1 00:00:00,150 --> 00:00:06,990 And the previous two lectures, we have derived the wife equation for E and B, and we have introduced 2 00:00:06,990 --> 00:00:10,140 the general solutions which were Wavves. 3 00:00:10,650 --> 00:00:17,340 So these were the waves for E and also for B, where we have these exponential functions here with the 4 00:00:17,340 --> 00:00:18,540 imaginary arguments. 5 00:00:19,260 --> 00:00:23,270 And also we have learned already something about R and Omega. 6 00:00:23,760 --> 00:00:31,590 But now in this lecture, I want to show you how the orientation of K is related to the orientation 7 00:00:31,590 --> 00:00:37,530 of B and E, so on comment so that you don't wonder. 8 00:00:38,010 --> 00:00:45,220 What I have done here is I have changed the sign of the of the face, which is something you can do. 9 00:00:45,270 --> 00:00:48,030 So this is something that should not confuse you. 10 00:00:48,420 --> 00:00:53,700 When you look up a textbook, for example, you may find that the face here is different. 11 00:00:53,700 --> 00:00:55,260 So there will be different sign here. 12 00:00:56,150 --> 00:01:04,790 So what this means is there is just when you have here a different, different sign here, it gives 13 00:01:04,790 --> 00:01:08,360 you a different face in the exponential function. 14 00:01:09,020 --> 00:01:14,110 So this is something that can be covered by this prefectly zero. 15 00:01:14,780 --> 00:01:23,150 So it's something that is physically not relevant because for measuring the electric field, for example, 16 00:01:23,600 --> 00:01:29,690 only the real part of this field is relevant, which is basically the cosine of this argument here. 17 00:01:30,110 --> 00:01:31,550 So it does not matter if you. 18 00:01:31,550 --> 00:01:34,880 Right, if you're a plus or negative, this argument here. 19 00:01:35,600 --> 00:01:42,080 But of course, when you solve one particular problem, you have to use the same convention all of the 20 00:01:42,080 --> 00:01:42,440 time. 21 00:01:43,070 --> 00:01:47,720 But if you have two independent problems that you want to solve, you don't really have to care about 22 00:01:47,720 --> 00:01:49,160 if you're right plus or negative. 23 00:01:49,160 --> 00:01:51,020 It's it's not that important, really. 24 00:01:51,860 --> 00:01:58,370 So here I have used this conversion where I have I and then K.R. minus only got previously. 25 00:01:58,370 --> 00:02:03,980 I had eight times on McAtee minus K.R., but the physics behind it is exactly the same. 26 00:02:05,300 --> 00:02:13,610 So now to understand how the orientation of K is related to eat, or B we must consider once again a 27 00:02:13,620 --> 00:02:14,660 Maxwell equation. 28 00:02:15,350 --> 00:02:18,760 This time we need the divergence of E is equal to zero. 29 00:02:19,670 --> 00:02:21,860 So yeah, I will calculate the divergence of E. 30 00:02:23,630 --> 00:02:32,240 That's what we get is once again, please think of this divergence maybe in terms of a one dimensional 31 00:02:32,240 --> 00:02:37,910 problem where this is just the derivative with respect to the coordinate X and here we then would have 32 00:02:38,180 --> 00:02:49,370 K Times X. So what we get is then another term, eight times K, so we get eight times K times E, zero 33 00:02:49,370 --> 00:02:49,820 times. 34 00:02:49,820 --> 00:02:51,980 This exponential function is equal to zero. 35 00:02:53,120 --> 00:02:57,910 And now since this is the exponential function and here we have a zero, we can just divide by it. 36 00:02:57,920 --> 00:03:04,000 So the condition that we are left with is that E zero times K is equal to zero. 37 00:03:04,760 --> 00:03:15,680 And if you remember when you have two vectors that are or whose scalar product must be zero, this means 38 00:03:15,680 --> 00:03:22,370 the vectors must either be zero themselves or they must be perpendicular to each other. 39 00:03:23,450 --> 00:03:27,860 So here the solution would be that e0 is perpendicular to K. 40 00:03:29,390 --> 00:03:32,070 And in a similar way, you can practice this yourself. 41 00:03:32,360 --> 00:03:36,410 You can show that when you start from another Maxwell equation, where you start from the divergence 42 00:03:36,410 --> 00:03:37,590 of B is equal to zero. 43 00:03:37,610 --> 00:03:45,230 You will find that B zero is perpendicular to K, and you may notice this is not a vector, it is just 44 00:03:45,230 --> 00:03:45,850 a scalar. 45 00:03:46,130 --> 00:03:49,900 So the electric field is really given by the orientation. 46 00:03:49,910 --> 00:03:55,400 I mean I mean the orientation of E is basically given by the orientation of E zero. 47 00:03:55,880 --> 00:04:03,890 And the same is true for B, so we know that the the electric field is perpendicular to K and also the 48 00:04:03,890 --> 00:04:05,960 magnetic field is perpendicular to K. 49 00:04:06,920 --> 00:04:12,860 Now the only thing that we don't know is how R, E and B related to each other. 50 00:04:13,940 --> 00:04:19,400 And to find out about this, we consider another mechanical equation, which is the rotation of E is 51 00:04:19,400 --> 00:04:21,720 equal to minus the time derivative of B. 52 00:04:22,580 --> 00:04:28,850 So once again, we calculate both of these equations here and we get something like this. 53 00:04:29,840 --> 00:04:34,100 So on the left hand side, it's pretty similar to this one here. 54 00:04:34,100 --> 00:04:41,750 But we get here a vector product and on the right hand side we get from the time derivative effect or 55 00:04:41,750 --> 00:04:46,570 items minus Omega, the minus sign cancels with this minus sign. 56 00:04:46,580 --> 00:04:48,170 So we just get Omega. 57 00:04:49,860 --> 00:05:01,080 And now we can reintroduce E and B, so e e zero times just exponential function and B is B zero times 58 00:05:01,090 --> 00:05:02,330 this exponential function. 59 00:05:02,760 --> 00:05:09,630 So we get this relation here and also, I mean, it doesn't really matter that much. 60 00:05:09,780 --> 00:05:16,200 We could also use here these zeroes is Provectus, because you as I told you, these are scalars. 61 00:05:16,380 --> 00:05:20,520 So the orientation of B is determined by the orientation of B zero. 62 00:05:21,930 --> 00:05:28,470 And also what I have done here is I have used the relation that Omega of K is equal to the velocity 63 00:05:28,470 --> 00:05:28,950 of light. 64 00:05:31,320 --> 00:05:40,410 And now what you can see here is that we know already that it is perpendicular to K, but we don't know 65 00:05:40,410 --> 00:05:44,820 how it can be are related to each other, but now we can see how they are related. 66 00:05:45,270 --> 00:05:47,010 So we have noticed two vectors here. 67 00:05:47,010 --> 00:05:56,580 K and E and there are vector products is proportional to B, and we know that the vector product of 68 00:05:56,580 --> 00:05:59,700 two vectors is always perpendicular to both vectors. 69 00:06:00,270 --> 00:06:08,250 So this means K is perpendicular to E, E is perpendicular to be and B is perpendicular to K and the 70 00:06:08,250 --> 00:06:12,530 same holds also for these zero and B zero constants. 71 00:06:13,050 --> 00:06:17,670 So all of the three vectors, K, B and E. are perpendicular to each other. 72 00:06:18,630 --> 00:06:25,080 So in an example that solves this problem is the following electromagnetic wave where you could say 73 00:06:25,080 --> 00:06:33,780 that, for example, the magnetic field fluctuates and space and time in the X Y plane, the electric 74 00:06:33,780 --> 00:06:44,220 field fluctuates in the Y Z plane, and then the vector is oriented perpendicular to this be zero and 75 00:06:44,220 --> 00:06:45,000 this is zero. 76 00:06:46,440 --> 00:06:50,400 And this is not the only solution for this problem. 77 00:06:50,970 --> 00:06:55,380 So in the following lecture, we will discuss the different polarizations of light.