WEBVTT 00:01.840 --> 00:05.310 Prof: Okay, this is the final lecture on 00:05.314 --> 00:06.754 electromagnetism. 00:06.750 --> 00:11.290 When you come back from the break we will start doing 00:11.293 --> 00:12.083 optics. 00:12.080 --> 00:15.250 Of course you know by now that's really electromagnetism, 00:15.250 --> 00:19.090 but it's going to be somewhat easier on your imagination 00:19.094 --> 00:22.664 because it doesn't involve many abstract things, 00:22.660 --> 00:24.950 pretty concrete things. 00:24.950 --> 00:28.430 Then it'll take a turn for the worse when we do quantum 00:28.434 --> 00:31.984 mechanics, but I think the mathematics of that should be 00:31.983 --> 00:33.083 quite simple. 00:33.080 --> 00:36.180 Before you do that you should make sure you know all about 00:36.179 --> 00:39.329 elementary complex numbers like what is e^(i) 00:39.333 --> 00:40.533 ^(θ). 00:40.530 --> 00:42.850 You should get that completely into your system. 00:42.850 --> 00:46.900 It's going to be all about that, then very elementary 00:46.902 --> 00:49.242 calculus, just one variable. 00:49.240 --> 00:53.490 So last time I told you that if you want to solve Maxwell's 00:53.490 --> 00:58.030 equation in free space you can write down any function you want 00:58.033 --> 01:00.563 for E and B that 01:00.564 --> 01:03.264 satisfy the four Maxwell equations. 01:03.259 --> 01:08.249 Two involve surface integrals, and two involve line integrals, 01:08.253 --> 01:11.613 and the answer we got looked like this. 01:11.610 --> 01:16.510 This is x, y and z. 01:16.510 --> 01:19.070 The electric field is going to point that way, 01:19.069 --> 01:21.269 magnetic field is going to point that way, 01:21.269 --> 01:25.649 and the functional form E was k (k is 01:25.653 --> 01:28.503 unit vector in the z direction) 01:28.498 --> 01:32.418 E_0sin(ky - ωt), 01:32.420 --> 01:41.710 and B was iB_0sin(ky - 01:41.712 --> 01:44.812 ωt). 01:44.810 --> 01:48.830 When you take this assumed form with this y and this 01:48.827 --> 01:51.387 t dependence you will find, 01:51.390 --> 01:55.080 as written, they automatically satisfy the surface integral, 01:55.080 --> 01:58.780 and the surface integral of these fields on any tiny cube 01:58.777 --> 01:59.567 will be 0. 01:59.569 --> 02:03.109 When you do line integrals you can take a loop in this plane, 02:03.108 --> 02:04.428 and that plane, and the third plane, 02:04.430 --> 02:07.170 and you fool around you get some additional constraints 02:07.174 --> 02:09.484 which I derived, but I don't want to do that 02:09.482 --> 02:09.832 again. 02:09.830 --> 02:14.500 But the conditions you derive are that ω should 02:14.495 --> 02:17.575 be equal to k times c, 02:17.580 --> 02:23.180 and remember the velocity of light emerged at this following 02:23.180 --> 02:24.510 combination. 02:24.508 --> 02:29.468 Then you will find the magnetic field should be electric field 02:29.470 --> 02:31.260 divided by c. 02:31.258 --> 02:35.848 These are the two conditions you get from writing the two 02:35.853 --> 02:36.513 other. 02:36.508 --> 02:37.958 There are three equations for loops. 02:37.960 --> 02:41.020 One of them is trivially satisfied 0 = 0, 02:41.021 --> 02:43.091 the others give you this. 02:43.090 --> 02:47.430 Now I want you to be able to at least visualize these fields. 02:47.430 --> 02:49.750 You can write them on a piece of paper, but E is not 02:49.750 --> 02:50.630 the letter E. 02:50.628 --> 02:53.118 E stands for something physical. 02:53.120 --> 02:55.610 So what do these things do? 02:55.610 --> 02:59.290 First of all what is k and what is ω? 02:59.288 --> 03:02.638 I realize that some of you are not familiar with what k 03:02.637 --> 03:03.677 and ω mean. 03:03.680 --> 03:07.190 First of all you know it's a periodic function in y at 03:07.189 --> 03:08.769 fixed t, if you like, 03:08.770 --> 03:11.170 because sine of anything is periodic. 03:11.169 --> 03:15.519 That means at a given time if you take a snapshot of this 03:15.515 --> 03:19.195 E it'll be oscillating as you vary y. 03:19.199 --> 03:22.399 And at a given point in space, if you study it as a function 03:22.401 --> 03:25.131 of time, it'll be also oscillating in 03:25.131 --> 03:28.711 time with an angular frequency ω. 03:28.710 --> 03:32.340 So let me relate these quantities to more familiar 03:32.342 --> 03:33.012 things. 03:33.008 --> 03:38.758 If the wave looks like this, and E is going to point 03:38.764 --> 03:42.524 in that direction, you know that if you go a full 03:42.518 --> 03:44.908 wavelength you come back to where you are. 03:44.910 --> 03:47.050 So let's call the wavelength λ. 03:47.050 --> 03:50.830 It's clear then whatever happens at 0 happens at 03:50.834 --> 03:51.724 λ. 03:51.720 --> 03:54.380 At zero, ky is 0, at λ, 03:54.375 --> 03:56.435 ky should be 2Π. 03:56.440 --> 04:01.820 When you go a distance lambda it should be equal to a 2Π 04:01.824 --> 04:02.664 change. 04:02.658 --> 04:06.268 So λ, which is more familiar to you 04:06.270 --> 04:10.230 as the wavelength, is the reciprocal up to the 04:10.232 --> 04:13.492 factor of 2Π of this k. 04:13.490 --> 04:18.440 k is called the wave number. 04:18.439 --> 04:22.479 The bigger the k the more rapidly the wave 04:22.483 --> 04:23.583 oscillates. 04:23.579 --> 04:28.029 That is bigger the λ the long wavelength means more 04:28.033 --> 04:30.303 slowly the wave oscillates. 04:30.300 --> 04:34.850 Similarly you can come to the second guy and you can ask 04:34.851 --> 04:37.171 yourself--let me continue. 04:37.170 --> 04:38.820 So that's what E is. 04:38.819 --> 04:41.469 B, again, is a function of space at a 04:41.471 --> 04:44.421 given time if you draw it, but the B vector's 04:44.420 --> 04:50.400 pointing this way, so it'll look like this, 04:50.401 --> 04:57.351 and you can keep repeating this. 04:57.350 --> 04:58.870 So E is going back. 04:58.870 --> 05:01.290 So now this is the thing to understand with this graph. 05:01.290 --> 05:04.060 You can see this graph, but you should know what it 05:04.064 --> 05:04.514 means. 05:04.509 --> 05:09.889 This whole graph describes the situation at one given time only 05:09.894 --> 05:12.764 at points on the y-axis. 05:12.759 --> 05:16.049 I've drawn it only at points on the y-axis. 05:16.050 --> 05:17.800 So this arrow, even though it sticks out here 05:17.800 --> 05:19.750 and there, is not talking about a whole region. 05:19.750 --> 05:22.300 It's talking about that one point. 05:22.300 --> 05:24.550 For example, at this one point the electric 05:24.550 --> 05:27.070 field points this way and has that magnitude. 05:27.069 --> 05:30.199 At this point the electric field points down and has that 05:30.202 --> 05:32.182 magnitude, and the magnetic field points 05:32.180 --> 05:34.430 away from the board, here the magnetic field into 05:34.432 --> 05:36.112 the board and away from the board, 05:36.110 --> 05:37.320 okay? 05:37.319 --> 05:40.689 That's the situation at one time. 05:40.690 --> 05:44.530 Then you can ask yourself what happens later. 05:44.529 --> 05:45.779 You can do it in two ways. 05:45.779 --> 05:48.059 First of all let me tell you the relation between 05:48.055 --> 05:49.805 ω and the time period. 05:49.810 --> 05:53.240 If you wait a full time period everything should come back to 05:53.244 --> 05:55.704 where it was, so the time period T by 05:55.704 --> 05:57.884 definition will be this condition. 05:57.879 --> 06:04.089 Therefore, the time period that you guys are all familiar with 06:04.088 --> 06:08.668 is also related to ω by a similar 06:08.668 --> 06:09.888 formula. 06:09.889 --> 06:15.419 So one way to write this wave that is in terms of things 06:15.420 --> 06:20.850 you're more familiar with would be to say it looks like 06:20.853 --> 06:25.083 E_0 sin(2Πy/λ 06:25.077 --> 06:27.287 - 2Πt/T). 06:27.290 --> 06:30.240 This is completely equivalent to what I wrote because these 06:30.242 --> 06:32.892 are the definitions of ω and k in 06:32.891 --> 06:35.031 terms of λ and T. 06:35.029 --> 06:38.199 Maybe writing it this way is more transparent - that if you 06:38.202 --> 06:40.832 add a λ to y on the top, 06:40.829 --> 06:43.219 if you say y goes to y λ 06:43.218 --> 06:44.888 you're adding a 2Π. 06:44.889 --> 06:47.749 If you add capital T to small t upstairs you're 06:47.747 --> 06:48.887 subtracting the 2Π. 06:48.889 --> 06:49.849 That doesn't matter. 06:49.850 --> 06:55.160 This shows you the periodicity in space of λ and 06:55.158 --> 06:57.768 in time of capital T. 06:57.769 --> 07:00.459 You can also write ω as 07:00.461 --> 07:03.231 2Π/T, but 1/T is the 07:03.232 --> 07:04.312 frequency. 07:04.310 --> 07:06.720 This is what we normally think of as frequency, 07:06.718 --> 07:08.758 so many hertz, megahertz or whatever. 07:08.759 --> 07:09.979 That's the frequency. 07:09.980 --> 07:14.010 The angular frequency omega we use is 2Π times that, 07:14.009 --> 07:17.269 because every time the thing completes a cycle it goes by 07:17.266 --> 07:19.146 2Π, and if it does f 07:19.148 --> 07:21.858 revolutions per second the angular frequency is 07:21.863 --> 07:23.993 2Πf radians per second. 07:23.990 --> 07:27.130 That's the difference between revolutions per second and 07:27.129 --> 07:28.329 radians per second. 07:28.329 --> 07:33.199 Okay, so this is another way to write it, and B will be 07:33.202 --> 07:34.562 the same thing. 07:34.560 --> 07:37.000 So you should understand what this wave does. 07:37.000 --> 07:42.270 So imagine the plane wave is coming from the blackboard 07:42.271 --> 07:43.641 towards you. 07:43.639 --> 07:45.949 If it's polarized this way--now I should tell you what 07:45.946 --> 07:46.726 polarization is. 07:46.730 --> 07:49.870 Polarization is the direction of E. 07:49.870 --> 07:52.500 B is understood to be perpendicular to it, 07:52.504 --> 07:55.314 and the direction of propagation is perpendicular to 07:55.305 --> 07:56.235 both of them. 07:56.240 --> 08:00.580 You can see E x B is the direction of propagation 08:00.579 --> 08:01.559 of the wave. 08:01.560 --> 08:04.530 You do the cross product from E to B and that's 08:04.529 --> 08:05.619 the way it advances. 08:05.620 --> 08:09.510 So what you will see is that if these rows in this classroom 08:09.514 --> 08:11.864 were not curved, but straight, 08:11.855 --> 08:16.535 then all the people in the first row will see exactly the 08:16.536 --> 08:18.456 same electric field. 08:18.459 --> 08:21.029 So maybe it's pointing up. 08:21.028 --> 08:24.628 Then people in the second row may see no electric field 08:24.627 --> 08:28.157 because they happen to be sitting at that distance. 08:28.160 --> 08:32.640 People in the third row may see a negative electric field. 08:32.639 --> 08:37.069 So every half-wave length it'll go from 0 to maximum to 0 to 08:37.067 --> 08:40.437 minimum back to 0, and so it oscillates at one 08:40.443 --> 08:41.123 time. 08:41.120 --> 08:44.380 If you wait a little bit the whole pattern will shift towards 08:44.384 --> 08:45.694 the back of the class. 08:45.690 --> 08:49.140 So what these guys saw now you will saw a little later. 08:49.139 --> 08:50.399 How much later? 08:50.399 --> 08:54.019 It's the time light takes to go from this row to your row. 08:54.019 --> 08:56.859 That's the delay, ωt, 08:56.860 --> 09:03.660 in fact I told you, you can write ky - ωt 09:03.660 --> 09:08.510 as-- let's see, as k times 09:08.514 --> 09:14.704 y - ct because ω = kc. 09:14.700 --> 09:18.520 And I told you very early on that any function of y - ct 09:18.524 --> 09:21.034 is a solution to wave equation, 09:21.028 --> 09:24.498 and this tells you there's a wave propagating to the right at 09:24.499 --> 09:27.439 speed c, because whatever it does at one 09:27.440 --> 09:29.790 time, say at t = 0, 09:29.791 --> 09:34.691 at later time it's shifted by an amount ct. 09:34.690 --> 09:38.290 Now if I send an electromagnetic wave towards 09:38.287 --> 09:40.907 you, first thing you should notice 09:40.913 --> 09:44.743 is the electric field is perpendicular to the plane, 09:44.740 --> 09:49.360 in which things are happening are normal to the direction of 09:49.364 --> 09:51.644 propagation, so the field vector is 09:51.644 --> 09:52.404 perpendicular. 09:52.399 --> 09:56.069 Both E and B are in this plane and the wave is 09:56.071 --> 09:57.131 going that way. 09:57.129 --> 09:59.989 For this reason it's called a transverse wave. 09:59.990 --> 10:05.310 So transverse means all the action is in a plane 10:05.312 --> 10:08.712 perpendicular to the motion. 10:08.710 --> 10:13.100 So one example of transverse waves is if you hold a string; 10:13.100 --> 10:16.120 I give it to the people in the last row and I hold it and then 10:16.116 --> 10:16.756 I shake it. 10:16.759 --> 10:19.939 These ripples will go from me towards the end of the room. 10:19.940 --> 10:23.780 The ripples are going like this but the wave is going that way. 10:23.779 --> 10:25.779 That's a transverse wave. 10:25.778 --> 10:28.318 Or if I shake it this way, same thing, it's going from 10:28.320 --> 10:29.040 side to side. 10:29.038 --> 10:32.248 The wave is going from me to you. 10:32.250 --> 10:34.830 A longitudinal wave is a sound wave. 10:34.830 --> 10:37.230 For example, when I talk now some diaphragm 10:37.230 --> 10:40.320 in the throat pushes the air out and compresses it, 10:40.320 --> 10:42.670 and maybe when it goes the other way decompress it, 10:42.668 --> 10:45.168 and the motion of the air is back and forth, 10:45.168 --> 10:47.818 namely in the same direction as the propagation. 10:47.820 --> 10:49.630 So sound waves are longitudinal, 10:49.633 --> 10:51.333 light waves are transverse. 10:51.330 --> 10:52.310 Yes? 10:52.308 --> 10:53.538 Student: You said the direction of 10:53.538 --> 10:56.248 E x B gives the direction of propagation. 10:56.250 --> 10:59.940 Is there a significance to the magnitude of that cross product? 10:59.940 --> 11:00.500 Prof: Yes. 11:00.500 --> 11:03.140 We'll come to that later today. 11:03.139 --> 11:03.959 That's correct. 11:03.960 --> 11:06.620 So E x B not only tells you the direction it's 11:06.620 --> 11:08.970 reasonable to ask "If it is a physical quantity, 11:08.966 --> 11:09.956 what does it mean? 11:09.960 --> 11:15.460 And we'll find out shortly what it means. 11:15.460 --> 11:19.630 Now polarization is the direction of the electric field 11:19.628 --> 11:23.098 in the solution, but we've got to realize that 11:23.101 --> 11:27.041 not every electric field has to look like this. 11:27.038 --> 11:33.538 I can make a new problem where I have solution looking like 11:33.543 --> 11:34.333 this. 11:34.330 --> 11:36.980 What is this guy doing? 11:36.980 --> 11:41.540 Can you tell it's going the other way, because I just 11:41.542 --> 11:43.562 changed that to that. 11:43.558 --> 11:47.208 So that's the plane wave going to the left. 11:47.210 --> 11:49.490 Now what should I do with the B field? 11:49.490 --> 11:53.530 Its magnitude should be E/c. 11:53.529 --> 11:56.389 I can also change this to going backward, otherwise they don't 11:56.392 --> 11:57.052 form a pair. 11:57.048 --> 11:59.368 E and B have to form a pair. 11:59.370 --> 12:02.350 But I claim this is not enough. 12:02.350 --> 12:04.670 If you took this pair it'll, of course, 12:04.668 --> 12:07.538 satisfy the wave equation because any function of y 12:07.538 --> 12:09.758 ct will satisfy the wave equation, 12:09.759 --> 12:12.689 but I want you to remember there are four Maxwell 12:12.687 --> 12:13.417 equations. 12:13.418 --> 12:15.198 You jiggle them all you get the wave equation. 12:15.200 --> 12:17.150 It's just one consequence of the four. 12:17.149 --> 12:20.549 You've got to satisfy all the Maxwell equations, 12:20.551 --> 12:24.751 and some of them won't be happy with this choice in which I 12:24.748 --> 12:26.918 reverse just the velocity. 12:26.918 --> 12:31.548 Can you think about what's the way to fix that or what's wrong 12:31.548 --> 12:32.458 with that? 12:32.460 --> 12:35.510 I'm just saying that if the wave traveled to the left 12:35.514 --> 12:38.634 instead of to the right that's what I've done here. 12:38.629 --> 12:40.989 I would just reverse the direction of propagation. 12:40.990 --> 12:42.550 That's not a good answer. 12:42.549 --> 12:44.749 Yes, you've got an answer? 12:44.750 --> 12:46.500 Student: Does it have to do with the fact that they're 12:46.495 --> 12:47.005 a cross product? 12:47.009 --> 12:48.639 Prof: Yeah, so what's about the cross 12:48.639 --> 12:49.019 product? 12:49.019 --> 12:51.649 Student: To make an inverse cross product you have 12:51.653 --> 12:52.673 to make it negative. 12:52.668 --> 12:55.558 Prof: You want the cross product to dictate the direction 12:55.558 --> 12:56.108 of motion. 12:56.110 --> 12:58.680 So if E and B are like this the cross product 12:58.677 --> 12:59.427 comes this way. 12:59.428 --> 13:02.518 If you want them to go the other way you've got to reverse 13:02.524 --> 13:05.244 the direction of B so it points that way. 13:05.240 --> 13:07.800 That is another pair. 13:07.798 --> 13:11.558 So this, in other words, you change the velocity here, 13:11.558 --> 13:13.748 but you forgot to change it--not you, 13:13.750 --> 13:17.020 I mean, I did not change it because I wanted to show you 13:17.015 --> 13:19.925 that you just cannot put together any E and 13:19.926 --> 13:21.526 B that you like. 13:21.528 --> 13:23.348 You've got to make sure that these form a pair. 13:23.350 --> 13:25.710 This is the pair that obeys Maxwell equation. 13:25.710 --> 13:28.400 Now I know this will work for another reason. 13:28.399 --> 13:30.689 If you took this long wave, okay? 13:30.690 --> 13:32.430 I want you to think in your head. 13:32.428 --> 13:36.108 Rotate it around this axis by 180 degrees. 13:36.110 --> 13:37.850 Can you do that in your head? 13:37.850 --> 13:40.830 Take that axis turn the whole pattern and it's going the 13:40.831 --> 13:43.471 opposite way, but in the process you can see 13:43.466 --> 13:46.936 this B will rotate from pointing this way to pointing 13:46.942 --> 13:47.652 that way. 13:47.649 --> 13:51.799 And one of the properties of nature is that if something is a 13:51.798 --> 13:55.598 solution, then you rotate the whole thing, that's also a 13:55.601 --> 13:56.501 solution. 13:56.500 --> 13:58.820 You've got to be careful when you say rotate. 13:58.820 --> 14:01.390 For example, if you have a grandfather clock 14:01.390 --> 14:04.920 and you rotate only the clock it won't work the same way. 14:04.918 --> 14:07.538 That's because the clock is very sensitive to the earth, 14:07.535 --> 14:10.285 but if you took the earth and the clock, you rotate both of 14:10.294 --> 14:11.584 them it doesn't matter. 14:11.580 --> 14:14.700 In fact, that's happening all the time. 14:14.700 --> 14:17.670 So all the things relevant should be rotated, 14:17.668 --> 14:21.648 but for electromagnetic field in vacuum there's nothing else 14:21.649 --> 14:24.619 to rotate that matters so you rotate it. 14:24.620 --> 14:27.450 Now another thing you can do instead of rotating around that 14:27.445 --> 14:30.215 axis you can rotate it around this axis, around the axis of 14:30.222 --> 14:30.992 propagation. 14:30.990 --> 14:34.450 You can turn E to that or you can turn B to 14:34.447 --> 14:34.877 that. 14:34.879 --> 14:37.859 I'm not good at drawing pictures, but in the E-B 14:37.855 --> 14:41.125 plane you had them like this, E and B, 14:41.126 --> 14:43.266 the thing is coming towards you. 14:43.269 --> 14:46.229 I'm saying you can turn this by some angle and turn that by the 14:46.229 --> 14:46.849 same angle. 14:46.850 --> 14:48.890 That'll also form a good solution. 14:48.889 --> 14:49.799 It must. 14:49.799 --> 14:50.499 I mean, what's wrong? 14:50.500 --> 14:53.160 If you look at a wave this way it should still obey Maxwell 14:53.158 --> 14:54.028 equations, right? 14:54.029 --> 14:58.199 The titled frame of reference is as good as the original one. 14:58.200 --> 15:02.800 All right, so now we've got a few properties of waves, 15:02.798 --> 15:07.748 and one thing maybe worth mentioning is the light from the 15:07.750 --> 15:12.960 light bulbs in this room is a chaotic set of waves being sent 15:12.964 --> 15:13.924 to you. 15:13.918 --> 15:16.918 Each atom emits light at a different polarization not in 15:16.919 --> 15:19.869 sync with the other atom, so it doesn't have a definite 15:19.865 --> 15:20.625 frequency. 15:20.629 --> 15:24.019 It's a big jumble, a big mess, but it's a 15:24.019 --> 15:27.579 superposition of elementary plane waves. 15:27.580 --> 15:29.360 Actually, I should amend myself. 15:29.360 --> 15:32.260 Plane waves are an idealization. 15:32.259 --> 15:34.979 Every wave you have, like if you turn on the light 15:34.982 --> 15:37.932 bulb the waves go out spherically from the center, 15:37.928 --> 15:40.688 but far from the center when this sphere is 10,000 miles in 15:40.691 --> 15:43.401 radius if you're a little creature at the end it will look 15:43.404 --> 15:46.244 like a plane wave, just like a sphere would look 15:46.241 --> 15:50.031 like a sheet to somebody who is very near like the earth does to 15:50.029 --> 15:50.389 us. 15:50.389 --> 15:52.439 Yep? 15:52.440 --> 15:54.040 Student: Professor, could you just say again what 15:54.043 --> 15:55.823 you said about switching the cross product or just ________? 15:55.820 --> 15:57.680 Prof: Yeah, I was saying that the property 15:57.676 --> 16:00.526 of the electromagnetic field that you saw here is that if you 16:00.528 --> 16:02.858 take the cross product, E x B... 16:02.860 --> 16:03.550 Student: You get the direction. 16:03.548 --> 16:05.018 Prof: You get the direction. 16:05.019 --> 16:09.299 I then took an opposite direction of propagation, 16:09.302 --> 16:13.052 but took the same E and B. 16:13.049 --> 16:14.379 They don't fit. 16:14.379 --> 16:16.579 If you reverse the velocity here you've got to change this 16:16.583 --> 16:18.903 so that the cross product of E and B vectors is 16:18.903 --> 16:19.913 pointing the opposite. 16:19.908 --> 16:24.768 Student: Okay, thank you. 16:24.769 --> 16:28.299 Prof: Now when you buy Polaroid glasses I think you 16:28.298 --> 16:29.968 know what they do, right? 16:29.970 --> 16:33.980 The polarizers in the glasses will allow light to travel in 16:33.980 --> 16:36.920 only one polarization, so even though the stuff coming 16:36.918 --> 16:40.118 in can be randomly polarized, what comes out of the other 16:40.119 --> 16:43.729 side towards your eye is pointing only one way, 16:43.730 --> 16:46.720 so you cut down about fifty percent of the light if you 16:46.715 --> 16:47.485 polarize it. 16:47.490 --> 16:50.830 In fact, when light reflects off a shiny surface and comes to 16:50.827 --> 16:53.607 your eyes it tends to be polarized horizontally. 16:53.610 --> 16:56.780 Therefore, if you choose your Polaroid to be vertical in your 16:56.782 --> 16:59.692 lens, that's the most effective way to cut the glare. 16:59.690 --> 17:04.180 Also, you can take two lenses, Polaroid lenses, 17:04.180 --> 17:07.670 and you turn them one relative to the other you can cancel the 17:07.673 --> 17:10.193 light completely, because the first one may let 17:10.188 --> 17:12.838 it go this way, the second one will only let it 17:12.835 --> 17:13.605 go that way. 17:13.608 --> 17:16.938 So if you chop it like this first then it won't go through 17:16.943 --> 17:17.883 the other way. 17:17.880 --> 17:20.090 Anyway, light is a very extensive and interesting 17:20.090 --> 17:22.810 subject which probably is more important to you than some of 17:22.809 --> 17:25.869 these fields, but we don't have time to go 17:25.869 --> 17:28.259 into all the aspects of that. 17:28.259 --> 17:32.339 I do want to mention to you that the light that you and I 17:32.340 --> 17:36.790 see has a very limited range of the possible wave lengths like 17:36.787 --> 17:39.627 400 nanometers or 4,000 angstroms, 17:39.630 --> 17:42.190 namely 400 times 10 to the minus 9 meters. 17:42.190 --> 17:44.820 To roughly double that is all you can see. 17:44.818 --> 17:47.348 Stuff on the other side is ultra violet, 17:47.348 --> 17:49.068 then you've x-rays, on the other side you've got 17:49.069 --> 17:50.919 infrared, you've got radio waves, 17:50.923 --> 17:53.323 but they're all electromagnetic waves. 17:53.318 --> 17:55.908 All you're doing is varying ω, 17:55.910 --> 17:58.580 but nature designed our eyes to respond only to a range of 17:58.582 --> 18:01.122 ω's, because that's where our 18:01.124 --> 18:02.064 enemies were. 18:02.058 --> 18:05.728 If you've got different enemies maybe you will have different 18:05.727 --> 18:06.397 eyesight. 18:06.400 --> 18:10.080 Maybe if you've got a lot of enemies you'll have eyes all 18:10.083 --> 18:13.573 over your head like some animals do, so we got two. 18:13.568 --> 18:17.398 Anyway, that's another interesting thing on how nature 18:17.403 --> 18:19.723 adapts to properties of light. 18:19.720 --> 18:24.540 All right, so let me tell you now about the energy contained 18:24.535 --> 18:25.755 in this wave. 18:25.759 --> 18:29.469 You've got to agree that when you have an electromagnetic wave 18:29.470 --> 18:32.330 you have energy that you did not have before. 18:32.329 --> 18:34.359 Let me ask you a first question. 18:34.358 --> 18:38.538 If I'm sending an electromagnetic wave towards 18:38.538 --> 18:42.718 you, and I ask you, which way is it polarized, 18:42.715 --> 18:47.075 what will you need to check that, any idea? 18:47.079 --> 18:47.779 Look, I've told you. 18:47.779 --> 18:49.909 If you cannot measure something or tell me in principle how 18:49.911 --> 18:51.861 we'll measure it you don't know what you're saying. 18:51.859 --> 18:52.479 Yes? 18:52.480 --> 18:54.160 You have an idea? 18:54.160 --> 18:54.770 Anybody? 18:54.769 --> 18:55.789 Yeah, go ahead. 18:55.788 --> 18:57.928 Student: Was there anything lost maybe? 18:57.930 --> 18:58.660 Prof: Pardon me? 18:58.660 --> 19:00.560 Student: Was there anything lost? 19:00.558 --> 19:02.428 Prof: Yeah, but forget the polarizing. 19:02.430 --> 19:08.490 Go back to basic definition. 19:08.490 --> 19:13.370 How will you know there's an electric field somewhere? 19:13.368 --> 19:16.668 You think you'd see a little arrow sticking out in space? 19:16.670 --> 19:17.390 Yes? 19:17.390 --> 19:18.910 Student: You take a test charge. 19:18.910 --> 19:22.210 Prof: Take a charge and you put it there and you see 19:22.205 --> 19:23.395 which way it moves. 19:23.400 --> 19:26.010 If it moves like this that's the polarization, 19:26.009 --> 19:28.849 if it moves like that, that's the polarization. 19:28.848 --> 19:31.818 In fact, the antenna on your radio has got a wire and the 19:31.821 --> 19:35.011 electric field from the radio station comes and starts moving 19:35.006 --> 19:37.816 the charges in the direction of the polarization. 19:37.819 --> 19:38.759 So you guys should know this. 19:38.759 --> 19:42.029 I'm going to assume that you didn't answer me because you're 19:42.029 --> 19:44.519 unusually modest, because you must know how to 19:44.522 --> 19:46.022 measure electric field. 19:46.019 --> 19:48.429 I could even ask you, how would you find the magnetic 19:48.426 --> 19:51.246 field," But I don't want to do that now because we should 19:51.251 --> 19:53.291 always know how to measure these things. 19:53.288 --> 19:56.258 So anyway, we know that when you have an electric field it 19:56.262 --> 19:57.152 has got energy. 19:57.150 --> 19:58.970 For example, when you took a capacitor, 19:58.968 --> 20:01.838 it took some energy to charge it, to rip the charges from one 20:01.838 --> 20:03.848 plate and ram them in the other plate. 20:03.848 --> 20:06.918 We saw that as an energy stored between the plates in the form 20:06.921 --> 20:11.061 of an electric field, and the energy of the electric 20:11.064 --> 20:14.294 field was ε_0 20:14.291 --> 20:15.761 E^(2)/2. 20:15.759 --> 20:19.649 Now you may ask me, you found the energy density in 20:19.652 --> 20:22.292 a capacitor, now we're talking about the 20:22.288 --> 20:24.958 electric field in vacuum traveling from some radio 20:24.955 --> 20:27.955 station, how do I know this formula is 20:27.962 --> 20:28.452 good? 20:28.450 --> 20:32.380 And the remarkable thing about the field is that any expression 20:32.384 --> 20:35.054 you derive for it is a local expression. 20:35.048 --> 20:37.638 It only cares about what the field is at this point. 20:37.640 --> 20:39.260 It doesn't care what the origin is. 20:39.259 --> 20:42.689 It doesn't matter if this is produced by static charges or 20:42.689 --> 20:46.059 maybe it's an electric field produce by changing magnetic 20:46.057 --> 20:46.657 field. 20:46.660 --> 20:48.200 It does not matter. 20:48.200 --> 20:50.390 The answer does not depend on the context. 20:50.390 --> 20:53.410 This is the energy density, energy per unit volume. 20:53.410 --> 20:56.750 For a magnetic field it looks like 20:56.749 --> 21:01.759 B^(2)/2μ_0, and you all know by now that 21:01.756 --> 21:05.056 whatever ε_0 does μ_0 does in 21:05.057 --> 21:06.117 the other place. 21:06.118 --> 21:09.918 If this guy's up that guy's down, or in the field laws 21:09.922 --> 21:14.152 μ_0/2Π's up and 1/4Πε_0 21:14.154 --> 21:15.594 epsilon is down. 21:15.588 --> 21:17.448 Therefore, if you have a region where there is, 21:17.450 --> 21:21.790 let's say, no electromagnetic field and suddenly a wave goes 21:21.788 --> 21:24.508 by that region has now got energy, 21:24.509 --> 21:26.459 and how much energy do we have? 21:26.460 --> 21:29.170 The electrical energy, you can see, 21:29.170 --> 21:34.660 is ε_0 over 2E_0^(2) 21:34.663 --> 21:38.623 sin^(2)(ky - ωt). 21:38.618 --> 21:45.248 And the magnetic energy is 1 over 2μ_0 21:45.246 --> 21:50.406 B_0^(2) sin^(2)(ky - 21:50.413 --> 21:52.803 ωt). 21:52.798 --> 21:56.508 I will now show you that these energy densities are actually 21:56.507 --> 21:57.007 equal. 21:57.009 --> 22:01.319 They're equal because ½μ_0-- 22:01.318 --> 22:05.348 B_0 is E/c, 22:05.348 --> 22:11.058 so let's write it as E_0^(2)/c^(2). 22:11.058 --> 22:13.888 From the definition of the velocity of light and 22:13.894 --> 22:16.854 μ_0 and ε_0 this is 22:16.848 --> 22:20.708 simply ε_0 over 2E_0^(2). 22:20.710 --> 22:24.840 Sorry, E_0^(2) plus all the sine squares, 22:24.837 --> 22:26.067 which I forgot. 22:26.068 --> 22:33.488 Therefore, the total energy is equal to E_0^(2) 22:33.491 --> 22:38.531 ε_0 sin^(2)(ky - 22:38.525 --> 22:40.785 ωt). 22:40.788 --> 22:43.298 You got a half from this and you got a half from that. 22:43.298 --> 22:46.038 Even though the formula looks different by the time you put in 22:46.040 --> 22:47.840 the relation between μ_0 and 22:47.838 --> 22:50.038 ε_0 and B_0 and 22:50.041 --> 22:52.381 E_0 it turns out to be equal. 22:52.380 --> 22:54.910 So even though the magnetic field is weaker than the 22:54.909 --> 22:57.879 electric field by a factor of 1/c you might think it's 22:57.884 --> 23:01.654 negligible in terms of energy, but it's got same energy 23:01.646 --> 23:05.276 density, and you add them up you get that. 23:05.278 --> 23:09.158 Now you can see this energy density is time dependent and 23:09.160 --> 23:13.320 space dependent because it's oscillating with time at a given 23:13.317 --> 23:16.197 point, and oscillating with space at a 23:16.199 --> 23:17.019 given time. 23:17.019 --> 23:19.649 So you can ask yourself, let me sit at one place and ask 23:19.651 --> 23:21.571 myself, "What does the average 23:21.573 --> 23:24.143 energy density average over a full cycle?" 23:24.140 --> 23:26.940 If something is up sometime and down sometime what's the 23:26.940 --> 23:27.450 average? 23:27.450 --> 23:30.510 The average will be ε_0 23:30.511 --> 23:35.111 E_0^(2)/2, because the average value of 23:35.111 --> 23:39.391 sin^(2)θ over a full cycle is 0 to 2Π 23:39.385 --> 23:42.505 divided by 2Π which is ½. 23:42.509 --> 23:44.629 This is something we have done before in circuits. 23:44.630 --> 23:45.830 You should know that. 23:45.828 --> 23:49.048 Average of sine squared is half, average of cosine squared 23:49.048 --> 23:51.228 is half, and the check is that sine 23:51.231 --> 23:54.601 squared plus cosine squared average is 1 because that's 1 23:54.597 --> 23:55.497 identically. 23:55.500 --> 23:56.970 So try to remember this. 23:56.970 --> 24:00.960 So this is the average energy density. 24:00.960 --> 24:05.040 Next I want to ask the following question. 24:05.038 --> 24:10.938 What is the rate at which energy is coming at me? 24:10.940 --> 24:16.600 And that is called intensity, I, and it is equal to the watts 24:16.596 --> 24:20.746 per meter squared, just like a flow of liquid 24:20.746 --> 24:24.326 except this is the flow of energy. 24:24.328 --> 24:27.388 So what I want to do is take a square meter, 24:27.390 --> 24:31.570 stand in the way of the beam and ask how many joules cross 24:31.571 --> 24:34.201 per second, or if you want, 24:34.202 --> 24:37.732 how many watts per square meter. 24:37.730 --> 24:42.230 That's easily calculated from the energy density and I'll give 24:42.233 --> 24:43.493 you the reason. 24:43.490 --> 24:47.060 It's identical to the reasoning for currents traveling in a 24:47.063 --> 24:49.223 wire, or fluid flowing in a tube. 24:49.220 --> 24:52.840 Suppose you take a region, a cylindrical cross section 24:52.838 --> 24:56.388 through which electromagnetic waves are traveling? 24:56.390 --> 25:02.740 This is cross section A and I wait 1 second it'll go a 25:02.741 --> 25:05.141 distance c. 25:05.140 --> 25:09.270 So all the stuff in a cylinder of base A and length 25:09.265 --> 25:13.025 c that energy will go past this checkpoint. 25:13.029 --> 25:13.689 Can you see that? 25:13.690 --> 25:14.360 It's like toothpaste. 25:14.358 --> 25:18.128 You squeeze the toothpaste these guys get passed in 1 25:18.126 --> 25:18.776 second. 25:18.778 --> 25:23.748 If I want per square meter we'll just call that 1. 25:23.750 --> 25:29.490 Therefore the intensity is simply the energy density 25:29.486 --> 25:33.756 multiplied by the velocity of light. 25:33.759 --> 25:39.359 That's the rate at which the energy flows. 25:39.358 --> 25:43.368 So let us now calculate that and you'll get a very 25:43.371 --> 25:45.091 interesting result. 25:45.088 --> 25:52.428 The intensity is equal to the energy density times c. 25:52.430 --> 25:57.390 The energy density was ε_0 25:57.386 --> 26:03.956 E_0^(2) sin^(2)(ky - ωt). 26:03.960 --> 26:05.180 This is not the average energy. 26:05.180 --> 26:06.960 This is instantaneous. 26:06.960 --> 26:09.370 I'll average it in a moment. 26:09.368 --> 26:15.818 Are you guys with me now somewhere here, 26:15.820 --> 26:16.980 here? 26:16.980 --> 26:21.220 Now I'm going to write it as follows, ε_0 26:21.224 --> 26:26.204 and E_0 then at B_0--I 26:26.204 --> 26:28.414 forgot a c here. 26:28.410 --> 26:29.250 You realize that? 26:29.250 --> 26:31.530 That's another c. 26:31.528 --> 26:35.898 Then E_0 is B_0 times another c 26:35.904 --> 26:39.234 times sin^(2)(ky - ωt). 26:39.230 --> 26:40.920 And what is that? 26:40.920 --> 26:43.070 c^(2) ε_0 is 26:43.067 --> 26:44.317 1/μ_0. 26:44.318 --> 26:50.068 So E_0 B_0 over 26:50.068 --> 26:56.938 μ_0 sin^(2)(ky - ωt). 26:56.940 --> 27:01.540 But if you now define a vector S, it's called a 27:01.536 --> 27:06.506 Poynting vector, which is spelled with a p, 27:06.513 --> 27:12.563 to be E x B over μ_0. 27:12.558 --> 27:16.498 The magnitude of that vector is precisely the intensity. 27:16.500 --> 27:20.420 The magnitude of S, which you can either call 27:20.422 --> 27:24.732 S or you can call it I, is exactly this. 27:24.730 --> 27:28.220 So E x B except for a factor 1/μ_0 27:28.224 --> 27:31.724 not only gives you the direction of the way of propagation it 27:31.721 --> 27:35.451 tells you how many watts are going to cross a square meter, 27:35.450 --> 27:41.220 or how many joules are going to cross square meter in 1 second. 27:41.220 --> 27:44.060 Now these are things you'll find in all the textbooks, 27:44.060 --> 27:44.330 so. 27:44.328 --> 27:47.348 All I've done is take this u c, put it in the formulas with 27:47.346 --> 27:50.516 E and bring B back into the picture so that it's 27:50.519 --> 27:52.859 symmetric between E and B, 27:52.858 --> 27:54.728 because this gives the impression it's all electric. 27:54.730 --> 27:56.350 Remember, this is electric and magnetic. 27:56.349 --> 27:57.909 They just happen to be equal. 27:57.910 --> 28:01.840 This way you can see the role played by E and B. 28:01.838 --> 28:05.938 The Poynting vector tells you the flux of energy. 28:05.940 --> 28:07.110 So here's one example. 28:07.108 --> 28:12.048 At the surface of the earth if you take a square meter, 28:12.047 --> 28:15.427 and here's the sun, emitting light. 28:15.430 --> 28:18.340 You can ask, "What's the intensity of 28:18.342 --> 28:19.552 sunlight?" 28:19.548 --> 28:25.038 Anybody have an idea how many watts per square meter from the 28:25.039 --> 28:25.589 sun. 28:25.589 --> 28:28.429 Student: A thousand? 28:28.430 --> 28:30.690 Prof: Yes, very close to a thousand. 28:30.690 --> 28:34.530 I think there's some other numbers, but for our purposes 28:34.529 --> 28:38.089 it's very close to 1,000 watts per meter squared. 28:38.088 --> 28:42.658 That's pretty amazing over the entire surface of the earth 28:42.660 --> 28:46.670 every second the sun is pumping in 1,000 joules. 28:46.670 --> 28:49.430 And you've got to remember the context of the sun. 28:49.430 --> 28:53.390 I mean, here is the sun and here we are ninety-three million 28:53.394 --> 28:57.834 miles away, and the light energy is going and this is our share. 28:57.828 --> 29:02.158 A tiny circle like 7,000 miles in diameter you're intercepting 29:02.163 --> 29:06.003 the light, and every square meter of it gets a thousand 29:05.999 --> 29:06.709 watts. 29:06.710 --> 29:09.000 You can see the stuff coming out of the sun, 29:08.997 --> 29:10.857 its a prodigious amount of light. 29:10.858 --> 29:13.618 Anyway, that's the electric field. 29:13.618 --> 29:17.168 I mean, that's the intensity, so let me write the following 29:17.170 --> 29:17.660 thing. 29:17.660 --> 29:21.230 This is oscillating rapidly with time, so let's define an 29:21.230 --> 29:24.740 average intensity as the average of the sine squared. 29:24.740 --> 29:26.980 That's E_0B _0 29:26.981 --> 29:28.161 /2μ_0. 29:28.160 --> 29:31.960 All these averages are simply half, anything involving in sine 29:31.957 --> 29:32.577 squared. 29:32.578 --> 29:34.138 I'm not going to worry about the half, 29:34.140 --> 29:37.040 but if you took this to be the average intensity you can ask, 29:37.038 --> 29:40.908 "How big is the electric field that comes with it," 29:40.913 --> 29:44.723 because that light is going to be this electric and magnetic 29:44.723 --> 29:45.373 field. 29:45.369 --> 29:47.189 That's all light is. 29:47.190 --> 29:49.400 So the sunlight produces electromagnetic field, 29:49.402 --> 29:52.482 is electromagnetic waves and I'm asking, "How big is the 29:52.480 --> 29:53.780 E vector?" 29:53.779 --> 29:56.449 And all you have to do is stick that into this number. 29:56.450 --> 29:59.230 If you want you can get rid of B and go back to E 29:59.226 --> 30:01.506 because E is just B times c. 30:01.509 --> 30:04.029 You'll find a pretty surprising amount. 30:04.028 --> 30:07.878 It's roughly 1,000 volts per meter. 30:07.880 --> 30:11.510 Remember electric field is measured in volts per meter. 30:11.509 --> 30:14.779 What that means is if you took the field, 30:14.778 --> 30:17.688 and it's uniform in space, between one place and another 30:17.690 --> 30:20.600 place there's a potential difference of 1,000 volts, 30:20.598 --> 30:23.888 or it'll take 1,000 joules to shove a coulomb from lower 30:23.886 --> 30:25.796 potential to higher potential. 30:25.798 --> 30:29.788 That's a pretty strong field, but it's very incoherent in 30:29.794 --> 30:30.654 direction. 30:30.650 --> 30:33.230 It's doing this for a while and doing that for a while, 30:33.230 --> 30:36.230 but if you roughly approximate it and just ask, 30:36.230 --> 30:41.280 "What is the average field strength," 30:41.277 --> 30:44.457 this is the number you get. 30:44.460 --> 30:49.110 So let's talk about one thing which I've not discussed at all, 30:49.107 --> 30:53.217 which is, where are these electromagnetic fields coming 30:53.223 --> 30:53.913 from? 30:53.910 --> 30:56.870 I said you don't need ρ, you don't need I, 30:56.868 --> 31:00.348 you don't need the current, you don't need the charge, 31:00.354 --> 31:02.594 these can exist in free space. 31:02.588 --> 31:04.848 But what's the origin of the electromagnetic waves? 31:04.849 --> 31:05.979 Anybody know? 31:05.980 --> 31:07.990 When can you get them? 31:07.990 --> 31:10.040 It didn't happen in anything we studied. 31:10.039 --> 31:10.839 Yes? 31:10.838 --> 31:12.848 Student: Electron changes energy level? 31:12.848 --> 31:14.838 Prof: Well, you can say in the atom the 31:14.836 --> 31:17.436 electron changes energy level, but before you do the quantum 31:17.441 --> 31:18.061 mechanics. 31:18.058 --> 31:22.058 In classical theory one can ask, what is the electric field? 31:22.058 --> 31:26.598 When does the electric field produce these waves? 31:26.598 --> 31:30.478 You take a static charge it has a 1/r^(2) field which is 31:30.480 --> 31:31.920 pinned to the charge. 31:31.920 --> 31:33.560 If you go too far from the charge you don't see it, 31:33.558 --> 31:37.148 but these guys can go off into space without being anywhere 31:37.146 --> 31:43.956 near a charge or current, but what produces them? 31:43.960 --> 31:45.540 Yes? 31:45.539 --> 31:46.389 You don't know? 31:46.390 --> 31:46.970 Yep? 31:46.970 --> 31:48.380 Student: When it oscillates. 31:48.380 --> 31:48.850 Prof: Pardon me? 31:48.848 --> 31:49.688 Student: When is oscillates. 31:49.690 --> 31:51.000 Prof: Which oscillates? 31:51.000 --> 31:52.520 Student: The electric field. 31:52.519 --> 31:55.279 Prof: The electric field is itself oscillating, 31:55.276 --> 31:57.456 but what's producing the electric field? 31:57.460 --> 32:00.830 What's the cause of the electric field? 32:00.829 --> 32:01.839 Electric field fairy? 32:01.839 --> 32:02.429 Yes? 32:02.430 --> 32:04.410 Student: When a charge oscillates? 32:04.410 --> 32:06.530 Prof: Yes. 32:06.528 --> 32:09.638 In other words, rather than oscillates the more 32:09.644 --> 32:13.034 general answer is whenever a charge accelerates. 32:13.029 --> 32:16.349 This is a very important result. 32:16.348 --> 32:29.808 Waves are produced by accelerating charges. 32:29.808 --> 32:32.708 If they travel at uniform velocity like in a wire they 32:32.714 --> 32:35.514 don't produce oscillations, electromagnetic waves, 32:35.510 --> 32:38.290 or if they're going in a circuit at constant rate that 32:38.285 --> 32:39.275 doesn't produce. 32:39.279 --> 32:42.209 But if you have charges which are say oscillating is one 32:42.209 --> 32:44.339 example when things are accelerating, 32:44.338 --> 32:48.088 right, because you're going back and forth when you radiate 32:48.092 --> 32:48.612 light. 32:48.608 --> 32:52.598 So every single source of electromagnetic wave is 32:52.599 --> 32:54.429 oscillating charges. 32:54.430 --> 32:56.640 You can sort of imagine why that will happen. 32:56.640 --> 32:59.380 I mean, if you took a capacitor plate, 32:59.380 --> 33:01.460 and you connect it to an AC source, 33:01.460 --> 33:04.790 let's say, there may be other things to keep it from burning 33:04.785 --> 33:06.895 out, then what will happen is 33:06.898 --> 33:11.128 charges will be like this for a while then they've got to go 33:11.126 --> 33:14.706 back and forth in order to change polarity with the 33:14.710 --> 33:18.200 alternating field, therefore the charges are going 33:18.199 --> 33:20.419 back and forth, and you have a time dependent 33:20.422 --> 33:21.332 electric field here. 33:21.328 --> 33:24.558 When you have a time dependent electric field you'll have a 33:24.561 --> 33:27.911 magnetic field going around it because the line integral of B 33:27.905 --> 33:30.855 will involve the rate of change of electric flux. 33:30.858 --> 33:33.458 And that will also be time dependent, but if that gets time 33:33.460 --> 33:35.750 dependent there'll be an electric field going around 33:35.746 --> 33:36.146 that. 33:36.150 --> 33:39.190 So basically these will curl around each other whenever 33:39.193 --> 33:42.073 they're dependent on time, and they can then free 33:42.073 --> 33:44.903 themselves loose from the capacitor and take off. 33:44.900 --> 33:47.870 All you need is two plates, and an AC source, 33:47.874 --> 33:51.254 and you connect them, you will make electromagnetic 33:51.253 --> 33:51.933 waves. 33:51.930 --> 33:53.920 You'll make them at the frequency of the source, 33:53.921 --> 33:55.321 so you won't be able to see it. 33:55.318 --> 33:58.438 Your dog won't be able to see, but some gadget will be able to 33:58.443 --> 33:59.113 pick it up. 33:59.108 --> 34:03.048 That's all you need, oscillating charges. 34:03.048 --> 34:07.348 So what happens in a radio station is you could imagine a 34:07.354 --> 34:10.664 simple radio station with an LC circuit, 34:10.659 --> 34:12.389 the current is oscillating at some rate, 34:12.389 --> 34:14.619 and part of the circuit is in the antenna, 34:14.619 --> 34:17.499 and the charges are going up and down as the current goes 34:17.498 --> 34:19.708 back and forth that sends out the waves, 34:19.710 --> 34:20.760 and the waves come to your house. 34:20.760 --> 34:22.120 So here's the picture. 34:22.119 --> 34:26.299 Here's the radio station, and the waves are emitted in 34:26.295 --> 34:30.855 big circles, and this is your house, and here's your little 34:30.864 --> 34:31.814 antenna. 34:31.809 --> 34:35.489 It's a piece of wire, and the electric field, 34:35.485 --> 34:40.075 if it's polarized this way, will move the charges up and 34:40.079 --> 34:44.339 down, and charges can be part of an LC circuit. 34:44.340 --> 34:46.170 So here are the antennas, if you like; 34:46.170 --> 34:47.360 part of the circuit. 34:47.360 --> 34:51.480 As the charge goes up and down the AC current will try to flow 34:51.476 --> 34:51.946 here. 34:51.949 --> 34:55.319 And if you tune this capacitor so that it resonates with the 34:55.322 --> 34:58.812 frequency you'll get a hefty signal from the radio station. 34:58.809 --> 35:00.779 So in the end it's all charges. 35:00.780 --> 35:02.310 Charges produce the field. 35:02.309 --> 35:03.759 Charges respond to the field. 35:03.760 --> 35:05.040 That was true in the static case. 35:05.039 --> 35:07.009 That's true in the time dependent case. 35:07.010 --> 35:08.000 Yes? 35:08.000 --> 35:10.780 Student: When you say that the field if it's loose 35:10.777 --> 35:13.167 from the capacitor, what do you mean free from the 35:13.166 --> 35:13.846 capacitor? 35:13.849 --> 35:17.299 Prof: It means it can go thousands of miles from the 35:17.295 --> 35:18.005 capacitor. 35:18.010 --> 35:20.760 And if you simply look at the Coulomb force due to the 35:20.764 --> 35:23.314 charges, the plus and minus charges, they die like 35:23.311 --> 35:24.301 1/r^(2). 35:24.300 --> 35:27.360 So if you calculate 1/r^(2) you will get a 35:27.360 --> 35:31.060 negligible number compared to the actual electric field. 35:31.059 --> 35:33.299 So it's really like you and your parents. 35:33.300 --> 35:37.120 I mean, at some point you are free from your parents. 35:37.119 --> 35:39.669 You are able to manage on your own, but you had parents 35:39.668 --> 35:40.988 somewhere sometime, right? 35:40.989 --> 35:41.819 That's what it is. 35:41.820 --> 35:44.090 The electromagnetic waves can go on their own, 35:44.090 --> 35:46.160 but they are not produced on their own. 35:46.159 --> 35:47.909 They're produced by charges. 35:47.909 --> 35:49.649 It is just that unlike the static fields, 35:49.650 --> 35:52.800 which are very near the currents and charges that 35:52.800 --> 35:55.760 produce them, the time dependent fields 35:55.762 --> 35:57.602 propagate on their own. 35:57.599 --> 36:00.499 E keeps B alive and B keeps E 36:00.496 --> 36:00.906 alive. 36:00.909 --> 36:02.209 If E tries to die there's a 36:02.210 --> 36:04.340 dE/dT that produces a B. 36:04.340 --> 36:06.490 If the B tries to go down it produces at 36:06.485 --> 36:08.905 dB/dT that produces an E, 36:08.909 --> 36:10.309 so they go back and forth. 36:10.309 --> 36:12.709 It's really like oscillations in which you have kinetic to 36:12.706 --> 36:13.586 potential transfer. 36:13.590 --> 36:15.940 You can have energy transfer. 36:15.940 --> 36:17.880 So the fields cannot die. 36:17.880 --> 36:21.230 They are self-sustaining, but to get all of that physics 36:21.226 --> 36:24.386 you had to put the term that Mr. Maxwell put in. 36:24.389 --> 36:27.189 Without that term you don't have this phenomenon. 36:27.190 --> 36:29.340 You don't get it from statics. 36:29.340 --> 36:33.160 So one typical problem you have is this radio station is, 36:33.159 --> 36:37.679 let's say, 100 kilowatts and you're sitting here at some 36:37.677 --> 36:41.207 distance r from the radio station. 36:41.210 --> 36:44.880 Then the intensity in your house will be 100 kilowatts 36:44.875 --> 36:47.775 spread over a sphere of radius R. 36:47.780 --> 36:51.650 That will be your intensity. 36:51.650 --> 36:54.050 Then you can go from the intensity and translate it to an 36:54.052 --> 36:56.252 electric field and say, "The electric field 36:56.246 --> 36:58.846 produced by the radio station oscillates with the following 36:58.847 --> 36:59.427 amplitude. 36:59.429 --> 37:05.469 I've got to build a circuit that's smart enough to pick up 37:05.474 --> 37:08.024 that tiny field." 37:08.018 --> 37:13.308 Okay, so I want to switch now to my favorite theme. 37:13.309 --> 37:18.639 The remarkable thing about electromagnetism is that you can 37:18.635 --> 37:23.865 ask what happened when physics went through the Einstein's 37:23.869 --> 37:27.449 revolution with special relativity. 37:27.449 --> 37:30.459 We know everything changed after Einstein, 37:30.463 --> 37:34.363 and all the Newtonian mechanics had to be modified. 37:34.360 --> 37:37.040 And so far I never mentioned the word relativity, 37:37.041 --> 37:40.121 so you can ask yourself, "How are these modified by 37:40.115 --> 37:41.565 Einstein's work?" 37:41.570 --> 37:46.450 So first let me tell you, remind you, now you guys did 37:46.454 --> 37:49.224 relativity last term, right? 37:49.219 --> 37:53.649 Is there anybody who's never seen it before? 37:53.650 --> 37:55.980 Okay, well you don't have to know a whole lot, 37:55.976 --> 37:57.836 but let me just say the following. 37:57.840 --> 38:02.240 In the Newtonian world there was a principle of relativity 38:02.239 --> 38:07.409 according to which the equations like F = ma you can ask, 38:07.409 --> 38:11.719 "Who is allowed to use this equation?" 38:11.719 --> 38:17.459 It says by definition an inertial observer can use this 38:17.460 --> 38:18.630 equation. 38:18.630 --> 38:20.400 And you say, "Who is an inertial 38:20.402 --> 38:21.242 observer?" 38:21.239 --> 38:23.739 You say, "Inertial observer is anybody who can use 38:23.740 --> 38:24.760 this equation." 38:24.760 --> 38:28.340 Sort of seems to be meaningless tautology. 38:28.340 --> 38:31.290 What makes it meaningful is the following. 38:31.289 --> 38:34.199 There are some people, at least, for whom this 38:34.199 --> 38:35.299 equation works. 38:35.300 --> 38:37.910 So there are at least some inertial observers. 38:37.909 --> 38:40.189 For example, we are an inertial observer 38:40.190 --> 38:43.290 because if you want to test this equation you can say, 38:43.289 --> 38:45.629 "I leave a piece of chalk here. 38:45.630 --> 38:46.990 I don't apply force. 38:46.989 --> 38:48.579 Does it have an acceleration?" 38:48.579 --> 38:50.149 It doesn't. 38:50.150 --> 38:53.530 Okay, so I'm obeying at least the first of the three laws of 38:53.532 --> 38:54.052 Newton. 38:54.050 --> 38:57.500 On the other hand, if I leave my iPod in Grand 38:57.503 --> 39:01.653 Central Station and I come back it's gone, that's not a 39:01.650 --> 39:04.030 violation of Newton's Laws. 39:04.030 --> 39:08.070 That just means I'm stupid because there are other forces 39:08.065 --> 39:11.375 acting on that iPod, and those forces moved it, 39:11.380 --> 39:13.470 so I can understand that. 39:13.469 --> 39:16.789 So I won't be that traumatized by the loss of the iPod because 39:16.791 --> 39:19.081 I feel as I know there's an explanation. 39:19.079 --> 39:22.929 But if you go to a plane and the plane is about to take off 39:22.929 --> 39:26.909 you leave stuff on the floor it will slide to the rear end of 39:26.909 --> 39:27.839 the plane. 39:27.840 --> 39:30.280 You have no F, but you have an a. 39:30.280 --> 39:33.700 That means an accelerating plane is a reference frame in 39:33.695 --> 39:36.175 which people cannot use F = ma. 39:36.179 --> 39:38.869 Stuff will accelerate for no apparent reason, 39:38.867 --> 39:40.697 so not everybody's inertial. 39:40.699 --> 39:43.309 So you can ask, "If I have at least one 39:43.309 --> 39:46.099 inertial observer in the universe does it imply 39:46.101 --> 39:48.591 others?," and the answer is yes. 39:48.590 --> 39:53.060 If I'm inertial and you move relative to me at constant 39:53.059 --> 39:55.709 velocity you're also inertial. 39:55.710 --> 39:57.890 There's a large number of people in the universe all 39:57.894 --> 40:00.384 allowed to use Newton's Laws if you've got one of them, 40:00.380 --> 40:03.810 and they differ by velocity, constant velocity, 40:03.809 --> 40:06.339 and we can understand that from Newton's Laws. 40:06.340 --> 40:09.450 Here is a mass and spring system. 40:09.449 --> 40:14.779 Newton's Law takes the form -kx = ma, 40:14.775 --> 40:19.245 let's say, or let's write mdv/dt. 40:19.250 --> 40:23.800 Now if you go to a train or you carry this on a train and you go 40:23.797 --> 40:28.197 at constant velocity I know it will obey this equation because 40:28.199 --> 40:30.149 I've not done anything. 40:30.150 --> 40:32.260 It's not my fault you're going on a train and this is riding 40:32.260 --> 40:33.120 with you on the train. 40:33.119 --> 40:35.179 It will obey this equation. 40:35.179 --> 40:38.769 By the way, this x is really x - x_0, 40:38.768 --> 40:41.188 where x_0 is the rest length of the spring 40:41.190 --> 40:43.610 and x - x_0 is the deviation from that. 40:43.610 --> 40:48.410 That's the equation. 40:48.409 --> 40:51.109 Now you and I differ by what? 40:51.110 --> 40:53.780 You and I differ by a constant velocity. 40:53.780 --> 40:59.410 The constant velocity just means x prime is equal to x - 40:59.405 --> 41:00.275 ut. 41:00.280 --> 41:02.920 Our origins are differing by an amount ut after time 41:02.922 --> 41:05.972 T, so if an event occurs here for 41:05.965 --> 41:10.705 me, you have moved a distance ut and it occurs at a 41:10.706 --> 41:14.446 distance x' which is x - ut. 41:14.449 --> 41:17.739 Then you can see the laws are going to work for you also 41:17.735 --> 41:21.495 because even though we disagree on velocity we don't disagree on 41:21.498 --> 41:23.588 the rate of change of velocity. 41:23.590 --> 41:25.940 Your velocity and mine differ by a constant, 41:25.943 --> 41:28.413 which has no derivative, so they have the same 41:28.409 --> 41:29.339 acceleration. 41:29.340 --> 41:31.620 As far as the spring is concerned if I think it's 41:31.621 --> 41:34.671 stretched by two inches you will also think it's stretched by two 41:34.666 --> 41:35.186 inches. 41:35.190 --> 41:37.430 I mean, your x is zooming to the right according 41:37.434 --> 41:39.184 to me and your x_0 is also 41:39.179 --> 41:42.299 zooming to the right, but the extension of the spring 41:42.295 --> 41:43.145 is the same. 41:43.150 --> 41:47.110 So F = ma doesn't have to be modified by going to a 41:47.108 --> 41:48.148 moving frame. 41:48.150 --> 41:51.260 And that is the relativity of Newtonian mechanics. 41:51.260 --> 41:54.710 If these are the laws of motion we can understand why if you're 41:54.708 --> 41:57.088 inside a train, which is completely closed, 41:57.085 --> 41:59.985 you cannot look outside and it's going at uniform velocity 41:59.989 --> 42:03.609 with respect to the ground, you cannot tell. 42:03.610 --> 42:07.150 You cannot tell because nothing you do will be different. 42:07.150 --> 42:09.350 Because everything you observe in the world is controlled by 42:09.353 --> 42:11.033 Newton's Laws, and Newton's Laws are 42:11.034 --> 42:13.824 unaffected by adding a constant velocity to everything. 42:13.820 --> 42:16.310 So when you wake up and I say, "Is the train 42:16.309 --> 42:17.969 moving," you cannot tell. 42:17.969 --> 42:24.569 Okay, so that's by doing physics experiments. 42:24.570 --> 42:26.890 You cannot say, "It says Amtrak so I know 42:26.891 --> 42:28.131 it's not moving." 42:28.130 --> 42:31.610 That kind of argument is based on sociological axioms. 42:31.610 --> 42:34.510 I'm just saying can you--after all it's possible, 42:34.505 --> 42:36.735 theoretically, Amtrak trains can move, 42:36.737 --> 42:39.027 so you must admit the possibility. 42:39.030 --> 42:43.970 Okay, now you come to Maxwell's equations and electromagnetic 42:43.969 --> 42:44.709 theory. 42:44.710 --> 42:46.670 Let me write down what we have. 42:46.670 --> 42:51.560 The first thing we have is q equals--I'm sorry, 42:51.563 --> 42:56.093 q times E v x B is the 42:56.085 --> 42:57.005 force. 42:57.010 --> 42:59.760 Then let me write down one other consequence, 42:59.760 --> 43:06.150 d^(2)E/dx^(2) − (1/c^(2))d^(2)E 43:06.148 --> 43:09.758 /dt^(2) is 0. 43:09.760 --> 43:13.750 These are some of the results we got from electromagnetic 43:13.748 --> 43:14.388 theory. 43:14.389 --> 43:18.049 Now comes the important question. 43:18.050 --> 43:22.610 This is the velocity of a particle according to whom? 43:22.610 --> 43:25.820 That question came up earlier in the class. 43:25.820 --> 43:29.300 Who is supposed to use it? 43:29.300 --> 43:32.770 I may assume it worked for me, but how do I know then when you 43:32.766 --> 43:36.006 see it it's got a different velocity will you get the same 43:36.005 --> 43:37.705 physical world that I get. 43:37.710 --> 43:41.100 Now let's look at this equation, this also came from 43:41.101 --> 43:44.431 Maxwell's equations, and compare the equation for a 43:44.427 --> 43:45.157 string. 43:45.159 --> 43:48.389 Let me call ψ as the displacement of the 43:48.385 --> 43:53.625 string rather than y, d^(2)ψ/dx^(2) is 43:53.628 --> 43:57.518 (1/v^(2))d^(2 )ψ/dt^(2). 43:57.518 --> 44:00.388 I just wrote the equation shifting to the other side. 44:00.389 --> 44:03.929 They look very similar. 44:03.929 --> 44:12.829 Here v is the velocity of the waves according to a person for 44:12.827 --> 44:17.127 whom the string is at rest. 44:17.130 --> 44:20.050 Okay, according to whom the string is at rest. 44:20.050 --> 44:21.250 You understand that? 44:21.250 --> 44:24.310 This velocity, because the waves are traveling 44:24.311 --> 44:26.831 in the string, a at speed v. 44:26.829 --> 44:32.549 So this equation is to be used in its present form only by a 44:32.550 --> 44:36.430 person for whom the string is at rest. 44:36.429 --> 44:40.469 If you want to see the string from a moving frame then 44:40.472 --> 44:44.082 x' is x - ut, and in classical mechanics 44:44.083 --> 44:46.613 t' is equal to t. 44:46.610 --> 44:49.240 You can do the change of variables with these partial 44:49.240 --> 44:49.950 derivatives. 44:49.949 --> 44:53.599 I don't want to do that, but you can say d/dx of 44:53.603 --> 44:57.053 ψ is equal to d/dx' of ψ times dx' 44:57.054 --> 45:00.554 over dx, then dψ/dt', 45:00.548 --> 45:03.148 then dt' over dx. 45:03.150 --> 45:05.870 Now t' is the same as t, but formally we can 45:05.865 --> 45:08.375 change variables, and we can take this equation 45:08.376 --> 45:11.916 and rewrite it in a frame moving to the right at speed u, 45:11.920 --> 45:14.580 and I promise you it won't look like this. 45:14.579 --> 45:18.069 It will look very different. 45:18.070 --> 45:21.980 More importantly at least understand conceptually that the 45:21.983 --> 45:25.973 velocity u of the moving observer will appear in the 45:25.967 --> 45:28.857 final equations, because these derivatives, 45:28.864 --> 45:31.834 x' over x and so on contain the velocity 45:31.827 --> 45:32.497 u. 45:32.500 --> 45:35.160 Can you see that, dx'/dt has the velocity 45:35.157 --> 45:35.777 u? 45:35.780 --> 45:39.230 So when you make all these changes and put them in you'll 45:39.233 --> 45:42.933 get a new equation involving x' and t' in which 45:42.934 --> 45:45.284 the velocity u will appear. 45:45.280 --> 45:49.310 So if you are that person you have to decide what's the 45:49.313 --> 45:53.203 velocity to use for you, and the answer is unique. 45:53.199 --> 45:56.079 It's your velocity relative to the string. 45:56.079 --> 45:57.619 The string is anchored in the lab. 45:57.619 --> 46:00.699 If you happen to have a speed u relative to that that's 46:00.702 --> 46:02.222 the speed you should put in. 46:02.219 --> 46:05.089 Therefore, the equation is not the same for everybody. 46:05.090 --> 46:09.340 There is a special observer for whom the string is at rest, 46:09.340 --> 46:11.790 namely in the laboratory frame for whom this equation works, 46:11.789 --> 46:14.149 and v is the velocity for that person. 46:14.150 --> 46:18.570 Now we come to this equation. 46:18.570 --> 46:20.940 It has no reference to the velocity of the observer. 46:20.940 --> 46:25.320 It's got a velocity of light, and you can ask who is supposed 46:25.315 --> 46:26.185 to use it. 46:26.190 --> 46:29.420 Whereas in the string we know the privileged frame of 46:29.420 --> 46:33.210 reference is where the string is nailed down, but the light is 46:33.208 --> 46:34.698 traveling in vacuum. 46:34.699 --> 46:37.369 There is no frame of reference. 46:37.369 --> 46:41.419 People thought maybe even the vacuum contains a medium called 46:41.420 --> 46:41.960 ether. 46:41.960 --> 46:44.770 Because everything needs a medium to travel they said there 46:44.771 --> 46:45.451 is an ether. 46:45.449 --> 46:48.079 Then, of course, this is to be used only by 46:48.081 --> 46:50.651 people who live in that ether at rest, 46:50.650 --> 46:54.560 but we are moving relative to the ether because we are on the 46:54.561 --> 46:57.041 earth which is going around the sun. 46:57.039 --> 46:58.679 You may say, "Well, maybe today I just 46:58.681 --> 47:00.441 happen to be at rest relative to the ether. 47:00.440 --> 47:02.760 That's possible, but then tomorrow I cannot be 47:02.762 --> 47:04.572 because I'm going around the sun. 47:04.570 --> 47:07.610 Six months from now I'm going the opposite way around the sun 47:07.606 --> 47:10.806 at a huge speed, but what I find is every single 47:10.809 --> 47:14.349 day of the year I'm able to use these equations. 47:14.349 --> 47:18.699 That means they apply to me no matter what my velocity is. 47:18.699 --> 47:21.799 So these equations, it turns out, 47:21.802 --> 47:27.822 are valid for any observer who is inertial, namely one for whom 47:27.815 --> 47:31.885 at low velocities Newton's Laws apply. 47:31.889 --> 47:35.339 Now what people were worried about in the old days they said, 47:35.340 --> 47:39.250 "Look, let's take x'x - ut and t' equal to 47:39.253 --> 47:42.323 t and put it into this equation." 47:42.320 --> 47:45.870 Then they found the equation changed their form because it's 47:45.865 --> 47:48.025 just like this wave equation here. 47:48.030 --> 47:50.300 Then they said, "We've got to change the 47:50.297 --> 47:53.027 equation because it depends on our speed u, 47:53.030 --> 47:56.220 but there is no valid choice for what our u is. 47:56.219 --> 47:59.539 What is our speed relative to this magical ether?" 47:59.539 --> 48:01.789 That was what they were worried about until Einstein came and 48:01.789 --> 48:04.029 said, "There is no either and 48:04.034 --> 48:07.374 this is the wrong set of transformations." 48:07.369 --> 48:11.739 If you use x' equal to x - ut divided by this 1 48:11.742 --> 48:15.682 − u^(2) over c^(2) and t' is 48:15.679 --> 48:20.709 t - ux over c^(2) divided by the same square root, 48:20.710 --> 48:25.560 if you change coordinates this way you'll find amazingly, 48:25.559 --> 48:29.799 if you change all the d/dx's to d/dx' 48:29.798 --> 48:34.458 and did the whole partial derivatives and chain rule and 48:34.461 --> 48:39.551 so on you will find that in the new frame of reference you'll 48:39.548 --> 48:43.278 find the equation will look like this. 48:43.280 --> 48:48.220 It will look the same for anybody moving relative to me at 48:48.219 --> 48:49.259 any speed. 48:49.260 --> 48:53.050 That's why it's not clear that I'm the privileged user. 48:53.050 --> 48:57.540 All people at uniform relative motion can use the very same 48:57.541 --> 49:01.801 equation with the very same number c entering. 49:01.800 --> 49:05.360 So this was the great triumph of the Maxwell theory. 49:05.360 --> 49:08.890 It was that it was already consist with the relativity. 49:08.889 --> 49:12.249 In fact it is what led to Einstein's revolution because 49:12.251 --> 49:16.111 this equation said no matter who you are a light pulse is going 49:16.112 --> 49:18.792 to travel at a speed c for you, 49:18.789 --> 49:21.219 no matter who you are. 49:21.219 --> 49:23.969 That's very strange because every signal we know has a 49:23.974 --> 49:27.254 property that if you move along the signal its speed is reduced, 49:27.250 --> 49:27.770 right? 49:27.768 --> 49:31.408 If you've got a bullet going at 700 miles per second if you 49:31.414 --> 49:34.624 travel at 400 you will think it is going at 300, 49:34.619 --> 49:37.279 but if it's a beam of light it's supposed to have the same 49:37.277 --> 49:39.657 velocity for everybody, even those moving in the same 49:39.661 --> 49:41.201 direction and the opposite direction. 49:41.199 --> 49:43.159 It doesn't matter. 49:43.159 --> 49:46.769 Therefore, something had to change with our definition of 49:46.766 --> 49:48.756 space, and time, and velocities, 49:48.762 --> 49:52.242 and Einstein replaced it with these new equations. 49:52.239 --> 49:55.809 And one of the consequences of that equation is that if I have 49:55.806 --> 49:59.196 an object that is going at a speed v and they're moving the 49:59.195 --> 50:02.815 same direction at the speed u in the old days you will subtract 50:02.820 --> 50:05.160 your speed in the same direction, 50:05.159 --> 50:09.019 but the correct answer is--this is 1 - uv over 50:09.016 --> 50:10.126 c^(2). 50:10.130 --> 50:12.820 This is the relativistic equation about how to change 50:12.818 --> 50:13.488 velocities. 50:13.489 --> 50:16.219 v is the speed of an object according to me. 50:16.219 --> 50:18.779 You are moving in the same direction at speed u. 50:18.780 --> 50:22.840 You will measure the speed w given by this. 50:22.840 --> 50:25.400 If u and v are much smaller than c you 50:25.400 --> 50:27.830 can forget this and it looks like the good old Newtonian 50:27.827 --> 50:29.837 days, but if u and v 50:29.835 --> 50:33.075 are comparable to c then the denominator is one less 50:33.076 --> 50:34.566 than, one minus something, 50:34.572 --> 50:37.302 so the whole thing will be a little bigger than what you 50:37.300 --> 50:40.080 thought because you're dividing by 1 minus something. 50:40.079 --> 50:43.059 And finally, if what I was looking at was a 50:43.056 --> 50:45.486 light pulse, that means v is equal to 50:45.492 --> 50:48.842 c, I get c - u divided by 1 50:48.835 --> 50:51.685 - uc over c^(2). 50:51.690 --> 50:55.950 And if you fiddle with that you'll find it's c. 50:55.949 --> 50:58.869 So this law of transformation of velocity has the amazing 50:58.871 --> 51:02.111 property that if I'm observing a light pulse that's got a speed 51:02.105 --> 51:04.605 c you will also get a speed c. 51:04.610 --> 51:08.120 So it's a very beautiful way in which the mystery was resolved. 51:08.119 --> 51:11.349 So Maxwell gave these equations, did not give a 51:11.351 --> 51:13.531 preferred frame of reference. 51:13.530 --> 51:15.790 Then you've got to ask yourself, "What coordinate 51:15.789 --> 51:18.349 transformation should exist so that the equation has the same 51:18.347 --> 51:20.727 form for everybody," because it doesn't tell you who 51:20.733 --> 51:21.803 supposed to use it. 51:21.800 --> 51:23.490 Then you get this equation. 51:23.489 --> 51:26.499 You can get to this equation simply by demanding that two 51:26.503 --> 51:29.413 observers looking at a light pulse somehow get the same 51:29.411 --> 51:29.951 speed. 51:29.949 --> 51:32.859 If you fiddle with that and the symmetry between the two 51:32.855 --> 51:34.435 observers you will get this. 51:34.440 --> 51:37.970 Anyway, I just wanted to tell you that there are many things 51:37.967 --> 51:40.777 you have to change, but you don't have to change 51:40.777 --> 51:42.807 any of electromagnetic theory. 51:42.809 --> 51:47.379 The equations I wrote down are correct. 51:47.380 --> 51:52.950 Okay, the last thing I want to do is something I promised long 51:52.945 --> 51:56.225 ago, which is the following thing. 51:56.230 --> 52:00.180 If you believe in relativity, namely if you believe that the 52:00.177 --> 52:04.317 laws of physics should have the same form for people in uniform 52:04.324 --> 52:07.844 relative motion, you can deduce the presence of 52:07.840 --> 52:11.570 magnetism given just the presence of electrostatics. 52:11.570 --> 52:13.860 In other words, suppose you never heard of 52:13.858 --> 52:14.528 magnetism. 52:14.530 --> 52:17.420 I can show you that it must exist. 52:17.420 --> 52:22.210 magnetic forces must exist, and I show that as follows. 52:22.210 --> 52:25.200 Before I show that you need a couple of results that you guys 52:25.202 --> 52:26.802 may not remember all the time. 52:26.800 --> 52:31.950 First result is if you've got a wire and it's got n is 52:31.947 --> 52:37.037 the number of carriers, charge carriers per unit 52:37.039 --> 52:43.709 volume, and A is the cross section of the wire, 52:43.710 --> 52:48.690 and e is the charge of the carrier, 52:48.690 --> 52:52.970 and v is the velocity of the carrier, 52:52.969 --> 52:57.039 then the current is equal to n, 52:57.039 --> 53:03.059 sorry, nAve. 53:03.059 --> 53:06.039 Let me tell you why that is true. 53:06.039 --> 53:07.489 Look at the wire. 53:07.489 --> 53:09.909 I'm going to stretch it a little bit so it looks like a 53:09.907 --> 53:11.517 thing with a finite cross section. 53:11.518 --> 53:13.258 It's just like what I did earlier on. 53:13.260 --> 53:17.240 If you wait 1 second the carriers in that cylinder will 53:17.240 --> 53:21.440 have crossed the checkpoint, and the volume of that region 53:21.443 --> 53:23.733 is A times v. 53:23.730 --> 53:26.760 This is the number of carriers per unit volume, 53:26.760 --> 53:28.700 and each one carries charge e, 53:28.699 --> 53:32.349 that many coulombs would have gone past this point and that's 53:32.347 --> 53:35.387 the meaning of current, first thing to know. 53:35.389 --> 53:40.289 Second thing to know, if I took a rod of length 53:40.286 --> 53:44.646 L and I put some charges on it, 53:44.650 --> 53:48.000 and they had a certain density n_0, 53:48.000 --> 53:52.560 if the rod moves at the velocity v you know it 53:52.563 --> 53:56.193 will shrink, therefore these plus signs will 53:56.188 --> 53:59.718 be compressed, and to a person seeing the 53:59.722 --> 54:05.042 moving rod the density n will be n_0 54:05.039 --> 54:10.359 (the density at rest) divided by 1 − v^(2) over 54:10.356 --> 54:11.976 c^(2). 54:11.980 --> 54:16.040 This is a relativistic effect - that the number of charges get 54:16.036 --> 54:19.226 squeezed because the rod itself gets squeezed. 54:19.230 --> 54:23.500 So a moving rod which is charged will appear to have a 54:23.503 --> 54:25.443 higher charge density. 54:25.440 --> 54:28.280 And the final result I'm going to invoke is the following. 54:28.280 --> 54:36.110 What is the charge per unit length? 54:36.110 --> 54:40.520 If the charge per unit volume is n then I claim the 54:40.516 --> 54:43.296 answer is n times a. 54:43.300 --> 54:45.210 That's also easy to understand. 54:45.210 --> 54:48.440 If you took a unit length of this wire, 54:48.440 --> 54:50.760 unit length, the volume of that is just 54:50.757 --> 54:54.357 A times 1 and that times the density is the amount of 54:54.356 --> 54:55.816 charge that's there. 54:55.820 --> 54:58.430 So if you have a very thin wire you may like to think about 54:58.431 --> 55:01.181 charge per unit length rather than charge per unit volume, 55:01.179 --> 55:04.759 and this is the way to go from one to the other. 55:04.760 --> 55:06.070 Now we are ready. 55:06.070 --> 55:11.120 Now I'm ready to show you how just by thinking you can deduce 55:11.121 --> 55:13.481 the presence of magnetism. 55:13.480 --> 55:16.060 And I like this argument because quite often this is how 55:16.059 --> 55:18.029 people make, theorists make discoveries. 55:18.030 --> 55:20.300 They will take something that's known. 55:20.300 --> 55:22.750 They'll appeal to a principle like symmetry, 55:22.751 --> 55:24.351 or relativity or whatever. 55:24.349 --> 55:26.339 Then they will say, "This implies that there 55:26.342 --> 55:28.212 is a new force, and I'm going to tell you what 55:28.208 --> 55:29.328 the new force is." 55:29.329 --> 55:32.469 So here's what I want you to imagine. 55:32.469 --> 55:39.209 There's a very long infinite line of charge and right on top 55:39.208 --> 55:45.718 of it there's another infinite line of negative charge. 55:45.719 --> 55:49.379 If they're just sitting there there's nothing interesting. 55:49.380 --> 55:58.110 Now what I want to do is I want to have the upper thing going at 55:58.106 --> 56:00.736 a speed v. 56:00.739 --> 56:12.409 Now n_0^( )^( )is charge density of plus 56:12.405 --> 56:23.445 charges in the rest frame of the rod, of the plus rod, 56:23.447 --> 56:28.237 and likewise minus. 56:28.239 --> 56:29.009 Are you with me? 56:29.010 --> 56:31.800 Each rod has a certain density when it's at rest. 56:31.800 --> 56:35.180 I'm going to call that with a subscript zero. 56:35.179 --> 56:39.419 Now I want to arrange this wire to be electrically neutral. 56:39.420 --> 56:42.450 I want it to be neutral, and I'm producing the current 56:42.454 --> 56:44.864 by moving the plus charges to the right. 56:44.860 --> 56:45.850 I'm dragging them. 56:45.849 --> 56:48.229 The minus charges are not doing anything. 56:48.230 --> 56:49.900 So what is the current? 56:49.900 --> 56:54.610 The current is going to be the density of plus charges, 56:54.610 --> 56:58.880 times the velocity, times area, times the value of 56:58.884 --> 57:00.284 each charge. 57:00.280 --> 57:03.880 But for the wire to be neutral--this is where I want 57:03.876 --> 57:05.776 you to follow me closely. 57:05.780 --> 57:09.390 The two of them cannot have the same densities at rest because 57:09.389 --> 57:12.879 if the plus charge had the same density at rest as the minus 57:12.880 --> 57:16.550 charge when it starts moving and it starts compressing the plus 57:16.550 --> 57:19.920 density will exceed the minus density and the wire will be 57:19.922 --> 57:20.872 neutral. 57:20.869 --> 57:26.079 So I cook it up so that n_0^( ) with 57:26.079 --> 57:31.489 this factor is the density of minus charges at rest. 57:31.489 --> 57:32.619 Are you with me? 57:32.619 --> 57:36.029 I take a rod with positive charge somewhat less than the 57:36.025 --> 57:39.425 density that the negative charges are going to have, 57:39.429 --> 57:44.219 but by moving it at the suitable speed I bring these two 57:44.217 --> 57:46.827 densities to the same value. 57:46.829 --> 57:50.009 So I want you to follow this in some detail because it's not so 57:50.005 --> 57:50.615 difficult. 57:50.619 --> 57:53.339 This is the neutrality condition for the wire. 57:53.340 --> 57:58.440 I want the wire to be neutral. 57:58.440 --> 58:01.110 See normally when you don't think in relativist terms you'll 58:01.112 --> 58:02.872 say, "Take a plus charge rod 58:02.869 --> 58:05.999 and a minus charge rod and make sure the charges are equal per 58:06.003 --> 58:07.363 length, and drag one of them, 58:07.360 --> 58:08.530 or you've got a current." 58:08.530 --> 58:10.470 That works in a non-relativistic limit, 58:10.469 --> 58:14.599 but if you take into account length contraction you won't get 58:14.603 --> 58:18.393 neutrality unless the plus charge had a slightly smaller 58:18.394 --> 58:22.534 density in its own rest frame that got boosted by this factor 58:22.528 --> 58:25.698 to equal the density of negative charge. 58:25.699 --> 58:34.059 Okay, now imagine a particle at rest here, and you know all 58:34.063 --> 58:37.383 about electrostatics. 58:37.380 --> 58:49.720 What do you think it will do? 58:49.719 --> 58:51.079 You've never heard of magnetism. 58:51.079 --> 58:52.419 You've heard of electrostatics. 58:52.420 --> 58:54.250 What will this charge do? 58:54.250 --> 58:55.090 Yep? 58:55.090 --> 58:56.350 Student: It is positive. 58:56.349 --> 58:58.339 It'd probably be attracted. 58:58.340 --> 59:00.210 Prof: Why would it be attracted? 59:00.210 --> 59:05.690 Student: Because the negative wire has more density 59:05.693 --> 59:06.453 and... 59:06.449 --> 59:06.879 Prof: No, no. 59:06.880 --> 59:08.480 That's what I said. 59:08.480 --> 59:09.130 No. 59:09.130 --> 59:11.970 I arranged it so that the positive wire, 59:11.965 --> 59:16.175 the moving positive wire has the same density as the static 59:16.181 --> 59:17.491 negative wire. 59:17.489 --> 59:19.429 You understand? 59:19.429 --> 59:21.329 I cooked it up so that the wire is neutral. 59:21.329 --> 59:23.629 All I'm telling you is, if one set of charges are 59:23.626 --> 59:26.356 sliding to the right at a certain speed that rod must have 59:26.355 --> 59:29.175 a density that's somewhat lower in its own rest frame, 59:29.179 --> 59:32.169 but by the time you translate it to the laboratory frame it's 59:32.172 --> 59:33.322 electrically neutral. 59:33.320 --> 59:35.950 So what'll this charge do is my question. 59:35.949 --> 59:37.429 Student: > 59:37.429 --> 59:39.509 Prof: Pardon me? 59:39.510 --> 59:41.610 It should not experience any electrical force, 59:41.605 --> 59:42.625 you understand that? 59:42.630 --> 59:47.680 Our prediction is that it will just stay there because it's 59:47.679 --> 59:51.509 neutral and all we know is electrostatics. 59:51.510 --> 59:54.720 It can have any velocity, but I want it to have a 59:54.721 --> 59:56.061 velocity v. 59:56.059 --> 59:59.889 Electrostatics doesn't care what the velocity of the charge 59:59.889 --> 1:00:00.219 is. 1:00:00.219 --> 1:00:07.019 It's still supposed to go in a straight line because the wire 1:00:07.016 --> 1:00:08.486 is neutral. 1:00:08.489 --> 1:00:12.989 But now I say I'm allowed to view the physics from any frame 1:00:12.990 --> 1:00:14.670 of reference I want. 1:00:14.670 --> 1:00:18.900 Let me go to the frame of reference of this charge. 1:00:18.900 --> 1:00:20.480 I move to the right. 1:00:20.480 --> 1:00:23.100 So in general, if you want to analyze the 1:00:23.101 --> 1:00:25.791 problem carefully, you can have this wire moving 1:00:25.791 --> 1:00:28.461 at one speed and this charge moving at a different speed, 1:00:28.460 --> 1:00:30.740 but I want to do the easier case where they're both the 1:00:30.735 --> 1:00:31.025 same. 1:00:31.030 --> 1:00:33.260 It makes life easier. 1:00:33.260 --> 1:00:41.780 So go to the rest frame of charge. 1:00:41.780 --> 1:00:44.570 You can already see what's going to happen in the rest 1:00:44.570 --> 1:00:45.730 frame of the charge. 1:00:45.730 --> 1:00:49.330 If you go to the frame moving with the particle the plus wire 1:00:49.329 --> 1:00:52.029 comes to rest, so the density of the plus 1:00:52.032 --> 1:00:55.662 charges will just become n_0^( )^( ), 1:00:55.659 --> 1:00:57.719 the rest density of the plus charge, 1:00:57.719 --> 1:01:02.479 but the minus charges now they will be boosted by this factor 1:01:02.478 --> 1:01:05.808 because they're going the opposite way, 1:01:05.809 --> 1:01:07.429 and it doesn't matter which way they're moving. 1:01:07.429 --> 1:01:08.759 Things depend on v squared. 1:01:08.760 --> 1:01:12.180 The minus charges will appear compressed. 1:01:12.179 --> 1:01:15.409 So if previously they're balanced now they won't balance. 1:01:15.409 --> 1:01:18.649 In fact, n_- is n_0^(- ) 1:01:18.650 --> 1:01:22.410 over this, which is n_0^( 1:01:22.407 --> 1:01:26.987 ) divided by 1 - v^(2)/c^(2). 1:01:26.989 --> 1:01:31.009 Therefore the wire will appear to be negatively charged because 1:01:31.010 --> 1:01:33.540 the negative charges are that value, 1:01:33.539 --> 1:01:36.759 the positive charges are that value, 1:01:36.760 --> 1:01:43.480 so net charge on the wire, net charge density will be 1:01:43.476 --> 1:01:49.286 equal to -n_0^( )^( )over 1 - 1:01:49.289 --> 1:01:55.489 v^(2)/c^(2) n_0^( ). 1:01:55.489 --> 1:01:57.659 If you like, this is the plus charge and 1:01:57.661 --> 1:02:00.671 that's the minus charge times e, if you like. 1:02:00.670 --> 1:02:02.770 And I'm going to make an approximation. 1:02:02.768 --> 1:02:06.698 I'm going to approximate this is n_0^( )^( 1:02:06.697 --> 1:02:08.727 )times 1 v^(2)/c^(2). 1:02:08.730 --> 1:02:12.150 There are more terms, but in the binomial expansion 1:02:12.152 --> 1:02:14.892 I'm going to stop with the first term. 1:02:14.889 --> 1:02:18.039 So this is the result that's very good in the limit of small 1:02:18.038 --> 1:02:20.758 v/c, but v/c is not set equal to 0. 1:02:20.760 --> 1:02:23.090 It is set equal to a finite but small amount. 1:02:23.090 --> 1:02:28.710 So if you compare those two you will find the net charge is - 1:02:28.708 --> 1:02:32.358 n_0^( ) v^(2)/c^(2). 1:02:32.360 --> 1:02:37.910 That's going to be the charge on the wire. 1:02:37.909 --> 1:02:41.069 So to the person moving with this charge the wire looks 1:02:41.065 --> 1:02:41.645 charged. 1:02:41.650 --> 1:02:46.240 In fact it looks negatively charged. 1:02:46.239 --> 1:02:48.919 So you know, knowing just electrostatics, 1:02:48.918 --> 1:02:50.658 what the charge will do. 1:02:50.659 --> 1:02:55.619 The charge will be drawn to the wire, start moving towards the 1:02:55.619 --> 1:02:56.189 wire. 1:02:56.190 --> 1:02:58.480 That's a fact. 1:02:58.480 --> 1:03:01.680 Well, if it's moving towards the wire for me it better be 1:03:01.679 --> 1:03:05.049 moving towards the wire for the original person also because 1:03:05.052 --> 1:03:08.082 even if you and I move horizontally the fact that your 1:03:08.081 --> 1:03:11.631 moving transverse to a velocity is going to be a true statement 1:03:11.625 --> 1:03:12.935 in both frames. 1:03:12.940 --> 1:03:16.180 From that you will conclude that even in the original 1:03:16.184 --> 1:03:19.934 laboratory frame this moving charge would have been attracted 1:03:19.927 --> 1:03:23.717 to that wire, and it's attracted by virtue of 1:03:23.722 --> 1:03:24.932 its velocity. 1:03:24.929 --> 1:03:29.919 So that means there is a new force in which a charge is 1:03:29.922 --> 1:03:34.182 attracted to the current in the same direction, 1:03:34.175 --> 1:03:35.095 right? 1:03:35.099 --> 1:03:37.349 So let's ask, "What is the force of 1:03:37.351 --> 1:03:38.451 attraction?" 1:03:38.449 --> 1:03:41.399 In Newtonian mechanics--I mean, at this point once you've got 1:03:41.402 --> 1:03:44.512 this length contraction you can just think in terms of Newtonian 1:03:44.505 --> 1:03:47.455 formulas because the errors you make involve higher powers of 1:03:47.458 --> 1:03:48.688 v^(2)/c^(2). 1:03:48.690 --> 1:03:50.290 The leading term is this. 1:03:50.289 --> 1:03:51.049 So look at that. 1:03:51.050 --> 1:03:57.400 You have a net charge of n_ ^(0 )v^(2) 1:03:57.398 --> 1:03:59.048 /c^(2). 1:03:59.050 --> 1:04:05.180 You have a charge per unit length which is the area of the 1:04:05.182 --> 1:04:07.122 wire times that. 1:04:07.119 --> 1:04:09.259 I sorry, that's charge e. 1:04:09.260 --> 1:04:10.530 That's the density. 1:04:10.530 --> 1:04:12.710 That's the actual charge. 1:04:12.710 --> 1:04:14.920 That's the λ. 1:04:14.920 --> 1:04:17.830 So what's the electric field of a wire with charge per unit 1:04:17.827 --> 1:04:18.377 λ? 1:04:18.380 --> 1:04:24.320 I remind you it's λ /2Πε_0 1:04:24.315 --> 1:04:26.805 r. 1:04:26.809 --> 1:04:33.699 So for this particular lambda it's n_ ^(0 )v^(2) 1:04:33.704 --> 1:04:39.574 over c^(2) eA times ½Πε 1:04:39.565 --> 1:04:42.435 _0r. 1:04:42.440 --> 1:04:46.090 In other words in its particle rest frame there'll be an 1:04:46.088 --> 1:04:49.068 electrical field because there's a net charge, 1:04:49.072 --> 1:04:52.392 negative charge, of that strength on the wire. 1:04:52.389 --> 1:04:54.139 Once you've got a charge per unit length 1:04:54.139 --> 1:04:56.379 λ/2Πε _0r is the 1:04:56.380 --> 1:04:58.310 field attracting it toward the center, 1:04:58.309 --> 1:05:00.339 so that's the force. 1:05:00.340 --> 1:05:01.820 So I'm going to write it as follows. 1:05:01.820 --> 1:05:04.950 I'm going to write it as n_ 1:05:04.949 --> 1:05:10.069 ^(0)veA_1 over 2Πε_0 1:05:10.072 --> 1:05:16.432 c^(2) times another e and another v. 1:05:16.429 --> 1:05:27.289 This is the force, this e times E. 1:05:27.289 --> 1:05:30.559 In Newtonian approximation the force perpendicular to the 1:05:30.561 --> 1:05:32.901 motion is the same for all observers, 1:05:32.900 --> 1:05:36.010 therefore I ask you--and this force must be present even in 1:05:36.010 --> 1:05:38.370 the lab, but look at this force. 1:05:38.369 --> 1:05:45.939 This guy is the current, nevA is the current. 1:05:45.940 --> 1:05:49.740 This guy 1/ε_0c^(2) is 1:05:49.735 --> 1:05:52.165 μ_0/2Π. 1:05:52.170 --> 1:05:54.070 This is ev. 1:05:54.070 --> 1:05:57.470 You see, this is the exactly the magnetic force we got, 1:05:57.472 --> 1:06:00.122 μ_0 I/2Πr. 1:06:00.119 --> 1:06:03.329 I forgot a 1/r here. 1:06:03.329 --> 1:06:10.179 That is the azimuthal magnetic field around the wire that times 1:06:10.177 --> 1:06:12.937 ev is the force. 1:06:12.940 --> 1:06:15.220 Because you can tell that no matter where the charge was, 1:06:15.219 --> 1:06:17.489 whether it was here, or whether it was there or 1:06:17.489 --> 1:06:20.349 anywhere around the wire it'll have the same attraction. 1:06:20.349 --> 1:06:23.239 So you can deduce in the laboratory frame there is an 1:06:23.242 --> 1:06:25.302 attractive force towards the wire, 1:06:25.300 --> 1:06:28.940 and that agrees with the exact formula you got for the 1:06:28.943 --> 1:06:29.773 magnetism. 1:06:29.768 --> 1:06:33.238 If you don't ignore the higher powers of v^(2)/c^(2) you 1:06:33.235 --> 1:06:36.635 will find that the force in the lab frame and the force in the 1:06:36.643 --> 1:06:38.883 moving frame are slightly different. 1:06:38.880 --> 1:06:42.150 That's because in relativistic theories the force is not the 1:06:42.146 --> 1:06:43.306 same for everybody. 1:06:43.309 --> 1:06:45.909 The force for me and the force for you do not actually agree 1:06:45.907 --> 1:06:47.887 because time is not the same for everybody. 1:06:47.889 --> 1:06:52.699 But the leading order in v^(2)/c^(2) you can see 1:06:52.704 --> 1:06:53.334 this. 1:06:53.329 --> 1:06:55.789 There's only one thing I did which is a little fudge, 1:06:55.789 --> 1:06:57.759 which you guys may not have noticed, 1:06:57.760 --> 1:07:01.660 is that the density here is n_ ^(0)^( 1:07:01.661 --> 1:07:05.341 )whereas the actual current in the lab is n_ 1:07:05.342 --> 1:07:09.392 _ and the two of them differ by this 1:07:09.391 --> 1:07:10.351 factor. 1:07:10.349 --> 1:07:13.829 However, if you have a v^(2)/c^(2) in front of 1:07:13.833 --> 1:07:17.123 your expression and you've ignored v^(4) or 1:07:17.117 --> 1:07:21.267 c^(4) there is no point in keeping this guy here because 1:07:21.271 --> 1:07:24.891 that makes an error of order v^(4)/c^(4). 1:07:24.889 --> 1:07:27.579 If I'm going to keep that I should go back right here and 1:07:27.583 --> 1:07:28.453 keep such terms. 1:07:28.449 --> 1:07:31.989 So the leading order you don't have to worry about the 1:07:31.994 --> 1:07:33.604 difference in density. 1:07:33.599 --> 1:07:35.579 This is a very subtle calculation because sometimes 1:07:35.576 --> 1:07:37.826 you worry about the difference and sometimes you don't. 1:07:37.829 --> 1:07:41.039 So the rule always is if you're trying to do calculations to a 1:07:41.036 --> 1:07:43.306 certain order, namely v/c the whole 1:07:43.311 --> 1:07:46.131 thing squared then things that make corrections of order 1:07:46.128 --> 1:07:48.328 v/c to the fourth can be dropped. 1:07:48.329 --> 1:07:50.859 Anyway, I'm not that worried about the details, 1:07:50.855 --> 1:07:54.035 but I want you to understand at least what the logic is. 1:07:54.039 --> 1:07:56.109 That's more important. 1:07:56.110 --> 1:07:57.870 I know about electrostatics. 1:07:57.869 --> 1:08:01.479 I cleared the neutral current carrying wire. 1:08:01.480 --> 1:08:04.910 It's neutral because the negative charges are at rest at 1:08:04.913 --> 1:08:05.853 some density. 1:08:05.849 --> 1:08:09.259 The positive charges in the moving rod are also at the same 1:08:09.264 --> 1:08:12.214 density, but the rest density is somewhat lower. 1:08:12.210 --> 1:08:16.300 Then I predicted the particle won't move because it's got no 1:08:16.296 --> 1:08:17.816 electric attraction. 1:08:17.819 --> 1:08:20.169 Then I go to the rest frame of the particle, 1:08:20.172 --> 1:08:23.182 then I find that the positive charges have come to rest, 1:08:23.184 --> 1:08:25.214 and therefore to a lower density. 1:08:25.210 --> 1:08:27.840 Negative charges are moving the other way, therefore at the 1:08:27.838 --> 1:08:28.608 higher density. 1:08:28.609 --> 1:08:30.099 They no longer cancel. 1:08:30.100 --> 1:08:31.350 The wire is charged. 1:08:31.350 --> 1:08:34.010 I expect this charge to be attracted to the wire. 1:08:34.010 --> 1:08:36.830 That means going back to the lab I expect the moving charge 1:08:36.832 --> 1:08:39.702 also to be attracted to the wire because when something goes 1:08:39.704 --> 1:08:42.384 towards the wire that goes towards the wire according to 1:08:42.381 --> 1:08:43.211 all people. 1:08:43.210 --> 1:08:45.990 But you can go beyond and actually compute the force and 1:08:45.993 --> 1:08:48.983 equate the force and deduce that that is a new force now. 1:08:48.979 --> 1:08:51.439 Whenever you have a current carrying wire with current 1:08:51.442 --> 1:08:54.572 I, and that's a particle of charge 1:08:54.565 --> 1:08:59.615 e moving at speed v it will be attracted to the wire by 1:08:59.617 --> 1:09:00.857 this amount. 1:09:00.859 --> 1:09:06.059 Okay, so I'll see you guys after the break, 1:09:06.060 --> 1:09:11.080 and my suggestion to you on what to do with the break is to 1:09:11.082 --> 1:09:14.462 try to carry your textbook with you, 1:09:14.460 --> 1:09:16.640 and the textbook has got lots of problems, 1:09:16.640 --> 1:09:20.210 some of which are a lot simpler than the one I gave in the early 1:09:20.207 --> 1:09:22.037 days, but roughly the same level of 1:09:22.042 --> 1:09:23.332 difficulty as the midterm. 1:09:23.328 --> 1:09:25.778 And do as many problems as you can. 1:09:25.779 --> 1:09:28.839 Look at the worked examples and try to solve them. 1:09:28.840 --> 1:09:34.000