1 00:00:00 --> 00:00:00,341 2 00:00:00,341 --> 00:00:05,085 When I expose material to an external magnetic field, 3 00:00:05,085 --> 00:00:10,558 then we learned last time that the field inside that material 4 00:00:10,558 --> 00:00:14,39 is modified. And we expressed that in terms 5 00:00:14,39 --> 00:00:18,039 of an equation, that the field inside the 6 00:00:18,039 --> 00:00:22,692 material is kappa of M, which is called the relative 7 00:00:22,692 --> 00:00:26,159 permeability, times the external field, 8 00:00:26,159 --> 00:00:30,265 and I will refer to that all the 9 00:00:30,265 --> 00:00:34,154 time as the vacuum field. And when we have diamagnetic 10 00:00:34,154 --> 00:00:37,749 material, kappa M is just a hair smaller than one; 11 00:00:37,749 --> 00:00:41,859 with paramagnetic material, it is a hair larger than one; 12 00:00:41,859 --> 00:00:45,822 but when we have ferromagnetic material it can be huge. 13 00:00:45,822 --> 00:00:48,317 It can be thousands, ten thousands, 14 00:00:48,317 --> 00:00:51,545 and even higher. Now in the case of para- and 15 00:00:51,545 --> 00:00:55,508 ferromagnetic material, the kappa of M is the result of 16 00:00:55,508 --> 00:00:59,544 the fact that the intrinsic dipoles of the atoms and the 17 00:00:59,544 --> 00:01:03,669 molecules are going to be aligned by the 18 00:01:03,669 --> 00:01:06,923 external field. And today I want to raise the 19 00:01:06,923 --> 00:01:10,917 question, how large can the magnetic dipole moment of a 20 00:01:10,917 --> 00:01:13,949 single atom be? And then comes the logical 21 00:01:13,949 --> 00:01:17,056 question, how strong can we actually, then, 22 00:01:17,056 --> 00:01:20,162 have a field inside ferromagnetic material? 23 00:01:20,162 --> 00:01:24,6 That means if we were able to align all the dipole moments of 24 00:01:24,6 --> 00:01:27,929 all the atoms, what is the maximum that we can 25 00:01:27,929 --> 00:01:30,369 achieve? To calculate the magnetic 26 00:01:30,369 --> 00:01:33,307 dipole moment of an atom, 27 00:01:33,307 --> 00:01:37,116 you have to do some quantum mechanics, and that's beyond the 28 00:01:37,116 --> 00:01:40,28 scope of this course. And so I will derive it in a 29 00:01:40,28 --> 00:01:43,185 classical way, and then at the very end I will 30 00:01:43,185 --> 00:01:46,801 add a little pepper and salt, which is quantum mechanics, 31 00:01:46,801 --> 00:01:50,223 just to make the result right. But it can be done in a 32 00:01:50,223 --> 00:01:52,806 classical way, and it can give you a very 33 00:01:52,806 --> 00:01:55,518 good, good idea. If I have a hydrogen atom, 34 00:01:55,518 --> 00:01:59,198 which has a proton at the center, has a charge plus E -- E 35 00:01:59,198 --> 00:02:02,878 is the charge of the electron but this is the plus charge. 36 00:02:02,878 --> 00:02:07,122 And let this have an orbit R, circular orbit. 37 00:02:07,122 --> 00:02:10,942 And the electron here E, I'll give it the minus sign to 38 00:02:10,942 --> 00:02:13,912 make sure that you know that it's negative. 39 00:02:13,912 --> 00:02:17,166 Say the electron goes around in this direction. 40 00:02:17,166 --> 00:02:19,713 This is the velocity of the electron. 41 00:02:19,713 --> 00:02:23,957 That means that of course the current around the proton would 42 00:02:23,957 --> 00:02:27,918 then be in this direction. If an electron goes like this, 43 00:02:27,918 --> 00:02:31,667 the current goes like that, that's just by convention. 44 00:02:31,667 --> 00:02:35,841 The mass of an electron -- you should know that by now -- is 45 00:02:35,841 --> 00:02:40,743 approximately nine point one times ten to the minus 46 00:02:40,743 --> 00:02:44,34 thirty-one kilograms. The charge of the electron, 47 00:02:44,34 --> 00:02:48,386 one point six times ten to the minus nineteen coulombs. 48 00:02:48,386 --> 00:02:52,807 And the radius of the orbit in a hydrogen atom -- it's often 49 00:02:52,807 --> 00:02:56,779 called the Bohr radius, by the way -- is approximately 50 00:02:56,779 --> 00:02:59,851 five times ten to the minus eleven meters. 51 00:02:59,851 --> 00:03:02,324 We're going to need these numbers. 52 00:03:02,324 --> 00:03:05,021 That's why I write them down for you. 53 00:03:05,021 --> 00:03:10,266 If you look at this current running around the proton, 54 00:03:10,266 --> 00:03:13,582 it's really a current which the current, say, 55 00:03:13,582 --> 00:03:17,275 goes in this direction. And here is that proton -- 56 00:03:17,275 --> 00:03:20,441 trying to make you see three dimensionally. 57 00:03:20,441 --> 00:03:24,209 Then it creates a magnetic field in this direction, 58 00:03:24,209 --> 00:03:27,374 and so the magnetic dipole moment mu is up. 59 00:03:27,374 --> 00:03:30,991 And the magnitude of that magnetic dipole moment, 60 00:03:30,991 --> 00:03:35,061 as we learned last time, is simply this current I times 61 00:03:35,061 --> 00:03:40,562 the area A of this current loop. Now the area A is trivial 62 00:03:40,562 --> 00:03:43,36 to calculate. That's pi R square, 63 00:03:43,36 --> 00:03:46,071 R being the radius of the orbit. 64 00:03:46,071 --> 00:03:50,005 And so A, that's the easiest, is pi R squared. 65 00:03:50,005 --> 00:03:54,376 And if I use my five times ten to the minus eleven, 66 00:03:54,376 --> 00:03:59,36 then I find that this area is eight times ten to the minus 67 00:03:59,36 --> 00:04:02,857 twenty-one square meters. So that's easy. 68 00:04:02,857 --> 00:04:06,092 But now comes the question, what is I? 69 00:04:06,092 --> 00:04:11,163 What is the current? So now we have to do a 70 00:04:11,163 --> 00:04:14,466 little bit more work. And we have to combine our 71 00:04:14,466 --> 00:04:17,066 knowledge of, uh eight oh two with our 72 00:04:17,066 --> 00:04:20,931 knowledge of eight oh one. If this electron goes around, 73 00:04:20,931 --> 00:04:24,796 the reason why it goes around is that the proton and the 74 00:04:24,796 --> 00:04:28,943 electron attract each other. And so there is a force in this 75 00:04:28,943 --> 00:04:31,262 direction. And we know that force, 76 00:04:31,262 --> 00:04:34,635 that's the Coulomb force. It's an electric force. 77 00:04:34,635 --> 00:04:38,5 That force is this charge times this charge, so that's E 78 00:04:38,5 --> 00:04:42,365 squared, divided by our famous four pi 79 00:04:42,365 --> 00:04:45,559 epsilon zero, and then we have to divide it 80 00:04:45,559 --> 00:04:49,057 by the radius squared. So that's Coulomb's Law. 81 00:04:49,057 --> 00:04:52,631 But from eight oh one, from Newtonian mechanics, 82 00:04:52,631 --> 00:04:57,118 we know that this is what we call the centripetal force that 83 00:04:57,118 --> 00:04:59,399 holds it in orbit, so to speak, 84 00:04:59,399 --> 00:05:02,973 and that is M V squared, M being the mass of the 85 00:05:02,973 --> 00:05:06,243 electron, V being the speed of the electron, 86 00:05:06,243 --> 00:05:10,121 and V squared divided by R. And so this allows me to 87 00:05:10,121 --> 00:05:13,399 calculate as a first step, 88 00:05:13,399 --> 00:05:17,95 before we get into the current, what the velocity of this 89 00:05:17,95 --> 00:05:20,225 electron is. It's phenomenal. 90 00:05:20,225 --> 00:05:24,776 It's an incredible speed. So V then becomes -- I lose one 91 00:05:24,776 --> 00:05:29,408 R -- so I get the square root, I get an E squared upstairs 92 00:05:29,408 --> 00:05:33,878 here, my M goes downstairs, I have four pi epsilon zero, 93 00:05:33,878 --> 00:05:37,941 and I have here an R. And I know all these numbers. 94 00:05:37,941 --> 00:05:41,354 I know what E is, I know what capital R is, 95 00:05:41,354 --> 00:05:45,742 I know what four pi epsilon zero is -- 96 00:05:45,742 --> 00:05:49,944 one over four pi epsilon zero is the famous nine to the power 97 00:05:49,944 --> 00:05:52,395 -- nine times ten to the power nine. 98 00:05:52,395 --> 00:05:54,636 And so I can calculate what V is. 99 00:05:54,636 --> 00:05:58,908 And if I stick in the numbers and if I did not make a mistake, 100 00:05:58,908 --> 00:06:02,62 then I find about two point three times ten to the six 101 00:06:02,62 --> 00:06:05,911 meters per second. It's an immensely high speed, 102 00:06:05,911 --> 00:06:09,903 five million miles per hour. If this were a straight line, 103 00:06:09,903 --> 00:06:13,124 you would make it to the moon in three minutes. 104 00:06:13,124 --> 00:06:17,046 Five million miles per hour this electron goes around the 105 00:06:17,046 --> 00:06:20,256 proton. Now I have to go to the 106 00:06:20,256 --> 00:06:22,37 current. I have to find out what the 107 00:06:22,37 --> 00:06:24,847 current is. So the question that I'm going 108 00:06:24,847 --> 00:06:28,351 to ask now is how long does it take for this electron to go 109 00:06:28,351 --> 00:06:29,741 around. Well, that time, 110 00:06:29,741 --> 00:06:32,943 capital T, is of course the circumference of my circle 111 00:06:32,943 --> 00:06:35,117 divided by the speed of the electron. 112 00:06:35,117 --> 00:06:37,353 Trivial. Even the high school students 113 00:06:37,353 --> 00:06:40,192 in my audience will understand that one, I hope. 114 00:06:40,192 --> 00:06:43,817 And so I know what two pi R is, because I know R and I know V 115 00:06:43,817 --> 00:06:47,2 and so I can calculate that time, just by sticking in the 116 00:06:47,2 --> 00:06:50,946 numbers. And I find that it is about one 117 00:06:50,946 --> 00:06:55,04 point one four times ten to the minus sixteen seconds. 118 00:06:55,04 --> 00:06:57,743 Just imagine how small that time is. 119 00:06:57,743 --> 00:07:02,069 You cannot even -- we cannot even imagine what it's like. 120 00:07:02,069 --> 00:07:05,932 It goes ten to the sixteen times per second around, 121 00:07:05,932 --> 00:07:10,49 because it has this huge speed. The one point one four times 122 00:07:10,49 --> 00:07:15,279 ten to the minus sixteen really should have been one point four 123 00:07:15,279 --> 00:07:20,069 times ten to the minus sixteen. Of course, it doesn't make much 124 00:07:20,069 --> 00:07:24,704 difference, but in case you substitute in the 125 00:07:24,704 --> 00:07:29,264 numbers, it is one point four times ten to the minus sixteen. 126 00:07:29,264 --> 00:07:34,053 Now, we still haven't found the current, but we're almost there. 127 00:07:34,053 --> 00:07:38,234 Because when you look here, there is this electron going 128 00:07:38,234 --> 00:07:42,491 by, and every one point one four ten to the minus sixteen 129 00:07:42,491 --> 00:07:46,139 seconds, that electron goes by. So the current I, 130 00:07:46,139 --> 00:07:50,32 that's the definition of current, is the charge per unit 131 00:07:50,32 --> 00:07:53,057 time. And so every capital T seconds, 132 00:07:53,057 --> 00:07:56,934 the charge E goes by, and so this is 133 00:07:56,934 --> 00:08:00,846 per definition the current. And so this current, 134 00:08:00,846 --> 00:08:04,676 then, that you have, which is simply due to the 135 00:08:04,676 --> 00:08:09,255 electron going around the proton, is about one point one 136 00:08:09,255 --> 00:08:12,252 times ten to the minus three amperes. 137 00:08:12,252 --> 00:08:15,583 And that is mind-boggling. A milliampere. 138 00:08:15,583 --> 00:08:20,495 One electron going around a proton represents a current of a 139 00:08:20,495 --> 00:08:23,825 milliampere. And now of course I have the 140 00:08:23,825 --> 00:08:28,238 magnetic moment mu, that is I times A. 141 00:08:28,238 --> 00:08:31,881 We already calculated A, and now we also have the 142 00:08:31,881 --> 00:08:36,208 current I, and so we now get that mu is approximately nine 143 00:08:36,208 --> 00:08:40,23 point three, if you put in all the decimals correctly, 144 00:08:40,23 --> 00:08:44,253 times ten to the minus twenty-four, and the unit is of 145 00:08:44,253 --> 00:08:47,441 course amperes square meters. This is area, 146 00:08:47,441 --> 00:08:51,161 and this is current. This A has nothing to do with 147 00:08:51,161 --> 00:08:53,286 that A, hey. This is amperes. 148 00:08:53,286 --> 00:08:56,095 Be careful. And this is square meters. 149 00:08:56,095 --> 00:08:59,434 But these are the units. And this has a name. 150 00:08:59,434 --> 00:09:03,381 This is called the Bohr magneton. 151 00:09:03,381 --> 00:09:07,292 Bohr magneton. What we cannot understand with 152 00:09:07,292 --> 00:09:11,38 our knowledge now, but you can if you ever take 153 00:09:11,38 --> 00:09:15,735 quantum mechanics, that the magnetic moment of all 154 00:09:15,735 --> 00:09:20,712 electrons in orbit can only be a multiple of this number, 155 00:09:20,712 --> 00:09:24 nothing in between. Quantum mechanics, 156 00:09:24 --> 00:09:26,844 the word says it is quantization. 157 00:09:26,844 --> 00:09:29,954 It's not in between. It's either or. 158 00:09:29,954 --> 00:09:34,982 It includes even zero, which is even harder to 159 00:09:34,982 --> 00:09:37,855 understand, that it could even be zero. 160 00:09:37,855 --> 00:09:42,543 In addition to a dipole moment due to the electron going around 161 00:09:42,543 --> 00:09:47,08 the proton, the electron itself is a charge which spins about 162 00:09:47,08 --> 00:09:50,029 its own axis, and that also means that a 163 00:09:50,029 --> 00:09:54,566 charge is going around on the spinning scale of the electron. 164 00:09:54,566 --> 00:09:58,498 And that magnetic dipole moment is always this value. 165 00:09:58,498 --> 00:10:03,11 And so the net magnetic dipole moment of an atom or a molecule 166 00:10:03,11 --> 00:10:07,118 is now the vectorial sum of all these 167 00:10:07,118 --> 00:10:10,081 dipole moments, all these electrons going 168 00:10:10,081 --> 00:10:14,23 around, means orbital dipole moments, and you have to add 169 00:10:14,23 --> 00:10:17,786 these spin dipoles. Some of these pair each other 170 00:10:17,786 --> 00:10:20,083 out. One electron would have its 171 00:10:20,083 --> 00:10:24,75 dipole moment in this direction and the other in this direction, 172 00:10:24,75 --> 00:10:27,269 and then the vectorial sum is zero. 173 00:10:27,269 --> 00:10:31,566 The net result is that most atoms and molecules have dipole 174 00:10:31,566 --> 00:10:35,493 moments which are either one Bohr magneton or two Bohr 175 00:10:35,493 --> 00:10:37,716 magnetons. That is very common. 176 00:10:37,716 --> 00:10:41,864 And that's what I will need today to discuss with you how 177 00:10:41,864 --> 00:10:47,071 strong a field we can create if we align all 178 00:10:47,071 --> 00:10:51,488 those magnetic dipoles. The magnetic field that is 179 00:10:51,488 --> 00:10:56,625 produced inside a material when I expose it to an external 180 00:10:56,625 --> 00:11:01,853 field, that magnetic field B is the vacuum field that I can 181 00:11:01,853 --> 00:11:07,171 create with a solenoid -- we will discuss that further today 182 00:11:07,171 --> 00:11:12,218 -- plus the field which I will call B prime which is that 183 00:11:12,218 --> 00:11:17,807 magnetic field that is the result of the fact that 184 00:11:17,807 --> 00:11:20,312 we're going to align these dipoles. 185 00:11:20,312 --> 00:11:23,775 The external field wants to align these dipoles, 186 00:11:23,775 --> 00:11:27,828 and the degree of success depends on the strength of the 187 00:11:27,828 --> 00:11:31,291 external field and of course on the temperature. 188 00:11:31,291 --> 00:11:35,123 If the temperature is low, it's easier to align them, 189 00:11:35,123 --> 00:11:37,997 because there is less thermal agitation. 190 00:11:37,997 --> 00:11:42,05 If, and that's a big if -- today you will see why it's a 191 00:11:42,05 --> 00:11:46,25 big if -- if B prime is linearly proportional to B vacuum, 192 00:11:46,25 --> 00:11:49,86 if that is the case -- today you 193 00:11:49,86 --> 00:11:54,995 will see that there are situations where that's not the 194 00:11:54,995 --> 00:12:00,605 case -- then I can write down that B prime equals xi of M -- 195 00:12:00,605 --> 00:12:06,595 we called that last lecture the magnetic susceptibility -- times 196 00:12:06,595 --> 00:12:09,923 B vacuum. The linear proportionality 197 00:12:09,923 --> 00:12:12,395 constant. If I can do that, 198 00:12:12,395 --> 00:12:17,72 then of course B is also proportional to B vacuum because 199 00:12:17,72 --> 00:12:22,664 now I can write down that B is one plus 200 00:12:22,664 --> 00:12:26,787 xi M times B vacuum. And that, for that we write 201 00:12:26,787 --> 00:12:31,523 kappa of M times B vacuum, which is the equation that I 202 00:12:31,523 --> 00:12:35,383 started out with today. And so that is only a 203 00:12:35,383 --> 00:12:40,558 meaningful equation if the sum of the alignment of all these 204 00:12:40,558 --> 00:12:45,908 dipoles can be written as being linearly proportional with the 205 00:12:45,908 --> 00:12:49,505 external field. And this is what I want to 206 00:12:49,505 --> 00:12:55,186 explore today in more detail. With paramagnetic material, 207 00:12:55,186 --> 00:12:59,328 there is never any worry that the linearity doesn't hold. 208 00:12:59,328 --> 00:13:03,248 But with ferromagnetic material, that is not the case, 209 00:13:03,248 --> 00:13:07,464 because with ferromagnetic material, it is relatively easy 210 00:13:07,464 --> 00:13:11,31 to align these dipoles, because they already group in 211 00:13:11,31 --> 00:13:15,6 domains, as we discussed last time, and the domains flip in 212 00:13:15,6 --> 00:13:17,967 unison. And so with ferromagnetic 213 00:13:17,967 --> 00:13:22,627 material, as you will see today, we can actually go into what we 214 00:13:22,627 --> 00:13:25,924 call saturation, that all the 215 00:13:25,924 --> 00:13:29,029 dipoles are aligned in the same direction. 216 00:13:29,029 --> 00:13:33,194 And now the question is, how strong would that field be? 217 00:13:33,194 --> 00:13:37,207 I'm going to make a rough calculation that gives you a 218 00:13:37,207 --> 00:13:39,857 pretty good feeling for the numbers. 219 00:13:39,857 --> 00:13:42,583 It depends on what material you have. 220 00:13:42,583 --> 00:13:46,445 I will choose a material whereby the magnetic dipole 221 00:13:46,445 --> 00:13:50,837 moment is two Bohr magnetons, so this is -- I told you it's 222 00:13:50,837 --> 00:13:55,23 either one or two or three, I pick one for 223 00:13:55,23 --> 00:13:58,85 which it is two. And I have them all aligned. 224 00:13:58,85 --> 00:14:02,8 So I take the situation that they're all aligned. 225 00:14:02,8 --> 00:14:06,668 So here is the current going around the nucleus, 226 00:14:06,668 --> 00:14:09,795 here's another one, here's another one, 227 00:14:09,795 --> 00:14:13,415 here's another one. This is a solid material, 228 00:14:13,415 --> 00:14:17,612 so these atoms or these molecules are nicely packed. 229 00:14:17,612 --> 00:14:21,48 And here we see all these currents going around, 230 00:14:21,48 --> 00:14:27,404 and all these magnetic dipole moments are nicely aligned. 231 00:14:27,404 --> 00:14:31,086 And so these magnetic fields are supporting each other. 232 00:14:31,086 --> 00:14:34,495 And the question now is, what is the magnetic field 233 00:14:34,495 --> 00:14:36,677 inside here? Well, that's an easy 234 00:14:36,677 --> 00:14:40,359 calculation, because this really looks like a solenoid, 235 00:14:40,359 --> 00:14:43,836 like you have windings, and you have a current going 236 00:14:43,836 --> 00:14:45,472 around. And you remember, 237 00:14:45,472 --> 00:14:48,813 or should remember, that if we have a solenoid and 238 00:14:48,813 --> 00:14:52,767 we run a current through a solenoid that the magnetic field 239 00:14:52,767 --> 00:14:57,88 in the solenoid is mu zero -- this mu zero is not this mu, 240 00:14:57,88 --> 00:15:00,803 this mu zero is the same one that -- oh no, 241 00:15:00,803 --> 00:15:03,795 it's no- we don't have it on the blackboard. 242 00:15:03,795 --> 00:15:07,761 You know, that's the famous four pi times ten to the minus 243 00:15:07,761 --> 00:15:10,335 seven. And then we have the current I, 244 00:15:10,335 --> 00:15:13,258 and then we have N, if that's the number of 245 00:15:13,258 --> 00:15:17,224 windings that we have in the solenoid, and then we have L, 246 00:15:17,224 --> 00:15:19,659 which is the length of the solenoid. 247 00:15:19,659 --> 00:15:23,277 So this is the number of windings of the solenoid per 248 00:15:23,277 --> 00:15:26,408 unit length, the number of windings per meter. 249 00:15:26,408 --> 00:15:29,748 So if we could figure out, with this arrangement, 250 00:15:29,748 --> 00:15:32,743 what this quantity is, 251 00:15:32,743 --> 00:15:35,842 then we're in business. I take a material, 252 00:15:35,842 --> 00:15:39,999 which is not unreasonable, whereby the number density of 253 00:15:39,999 --> 00:15:44,383 atoms, I call that capital N, written in the subscript way, 254 00:15:44,383 --> 00:15:46,726 is about ten to the twenty-nine. 255 00:15:46,726 --> 00:15:50,883 So this is atoms or molecules, whatever may be the case, 256 00:15:50,883 --> 00:15:53,907 per cubic meter. That's not unreasonable. 257 00:15:53,907 --> 00:15:57,535 And now I have to somehow manipulate, massage the 258 00:15:57,535 --> 00:16:02,373 mathematics, so that I get in here this magnetic 259 00:16:02,373 --> 00:16:06,405 moment, this Bohr magneton. The two Bohr magnetons. 260 00:16:06,405 --> 00:16:09,631 And there are several ways of doing that. 261 00:16:09,631 --> 00:16:13,421 I have chosen one way, and that's the following. 262 00:16:13,421 --> 00:16:16,083 I take here a length of one meter. 263 00:16:16,083 --> 00:16:19,954 So this is a solenoid, and I take only one meter. 264 00:16:19,954 --> 00:16:24,631 Could have taken three or five meters, makes no difference. 265 00:16:24,631 --> 00:16:28,18 I take one meter. And each one of these loops 266 00:16:28,18 --> 00:16:31,164 here has an area A. So we would agree, 267 00:16:31,164 --> 00:16:36,127 I hope, that the area -- that the volume, 268 00:16:36,127 --> 00:16:42,337 the volume of this solenoid -- this has a length one meter -- 269 00:16:42,337 --> 00:16:46,995 that that volume is A square meters times one. 270 00:16:46,995 --> 00:16:50,618 And so the volume is A cubic meters. 271 00:16:50,618 --> 00:16:54,241 A times one meter is A cubic meters. 272 00:16:54,241 --> 00:16:59,623 But the number of atoms per cubic meter is ten to the 273 00:16:59,623 --> 00:17:04,178 twenty-ninth, and so the number of atoms that 274 00:17:04,178 --> 00:17:08,587 I have in this solenoid per meter 275 00:17:08,587 --> 00:17:12,917 is this A times that N. So this is the number of 276 00:17:12,917 --> 00:17:18,352 windings, if I call this one winding, the number of windings 277 00:17:18,352 --> 00:17:21,669 per meter. Or you can think of it the 278 00:17:21,669 --> 00:17:26,368 number of atoms per meter, the way they're lined up. 279 00:17:26,368 --> 00:17:31,158 And so now I am in business, because this now is my N 280 00:17:31,158 --> 00:17:35,12 divided by L. And so I can write now mu zero 281 00:17:35,12 --> 00:17:39,88 times the current I times that area A 282 00:17:39,88 --> 00:17:43,764 times N, which is ten to the twenty-nine. 283 00:17:43,764 --> 00:17:47,842 But look now. Now you see why I did it this 284 00:17:47,842 --> 00:17:53,376 way, because I times A is the magnetic dipole moment of my 285 00:17:53,376 --> 00:17:55,998 atoms. And that was two Bohr 286 00:17:55,998 --> 00:17:59,882 magnetons. And so this now also equals mu 287 00:17:59,882 --> 00:18:02,989 zero times twice mu Bohr times N. 288 00:18:02,989 --> 00:18:09,105 And I'm finished, because I know what mu zero is, 289 00:18:09,105 --> 00:18:12,438 and I know what moo- mu Bohr is -- we calculated that, 290 00:18:12,438 --> 00:18:15,33 it's still here, this number -- and I know what 291 00:18:15,33 --> 00:18:18,725 my capital subscript N is, ten to the twenty-nine atoms 292 00:18:18,725 --> 00:18:21,303 per cubic meter. And so I just shove those 293 00:18:21,303 --> 00:18:24,196 numbers in my equation, and I find that this is 294 00:18:24,196 --> 00:18:27,78 approximately two point three tesla for the numbers that I 295 00:18:27,78 --> 00:18:30,483 have chosen. It's not for all materials this 296 00:18:30,483 --> 00:18:34,381 way, because I have adopted two Bohr magnetons for the magnetic 297 00:18:34,381 --> 00:18:38,028 dipole moment of each atom, and I have adopted a density of 298 00:18:38,028 --> 00:18:41,898 ten to the twenty-nine atoms per 299 00:18:41,898 --> 00:18:45,007 cubic meter. And for that situation, 300 00:18:45,007 --> 00:18:50,247 you get two point three tesla. Now I want to use this number 301 00:18:50,247 --> 00:18:54,954 and understand what's going to happen in ferromagnetic 302 00:18:54,954 --> 00:18:58,329 material. I take ferromagnetic material 303 00:18:58,329 --> 00:19:03,391 and I expose it to an external field, I call it the vacuum 304 00:19:03,391 --> 00:19:06,678 field. So I stick it in a solenoid and 305 00:19:06,678 --> 00:19:11,568 I can choose the current through my solenoid. 306 00:19:11,568 --> 00:19:15,055 And I'm going to plot for you the vacuum field. 307 00:19:15,055 --> 00:19:19,299 This vacuum field is linearly proportional to the current 308 00:19:19,299 --> 00:19:22,785 through my solenoid. This is an actual physical 309 00:19:22,785 --> 00:19:24,528 solenoid. I have a wire, 310 00:19:24,528 --> 00:19:28,394 it goes around like this. I don't have one here now. 311 00:19:28,394 --> 00:19:30,895 And I run a current through there. 312 00:19:30,895 --> 00:19:35,518 And that vacuum B field equals mu zero times I times N divided 313 00:19:35,518 --> 00:19:39,838 by L, except that this N divided by L is now the number of 314 00:19:39,838 --> 00:19:45,012 windings of my current wire, and this is the length of my 315 00:19:45,012 --> 00:19:47,242 solenoid. So don't confuse it with the 316 00:19:47,242 --> 00:19:50,015 one we had there, because that was on an atomic 317 00:19:50,015 --> 00:19:52,426 scale and this is on a macroscopic scale. 318 00:19:52,426 --> 00:19:55,861 How many windings you have could be -- where you had a big 319 00:19:55,861 --> 00:19:58,754 one here in class, twenty-eight hundred windings, 320 00:19:58,754 --> 00:20:02,19 and we had sixty centimeters long, and so that's what this 321 00:20:02,19 --> 00:20:05,204 number is all about. So the moment that I know what 322 00:20:05,204 --> 00:20:08,398 the current is through my solenoid, I immediately know 323 00:20:08,398 --> 00:20:11,05 what my vacuum field is. There's a one-to-one 324 00:20:11,05 --> 00:20:13,642 correspondence. And now I'm going to measure 325 00:20:13,642 --> 00:20:17,365 inside the ferromagnetic material, 326 00:20:17,365 --> 00:20:21,889 which I stick in the solenoid. I'm going to measure there the 327 00:20:21,889 --> 00:20:25,358 magnetic field. And what's going to happen now? 328 00:20:25,358 --> 00:20:29,429 Well, first of all I cannot at all plot this curve on a 329 00:20:29,429 --> 00:20:32,973 one-to-one scale, the reason being that kappa of 330 00:20:32,973 --> 00:20:37,723 M for ferromagnetic material is so large -- let us adopt for now 331 00:20:37,723 --> 00:20:39,91 a number of one thousand, say. 332 00:20:39,91 --> 00:20:41,945 Or it could be larger, even. 333 00:20:41,945 --> 00:20:46,055 Let's say kappa of M is one thousand. 334 00:20:46,055 --> 00:20:51,836 That means that if the magnetic field in terms of a vector is 335 00:20:51,836 --> 00:20:56,846 this big, one centimeter in the length of the vector, 336 00:20:56,846 --> 00:21:01,856 that the field inside the ferromagnetic material is a 337 00:21:01,856 --> 00:21:06,674 thousand times higher. If this is one centimeter in 338 00:21:06,674 --> 00:21:11,01 length, a thousand times higher is ten meters. 339 00:21:11,01 --> 00:21:15,538 So this is the length of the vector 340 00:21:15,538 --> 00:21:20,007 of the magnetic field inside the ferromagnetic material. 341 00:21:20,007 --> 00:21:22,77 That's why I cannot do it to scale. 342 00:21:22,77 --> 00:21:27,563 So when I draw this line here, keep in mind that if I did it 343 00:21:27,563 --> 00:21:30,082 to scale, if I did, but I can't, 344 00:21:30,082 --> 00:21:33,738 then the tangent of alpha would be kappa of M, 345 00:21:33,738 --> 00:21:37,313 which in this case would be ten to the third. 346 00:21:37,313 --> 00:21:42,025 And so this angle alpha is something like eighty-nine point 347 00:21:42,025 --> 00:21:47,387 nine something degrees. So I cannot do it to scale. 348 00:21:47,387 --> 00:21:50,903 Keep that in mind. So in the beginning I get a 349 00:21:50,903 --> 00:21:54,732 nice linear curve, but now slowly I'm beginning to 350 00:21:54,732 --> 00:21:58,091 reach saturation, that all these dipoles are 351 00:21:58,091 --> 00:22:02,076 going to be aligned, and what you're going to see is 352 00:22:02,076 --> 00:22:06,451 that this curve bends over and bends over and bends over, 353 00:22:06,451 --> 00:22:10,982 and the magnetic field that you finally achieve here is the 354 00:22:10,982 --> 00:22:15,357 famous two point three tesla, which I calculated for that 355 00:22:15,357 --> 00:22:17,936 imaginary material, plus B vacuum. 356 00:22:17,936 --> 00:22:22,077 The two point three is now the field 357 00:22:22,077 --> 00:22:25,977 which I will call B prime. This is the field that is the 358 00:22:25,977 --> 00:22:29,098 result of the alignment of all those dipoles. 359 00:22:29,098 --> 00:22:31,864 And so when I increase the vacuum field, 360 00:22:31,864 --> 00:22:35,907 this goes into saturation, and settles for two point three 361 00:22:35,907 --> 00:22:39,666 and can no longer increase, because I have aligned all 362 00:22:39,666 --> 00:22:43,425 these magnetic dipoles. And so if your vacuum field is 363 00:22:43,425 --> 00:22:47,325 this strong, then this field is no longer thousand times 364 00:22:47,325 --> 00:22:51,652 stronger than the vacuum field. You're no longer in the linear 365 00:22:51,652 --> 00:22:54,063 part. So you could also think of it 366 00:22:54,063 --> 00:22:57,647 as kappa of M being smaller than a 367 00:22:57,647 --> 00:23:00,079 thousand. Whichever way you prefer is 368 00:23:00,079 --> 00:23:02,511 fine. But it's no longer proportional 369 00:23:02,511 --> 00:23:06,497 to the value of one thousand. If the temperature is lower of 370 00:23:06,497 --> 00:23:09,132 the material, it's easier to align them, 371 00:23:09,132 --> 00:23:11,97 and so you will achieve saturation earlier, 372 00:23:11,97 --> 00:23:14,402 and so your curve would go like this. 373 00:23:14,402 --> 00:23:17,509 So the curve is also a function of temperature. 374 00:23:17,509 --> 00:23:21,36 So this is if the temperature is low relative to this one. 375 00:23:21,36 --> 00:23:25,954 So these curves depend on, on temperature as well. 376 00:23:25,954 --> 00:23:30,615 The lower the temperature, the easier it is to align them. 377 00:23:30,615 --> 00:23:34,866 If I reach this point here, when my B prime goes into 378 00:23:34,866 --> 00:23:39,772 saturation, I can only increase the field, the B field in the 379 00:23:39,772 --> 00:23:43,042 material, by increasing the vacuum field, 380 00:23:43,042 --> 00:23:46,558 because B prime is not going to go up again. 381 00:23:46,558 --> 00:23:51,627 And so I can only get a higher field by increasing this current 382 00:23:51,627 --> 00:23:55,797 so that this B, this B vacuum goes up. 383 00:23:55,797 --> 00:24:00,304 And that goes up very slowly, because this huge magnification 384 00:24:00,304 --> 00:24:02,858 factor of one thousand is gone now. 385 00:24:02,858 --> 00:24:04,887 So the slow, it's very slow, 386 00:24:04,887 --> 00:24:09,319 the growth, and that's why you see that I drew it like this, 387 00:24:09,319 --> 00:24:13,602 that it increases very slowly. But my plot is not to scale 388 00:24:13,602 --> 00:24:16,381 anyhow. Now I want to discuss with you 389 00:24:16,381 --> 00:24:20,588 what happens with the material once I have driven it into 390 00:24:20,588 --> 00:24:23,518 saturation. What happens if now I change 391 00:24:23,518 --> 00:24:28,704 the current and I make my vacuum field zero again? 392 00:24:28,704 --> 00:24:32,155 And now you get a very unusual behavior. 393 00:24:32,155 --> 00:24:35,429 Let me do that here on the blackboard. 394 00:24:35,429 --> 00:24:40,208 So I'll make a new drawing. I could have continued with 395 00:24:40,208 --> 00:24:43,305 that one, but let me make a new one. 396 00:24:43,305 --> 00:24:48,084 So I'm going to do the following experiment in my head. 397 00:24:48,084 --> 00:24:53,305 I have a solenoid and I run a current through that solenoid. 398 00:24:53,305 --> 00:24:58,526 And if the current is in clockwise direction, 399 00:24:58,526 --> 00:25:01,154 my vacuum field will be in this direction. 400 00:25:01,154 --> 00:25:04,552 And when the current is in counterclockwise direction, 401 00:25:04,552 --> 00:25:08,078 I will assume that my vacuum field is in this direction. 402 00:25:08,078 --> 00:25:10,77 So there is ferromagnetic material in here. 403 00:25:10,77 --> 00:25:13,527 If the current flows in clockwise direction, 404 00:25:13,527 --> 00:25:15,899 the vacuum field is in this direction. 405 00:25:15,899 --> 00:25:19,81 If I run it in counterclockwise direction, the vacuum field is 406 00:25:19,81 --> 00:25:22,63 in that direction. And here is going to be my 407 00:25:22,63 --> 00:25:25,323 vacuum field. Easy for me to know what that 408 00:25:25,323 --> 00:25:28,656 is, because if I know the current 409 00:25:28,656 --> 00:25:31,84 through my solenoid, this equation will immediately 410 00:25:31,84 --> 00:25:33,877 tell me what the vacuum field is. 411 00:25:33,877 --> 00:25:37,06 So I have never any problems with the vacuum field. 412 00:25:37,06 --> 00:25:40,562 I stick a probe in here and I measure the magnetic field 413 00:25:40,562 --> 00:25:43,808 inside that material. How we do that is not so easy, 414 00:25:43,808 --> 00:25:46,737 but we can do it. There are a few things that I 415 00:25:46,737 --> 00:25:48,965 can't tell you. This is one of them. 416 00:25:48,965 --> 00:25:52,085 So here is the magnetic field inside the material. 417 00:25:52,085 --> 00:25:55,713 All right, so there we start. We do the same thing that we 418 00:25:55,713 --> 00:25:59,47 did here. So we approach the saturation. 419 00:25:59,47 --> 00:26:02,712 But now when I'm here, I am reducing the current and 420 00:26:02,712 --> 00:26:05,699 go back to zero. Remember that when we are here, 421 00:26:05,699 --> 00:26:09,069 all these domains that we discussed last time have all 422 00:26:09,069 --> 00:26:11,866 flipped in the direction of the vacuum field. 423 00:26:11,866 --> 00:26:15,362 So this field is enormous. But now I make the current go 424 00:26:15,362 --> 00:26:18,032 back to zero. And what happens now is I end 425 00:26:18,032 --> 00:26:21,465 up here, at this point P here. The current now is zero. 426 00:26:21,465 --> 00:26:24,389 There is no current going through the solenoid. 427 00:26:24,389 --> 00:26:26,36 Notice the vacuum field is zero. 428 00:26:26,36 --> 00:26:31,127 I can take the material out, the ferromagnetic material. 429 00:26:31,127 --> 00:26:33,944 The material itself is now magnetic. 430 00:26:33,944 --> 00:26:37,726 And you see there is a magnetic field inside it. 431 00:26:37,726 --> 00:26:41,025 Why is that? Because some of those domains 432 00:26:41,025 --> 00:26:44,002 remain aligned, so they don't go back. 433 00:26:44,002 --> 00:26:47,381 And so we have created permanent magnetism. 434 00:26:47,381 --> 00:26:50,68 And so in the location, at the location P, 435 00:26:50,68 --> 00:26:53,657 we have B vacuum is zero, but B prime, 436 00:26:53,657 --> 00:26:57,922 which is the result of those aligned magnetic moments, 437 00:26:57,922 --> 00:27:02,106 is still in this direction. Nothing is to scale here, 438 00:27:02,106 --> 00:27:05,98 of course. And so you still have a 439 00:27:05,98 --> 00:27:08,726 magnetic field. Now I reverse the current. 440 00:27:08,726 --> 00:27:11,94 I go counterclockwise. So I'm creating a magnetic 441 00:27:11,94 --> 00:27:14,217 field vacuum now in this direction. 442 00:27:14,217 --> 00:27:17,097 And so now what will happen with this curve? 443 00:27:17,097 --> 00:27:19,307 I come up here. And look now here, 444 00:27:19,307 --> 00:27:21,852 at this location Q. What do I have now? 445 00:27:21,852 --> 00:27:25,87 I have something very bizarre. I have now a situation whereby 446 00:27:25,87 --> 00:27:29,352 the vacuum field is in this direction but there is no 447 00:27:29,352 --> 00:27:31,629 magnetic field inside the material. 448 00:27:31,629 --> 00:27:35,246 The magnetic field inside is zero. 449 00:27:35,246 --> 00:27:39,367 So when we have point Q, so we have B vacuum is in this 450 00:27:39,367 --> 00:27:42,497 direction, but B inside, B prime -- oh no, 451 00:27:42,497 --> 00:27:44,328 it's not B prime, it's B. 452 00:27:44,328 --> 00:27:47,076 It's the total field inside, is zero. 453 00:27:47,076 --> 00:27:51,351 The reason being that B prime is still in this direction. 454 00:27:51,351 --> 00:27:55,396 The reason being that the domains are still aligned in 455 00:27:55,396 --> 00:27:58,678 this direction, and so the vacuum field plus 456 00:27:58,678 --> 00:28:02,113 the B prime field, which has to be vectorially 457 00:28:02,113 --> 00:28:06,234 added, adds up with a net field zero. 458 00:28:06,234 --> 00:28:07,999 Quite bizarre, isn't it. 459 00:28:07,999 --> 00:28:11,298 Now I increase the current, but I keep going 460 00:28:11,298 --> 00:28:14,75 counterclockwise, and so the magnetic field of 461 00:28:14,75 --> 00:28:17,512 the vacuum remains in this direction. 462 00:28:17,512 --> 00:28:21,731 I go into saturation again, in a similar way that I went 463 00:28:21,731 --> 00:28:25,337 into saturation here. And now I stop here -- OH, 464 00:28:25,337 --> 00:28:29,94 I don't want to lose my brooch. And now I stop here and I say 465 00:28:29,94 --> 00:28:32,778 to the current, go back to zero again. 466 00:28:32,778 --> 00:28:35,463 So my current now goes back to zero. 467 00:28:35,463 --> 00:28:39,273 There we go. And now I arrive here, 468 00:28:39,273 --> 00:28:41,68 point S. And again, I have a situation 469 00:28:41,68 --> 00:28:45,453 that my vacuum field is zero. I could take the material out 470 00:28:45,453 --> 00:28:48,511 of the solenoid, just walk around with it on the 471 00:28:48,511 --> 00:28:50,918 street. It will be a permanent magnet. 472 00:28:50,918 --> 00:28:54,626 But now the magnetic field inside this material in point P 473 00:28:54,626 --> 00:28:57,684 it was in this direction. If I take it out here, 474 00:28:57,684 --> 00:29:01,588 then it is in this direction. Now some domains say aligned in 475 00:29:01,588 --> 00:29:04,19 this direction. The reason was that I had 476 00:29:04,19 --> 00:29:07,638 counterclockwise current, and so those domains flipped 477 00:29:07,638 --> 00:29:11,634 over. And they're not all willing to 478 00:29:11,634 --> 00:29:15,189 flip back again. So I've also made a permanent 479 00:29:15,189 --> 00:29:17,558 magnet here. Vacuum field zero, 480 00:29:17,558 --> 00:29:21,034 but B prime is now in the opposite direction. 481 00:29:21,034 --> 00:29:25,773 And then if I continue now to go clockwise with current again 482 00:29:25,773 --> 00:29:28,932 and increase the current, I end up there. 483 00:29:28,932 --> 00:29:33,277 And this is a bizarre curve. We call this the hysteresis 484 00:29:33,277 --> 00:29:35,804 curve. If you look at this curve, 485 00:29:35,804 --> 00:29:39,122 it's really amazing. It's actually hard to, 486 00:29:39,122 --> 00:29:43,567 to digest this. You, you have to give it a 487 00:29:43,567 --> 00:29:46,961 little bit of thought. Because for one particular 488 00:29:46,961 --> 00:29:49,718 value of the current, for instance here, 489 00:29:49,718 --> 00:29:52,9 once I have a particular value of the current, 490 00:29:52,9 --> 00:29:56,859 I know that B vacuum is a given, I have two possibilities 491 00:29:56,859 --> 00:30:00,607 here for the magnetic field. And for the current here, 492 00:30:00,607 --> 00:30:03,93 I have two possibilities for the magnetic field. 493 00:30:03,93 --> 00:30:07,889 And so I cannot even know when I take this material and I 494 00:30:07,889 --> 00:30:14,514 expose it to an external field, I can't even calculate what the 495 00:30:14,514 --> 00:30:20,692 magnetic field inside will be. It depends on the history of 496 00:30:20,692 --> 00:30:25,379 this material. Look at this point here and at 497 00:30:25,379 --> 00:30:30,492 this point there. If I asked you what is kappa of 498 00:30:30,492 --> 00:30:34,965 M, *pff* it's almost a ridiculous question. 499 00:30:34,965 --> 00:30:40,717 Because what is kappa of M? I have, um, I have a vacuum 500 00:30:40,717 --> 00:30:44,371 field, but I have no field inside. 501 00:30:44,371 --> 00:30:46,944 So the field inside, which is B, is zero, 502 00:30:46,944 --> 00:30:49,002 and the vacuum field is not zero. 503 00:30:49,002 --> 00:30:51,832 So you would have to answer, kappa M is zero. 504 00:30:51,832 --> 00:30:54,084 That's the only thing you could say. 505 00:30:54,084 --> 00:30:55,37 Quite bizarre, right. 506 00:30:55,37 --> 00:30:58,264 But right here, there is a vacuum field but no 507 00:30:58,264 --> 00:31:00,837 field inside. So kappa M is zero here and 508 00:31:00,837 --> 00:31:03,024 kappa M is zero here. And remember, 509 00:31:03,024 --> 00:31:06,047 here it was one thousand. Look at the situation, 510 00:31:06,047 --> 00:31:09,649 take this point of the curve and this point of the curve. 511 00:31:09,649 --> 00:31:12,093 Kappa M is less than zero, is negative, 512 00:31:12,093 --> 00:31:16,895 because here the vacuum field is in this direction, 513 00:31:16,895 --> 00:31:19,991 but B prime is in that direction, so they are in 514 00:31:19,991 --> 00:31:23,088 opposite directions, so the net field is in that 515 00:31:23,088 --> 00:31:25,986 direction, but B vacuum is in this direction. 516 00:31:25,986 --> 00:31:29,807 And here it's also reversed. So there's a bizarre situation 517 00:31:29,807 --> 00:31:33,957 that you are effectively having situations whereby kappa of M is 518 00:31:33,957 --> 00:31:37,844 zero and kappa of M can also be negative for those s- points 519 00:31:37,844 --> 00:31:41,006 that I have there. I can show you this hysteresis 520 00:31:41,006 --> 00:31:45,946 curve, and I do it exactly the way that I explained to you, 521 00:31:45,946 --> 00:31:50,591 except that I will not be able to run this current very slowly 522 00:31:50,591 --> 00:31:53,332 up and down. I do it with sixty hertz 523 00:31:53,332 --> 00:31:56,605 alternating current, just get it out of the, 524 00:31:56,605 --> 00:31:59,194 the wall. And so I run through this 525 00:31:59,194 --> 00:32:02,468 solenoid n- sixty hertz alternating current. 526 00:32:02,468 --> 00:32:06,731 That means we go through this curve very quickly back and 527 00:32:06,731 --> 00:32:09,168 forth. Between this point maximum 528 00:32:09,168 --> 00:32:12,061 current and this point maximum current. 529 00:32:12,061 --> 00:32:17,009 The current clockwise counterclockwise clockwise 530 00:32:17,009 --> 00:32:20,25 counterclockwise, and we change that sixty times 531 00:32:20,25 --> 00:32:23,008 per second. And then I will show you this 532 00:32:23,008 --> 00:32:25,076 curve, again, highly distorted. 533 00:32:25,076 --> 00:32:28,523 I cannot plot it one-to-one, for the reasons that I 534 00:32:28,523 --> 00:32:31,419 explained to you. And you will see then the 535 00:32:31,419 --> 00:32:34,521 hysteresis curve. That's called the hysteresis 536 00:32:34,521 --> 00:32:35,831 curve. And for that, 537 00:32:35,831 --> 00:32:39,899 I have to do several things. And I always forget what I have 538 00:32:39,899 --> 00:32:43,553 to do, but just don't worry about it, I will find out. 539 00:32:43,553 --> 00:32:46,449 My TV goes on. This light goes off and this 540 00:32:46,449 --> 00:32:50,723 light goes off. Look there on the screen, 541 00:32:50,723 --> 00:32:55,03 and you will see within seconds there is the hysteresis curve. 542 00:32:55,03 --> 00:32:58,489 And as I said to you earlier, I cannot start here, 543 00:32:58,489 --> 00:33:01,454 unfortunately, because we switch it so fast 544 00:33:01,454 --> 00:33:05,266 back and forth that this part of the curve, by the way, 545 00:33:05,266 --> 00:33:08,513 which is called the virginal curve -- virginal, 546 00:33:08,513 --> 00:33:10,842 because it's once and never again. 547 00:33:10,842 --> 00:33:14,584 Once you have reached that point, from that moment on, 548 00:33:14,584 --> 00:33:18,537 you always stick to this -- I know, you should know about 549 00:33:18,537 --> 00:33:19,666 that. [laughter]. 550 00:33:19,666 --> 00:33:24,963 And so here you see a striking example of a hysteresis curve. 551 00:33:24,963 --> 00:33:27,658 And so you can ask yourself now the question, 552 00:33:27,658 --> 00:33:30,107 can we make this material virginal again? 553 00:33:30,107 --> 00:33:33,598 And then the answer is yes. There are various ways you can 554 00:33:33,598 --> 00:33:35,864 do that. One way is you could take the 555 00:33:35,864 --> 00:33:38,498 material out, so that means you take it out, 556 00:33:38,498 --> 00:33:41,193 for instance, when there is remnant magnetism 557 00:33:41,193 --> 00:33:44,316 here, so it's a magnet, and now you heat it up above 558 00:33:44,316 --> 00:33:46,827 the Curie point, as we did last time in my 559 00:33:46,827 --> 00:33:49,951 lecture, and then the domains completely fall apart, 560 00:33:49,951 --> 00:33:54,176 and then you cool it again below the Curie point, 561 00:33:54,176 --> 00:33:56,212 and then it is virginal material again. 562 00:33:56,212 --> 00:33:58,088 And then you could start here again. 563 00:33:58,088 --> 00:34:01,356 That's one way you could do it. There is another way you could 564 00:34:01,356 --> 00:34:03,66 try to do it. You take a hammer and you just 565 00:34:03,66 --> 00:34:05,91 bang on it. So you take it out and you have 566 00:34:05,91 --> 00:34:08,106 permanent magnetism, either here or there, 567 00:34:08,106 --> 00:34:10,517 and you bang on it, and you hope for the best. 568 00:34:10,517 --> 00:34:12,5 And maybe you can get it back to here. 569 00:34:12,5 --> 00:34:15,232 There is another way, which we call demagnetization, 570 00:34:15,232 --> 00:34:18,393 and I think that's what happens when you steal a book in the 571 00:34:18,393 --> 00:34:21,768 library and you try to get away with it, and the alarm goes off. 572 00:34:21,768 --> 00:34:26,243 Someone hasn't demagnetized the magnetic strip in the book. 573 00:34:26,243 --> 00:34:28,938 You may have noticed that when you take it out, 574 00:34:28,938 --> 00:34:31,457 that someone under the table goes like this. 575 00:34:31,457 --> 00:34:34,855 And what they are doing then is they are demagnetizing that 576 00:34:34,855 --> 00:34:36,905 strip. And the way you do that is as 577 00:34:36,905 --> 00:34:39,132 follows. Here the current goes back and 578 00:34:39,132 --> 00:34:41,416 forth between this value and this value. 579 00:34:41,416 --> 00:34:45,048 That means the vacuum and I are directly coupled to each other. 580 00:34:45,048 --> 00:34:47,099 So I think of this as being current. 581 00:34:47,099 --> 00:34:50,555 And now I go up to here and then back and then here and here 582 00:34:50,555 --> 00:34:54,129 and here and here and here and here and here and here and here 583 00:34:54,129 --> 00:34:57,062 and I end up there again. 584 00:34:57,062 --> 00:35:00,769 And I can show you that. So again you have alternating 585 00:35:00,769 --> 00:35:04,755 current, but the amplitude of the current you decrease and 586 00:35:04,755 --> 00:35:08,252 decrease and decrease, and you're going to see that 587 00:35:08,252 --> 00:35:10,56 there. And you see that I can turn 588 00:35:10,56 --> 00:35:13,986 this material -- see the hysteresis curve changes. 589 00:35:13,986 --> 00:35:17,273 So the amplitude of the current is not as large, 590 00:35:17,273 --> 00:35:20,63 so the amplitude of the B vacuum is not as large, 591 00:35:20,63 --> 00:35:24,127 and I slowly go back. And I can change a non-virgin 592 00:35:24,127 --> 00:35:30,268 back into a virgin. And that's the way we do it as 593 00:35:30,268 --> 00:35:33,466 physicists. Demagnetization. 594 00:35:33,466 --> 00:35:37,138 Did I turn that off? Yes, I did. 595 00:35:37,138 --> 00:35:39,626 I have here a, a coil. 596 00:35:39,626 --> 00:35:45,075 You can't see that there is a coil inside here, 597 00:35:45,075 --> 00:35:49,813 but there is. I can, uh, power this coil, 598 00:35:49,813 --> 00:35:57,512 putting a current through it. Ferromagnetic material, 599 00:35:57,512 --> 00:36:01,226 I don't know what kappa is, but, uh, at least a thousand. 600 00:36:01,226 --> 00:36:04,143 I get an enormously strong field inside here. 601 00:36:04,143 --> 00:36:08,055 And this field is so strong that this piece of ferromagnetic 602 00:36:08,055 --> 00:36:11,436 material will be attracted. The field is non-uniform 603 00:36:11,436 --> 00:36:13,956 outside. We discussed last time that it 604 00:36:13,956 --> 00:36:16,21 will go clunk, it will stick there. 605 00:36:16,21 --> 00:36:19,26 So let's do that. So I power this electromagnet 606 00:36:19,26 --> 00:36:22,973 -- you can't see that I did, you have to take my word for 607 00:36:22,973 --> 00:36:25,692 it, but you believe it now. There it goes. 608 00:36:25,692 --> 00:36:28,012 Oh boy. The force is so large that I 609 00:36:28,012 --> 00:36:31,791 would ask two people to come and see 610 00:36:31,791 --> 00:36:35,487 whether they can pull it apart. The force that the two are 611 00:36:35,487 --> 00:36:39,442 together is so large that you may not even be able to separate 612 00:36:39,442 --> 00:36:41,517 it. Do we have two strong people? 613 00:36:41,517 --> 00:36:43,591 One strong woman, one strong man. 614 00:36:43,591 --> 00:36:46,055 You look very strong to me. [laughter]. 615 00:36:46,055 --> 00:36:48,454 Come on. It's not going to be a tug of 616 00:36:48,454 --> 00:36:51,436 war between the two of you. That's not my plan. 617 00:36:51,436 --> 00:36:54,16 Be very careful, because if you touch this, 618 00:36:54,16 --> 00:36:56,623 you get electrocuted. So don't do that. 619 00:36:56,623 --> 00:36:59,346 [laughter]. But make sure that everyone can 620 00:36:59,346 --> 00:37:02,789 see you. Because there is a current 621 00:37:02,789 --> 00:37:05,132 running through this solenoid now, yeah? 622 00:37:05,132 --> 00:37:06,814 OK. Now just in case that you 623 00:37:06,814 --> 00:37:09,037 succeed, I don't want you to get hurt. 624 00:37:09,037 --> 00:37:11,44 Student: OK. So make sure that you secure 625 00:37:11,44 --> 00:37:12,641 yourself. [laughter]. 626 00:37:12,641 --> 00:37:16,186 Because suppose the current stopped running all of a sudden. 627 00:37:16,186 --> 00:37:18,589 It's possible, it's MIT [laughter] right, 628 00:37:18,589 --> 00:37:20,872 anything could happen. Then, of course, 629 00:37:20,872 --> 00:37:24,416 it's no longer a strong magnet. It's only a strong magnet as 630 00:37:24,416 --> 00:37:27,48 long as the current is running through the solenoid. 631 00:37:27,48 --> 00:37:30,664 So you secure yourself too. OK, three two one zero go. 632 00:37:30,664 --> 00:37:32,046 [laughter]. Don't worry. 633 00:37:32,046 --> 00:37:35,504 I knew that in advance. 634 00:37:35,504 --> 00:37:38,862 But thank you very much. [laughter]. 635 00:37:38,862 --> 00:37:41,548 Very kind of you. [applause]. 636 00:37:41,548 --> 00:37:45,77 Now comes something that you will understand. 637 00:37:45,77 --> 00:37:51,431 If I make the current go to zero, then the vacuum field goes 638 00:37:51,431 --> 00:37:54,885 to zero. That means the field that is 639 00:37:54,885 --> 00:37:58,531 generated by the solenoid goes to zero. 640 00:37:58,531 --> 00:38:02,753 I will do that now in front of your own eyes. 641 00:38:02,753 --> 00:38:06,494 No more current, right? 642 00:38:06,494 --> 00:38:08,813 Why is this still hanging there? 643 00:38:08,813 --> 00:38:10,982 Yeah? The solenoid [inaudible] 644 00:38:10,982 --> 00:38:13,45 magnetize the ferrous [inaudible]. 645 00:38:13,45 --> 00:38:16,292 You hear? You've taken the vacuum field 646 00:38:16,292 --> 00:38:19,957 out, but the domains to a certain degree are still 647 00:38:19,957 --> 00:38:23,622 aligned, and so the whole thing is still a magnet. 648 00:38:23,622 --> 00:38:28,184 Not as strong as magnet -- if I were to invite you to come now 649 00:38:28,184 --> 00:38:30,279 and take it apart, you could. 650 00:38:30,279 --> 00:38:33,345 But it still takes sub- substantial force. 651 00:38:33,345 --> 00:38:38,218 And I will show you how large that force is. 652 00:38:38,218 --> 00:38:42,895 I'm going to load it down now. There's now one kilogram 653 00:38:42,895 --> 00:38:46,705 hanging on it. Ooh, let me make sure I secure 654 00:38:46,705 --> 00:38:50,602 that, otherwise I can get killed for a change. 655 00:38:50,602 --> 00:38:54,066 OK, three kilograms is hanging on it now. 656 00:38:54,066 --> 00:38:57,097 Five kilograms is hanging on it now. 657 00:38:57,097 --> 00:39:00,214 Seven kilograms is hanging on it now. 658 00:39:00,214 --> 00:39:03,245 Nine kilograms is hanging on it now. 659 00:39:03,245 --> 00:39:08,182 Oh boy, we may never make it. Ten kilograms is -- there it 660 00:39:08,182 --> 00:39:11,213 goes. [laughter]. 661 00:39:11,213 --> 00:39:15,909 Ten kilograms. Now the show is not over yet. 662 00:39:15,909 --> 00:39:21,808 What is very interesting, and I want you to think about 663 00:39:21,808 --> 00:39:27,925 it, that if now I take these two pieces of -- [laughter]. 664 00:39:27,925 --> 00:39:33,605 If I take these two pieces of ferromagnetic material, 665 00:39:33,605 --> 00:39:37,1 that -- nothing. Do you know why? 666 00:39:37,1 --> 00:39:42,452 I dropped it on the floor. [laughter]. 667 00:39:42,452 --> 00:39:45,109 That's really why. I dropped it on the floor, 668 00:39:45,109 --> 00:39:47,584 and that is like banging it with a hammer, 669 00:39:47,584 --> 00:39:50,964 and then the domains go away. Had I not dropped it on the 670 00:39:50,964 --> 00:39:53,862 floor [laughter], there would have been something 671 00:39:53,862 --> 00:39:57,061 left, but very little, which is interesting by itself. 672 00:39:57,061 --> 00:39:59,899 The shock of the separation, when that happened, 673 00:39:59,899 --> 00:40:02,495 already makes many of the domains flip back, 674 00:40:02,495 --> 00:40:04,607 and there would be very little left. 675 00:40:04,607 --> 00:40:08,109 Not enough to carry this weight anymore, but that's largely 676 00:40:08,109 --> 00:40:10,101 because I dropped it on the floor. 677 00:40:10,101 --> 00:40:13,481 I did that purposely so you can see 678 00:40:13,481 --> 00:40:17,56 when you drop [laughter] things on the floor. 679 00:40:17,56 --> 00:40:22,472 If I bring ferromagnetic material in the vicinity of a 680 00:40:22,472 --> 00:40:27,014 magnet, I change the magnetic field configuration, 681 00:40:27,014 --> 00:40:30,536 and that's very easy to understand now. 682 00:40:30,536 --> 00:40:35,356 Suppose I have here a magnet, north pole, south pole, 683 00:40:35,356 --> 00:40:40,176 and the magnetic field, magnetic dipole field is sort 684 00:40:40,176 --> 00:40:44,254 of like this. And now I bring in the vicinity 685 00:40:44,254 --> 00:40:47,583 here a piece of ferromagnetic 686 00:40:47,583 --> 00:40:49,347 material. Could be a wrench. 687 00:40:49,347 --> 00:40:53,266 What happens now is that this ferromagnetic material will see 688 00:40:53,266 --> 00:40:56,663 this vacuum field -- this is called the vacuum field, 689 00:40:56,663 --> 00:40:59,929 is an external field. And so these domains in there 690 00:40:59,929 --> 00:41:03,653 trying to align a little bit. Degree of success depends on 691 00:41:03,653 --> 00:41:06,984 how strong the field is, depends on the temperature, 692 00:41:06,984 --> 00:41:09,532 depends on the kappa M of that material. 693 00:41:09,532 --> 00:41:12,863 But certainly this will become sort of a south pole, 694 00:41:12,863 --> 00:41:16,456 this will become a north pole. That's the way that these 695 00:41:16,456 --> 00:41:20,11 dipoles are going to align themselves. 696 00:41:20,11 --> 00:41:23,261 They are going to create themselves a field in this 697 00:41:23,261 --> 00:41:25,719 direction. They're going to support that 698 00:41:25,719 --> 00:41:27,925 field. And so the net result is that 699 00:41:27,925 --> 00:41:30,509 the field inside here becomes very strong. 700 00:41:30,509 --> 00:41:34,353 And so what happens with these field lines, they go like this. 701 00:41:34,353 --> 00:41:37,694 They're being sucked into this ferromagnetic material. 702 00:41:37,694 --> 00:41:40,34 It's very hard to know exactly how they go. 703 00:41:40,34 --> 00:41:44,059 And the field here will weaken. And I'm going to demonstrate 704 00:41:44,059 --> 00:41:46,643 that to you. This is actually very easy to 705 00:41:46,643 --> 00:41:50,816 demonstrate. And the way I'm going to 706 00:41:50,816 --> 00:41:56,388 demonstrate that is as follows. I have there a setup whereby we 707 00:41:56,388 --> 00:42:01,42 have a, a magnet and we have a nail and we have a strong. 708 00:42:01,42 --> 00:42:04,835 And the nail wants to go to the magnet. 709 00:42:04,835 --> 00:42:09,418 The mail its- the nail itself is, uh, ferromagnetic. 710 00:42:09,418 --> 00:42:13,551 So the nail would love to go in this direction, 711 00:42:13,551 --> 00:42:16,696 but it can't. So it just sits there, 712 00:42:16,696 --> 00:42:21,099 hangs there in space. First of all, 713 00:42:21,099 --> 00:42:25,018 what I want to show you is that if I bring paramagnetic material 714 00:42:25,018 --> 00:42:27,505 in the vicinity, that that magnetic field 715 00:42:27,505 --> 00:42:30,49 configuration here is not going to change at all. 716 00:42:30,49 --> 00:42:34,097 Paramagnetic material has a kappa of M so close to one that 717 00:42:34,097 --> 00:42:37,456 nothing is going to happen. But the moment that I bring 718 00:42:37,456 --> 00:42:39,695 ferromagnetic material, for instance, 719 00:42:39,695 --> 00:42:42,618 here, then you get a field configuration change, 720 00:42:42,618 --> 00:42:46,349 and if I do that just the right way, then the nail will fall. 721 00:42:46,349 --> 00:42:49,023 In other words, there is not enough magnetic 722 00:42:49,023 --> 00:42:53,626 field here in order to hold the nail in that direction. 723 00:42:53,626 --> 00:42:58,14 And I'm going to show that for you, to you there. 724 00:42:58,14 --> 00:43:02,937 Need some power here, I believe, and we have to make 725 00:43:02,937 --> 00:43:06,981 it dark here. I'm going to shadow project it 726 00:43:06,981 --> 00:43:10,461 for you. And the shadow projection you 727 00:43:10,461 --> 00:43:15,446 will see coming up very shortly. This is a carbon arc. 728 00:43:15,446 --> 00:43:20,055 You have to give it a little bit of time to start. 729 00:43:20,055 --> 00:43:26,074 There is the carbon arc. So there you see the nail. 730 00:43:26,074 --> 00:43:29,936 And here you see the magnet. You see that? 731 00:43:29,936 --> 00:43:33,797 That's exactly the way I drew the picture. 732 00:43:33,797 --> 00:43:38,506 And here, you hav- I have here a piece of aluminum, 733 00:43:38,506 --> 00:43:43,31 which is, paramagnetic. I can bring that through the 734 00:43:43,31 --> 00:43:45,853 field here. Nothing happens. 735 00:43:45,853 --> 00:43:50,185 My hands, believe it or not, are definitely not 736 00:43:50,185 --> 00:43:54,141 ferromagnetic, so I can also bring my hands 737 00:43:54,141 --> 00:43:55,365 here. Nothing. 738 00:43:55,365 --> 00:43:59,29 Nothing. So magnetic fields is not 739 00:43:59,29 --> 00:44:01,802 disturbed in any way, in any serious way, 740 00:44:01,802 --> 00:44:04,377 either by paramagnetic material, aluminum, 741 00:44:04,377 --> 00:44:06,513 or my hands, which I think are also 742 00:44:06,513 --> 00:44:08,397 paramagnetic, but I'm not sure. 743 00:44:08,397 --> 00:44:11,663 I'm not sure whether I'm diamagnetic or paramagnetic, 744 00:44:11,663 --> 00:44:14,803 but it doesn't make any difference, because in both 745 00:44:14,803 --> 00:44:18,446 cases there is no significant change of the magnetic field. 746 00:44:18,446 --> 00:44:21,712 But now I have a wrench here. Here, there's a wrench. 747 00:44:21,712 --> 00:44:23,094 You see it? [laughter]. 748 00:44:23,094 --> 00:44:25,104 OK. And now I'll bring the wrench 749 00:44:25,104 --> 00:44:30,474 close to that magnet. My major worry is that magnetic 750 00:44:30,474 --> 00:44:35,899 field is so strong that once the wrench go- there's no way I can 751 00:44:35,899 --> 00:44:39,86 get it off again. So I get only one shot at it. 752 00:44:39,86 --> 00:44:44,425 And there goes the nail. So what I -- what you saw now 753 00:44:44,425 --> 00:44:49,333 in front of your eyes is that I changed the magnetic field 754 00:44:49,333 --> 00:44:54,758 configuration in such a way that the field was not strong enough 755 00:44:54,758 --> 00:44:57,945 to pull in the, the nail. 756 00:44:57,945 --> 00:45:03,928 Now comes an important question, a big moment in our 757 00:45:03,928 --> 00:45:05,923 life. And that is, 758 00:45:05,923 --> 00:45:12,376 what now is the effect of magnetic material on Maxwell's 759 00:45:12,376 --> 00:45:16,365 equations? And let's take a look at 760 00:45:16,365 --> 00:45:21,293 Maxwell's equations. Here we have Maxwell's 761 00:45:21,293 --> 00:45:26,221 equations the way we know them. [laughter]. 762 00:45:26,221 --> 00:45:31,853 And let's first look at number one. 763 00:45:31,853 --> 00:45:35,044 That's Gauss's Law. Gauss's Law says that the 764 00:45:35,044 --> 00:45:39,541 closed surface integral of E dot DA -- that's the electric flux 765 00:45:39,541 --> 00:45:44,038 through a closed surface -- E is equal to all the charge inside 766 00:45:44,038 --> 00:45:47,882 divided by epsilon zero, but you have to allow for the 767 00:45:47,882 --> 00:45:51,074 kappa, for the electric, dielectric constant. 768 00:45:51,074 --> 00:45:54,845 The kappa, by the way, always lowers the field inside 769 00:45:54,845 --> 00:45:58,182 the material when we deal with electric fields. 770 00:45:58,182 --> 00:46:01,809 It never increases it like magnetic 771 00:46:01,809 --> 00:46:03,489 fields. It always lowers it. 772 00:46:03,489 --> 00:46:06,164 Kappa is normally a few -- except for water, 773 00:46:06,164 --> 00:46:08,031 it is eighty, it's quite large, 774 00:46:08,031 --> 00:46:11,765 and there are some ridiculous substances whereby kappa can be 775 00:46:11,765 --> 00:46:15,312 as large as three hundred. I think strontium titanate -- I 776 00:46:15,312 --> 00:46:18,921 just looked it up this morning -- has a ridiculous value of 777 00:46:18,921 --> 00:46:21,721 kappa of three hundred. So that's Gauss's Law. 778 00:46:21,721 --> 00:46:24,894 Nothing's going to change there as far as I can see. 779 00:46:24,894 --> 00:46:26,823 And then we have the second one. 780 00:46:26,823 --> 00:46:31,242 The closed surface integral of B dot DL equals zero. 781 00:46:31,242 --> 00:46:33,039 Oh, it says an, it says an L. 782 00:46:33,039 --> 00:46:36,57 That shouldn't even be an L. Ho, I hope you caught that. 783 00:46:36,57 --> 00:46:38,047 This is an A, of course. 784 00:46:38,047 --> 00:46:40,487 How could I? This is the closed surface 785 00:46:40,487 --> 00:46:44,339 integral of B dot DA is zero. What this is telling me is that 786 00:46:44,339 --> 00:46:47,1 magnetic ma- magnetic monopoles don't exist. 787 00:46:47,1 --> 00:46:49,283 At least we think they don't exist. 788 00:46:49,283 --> 00:46:52,557 Don't think that people are not trying to find them. 789 00:46:52,557 --> 00:46:56,088 And if you find a magnetic monopole, and if you put that 790 00:46:56,088 --> 00:46:58,4 inside a box, then the closed surface 791 00:46:58,4 --> 00:47:02,702 integral of the magnetic flux through that box 792 00:47:02,702 --> 00:47:06,248 would not be zero. And so then this is not true. 793 00:47:06,248 --> 00:47:09,191 But as far as we know, it's always true, 794 00:47:09,191 --> 00:47:13,643 because we don't think that magnetic poles- monopoles exist. 795 00:47:13,643 --> 00:47:16,661 And so then we come to the Faraday's Law. 796 00:47:16,661 --> 00:47:21,113 Faraday's Law runs our economy. Faraday's Law tells you when 797 00:47:21,113 --> 00:47:25,565 you move conducting loops in magnetic fields that you create 798 00:47:25,565 --> 00:47:28,81 electricity. This equation runs our economy. 799 00:47:28,81 --> 00:47:33,338 And now we come -- none of these, by the way, 800 00:47:33,338 --> 00:47:36,772 require any adjustment in terms of kappa of M. 801 00:47:36,772 --> 00:47:39,138 But now we come to Ampere's Law. 802 00:47:39,138 --> 00:47:42,267 Ampere's Law, which was amended by Maxwell 803 00:47:42,267 --> 00:47:45,625 himself, tells me what the magnetic field is, 804 00:47:45,625 --> 00:47:48,372 and all these results are for vacuum. 805 00:47:48,372 --> 00:47:51,73 But now we know that that's not true anymore. 806 00:47:51,73 --> 00:47:56,004 So this has to be adjusted now by a factor of kappa of M, 807 00:47:56,004 --> 00:47:58,599 which is the relative permeability. 808 00:47:58,599 --> 00:48:02,72 And kappa of M is perfectly kosher for paramagnetic and 809 00:48:02,72 --> 00:48:05,69 diamagnetic materials. 810 00:48:05,69 --> 00:48:07,798 There's never any problem there. 811 00:48:07,798 --> 00:48:10,722 So here it comes. For diamagnetic materials, 812 00:48:10,722 --> 00:48:14,462 it's a little less than one. For paramagnetic materials, 813 00:48:14,462 --> 00:48:17,59 a little larger than one. But when we deal with 814 00:48:17,59 --> 00:48:21,126 ferromagnetic materials, you have to be very careful, 815 00:48:21,126 --> 00:48:24,73 because we have seen today this hysteresis phenomenon, 816 00:48:24,73 --> 00:48:28,878 that there are even situations whereby kappa of M is negative, 817 00:48:28,878 --> 00:48:32,278 whereby kappa M is zero, and whereby kappa M can be 818 00:48:32,278 --> 00:48:37,583 huge, can be ten to the third. So there you have to be very, 819 00:48:37,583 --> 00:48:43,43 very careful when you apply this equation without thinking. 820 00:48:43,43 --> 00:48:49,277 Maxwell's equations are so important that I'm sure you want 821 00:48:49,277 --> 00:48:54,115 to see more of them. So you see them there again. 822 00:48:54,115 --> 00:48:58,047 And maybe that's not enough. [laughter]. 823 00:48:58,047 --> 00:49:01,979 Maybe you want to see even more of them. 824 00:49:01,979 --> 00:49:04,801 So look at them. Inhale them. 825 00:49:04,801 --> 00:49:09,099 Let them penetrate your brains. 826 00:49:09,099 --> 00:49:13,108 [laughter]. I don't care in which direction 827 00:49:13,108 --> 00:49:16,83 you look now. It's hard not to see them. 828 00:49:16,83 --> 00:49:21,889 Today is therefore very special, because today we have 829 00:49:21,889 --> 00:49:25,421 all four Maxwell's equations in place. 830 00:49:25,421 --> 00:49:30,671 And this was one of the main objectives of eight oh two. 831 00:49:30,671 --> 00:49:36,207 So we have completed the long journey, and on April five we 832 00:49:36,207 --> 00:49:41,956 have reached the summit. Now I realize that the view is 833 00:49:41,956 --> 00:49:46,341 not spectacular for all of you yet, because often at the summit 834 00:49:46,341 --> 00:49:49,241 there is some fog. But the fog will clear. 835 00:49:49,241 --> 00:49:53,13 And I can assure you that from here on on, it's climbing 836 00:49:53,13 --> 00:49:55,747 downhill. I think this moment is worth 837 00:49:55,747 --> 00:49:59,99 celebrating, and therefore I bought six hundred daffodils for 838 00:49:59,99 --> 00:50:01,758 this occasion. [laughter]. 839 00:50:01,758 --> 00:50:06,072 And I would like you to come at the, at the end of the lecture 840 00:50:06,072 --> 00:50:09,821 and pick up one of these daffodils and take it back to 841 00:50:09,821 --> 00:50:14,034 your dormitory. I don't know whether I have 842 00:50:14,034 --> 00:50:18,419 enough for all these high school students and all their parents, 843 00:50:18,419 --> 00:50:20,299 but why not, give it a shot, 844 00:50:20,299 --> 00:50:23,362 and take one. And when you look at it tonight 845 00:50:23,362 --> 00:50:26,912 at home and tomorrow, remember that you only once in 846 00:50:26,912 --> 00:50:30,74 your life go through this experience, that for the first 847 00:50:30,74 --> 00:50:34,221 time you see all four Maxwell's equations complete, 848 00:50:34,221 --> 00:50:38,05 all of it, and that you're capable of appreciating them, 849 00:50:38,05 --> 00:50:41,6 at least in principle. This will never happen again. 850 00:50:41,6 --> 00:50:45,071 You will never be the same. 851 00:50:45,071 --> 00:50:50,665 [laughter]. To put it in simple terms, 852 00:50:50,665 --> 00:50:58,679 as far as eight oh two is concerned, you are no longer 853 00:50:58,679 --> 00:51:04,425 unspoiled virginal material [laughter]. 854 00:51:04,425 --> 00:51:10,171 You've lost your virginity. [laughter]. 855 00:51:10,171 --> 00:51:14,254 Congratulations! [laughter]. 856 00:51:14,254 --> 00:51:20 [applause]. [crowd noise]. 857 00:51:20 --> 00:51:38,666 [crowd noise]. [crowd noise]. 858 00:51:38,666 --> 00:52:00 [crowd noise]. [crowd noise]. 859 00:52:00 --> 00:52:18,666 [crowd noise]. [crowd noise]. 860 00:52:18,666 --> 00:52:40 [crowd noise]. [crowd noise]. 861 00:52:40 --> 00:52:58,666 [crowd noise]. [crowd noise]. 862 00:52:58,666 --> 00:53:20 [crowd noise]. [crowd noise]. 863 00:53:20 --> 00:53:38,671 [crowd noise]. [crowd noise]. 864 00:53:38,671 --> 53:44 [crowd noise]. [crowd noise].