1 00:00:00 --> 00:00:07 2 00:00:07 --> 00:00:11,842 Today, I will quantify the ability of a circuit to fight a 3 00:00:11,842 --> 00:00:16,684 magnetic flux that is produced by the circuits themselves. 4 00:00:16,684 --> 00:00:21,271 If you have a circuit and you run a current through the 5 00:00:21,271 --> 00:00:25,093 circuit, then you create some magnetic fields, 6 00:00:25,093 --> 00:00:29,256 and if the currents are changing then the magnetic 7 00:00:29,256 --> 00:00:33,588 fields are changing. And so there will be an induced 8 00:00:33,588 --> 00:00:37,666 EMF in that circuit that fights the 9 00:00:37,666 --> 00:00:42,94 change, and we express that in terms of a self-inductance: 10 00:00:42,94 --> 00:00:47,289 L self-inductance , and the word self speaks for 11 00:00:47,289 --> 00:00:50,158 itself. It's doing it to itself. 12 00:00:50,158 --> 00:00:55,155 Magnetic flux that is produced by the circuit is always 13 00:00:55,155 --> 00:00:59,875 proportional to the current. You double the current, 14 00:00:59,875 --> 00:01:03,761 the magnetic flux doubles. And so it is the 15 00:01:03,761 --> 00:01:10,331 proportionality constant that we call L that is the 16 00:01:10,331 --> 00:01:15,631 self-inductance, and so therefore the induced 17 00:01:15,631 --> 00:01:21,533 EMF equals minus D five ET, that is Faraday's Law. 18 00:01:21,533 --> 00:01:25,507 And so that becomes minus L DI DT. 19 00:01:25,507 --> 00:01:32,372 L is only a matter of geometry. L is not a function of the 20 00:01:32,372 --> 00:01:37,913 current itself. I will calculate for you a very 21 00:01:37,913 --> 00:01:43,333 simple case of the self-inductance 22 00:01:43,333 --> 0. of a solenoid. 23 0. --> 00:01:44,687 yes, there you see this, 24 00:01:44,687 --> 00:01:49,292 Let this be a solenoid and this is a closed circuit, 25 00:01:49,292 --> 00:01:53,265 and we want a current I through the solenoid, 26 00:01:53,265 --> 00:01:57,238 and the radius of these windings is little R. 27 00:01:57,238 --> 00:02:02,475 Let's say they're N windings and the length of the solenoid 28 00:02:02,475 --> 00:02:06,358 is little L. Perhaps you'll remember that we 29 00:02:06,358 --> 00:02:09,609 earlier derived, using *Empea's* Law, 30 00:02:09,609 --> 00:02:14,665 that the magnetic field inside the 31 00:02:14,665 --> 00:02:19,006 solenoids is new zero times I times capital N divided by L. 32 00:02:19,006 --> 00:02:21,999 This is the number of windings per meter. 33 00:02:21,999 --> 00:02:25,591 If we attach an open surface to this closed loop, 34 00:02:25,591 --> 00:02:30,156 very difficult to imagine what that open surface looks like -- 35 00:02:30,156 --> 00:02:34,646 we discussed that many times -- inside this solenoid you have 36 00:02:34,646 --> 00:02:37,265 sort of a staircase-like of surface. 37 00:02:37,265 --> 00:02:41,904 That magnetic field penetrates that surface N times because you 38 00:02:41,904 --> 00:02:45,903 have N loops. And so the magnetic flux, 39 00:02:45,903 --> 00:02:49,407 file B, is simply the area by little R squared, 40 00:02:49,407 --> 00:02:53,9 which is the surface area of one loop, because I assume that 41 00:02:53,9 --> 00:02:57,708 the magnetic field is constant inside the solenoid, 42 00:02:57,708 --> 00:03:02,049 and I assume that it is zero outside, which is a very good 43 00:03:02,049 --> 00:03:05,324 approximation. So we get pi little R squared 44 00:03:05,324 --> 00:03:09,437 surface area of one loop, then we have N loops and then 45 00:03:09,437 --> 00:03:13,549 we have to multiply it by that constant magnetic field. 46 00:03:13,549 --> 00:03:18,5 So we get an N squared because we have an N here, 47 00:03:18,5 --> 00:03:23,774 new zero I divided by L. And this we call L times I. 48 00:03:23,774 --> 00:03:27,91 That's our definition for self-inductors. 49 00:03:27,91 --> 00:03:32,77 And so the self-inductions L is purely geometry. 50 00:03:32,77 --> 00:03:38,355 It's pi level R squared, capital N squared divided by L 51 00:03:38,355 --> 0. times new zero. 52 0. --> 00:03:40,009 yes, there you see this, 53 00:03:40,009 --> 00:03:43,939 Let me check this. Pi little R squared, 54 00:03:43,939 --> 00:03:51,169 I have a capital N squared, new zero, that's correct, 55 00:03:51,169 --> 00:03:55,338 divided by little L. And so we can calculate, 56 00:03:55,338 --> 00:03:59,224 for instance, what this self-inductance is 57 00:03:59,224 --> 00:04:04,436 for a solenoid that we have used in class several times. 58 00:04:04,436 --> 00:04:09,648 We had one whereby we had twenty eight hundred windings. 59 00:04:09,648 --> 00:04:15,24 R I think was something like five centimeters -- you have to 60 00:04:15,24 --> 00:04:19,409 work SI of course, be careful -- and we had a 61 00:04:19,409 --> 00:04:23,412 length, was 0 point six meters. 62 00:04:23,412 --> 00:04:28,136 We had it several times out here, and if you substitute 63 00:04:28,136 --> 00:04:32,073 those numbers in there, you will find that the 64 00:04:32,073 --> 00:04:37,585 self-inductance of that solenoid is 0 point 1 in SI units and we 65 00:04:37,585 --> 00:04:40,559 call the SI units Henry, capital H. 66 00:04:40,559 --> 00:04:44,584 It would be the same as volt seconds or ampere, 67 00:04:44,584 --> 00:04:48,958 but no one would ever use that. We call that Henry. 68 00:04:48,958 --> 00:04:53,333 Every circuit has a finite value for 69 00:04:53,333 --> 00:04:56,765 the self-inductance, however small that may be. 70 00:04:56,765 --> 00:05:01,167 Sometimes it's so small that we ignore it, but if you take a 71 00:05:01,167 --> 00:05:03,778 simple loop, a sum-, simple current, 72 00:05:03,778 --> 00:05:08,404 just one wire that goes around -- whether it's this,a rectangle 73 00:05:08,404 --> 00:05:12,731 or whether it is a circle it doesn't make any difference -- 74 00:05:12,731 --> 00:05:15,343 it always produces a magnetic field. 75 00:05:15,343 --> 00:05:18,999 It always produces a magnetic flux to the surface, 76 00:05:18,999 --> 00:05:22,356 and so it always has a finite self-inductance. 77 00:05:22,356 --> 00:05:27,986 Maybe only nine nano Henrys, maybe only micro Henrys, 78 00:05:27,986 --> 0. but it's never zero. 79 0. --> 00:05:29,974 yes, there you see this, 80 00:05:29,974 --> 00:05:35,37 And so now what I want to do is to show you the remarkable 81 00:05:35,37 --> 00:05:41,05 consequences of the presence of a self-inductor in a circuit, 82 00:05:41,05 --> 00:05:46,256 and I start very simple. I have here a battery which has 83 00:05:46,256 --> 00:05:48,718 EMFV. I have here a switch, 84 00:05:48,718 --> 00:05:54,493 and here are the self-inductor. We always draw a self-inductor 85 00:05:54,493 --> 00:05:59,983 in a circuit with these coils, and we also have in series a 86 00:05:59,983 --> 00:06:04,149 resistor, which we always indicate with this, 87 00:06:04,149 --> 00:06:07,935 these teeth. And I close this switch when 88 00:06:07,935 --> 00:06:12,006 there is no current running. In other words, 89 00:06:12,006 --> 00:06:16,171 a time T equals zero when I close the switch, 90 00:06:16,171 --> 00:05:58,685 there is no current. When I close the switch the 91 00:05:58,685 --> 00:05:14,056 current wants to increase, but the self-inductance says 92 00:05:14,056 --> 00:04:50,915 uh-huh, uh-huh, take it easy, 93 00:04:50,915 --> 00:04:07,112 land slow, I don't like the change of such occurrence. 94 00:04:07,112 --> 00:03:20,83 So the self-inductance is fighting the current that wants 95 00:03:20,83 --> 00:02:44,466 to go through it. There comes a time that the 96 00:02:44,466 --> 00:01:58,184 self-inductance loses the fight, if you wait long enough, 97 00:01:58,184 --> 00:01:09,422 and then of course the current has reached a maximum volume, 98 00:01:09,422 --> 00:00:19,008 which you can find with Ohm's Law, because the self-inductance 99 00:00:19,008 --> 00:00:00 itself 100 00:00:00 --> 00:06:54 101 00:06:54 --> 00:06:58,265 has no resistor. Think of the self-inductance as 102 00:06:58,265 --> 00:07:01,26 made of super conducting material. 103 00:07:01,26 --> 00:07:05,707 There's no resistance. And so without knowing much 104 00:07:05,707 --> 00:07:09,609 about physics, you can make a plot about the 105 00:07:09,609 --> 00:07:14,237 current that is going to flow as a function of time. 106 00:07:14,237 --> 00:07:18,14 You start out with zero and then ultimately, 107 00:07:18,14 --> 00:07:22,768 if you wait long enough, you reach a maximum current 108 00:07:22,768 --> 00:07:26,277 which is given by Ohm's Law, 109 00:07:26,277 --> 00:07:30,692 which is simply V divided by R. And you slowly approach that 110 00:07:30,692 --> 00:07:33,312 value. And how slowly depends on the 111 00:07:33,312 --> 00:07:37,727 value of the self-inductance. If the self-inductance is very 112 00:07:37,727 --> 00:07:41,992 high, it might climb up like this, so this is a high value 113 00:07:41,992 --> 00:07:44,686 for L. If the self-inductance is very 114 00:07:44,686 --> 00:07:47,305 low, then there's a low value for L. 115 00:07:47,305 --> 00:07:50,972 If the self-inductance were zero, it would come up 116 00:07:50,972 --> 00:07:54,339 instantaneously, but I just convinced you that 117 00:07:54,339 --> 00:07:39,696 there I no such thing as zero self-inductance. 118 00:07:39,696 --> 00:06:40,673 There's always something finite, no matter how small. 119 00:06:40,673 --> 00:05:32,57 And so this is qualitatively what you would expect if you use 120 00:05:32,57 --> 00:04:36,952 your stomach and if you don't use your brains yet. 121 00:04:36,952 --> 00:03:31,119 There's nothing wrong with using your stomach occasionally, 122 00:03:31,119 --> 00:02:35,502 but now I want to do this in a move civilized way, 123 00:02:35,502 --> 00:01:28,534 and I want to use my brains, and when I use my brains I have 124 00:01:28,534 --> 00:00:46,537 to set up an equation for the circuit. 125 00:00:46,537 --> 00:00:00 And if you read your book, 126 00:00:00 --> 00:08:29 127 00:08:29 --> 00:08:32,792 you will find that Mister John *Coley* tells you to use 128 00:08:32,792 --> 00:08:36,514 *Kipshof's* Loop Rule. But Mister John *Coley* doesn't 129 00:08:36,514 --> 00:08:40,095 understand Faraday's Law, and he's not the only one. 130 00:08:40,095 --> 00:08:44,098 Almost every college book that you read on physics do this 131 00:08:44,098 --> 00:08:46,064 wrong. They advise you to use 132 00:08:46,064 --> 00:08:49,786 *Kipshof's* Loop Rule, which says that the closed loop 133 00:08:49,786 --> 00:08:52,244 integral around the circuit is zero. 134 00:08:52,244 --> 00:08:54,632 That, of course, is utter nonsense. 135 00:08:54,632 --> 00:08:57,932 How can it be zero? Because there is a change in 136 00:08:57,932 --> 00:09:03,06 magnetic flux, and so it can only be minus D 137 00:09:03,06 --> 00:09:06,84 five ET. I advise you to go to the eight 138 00:09:06,84 --> 00:09:12,656 0 two website and download a lecture supplement that you will 139 00:09:12,656 --> 00:09:17,987 find in which I address this issue and hit it very hard. 140 00:09:17,987 --> 00:09:23,802 So the closed loop integral of E dot EL, if you go around the 141 00:09:23,802 --> 00:09:28,842 circuit, is not zero, is minus D five ET -- Faraday's 142 00:09:28,842 --> 00:09:34,367 Law -- so it's minus L the IDT. So we have to go around to 143 00:09:34,367 --> 00:09:38,806 circuits and we have to apply Faraday's 144 00:09:38,806 --> 00:09:43,341 law and not *Kipshof's* Loop Rule, which doesn't apply here. 145 00:09:43,341 --> 00:09:47,722 This is the plus side of the battery and this is the minus 146 00:09:47,722 --> 00:09:51,72 side, so the electric field in the battery is in this 147 00:09:51,72 --> 00:09:54,41 direction. The electric field in the 148 00:09:54,41 --> 00:09:58,792 self-inductance is zero because the self-inductance has no 149 00:09:58,792 --> 00:10:02,02 resistance, it's super conducting material, 150 00:10:02,02 --> 00:10:06,555 and so the electric fields in the resistor -- if the current 151 00:10:06,555 --> 00:10:11,656 is in this direction, which it will be -- then the 152 00:10:11,656 --> 00:10:16,17 electric field in the resistor will be in this direction. 153 00:10:16,17 --> 00:10:21,087 So now I am equipped to write down the closed loop integral of 154 00:10:21,087 --> 00:10:24,311 E dot VL. I start here and I always go in 155 00:10:24,311 --> 00:10:28,744 the direction of the current, and I advise you to do the 156 00:10:28,744 --> 00:10:31,484 same. I don't care if you guess the 157 00:10:31,484 --> 00:10:35,031 wrong direction for the current. That's fine. 158 00:10:35,031 --> 00:10:39,142 Later, minus signs will correct that, 159 00:10:39,142 --> 00:09:21,899 they will tell you that you really guessed the wrong 160 00:09:21,899 --> 00:07:46,482 direction, but always go around the loop in the direction of the 161 00:07:46,482 --> 00:06:29,24 current, because then the EMF is always minus LVIDT. 162 00:06:29,24 --> 00:05:15,027 If you go in the direction opposed to the current, 163 00:05:15,027 --> 00:03:54,755 then it is plus LVIDT and that could become confusing. 164 00:03:54,755 --> 00:02:45,086 So I always go in the direction of the current, 165 00:02:45,086 --> 00:01:36,931 and so I first go through the self-inductance. 166 00:01:36,931 --> 00:00:00 There is no electric field in the self-inductance, 167 00:00:00 --> 00:11:10 168 00:11:10 --> 00:11:14,927 so the integral E dot EL -- in going from here to here -- is 169 00:11:14,927 --> 00:11:17,599 zero. This is where the books are 170 00:11:17,599 --> 00:11:19,019 wrong. It is zero. 171 00:11:19,019 --> 00:11:24,113 Now, I go through the resistor, and so now I get plus IR E and 172 00:11:24,113 --> 00:11:29,04 the L are in the same direction. Ohm's Law tells me it's IR. 173 00:11:29,04 --> 00:11:32,38 In the battery, I go against the electric 174 00:11:32,38 --> 00:11:36,806 fields, and so I get minus V. That now equals minus L, 175 00:11:36,806 --> 00:11:41,483 DIDT, and this is the only thing and the only correct way 176 00:11:41,483 --> 00:11:47,327 to apply Faraday's Law in this circuit. 177 00:11:47,327 --> 00:11:54,34 You can write it a little differently, which may give you 178 00:11:54,34 --> 00:11:57,596 some insight. For instance, 179 00:11:57,596 --> 00:12:03,106 you could write that I can bring V and the L, 180 00:12:03,106 --> 00:12:10,495 and L, to one side -- so I can write down that V minus LDIDT 181 00:12:10,495 --> 00:12:15,402 equals IR. It's the same equation when you 182 00:12:15,402 --> 00:12:19,82 look, and the nice thing about writing it this way is that 183 00:12:19,82 --> 00:12:24,47 since the IDT is positive here -- it's growing in time -- the 184 00:12:24,47 --> 00:12:29,274 induced EMF, which is this value -- notice it's always opposing 185 00:12:29,274 --> 00:12:34,157 the voltage of my battery -- and that's what *Land's* Law is all 186 00:12:34,157 --> 00:12:36,637 about. It's not until the IDT has 187 00:12:36,637 --> 00:12:40,202 become zero that V equals IR, and that happens, 188 00:12:40,202 --> 00:12:44 of course, if you wait long enough. 189 00:12:44 --> 00:12:48,266 And so we have to solve the differential equation, 190 00:12:48,266 --> 00:12:53,665 and what is often done that you bring all the terms to the left 191 00:12:53,665 --> 00:12:57,845 side and that you get an, zero on the right side. 192 00:12:57,845 --> 00:13:02,547 And so what you often see is that LVIDT plus IR minus V 193 00:13:02,547 --> 00:13:05,856 equals zero. And because we have a zero 194 00:13:05,856 --> 00:13:10,82 here, some physicist thinks that this is an application of 195 00:13:10,82 --> 00:13:15 *Kipshof's* Rule. This is nonsense. 196 00:13:15 --> 00:13:18,658 You can always make it zero here by bringing all the terms 197 00:13:18,658 --> 00:13:21,354 to this side. The closed loop integral of E 198 00:13:21,354 --> 00:13:24,434 DEL TL is not zero, the closed loop integral of E 199 00:13:24,434 --> 00:13:27,836 DEL TL is minus LDIDT, but when I shift minus LDIDT to 200 00:13:27,836 --> 00:13:31,302 this side I get zero here. And of course the people who 201 00:13:31,302 --> 00:13:35,345 write these books know that this is the right answer and so they 202 00:13:35,345 --> 00:13:39,004 manipulate it so they get this equation and they call that 203 00:13:39,004 --> 00:13:41,828 *Kipshof's* Rule. Sad, and also embarrassing. 204 00:13:41,828 --> 00:13:46 So, this is the equation that you have to solve. 205 00:13:46 --> 00:13:49,826 Some of you may have solved this equation in eight 0 one 206 00:13:49,826 --> 00:13:52,26 already. Surely you didn't have an I 207 00:13:52,26 --> 00:13:54,695 here. You may have had an X here for 208 00:13:54,695 --> 00:13:57,617 the position, but you probably solved it or 209 00:13:57,617 --> 00:13:59,982 you had friction. Maybe you didn't. 210 00:13:59,982 --> 00:14:04,017 I will give you the solution to that differential equation. 211 00:14:04,017 --> 00:14:07,773 It's a very easy solution. The current as a function of 212 00:14:07,773 --> 00:14:11,6 time is a maximum value times one minus E to the minus R 213 00:14:11,6 --> 00:14:16,399 divided by L times T, and IMX -- that is the maximum 214 00:14:16,399 --> 00:14:21,489 current -- is V divided by R. And let's look at this in a 215 00:14:21,489 --> 00:14:25,852 little bit more detail. First, notice that when T 216 00:14:25,852 --> 00:14:30,032 equals zero that indeed you find I equals zero. 217 00:14:30,032 --> 00:14:34,94 Substituting here T equals zero, you get one minus one. 218 00:14:34,94 --> 00:14:38,485 So you find, indeed, that I equals zero. 219 00:14:38,485 --> 00:14:43,756 Substituting there T goes to infinity, then you find that I 220 00:14:43,756 --> 00:14:49,668 indeed becomes V divided by R, which is exactly what you 221 00:14:49,668 --> 00:14:51,993 expect. If T becomes infinity, 222 00:14:51,993 --> 00:14:55,52 then clearly the self-inductance has lost all 223 00:14:55,52 --> 00:14:59,608 it's power, so to speak, and the current is simply V 224 00:14:59,608 --> 00:15:02,894 divided by R, the maximum current that you 225 00:15:02,894 --> 00:15:05,299 can have. And so that's a must, 226 00:15:05,299 --> 00:15:09,467 that's a requirement. If you wait L over R seconds -- 227 00:15:09,467 --> 00:15:12,994 and believe it or not, if you have some time, 228 00:15:12,994 --> 00:15:18,284 convince yourself that L over R indeed, as a unit, 229 00:15:18,284 --> 00:15:24,023 is seconds -- then the current I is about sixty three percent 230 00:15:24,023 --> 00:15:29,665 of IMX, because if T is L over R, then you get one minus one 231 00:15:29,665 --> 00:15:33,682 divided by E, and that is about 0 point six 232 00:15:33,682 --> 00:15:36,838 three. And if you wait double this 233 00:15:36,838 --> 00:15:42,384 time, then you have about eighty six percent of the maximum 234 00:15:42,384 --> 00:15:44,584 current. In other words, 235 00:15:44,584 --> 00:15:48,888 right here -- if I wait L over R 236 00:15:48,888 --> 00:12:55,274 seconds -- this value here is about 0 point six three times 237 00:12:55,274 --> 00:10:52,547 the maximum value possible, and it's very, 238 00:10:52,547 --> 00:08:19,887 it's climbing up and asymptotically approaches then, 239 00:08:19,887 --> 00:05:35,253 ultimately, the maximum current which is V divided by R. 240 00:05:35,253 --> 00:02:44,633 Make sure you download that lecture supplement that you'll 241 00:02:44,633 --> 00:00:00 find on the Web. Now, what I'm going to do is 242 00:00:00 --> 00:16:18 243 00:16:18 --> 00:13:52,157 all of a sudden I'm going to make this voltage zero. 244 00:13:52,157 --> 00:11:23,456 The way I could do that is by simply shorting it out. 245 00:11:23,456 --> 00:08:57,614 Of course on the blackboard, I can simply remove it. 246 00:08:57,614 --> 00:06:46,07 So it's not there. The current is still running 247 00:06:46,07 --> 00:04:43,105 and all of a sudden at a new time -- T zero, 248 00:04:43,105 --> 00:02:11,543 I define the time T zero again -- the voltage is zero. 249 00:02:11,543 --> 00:00:00 And now comes the question what is 250 00:00:00 --> 00:16:48 251 00:16:48 --> 00:14:40,225 now going to happen? Well, the self-inductance 252 00:14:40,225 --> 00:12:12,574 doesn't like the fact that the current is going down, 253 00:12:12,574 --> 00:09:30,726 so it's going to fight that change, and so you expect that 254 00:09:30,726 --> 00:07:20,112 the current is not going to die off right away, 255 00:07:20,112 --> 00:04:21,228 but you expect that the current is going to go down sort of like 256 00:04:21,228 --> 00:03:35,797 so. And you want, 257 00:03:35,797 --> 00:01:42,219 when you wait long enough, you want that, 258 00:01:42,219 --> 00:00:00 at T goes to infinity, 259 00:00:00 --> 00:17:19 260 00:17:19 --> 00:17:22,928 where T equals zero the current is still maximum, 261 00:17:22,928 --> 00:17:27,265 so it's still V over R, but when you go to infinity -- 262 00:17:27,265 --> 00:17:32,258 if you wait long enough -- then, of course, the current has to 263 00:17:32,258 --> 00:17:35,531 become zero. And as the current dies out, 264 00:17:35,531 --> 00:17:40,606 heat is being produced in that resistor at a rate of I square R 265 00:17:40,606 --> 00:17:44,289 joules per second, and then there comes a time 266 00:17:44,289 --> 00:17:49,363 that the current becomes almost zero and then the whole show is 267 00:17:49,363 --> 00:17:53,687 over. And so we can also calculate 268 00:17:53,687 --> 00:17:59,062 the exact time behavior by going back to our differential 269 00:17:59,062 --> 00:18:03,958 equation and make V zero. Where is that differential 270 00:18:03,958 --> 0. equation? 271 0. --> 00:18:04,917 yes, there you see this, 272 00:18:04,917 --> 00:18:07,701 Is it hiding? Oh, there it is. 273 00:18:07,701 --> 00:18:13,46 So I solved this differential equation, but this is now zero. 274 00:18:13,46 --> 00:18:18,739 And the solution to that differential equation is that I 275 00:18:18,739 --> 00:18:24,499 as a function of time is IMX times E to 276 00:18:24,499 --> 00:18:29,126 the minus R over L times T, and that exactly has all the 277 00:18:29,126 --> 00:18:34,175 quantities that you want it to have, because notice that at T 278 00:18:34,175 --> 00:18:39,307 equals zero -- when you put in T equals zero -- the current is 279 00:18:39,307 --> 00:18:42,925 indeed maximum, and that's what you require. 280 00:18:42,925 --> 00:18:47,217 That's the moment that you make that capital V zero, 281 00:18:47,217 --> 00:18:52,265 the current was still running. But notice that when T goes to 282 00:18:52,265 --> 00:18:56,822 infinity -- if you wait long enough -- that 283 00:18:56,822 --> 00:19:01,406 indeed the current goes to zero. And if you wait L over R 284 00:19:01,406 --> 00:19:06,154 seconds, then you are down to about thirty seven percent of 285 00:19:06,154 --> 00:19:09,674 your maximum current. So if you now go to I, 286 00:19:09,674 --> 00:19:14,258 I re-define T equals zero here, so if now I wait L over R 287 00:19:14,258 --> 00:19:19,17 seconds then this value here is about thirty seven percent of 288 00:19:19,17 --> 00:19:22,199 that value. So you've lost sixty three 289 00:19:22,199 --> 00:19:25,085 percent. And so you see, 290 00:19:25,085 --> 00:19:29,573 this is the consequence of the fact that the circuit is capable 291 00:19:29,573 --> 00:19:33,409 of fighting its own magnetic flux that it is creating. 292 00:19:33,409 --> 00:19:36,377 When the current was running happily here, 293 00:19:36,377 --> 00:19:39,417 with the battery in place, the current was, 294 00:19:39,417 --> 00:19:43,543 let's say, all the time IMX -- at least very close to that 295 00:19:43,543 --> 00:19:46,293 value -- V over R. And so all the time, 296 00:19:46,293 --> 00:19:49,116 there was heat produced in the resistor. 297 00:19:49,116 --> 00:19:52,88 I square R joules per second. Who was providing that, 298 00:19:52,88 --> 00:19:54,835 uh, energy? Well, of course, 299 00:19:54,835 --> 00:19:57,082 the battery. 300 00:19:57,082 --> 00:20:00,797 But now, when I take the battery out, there is still 301 00:20:00,797 --> 00:20:03,711 current running, and that means while the 302 00:20:03,711 --> 00:20:08,227 current is dying there is still heat produced in that resistor, 303 00:20:08,227 --> 00:20:12,452 and that heat slowly comes out until the current ultimately 304 00:20:12,452 --> 00:20:15,657 becomes zero. Now where does that energy come 305 00:20:15,657 --> 00:20:18,279 from? Well that energy must come from 306 00:20:18,279 --> 00:20:21,921 the magnetic field that is present in the solenoid, 307 00:20:21,921 --> 00:20:26,073 and this idea -- that we have energy that comes out in the 308 00:20:26,073 --> 00:20:31,102 form of heat, which really was there ear-, 309 00:20:31,102 --> 00:20:36,444 earlier in the form of a magnetic field -- allows us to 310 00:20:36,444 --> 00:20:41,885 evaluate what we call the magnetic energy field density. 311 00:20:41,885 --> 00:20:48,018 Let me first calculate how much heat is produced as the current 312 00:20:48,018 --> 00:20:51,778 goes from a maximum value down to zero. 313 00:20:51,778 --> 00:20:55,24 Hmmmm, I'll have to erase something. 314 00:20:55,24 --> 00:21:01,779 I'll erase this part here. So at any moment in time, 315 00:21:01,779 --> 00:21:06,781 the current is producing heat in the resistor, 316 00:21:06,781 --> 00:21:11,339 and so if I, if my voltage becomes zero at 317 00:21:11,339 --> 00:21:16,342 time T equals zero, then this is the amount of 318 00:21:16,342 --> 00:21:20,899 heat, uh-uh, no square here. I square RDT, 319 00:21:20,899 --> 00:21:27,014 integrated from zero to infinity, is the total heat that 320 00:21:27,014 --> 00:21:31,713 is produced as the current dies out, 321 00:21:31,713 --> 00:21:36,356 but I know what this current was -- I just erased it, 322 00:21:36,356 --> 00:21:41,803 if I still remember it -- so I can bring IMX outside and I can 323 00:21:41,803 --> 00:21:47,517 bring the resistance outside and the I get the integral from zero 324 00:21:47,517 --> 00:21:51,803 to infinity of E to the minus R over L times TDT. 325 00:21:51,803 --> 00:21:56,089 And this is a trivial integral. This integral is, 326 00:21:56,089 --> 00:21:59,571 uhm, L divided by two R. Oh, by the way, 327 00:21:59,571 --> 00:22:04,331 it is I squared, so I have a two here. 328 00:22:04,331 --> 00:22:08,394 It's very important. Don't forget the two. 329 00:22:08,394 --> 00:22:12,556 And so that integral is L divided by two R, 330 00:22:12,556 --> 00:22:18,304 and so if now I look at the product of I square maximum R L 331 00:22:18,304 --> 00:22:22,367 divided by two R, I get one half L times I 332 00:22:22,367 --> 00:22:26,826 maximum squared. So this comes out in the form 333 00:22:26,826 --> 00:22:32,772 of heat, and IMX is then the maximum current that we had when 334 00:22:32,772 --> 00:22:38,196 the current was flowing after a long time. 335 00:22:38,196 --> 00:22:41,829 I can now by, by manipulating numbers, 336 00:22:41,829 --> 00:22:47,526 I can now calculate how much energy there was in that field 337 00:22:47,526 --> 00:22:52,043 per cubic meter, because the magnetic field was 338 00:22:52,043 --> 00:22:55,186 exclusively inside that solenoid. 339 00:22:55,186 --> 00:23:01,177 And if I know that the energy that is ultimately coming out is 340 00:23:01,177 --> 00:23:06,284 one half LI squared, then I have here I, 341 00:23:06,284 --> 00:23:10,079 and so I can replace that I there by B divided by new zero 342 00:23:10,079 --> 00:23:12,61 times L divided by N and here I have L. 343 00:23:12,61 --> 00:23:16,672 And so if I substitute in here the value for L that we have on 344 00:23:16,672 --> 00:23:19,869 the blackboard there, and we substitute for I the 345 00:23:19,869 --> 00:23:23,997 value that we have here -- you can drop the maximum now -- this 346 00:23:23,997 --> 00:23:27,061 simply tells you, then, that any moment in time 347 00:23:27,061 --> 00:23:30,39 that I have a current I running through a solenoid, 348 00:23:30,39 --> 00:23:34,053 that the energy that is available in the solenoid in the 349 00:23:34,053 --> 00:22:15,681 forms of magnetic energy is one half LR squared. 350 00:22:15,681 --> 00:17:50,584 And so when you do that, you substitute capital L and 351 00:17:50,584 --> 00:12:39,605 capital I, you will find that one half LI squared then becomes 352 00:12:39,605 --> 00:07:43,919 B squared over two new zero times pi little R squared times 353 00:07:43,919 --> 00:05:36,469 L. You check that at home. 354 00:05:36,469 --> 00:00:00 It's simply a substitution. But this is the volume of the 355 00:00:00 --> 00:24:07 356 00:24:07 --> 00:24:10,485 solenoid where the magnetic field exists, 357 00:24:10,485 --> 00:24:14,843 and we have assumed that the magnetic field is zero 358 00:24:14,843 --> 00:24:18,503 everywhere outside. And if you accept that, 359 00:24:18,503 --> 00:24:23,732 then you see that we now have a result for the magnetic field 360 00:24:23,732 --> 00:24:28,874 energy density -- that is how much energy there is per cubic 361 00:24:28,874 --> 00:24:32,272 meter -- that is, of course, this value. 362 00:24:32,272 --> 00:24:38,199 Because this is the total energy of the magnetic field, 363 00:24:38,199 --> 00:24:41,842 if we know the current, and this is the volume of the 364 00:24:41,842 --> 00:24:43,874 magnetic field. So the magne-, 365 00:24:43,874 --> 00:24:48,148 magnetic field energy density is then B squared divided by two 366 00:24:48,148 --> 00:24:51,441 new zero, and this is in joules per cubic meter. 367 00:24:51,441 --> 00:24:54,664 So in principle, if you knew the magnetic field 368 00:24:54,664 --> 00:24:58,237 everywhere in space, then you can integrate over all 369 00:24:58,237 --> 00:25:01,81 space and you can then calculate how much energy is, 370 00:25:01,81 --> 00:25:04,193 uhm, present in the magnetic field. 371 00:25:04,193 --> 00:25:08,046 And earlier in this course, we did something similar for 372 00:25:08,046 --> 00:25:12,474 electric fields. We calculated the electric 373 00:25:12,474 --> 00:25:16,121 field energy density. Perhaps you remember whatit 374 00:25:16,121 --> 00:25:18,553 was. It was one half epsilon zero 375 00:25:18,553 --> 00:25:22,58 kappa times E squared. It was also in joules per cubic 376 00:25:22,58 --> 00:25:25,315 meter. Now in the case of an electric 377 00:25:25,315 --> 00:25:30,026 field, this represents the work that I had to do to arrange the 378 00:25:30,026 --> 00:25:32,61 charges in a certain configuration. 379 00:25:32,61 --> 00:25:36,713 In the case of a magnetic field, it represents the work 380 00:25:36,713 --> 00:25:41,5 that I have to do to get a current going inside 381 00:25:41,5 --> 00:25:45,762 a pure self-inductor. That means the resistance of 382 00:25:45,762 --> 00:25:50,547 the self-inductor is zero, and it takes work because the 383 00:25:50,547 --> 00:25:54,984 solenoid will oppose the building up of the current, 384 00:25:54,984 --> 00:25:59,681 and so I have to do work. So there's a parallel between 385 00:25:59,681 --> 00:26:02,03 the two. I can make you see, 386 00:26:02,03 --> 00:26:06,815 in a quite dramatic way, how strong self-inductances can 387 00:26:06,815 --> 00:26:11,252 fight their own current, and the way I'm going to do 388 00:26:11,252 --> 00:24:07,312 that is with in-, inside of there, 389 00:24:07,312 --> 00:19:16,882 whereby I have a twelve volt car battery and I have two light 390 00:19:16,882 --> 00:16:56,507 bulbs. I have here an enormous 391 00:16:56,507 --> 00:14:26,451 self-inductance L, thirty Henry. 392 00:14:26,451 --> 00:09:50,542 We will learn later in the course how you make such a high 393 00:09:50,542 --> 00:06:07,878 self-inductance, and then here is a light bulb. 394 00:06:07,878 --> 00:02:44,577 The light bulb has a resistance of six ohm. 395 00:02:44,577 --> 00:00:00 This self-inductance, 396 00:00:00 --> 00:26:45 397 00:26:45 --> 00:26:47,909 there is nothing we can do about it. 398 00:26:47,909 --> 00:26:51,068 It happens to have four ohm resistance. 399 00:26:51,068 --> 00:26:54,227 We don't have a self, we don't have s-, 400 00:26:54,227 --> 00:26:58,965 superconducting wires here, so it also has a resistance of 401 00:26:58,965 --> 00:27:02,207 four ohms. Forgive me for that but there 402 00:27:02,207 --> 00:27:07,112 is nothing I can do about it. I have here another resistance 403 00:27:07,112 --> 00:27:11,933 of our ohms and a light bulb, which is the same one as that 404 00:27:11,933 --> 00:27:17,004 one -- also six ohms -- and then here is my car battery plus a 405 00:27:17,004 --> 00:27:21,152 switch. And the car battery is twelve 406 00:27:21,152 --> 00:27:25,011 volts, and I'm going to throw the switch, turn it on. 407 00:27:25,011 --> 00:27:28,574 You will see that this light bulb comes on almost 408 00:27:28,574 --> 00:27:31,988 instantaneously. There is no self-inductance in 409 00:27:31,988 --> 00:27:35,625 this loop -- well, maybe a few micro Henry or even 410 00:27:35,625 --> 00:27:39,114 less -- but in this loop here there is this huge 411 00:27:39,114 --> 00:27:42,602 self-inductance, and so the self-inductance says 412 00:27:42,602 --> 00:27:45,645 take it easy to the current, take it easy, 413 00:27:45,645 --> 00:27:50,098 just wait, and you will very slowly see that 414 00:27:50,098 --> 00:27:54,093 light bulb come on. And we can calculate how long 415 00:27:54,093 --> 00:27:59,002 it takes, because we have here ten ohms -- six ohms and four 416 00:27:59,002 --> 00:28:03,412 ohms, so L divided by R, we have thirty divided by ten 417 00:28:03,412 --> 00:28:08,321 -- so that is three seconds. So what that means is that even 418 00:28:08,321 --> 00:28:11,899 after six seconds, which is twice this time, 419 00:28:11,899 --> 00:28:16,808 even then the current through this light bulb is only eighty 420 00:28:16,808 --> 00:28:22,799 six percentof it's maximum. But that means since the light, 421 00:28:22,799 --> 00:28:26,305 of course, is proportional to I square R in the light bulb, 422 00:28:26,305 --> 00:28:29,569 that the light is only seventy five percent of maximum, 423 00:28:29,569 --> 00:28:32,772 and even if you wait nine seconds, then the light that 424 00:28:32,772 --> 00:28:36,338 comes out here is still only ninety percent of it's maximum, 425 00:28:36,338 --> 00:28:38,575 whereas this one comes on immediately. 426 00:28:38,575 --> 00:28:42,201 The reason why we put the four ohm here, we want this part to 427 00:28:42,201 --> 00:28:45,949 be -- from an ohmic resistance point of view -- to be identical 428 00:28:45,949 --> 00:28:48,547 as this part. So that's why you artificially 429 00:28:48,547 --> 00:28:51,811 added here the four ohm, because the four ohm is always 430 00:28:51,811 --> 00:27:06,812 there in the self-inductance, there is nothing we can do 431 00:27:06,812 --> 00:24:29,111 about it. And so you are going to see a 432 00:24:29,111 --> 00:21:34,809 remarkable example. This is one light bulb. 433 00:21:34,809 --> 00:17:25,807 That is the one that is here, and this is the light bulb that 434 00:17:25,807 --> 00:14:27,356 is there, and the self-inductance is in this 435 00:14:27,356 --> 00:12:47,755 incredible monster, here. 436 00:12:47,755 --> 00:09:32,704 I'll make the lights, change the light setting a 437 00:09:32,704 --> 00:05:56,902 little bit so that we can see the lights of the bulb. 438 00:05:56,902 --> 00:02:16,951 The bulbs are only eight-watt bulbs, they're not very, 439 00:02:16,951 --> 00:00:00 very strong, not 440 00:00:00 --> 00:29:25 441 00:29:25 --> 00:29:28,492 very bright bulbs, but the effect will be very 442 00:29:28,492 --> 00:29:31,286 clear. So if you're ready for this -- 443 00:29:31,286 --> 00:29:35,322 so this is the one that I think comes on immediately, 444 00:29:35,322 --> 00:29:39,125 and this is the one that takes it's time -- three, 445 00:29:39,125 --> 00:29:41,531 two, one, zero. One, two, three, 446 00:29:41,531 --> 00:29:44,092 four, five, six, you see how slow? 447 00:29:44,092 --> 00:29:48,361 It's still not very bright, it's still getting brighter. 448 00:29:48,361 --> 00:29:51,155 It's still not as bright as this one. 449 00:29:51,155 --> 00:29:55,036 It's getting there, you can actually do some timing 450 00:29:55,036 --> 00:29:57,397 by counting. 451 00:29:57,397 --> 00:30:02,19 By the way, the values of the resistance that I gave you are 452 00:30:02,19 --> 00:30:05,845 when the light bulbs are hot. Three, two, one, 453 00:30:05,845 --> 00:30:08,039 zero, one, two, three, four, 454 00:30:08,039 --> 00:30:10,395 five, six, seven, eight, nine, 455 00:30:10,395 --> 00:30:13,563 ten, eleven, twelve, I'm still seeing it 456 00:30:13,563 --> 00:30:17,219 getting brighter. What you're seeing here is a 457 00:30:17,219 --> 00:30:21,605 remarkable example that the self-inductance is fighting 458 00:30:21,605 --> 00:30:23,718 itself. That's why the name 459 00:30:23,718 --> 00:30:27,918 self-inductance is so nice. 460 00:30:27,918 --> 00:30:35,651 Now I want to go one step further and I want to power the 461 00:30:35,651 --> 00:30:40,208 LR circuit with a AC power supply. 462 00:30:40,208 --> 00:30:48,079 If you have an AC power supply -- so it's changing all the 463 00:30:48,079 --> 00:30:55,674 time, the voltage -- now, of course, the self-inductance 464 00:30:55,674 --> 00:31:00,07 is fighting back all the time, 465 00:31:00,07 --> 00:31:05,85 not just only in the beginning as you saw in this circuit, 466 00:31:05,85 --> 00:31:10,515 but now, of course, it is active almost all the 467 00:31:10,515 --> 00:31:13,861 time. So we can do away with this, 468 00:31:13,861 --> 00:31:19,337 and so now we replace the battery by a AC power supply, 469 00:31:19,337 --> 00:31:23,495 which we normally put just a wiggle there, 470 00:31:23,495 --> 00:31:31 and here is the self-inductance L, and here is the resistor R. 471 00:31:31 --> 00:31:36,907 And let the voltage provided by this power supply be V zero 472 00:31:36,907 --> 00:31:42,101 times the cosine of omega T, omega being the angular 473 00:31:42,101 --> 00:31:45,564 frequency. And now, I have to apply 474 00:31:45,564 --> 00:31:49,332 Faraday's Law, not *Kipshof's* Rule -- 475 00:31:49,332 --> 00:31:55,24 Faraday's Law -- when I go around this circuit and I set up 476 00:31:55,24 --> 00:32:02,165 the differential equation. And of course the differential 477 00:32:02,165 --> 00:32:06,722 equation is going to be exactly like this, except that V, 478 00:32:06,722 --> 00:32:10,139 now, is V zero times the cosine of omega T. 479 00:32:10,139 --> 00:32:14,533 And now I have to solve for this differential equation. 480 00:32:14,533 --> 00:32:19,09 That's the only difference, so I don't have to start from 481 00:32:19,09 --> 00:32:22,019 ground zero. And the solution to this 482 00:32:22,019 --> 00:32:27,063 differential equation is quite remarkable and not so intuitive. 483 00:32:27,063 --> 00:32:32,027 The current -- as a function of time, now -- is V zero divided 484 00:32:32,027 --> 00:32:36,664 by the square root of R squared plus omega 485 00:32:36,664 --> 00:32:41,319 L squared times the cosine of omega T minus phi, 486 00:32:41,319 --> 00:32:45,082 and the tangent of phi, that angle phi, 487 00:32:45,082 --> 00:32:50,529 is omega L divided by R. And of course we need some time 488 00:32:50,529 --> 00:32:55,085 to digest this. The first thing that you notice 489 00:32:55,085 --> 00:33:00,532 is that there is a phase lag between the current and the 490 00:33:00,532 --> 00:33:06,375 driving voltage. If phi, as you're going to see, 491 00:33:06,375 --> 00:33:10,065 is, uhm, ninety degrees, then the current is delayed by 492 00:33:10,065 --> 00:33:13,277 one quarter of a cycle, though the fact that the 493 00:33:13,277 --> 00:33:16,967 current comes later than the driving voltage perhaps is 494 00:33:16,967 --> 00:33:20,589 intuitive, because the self-inductance is fighting the 495 00:33:20,589 --> 00:33:23,596 change in the current. So it's perhaps not so 496 00:33:23,596 --> 00:33:27,696 surprising that there's going to be a delay, that the current 497 00:33:27,696 --> 00:33:31,044 comes a little later. If you look at this equation 498 00:33:31,044 --> 00:33:34,735 here, then what you have in front of the cosine term is 499 00:33:34,735 --> 00:32:09,621 obviously the maximum possible current, 500 00:32:09,621 --> 00:26:44,578 because the cosine term is just oscillating between plus one and 501 00:26:44,578 --> 00:21:24,694 minus one, and so this here is the maximum current that you can 502 00:21:24,694 --> 00:18:29,274 ever get. In one full cycle you get 503 00:18:29,274 --> 00:15:08,057 positive and you get negative net value. 504 00:15:08,057 --> 00:10:03,651 And notice here the role of omega L really plays the role of 505 00:10:03,651 --> 00:06:32,115 a resistance, and in fact the dimension of 506 00:06:32,115 --> 00:03:47,014 omega L is ome-, is, is, is ohms. 507 00:03:47,014 --> 00:00:00 It really plays the role of a 508 00:00:00 --> 00:34:07 509 00:34:07 --> 00:34:11,081 resistance, and if omega is very high then the resistance 510 00:34:11,081 --> 00:34:15,382 here becomes very high and so your current becomes very low. 511 00:34:15,382 --> 00:34:19,172 Well, that's intuitively pleasing because if omega is 512 00:34:19,172 --> 00:34:22,671 high, then the changes -- the DIDT's -- are very, 513 00:34:22,671 --> 00:34:26,899 very high, and therefore if there are very fast changes the 514 00:34:26,899 --> 00:34:31,491 induced EMF is going to be high, and so the current will be low. 515 00:34:31,491 --> 00:34:35,5 Also, if L is very high, then the system also is capable 516 00:34:35,5 --> 00:34:39,291 of fighting back very hard, and so it puts up a large 517 00:34:39,291 --> 00:32:25 resistance. So it's also pleasing that you 518 00:32:25 --> 00:30:22,301 see the L there, downstairs. 519 00:30:22,301 --> 00:26:25,993 If omega is very low, in your mind you can make omega 520 00:26:25,993 --> 00:23:42,395 zero. You don't even have alternating 521 00:23:42,395 --> 00:19:46,086 current when omega is zero, then you have DC which is 522 00:19:46,086 --> 00:16:30,677 direct current. So when you make omega zero, 523 00:16:30,677 --> 00:13:33,446 you simply get I is V zero divided by R. 524 00:13:33,446 --> 00:10:13,493 That's Ohm's Law, that's obvious that you get 525 00:10:13,493 --> 00:07:48,072 that. Let's now look at the phase 526 00:07:48,072 --> 00:05:09,018 angle. The tangent of phi is omega L 527 00:05:09,018 --> 00:02:02,698 divided by R. If the self-inductor is very 528 00:02:02,698 --> 00:00:00 large, 529 00:00:00 --> 00:35:14 530 00:35:14 --> 00:35:17,897 then the system has a strong ability to fight back, 531 00:35:17,897 --> 00:35:21,483 so it can delay that current by a large amount, 532 00:35:21,483 --> 00:35:24,368 and the same is true is omega is high. 533 00:35:24,368 --> 00:35:28,032 If omega is high then the time changes are very, 534 00:35:28,032 --> 00:35:32,553 changes occur on a very small time scale, and so the system 535 00:35:32,553 --> 00:35:35,048 can fight back. Because remember, 536 00:35:35,048 --> 00:35:38,556 you always have the EMF proportion to the IDT. 537 00:35:38,556 --> 00:35:42,609 And so it's also pleasing to see that omega and L are 538 00:35:42,609 --> 00:35:45,883 upstairs here. Either one of being large it 539 00:35:45,883 --> 00:35:50,215 can fight back and it can hold back 540 00:35:50,215 --> 0. the, the current. 541 0. --> 00:35:51,734 yes, there you see this, 542 00:35:51,734 --> 00:35:56,882 I have worked out a situation whereby we have an LR circuit -- 543 00:35:56,882 --> 00:36:00,848 this is on the Web, you can download that so you 544 00:36:00,848 --> 00:36:06,081 don't have to copy it -- and the reason why I have these values 545 00:36:06,081 --> 00:36:11,144 is because directly coupled to a demonstration that I will do 546 00:36:11,144 --> 00:36:14,436 shortly. You see, an L in series with an 547 00:36:14,436 --> 00:36:19,5 R, the L is ten *milli-Henry* and the R 548 00:36:19,5 --> 00:32:58,16 is ten ohms, and let V zero be ten volts. 549 00:32:58,16 --> 00:28:56,553 And here you see three frequencies hundred hertz, 550 00:28:56,553 --> 00:24:34,811 thousand and ten thousand hertz, and here you see the 551 00:24:34,811 --> 00:20:33,204 values for omega. You have to multiply hertz with 552 00:20:33,204 --> 00:17:57,166 two pi. And look now at omega L. 553 00:17:57,166 --> 00:13:05,224 At low frequency -- a hundred hertz -- omega L is six point 554 00:13:05,224 --> 00:09:33,817 three ohms. Compare that with the ten ohms. 555 00:09:33,817 --> 00:05:27,176 They'd be comparable, but now look for instance at 556 00:05:27,176 --> 00:02:10,87 ten thousand hertz. The omega L is huge. 557 00:02:10,87 --> 00:00:00 It's six 558 00:00:00 --> 00:36:51 559 00:36:51 --> 00:36:54,587 hundred and thirty ohms. So it entirely determines -- so 560 00:36:54,587 --> 00:36:57,326 to speak -- the resistance of that circuit. 561 00:36:57,326 --> 00:37:01,109 And so the current that is going to run -- at least this is 562 00:37:01,109 --> 00:37:04,957 the maximum current is this value, which we also saw here on 563 00:37:04,957 --> 00:37:09,001 the blackboard -- that current at high frequently is enormously 564 00:37:09,001 --> 00:37:11,284 reduced. It's fifty times lower than 565 00:37:11,284 --> 00:37:15,197 this current at low frequency, even though they have the same 566 00:37:15,197 --> 00:37:18,328 value for P zero. And then you see here the phase 567 00:37:18,328 --> 00:37:20,545 angles. And the reason why you have 568 00:37:20,545 --> 00:37:23,858 these values is that I can make you 569 00:37:23,858 --> 00:37:26,269 listen to this, I can make you hear this, 570 00:37:26,269 --> 00:37:29,283 because your hearing is very good at hundred hertz, 571 00:37:29,283 --> 00:37:32,899 and since all of you are young you can probably hear even ten 572 00:37:32,899 --> 00:37:35,672 thousand hertz, maybe some of you can even hear 573 00:37:35,672 --> 00:37:38,324 twenty kilohertz. When you get older you lose 574 00:37:38,324 --> 00:37:41,398 your high frequencies. In fact, my frequency cut off 575 00:37:41,398 --> 00:37:43,628 is somewhere near four thousand hertz. 576 00:37:43,628 --> 00:37:47,425 I'm going to make you listen to music, and there will be violins 577 00:37:47,425 --> 00:37:50,74 which produce probably three, four, five thousand hertz, 578 00:37:50,74 --> 00:37:53,474 and then I will turn on, 579 00:37:53,474 --> 00:37:56,125 all of a sudden, the ten *milli-Henry*. 580 00:37:56,125 --> 00:38:00,52 So first I will make you listen to music whereby there is no ten 581 00:38:00,52 --> 00:38:04,287 *milli-Henry* in there, and then I will turn on the ten 582 00:38:04,287 --> 00:38:06,868 *milli-Henry*, and what you will hear, 583 00:38:06,868 --> 00:38:10,915 that the violins disappear because the current reduction is 584 00:38:10,915 --> 00:38:15,031 now huge on the high frequency but is very little on the low 585 00:38:15,031 --> 00:38:19,216 frequency, and that's the idea of what a self-inductor can do 586 00:38:19,216 --> 00:38:21,449 for you. So if you listen to this 587 00:38:21,449 --> 00:38:27,815 [playback of classical music with violin] 588 00:38:27,815 --> 00:38:35,768 there's no self-inductor in now. There's no music either 589 00:38:35,768 --> 00:38:41,841 [laughter] OK. [Playback of classical music 590 00:38:41,841 --> 00:38:47,047 continues, without violin notes.] OK. 591 00:38:47,047 --> 00:38:55 It's different music. No self-inductor. 592 00:38:55 --> 00:38:59,973 [Click] Self-inductor. The high frequencies are gone. 593 00:38:59,973 --> 00:39:05,33 [Playback of classical music continues, again with violin 594 00:39:05,33 --> 00:39:10,4 notes] No self-inductor. [Click] You can turn a violin 595 00:39:10,4 --> 00:39:14,417 concerto into a cello concerto. [laughter]. 596 00:39:14,417 --> 00:39:17,478 You just cut the violins out. OK. 597 00:39:17,478 --> 00:39:23,121 I cannot make you listen to the phase shift, not even in the 598 00:39:23,121 --> 00:39:28 case of the ninety degree phase shift, 599 00:39:28 --> 00:39:31,242 and that is quite obvious because what does it mean that 600 00:39:31,242 --> 00:39:33,364 there is a ninety degree phase shift? 601 00:39:33,364 --> 00:39:36,902 It means that during one cycle of ten thousand hertz -- which 602 00:39:36,902 --> 00:39:40,262 takes only one ten-thousandth of a second -- that the high 603 00:39:40,262 --> 00:39:43,563 frequencies are shifted by only twenty-five microseconds, 604 00:39:43,563 --> 00:39:46,983 and there's no way that your ears, your ears can hear that. 605 00:39:46,983 --> 00:39:50,343 The fact that the composer wanted those violins to come in 606 00:39:50,343 --> 00:39:54,057 twenty-five microseconds earlier than, than they do of course is 607 00:39:54,057 --> 00:39:57,359 something you cannot hear. So I cannot make you listen to 608 00:39:57,359 --> 00:40:01,25 the phase shift, but for the phase shift I have 609 00:40:01,25 --> 00:40:05,581 something else. And for that something else, 610 00:40:05,581 --> 00:40:10,214 I'm going to return to the, to my last lecture, 611 00:40:10,214 --> 00:40:15,452 in which we levitated a woman -- magnetic levitation. 612 00:40:15,452 --> 00:40:21,193 And so I'm going to return to that idea and grind a little 613 00:40:21,193 --> 00:40:25,524 deeper than we did when I gave that lecture, 614 00:40:25,524 --> 00:40:31,064 just before spring break. I had a coil and I was driving 615 00:40:31,064 --> 00:40:36 that coal, coil with sixty hertz AC. 616 00:40:36 --> 00:34:17,347 And let's assume that looking from above that the current was 617 00:34:17,347 --> 00:27:46,072 running in clockwise direction, which is exactly what I assumed 618 00:27:46,072 --> 00:22:05,284 when I discussed with you, and so the magnetic field is 619 00:22:05,284 --> 00:17:02,362 coming down like this -- magnetic dipole field -- 620 00:17:02,362 --> 00:12:24,683 produced by this coil. And then we have here, 621 00:12:24,683 --> 00:06:06,031 we had a conducting place under there, and I said to you when 622 00:06:06,031 --> 00:00:00 this magnetic field is increasing in strength, 623 00:00:00 --> 00:41:06 624 00:41:06 --> 00:41:09,116 then there's going to be an induced EMF here, 625 00:41:09,116 --> 00:41:11,879 which tries to imp-, oppose that change, 626 00:41:11,879 --> 00:41:15,208 and so the induced current that is going to run, 627 00:41:15,208 --> 00:41:18,82 which we call eddy currents, is going to run in this 628 00:41:18,82 --> 00:41:21,016 direction. If this is clockwise, 629 00:41:21,016 --> 00:41:24,204 this current is going to be counter clockwise, 630 00:41:24,204 --> 00:41:27,604 so it's going to produce a magnetic field in this 631 00:41:27,604 --> 00:41:30,295 direction. It opposes the change of the 632 00:41:30,295 --> 00:41:33,058 increase in magnetic field: *Lendz* Law. 633 00:41:33,058 --> 00:41:36,741 And since the two currents are in opposite direction, 634 00:41:36,741 --> 00:41:41,665 the two repel each other. And you bought off on that, 635 00:41:41,665 --> 00:41:44,935 and we levitated a woman. However, no one asked me the 636 00:41:44,935 --> 00:41:48,76 question, what happens a little later in time when the magnetic 637 00:41:48,76 --> 00:41:52,523 field is still in the downward direction but it is decreasing, 638 00:41:52,523 --> 00:41:55,793 since it is an AC current, there comes a time that the 639 00:41:55,793 --> 00:41:58,322 magnetic field will be decreasing in time. 640 00:41:58,322 --> 00:42:01,777 Now the EMF here must flip over because Lendz says sorry, 641 00:42:01,777 --> 00:42:05,231 we don't like the decrease. The moment that the EMF flips 642 00:42:05,231 --> 00:42:07,267 over, this current will flip over. 643 00:42:07,267 --> 00:42:10,599 The two currents are now in the same 644 00:42:10,599 --> 00:37:10,002 direction, and they will attract each other, 645 00:37:10,002 --> 00:32:16,397 and so there goes your magnetic levitation. 646 00:32:16,397 --> 00:25:58,904 Half the time attraction, half the time they repel each 647 00:25:58,904 --> 00:21:40,252 other. But yet we did elev- levitate a 648 00:21:40,252 --> 00:15:57,712 woman, and the secret lies in the self-inductance. 649 00:15:57,712 --> 00:09:33,229 This current that runs here runs over a pass -- which is 650 00:09:33,229 --> 00:03:50,689 very difficult for me to anticipate -- which has a 651 00:03:50,689 --> 00:00:00 certain resistance, 652 00:00:00 --> 00:42:42 653 00:42:42 --> 00:42:45,405 R; and it has a certain self-inductance, 654 00:42:45,405 --> 00:42:47,5 L. We know what omega is; 655 00:42:47,5 --> 00:42:50,294 that's about three hundred sixty. 656 00:42:50,294 --> 00:42:55,009 And so we do get in this conductor, we get the current, 657 00:42:55,009 --> 00:42:59,549 the induced current here, is delayed by a phase angle 658 00:42:59,549 --> 00:43:04,002 driven by this equation, is delayed over the induced 659 00:43:04,002 --> 00:43:06,971 EMF. The EMF immediately coupled to 660 00:43:06,971 --> 00:43:11,336 what this coil is doing, but the induced current is 661 00:43:11,336 --> 00:43:14,654 delayed. And I have something that will 662 00:43:14,654 --> 00:43:19,528 allow that, allow you to see that, 663 00:43:19,528 --> 00:43:24,59 perhaps in even more detail. This *raf* curve is the 664 00:43:24,59 --> 00:43:29,353 current for the coil, the coil that you see there 665 00:43:29,353 --> 00:43:32,827 above. And when the current is above 666 00:43:32,827 --> 00:43:38,682 the black line it's clockwise and when it is below the black 667 00:43:38,682 --> 00:43:43,743 line it's counter clockwise. The vertical scales are 668 00:43:43,743 --> 00:43:49,5 arbitrary. The green curve is the EMF, 669 00:43:49,5 --> 00:43:51,991 which is induced in the conductor. 670 00:43:51,991 --> 00:43:56,673 Notice when the magnetic field increases, when the current goes 671 00:43:56,673 --> 00:43:59,996 up in the coil, that the EMF in the conductor 672 00:43:59,996 --> 00:44:04,225 is in such a direction that it opposes the change of that 673 00:44:04,225 --> 00:44:07,245 magnetic field. But now when the magnetic 674 00:44:07,245 --> 00:44:10,493 fields go down, when the current in the coil 675 00:44:10,493 --> 00:44:15,175 goes down, immediately the EMF flips over, which is what I just 676 00:44:15,175 --> 00:44:18,573 mentioned to you, and therefore if the induced 677 00:44:18,573 --> 00:44:22,5 current and the induced EMF were in 678 00:44:22,5 --> 00:44:26,673 phase with each other, half the time you would have 679 00:44:26,673 --> 00:44:31,764 attraction and half the time you would have that the two repel 680 00:44:31,764 --> 00:44:36,355 each other, and that won't give you magnetic levitation. 681 00:44:36,355 --> 00:44:40,862 Here, what you see is a blue curve which represents the 682 00:44:40,862 --> 00:44:44,451 induced current. I call it the eddy current. 683 00:44:44,451 --> 00:44:49,208 If there is no phase shift between the induced EMF and the 684 00:44:49,208 --> 00:44:53,798 induced current, notice that half the time 685 00:44:53,798 --> 00:44:57,784 the blue curve and the red curve are in opposite direction. 686 00:44:57,784 --> 00:45:01,7 When they are in opposite direction they repel each other. 687 00:45:01,7 --> 00:45:05,273 When they are in the same direction they attract each 688 00:45:05,273 --> 00:45:07,471 other. But now, if I have a phase 689 00:45:07,471 --> 00:45:11,388 delay so that the induced current comes later than the EMF 690 00:45:11,388 --> 00:45:15,716 -- and I'm going to do something dramatic, I'm going to shift it 691 00:45:15,716 --> 00:45:19,908 by ninety degrees so the current is now ninety degrees delayed 692 00:45:19,908 --> 00:45:24,929 relative to the induced EMF -- look now that the red curve and 693 00:45:24,929 --> 00:45:27,854 the blue curve are always in opposite direction. 694 00:45:27,854 --> 00:45:31,464 And so now there is hundred percent of the time a repelling 695 00:45:31,464 --> 00:45:33,643 force. The coil repels the conductor 696 00:45:33,643 --> 00:45:35,697 and the conductor repels the coil. 697 00:45:35,697 --> 00:45:38,373 Now in the case when we levitated the woman, 698 00:45:38,373 --> 00:45:42,232 I am sure that the phase delay was not ninety degrees but maybe 699 00:45:42,232 --> 00:45:45,78 it was only thirty or forty degrees, but the net result is 700 00:45:45,78 --> 00:45:49,452 here the shift is not ninety degrees, the net result is that 701 00:45:49,452 --> 00:45:53 you get, on average, a repelling force. 702 00:45:53 --> 00:45:55,905 And so, the secret of the repelling force, 703 00:45:55,905 --> 00:46:00,016 in the case of the levitation of this coil and therefore of 704 00:46:00,016 --> 00:46:04,055 the levitation of the woman, lies in the effect that there 705 00:46:04,055 --> 00:46:07,67 is a finite self-inductance [tapping noise] in here. 706 00:46:07,67 --> 00:46:10,717 If R is zero, then of course we have a super 707 00:46:10,717 --> 00:46:13,977 conductor, then *phi* is always ninety degrees. 708 00:46:13,977 --> 00:46:16,741 When R is zero, this is infinitely high, 709 00:46:16,741 --> 00:46:20,498 so now we get a ninety-degree phase shift, and I did a 710 00:46:20,498 --> 00:46:25,373 demonstration whereby I had a little magnet floating 711 00:46:25,373 --> 00:46:28,799 above a super conductor. That was an ideal case. 712 00:46:28,799 --> 00:46:32,663 Phi was then ninety degrees, so they always repel each 713 00:46:32,663 --> 00:46:34,922 other. Today I want to do a more 714 00:46:34,922 --> 00:46:38,348 controlled demonstration, whereby I can actually 715 00:46:38,348 --> 00:46:42,503 calculate the self-inductance and I also can calculate the 716 00:46:42,503 --> 00:46:45,419 resistance. And what I will do today is I 717 00:46:45,419 --> 00:46:49,355 will have a coil which is stationary, and I will have a 718 00:46:49,355 --> 00:46:53 conductor which is not stationary. 719 00:46:53 --> 00:46:55,906 Here is my coil. AC, sixty hertz. 720 00:46:55,906 --> 00:46:59,084 There's the coil. And I have a ring, 721 00:46:59,084 --> 00:47:04,08 and the ring is made of aluminum, and I know exactly the 722 00:47:04,08 --> 00:47:09,075 dimensions of this ring. I know the radius -- it's about 723 00:47:09,075 --> 00:47:14,433 five centimeters -- I know the thickness, I know everything. 724 00:47:14,433 --> 00:47:19,065 It's an aluminum ring and has a radius of about five 725 00:47:19,065 --> 00:47:21,971 centimeters. Since I know all the 726 00:47:21,971 --> 00:47:26,225 dimensions, I can calculate the resistance 727 00:47:26,225 --> 00:47:28,939 of that ring. You should be able to do that, 728 00:47:28,939 --> 00:47:31,021 too, if I gave you the dimensions. 729 00:47:31,021 --> 00:47:34,87 And so the resistance of that ring very roughly is about seven 730 00:47:34,87 --> 00:47:36,889 times ten to the minus five ohms. 731 00:47:36,889 --> 00:47:39,602 It's very small. I can all-, this is at room 732 00:47:39,602 --> 00:47:42,693 temperature, by the way, the lower temperature the 733 00:47:42,693 --> 00:47:45,533 resistance is lower. I can also calculate very 734 00:47:45,533 --> 00:47:48,561 roughly what the self-inductance is of that ring. 735 00:47:48,561 --> 00:47:51,905 Now that's not so easy, because here when I calculated 736 00:47:51,905 --> 00:47:55,25 the self-inductance, the magnetic 737 00:47:55,25 --> 00:47:58,227 field was constant. I assumed it was constant, 738 00:47:58,227 --> 00:48:01,072 uniform inside. That's not the case when you 739 00:48:01,072 --> 00:48:03,454 have a ring. You have a dipole field. 740 00:48:03,454 --> 00:48:07,358 However, I just assumed that the magnetic field was the same 741 00:48:07,358 --> 00:48:11,593 everywhere at the surface of the ring -- and with that assumption 742 00:48:11,593 --> 00:48:15,629 admittedly I could be off maybe by twenty or thirty percent -- 743 00:48:15,629 --> 00:48:19,665 with that assumption I find that the self-inductance is ten to 744 00:48:19,665 --> 00:48:22,775 the minus seven Henry. I know what the omega is. 745 00:48:22,775 --> 00:48:26,216 It's three hundred sixty, roughly, and so I find that 746 00:48:26,216 --> 00:48:30,704 omega L over R for this ring is about one 747 00:48:30,704 --> 00:48:33,629 half, run at that frequency omega. 748 00:48:33,629 --> 00:48:38,771 And that gives me a phase angle phi of twenty five degrees, 749 00:48:38,771 --> 00:48:44,178 and therefore the ring is going to be repelled by the coil and 750 00:48:44,178 --> 00:48:48,965 of course the coil is going to be repelled by the ring. 751 00:48:48,965 --> 00:48:53,485 I'll put the ring here, and the ring is supported by 752 00:48:53,485 --> 00:48:56,588 this, so this ring cannot fall over. 753 00:48:56,588 --> 00:49:00,666 The only difference between this 754 00:49:00,666 --> 00:43:02,552 experiment and that one is first of all I can be very 755 00:43:02,552 --> 00:37:38,872 quantitative there, I can actually calculate the 756 00:37:38,872 --> 00:32:08,305 phase angle, where here that's almost impossible. 757 00:32:08,305 --> 00:25:15,097 Here, it is the conductor that I'm going to make levitate and 758 00:25:15,097 --> 00:19:23,87 the coil is stationary. In here it was the conductor 759 00:19:23,87 --> 00:14:13,963 that was stationary, and the coil is floating, 760 00:14:13,963 --> 00:09:24,718 but of course the idea is exactly the same. 761 00:09:24,718 --> 00:03:40,377 And so what I want to do now is make you see there, 762 00:03:40,377 --> 00:00:00 actually, hmmmm, 763 00:00:00 --> 00:49:33 764 00:49:33 --> 00:49:36,709 and we'll have to ... it should come up there . 765 00:49:36,709 --> 0. 766 0. --> 00:49:38,682 yes, there you see this, 767 00:49:38,682 --> 00:49:41,76 this ring. Maybe I should first show you 768 00:49:41,76 --> 00:49:45,47 the whole set up. So this ring goes over here -- 769 00:49:45,47 --> 00:49:50,442 this is an aluminum ring -- and I'm going to make it levitate by 770 00:49:50,442 --> 00:49:55,414 simply running sixty hertz under ten volts through this coil and 771 00:49:55,414 --> 00:49:59,913 I hope I do nothing wrong. Oh no, I have to turn on my AC. 772 00:49:59,913 --> 00:50:04,333 Oh my God, that was not, gee, that 773 00:50:04,333 --> 00:50:07,638 was not my intention. A good thing we don't have a 774 00:50:07,638 --> 00:50:11,147 woman sitting on the ring now. My idea was to have it 775 00:50:11,147 --> 00:50:13,846 levitate. By the way, you did see that it 776 00:50:13,846 --> 00:50:16,14 was repelled, that was quite clear. 777 00:50:16,14 --> 00:50:19,85 I had the current too high. [laughter] I had the current 778 00:50:19,85 --> 00:50:22,549 too high so we'll have it a little lower, 779 00:50:22,549 --> 00:50:26,395 and I will make the current come up very slowly and then I 780 00:50:26,395 --> 00:50:28,621 want you to see that it levitates. 781 00:50:28,621 --> 00:50:30,173 There it is, levitating. 782 00:50:30,173 --> 00:50:32,535 Oh, oh, off the screen. There it is. 783 00:50:32,535 --> 00:50:35,773 And I can turn it over and it's still levitating, 784 00:50:35,773 --> 00:47:32,675 of course. And the secret is this phase 785 00:47:32,675 --> 00:43:14,544 delay introduced by the self-inductance. 786 00:43:14,544 --> 00:36:37,42 I have another ring here which has a, a slot -- also aluminum 787 00:36:37,42 --> 00:33:12,239 -- same ring, but it has a slot. 788 00:33:12,239 --> 00:27:34,684 Well, the EMF in this ring is going to be identical. 789 00:27:34,684 --> 00:21:43,891 There's no difference. In fact, the self-inductance of 790 00:21:43,891 --> 00:15:46,479 this ring is identical, but the resistance of this ring 791 00:15:46,479 --> 00:09:09,355 is huge because there's a slot in there and the resistance is 792 00:09:09,355 --> 00:03:38,418 almost infinitely high. And so if the resistance is 793 00:03:38,418 --> 00:00:00 infinitely high, 794 00:00:00 --> 00:51:11 795 00:51:11 --> 00:44:59,561 no matter what L is and what omega is, phi is going to be 796 00:44:59,561 --> 00:42:46,904 zero. It won't repel. 797 00:42:46,904 --> 00:37:15,263 Half the time it attracts, half the time it repels. 798 00:37:15,263 --> 00:31:30,356 [metallic clanking sound] That means nothing happens, 799 00:31:30,356 --> 00:25:32,183 no magnetic levitation. Same EMF, same self-inductance, 800 00:25:32,183 --> 00:19:53,909 but an infinite resistance and here you see magnetic 801 00:19:53,909 --> 00:15:35,228 levitation. Since the induced current in 802 00:15:35,228 --> 00:09:50,321 the ring is extremely small because of it's very high 803 00:09:50,321 --> 00:03:25,617 resistance, the force on the ring -- whether it's repelling 804 00:03:25,617 --> 00:00:00 or attracting 805 00:00:00 --> 00:51:41 806 00:51:41 --> 01:46:53,431 -- in any case is practically zero. 807 01:46:53,431 --> 03:27:33,748 So that alone is enough reason for the ring not to move at all. 808 03:27:33,748 --> 05:09:51,489 All right, I hope to see all of you tomorrow during our exciting 809 05:09:51,489 --> 309:56 testing of the motors.