1 00:00:00 --> 00:00:00,354 2 00:00:00,354 --> 00:00:03,587 Here are the topics the way I see them. 3 00:00:03,587 --> 00:00:07,756 They're on the web. Look on the lecture supplement 4 00:00:07,756 --> 00:00:10,65 of today and you can download them. 5 00:00:10,65 --> 00:00:14,819 I want to point out that, uh, the discount is very 6 00:00:14,819 --> 00:00:20,265 reasonable because these magnets are broken, and when you break a 7 00:00:20,265 --> 00:00:24,86 magnet, you end up with two monopoles, so you get fifty 8 00:00:24,86 --> 00:00:28,008 percent off. You should know better by 9 00:00:28,008 --> 00:00:32,152 looking at the key equation there, 10 00:00:32,152 --> 00:00:37,061 that magnetic monopoles don't exist, but that's a detail. 11 00:00:37,061 --> 00:00:42,145 I want you to appreciate that I cannot possibly cover today 12 00:00:42,145 --> 00:00:46,879 these topics in any depth, nor can I cover all of these 13 00:00:46,879 --> 00:00:51,875 during a fifty minute exam. So please understand that this 14 00:00:51,875 --> 00:00:56,872 review is highly incomplete, and what is not covered today 15 00:00:56,872 --> 00:01:02,682 can and will be on the exam. I'm interested in concepts. 16 00:01:02,682 --> 00:01:07,46 I'm not interested in math. There will be seven problems. 17 00:01:07,46 --> 00:01:11,641 Five of the seven problems have only one question. 18 00:01:11,641 --> 00:01:14,286 Two problems have two questions. 19 00:01:14,286 --> 00:01:19,32 I don't think that time with the length of the exam is going 20 00:01:19,32 --> 00:01:23,159 to be an issue. This exam was taken by several 21 00:01:23,159 --> 00:01:26,146 instructors. It took them fifteen to 22 00:01:26,146 --> 00:01:29,473 eighteen minutes, and that's normally my 23 00:01:29,473 --> 00:01:33,983 objective. Courtesy of Professor Belcher, 24 00:01:33,983 --> 00:01:36,428 we have old tests on the website. 25 00:01:36,428 --> 00:01:40,782 I don't have the solutions. There's only so much I can do. 26 00:01:40,782 --> 00:01:45,518 I'm very grateful to Professor Belcher that he made these exams 27 00:01:45,518 --> 00:01:48,344 available. If you can't do some of the 28 00:01:48,344 --> 00:01:52,927 problems, I would suggest you see your tutors or you see your 29 00:01:52,927 --> 00:01:55,907 instructor. I will also be available all 30 00:01:55,907 --> 00:01:57,969 day afternoon, not tomorrow, 31 00:01:57,969 --> 00:02:01,789 because tomorrow afternoon I'll be 32 00:02:01,789 --> 00:02:06,916 in twenty-six one hundred all afternoon to work on 33 00:02:06,916 --> 00:02:10,265 demonstrations for the next week. 34 00:02:10,265 --> 00:02:14,659 Many of these problems are straightforward. 35 00:02:14,659 --> 00:02:19,473 You may also want to consult your study guides. 36 00:02:19,473 --> 00:02:22,822 All right. Let's first start with 37 00:02:22,822 --> 00:02:27,321 Biot-Savart. There are not too many problems 38 00:02:27,321 --> 00:02:30,67 that one can do with Biot-Savart. 39 00:02:30,67 --> 00:02:36,007 DB equals mu zero divided by four pi 40 00:02:36,007 --> 00:02:41,544 times the current DL cross R divided by R squared. 41 00:02:41,544 --> 00:02:46,856 That's the formalism. A classic problem that you 42 00:02:46,856 --> 00:02:51,829 probably have done. We have point P here at a 43 00:02:51,829 --> 00:02:57,48 distance D from a wire, and the current through the 44 00:02:57,48 --> 00:03:01,774 wire is I. If you want to know what the 45 00:03:01,774 --> 00:03:08,443 magnetic field at P is, you can use Biot-Savart. 46 00:03:08,443 --> 00:03:12,833 It would be a stupid thing to do, but you can do it. 47 00:03:12,833 --> 00:03:17,052 You take then a small element DL here of the wire. 48 00:03:17,052 --> 00:03:20,926 This distance is R. This is the unit vector R, 49 00:03:20,926 --> 00:03:23,595 which you have in this equation. 50 00:03:23,595 --> 00:03:28,072 And you can calculate, then, what the contribution to 51 00:03:28,072 --> 00:03:32,721 the magnetic field right here is -- it comes out of the 52 00:03:32,721 --> 00:03:35,82 blackboard -- due to this section DL. 53 00:03:35,82 --> 00:03:39,78 This angle is theta, and the sine of theta is D 54 00:03:39,78 --> 00:03:42,351 divided by R. 55 00:03:42,351 --> 00:03:47,897 And then you have to do an integral over the whole wire, 56 00:03:47,897 --> 00:03:52,535 theta zero to pi. And then you get the magnetic 57 00:03:52,535 --> 00:03:56,367 field here. Not very smart thing to do. 58 00:03:56,367 --> 00:04:00,905 A waste of time. Because clearly the way to do 59 00:04:00,905 --> 00:04:06,754 this is to use Ampere's Law, which is the one at the bottom 60 00:04:06,754 --> 00:04:08,77 there. In which case, 61 00:04:08,77 --> 00:04:15,022 you would construct a closed loop with radius D. 62 00:04:15,022 --> 00:04:18,112 This loop is perpendicular to the blackboard. 63 00:04:18,112 --> 00:04:21,412 I'll try to make you see it three dimensionally. 64 00:04:21,412 --> 00:04:25,203 You have to attach an open surface to that closed loop. 65 00:04:25,203 --> 00:04:28,854 Any open surface will do. Let's make the open surface 66 00:04:28,854 --> 00:04:31,382 flat. And then we apply Ampere's Law, 67 00:04:31,382 --> 00:04:33,909 which is the one at the bottom there. 68 00:04:33,909 --> 00:04:37,49 We don't have the second portion because there is no 69 00:04:37,49 --> 00:04:41,141 changing electric flux. We don't deal with kappa M at 70 00:04:41,141 --> 00:04:43,528 all. So we simply have that B times 71 00:04:43,528 --> 00:04:47,531 two pi D, that is going around in this 72 00:04:47,531 --> 00:04:52,552 circle, because I know that B is tangentially to that circle. 73 00:04:52,552 --> 00:04:57,323 I also even know the direction according to the right-hand 74 00:04:57,323 --> 00:05:00,922 corkscrew rule, coming out of the blackboard 75 00:05:00,922 --> 00:05:03,768 here. So as I go around the circle, 76 00:05:03,768 --> 00:05:08,789 B and the DL that you see there at the bottom are in the same 77 00:05:08,789 --> 00:05:11,719 direction. So I get two pi B times D 78 00:05:11,719 --> 00:05:16,824 equals mu zero times the current that penetrates that surface, 79 00:05:16,824 --> 00:05:21,418 and that is I. And so the answer is very 80 00:05:21,418 --> 00:05:24,625 simple, mu zero I divided by two pi D. 81 00:05:24,625 --> 00:05:28,091 That's the way you would do this problem, 82 00:05:28,091 --> 00:05:31,558 and you would stay away from Biot-Savart. 83 00:05:31,558 --> 00:05:36,931 There is one particular problem whereby Ampere's Law will fail. 84 00:05:36,931 --> 00:05:41,438 Of course, Ampere's Law in general works when we have 85 00:05:41,438 --> 00:05:45,771 cylindrical symmetry. This is cylindrical symmetry. 86 00:05:45,771 --> 00:05:50,017 There is one problem where Ampere's 87 00:05:50,017 --> 00:05:54,134 Law bitterly fails and where Biot-Savart is highly superior. 88 00:05:54,134 --> 00:05:57,203 I have here a conducting loop. It's a circle. 89 00:05:57,203 --> 00:06:00,761 And you're being asked, it runs a certain current I, 90 00:06:00,761 --> 00:06:04,18 and you're being asked, what is the magnetic field 91 00:06:04,18 --> 00:06:07,599 right at the center? It only works for the center. 92 00:06:07,599 --> 00:06:11,087 You could not find what the magnetic field is here. 93 00:06:11,087 --> 00:06:14,645 We did that in class. You probably also did that for 94 00:06:14,645 --> 00:06:17,575 your homework. Biot-Savart will immediately 95 00:06:17,575 --> 00:06:23,476 give you the answer. And I will leave you with that. 96 00:06:23,476 --> 00:06:29,58 And Ampere's Law won't work. So let's now turn to Ampere's 97 00:06:29,58 --> 00:06:34,185 Law and do a few problems with Ampere's Law. 98 00:06:34,185 --> 00:06:39,968 We need cylindrical symmetry, with very few exceptions. 99 00:06:39,968 --> 00:06:44,251 I have a hollow cylinder here, radius R1. 100 00:06:44,251 --> 00:06:48,749 Concentric another cylinder with radius R2. 101 00:06:48,749 --> 00:06:55,563 These are very long cylinders. And assume that there is a 102 00:06:55,563 --> 00:06:59,395 current flowing, I, in this direction on the 103 00:06:59,395 --> 00:07:04,742 surface of the inner cylinder, and the current I is returning 104 00:07:04,742 --> 00:07:09,821 on the surface of the outer cylinder, and the two currents 105 00:07:09,821 --> 00:07:14,811 are the same in magnitude. And I want to know what is the 106 00:07:14,811 --> 00:07:17,841 magnetic field everywhere in space. 107 00:07:17,841 --> 00:07:21,673 I will make a, a drawing whereby we only see 108 00:07:21,673 --> 00:07:27,197 the cross-section. So this is R1 and this is R2. 109 00:07:27,197 --> 00:07:31,495 And let's first calculate the magnetic field for R being 110 00:07:31,495 --> 00:07:34,933 larger than R2. It's immediately obvious that 111 00:07:34,933 --> 00:07:39,856 your closed loop that you choose itself is going to be a circle. 112 00:07:39,856 --> 00:07:41,732 That is a must. Radius R. 113 00:07:41,732 --> 00:07:46,186 We use a symmetry argument. Whatever the magnetic field is 114 00:07:46,186 --> 00:07:49,937 at that distance R, little R, it must be the same 115 00:07:49,937 --> 00:07:53,218 everywhere. It cannot be any different here 116 00:07:53,218 --> 00:07:58,766 in terms of magnitude than there because of the symmetry 117 00:07:58,766 --> 00:08:02,289 of the problem. We have cylindrical symmetry. 118 00:08:02,289 --> 00:08:06,131 We also know that if there is any magnetic field, 119 00:08:06,131 --> 00:08:09,973 that it is going to be tangential, either in this 120 00:08:09,973 --> 00:08:14,055 direction or in that direction. And so we go around. 121 00:08:14,055 --> 00:08:18,378 We make the closed loop integral, use the equation that 122 00:08:18,378 --> 00:08:22,94 we have there at the bottom, and so you get B times two pi 123 00:08:22,94 --> 00:08:27,023 little R equals mu zero. And now I have to attach an 124 00:08:27,023 --> 00:08:30,118 open surface to this loop. 125 00:08:30,118 --> 00:08:33,116 I will use the surface in the blackboard. 126 00:08:33,116 --> 00:08:37,164 I could use any surface, but I might as well use a flat 127 00:08:37,164 --> 00:08:40,012 surface. And now I have to know what is 128 00:08:40,012 --> 00:08:42,861 the current going through that surface. 129 00:08:42,861 --> 00:08:47,134 Right here, on this surface, the current is going into the 130 00:08:47,134 --> 00:08:50,357 blackboard, and right here, on this surface, 131 00:08:50,357 --> 00:08:53,58 the current is coming out of the blackboard. 132 00:08:53,58 --> 00:08:57,553 The two magnitudes are the same, so the net current is 133 00:08:57,553 --> 00:08:59,802 zero. And so the magnetic field 134 00:08:59,802 --> 00:09:04 outside the second cylinder is zero. 135 00:09:04 --> 00:09:10,418 So let's now look at the area in between the two cylinders. 136 00:09:10,418 --> 00:09:16,504 These are hollow cylinders, now, this is completely open 137 00:09:16,504 --> 00:09:20,046 here and this is completely open. 138 00:09:20,046 --> 00:09:23,144 They are thin, thin material, 139 00:09:23,144 --> 00:09:26,575 thin shells, are both cylinders. 140 00:09:26,575 --> 00:09:32,993 Now of course the closed loop is going to be one inside the 141 00:09:32,993 --> 00:09:37,993 opening between the two cylinders. 142 00:09:37,993 --> 00:09:42,387 And again, I pick radius R. And here I go. 143 00:09:42,387 --> 00:09:48,175 I get B times two pi R equals mu zero, but now there is 144 00:09:48,175 --> 00:09:54,391 current going through this surface, and the current that is 145 00:09:54,391 --> 00:09:57,821 going through is I. Not this one, 146 00:09:57,821 --> 00:10:00,929 but this one. And so we get I. 147 00:10:00,929 --> 00:10:07,36 And so now we get that B is mu zero times I divided by two pi 148 00:10:07,36 --> 00:10:10,35 R. One over R field. 149 00:10:10,35 --> 00:10:14,553 The direction you will find from the right-hand corkscrew 150 00:10:14,553 --> 00:10:17,029 rule. That's the way I normally do 151 00:10:17,029 --> 00:10:19,356 it. You take a corkscrew and you 152 00:10:19,356 --> 00:10:22,808 turn it clockwise, it goes into the blackboard, 153 00:10:22,808 --> 00:10:26,335 so the magnetic field here is in this direction. 154 00:10:26,335 --> 00:10:30,012 If you're not used to turning corkscrews in corks, 155 00:10:30,012 --> 00:10:33,689 think about when you try to tighten a screw with a 156 00:10:33,689 --> 00:10:36,09 screwdriver. If you go clockwise, 157 00:10:36,09 --> 00:10:41,193 the screw goes in. At least, that is the case with 158 00:10:41,193 --> 00:10:45,871 ninety-nine point nine nine percent of all screws that you 159 00:10:45,871 --> 00:10:49,564 find in this country. They have a right-handed 160 00:10:49,564 --> 00:10:52,519 thread. You could make one that has a 161 00:10:52,519 --> 00:10:57,196 left-handed threa- thread, but that's not done in general. 162 00:10:57,196 --> 00:11:00,725 So now we can go to the area R less than R1. 163 00:11:00,725 --> 00:11:04,09 So that's inside this thin shell cylinder, 164 00:11:04,09 --> 00:11:08,193 and again, you would take a surface, a closed loop, 165 00:11:08,193 --> 00:11:12,543 that is a circle with radius little R, 166 00:11:12,543 --> 00:11:17,773 and this is your surface attached, open surface attached 167 00:11:17,773 --> 00:11:22,527 to that closed loop. And you will find for the same 168 00:11:22,527 --> 00:11:28,328 reason that you found here that B is zero, because there is no 169 00:11:28,328 --> 00:11:31,561 current going through that surface. 170 00:11:31,561 --> 00:11:36,982 And so if you now make a plot of the magnetic field B as a 171 00:11:36,982 --> 00:11:40,5 function of R, here is R1, here is R2, 172 00:11:40,5 --> 00:11:45,67 then it is zero here, it's zero there, 173 00:11:45,67 --> 00:11:51,083 it has some value here, which is this value if you 174 00:11:51,083 --> 00:11:57,491 substitute for little R R1 and right here it is this value, 175 00:11:57,491 --> 00:12:01,136 if you substitute for little R R2. 176 00:12:01,136 --> 00:12:07,101 And this is a curve that is proportional to one over R. 177 00:12:07,101 --> 00:12:12,073 And so this is the magnetic field, and inside, 178 00:12:12,073 --> 00:12:17,859 seen from where you were sitting, it is clockwise. 179 00:12:17,859 --> 00:12:21,19 There is one case, and we did that in lectures, 180 00:12:21,19 --> 00:12:25,317 whereby we wanted to calculate the magnetic field inside a 181 00:12:25,317 --> 00:12:28,502 solenoid, where Ampere's Law works very well, 182 00:12:28,502 --> 00:12:32,918 even though we don't have then closed loops which are circles, 183 00:12:32,918 --> 00:12:36,394 we chose a rectangle. I want you to revisit that. 184 00:12:36,394 --> 00:12:40,448 It's undoubtedly in your book, and I covered it during my 185 00:12:40,448 --> 00:12:43,127 lectures. We assumed that the magnetic 186 00:12:43,127 --> 00:12:47,254 field was uniform inside the solenoid and 187 00:12:47,254 --> 00:12:53,476 zero outside the solenoid, which was not a bad assumption, 188 00:12:53,476 --> 00:12:58,17 and that allows you then, with Ampere's Law, 189 00:12:58,17 --> 00:13:03,301 even though you go rectangular and not circular, 190 00:13:03,301 --> 00:13:07,996 to get the magnetic field inside a solenoid. 191 00:13:07,996 --> 00:13:12,362 Very classic. Check your lecture notes or 192 00:13:12,362 --> 00:13:17,93 watch my lecture again, which you can do on the web. 193 00:13:17,93 --> 00:13:22,078 So now I want to turn to Lorentz force. 194 00:13:22,078 --> 00:13:29,174 The Lorentz force, F, is Q times E plus V cross B. 195 00:13:29,174 --> 00:13:33,616 This is the charge which is moving with velocity V in a 196 00:13:33,616 --> 00:13:38,222 magnetic field B and at the location of that charge there 197 00:13:38,222 --> 00:13:41,348 happens to be also an electric field E. 198 00:13:41,348 --> 00:13:45,131 Let's take a situation that I have an electron. 199 00:13:45,131 --> 00:13:49,737 I put a minus sign there to remind you that the charge is 200 00:13:49,737 --> 00:13:53,027 negative. And let this be the velocity of 201 00:13:53,027 --> 00:13:56,564 that electron. And let the magnetic field be 202 00:13:56,564 --> 00:14:02,65 uniform, perpendicular to the blackboard in this direction, 203 00:14:02,65 --> 00:14:07,233 going into the blackboard. There is no electric field, 204 00:14:07,233 --> 00:14:10,518 so we only deal with Q times V cross B. 205 00:14:10,518 --> 00:14:15,792 V cross B -- you should be able to do that cross product -- is 206 00:14:15,792 --> 00:14:19,336 in this direction, but since the charge is 207 00:14:19,336 --> 00:14:24,091 negative, the force on this charge is in this direction. 208 00:14:24,091 --> 00:14:29,451 And so the electron is going to turn over in this direction and 209 00:14:29,451 --> 00:14:35,59 is going to get a circle if the magnetic field is uniform. 210 00:14:35,59 --> 00:14:40,321 And the force is right at the center of that circle. 211 00:14:40,321 --> 00:14:44,867 A little later in time, when the electron is here, 212 00:14:44,867 --> 00:14:49,228 going with the same speed, the force is like so. 213 00:14:49,228 --> 00:14:54,145 The speed cannot change. The force cannot do any work, 214 00:14:54,145 --> 00:14:58,97 cannot change the kinetic energy, because F is always 215 00:14:58,97 --> 00:15:02,867 perpendicular to the plane through V and B, 216 00:15:02,867 --> 00:15:08,96 because it's a cross product. And since the force is always 217 00:15:08,96 --> 00:15:13,429 perpendicular to the velocity, you cannot change the kinetic 218 00:15:13,429 --> 00:15:16,686 energy, because you're never doing any work. 219 00:15:16,686 --> 00:15:21,155 The only thing that the force is doing is make it go around. 220 00:15:21,155 --> 00:15:24,185 It changes the direction of the velocity. 221 00:15:24,185 --> 00:15:27,745 It doesn't change the magnitude of the velocity. 222 00:15:27,745 --> 00:15:30,623 It doesn't change the speed. All right? 223 00:15:30,623 --> 00:15:35,017 So let this radius be little R. Combining eight oh two with 224 00:15:35,017 --> 00:15:39,068 eight oh one, we now have that M V 225 00:15:39,068 --> 00:15:44,428 squared over R -- M being the mass of the electron -- is that 226 00:15:44,428 --> 00:15:47,734 force. And since V is perpendicular to 227 00:15:47,734 --> 00:15:53,451 B, the sine of the angle between them is one, and so I simply get 228 00:15:53,451 --> 00:15:56,936 here Q V B. This is the magnitude of the 229 00:15:56,936 --> 00:16:00,152 force. We already know the direction. 230 00:16:00,152 --> 00:16:04,529 And so one V goes, and so I get that the radius of 231 00:16:04,529 --> 00:16:08,817 that circle is M V divided by Q B. 232 00:16:08,817 --> 00:16:11,745 This being the momentum of the electron. 233 00:16:11,745 --> 00:16:14,972 So V is the velocity of the electron, speed. 234 00:16:14,972 --> 00:16:17,975 Q is the charge. B is the magnetic field. 235 00:16:17,975 --> 00:16:22,553 M is the mass of the electron. And as long as this velocity is 236 00:16:22,553 --> 00:16:25,706 reasonably smaller than the speed of light, 237 00:16:25,706 --> 00:16:29,609 we don't have to make a relativistic correction here. 238 00:16:29,609 --> 00:16:33,211 If it's not much smaller than the speed of light, 239 00:16:33,211 --> 00:16:36,289 we have to make a relativistic correction. 240 00:16:36,289 --> 00:16:41,936 I discussed the relativistic correction in my lectures, 241 00:16:41,936 --> 00:16:46,91 but I also mentioned that that would not be part of exams, 242 00:16:46,91 --> 00:16:50,314 so I will not further elaborate on that. 243 00:16:50,314 --> 00:16:54,241 So we will just assume that this is completely 244 00:16:54,241 --> 00:16:58,256 non-relativistic. The time for this electron to 245 00:16:58,256 --> 00:17:02,271 go around, capital T, is two pi R divided by V. 246 00:17:02,271 --> 00:17:05,587 But I know R, so I get two pi times M V 247 00:17:05,587 --> 00:17:10,212 divided by Q B. And then I have this V, 248 00:17:10,212 --> 00:17:13,718 and I lose this V, and that is not so intuitive, 249 00:17:13,718 --> 00:17:18,044 but you see that the time is independent of the velocity of 250 00:17:18,044 --> 00:17:20,357 the electron, assuming that it's 251 00:17:20,357 --> 00:17:23,713 non-relativistic. And we discussed that in the 252 00:17:23,713 --> 00:17:27,741 framework of cyclotrons, time for these particles to go 253 00:17:27,741 --> 00:17:30,501 around is independent of the velocity. 254 00:17:30,501 --> 00:17:34,901 Suppose I have a magnetic field B which is seven point eight 255 00:17:34,901 --> 00:17:37,363 times ten to the minus four tesla. 256 00:17:37,363 --> 00:17:39,675 Now I know what you're thinking. 257 00:17:39,675 --> 00:17:44,495 You're thinking, why the hell do you take this 258 00:17:44,495 --> 00:17:47,466 crazy value? There is a reason for that, 259 00:17:47,466 --> 00:17:50,133 because I have a demonstration here. 260 00:17:50,133 --> 00:17:54,628 And the magnetic field is seven point eight times ten to the 261 00:17:54,628 --> 00:17:57,524 minus four tesla. If that is the field, 262 00:17:57,524 --> 00:18:01,028 I can calculate how long it takes to go around, 263 00:18:01,028 --> 00:18:05,6 because I know the mass of the electron, I know the charge of 264 00:18:05,6 --> 00:18:08,571 the electron, I know the magnetic field. 265 00:18:08,571 --> 00:18:11,314 So in that case, T is about forty-six 266 00:18:11,314 --> 00:18:14,59 nanoseconds. And so the number of times that 267 00:18:14,59 --> 00:18:17,691 it goes around, 268 00:18:17,691 --> 00:18:24,576 F, which is the frequency in hertz, is one over the period, 269 00:18:24,576 --> 00:18:30,275 is about twenty-two megahertz. So it goes around, 270 00:18:30,275 --> 00:18:35,261 this electron, twenty-two million times per 271 00:18:35,261 --> 00:18:39,534 second. The demonstration that I will 272 00:18:39,534 --> 00:18:44,046 show you. We have a, a glass container, 273 00:18:44,046 --> 00:18:49,031 spherical. It's about yea big. 274 00:18:49,031 --> 00:18:53,116 You can see it there. There is low pressure gas in 275 00:18:53,116 --> 00:18:57,617 there, so that some of the electrons can go around in a 276 00:18:57,617 --> 00:19:00,367 circle before they ionize the gas. 277 00:19:00,367 --> 00:19:04,368 We will accelerate the electrons over a potential 278 00:19:04,368 --> 00:19:08,869 difference, which I can vary from zero to three hundred 279 00:19:08,869 --> 00:19:11,37 volts. And so I can give them a 280 00:19:11,37 --> 00:19:15,371 certain velocity. Suppose I gave them a potential 281 00:19:15,371 --> 00:19:19,455 difference of hundred volts, then Q times delta V, 282 00:19:19,455 --> 00:19:23,829 which is the potential difference, 283 00:19:23,829 --> 00:19:29,045 would then be one half M V squared, and so if I make this 284 00:19:29,045 --> 00:19:33,144 one hundred volts, I can calculate what V is, 285 00:19:33,144 --> 00:19:38,08 and I've found that the velocity of the electrons then 286 00:19:38,08 --> 00:19:43,204 is about five point nine times ten to the six meters per 287 00:19:43,204 --> 00:19:46,371 second. So that's the velocity if I 288 00:19:46,371 --> 00:19:50,842 have a potential difference of one hundred volts. 289 00:19:50,842 --> 00:19:56,751 And so I can use that velocity, substitute that in this 290 00:19:56,751 --> 00:20:00,932 equation, and then I get the radius of these electrons in 291 00:20:00,932 --> 00:20:04,666 that magnetic field, and that radius turns out then 292 00:20:04,666 --> 00:20:07,652 to be about four point three centimeters. 293 00:20:07,652 --> 00:20:11,834 And that is a little smaller than the sphere that I have. 294 00:20:11,834 --> 00:20:15,418 How can I create a uniform magnetic field of that 295 00:20:15,418 --> 00:20:19,375 magnitude, that's indeed perpendicular to the orbit of 296 00:20:19,375 --> 00:20:22,362 the electrons? We do that using Helmholtz 297 00:20:22,362 --> 00:20:25,572 coals- uh, coils. We never discussed that in 298 00:20:25,572 --> 00:20:28,972 detail during our lectures. 299 00:20:28,972 --> 00:20:32,346 You see here coils, and you see here coils. 300 00:20:32,346 --> 00:20:35,962 And when you have two sets of coils like that, 301 00:20:35,962 --> 00:20:39,417 and you have a proper distance between them, 302 00:20:39,417 --> 00:20:44,158 you can create between them a magnetic field which is almost 303 00:20:44,158 --> 00:20:45,926 constant. In this case, 304 00:20:45,926 --> 00:20:49,702 this is the field, fifteen times larger than the 305 00:20:49,702 --> 00:20:53,8 Earth's magnetic field. Perhaps some of you remember 306 00:20:53,8 --> 00:20:58,058 that I linked you on the eight oh two 307 00:20:58,058 --> 00:21:01,715 website a few weeks ago to a very nice program that I 308 00:21:01,715 --> 00:21:06,004 received from Professor Belcher which allowed you to calculate 309 00:21:06,004 --> 00:21:09,379 magnetic field configurations with current loops. 310 00:21:09,379 --> 00:21:12,965 And one of the things I asked you on the website is, 311 00:21:12,965 --> 00:21:16,058 try to create what we call a magnetic bottle. 312 00:21:16,058 --> 00:21:19,574 This is a magnetic bottle. It's also referred to as 313 00:21:19,574 --> 00:21:22,808 Helmholtz coils, whereby you can create between 314 00:21:22,808 --> 00:21:25,551 two coils a near-uniform magnetic field. 315 00:21:25,551 --> 00:21:29,558 And so that's what we have here, and I'm going to show you 316 00:21:29,558 --> 00:21:33,939 now how beautiful these electrons go 317 00:21:33,939 --> 00:21:38,042 around in circles. They ionize the gas inside, 318 00:21:38,042 --> 00:21:42,418 and then through de-excitation of the ionization, 319 00:21:42,418 --> 00:21:46,977 you will see light. Just the same idea that you see 320 00:21:46,977 --> 00:21:49,621 from Aurora. So in that sense, 321 00:21:49,621 --> 00:21:54,544 you're seeing an artificial Aurora inside a glass tube. 322 00:21:54,544 --> 00:22:00,106 I have to make it very dark for this, because the light is not 323 00:22:00,106 --> 00:22:05,121 very bright. So we have to turn this off. 324 00:22:05,121 --> 00:22:07,998 And I have to turn all this off. 325 00:22:07,998 --> 00:22:13,29 And I have to turn this off. And some of you who are close 326 00:22:13,29 --> 00:22:17,467 can actually see the light immediately in the, 327 00:22:17,467 --> 00:22:21,088 in the sphere right here in front of me. 328 00:22:21,088 --> 00:22:25,915 But if not, then you will see it there on the screen. 329 00:22:25,915 --> 00:22:31,3 So this is the -- let me make sure that I can have my -- so 330 00:22:31,3 --> 00:22:37,545 this is the electron gun here. It has a potential difference 331 00:22:37,545 --> 00:22:40,498 which I set roughly at a hundred volts. 332 00:22:40,498 --> 00:22:43,994 And here you see these electrons going around. 333 00:22:43,994 --> 00:22:48,191 And they go around roughly twenty-two million times per 334 00:22:48,191 --> 00:22:51,066 second. And the gas pressure is low so 335 00:22:51,066 --> 00:22:55,65 that some of them make it all the way before they ionize the 336 00:22:55,65 --> 00:22:58,215 gas. And I can change the delta V. 337 00:22:58,215 --> 00:23:02,877 I now make the speed lower by making the potential difference 338 00:23:02,877 --> 00:23:08,782 over which I accelerate the electrons, by making that lower. 339 00:23:08,782 --> 00:23:13,592 The radius is proportional to the square root of that 340 00:23:13,592 --> 00:23:17,384 potential difference. So now you see here, 341 00:23:17,384 --> 00:23:22,933 when I increase the potential difference, the radius goes up. 342 00:23:22,933 --> 00:23:25,893 See how beautiful that circle is? 343 00:23:25,893 --> 00:23:30,887 It's absolutely amazing. Very, very nice demonstration. 344 00:23:30,887 --> 00:23:36,436 And then we reach the edge of the -- we are now very close to 345 00:23:36,436 --> 00:23:42,191 the diameter of the, of the glass sphere, 346 00:23:42,191 --> 00:23:49,892 so we can't go much further. So now I want to discuss with 347 00:23:49,892 --> 00:23:55,973 you Faraday's Law. Faraday's Law comes in many 348 00:23:55,973 --> 00:24:02,053 different ways. Faraday's Law -- I will get up 349 00:24:02,053 --> 00:24:07,593 the, get the Maxwell's equations up again. 350 00:24:07,593 --> 00:24:14,096 Faraday's Law -- Faraday's Law tells you that 351 00:24:14,096 --> 00:24:17,62 the induced EMF is minus D phi DT. 352 00:24:17,62 --> 00:24:23,921 D phi DT is the magnetic flux change through an open surface 353 00:24:23,921 --> 00:24:29,154 attached to a loop. The loop could be somewhere in 354 00:24:29,154 --> 00:24:33,212 your brains or it could be a real loop. 355 00:24:33,212 --> 00:24:36,736 You can imagine any loop in space. 356 00:24:36,736 --> 00:24:40,794 That is always correct, that statement. 357 00:24:40,794 --> 00:24:46,988 So the EMF induced equals minus D phi DT, minus D DT of the 358 00:24:46,988 --> 00:24:50,665 integral of B dot DA, 359 00:24:50,665 --> 00:24:54,562 open surface. You can think of many ways of 360 00:24:54,562 --> 00:24:59,666 changing the magnetic flux. One way is a stationary loop 361 00:24:59,666 --> 00:25:03,007 and having a changing magnetic field. 362 00:25:03,007 --> 00:25:07,739 Another way is of having B constant but changing the 363 00:25:07,739 --> 00:25:13,028 geometry so that the magnetic flux is changing through the 364 00:25:13,028 --> 00:25:18,411 surface, if you rotate it around or you move things around. 365 00:25:18,411 --> 00:25:25,036 Let's look at both today. Let's first take the stationary 366 00:25:25,036 --> 00:25:28,139 loop. I spent a major part of one 367 00:25:28,139 --> 00:25:32,503 lecture on that. Say this is a conducting wire 368 00:25:32,503 --> 00:25:35,704 and it has a resistance capital R. 369 00:25:35,704 --> 00:25:39,68 This is net resistance of the entire wire. 370 00:25:39,68 --> 00:25:44,432 And suppose here I have a changing magnetic field, 371 00:25:44,432 --> 00:25:49,572 and this area right here is A. And the magnetic field, 372 00:25:49,572 --> 00:25:55,391 let's say, is perpendicular to the blackboard. 373 00:25:55,391 --> 00:26:01,031 And the strength of the field is B, and so the magnetic flux 374 00:26:01,031 --> 00:26:04,185 phi of B is then simply A times B. 375 00:26:04,185 --> 00:26:07,723 And so D phi DT is then A times DB DT. 376 00:26:07,723 --> 00:26:12,694 Remember we were going to keep the geometry constant. 377 00:26:12,694 --> 00:26:17,76 It's a s- it's a stationary system, but we're going to 378 00:26:17,76 --> 00:26:22,636 change the magnetic field. So this magnetic field is 379 00:26:22,636 --> 00:26:27,702 changing. Perhaps it's coming out of the 380 00:26:27,702 --> 00:26:31,819 blackboard now, and maybe it's increasing. 381 00:26:31,819 --> 00:26:37,141 And so you induce then in this conducting wire an EMF, 382 00:26:37,141 --> 00:26:43,066 and the EMF has this magnitude. This is the magnitude of the 383 00:26:43,066 --> 00:26:46,882 induced EMF. And so the current that is 384 00:26:46,882 --> 00:26:51,099 going to flow, that is the induced current, 385 00:26:51,099 --> 00:26:56,823 is the induced EMF divided by the resistance of that whole 386 00:26:56,823 --> 00:27:01,658 conducting loop. The direction is never an 387 00:27:01,658 --> 00:27:05,044 issue, because you all can handle Lenz's Law. 388 00:27:05,044 --> 00:27:08,276 Notice I didn't even put a minus sign here, 389 00:27:08,276 --> 00:27:11,585 because the EMF is of course minus D phi DT. 390 00:27:11,585 --> 00:27:15,894 That's not important for me if you put a minus sign here, 391 00:27:15,894 --> 00:27:19,972 but I put these bars there, so get rid of minus signs. 392 00:27:19,972 --> 00:27:24,512 Because if the magnetic field is coming out of the board and 393 00:27:24,512 --> 00:27:28,206 if it were increasing, then Lenz's Law will run a 394 00:27:28,206 --> 00:27:32,131 current in this direction to oppose 395 00:27:32,131 --> 00:27:35,534 that increase. And so minus signs in general 396 00:27:35,534 --> 00:27:39,333 are for the birds. You can always reason in which 397 00:27:39,333 --> 00:27:43,924 direction the current is going. So that's a case whereby we 398 00:27:43,924 --> 00:27:48,278 have a stationary loop and whereby the magnetic field is 399 00:27:48,278 --> 00:27:52,948 changing but not the geometry. Now we'll have a case whereby 400 00:27:52,948 --> 00:27:55,797 the magnetic field, say, is constant. 401 00:27:55,797 --> 00:28:00,15 So we have here a conducting wire and we have a magnetic 402 00:28:00,15 --> 00:28:04,346 field, for instance, coming out of the 403 00:28:04,346 --> 00:28:09,037 blackboard, uniform -- uniform is always nice when we do 404 00:28:09,037 --> 00:28:14,069 integration to get this flux. It's always nice to have it be 405 00:28:14,069 --> 00:28:17,14 uniform. And we have a bar here which 406 00:28:17,14 --> 00:28:20,125 has length L. This is length L here. 407 00:28:20,125 --> 00:28:24,219 And we move the bar with velocity V to the right. 408 00:28:24,219 --> 00:28:29,336 So this is a very simple case. V is here perpendicular to the 409 00:28:29,336 --> 00:28:32,066 bar. That makes it always easier. 410 00:28:32,066 --> 00:28:37,78 And B is perpendicular to the plane through V and L, 411 00:28:37,78 --> 00:28:43,236 so that makes all sines of theta that we may have, 412 00:28:43,236 --> 00:28:46,353 or cosines of theta, all one. 413 00:28:46,353 --> 00:28:52,143 Magnetic field is constant, and so the magnetic flux, 414 00:28:52,143 --> 00:28:57,264 phi of B, is the area times the magnetic field. 415 00:28:57,264 --> 00:29:04,056 The angles are just wonderful, so that is B times X times L if 416 00:29:04,056 --> 00:29:09,511 this distance is X. So the magnetic flux change in 417 00:29:09,511 --> 00:29:14,354 time, D phi B DT -- notice I don't 418 00:29:14,354 --> 00:29:19,523 care about minus signs. I don't need n- minus signs. 419 00:29:19,523 --> 00:29:24,389 That D phi DT is there for L times B times DX DT, 420 00:29:24,389 --> 00:29:30,065 and DX DT is the velocity. So the magnitude of the EMF is 421 00:29:30,065 --> 00:29:33,309 L B V. And so the current that is 422 00:29:33,309 --> 00:29:36,755 going to flow, the induced current, 423 00:29:36,755 --> 00:29:40,303 is that value, L B V, divided by the 424 00:29:40,303 --> 00:29:44,958 resistance R, and that is the resistance, 425 00:29:44,958 --> 00:29:47,302 then, that is in this entire loop. 426 00:29:47,302 --> 00:29:50,57 Whatever there is and where it is I don't care, 427 00:29:50,57 --> 00:29:53,412 but that is the magnitude of the current. 428 00:29:53,412 --> 00:29:56,822 The direction is again easy, it's non-negotiable. 429 00:29:56,822 --> 00:30:00,516 If I move this bar to the right, the magnetic flux is 430 00:30:00,516 --> 00:30:02,79 increasing. It's pointing in this 431 00:30:02,79 --> 00:30:07,124 direction, the magnetic field, so the current is going to flow 432 00:30:07,124 --> 00:30:10,463 in such a direction that it opposes that change, 433 00:30:10,463 --> 00:30:15,08 and so the current will flow in this direction. 434 00:30:15,08 --> 00:30:20,049 That's the induced current, and this is the magnitude. 435 00:30:20,049 --> 00:30:24,267 In nineteen ninety-six, NASA attached a twenty 436 00:30:24,267 --> 00:30:29,516 kilometer conducting wire called a tether to the shuttle, 437 00:30:29,516 --> 00:30:34,953 so L was twenty kilometers. The magnetic field of the Earth 438 00:30:34,953 --> 00:30:39,359 is about half a gauss. Even at a distance of two 439 00:30:39,359 --> 00:30:43,577 hundred miles, that is not much different from 440 00:30:43,577 --> 00:30:47,307 here. So that is five times ten to 441 00:30:47,307 --> 00:30:50,005 the minus five tesla. And the shuttle, 442 00:30:50,005 --> 00:30:54,089 as you should know from eight oh one, like any near-Earth 443 00:30:54,089 --> 00:30:58,538 satellite, they all fly with a speed of about eight kilometers 444 00:30:58,538 --> 00:31:01,018 per second. If they go much faster, 445 00:31:01,018 --> 00:31:04,665 then they will leave the gravitational field of the 446 00:31:04,665 --> 00:31:06,123 Earth. So V is about, 447 00:31:06,123 --> 00:31:09,551 in circular orbit, is about eight kilometers per 448 00:31:09,551 --> 00:31:12,177 second. So here you have it in meters 449 00:31:12,177 --> 00:31:16,625 per second. If I calculate for this tether, 450 00:31:16,625 --> 00:31:19,306 moved around, if I calculate L B V, 451 00:31:19,306 --> 00:31:23,958 I would get eight kilovolts. However, keep in mind that that 452 00:31:23,958 --> 00:31:27,979 is only correct if B were perpendicular to the plane 453 00:31:27,979 --> 00:31:32,395 through L V, and also if the velocity of the shuttle were 454 00:31:32,395 --> 00:31:35,706 perpendicular to the direction of the wire. 455 00:31:35,706 --> 00:31:39,964 None of that is the case. The magnetic field where they 456 00:31:39,964 --> 00:31:43,512 were is not exactly perpendicular to the plane 457 00:31:43,512 --> 00:31:47,06 through V and L. And so what they observed was 458 00:31:47,06 --> 00:31:52,81 three and a half kilovolts, about half of the maximum that 459 00:31:52,81 --> 00:31:55,09 you could achieve. You may say, 460 00:31:55,09 --> 00:31:58,662 gee, this is strange, because if you just drag a 461 00:31:58,662 --> 00:32:03,07 conducting wire through space, you don't have a closed loop 462 00:32:03,07 --> 00:32:07,631 circuit, so you can't talk about the idea of a magnetic flux, 463 00:32:07,631 --> 00:32:12,115 because you have no surface. Well, at the altitude where the 464 00:32:12,115 --> 00:32:15,687 shuttle is flying, there is still a teeny-weenie 465 00:32:15,687 --> 00:32:18,651 little bit of air, very little but some, 466 00:32:18,651 --> 00:32:22,452 and that is highly ionized because 467 00:32:22,452 --> 00:32:25,512 of the ultraviolet light from the sun. 468 00:32:25,512 --> 00:32:28,985 We call that the ionosphere. It's a plasma. 469 00:32:28,985 --> 00:32:33,369 And so in the surrounding of this wire, here and here, 470 00:32:33,369 --> 00:32:38,331 you have a conducting medium. And so current can flow through 471 00:32:38,331 --> 00:32:43,046 that medium, this way or this way, and that's exactly what 472 00:32:43,046 --> 00:32:45,858 will happen. So that you don't know 473 00:32:45,858 --> 00:32:48,918 precisely the path. Current will flow, 474 00:32:48,918 --> 00:32:54,468 so you do have closed loops. And so it is meaningful to talk 475 00:32:54,468 --> 00:32:57,297 about magnetic flux change and about the EMF, 476 00:32:57,297 --> 00:33:00,704 the induced EMF that is generated as a result of that. 477 00:33:00,704 --> 00:33:04,176 The thing that NASA could not predict very well was the 478 00:33:04,176 --> 00:33:07,841 current, because you do not quite know what the resistance 479 00:33:07,841 --> 00:33:11,313 is of that closed loop. You know what the resistance is 480 00:33:11,313 --> 00:33:13,885 of the cable, but you don't know what the 481 00:33:13,885 --> 00:33:17,293 resistance is of the currents as they flow through the 482 00:33:17,293 --> 00:33:19,736 ionosphere. But the net result was that 483 00:33:19,736 --> 00:33:23,079 they had a current I in their wire 484 00:33:23,079 --> 00:33:26,555 of about one ampere. It was very tragic, 485 00:33:26,555 --> 00:33:31,545 because this current was so high that the conducting wire 486 00:33:31,545 --> 00:33:36,713 melted, and the tether broke off, and very early on in that 487 00:33:36,713 --> 00:33:41,08 experiment, the tether separated from the shuttle. 488 00:33:41,08 --> 00:33:46,515 But this is a marvelous example of where you have motional EMF 489 00:33:46,515 --> 00:33:50,08 in space. I want you to think about it at 490 00:33:50,08 --> 00:33:52,842 home. Where does the energy come 491 00:33:52,842 --> 00:33:56,594 from? Because you're generating 492 00:33:56,594 --> 00:33:59,271 current. You could have lit a light 493 00:33:59,271 --> 00:34:01,634 bulb. The energy must come from 494 00:34:01,634 --> 00:34:03,761 somewhere. Think about that. 495 00:34:03,761 --> 00:34:06,36 Conceptually interesting question. 496 00:34:06,36 --> 00:34:10,534 By the way, I linked you on the website to the tether. 497 00:34:10,534 --> 00:34:15,181 All you have to do is click on it and you will get some more 498 00:34:15,181 --> 00:34:18,41 information on this incredible experiment. 499 00:34:18,41 --> 00:34:21,403 All right. Let's continue with the most 500 00:34:21,403 --> 00:34:26,523 important contribution, Faraday to our economy. 501 00:34:26,523 --> 00:34:31,125 And that is the situation whereby we rotate a loop, 502 00:34:31,125 --> 00:34:34,991 a current loop, conducting loop -- could be 503 00:34:34,991 --> 00:34:40,514 rectangular, could be circular -- and we rotate it round with 504 00:34:40,514 --> 00:34:44,472 angular frequency omega. And for simplicity, 505 00:34:44,472 --> 00:34:48,707 let's have the magnetic field just straight up. 506 00:34:48,707 --> 00:34:52,389 This is supposed to be three dimensional. 507 00:34:52,389 --> 00:34:56,439 That's my idea. And let this side be length A 508 00:34:56,439 --> 00:35:02,043 and this B. So if you look at it from this 509 00:35:02,043 --> 00:35:06,003 direction, you will only see the line. 510 00:35:06,003 --> 00:35:12,103 B here is rotating like this, with angular velocity omega. 511 00:35:12,103 --> 00:35:16,383 And a little later in time would be here. 512 00:35:16,383 --> 00:35:22,697 And so this length here is B. And so this length is B cosine 513 00:35:22,697 --> 00:35:26,229 theta. And let's assume -- we call 514 00:35:26,229 --> 00:35:32,507 this angle theta -- let's assume that it is here at 515 00:35:32,507 --> 00:35:35,563 T equals zero when theta is zero. 516 00:35:35,563 --> 00:35:40,052 I can choose that arbitrarily any way I want to, 517 00:35:40,052 --> 00:35:43,967 of course. And there could be a resistance 518 00:35:43,967 --> 00:35:48,456 R in this entire loop. And so the magnetic flux, 519 00:35:48,456 --> 00:35:53,804 phi of B, through a surface attached to this loop is B -- 520 00:35:53,804 --> 00:35:58,102 that's constant, that's not changing -- and we 521 00:35:58,102 --> 00:36:04,142 have A times B, but remember since it is a dot 522 00:36:04,142 --> 00:36:10,792 product between B dot DA then you get a cosine theta in there, 523 00:36:10,792 --> 00:36:15,044 so you get A times B times cosine theta. 524 00:36:15,044 --> 00:36:20,168 So that's the flux. But cosine theta is changing 525 00:36:20,168 --> 00:36:23,112 with time. Theta is omega T, 526 00:36:23,112 --> 00:36:29,435 constant angular frequency omega, so the magnetic flux is A 527 00:36:29,435 --> 00:36:33,727 B B cosine omega T. So D phi DT. 528 00:36:33,727 --> 00:36:36,387 Take the derivative, I get an omega, 529 00:36:36,387 --> 00:36:40,036 I get A, I get B, I get B, I get sine of omega T. 530 00:36:40,036 --> 00:36:43,533 If you're really interested in that minus sign, 531 00:36:43,533 --> 00:36:47,866 you get that automatically, there's nothing I can do about 532 00:36:47,866 --> 00:36:50,147 it. You get a minus sign there, 533 00:36:50,147 --> 00:36:53,644 but I'm not too much interested in minus signs. 534 00:36:53,644 --> 00:36:57,445 But since I get it, I can't shove it under the rug. 535 00:36:57,445 --> 00:37:02,006 So I put a minus sign here and I change that into a plus one. 536 00:37:02,006 --> 00:37:06,263 The advantage is then that I have immediately the induced 537 00:37:06,263 --> 00:37:10,493 EMF. Notice that the induced EMF 538 00:37:10,493 --> 00:37:14,234 itself is linearly proportional with omega. 539 00:37:14,234 --> 00:37:19,668 And so my induced current that is going to run in that loop is 540 00:37:19,668 --> 00:37:24,478 the induced EMF divided by the resistance of that loop. 541 00:37:24,478 --> 00:37:28,932 And so if the induced EMF is proportional to omega, 542 00:37:28,932 --> 00:37:33,654 so is the induced current. So if you go faster around, 543 00:37:33,654 --> 00:37:38,909 you get a higher current for the same resistor. 544 00:37:38,909 --> 00:37:42,584 You built motors, and during the prize ceremony I 545 00:37:42,584 --> 00:37:45,876 told you that the faster your motor rotates, 546 00:37:45,876 --> 00:37:50,393 the larger an induced EMF is which is generated in the loop. 547 00:37:50,393 --> 00:37:54,833 You had a magnetic field which let's assume is more or less 548 00:37:54,833 --> 00:37:57,666 constant. Of course, it wasn't in your 549 00:37:57,666 --> 00:38:01,264 case, but let's assume for the sake of argument. 550 00:38:01,264 --> 00:38:05,704 And as your motor was going to rotate, there was an induced 551 00:38:05,704 --> 00:38:10,759 current generated by Mr. Faraday, which is proportional 552 00:38:10,759 --> 00:38:14,485 to omega, so the current is larger when your motor goes 553 00:38:14,485 --> 00:38:16,624 faster. And that induced current 554 00:38:16,624 --> 00:38:20,349 opposes the current which was produced by your battery. 555 00:38:20,349 --> 00:38:24,42 And perhaps you remember that I did a demonstration with the 556 00:38:24,42 --> 00:38:27,11 winning motor. I blocked the rotor and I 557 00:38:27,11 --> 00:38:30,974 showed you that the current through the rotor when it was 558 00:38:30,974 --> 00:38:34,768 blocked, when omega was zero, was one point six amperes. 559 00:38:34,768 --> 00:38:37,252 Simply Ohm's Law. Battery, voltage V, 560 00:38:37,252 --> 00:38:41,53 resistance R, the current is V divided by R. 561 00:38:41,53 --> 00:38:45,215 Then we ran the motor, and the current went down 562 00:38:45,215 --> 00:38:47,723 enormously, by a factor of forty. 563 00:38:47,723 --> 00:38:52,193 It was only forty milliamperes, the time-averaged current, 564 00:38:52,193 --> 00:38:56,348 when the motor was running. The reason is what you see 565 00:38:56,348 --> 00:38:58,543 here. You get an induced EMF, 566 00:38:58,543 --> 00:39:02,385 an induced current, which opposes the current from 567 00:39:02,385 --> 00:39:04,345 the battery. L R circuits. 568 00:39:04,345 --> 00:39:08,814 I prefer to stay on this center board, although I could go 569 00:39:08,814 --> 00:39:13,126 there, but I don't think we need these 570 00:39:13,126 --> 00:39:15,519 anymore. Can I cover this? 571 00:39:15,519 --> 00:39:18,485 Most of you are happy with that. 572 00:39:18,485 --> 00:39:22,886 I could work there, but I prefer to stay at the 573 00:39:22,886 --> 00:39:24,8 center. L R circuits. 574 00:39:24,8 --> 00:39:29,202 With L R circuits, we have the embarrassment -- 575 00:39:29,202 --> 00:39:34,847 not for us but for some others -- that almost all textbooks, 576 00:39:34,847 --> 00:39:38,962 college physics, do not understand Faraday's 577 00:39:38,962 --> 00:39:44,033 Law, and therefore they treat the subject incorrectly. 578 00:39:44,033 --> 00:39:49,574 Embarrassing, but of course since they know 579 00:39:49,574 --> 00:39:52,982 the answer, their answers are correct. 580 00:39:52,982 --> 00:39:56,113 But their physics is totally wrong. 581 00:39:56,113 --> 00:40:01,731 I will address a problem which I found on, on the website when 582 00:40:01,731 --> 00:40:05,139 I looked at Professor Belcher's exams. 583 00:40:05,139 --> 00:40:09,375 He has one very nice problem of an L R circuit. 584 00:40:09,375 --> 00:40:12,599 And I will go through that with you. 585 00:40:12,599 --> 00:40:17,94 We have an AC power supply, V equals V zero cosine omega T. 586 00:40:17,94 --> 00:40:25,503 The frequency is sixty hertz. It just comes out of the wall. 587 00:40:25,503 --> 00:40:29,035 And so omega, which is two pi F, 588 00:40:29,035 --> 00:40:34,391 is about three hundred seventy-seven radians per 589 00:40:34,391 --> 00:40:37,923 second. And V zero in this case, 590 00:40:37,923 --> 00:40:41,569 hundred volts. And we have here a 591 00:40:41,569 --> 00:40:46,583 self-inductor, and here we have a light bulb. 592 00:40:46,583 --> 00:40:52,394 And the light bulb has a resistance which is hundred 593 00:40:52,394 --> 00:40:55,813 ohms. And this self-inductor is 594 00:40:55,813 --> 00:41:01,637 variable. We don't know yet how you make 595 00:41:01,637 --> 00:41:06,794 a variable self-inductor, but we will learn that very 596 00:41:06,794 --> 00:41:11,256 shortly, either Friday or next week, I forgot. 597 00:41:11,256 --> 00:41:17,008 This self-inductance can be increased over a certain range. 598 00:41:17,008 --> 00:41:22,76 And the first question that Professor Belcher is asking is, 599 00:41:22,76 --> 00:41:27,62 what is the energy dissipation in this light bulb? 600 00:41:27,62 --> 00:41:33,67 The closed loop integral of E dot DL is not zero, 601 00:41:33,67 --> 00:41:38,144 because Kirchoff's loop rule does not apply here. 602 00:41:38,144 --> 00:41:43,178 There is a self-inductor. If you attach an open surface 603 00:41:43,178 --> 00:41:47,839 to this closed loop, there is a magnetic flux going 604 00:41:47,839 --> 00:41:52,034 through there, so you must deal with the third 605 00:41:52,034 --> 00:41:56,043 equation there. You must deal with Faraday's 606 00:41:56,043 --> 00:42:00,051 Law, no matter what your textbooks tell you. 607 00:42:00,051 --> 00:42:03,221 Kirchoff's loop rule does not hold. 608 00:42:03,221 --> 00:42:09,382 And if you do that correctly, you end up with the right 609 00:42:09,382 --> 00:42:13,502 differential equation, which happens to be the same 610 00:42:13,502 --> 00:42:18,446 one that those people find who do not understand the physics, 611 00:42:18,446 --> 00:42:23,39 but they get the same equation. They massage things in such a 612 00:42:23,39 --> 00:42:26,109 way that they get the same answer. 613 00:42:26,109 --> 00:42:30,888 And the I- answer then is that the current that is going to 614 00:42:30,888 --> 00:42:35,008 flow as a result of this voltage, variable voltage, 615 00:42:35,008 --> 00:42:39,704 that current I is a maximum value times the cosine omega T 616 00:42:39,704 --> 00:42:45,791 minus phi. That the maximum current itself 617 00:42:45,791 --> 00:42:52,917 is V zero divided by the s- square root of R squared plus 618 00:42:52,917 --> 00:42:58,77 omega L squared. And that the tangent of phi is 619 00:42:58,77 --> 00:43:05,387 omega L divided by R. I spent quite some time on this 620 00:43:05,387 --> 00:43:12,64 during one of my lectures. And so the maximum current that 621 00:43:12,64 --> 00:43:16,78 will flow depends on omega and 622 00:43:16,78 --> 00:43:20,647 depends on L, on the self-inductance -- of 623 00:43:20,647 --> 00:43:23,666 course on the resistance as well. 624 00:43:23,666 --> 00:43:29,137 In Professor Belcher's problem, he starts off with L equals 625 00:43:29,137 --> 00:43:34,514 zero, and he asks you what is the power dissipated in this 626 00:43:34,514 --> 00:43:37,155 light bulb. If L equals zero, 627 00:43:37,155 --> 00:43:41,966 this term is not there, and so you simply have Ohm's 628 00:43:41,966 --> 00:43:45,078 Law, I equals V zero divided by R. 629 00:43:45,078 --> 00:43:51,294 So I max is hundred divided by hundred, is one ampere. 630 00:43:51,294 --> 00:43:56,354 But then he asks you something that would give you the hiccups, 631 00:43:56,354 --> 00:44:01,005 and that is what is now the time-averaged power dissipated 632 00:44:01,005 --> 00:44:04,351 in that light bulb? And you will remember, 633 00:44:04,351 --> 00:44:08,186 or should remember, that one half I squared R is 634 00:44:08,186 --> 00:44:12,837 the power dissipated in the light bulb if I is the current 635 00:44:12,837 --> 00:44:17,57 through the light bulb and R is the resistance of the light 636 00:44:17,57 --> 00:44:21,204 bulb. And we will assume that the 637 00:44:21,204 --> 00:44:25,448 resistance is independent of the current, independent of 638 00:44:25,448 --> 00:44:28,535 temperature. But I is changing with time, 639 00:44:28,535 --> 00:44:32,856 in a cosinusoidal fashion. And so now you have to be able 640 00:44:32,856 --> 00:44:37,1 to evaluate the time average of a cosine squared omega T 641 00:44:37,1 --> 00:44:39,724 function. And the time average of a 642 00:44:39,724 --> 00:44:44,354 cosine squared or sine squared omega T function is always one 643 00:44:44,354 --> 00:44:46,901 half. You'd like to remember that, 644 00:44:46,901 --> 00:44:50,759 rather than spending five minutes on deriving that. 645 00:44:50,759 --> 00:44:54,231 And so this, time averaged, 646 00:44:54,231 --> 00:44:58,939 is then going to be one half times I max squared times R -- 647 00:44:58,939 --> 00:45:01,78 and by the way, sorry, the -- I was, 648 00:45:01,78 --> 00:45:06,731 I was ahead of myself with my, my -- I squared R is the energy 649 00:45:06,731 --> 00:45:09,409 that is dissipated, right, in the, 650 00:45:09,409 --> 00:45:12,737 in the resistor, not one half I squared R. 651 00:45:12,737 --> 00:45:16,714 It is I squared R. And because of the time-average 652 00:45:16,714 --> 00:45:21,096 of the cosine squared function I get my one half there. 653 00:45:21,096 --> 00:45:26,128 And so it's very easy now. I max is one ampere, 654 00:45:26,128 --> 00:45:29,821 R is one hundred, and this factor of one half, 655 00:45:29,821 --> 00:45:34,253 which comes from the time average of the square of this 656 00:45:34,253 --> 00:45:39,259 function -- the phi has nothing to do with that -- the average 657 00:45:39,259 --> 00:45:43,936 of co- of the cosine squared of this function is one half, 658 00:45:43,936 --> 00:45:48,367 and you get fifty watts. And now he calls this device a 659 00:45:48,367 --> 00:45:51,814 light dimmer. Now he is going to increase L 660 00:45:51,814 --> 00:45:56,738 to three hundred millihenries. So L now becomes three hundred 661 00:45:56,738 --> 00:46:01,723 millihenry. At three hundred millihenry, 662 00:46:01,723 --> 00:46:05,389 omega L is hundred and thirteen ohms. 663 00:46:05,389 --> 00:46:10,378 I multiply three hundred by the three seven seven. 664 00:46:10,378 --> 00:46:14,859 And now I find that this time-averaged value, 665 00:46:14,859 --> 00:46:20,867 which is now one half times this, I find that I max -- I put 666 00:46:20,867 --> 00:46:26,569 in for omega L hundred thirteen squared, hundred squared, 667 00:46:26,569 --> 00:46:31,769 I max is now oh point six seven amperes. 668 00:46:31,769 --> 00:46:37,792 I put that in this equation and I find that the power is now 669 00:46:37,792 --> 00:46:43,101 about twenty-two watts. So that's what a light dimmer 670 00:46:43,101 --> 00:46:46,674 is doing for you. So you turn up the 671 00:46:46,674 --> 00:46:51,268 self-inductance, and now your light is dimmer. 672 00:46:51,268 --> 00:46:55,249 Now a very deep, very deep, maybe nasty, 673 00:46:55,249 --> 00:47:00,252 conceptual question. Why would one want to build a 674 00:47:00,252 --> 00:47:03,965 light dimmer with a self-inductor? 675 00:47:03,965 --> 00:47:07,892 Why not put in here a variable resistor and then turning up the 676 00:47:07,892 --> 00:47:11,439 resistor so that the current, which, when the resistor is 677 00:47:11,439 --> 00:47:15,176 zero, the current is one ampere, so you get you fifty watts, 678 00:47:15,176 --> 00:47:18,596 but then you increase that resistor so that the current 679 00:47:18,596 --> 00:47:22,586 goes down to oh point six seven amperes, and then the light bulb 680 00:47:22,586 --> 00:47:25,31 will dissipate twenty-two joules per second. 681 00:47:25,31 --> 00:47:28,223 Why would you not put here a variable resistor, 682 00:47:28,223 --> 00:47:30,25 but why a variable self-inductor? 683 00:47:30,25 --> 00:47:33,48 If you can answer that question, that shows that you 684 00:47:33,48 --> 00:47:37,456 have a deep insight already in eight oh 685 00:47:37,456 --> 00:47:39,84 two. I promise you I will not ask 686 00:47:39,84 --> 00:47:43,865 this question Wednesday, but I may ask it on the final. 687 00:47:43,865 --> 00:47:47,665 Displacement current. Displacement current is always 688 00:47:47,665 --> 00:47:52,062 a little bit problematic in the sense that there are not too 689 00:47:52,062 --> 00:47:56,161 many problems that you can do with displacement current. 690 00:47:56,161 --> 00:47:59,366 This is here this displacement current term, 691 00:47:59,366 --> 00:48:02,645 that Amp- that, uh, Maxwell added to Ampere's 692 00:48:02,645 --> 00:48:05,03 Law. The only application at this 693 00:48:05,03 --> 00:48:09,427 stage in eight oh two that I can think of 694 00:48:09,427 --> 00:48:14,004 is the one that I hit very hard during my lectures. 695 00:48:14,004 --> 00:48:19,132 You have plate capacitor disks. They are circular plates. 696 00:48:19,132 --> 00:48:24,167 And you charge the capacitor or discharge the capacitor, 697 00:48:24,167 --> 00:48:27,921 and you can calculate now, using this law, 698 00:48:27,921 --> 00:48:33,232 what the magnetic field is in between the capacitor plates. 699 00:48:33,232 --> 00:48:38,542 I advise you to check your notes from that lecture or watch 700 00:48:38,542 --> 00:48:44,86 the lecture again on the web. The lecture is on the web. 701 00:48:44,86 --> 00:48:48,961 Then some fatherly advice when it comes to the exam itself. 702 00:48:48,961 --> 00:48:53,062 I would advise you to read a problem at least twice to make 703 00:48:53,062 --> 00:48:55,819 sure that you fully understand the text. 704 00:48:55,819 --> 00:48:59,567 If you read fast -- at least that happens to me -- you 705 00:48:59,567 --> 00:49:02,678 sometimes misread. At least, I often misread. 706 00:49:02,678 --> 00:49:06,638 I would then advise you to do the easiest problems first. 707 00:49:06,638 --> 00:49:10,951 What may be the easiest for you may not be, be the easiest for 708 00:49:10,951 --> 00:49:13,072 you. But do the easiest for you 709 00:49:13,072 --> 00:49:16,684 first. If you get stuck on a problem, 710 00:49:16,684 --> 00:49:19,861 never spend more than ten minutes on one problem. 711 00:49:19,861 --> 00:49:23,765 Immediately abandon in then, when you see that it's grinding 712 00:49:23,765 --> 00:49:26,876 yourself into a hole, and go to another problem. 713 00:49:26,876 --> 00:49:30,582 Between now and Wednesday, I would suggest you see tutors 714 00:49:30,582 --> 00:49:33,56 if you need them. You can see your instructor. 715 00:49:33,56 --> 00:49:35,81 I have office hours this afternoon. 716 00:49:35,81 --> 00:49:39,053 I'll make myself as much available today as I can. 717 00:49:39,053 --> 00:49:43,09 Tomorrow afternoon I will be in twenty-six one hundred working 718 00:49:43,09 --> 00:49:46,411 on the demonstrations for you for 719 00:49:46,411 --> 00:49:49,074 next week, so I'm not available tomorrow. 720 00:49:49,074 --> 49:54 I wish you luck, and I'll see you on Wednesday.