00:00:09,246 --> 00:00:14,795 fifty students and I had some interesting exchanges with some 5 00:00:14,795 --> 00:00:18,308 of you. Many of you are very happy with 6 00:00:18,308 --> 00:00:22,192 their recitation instructors. That's great. 7 00:00:22,192 --> 00:00:26,076 Many are moderately happy. Maybe that's OK. 8 00:00:26,076 --> 00:00:30,33 But there are quite a few who are 9 00:00:30,33 --> 00:00:33,805 very happy with their recitation instructors. 10 00:00:33,805 --> 00:00:38,15 If you are very unhappy with your recitation instructor, 11 00:00:38,15 --> 00:00:42,416 you are complete idiots if you stay in that recitation. 12 00:00:42,416 --> 00:00:46,839 We have thirteen recitation instructors, and I can assure 13 00:00:46,839 --> 00:00:51,737 you that it will be very easy to find one that agrees with you, 14 00:00:51,737 --> 00:00:54,976 and you can come and see me if that helps. 15 00:00:54,976 --> 00:01:00,743 Some are better than others. That's the way it goes in life. 16 00:01:00,743 --> 00:01:04,875 Some students would like to see more cut-and-dried problem 17 00:01:04,875 --> 00:01:08,355 solving in my lectures. I think that's really the 18 00:01:08,355 --> 00:01:11,981 domain of recitations. Lectures and recitations are 19 00:01:11,981 --> 00:01:13,866 complementary. In lectures, 20 00:01:13,866 --> 00:01:17,419 I prefer to go over the concepts and I always give 21 00:01:17,419 --> 00:01:21,769 numerical examples to support the concepts -- in a way that's 22 00:01:21,769 --> 00:01:25,684 problem solving -- and I show demonstrations to further 23 00:01:25,684 --> 00:01:28,222 support the concept, because seeing, 24 00:01:28,222 --> 00:01:33,512 obviously, is believing. I try to make you see through 25 00:01:33,512 --> 00:01:37,268 the dumb equations and admittedly my methods are 26 00:01:37,268 --> 00:01:42,063 sometimes somewhat different from what you're used to here at 27 00:01:42,063 --> 00:01:44,54 MIT. I try to inspire you and at 28 00:01:44,54 --> 00:01:47,737 times I try to make you wonder and think. 29 00:01:47,737 --> 00:01:52,052 And I want to keep it that way. I believe that hardcore 30 00:01:52,052 --> 00:01:56,208 probling- problem-solving is really the domain of the 31 00:01:56,208 --> 00:01:59,564 recitations. Many of you found the exam too 32 00:01:59,564 --> 00:02:03,984 easy, and many of you found the exam 33 00:02:03,984 --> 00:02:07,638 too hard. Some complained it was too hard 34 00:02:07,638 --> 00:02:12,116 because it was too easy. [audience laughter] Quite 35 00:02:12,116 --> 00:02:16,136 ironic, isn't it? They say we want more math, 36 00:02:16,136 --> 00:02:21,344 we want more standard problems. Look, who wants more math? 37 00:02:21,344 --> 00:02:24,999 I'm teaching physics. I test you physics, 38 00:02:24,999 --> 00:02:27,831 I don't test you math abilities. 39 00:02:27,831 --> 00:02:33,554 If you digest the homework, and that's very important that 40 00:02:33,554 --> 00:02:36,321 you make the homework part of your culture, 41 00:02:36,321 --> 00:02:40,077 that you study the solutions. The solutions that we put on 42 00:02:40,077 --> 00:02:41,922 the web, today, four fifteen, 43 00:02:41,922 --> 00:02:45,085 solutions through number four will go on the Web. 44 00:02:45,085 --> 00:02:48,115 Believe me, they are truly excellent solutions, 45 00:02:48,115 --> 00:02:50,619 not cut and dry. They give you a lot of 46 00:02:50,619 --> 00:02:53,321 background. If you digest those solutions, 47 00:02:53,321 --> 00:02:57,274 then the concepts will sink in. And now, at your fifty minute 48 00:02:57,274 --> 00:03:01,03 test, do you really want problems which 49 00:03:01,03 --> 00:03:04,226 are complicated maths? Clearly, not. 50 00:03:04,226 --> 00:03:07,331 I could try that, during next exam, 51 00:03:07,331 --> 00:03:12,628 but then I may have to buy myself a bullet-proof vest to be 52 00:03:12,628 --> 00:03:15,367 safe. Concepts is what matters. 53 00:03:15,367 --> 00:03:20,664 When I gave my exam review here, I highlighted the concept. 54 00:03:20,664 --> 00:03:25,322 Each little problem I did here was extremely simple. 55 00:03:25,322 --> 00:03:28,701 Conceptually, they were not so simple. 56 00:03:28,701 --> 00:03:33,444 But from a math point of view, trivial. 57 00:03:33,444 --> 00:03:38,192 Clearly, I can not cover all the subjects in a fifteen minute 58 00:03:38,192 --> 00:03:40,487 exam. I have to make a choice, 59 00:03:40,487 --> 00:03:43,653 so your preferred topic may not be there. 60 00:03:43,653 --> 00:03:48,243 Some of you think that the pace of this course is too slow. 61 00:03:48,243 --> 00:03:50,696 Some of you think it's too fast. 62 00:03:50,696 --> 00:03:54,732 The score, the average score, was three point eight. 63 00:03:54,732 --> 00:03:57,66 Four point zero would have been ideal. 64 00:03:57,66 --> 00:04:03,516 What do you want me to do? I can't accommodate all of you. 65 00:04:03,516 --> 00:04:08,629 Those who think it's too slow, and those who think it's too 66 00:04:08,629 --> 00:04:11,361 fast. Three point eight is close 67 00:04:11,361 --> 00:04:14,798 enough to ideal for me, four point zero. 68 00:04:14,798 --> 00:04:18,5 And so I'll have to leave it the way it is. 69 00:04:18,5 --> 00:04:22,026 Besides that, keep in mind you are now at 70 00:04:22,026 --> 00:04:24,494 MIT. You're no longer in high 71 00:04:24,494 --> 00:04:26,697 school. Now the good news. 72 00:04:26,697 --> 00:04:31,897 There were quite a few students who said the homework is too 73 00:04:31,897 --> 00:04:35,07 long. Not a single person said it was 74 00:04:35,07 --> 00:04:39,036 too short. I can fix that. 75 00:04:39,036 --> 00:04:43,668 I will reduce all future assignments by about twenty-five 76 00:04:43,668 --> 00:04:47,969 percent, effective tomorrow. I have already taken off 77 00:04:47,969 --> 00:04:50,946 assignment number five, two problems. 78 00:04:50,946 --> 00:04:54,586 You're down now to seven, and I will do that, 79 00:04:54,586 --> 00:04:57,398 all assignments that are coming up. 80 00:04:57,398 --> 00:05:00,871 My pleasure. Today, I'm going to cover with 81 00:05:00,871 --> 00:05:05,999 you something that conceptually is the most difficult of all of 82 00:05:05,999 --> 00:05:10,218 eight oh two. And you will need time 83 00:05:10,218 --> 00:05:13,813 to digest it. And if you think that what 84 00:05:13,813 --> 00:05:18,516 you're going to see is crazy, then you're not alone. 85 00:05:18,516 --> 00:05:22,112 The only good news is that conceptually, 86 00:05:22,112 --> 00:05:25,708 it's not going to become more difficult. 87 00:05:25,708 --> 00:05:31,332 Remember that Oersted in 1819 discovered that a steady current 88 00:05:31,332 --> 00:05:36,034 produces a steady magnetic field, and that connected 89 00:05:36,034 --> 00:05:41,197 electricity with magnetism. A little later, 90 00:05:41,197 --> 00:05:46,353 Faraday therefore suggested that maybe a steady magnetic 91 00:05:46,353 --> 00:05:52,164 field produces a steady current. And he did many experiments to 92 00:05:52,164 --> 00:05:55,257 show that. Turned to not to be so. 93 00:05:55,257 --> 00:05:58,912 And one way he tried that is as follows. 94 00:05:58,912 --> 00:06:02,099 He had here battery, with a switch, 95 00:06:02,099 --> 00:06:06,598 and here he had a solenoid. He closes the switch. 96 00:06:06,598 --> 00:06:11,003 A current will flow, and that creates a magnetic 97 00:06:11,003 --> 00:06:14,638 field in the solenoid, 98 00:06:14,638 --> 00:06:19,699 and that magnetic field, maybe it runs like so, 99 00:06:19,699 --> 00:06:23,99 depends on the direction of the current. 100 00:06:23,99 --> 00:06:29,162 And so now, he put around this solenoid a loop . 101 00:06:29,162 --> 00:06:34,993 Let's call this loop number two, and it was around the 102 00:06:34,993 --> 00:06:39,945 solenoid, and let's call this loop number one, 103 00:06:39,945 --> 00:06:48,611 of which the solenoid is part. Whenever there was a current in 104 00:06:48,611 --> 00:06:53,169 number one, he never managed to see a current in number two. 105 00:06:53,169 --> 00:06:57,882 If there is a current going in number one, there is a magnetic 106 00:06:57,882 --> 00:07:01,59 field and that magnetic field is seen, of course, 107 00:07:01,59 --> 00:07:04,68 by the conductor number two by that loop. 108 00:07:04,68 --> 00:07:08,31 Never any current. So he concluded that a steady 109 00:07:08,31 --> 00:07:11,632 magnetic field as produced by the solenoids, 110 00:07:11,632 --> 00:07:14,491 circuit one, does not produce a steady 111 00:07:14,491 --> 00:07:18,353 current in number two. But then, one day he noticed 112 00:07:18,353 --> 00:07:22,659 that as he closed the switch he saw a 113 00:07:22,659 --> 00:07:26,834 current in number two, and when he opened the switch 114 00:07:26,834 --> 00:07:31,5 again he saw a current in number two, and therefore he now 115 00:07:31,5 --> 00:07:35,757 concluded that a changing magnetic field is causing a 116 00:07:35,757 --> 00:07:38,705 current. Not a steady magnetic field, 117 00:07:38,705 --> 00:07:43,044 but a changing magnetic field. And this was a profound 118 00:07:43,044 --> 00:07:47,874 discovery which changed our world and it contributed largely 119 00:07:47,874 --> 00:07:54,259 to the technological revolution of the late nineteenth and early 120 00:07:54,259 --> 00:07:57,44 twenty century. The current, 121 00:07:57,44 --> 00:08:04,154 therefore an electric field, can be produced by a changing 122 00:08:04,154 --> 00:08:09,337 magnetic field, and that phenomenon is called 123 00:08:09,337 --> 00:08:15,697 electromagnetic induction, and that phenomenon runs our 124 00:08:15,697 --> 00:08:21,469 economy, as you will see in the next few lectures. 125 00:08:21,469 --> 00:08:27,476 I have here a conducting wire, a square. 126 00:08:27,476 --> 00:08:30,295 I could've chosen any other shape. 127 00:08:30,295 --> 00:08:33,625 Try to make you see three dimensionally. 128 00:08:33,625 --> 00:08:38,152 And I approach this conducting wire with a bar magnet. 129 00:08:38,152 --> 00:08:42,507 The bar magnet has a magnetic field running like so. 130 00:08:42,507 --> 00:08:46,35 As I approach that loop, that conducting wire, 131 00:08:46,35 --> 00:08:49,681 moving the bar magnet, that's essential. 132 00:08:49,681 --> 00:08:53,097 I can't hold it still. I have to move it. 133 00:08:53,097 --> 00:08:56,854 If I come down from above and I move it down, 134 00:08:56,854 --> 00:09:02,121 you're going to see a current going through this 135 00:09:02,121 --> 00:09:04,764 loop. And that current will go into 136 00:09:04,764 --> 00:09:09,274 such a direction that it opposes the change of the magnetic 137 00:09:09,274 --> 00:09:11,995 field. The magnetic field is in down 138 00:09:11,995 --> 00:09:16,504 direction and it is increasing as I move the bar magnet in. 139 00:09:16,504 --> 00:09:20,936 Then this current loop will produce a magnetic field which 140 00:09:20,936 --> 00:09:24,745 is in this direction, and when you look from below 141 00:09:24,745 --> 00:09:28,944 the current will go clockwise, producing a current -- a 142 00:09:28,944 --> 00:09:32,754 magnetic field in this direction. 143 00:09:32,754 --> 00:09:37,975 If you move the bar magnetic out, then the magnetic field is 144 00:09:37,975 --> 00:09:42,046 going down here, then the current will reverse. 145 00:09:42,046 --> 00:09:47,355 The current wants to oppose the change in the magnetic field, 146 00:09:47,355 --> 00:09:52,311 and that's called Lenz's Law. It is the most human law in 147 00:09:52,311 --> 00:09:56,293 physics, because there's inertia in all of us. 148 00:09:56,293 --> 00:09:59,214 We all fight change at some level. 149 00:09:59,214 --> 00:10:03,461 Lenz's Law is extremely powerful in 150 00:10:03,461 --> 00:10:08,746 always determining in which direction these induced currents 151 00:10:08,746 --> 00:10:12,149 will run. It is not a quantitative law. 152 00:10:12,149 --> 00:10:16,807 You can not calculate how strong the current will be, 153 00:10:16,807 --> 00:10:21,554 but it's very useful as you will see today to know the 154 00:10:21,554 --> 00:10:26,749 direction of that current that gets you out of all kinds of 155 00:10:26,749 --> 00:10:30,689 problems with minus signs. I now want to do a 156 00:10:30,689 --> 00:10:36,87 demonstration which is very much like what you see here. 157 00:10:36,87 --> 00:10:43,286 I have here a loop. That is the square that you see 158 00:10:43,286 --> 00:10:50,73 there except that it's not- not one loop, but it is many of 159 00:10:50,73 --> 00:10:54,58 them. Hundreds, doesn't matter. 160 00:10:54,58 --> 00:11:01,639 And what we're going to show you is an amp meter that is 161 00:11:01,639 --> 00:11:07,927 connected, so there is somewhere in this 162 00:11:07,927 --> 00:11:11,798 circuit an amp meter. I have a bar magnet and I'm 163 00:11:11,798 --> 00:11:15,668 going to approach this conducting loop with a bar 164 00:11:15,668 --> 00:11:20,828 magnet and you're going to see a current running in one direction 165 00:11:20,828 --> 00:11:25,343 and when I pull it out it will be running in the opposite 166 00:11:25,343 --> 00:11:30,181 direction, and when I hold my hand still so that the magnetic 167 00:11:30,181 --> 00:11:32,842 field is not changing, no current. 168 00:11:32,842 --> 00:11:38,163 You're going to see the current meter there, and here 169 00:11:38,163 --> 00:11:41,96 is the bar magnet. I come close to this conducting 170 00:11:41,96 --> 00:11:44,206 loop. Notice we see a current. 171 00:11:44,206 --> 00:11:47,228 I pull back, the current is in the other 172 00:11:47,228 --> 00:11:49,629 direction. Now I will go faster, 173 00:11:49,629 --> 00:11:53,968 so that the change of the magnetic field per unit time is 174 00:11:53,968 --> 00:11:56,447 stronger. [whistle] More current. 175 00:11:56,447 --> 00:11:59,313 I go out fast. [whistle] More current. 176 00:11:59,313 --> 00:12:03,961 So we see it's the change of the magnetic field that matters. 177 00:12:03,961 --> 00:12:07,06 If I come in very slowly, which I do now, 178 00:12:07,06 --> 00:12:11,244 very slowly, we almost see nothing. 179 00:12:11,244 --> 00:12:17,248 Right now the entire magnetic field is inside this loop. 180 00:12:17,248 --> 00:12:23,472 The strongest I can have it. Nothing happens because there 181 00:12:23,472 --> 00:12:27,184 is no change in the magnetic field. 182 00:12:27,184 --> 00:12:32,534 It's only when I do this that you see the current. 183 00:12:32,534 --> 00:12:38,539 So an induced current is clearly the result of a driving 184 00:12:38,539 --> 00:12:42,033 force. There must be, 185 00:12:42,033 --> 00:12:46,712 just like we had with batteries in the past, there must be an 186 00:12:46,712 --> 00:12:49,441 EMF. There must be an electric field 187 00:12:49,441 --> 00:12:52,56 that is produced in this conducting loop. 188 00:12:52,56 --> 00:12:57,084 And so I create now an induced EMF -- we used that word EMF 189 00:12:57,084 --> 00:13:01,139 earlier for batteries, so now we have an induced EMF, 190 00:13:01,139 --> 00:13:05,116 which is the result of this changing magnetic field, 191 00:13:05,116 --> 00:13:09,015 and that therefore is the induced current times the 192 00:13:09,015 --> 00:13:13,538 resistance of that entire closed conductor, 193 00:13:13,538 --> 00:13:16,598 whatever is in there. In this case, 194 00:13:16,598 --> 00:13:20,377 the total resistance of all these windings, 195 00:13:20,377 --> 00:13:23,976 of all the copper wire. That's Ohm's Law. 196 00:13:23,976 --> 00:13:29,105 So the induced EMF is always the induced current times the 197 00:13:29,105 --> 00:13:31,894 resistance. Faraday did a lot of 198 00:13:31,894 --> 00:13:37,203 experiments, and one of the experiments that he did was that 199 00:13:37,203 --> 00:13:43,501 he produced a magnetic field, so he ran a current through 200 00:13:43,501 --> 00:13:48,258 a loop of some kind, let's say he ran a current 201 00:13:48,258 --> 00:13:52,6 going around, creating therefore a magnetic 202 00:13:52,6 --> 00:13:58,804 field, and he was switching the current in and out so that he 203 00:13:58,804 --> 00:14:04,387 could change the current, and so it produces a magnetic 204 00:14:04,387 --> 00:14:10,901 and this magnetic field changes when you close and open the- the 205 00:14:10,901 --> 00:14:13,072 switch. And then here, 206 00:14:13,072 --> 00:14:18,759 he had his second conducting wire, 207 00:14:18,759 --> 00:14:23,398 just like we had there, and he measured in there the 208 00:14:23,398 --> 00:14:25,763 current. And what he found, 209 00:14:25,763 --> 00:14:29,22 experimentally, is that the EMF that is 210 00:14:29,22 --> 00:14:33,677 generated in here, which I will call EMF generated 211 00:14:33,677 --> 00:14:38,68 in my conducting loop number two, is proportional to the 212 00:14:38,68 --> 00:14:42,683 magnetic field change produced by number one, 213 00:14:42,683 --> 00:14:47,504 so the field goes through number two and this field is 214 00:14:47,504 --> 00:14:52,864 changing, so he knows that if the change 215 00:14:52,864 --> 00:14:57,947 is faster, as you just saw, you get a higher EMF. 216 00:14:57,947 --> 00:15:03,771 He also noticed that E two is proportional to this area, 217 00:15:03,771 --> 00:15:10,124 so to the area of number two. And that gave him the idea that 218 00:15:10,124 --> 00:15:16,689 the EMF really is the result of the change of the magnetic flux 219 00:15:16,689 --> 00:15:21,455 through this surface of number two. 220 00:15:21,455 --> 00:15:27,347 And I want to refresh your memory on the idea of magnetic 221 00:15:27,347 --> 00:15:31,134 flux. We do know, or we remember what 222 00:15:31,134 --> 00:15:34,817 electric flux is. And magnetic flux, 223 00:15:34,817 --> 00:15:38,394 very similar. If this is a surface, 224 00:15:38,394 --> 00:15:43,76 and the local vector perpendicular to the surface is 225 00:15:43,76 --> 00:15:49,547 like so, of course it could be in a different direction, 226 00:15:49,547 --> 00:15:54,597 and the local magnetic field is for 227 00:15:54,597 --> 00:15:59,461 instance like so, then a magnetic flux through 228 00:15:59,461 --> 00:16:04,216 this surface is defined. We call it phi of B, 229 00:16:04,216 --> 00:16:08,106 is the integral over an open surface. 230 00:16:08,106 --> 00:16:11,889 This is an open surface of B dot DA. 231 00:16:11,889 --> 00:16:18,157 And the electric fields we defined in exactly the same way, 232 00:16:18,157 --> 00:16:24,424 electric flux, except we had an E here. 233 00:16:24,424 --> 00:16:32,034 There was nothing there. So if this magnetic flux is 234 00:16:32,034 --> 00:16:40,986 changing, Faraday concluded that then you have an EMF in this 235 00:16:40,986 --> 00:16:47,7 conducting wire. So essential is the change of 236 00:16:47,7 --> 00:16:54,116 the magnetic flux. If we take some kind of a 237 00:16:54,116 --> 00:16:59,188 conducting wire, like so, 238 00:16:59,188 --> 00:17:04,083 let's make it in the blackboard for now to make it easy. 239 00:17:04,083 --> 00:17:09,244 And I attach to this wire a surface because the moment that 240 00:17:09,244 --> 00:17:14,227 you talk about flux you must always specify your surface. 241 00:17:14,227 --> 00:17:19,388 A flux can only go through a surface, so this is my surface 242 00:17:19,388 --> 00:17:23,659 now for simplicity. And there is a magnetic field 243 00:17:23,659 --> 00:17:29,443 coming out of the blackboard at me, and it is growing. 244 00:17:29,443 --> 00:17:33,905 It is increasing. I will now get an EMF, 245 00:17:33,905 --> 00:17:37,909 a current flowing in this direction. 246 00:17:37,909 --> 00:17:41,912 Lenz's law. If the magnetic field is 247 00:17:41,912 --> 00:17:48,776 increasing, then the current will be in such a direction that 248 00:17:48,776 --> 00:17:54,61 it opposes the change. It doesn't want that magnetic 249 00:17:54,61 --> 00:17:59,643 field to increase, and so it goes around like 250 00:17:59,643 --> 00:18:03,161 this, the current, 251 00:18:03,161 --> 00:18:10,818 sort of, it produces a magnetic field that is in the blackboard. 252 00:18:10,818 --> 00:18:18,476 And so it is the flux change of that magnetic field through this 253 00:18:18,476 --> 00:18:22,852 flat surface that determines the EMF. 254 00:18:22,852 --> 00:18:28,2 So the EMF is then the flux change, d phi dt, 255 00:18:28,2 --> 00:18:34,131 through that surface. To express Lenz's Law that it 256 00:18:34,131 --> 00:18:37,861 is always opposing the change of the magnetic flux, 257 00:18:37,861 --> 00:18:41,739 we have a minus sign here. But minus signs will never 258 00:18:41,739 --> 00:18:45,618 bother you, believe me, because you'll always know in 259 00:18:45,618 --> 00:18:49,572 which direction the EMF is. It's clear that the EMF is 260 00:18:49,572 --> 00:18:53,973 going to be in this direction. That's the direction in which 261 00:18:53,973 --> 00:18:58,449 it will make the current flow. But we have to put it there to 262 00:18:58,449 --> 00:19:04,304 be mathematically correct. That's really Lenz's Law. 263 00:19:04,304 --> 00:19:08,093 You're looking at Lenz's Law here. 264 00:19:08,093 --> 00:19:15,098 So you can also write down for this minus the surface integral 265 00:19:15,098 --> 00:19:21,53 of B dot DA over that open -- whoo, I hope you didn't see 266 00:19:21,53 --> 00:19:24,745 this. Over this open surface. 267 00:19:24,745 --> 00:19:30,257 That's the [break in tape] Oops, look what I did. 268 00:19:30,257 --> 00:19:36,577 I forgot the DDT in front of the integral sign. 269 00:19:36,577 --> 00:19:42,196 If you put yourself inside that conductor, and you marched 270 00:19:42,196 --> 00:19:48,309 around in the direction of the current, you will see everywhere 271 00:19:48,309 --> 00:19:52,252 in the wire an electric field, of course. 272 00:19:52,252 --> 00:19:56,59 Otherwise, there would be no current flowing. 273 00:19:56,59 --> 00:20:01,223 And so if you go once around this whole circuit, 274 00:20:01,223 --> 00:20:05,955 then that EMF must of course also be 275 00:20:05,955 --> 00:20:11,32 E dot DL over the closed loop. So you're marching inside the 276 00:20:11,32 --> 00:20:16,684 wire, you find everywhere an electric field and these little 277 00:20:16,684 --> 00:20:20,684 sections IDL. E and DL are always in the same 278 00:20:20,684 --> 00:20:26,048 direction if you stay in the wire, and so this should be the 279 00:20:26,048 --> 00:20:31,685 same and this is a closed loop. So this is all if you want what 280 00:20:31,685 --> 00:20:36,049 we call Faraday's Law. We never see it in so much 281 00:20:36,049 --> 00:20:40,35 detail. I will abbreviate it a little 282 00:20:40,35 --> 00:20:44,716 bit on the board there. But I want you to appreciate 283 00:20:44,716 --> 00:20:48,141 that there is no battery in this circuit. 284 00:20:48,141 --> 00:20:53,277 There is only a change in the magnetic flux through a surface 285 00:20:53,277 --> 00:20:57,386 that I have attached through the conducting wire, 286 00:20:57,386 --> 00:21:02,609 and then I get an induced EMF and the induced EMF will produce 287 00:21:02,609 --> 00:21:07,66 a current given by Ohm's Law. So I want to write down now on 288 00:21:07,66 --> 00:21:13,816 that blackboard there, Faraday's law in a somewhat 289 00:21:13,816 --> 00:21:20,99 abbreviated way because we have all Maxwell's equations here and 290 00:21:20,99 --> 00:21:26 so we now have that the closed loop integral, 291 00:21:26 --> 00:21:31,694 closed loop of E dot DL -- that's that induced EMF. 292 00:21:31,694 --> 00:21:38,071 You can take minus d phi dt or the time derivative of the 293 00:21:38,071 --> 00:21:44,513 integral B dot DA. That's the one I will take. 294 00:21:44,513 --> 00:21:50,9 Integral of B dot DA and this is over an open surface. 295 00:21:50,9 --> 00:21:57,287 And that open surface has to be attached to this loop, 296 00:21:57,287 --> 00:22:02,107 and that is Faraday. We have Gauss's Law, 297 00:22:02,107 --> 00:22:08,012 we have Ampere's Law. We have this one which tells 298 00:22:08,012 --> 00:22:13,622 you that magnetic monopoles don't exist. 299 00:22:13,622 --> 00:22:19,474 This would only not be zero if you had a magnetic monopole and 300 00:22:19,474 --> 00:22:24,656 put it in a closed surface. Come and see me if you find 301 00:22:24,656 --> 00:22:27,918 one. And this now is Faraday's Law, 302 00:22:27,918 --> 00:22:33,003 so you think that all four Maxwell's equations are now 303 00:22:33,003 --> 00:22:34,826 complete. Not quite. 304 00:22:34,826 --> 00:22:38,472 We're going to change this one shortly. 305 00:22:38,472 --> 00:22:42,501 So we can't celebrate yet. We have to wait. 306 00:22:42,501 --> 00:22:48,66 It's a big party. There's always a little bit of 307 00:22:48,66 --> 00:22:54,943 an issue about the direction of DA and I will explain to you how 308 00:22:54,943 --> 00:23:00,728 the convention goes but it really is not so crucial because 309 00:23:00,728 --> 00:23:06,713 Lenz's Law always helps you to find the direction of the EMF, 310 00:23:06,713 --> 00:23:12,498 but if we are trying to be purist, if this is my conducting 311 00:23:12,498 --> 00:23:16,786 loop and if I attach a flat surface to this, 312 00:23:16,786 --> 00:23:22,462 if I did that, and if I go around closed loop 313 00:23:22,462 --> 00:23:26,042 integral E dot DL, Faraday doesn't tell me which 314 00:23:26,042 --> 00:23:28,784 way I have to go. I can go clockwise. 315 00:23:28,784 --> 00:23:33,049 I can go counterclockwise. We will then do the same thing 316 00:23:33,049 --> 00:23:37,391 that we did before with Ampere's Law, apply the right-hand 317 00:23:37,391 --> 00:23:40,742 corkscrew rule, and that is that if you march 318 00:23:40,742 --> 00:23:43,713 around clockwise, then DA will be in the 319 00:23:43,713 --> 00:23:48,206 blackboard perpendicular to the blackboard, perpendicular to 320 00:23:48,206 --> 00:23:51,482 this surface, and if you go counterclockwise 321 00:23:51,482 --> 00:23:57,624 then DA will come towards you. The surface doesn't have to be 322 00:23:57,624 --> 00:23:59,282 flat. It can be flat. 323 00:23:59,282 --> 00:24:04,007 There's nothing wrong with it. But there can also be a bag 324 00:24:04,007 --> 00:24:07,157 attached to it, as we also had earlier. 325 00:24:07,157 --> 00:24:11,55 I have here the closed conducting wire and I could put 326 00:24:11,55 --> 00:24:16,275 a surface right here but I can also make it a [inaudible], 327 00:24:16,275 --> 00:24:20,419 like this, perfectly fine. Nothing wrong with that. 328 00:24:20,419 --> 00:24:23,984 That's a open surface attached to this loop. 329 00:24:23,984 --> 00:24:27,908 That's fine. You have a choice, 330 00:24:27,908 --> 00:24:32,451 and the convention with DA is then exactly the same, 331 00:24:32,451 --> 00:24:37,082 that if you go clockwise then the DA would be in this 332 00:24:37,082 --> 00:24:41,803 direction using the right-hand corkscrew locally here. 333 00:24:41,803 --> 00:24:46,435 If you went counterclockwise, the DA would flip over. 334 00:24:46,435 --> 00:24:50,8 So what is now the recipe that you have to follow? 335 00:24:50,8 --> 00:24:54,363 You have a circuit, electric circuit that 336 00:24:54,363 --> 00:24:57,733 determines then your loops, 337 00:24:57,733 --> 00:25:00,814 of course. You can take loops anywhere in 338 00:25:00,814 --> 00:25:04,896 space, but that's not too meaningful, so you take them 339 00:25:04,896 --> 00:25:08,439 into your circuits, and so you define the loop 340 00:25:08,439 --> 00:25:11,135 first. Then you define the direction 341 00:25:11,135 --> 00:25:14,678 in which you want to march around that circuit. 342 00:25:14,678 --> 00:25:18,221 You attach an open surface to that closed loop, 343 00:25:18,221 --> 00:25:22,92 and you can determine on that entire surface the integral of B 344 00:25:22,92 --> 00:25:25,462 dot DA. Everywhere on that surface 345 00:25:25,462 --> 00:25:29,095 locally you know the DA, 346 00:25:29,095 --> 00:25:32,966 locally you know the B, you do the integration and you 347 00:25:32,966 --> 00:25:36,764 get your magnetic flux, and then if you know the time 348 00:25:36,764 --> 00:25:40,489 change of that magnetic flux, then you know the EMF. 349 00:25:40,489 --> 00:25:43,63 If you go around in this conducting circuit, 350 00:25:43,63 --> 00:25:46,989 and you measure everywhere the electric fields, 351 00:25:46,989 --> 00:25:51,372 then the integral of E dot DL, if you go around the loop will 352 00:25:51,372 --> 00:25:55,096 give you the same answer, and that connects the two. 353 00:25:55,096 --> 00:25:59,406 The magnetic flux change is connected with the integral of E 354 00:25:59,406 --> 00:26:02,904 dot DL when you go around. 355 00:26:02,904 --> 00:26:07,28 And you have to take that minus sign into account. 356 00:26:07,28 --> 00:26:11,835 How come it doesn't matter whether you choose a flat 357 00:26:11,835 --> 00:26:14,961 surface or whether you choose a bag? 358 00:26:14,961 --> 00:26:19,962 Well, think of magnetic field lines as a flow of water or 359 00:26:19,962 --> 00:26:23,981 spaghetti, if you like that, or a flow of air. 360 00:26:23,981 --> 00:26:28,804 It is clear that if there is some kind of a flow of air 361 00:26:28,804 --> 00:26:33,29 through this opening, that it's got to come 362 00:26:33,29 --> 00:26:36,464 out somewhere, so it always comes out of this 363 00:26:36,464 --> 00:26:38,051 surface. And therefore, 364 00:26:38,051 --> 00:26:41,009 you're really free to choose that surface, 365 00:26:41,009 --> 00:26:45,121 so you always pick a surface that is the best one for you. 366 00:26:45,121 --> 00:26:47,719 Now, all this looks very complicated. 367 00:26:47,719 --> 00:26:50,027 But in practice, it really isn't, 368 00:26:50,027 --> 00:26:54,428 because your loop is always a conducting wire in your circuit, 369 00:26:54,428 --> 00:26:58,684 and the minus sign is never an issue because you always know 370 00:26:58,684 --> 00:27:03,23 with Lenz's law in which direction the EMF is. 371 00:27:03,23 --> 00:27:07,414 In fact, when I solved these problems, I don't even look at 372 00:27:07,414 --> 00:27:10,155 the minus sign. I ignore it completely. 373 00:27:10,155 --> 00:27:14,556 I def- I calculate the magnetic flux change, and then I always 374 00:27:14,556 --> 00:27:18,813 know in which direction the current is, so I don't even look 375 00:27:18,813 --> 00:27:21,843 at the minus sign. Now I want to show you a 376 00:27:21,843 --> 00:27:26,316 demonstration which is very much like what Faraday tried to do. 377 00:27:26,316 --> 00:27:29,923 I have here a solenoid. We've seen this one before. 378 00:27:29,923 --> 00:27:35,052 We can generate quite a strong magnetic field with that. 379 00:27:35,052 --> 00:27:38,819 And we're going to put around this solenoid one loop, 380 00:27:38,819 --> 00:27:41,282 like we had here, like Faraday did, 381 00:27:41,282 --> 00:27:45,556 and then we're going to close the switch, and so we're going 382 00:27:45,556 --> 00:27:49,685 to build up this magnetic field and we're going to see the 383 00:27:49,685 --> 00:27:53,452 current in that loop. And so if we look- if we make a 384 00:27:53,452 --> 00:27:57,508 cross-section straight through here, then it will look as 385 00:27:57,508 --> 00:27:59,609 follows. Then you see here the 386 00:27:59,609 --> 00:28:03,81 solenoids, so the magnetic field is really 387 00:28:03,81 --> 00:28:07,898 confined to the solenoid. Magnetic field outside the 388 00:28:07,898 --> 00:28:11,664 solenoid as we discussed earlier is almost zero, 389 00:28:11,664 --> 00:28:15,11 so there's only a magnetic field right here. 390 00:28:15,11 --> 00:28:17,755 Keep that in mind in what follows. 391 00:28:17,755 --> 00:28:21,201 And now we're going to put a wire around it, 392 00:28:21,201 --> 00:28:25,849 with an amp meter in there. If the magnetic field comes out 393 00:28:25,849 --> 00:28:28,093 of the board, and is growing, 394 00:28:28,093 --> 00:28:33,462 increasing, the current will flow in this direction. 395 00:28:33,462 --> 00:28:35,841 Lenz's Law. If it is decreasing, 396 00:28:35,841 --> 00:28:39,294 the current will go in the opposite direction. 397 00:28:39,294 --> 00:28:43,131 Now keep in mind that the magnetic flux though this 398 00:28:43,131 --> 00:28:46,584 surface, that is, my surface which I attach to 399 00:28:46,584 --> 00:28:50,268 this closed loop, that that magnetic flux remains 400 00:28:50,268 --> 00:28:55,025 the same whether I make the loop this big or whether I make the 401 00:28:55,025 --> 00:28:59,169 loop very crooked like so, because the magnetic flux is 402 00:28:59,169 --> 00:29:03,467 only confined to the inner portion of the 403 00:29:03,467 --> 00:29:07,036 solenoid and that's not changing. 404 00:29:07,036 --> 00:29:12,502 And so when I change the shape of this outer loop, 405 00:29:12,502 --> 00:29:17,188 you will not see any change in the current. 406 00:29:17,188 --> 00:29:20,646 I hope that doesn't confuse you. 407 00:29:20,646 --> 00:29:26,224 I'm going to purposely change the size of the loop, 408 00:29:26,224 --> 00:29:29,682 and so I'm going to do that now. 409 00:29:29,682 --> 00:29:35,929 You're going to see there a very sensitive amp meter and 410 00:29:35,929 --> 00:29:42,288 you're going to see here this loop, 411 00:29:42,288 --> 00:29:45,815 the big wire, and I'm going to just put it 412 00:29:45,815 --> 00:29:50,03 over this solenoid. Let me first make sure that my 413 00:29:50,03 --> 00:29:54,59 amp meter which is extremely sensitive, I can zero it. 414 00:29:54,59 --> 00:29:58,548 It's sign sensitive. If the current goes in one 415 00:29:58,548 --> 00:30:03,194 direction, you will see the needle go in one direction. 416 00:30:03,194 --> 00:30:06,807 If the current goes in the other direction, 417 00:30:06,807 --> 00:30:11,109 you will see the change. And so now I put this loop 418 00:30:11,109 --> 00:30:14,034 around here, crazy shape this loop. 419 00:30:14,034 --> 00:30:18,422 So it's around this solenoid once, 420 00:30:18,422 --> 00:30:22,24 so the magnetic field is inside the solenoids, 421 00:30:22,24 --> 00:30:27,502 and so think of a surface which is attached to this crazy loop, 422 00:30:27,502 --> 00:30:32,509 and now I'm going to turn the current on, and only while the 423 00:30:32,509 --> 00:30:37,431 current is changing will there be a changing magnetic flux. 424 00:30:37,431 --> 00:30:41,845 Only during that portion will you see a current flow. 425 00:30:41,845 --> 00:30:43,627 Three, two, one, zero. 426 00:30:43,627 --> 00:30:47,106 I will break the current, three, two, one, 427 00:30:47,106 --> 00:30:49,652 zero. Went the other direction. 428 00:30:49,652 --> 00:30:53,895 If I change the size of the loop, 429 00:30:53,895 --> 00:30:57,13 I'm making it now different, much smaller. 430 00:30:57,13 --> 00:31:01,153 Makes no difference, for reasons that I explained to 431 00:31:01,153 --> 00:31:05,886 you, because the magnetic flux is not determined in this case 432 00:31:05,886 --> 00:31:10,382 by the size of my loop but is determined by the solenoids, 433 00:31:10,382 --> 00:31:14,642 so if I do it again now, with a very different shape of 434 00:31:14,642 --> 00:31:17,324 the loop -- let me zero this again. 435 00:31:17,324 --> 00:31:18,981 Three, two, one, zero. 436 00:31:18,981 --> 00:31:20,637 Three, two, one, zero. 437 00:31:20,637 --> 00:31:24,029 No change. Almost the same 438 00:31:24,029 --> 00:31:29,143 which you saw before. Now comes something that may 439 00:31:29,143 --> 00:31:35,196 not be so intuitive to you. I'm now going to wrap this wire 440 00:31:35,196 --> 00:31:39,58 three times around. And so this outer loop, 441 00:31:39,58 --> 00:31:44,172 this outer conducting wire, is now like this. 442 00:31:44,172 --> 00:31:47,929 One, two, three. Something like that. 443 00:31:47,929 --> 00:31:55,548 Now I have to attach in my head a surface to this closed loop. 444 00:31:55,548 --> 00:31:59,702 My god, what does it look like? What a ridiculous surface. 445 00:31:59,702 --> 00:32:03,201 Well, that's your problem, not Faraday's problem. 446 00:32:03,201 --> 00:32:07,502 How can you imagine that there is a surface attached to this 447 00:32:07,502 --> 00:32:10,053 loop? Well, take the whole thing and 448 00:32:10,053 --> 00:32:13,188 dip it in soap. Take it out and see what you 449 00:32:13,188 --> 00:32:15,739 see. The soap will attach everywhere 450 00:32:15,739 --> 00:32:19,311 on the conducting loop. And if this loop were like 451 00:32:19,311 --> 00:32:23,684 this, going up like a spiral staircase, you're going to get a 452 00:32:23,684 --> 00:32:27,839 surface that goes up like this. But the magnetic fields go 453 00:32:27,839 --> 00:32:31,253 through all three of them. 454 00:32:31,253 --> 00:32:35,473 Therefore, the changing magnetic flux will go three 455 00:32:35,473 --> 00:32:39,692 times through the surface now, and so Faraday says, 456 00:32:39,692 --> 00:32:44,502 fah- that you're going to see three times the EMF that you 457 00:32:44,502 --> 00:32:47,624 would see if there were only one loop. 458 00:32:47,624 --> 00:32:52,519 And if you go thousand times around, you get thousand times 459 00:32:52,519 --> 00:32:55,641 the EMF of one loop. Not so intuitive. 460 00:32:55,641 --> 00:33:00,366 So I'm around now once. I go around twice. 461 00:33:00,366 --> 00:33:05,009 And I go around a third time. I have three loops around it 462 00:33:05,009 --> 00:33:06,638 now. I can zero that, 463 00:33:06,638 --> 00:33:10,222 but that's not so important. Three, two, one, 464 00:33:10,222 --> 00:33:13,398 zero, and you saw a much larger current. 465 00:33:13,398 --> 00:33:18,204 It's about three times larger because the EMF is three times 466 00:33:18,204 --> 00:33:20,403 larger. I break the current. 467 00:33:20,403 --> 00:33:24,964 We see it three times larger. And this is the idea behind 468 00:33:24,964 --> 00:33:28,222 transformers. You can get any EMF in that 469 00:33:28,222 --> 00:33:32,212 wire that you want to, by having 470 00:33:32,212 --> 00:33:36,617 many, many loops. You can get it up to thousands 471 00:33:36,617 --> 00:33:40,084 of volts, and that's not so intuitive. 472 00:33:40,084 --> 00:33:43,644 So Faraday's law is very non-intuitive. 473 00:33:43,644 --> 00:33:46,83 Kirchoff's Rule was very intuitive. 474 00:33:46,83 --> 00:33:52,172 Kirchoff said when you go around a circuit the closed-loop 475 00:33:52,172 --> 00:33:55,451 integral of E dot DL is always zero. 476 00:33:55,451 --> 00:33:59,668 Not true is you have a changing magnetic flux. 477 00:33:59,668 --> 00:34:04,54 If you have a changing magnetic flux, 478 00:34:04,54 --> 00:34:08,698 the electric fields inside the conducting wires now become 479 00:34:08,698 --> 00:34:12,053 non-conservative. Kirchoff's Rule only holds as 480 00:34:12,053 --> 00:34:15,262 long as the electric fields are conservative. 481 00:34:15,262 --> 00:34:18,981 If an electric field is conservative and you go from 482 00:34:18,981 --> 00:34:22,409 point one to point two, the integral E dot DL is 483 00:34:22,409 --> 00:34:26,42 independent of the path. That's the potential difference 484 00:34:26,42 --> 00:34:29,556 between two points, that's uniquely defined. 485 00:34:29,556 --> 00:34:33,713 That's no longer the case. If you go around once with this 486 00:34:33,713 --> 00:34:37,675 experiment, you get a certain EMF, 487 00:34:37,675 --> 00:34:42,212 you go three times around, you get a different value. 488 00:34:42,212 --> 00:34:47,185 Your path is now different, and that's very non-intuitive, 489 00:34:47,185 --> 00:34:51,897 because you're dealing with non-conservative fields for 490 00:34:51,897 --> 00:34:54,776 which we have very little feeling. 491 00:34:54,776 --> 00:34:57,568 Now, I'm going to blow your mind. 492 00:34:57,568 --> 00:35:01,843 I'm going to make you see something that you won't 493 00:35:01,843 --> 00:35:06,73 believe, and so try to follow step-by-step 494 00:35:06,73 --> 00:35:15,313 leading up to this unbelievable and very non-intuitive result. 495 00:35:15,313 --> 00:35:22,489 I have here a battery, and the battery has an EMF of 496 00:35:22,489 --> 00:35:26,428 one volt. Here is a resistor, 497 00:35:26,428 --> 00:35:33,745 R one, which is hundred ohms. And here is a resistor, 498 00:35:33,745 --> 00:35:39,795 R two, which is nine hundred ohms. 499 00:35:39,795 --> 00:35:44,978 And I'm asking you what is the current that is flowing around. 500 00:35:44,978 --> 00:35:49,565 And you will laugh at me. You will say that's almost an 501 00:35:49,565 --> 00:35:52,284 insult. I wish you had given that 502 00:35:52,284 --> 00:35:56,871 problem at the first exam, because E equals the current 503 00:35:56,871 --> 00:36:01,034 that is going to run, divided by R one plus R two. 504 00:36:01,034 --> 00:36:03,583 Oh, my goodness, what did I do. 505 00:36:03,583 --> 00:36:06,216 I forgot Ohm's Law. E equals IR, 506 00:36:06,216 --> 00:36:11,739 remember, not I over R. So R one plus R two should 507 00:36:11,739 --> 00:36:16,493 go upstairs. And everything that follows is 508 00:36:16,493 --> 00:36:21,7 correct, so you don't have to worry about that. 509 00:36:21,7 --> 00:36:25,661 This was just a big slip of the pen. 510 00:36:25,661 --> 00:36:31,774 And so the current I is ten to the minus three amperes. 511 00:36:31,774 --> 00:36:34,604 One milliampere. Big deal. 512 00:36:34,604 --> 00:36:38,452 Easy. Current is going to flow like 513 00:36:38,452 --> 00:36:39,584 this. Fine. 514 00:36:39,584 --> 00:36:46,603 Let's call this point D, and call this point A. 515 00:36:46,603 --> 00:36:51,696 And I asked you what is the potential difference between D 516 00:36:51,696 --> 00:36:54,823 and A. You will be equally insulted. 517 00:36:54,823 --> 00:36:57,682 VD minus VA, you apply Ohm's Law, 518 00:36:57,682 --> 00:37:01,167 you say that's this current times R two. 519 00:37:01,167 --> 00:37:03,401 Absolutely. I times R two. 520 00:37:03,401 --> 00:37:06,618 But that is plus oh point nine volts. 521 00:37:06,618 --> 00:37:10,907 Now I say to you, well, suppose you had gone this 522 00:37:10,907 --> 00:37:15,911 way, then you would've said, well, I find the same thing, 523 00:37:15,911 --> 00:37:19,842 of course. Kirchoff's Rule. 524 00:37:19,842 --> 00:37:24,371 So indeed, if you go VD minus VA, and you go this way, 525 00:37:24,371 --> 00:37:28,9 then notice this battery, this point is one volt above 526 00:37:28,9 --> 00:37:32,232 this point. But you have in the resistor 527 00:37:32,232 --> 00:37:36,675 here, you have a voltage drop according to Ohm's Law, 528 00:37:36,675 --> 00:37:41,375 and the current times hundred ohms gives you a one-tenth 529 00:37:41,375 --> 00:37:45,476 voltage drop here, so VD minus VA is the one volt 530 00:37:45,476 --> 00:37:51,458 from the battery minus I times R one, and that is plus oh 531 00:37:51,458 --> 00:37:55,061 point nine volts. What a waste of time that we 532 00:37:55,061 --> 00:37:58,345 did it twice and we found the same result. 533 00:37:58,345 --> 00:38:03,07 So I connect here a voltmeter. The voltmeter is connected to 534 00:38:03,07 --> 00:38:07,154 point D and to point A. And I asked you what are you 535 00:38:07,154 --> 00:38:10,358 going to see. The answer is plus oh point 536 00:38:10,358 --> 00:38:14,682 nine volts, and you will provided that the plus side of 537 00:38:14,682 --> 00:38:19,167 the voltmeter is connected here and the minus side of the 538 00:38:19,167 --> 00:38:36,754 voltmeter there. Voltmeters are polarity 539 00:38:36,754 --> 00:38:53,172 sensitive. This is fine. 540 00:38:53,172 --> 00:38:57,994 a solenoid which you see right here, and this solenoid when I 541 00:38:57,994 --> 00:39:02,254 switch it on is creating an increasing magnetic field. 542 00:39:02,254 --> 00:39:06,594 Only here, and let's assume that an increasing magnetic 543 00:39:06,594 --> 00:39:10,372 field is coming out of the board, and that it is 544 00:39:10,372 --> 00:39:13,426 increasing. Lenz's Law will immediately 545 00:39:13,426 --> 00:39:16,722 tell you in what direction the current is. 546 00:39:16,722 --> 00:39:20,58 If this magnetic field is increasing towards you, 547 00:39:20,58 --> 00:39:24,971 the current will be in this direction. 548 00:39:24,971 --> 00:39:30,051 The magnetic flux change, d phi dt, at a particular 549 00:39:30,051 --> 00:39:33,911 moment in time, happens to be one volt. 550 00:39:33,911 --> 00:39:37,162 An amazing coincidence, isn't it. 551 00:39:37,162 --> 00:39:41,327 E induced at a moment in time is one volt. 552 00:39:41,327 --> 00:39:44,883 Now, I ask you, what is the current? 553 00:39:44,883 --> 00:39:50,978 Well, you'll be surprised that I even have the courage to ask 554 00:39:50,978 --> 00:39:55,855 you that, because Ohm's Law holds. 555 00:39:55,855 --> 00:40:00,555 The induced EMF is one volt and R one plus R two is still a 556 00:40:00,555 --> 00:40:03,715 thousand ohms, so ten to the minus three 557 00:40:03,715 --> 00:40:06,551 amperes. I really make a nuisance of 558 00:40:06,551 --> 00:40:11,413 myself when I say what is VD minus VA, and you get annoyed at 559 00:40:11,413 --> 00:40:14,979 me and you say, look, the current I through R 560 00:40:14,979 --> 00:40:17,167 two, Ohm's Law, V equals IR, 561 00:40:17,167 --> 00:40:20,408 plus oh point nine volts. And then I say, 562 00:40:20,408 --> 00:40:25,514 but now suppose we go the other- the other side, 563 00:40:25,514 --> 00:40:32,66 and we want to know now what VD minus VA is, and now it's not so 564 00:40:32,66 --> 00:40:36,517 simple, because there's no battery. 565 00:40:36,517 --> 00:40:42,756 And so now when I go from D to A, I don't have this one, 566 00:40:42,756 --> 00:40:48,314 and therefore I now find minus oh point one volts. 567 00:40:48,314 --> 00:40:52,057 I find a totally different answer. 568 00:40:52,057 --> 00:40:59,213 I attach a voltmeter here. That voltmeter will show me 569 00:40:59,213 --> 00:41:05,24 plus oh point nine volts. Now I attach a voltmeter here, 570 00:41:05,24 --> 00:41:08,309 the same one. I flip it over. 571 00:41:08,309 --> 00:41:12,911 It's connected between point D and point A. 572 00:41:12,911 --> 00:41:16,966 It will read minus oh point one volts. 573 00:41:16,966 --> 00:41:21,678 This voltmeter, which is connected between D 574 00:41:21,678 --> 00:41:28,526 and A, reads plus oh point nine. This voltmeter which is 575 00:41:28,526 --> 00:41:32,491 connected to D and A reads minus oh point one. 576 00:41:32,491 --> 00:41:37,248 The two values are different, and I placed on the web a 577 00:41:37,248 --> 00:41:41,742 lecture supplement which goes through the derivation 578 00:41:41,742 --> 00:41:45,354 step-by-step, which will convince you that 579 00:41:45,354 --> 00:41:48,174 indeed this is what is happening. 580 00:41:48,174 --> 00:41:53,196 Why we can't digest this so easily is we don't know how to 581 00:41:53,196 --> 00:41:58,57 handle non-conservative fields. If you have a non-conservative 582 00:41:58,57 --> 00:42:03,357 field, then if you go from A to D of 583 00:42:03,357 --> 00:42:08,579 E dot DL or from D to A for that matter, doesn't matter, 584 00:42:08,579 --> 00:42:14,276 the answer depends on the path. It's no longer independent of 585 00:42:14,276 --> 00:42:17,029 the path. And so if here is D, 586 00:42:17,029 --> 00:42:20,163 and here is A, and we go this way, 587 00:42:20,163 --> 00:42:25,48 you find oh point nine volts, plus if you go this way you 588 00:42:25,48 --> 00:42:30,987 find minus oh point one volts. Faraday has no problems with 589 00:42:30,987 --> 00:42:35,758 that. Kirchoff has a problem with 590 00:42:35,758 --> 00:42:38,657 that, but who cares about Kirchoff? 591 00:42:38,657 --> 00:42:43,176 Faraday is the law that matters, because Faraday's Law 592 00:42:43,176 --> 00:42:46,672 always holds, because if d phi dt is zero, 593 00:42:46,672 --> 00:42:51,021 then you get Kirchoff's. Kirchoff's rule is simply a 594 00:42:51,021 --> 00:42:56,222 special case of Faraday's Law, and Faraday's Law always holds, 595 00:42:56,222 --> 00:43:00,315 so Kirchoff is for the birds, and Faraday is not. 596 00:43:00,315 --> 00:43:05,493 Suppose you go from D to A and back to D. 597 00:43:05,493 --> 00:43:12,651 Well, we know that VD minus VA, if we go through this- if we go 598 00:43:12,651 --> 00:43:18,539 this way, through R two, we know that VD minus VA is 599 00:43:18,539 --> 00:43:24,312 plus oh point nine volts. Now we are at A and we go 600 00:43:24,312 --> 00:43:31,123 through the left side back to D. So we now have VA minus VD. 601 00:43:31,123 --> 00:43:36,318 That of course is now plus oh point 602 00:43:36,318 --> 00:43:41,14 one volts, because remember, if VD minus VA is minus oh 603 00:43:41,14 --> 00:43:44,265 point one, then VA minus VD is plus. 604 00:43:44,265 --> 00:43:48,997 And so we add them up, and we find that VD minus VD is 605 00:43:48,997 --> 00:43:51,586 plus one volts. Kirchoff said, 606 00:43:51,586 --> 00:43:55,425 has to be zero, because I'm back at the same 607 00:43:55,425 --> 00:43:59,175 potential where I was before. Faraday says, 608 00:43:59,175 --> 00:44:02,3 uh-uh, I'm sorry, you can't do that. 609 00:44:02,3 --> 00:44:07,746 That one volt is exactly that EMF of one volt. 610 00:44:07,746 --> 00:44:11,678 That is the closed loop integral of E dot DL around that 611 00:44:11,678 --> 00:44:13,465 loop. It's no longer zero. 612 00:44:13,465 --> 00:44:16,538 And therefore, whenever you define potential 613 00:44:16,538 --> 00:44:20,898 difference, if you do that in the way of the integral of E dot 614 00:44:20,898 --> 00:44:24,472 DL, keep in mind that with non-conservative fields, 615 00:44:24,472 --> 00:44:28,331 it depends on the path, and that is very non-intuitive. 616 00:44:28,331 --> 00:44:31,476 And I'm going to demonstrate this now to you. 617 00:44:31,476 --> 00:44:35,193 I have a circuit which is exactly what you have here. 618 00:44:35,193 --> 00:44:39,553 I have nine hundred volts in a conducting copper wire here and 619 00:44:39,553 --> 00:44:43,413 I have a hundred volts here and 620 00:44:43,413 --> 00:44:47,687 here is the solenoid. W We can switch the current on 621 00:44:47,687 --> 00:44:52,631 in the solenoid and get a blast of magnetic field coming up, 622 00:44:52,631 --> 00:44:57,24 and the system is going to react by driving a current in 623 00:44:57,24 --> 00:44:59,921 the direction that you see there. 624 00:44:59,921 --> 00:45:04,447 And I'd like to be even a little bit more quantitative, 625 00:45:04,447 --> 00:45:08,469 so that you get a little bit more for your money. 626 00:45:08,469 --> 00:45:14,168 The magnetic field takes a little bit of time to reach 627 00:45:14,168 --> 00:45:16,526 the maximum value. In this course, 628 00:45:16,526 --> 00:45:20,67 we will be able to calculate the time that it takes for the 629 00:45:20,67 --> 00:45:24,457 magnetic field to build up. We didn't get to that yet, 630 00:45:24,457 --> 00:45:27,458 so forget that part. It's not so important. 631 00:45:27,458 --> 00:45:31,817 I just want you to appreciate the fact that the magnetic field 632 00:45:31,817 --> 00:45:36,319 as a function of time will come up like this and will then reach 633 00:45:36,319 --> 00:45:38,748 a maximum. It's no longer changing. 634 00:45:38,748 --> 00:45:41,249 It's constant. It's a maximum value. 635 00:45:41,249 --> 00:45:44,393 It's very high, seven, eight hundred Gauss or 636 00:45:44,393 --> 00:45:47,012 so for this unit. 637 00:45:47,012 --> 00:45:50,425 We are not interested in a magnetic field. 638 00:45:50,425 --> 00:45:54,837 We are interested in the change of the magnetic field, 639 00:45:54,837 --> 00:45:59,499 so the change of the magnetic field, dB dT is going to be 640 00:45:59,499 --> 00:46:03,411 something like this, just the derivative of this 641 00:46:03,411 --> 00:46:06,325 curve. And that is proportional with 642 00:46:06,325 --> 00:46:11,32 the induced EMF and that's in por- pro- proportional with the 643 00:46:11,32 --> 00:46:15,982 current, through Ohm's law. So if we now plot the voltage 644 00:46:15,982 --> 00:46:20,643 as a function of -- let me do that here, 645 00:46:20,643 --> 00:46:26,961 the voltage as a function of time, then that voltmeter on the 646 00:46:26,961 --> 00:46:31,489 right side, I call that V two, will do this. 647 00:46:31,489 --> 00:46:36,016 This is V two, which is I times R two at the 648 00:46:36,016 --> 00:46:40,755 maximum value. If those values were correct it 649 00:46:40,755 --> 00:46:46,862 would be oh point nine volts, and V one would go like this. 650 00:46:46,862 --> 00:46:50,231 V one equals minus I times R one. 651 00:46:50,231 --> 00:46:55,7 That gives me the minus oh point one volts. 652 00:46:55,7 --> 00:47:01,087 So the question now is what is the largest value of dB dT that 653 00:47:01,087 --> 00:47:04,443 we can expect. We also have to know the 654 00:47:04,443 --> 00:47:09,654 surface area of the solenoids so we can convert it to a flux 655 00:47:09,654 --> 00:47:12,746 change. Well, the change in magnetic 656 00:47:12,746 --> 00:47:18,222 fields is roughly at the fastest here is about hundred Gauss in 657 00:47:18,222 --> 00:47:20,783 one millisecond. Very roughly. 658 00:47:20,783 --> 00:47:25,641 So that would mean a field change, dB dT. 659 00:47:25,641 --> 00:47:29,458 That's the maximum value possible only in the beginning 660 00:47:29,458 --> 00:47:33,064 of about ten Tesla per second. And the surface area, 661 00:47:33,064 --> 00:47:37,094 which is that inner circle there through which the flux is 662 00:47:37,094 --> 00:47:40,982 changing, the fact that my surface has to be attached to 663 00:47:40,982 --> 00:47:43,952 that loop doesn't change the magnetic flux. 664 00:47:43,952 --> 00:47:47,275 The magnetic flux is only determined, of course, 665 00:47:47,275 --> 00:47:51,022 by that inner portion, and so if the inner portion has 666 00:47:51,022 --> 00:47:55,051 an area of say ten square centimeters, which is ten to the 667 00:47:55,051 --> 00:47:59,505 minus two square meters, then d phi dt will be 668 00:47:59,505 --> 00:48:02,691 approximately ten times ten to the minus two, 669 00:48:02,691 --> 00:48:06,021 so that's about oh point one, and that's volts. 670 00:48:06,021 --> 00:48:08,411 That's EMF. I don't care about the 671 00:48:08,411 --> 00:48:11,017 direction, because I know Lenz's Law. 672 00:48:11,017 --> 00:48:14,709 So you're going to see an experiment which is almost 673 00:48:14,709 --> 00:48:19,198 identical to what I have there, except all values are down by a 674 00:48:19,198 --> 00:48:21,298 factor of ten. But that's all. 675 00:48:21,298 --> 00:48:24,773 And you're going to see that demonstration there. 676 00:48:24,773 --> 00:48:27,741 And a few years ago, when I first did this 677 00:48:27,741 --> 00:48:31,795 experiment in twenty-six one hundred, 678 00:48:31,795 --> 00:48:36,075 there were several of my colleagues, professors of both 679 00:48:36,075 --> 00:48:40,434 the physics department and EE department in my audience. 680 00:48:40,434 --> 00:48:43,446 And some did not believe what they saw. 681 00:48:43,446 --> 00:48:48,122 In fact, it was so bad that after my lecture they came to me 682 00:48:48,122 --> 00:48:52,797 and some accused me for having cheated on the demonstration. 683 00:48:52,797 --> 00:48:55,571 This tells you something about them. 684 00:48:55,571 --> 00:49:00,168 Imagine, professors in physics and professors in electrical 685 00:49:00,168 --> 00:49:04,219 engineering department who did not believe 686 00:49:04,219 --> 00:49:06,872 what they were seeing. That tells you how 687 00:49:06,872 --> 00:49:10,388 non-intuitive this is. The simple fact that we had one 688 00:49:10,388 --> 00:49:14,236 voltmeter connected to point D and A, and another voltmeter 689 00:49:14,236 --> 00:49:18,018 connected to the same point, they were unwilling to accept 690 00:49:18,018 --> 00:49:21,6 that the two voltmeters read a totally different value. 691 00:49:21,6 --> 00:49:24,585 They were not used to non-conservative fields. 692 00:49:24,585 --> 00:49:26,642 Their brains couldn't handle it. 693 00:49:26,642 --> 00:49:30,224 But that's the way it is, and I'm going to show this to 694 00:49:30,224 --> 00:49:33,94 you now. You're going to see it there, 695 00:49:33,94 --> 00:49:36,467 and when you see this demonstration, 696 00:49:36,467 --> 00:49:40,726 it will be probably the only time in your life that you will 697 00:49:40,726 --> 00:49:43,614 ever see this, and I want you to remember 698 00:49:43,614 --> 00:49:46,069 this. You're going to see something 699 00:49:46,069 --> 00:49:49,463 that is very strange, and I want you to tell you 700 00:49:49,463 --> 00:49:53,289 grandchildren about it, that you have actually seen it 701 00:49:53,289 --> 00:49:56,755 with your own eyes. You're going to see there on 702 00:49:56,755 --> 00:49:59,643 the left side, you're going to see V one, 703 00:49:59,643 --> 00:50:04,842 and on the right side you're going to see V two. 704 00:50:04,842 --> 00:50:09,606 The vertical scale is such that very roughly from here to here 705 00:50:09,606 --> 00:50:13,825 is about oh point one volts. And the horizontal unit is 706 00:50:13,825 --> 00:50:17,808 about five milliseconds, and the whole voltage pulse 707 00:50:17,808 --> 00:50:22,261 lasts about ten milliseconds, because from here to here is 708 00:50:22,261 --> 00:50:26,323 about ten milliseconds. And the value that you expect 709 00:50:26,323 --> 00:50:30,463 for V two will be nine times higher than V one and the 710 00:50:30,463 --> 00:50:34,837 polarities will be reversed. If you're ready for this big 711 00:50:34,837 --> 00:50:37,492 moment in your life, 712 00:50:37,492 --> 00:50:39,13 three, two, one, zero. 713 00:50:39,13 --> 00:50:41,548 Look on the left. There's V one. 714 00:50:41,548 --> 00:50:44,668 Notice, it's negative. Look on the right. 715 00:50:44,668 --> 00:50:47,944 There's V two. It's about nine times larger 716 00:50:47,944 --> 00:50:51,22 than V one. Don't pay any attention to this 717 00:50:51,22 --> 00:50:54,419 [inaudible]. It has to do with the voltage 718 00:50:54,419 --> 00:50:57,539 that we apply, which is not exactly flat. 719 00:50:57,539 --> 00:51:01,751 And notice that the whole pulse goes from here to here, 720 00:51:01,751 --> 00:51:06,197 lasts about ten milliseconds. The moment that the magnetic 721 00:51:06,197 --> 00:51:10,418 field reaches a maximum and remains 722 00:51:10,418 --> 00:51:14,176 constant, there is no longer any induced current. 723 00:51:14,176 --> 00:51:17,307 Think about this. Give this some thought. 724 00:51:17,307 --> 51:22 This is not easy. And have a good weekend.