1 00:00:00,300 --> 00:00:07,560 Now, the idea of this whole lecture is to construct the Maxwell's equations based on symmetry arguments, 2 00:00:08,580 --> 00:00:18,420 and what we want to do is we want to take the electric field and magnetic fields and consider only first 3 00:00:18,420 --> 00:00:19,400 order derivatives. 4 00:00:19,560 --> 00:00:21,480 So we have limited time derivatives. 5 00:00:21,780 --> 00:00:25,650 We have to divergences and we have two rotations for them. 6 00:00:25,660 --> 00:00:26,640 On the previous slide. 7 00:00:26,640 --> 00:00:29,780 We already had to charge density and current density. 8 00:00:30,540 --> 00:00:39,210 And now we want to classify all of these different objects here in terms of their behavior on the time 9 00:00:39,210 --> 00:00:40,600 inversion and space inversion. 10 00:00:42,030 --> 00:00:49,080 So we have this we have these Maxwell's equations and these as well. 11 00:00:49,080 --> 00:00:50,940 As we have learned, these are four equations. 12 00:00:51,320 --> 00:00:53,880 Now we have two of these operations. 13 00:00:53,880 --> 00:00:58,380 So we can either get two plus, plus, minus, minus, plus, minus and minus plus. 14 00:00:58,710 --> 00:01:05,130 So we have four different categories and we have equation with the left and on the right side, I mean, 15 00:01:05,330 --> 00:01:10,400 it's clear the left side must be equal to the right side and therefore the left side must transform 16 00:01:10,410 --> 00:01:12,100 in the same way as the right side. 17 00:01:12,720 --> 00:01:20,580 So this means in a single Maxwell equation, we can only put objects that behave equally under time 18 00:01:20,760 --> 00:01:22,420 inversion and space inversion. 19 00:01:23,370 --> 00:01:28,800 So what we will do is we will go through all of these possible terms here that are, as I told you, 20 00:01:28,830 --> 00:01:33,960 first order and derivatives and then classify them into the four different categories. 21 00:01:34,350 --> 00:01:39,270 Or we will see is that this gives us the Maxwell's equations. 22 00:01:40,180 --> 00:01:47,070 But yeah, besides the prefectures, the prefectures, we cannot really develop or understand from this 23 00:01:47,190 --> 00:01:53,400 symmetry arguments, but we can understand why certain objects are in the same Maxwell equation. 24 00:01:54,510 --> 00:01:55,390 So let's go ahead. 25 00:01:55,410 --> 00:01:58,340 We already had the charge density, which is kind of trivial. 26 00:01:58,350 --> 00:02:02,790 It's just a scalar and it's symmetric on the time inversion and space inversion. 27 00:02:02,790 --> 00:02:04,260 So it means it does not change. 28 00:02:05,850 --> 00:02:11,660 Now, the current density is essentially the velocity, so it's a. symmetric under both of these operations. 29 00:02:11,670 --> 00:02:12,650 We have this as well. 30 00:02:14,060 --> 00:02:20,660 The electric field is asymmetric on the time inversion, so it doesn't really care about if you reverse 31 00:02:20,750 --> 00:02:25,610 time or not and it's a. symmetric under space inversion. 32 00:02:26,800 --> 00:02:28,760 So I think that's that's quite clear. 33 00:02:28,780 --> 00:02:34,480 It's just like a typical typical vector, so if you look in the mirror, it will revert. 34 00:02:34,990 --> 00:02:40,750 But if you change the time and does not really revert because it's just a vector, and now you might 35 00:02:40,750 --> 00:02:43,480 think that this is also true for the magnetic fields. 36 00:02:44,200 --> 00:02:51,070 And this is typically a misconception because the magnetic field and the electric field are typically 37 00:02:51,070 --> 00:02:54,010 both considered as being conventional vectors. 38 00:02:54,550 --> 00:02:58,390 But it turns out that a magnetic field is a bit different. 39 00:02:58,840 --> 00:03:01,600 I mean, you can, of course, ride it as a vector. 40 00:03:01,600 --> 00:03:03,760 And we do this throughout this whole course. 41 00:03:03,760 --> 00:03:06,850 Here you have a vector with the X, Y and Z component. 42 00:03:07,240 --> 00:03:13,600 But it's behavior on that is transformation's is very different to a typical vector, which is also 43 00:03:13,600 --> 00:03:18,070 why we call it a pseudo vector, or sometimes it's also called an actual vector. 44 00:03:19,570 --> 00:03:27,370 So the reason why the magnetic field is anti symmetric under time version is the following consideration. 45 00:03:27,850 --> 00:03:34,640 So let's take an electron starting from here and move it inside an area where we have a magnetic field. 46 00:03:35,260 --> 00:03:37,480 So in the beginning, we have no force at all. 47 00:03:37,480 --> 00:03:44,110 The electron will move with its velocity, but then we will have here the Lawrence Force, because this 48 00:03:44,110 --> 00:03:48,220 is a vector product of the velocity which goes into this direction. 49 00:03:48,730 --> 00:03:52,210 And the magnetic field, which you can see is located. 50 00:03:52,750 --> 00:03:56,320 Yeah, not towards us, but towards the other direction. 51 00:03:57,310 --> 00:04:02,610 So since we have a negative charge, this means we have a force acting in this direction. 52 00:04:02,620 --> 00:04:09,760 So the electron will move on such a circular trajectory through the magnetic field and then at some 53 00:04:09,760 --> 00:04:14,540 point it will hit this point where the magnetic field will go back to zero. 54 00:04:14,830 --> 00:04:17,950 So the electron will move in this direction. 55 00:04:19,660 --> 00:04:22,870 Now, imagine what happens if we reverse the time. 56 00:04:24,130 --> 00:04:28,930 Well, of course, as we have learned, the velocity will change as we change its sine. 57 00:04:29,240 --> 00:04:35,920 So you can, for example, film this trajectory here, this process, and then rewind this whole movie. 58 00:04:36,110 --> 00:04:38,560 Then the electron will go from here to here. 59 00:04:39,700 --> 00:04:46,450 And now you have to think in which direction must the magnetic fields be oriented so that we can move 60 00:04:46,450 --> 00:04:48,250 along this trajectory? 61 00:04:49,240 --> 00:04:54,400 Well, of course, the magnetic field must be reversed because if it is oriented like it's shown here, 62 00:04:54,670 --> 00:04:56,470 then the electron would move like this. 63 00:04:59,050 --> 00:05:02,200 But instead, it moves like this when we rewind. 64 00:05:02,440 --> 00:05:08,570 So this means the orientation of the magnetic field has to flip the sign when we apply time version. 65 00:05:09,220 --> 00:05:15,970 Therefore, it's a. symmetric under time inversion and it's instead symmetric on this space inversion. 66 00:05:18,220 --> 00:05:23,790 OK, now the rest is pretty trivial, I would say, because we now know how the electric field behaves, 67 00:05:24,220 --> 00:05:30,640 we know how the magnetic field behaves, and these derivatives are essentially just these two fields 68 00:05:31,780 --> 00:05:39,100 divided by tea or multiplied or divided by the space because it is a derivative versus vector space. 69 00:05:39,880 --> 00:05:46,690 So we just have to look up how the space and the time transformed as we had on the previous slide. 70 00:05:47,110 --> 00:05:50,390 And then we can fill out this table, as I have done here for you. 71 00:05:51,730 --> 00:05:58,210 So, for example, here, because we have the time here, we get here minus sign while we had here plus 72 00:05:58,210 --> 00:06:03,040 sine and here, the minus sign stays the same for the magnetic field. 73 00:06:03,040 --> 00:06:07,390 It's similar instead of the minus we get now plus and he had a plus stays the same. 74 00:06:08,230 --> 00:06:14,230 And then here, for example, for the electric field, the and the behavior and the time. 75 00:06:14,390 --> 00:06:17,980 Inversion remains the same similarly here. 76 00:06:18,610 --> 00:06:21,610 But the behavior and the space inversion flips to sign. 77 00:06:21,670 --> 00:06:24,370 So now instead of the minus side, we have a plus sine. 78 00:06:26,320 --> 00:06:31,840 So if you want to understand it yourself, it's probably better to post a video here and have a careful 79 00:06:31,840 --> 00:06:32,110 look. 80 00:06:32,560 --> 00:06:38,080 But I think once you have understood how the time transforms into a space transforms, which is really 81 00:06:38,080 --> 00:06:41,350 trivial because this is a time inversion, space inversion. 82 00:06:41,770 --> 00:06:47,790 And when you have understood how electric field and magnetic fields transform, this is actually difficult 83 00:06:47,800 --> 00:06:48,030 here. 84 00:06:48,040 --> 00:06:50,260 You have to understand this whole cartoon. 85 00:06:51,040 --> 00:06:54,880 If you understand that all of these aspects, you can fill out the whole table.