WEBVTT 00:00:00.050 --> 00:00:01.770 The following content is provided 00:00:01.770 --> 00:00:04.010 under a Creative Commons license. 00:00:04.010 --> 00:00:06.860 Your support will help MIT OpenCourseWare continue 00:00:06.860 --> 00:00:10.720 to offer high-quality educational resources for free. 00:00:10.720 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.209 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.209 --> 00:00:17.834 at ocw.mit.edu. 00:00:22.900 --> 00:00:24.210 PROFESSOR: Welcome back. 00:00:24.210 --> 00:00:32.836 And today we are going to look at a harder situation. 00:00:32.836 --> 00:00:37.930 At oscillations waves in the electromagnetic field. 00:00:42.890 --> 00:00:46.200 Why I say it's harder, for many reasons. 00:00:46.200 --> 00:00:49.150 First of all, so far we've always 00:00:49.150 --> 00:00:53.000 considered situations which we could either visualize or had 00:00:53.000 --> 00:00:59.100 some sensual way of getting a feel for what 00:00:59.100 --> 00:01:01.790 the physical situation is. 00:01:01.790 --> 00:01:05.069 When it comes to the electromagnetic field, 00:01:05.069 --> 00:01:09.570 as you well know, we can't see it, sense it, at all. 00:01:09.570 --> 00:01:14.520 And the only way to describe it is, in fact, 00:01:14.520 --> 00:01:16.230 in terms of mathematics. 00:01:16.230 --> 00:01:19.470 So there isn't, first, a word-- a description 00:01:19.470 --> 00:01:21.395 by analogy with what we see around. 00:01:24.470 --> 00:01:28.840 Secondly, it's more complicated. 00:01:28.840 --> 00:01:34.090 These are oscillations in three dimensions. 00:01:34.090 --> 00:01:37.570 And, as you well know, there both electric and magnetic 00:01:37.570 --> 00:01:38.630 fields. 00:01:38.630 --> 00:01:43.140 Overall, it is just much more difficult situation. 00:01:43.140 --> 00:01:51.760 So first of all, I start by a mathematical description 00:01:51.760 --> 00:01:53.190 of this system. 00:01:53.190 --> 00:01:55.760 Because, as I say, there is no other way 00:01:55.760 --> 00:01:58.770 we know of discussing it. 00:01:58.770 --> 00:02:05.710 And the mathematical description of the electromagnetic field, 00:02:05.710 --> 00:02:10.130 as you all know, are the so-called Maxwell's equations. 00:02:10.130 --> 00:02:17.750 I've written here the four Maxwell's equations for vacuum. 00:02:17.750 --> 00:02:23.230 So this is what the electric and magnetic fields 00:02:23.230 --> 00:02:26.240 have to satisfy. 00:02:26.240 --> 00:02:32.250 And I'm just reminding you that the definition 00:02:32.250 --> 00:02:34.440 of what electric and magnetic field-- 00:02:34.440 --> 00:02:38.510 the operational definition comes from the Lorentz force. 00:02:38.510 --> 00:02:41.700 Basically, this is just quickly to remind you, 00:02:41.700 --> 00:02:46.980 if I have a charge in vacuum and if it experiences a force, 00:02:46.980 --> 00:02:50.110 I know there is an electric field there. 00:02:50.110 --> 00:02:52.620 On the other hand, if it experiences 00:02:52.620 --> 00:02:55.640 a force when it's moving, then I know 00:02:55.640 --> 00:02:57.650 that there is a magnetic field. 00:02:57.650 --> 00:03:00.990 So this tells us that here, although we can't see it, 00:03:00.990 --> 00:03:05.140 there is an electromagnetic field. 00:03:05.140 --> 00:03:09.980 If one looks at these equations and plays around with them, 00:03:09.980 --> 00:03:15.500 one find that the electromagnetic field actually 00:03:15.500 --> 00:03:19.530 satisfy wave equations. 00:03:19.530 --> 00:03:22.840 This is the wave equation for the-- three-dimensional wave 00:03:22.840 --> 00:03:25.140 equation for the electric field. 00:03:25.140 --> 00:03:28.590 And this is for the magnetic field, where c is the phase 00:03:28.590 --> 00:03:32.630 velocity as always, and in the case of electromagnetism 00:03:32.630 --> 00:03:34.220 c is given by that. 00:03:34.220 --> 00:03:36.140 That's the speed of light, or the speed 00:03:36.140 --> 00:03:40.110 of electromagnetic waves. 00:03:40.110 --> 00:03:47.500 Now, so what this tells us, is that in vacuum, you 00:03:47.500 --> 00:03:51.170 can have excitations, oscillations 00:03:51.170 --> 00:03:54.000 of the electromagnetic and magnetic fields, 00:03:54.000 --> 00:03:55.920 which propagate. 00:03:55.920 --> 00:03:59.020 And we have all of the wave phenomena 00:03:59.020 --> 00:04:02.950 we've learned for other systems. 00:04:02.950 --> 00:04:08.780 The thing to keep in mind is that whatever 00:04:08.780 --> 00:04:11.960 the solution of the system is, whatever is propagating, 00:04:11.960 --> 00:04:17.950 it must satisfy all of these equations. 00:04:17.950 --> 00:04:21.329 Not every situation has to satisfy this. 00:04:21.329 --> 00:04:26.760 This is a subset of the infinite possibilities that 00:04:26.760 --> 00:04:29.970 are allowed by Maxwell's equations. 00:04:29.970 --> 00:04:31.300 OK. 00:04:31.300 --> 00:04:38.260 So now, instead of doing solutions 00:04:38.260 --> 00:04:42.870 to some specific situations with a specific boundary condition, 00:04:42.870 --> 00:04:46.660 et cetera, since it's already much more difficult, 00:04:46.660 --> 00:04:51.850 all I will do today is see how we 00:04:51.850 --> 00:04:56.280 can identify solutions of these equations. 00:04:56.280 --> 00:04:58.720 What kind of waves they correspond to. 00:04:58.720 --> 00:05:02.330 Or vice versa, if you want to describe 00:05:02.330 --> 00:05:05.310 in terms of mathematics some particular wave, 00:05:05.310 --> 00:05:08.070 how do we do that? 00:05:08.070 --> 00:05:12.820 That is the kind of problems I will be discussing today. 00:05:12.820 --> 00:05:15.920 So, let me come to the first problem. 00:05:22.830 --> 00:05:25.800 And probably using the word problem is a misnomer. 00:05:25.800 --> 00:05:28.380 The description. 00:05:28.380 --> 00:05:33.700 I'll consider first progressive wave 00:05:33.700 --> 00:05:35.165 solutions of these equations. 00:05:38.680 --> 00:05:47.060 Suppose we know that there is an electric field, which 00:05:47.060 --> 00:05:51.410 is a propagating electric field, sinusoidal. 00:05:51.410 --> 00:05:52.760 All right? 00:05:52.760 --> 00:05:59.820 I assure you, this does not contradict Maxwell's equations. 00:05:59.820 --> 00:06:01.440 You can try it. 00:06:01.440 --> 00:06:03.750 All right? 00:06:03.750 --> 00:06:06.195 It's not complete, as you'll see in a moment. 00:06:09.100 --> 00:06:12.360 The question here is, if you have an electric field 00:06:12.360 --> 00:06:17.460 like that, can we describe as well as 00:06:17.460 --> 00:06:21.860 possible in words, what kind of a wave this corresponds to? 00:06:21.860 --> 00:06:25.920 And secondly, answer the question 00:06:25.920 --> 00:06:29.980 if this is a real electromagnetic wave 00:06:29.980 --> 00:06:35.320 in the vacuum, what must be the corresponding magnetic field? 00:06:35.320 --> 00:06:38.610 By itself, this equation does not 00:06:38.610 --> 00:06:41.580 satisfy all the Maxwell's equations. 00:06:41.580 --> 00:06:46.930 You need a corresponding magnetic field. 00:06:46.930 --> 00:06:49.810 So, let's look at that. 00:06:49.810 --> 00:06:55.820 First of all, we know that any function, which 00:06:55.820 --> 00:07:03.080 is a function of x plus or minus vt describes 00:07:03.080 --> 00:07:06.300 a progressive wave. 00:07:06.300 --> 00:07:08.880 It satisfies the classical wave equation, 00:07:08.880 --> 00:07:12.090 you can try it and see. 00:07:12.090 --> 00:07:17.040 If these two terms-- the x and the t terms-- 00:07:17.040 --> 00:07:23.220 are of opposite sign, then this describes a progressive wave, 00:07:23.220 --> 00:07:27.030 which goes in the plus-x direction. 00:07:27.030 --> 00:07:29.580 If they are the same sign, then it 00:07:29.580 --> 00:07:32.720 goes in the opposite direction. 00:07:32.720 --> 00:07:37.100 And again I say, plot any function like this, 00:07:37.100 --> 00:07:41.270 and see what happens as you change t. 00:07:41.270 --> 00:07:44.660 The shape of the function will not change. 00:07:44.660 --> 00:07:48.180 But it will move either to the left or to the right 00:07:48.180 --> 00:07:49.300 as you change the time. 00:07:52.110 --> 00:07:54.610 So we immediately see, since this 00:07:54.610 --> 00:08:00.800 is a cosine of this, which is of this form, if I divide by a, 00:08:00.800 --> 00:08:03.240 this is x minus b over a. 00:08:03.240 --> 00:08:05.120 And I could take the a outside that. 00:08:05.120 --> 00:08:08.180 So this is a progressive wave. 00:08:08.180 --> 00:08:11.650 These two have opposite signs. 00:08:11.650 --> 00:08:13.310 It's a function of x and t. 00:08:13.310 --> 00:08:16.190 So this is a progressive wave, which is moving or progressing 00:08:16.190 --> 00:08:17.710 in the x direction. 00:08:17.710 --> 00:08:21.250 They're opposite signs, so in the plus-x direction. 00:08:21.250 --> 00:08:25.610 So immediately I know that this is a progressive wave. 00:08:25.610 --> 00:08:27.320 It is a sinusoidal one. 00:08:27.320 --> 00:08:29.690 Well, this is a cosine function, right? 00:08:29.690 --> 00:08:35.970 It's a sinusoidal wave. 00:08:35.970 --> 00:08:43.510 Now we know if I divide by a, I get minus-B over a t. 00:08:43.510 --> 00:08:45.670 So it becomes of this form. 00:08:45.670 --> 00:08:53.090 So the phase velocity of this wave will be B over a. 00:08:53.090 --> 00:08:57.102 And this we call normally, by the letter c, 00:08:57.102 --> 00:09:00.170 that's the phase velocity of electromagnetic waves 00:09:00.170 --> 00:09:00.840 in vacuum. 00:09:00.840 --> 00:09:03.540 Or commonly known as speed of light. 00:09:07.800 --> 00:09:13.340 That identifies it so far, as best as we can in words. 00:09:13.340 --> 00:09:20.930 This is, as I say, a progressive sinusoidal electric field 00:09:20.930 --> 00:09:22.410 moving in the plus-x direction. 00:09:27.850 --> 00:09:29.990 What are the a's and b's? 00:09:33.750 --> 00:09:40.450 By how much must you change x so that the wave gets 00:09:40.450 --> 00:09:43.130 the same amplitude as where you started? 00:09:43.130 --> 00:09:47.510 And the answer of that is that, of course, 00:09:47.510 --> 00:09:50.280 a must be 2 pi over lambda. 00:09:53.130 --> 00:09:57.060 because then if x changes by lambda, 00:09:57.060 --> 00:10:00.740 your cosine changes by 2 pi. 00:10:00.740 --> 00:10:05.290 So a in that equation must be 2 pi over lambda. 00:10:05.290 --> 00:10:08.840 That quantity is normally given the symbol k, 00:10:08.840 --> 00:10:12.810 it's called the wave number. 00:10:12.810 --> 00:10:19.720 Similarly, if I look at the time turn, 00:10:19.720 --> 00:10:23.860 b must be equal to 2 pi divided by the period. 00:10:26.810 --> 00:10:29.680 Because if t changes by the period, 00:10:29.680 --> 00:10:33.280 then that cosine-- the angle of that cosine, 00:10:33.280 --> 00:10:36.720 the phase of that function-- changes by 2 pi. 00:10:36.720 --> 00:10:39.350 And you're back where you started. 00:10:39.350 --> 00:10:41.410 So b must be 2 pi over t. 00:10:41.410 --> 00:10:45.980 So this tells you for that particular wave 00:10:45.980 --> 00:10:49.210 what the a must be, what the b is. 00:10:49.210 --> 00:10:51.980 2 pi over t is, of course, the same as 2 pi times 00:10:51.980 --> 00:10:54.250 the frequency, which we normally call 00:10:54.250 --> 00:10:57.400 the angular frequency, omega. 00:10:57.400 --> 00:11:01.000 So a is k, and b is omega. 00:11:05.590 --> 00:11:07.950 Next. 00:11:07.950 --> 00:11:16.310 I said that any solution that is real 00:11:16.310 --> 00:11:19.640 of the electromagnetic field must 00:11:19.640 --> 00:11:23.220 satisfy Maxwell's equations. 00:11:23.220 --> 00:11:26.000 So the same must be true of this. 00:11:26.000 --> 00:11:32.810 If this is the wave of the electron, electric field, 00:11:32.810 --> 00:11:42.140 there must be associated with it a magnetic field such that all 00:11:42.140 --> 00:11:46.450 of Maxwell's equations are satisfied. 00:11:46.450 --> 00:11:51.460 In particular, if we take this one-- Faraday's law-- 00:11:51.460 --> 00:11:55.850 we know that the rate of change of the magnetic field 00:11:55.850 --> 00:11:59.940 must be equal to minus the curl of the electric field. 00:12:03.350 --> 00:12:06.580 This you can look up in books of mathematics. 00:12:06.580 --> 00:12:09.940 If you look at all the components, the way I always 00:12:09.940 --> 00:12:13.570 remember it, it is the determinant 00:12:13.570 --> 00:12:16.760 where here you have the unit x direction yz. 00:12:16.760 --> 00:12:18.570 This is dx, dy, dz. 00:12:18.570 --> 00:12:21.634 And here is the x component of electric field, y component, 00:12:21.634 --> 00:12:22.300 and z component. 00:12:24.840 --> 00:12:29.600 For our particular electric field, 00:12:29.600 --> 00:12:33.120 I only have the z component. 00:12:33.120 --> 00:12:36.030 And it's only a function of x. 00:12:36.030 --> 00:12:39.090 So most of the terms of this expansion 00:12:39.090 --> 00:12:47.360 are 0, except the one-- the rate of change of with x of Ez. 00:12:47.360 --> 00:12:50.230 And that will be in the y direction. 00:12:50.230 --> 00:12:57.260 So this db dt must be equal to that if that is a solution 00:12:57.260 --> 00:12:59.450 the Maxwell's equations. 00:12:59.450 --> 00:13:07.940 If I take the x derivative of E up there. 00:13:07.940 --> 00:13:10.050 I end up-- and you could almost do it 00:13:10.050 --> 00:13:14.100 in your head-- db dt is minus this quantity. 00:13:17.960 --> 00:13:20.250 But if this is the rate of change of t, 00:13:20.250 --> 00:13:21.890 I can integrate this. 00:13:21.890 --> 00:13:28.360 And if I integrated it, B must be equal to-- the a 00:13:28.360 --> 00:13:34.320 comes from here, a minus a, a over-- sorry. 00:13:34.320 --> 00:13:37.820 The b comes from here, I misspoke. 00:13:37.820 --> 00:13:41.430 That comes out, and the integral of sine 00:13:41.430 --> 00:13:46.050 gives you cosine, so that must be satisfied. 00:13:46.050 --> 00:13:48.000 But since we integrated B, there will 00:13:48.000 --> 00:13:50.820 be a constant of integration. 00:13:50.820 --> 00:13:54.190 So if I add to this any constant B, 00:13:54.190 --> 00:13:56.800 this will still satisfy this equation. 00:14:02.300 --> 00:14:06.240 All of this is telling me is that if I 00:14:06.240 --> 00:14:09.660 have that electric field-- propagating electric field-- 00:14:09.660 --> 00:14:14.750 I must simultaneously have this propagating magnetic field. 00:14:14.750 --> 00:14:18.895 And on top of that, I can have any constant magnetic field. 00:14:23.610 --> 00:14:27.410 It means that is a more general situation where 00:14:27.410 --> 00:14:29.960 this electric field and these magnetic fields 00:14:29.960 --> 00:14:36.020 can exist with any constant B. I'll just call it 0. 00:14:36.020 --> 00:14:38.400 It's not an interesting part of this, 00:14:38.400 --> 00:14:40.350 it's not a propagating field. 00:14:40.350 --> 00:14:45.480 And so we end up that if you have that electric field 00:14:45.480 --> 00:14:49.460 propagating, and in with this magnetic field, 00:14:49.460 --> 00:14:55.270 then that system satisfies all Maxwell's equations. 00:14:55.270 --> 00:14:59.310 Both the E and B will satisfy these wave equations. 00:14:59.310 --> 00:15:02.080 Try it for yourself, and you'll see. 00:15:02.080 --> 00:15:07.153 So the answer to this is, what this 00:15:07.153 --> 00:15:13.000 is, this is a polarized-- plane-polarized electromagnetic 00:15:13.000 --> 00:15:17.430 wave, where we identified the wavelength, the frequency, 00:15:17.430 --> 00:15:20.680 it's propagating in the x direction. 00:15:20.680 --> 00:15:27.220 And the electric field is polarized in the z direction. 00:15:27.220 --> 00:15:29.640 One of the things we will learn from this so we don't have 00:15:29.640 --> 00:15:33.360 to repeat over and over again when we're looking at different 00:15:33.360 --> 00:15:38.950 formulae, which describe ways to help us to identify , them is-- 00:15:38.950 --> 00:15:43.390 notice that what we have found was that the electric 00:15:43.390 --> 00:15:47.640 and the magnetic fields are perpendicular to each other. 00:15:47.640 --> 00:15:49.620 The electric field in the z direction, 00:15:49.620 --> 00:15:51.840 the magnetic in the y direction. 00:15:51.840 --> 00:15:57.730 But the sinusoidal part and the phase velocity and everything 00:15:57.730 --> 00:16:02.100 else-- wavelength, frequency-- are exactly the same and phase. 00:16:02.100 --> 00:16:04.560 This is completely in general. 00:16:04.560 --> 00:16:10.940 If you have a progressive electromagnetic wave in vacuum, 00:16:10.940 --> 00:16:15.050 you find that the only way it can 00:16:15.050 --> 00:16:19.050 exist if you have simultaneously an electric 00:16:19.050 --> 00:16:21.620 and a magnetic field propagating. 00:16:21.620 --> 00:16:24.390 They are always at right angles to each other. 00:16:24.390 --> 00:16:27.960 This is the electric field, this will be the magnetic field. 00:16:27.960 --> 00:16:31.380 If it's propagating in that direction. 00:16:31.380 --> 00:16:36.860 It's always from e to b in a clockwise rotation, 00:16:36.860 --> 00:16:42.140 if they're propagating in that direction. 00:16:42.140 --> 00:16:44.840 So I drew a general sketch here. 00:16:44.840 --> 00:16:50.300 This is true for any progressive wave, 00:16:50.300 --> 00:16:52.890 electromagnetic progressive wave. 00:16:52.890 --> 00:16:57.310 And you have the electric field, magnetic field perpendicular 00:16:57.310 --> 00:17:01.770 to it, and the two propagate in that direction, 00:17:01.770 --> 00:17:03.830 given by this vector equation. 00:17:03.830 --> 00:17:08.000 Furthermore, if they satisfy Maxwell's equation the ratio 00:17:08.000 --> 00:17:11.740 of E to B, the magnitude, is equal to c. 00:17:11.740 --> 00:17:13.400 This is completely general. 00:17:13.400 --> 00:17:16.069 It is worth remembering when we're 00:17:16.069 --> 00:17:19.829 analyzing different situations. 00:17:19.829 --> 00:17:21.950 So that I went slowly through this, 00:17:21.950 --> 00:17:25.780 but that is one example where we see 00:17:25.780 --> 00:17:28.770 this mathematical description of something which 00:17:28.770 --> 00:17:32.730 we can recognize what it is, and which 00:17:32.730 --> 00:17:38.090 is a solution to Maxwell's equations in vacuum. 00:17:38.090 --> 00:17:41.210 What actually happens in the physical situation 00:17:41.210 --> 00:17:44.250 depends, as always, on all the boundary conditions, 00:17:44.250 --> 00:17:46.900 the initial conditions, et cetera. 00:17:46.900 --> 00:17:49.960 This doesn't address all those questions. 00:17:49.960 --> 00:17:56.800 All this says is this is one of the infinite possible solutions 00:17:56.800 --> 00:17:59.820 of Maxwell's equation. 00:17:59.820 --> 00:18:03.940 In other words, for electromagnetic fields 00:18:03.940 --> 00:18:10.450 corresponding to the plane wave propagating in one direction. 00:18:10.450 --> 00:18:11.845 Let's take a harder example. 00:18:16.260 --> 00:18:19.140 The question is the following. 00:18:19.140 --> 00:18:21.370 Can we now do the opposite? 00:18:21.370 --> 00:18:24.270 Not someone tells us the equation. 00:18:24.270 --> 00:18:29.510 Can we actually describe in mathematical forms 00:18:29.510 --> 00:18:33.510 a electromagnetic wave whose properties 00:18:33.510 --> 00:18:36.150 we know what we want and would like 00:18:36.150 --> 00:18:38.730 to write it mathematically. 00:18:38.730 --> 00:18:40.370 And I took a slightly harder one, 00:18:40.370 --> 00:18:45.350 so I said we would like to describe both 00:18:45.350 --> 00:18:49.630 the electric and the magnetic fields, 00:18:49.630 --> 00:18:54.890 which describes a monochromatic electromagnetic wave-- 00:18:54.890 --> 00:19:00.290 monochromatic means a single frequency, single wavelength-- 00:19:00.290 --> 00:19:05.610 with wavelength lambda, which propagates now 00:19:05.610 --> 00:19:10.350 not along the x or y or z axes that makes life easy. 00:19:10.350 --> 00:19:11.900 Let's say it goes at some angle. 00:19:11.900 --> 00:19:18.930 It goes at 45 degrees to the x-axis and y-axis. 00:19:18.930 --> 00:19:20.750 And the z is out of the board. 00:19:20.750 --> 00:19:23.570 So the wave-- we want the wave, which 00:19:23.570 --> 00:19:35.910 is propagating like this, where the wave front is-- let me come 00:19:35.910 --> 00:19:39.590 to it in a second-- where the vector 00:19:39.590 --> 00:19:43.750 perpendicular to the wave front is at 45 degrees 00:19:43.750 --> 00:19:45.696 to both the x-axis and y-axis. 00:19:48.760 --> 00:19:52.430 We want it plane-polarized, meaning 00:19:52.430 --> 00:19:58.730 that the electric vector is always in a plane 00:19:58.730 --> 00:20:01.560 and it's linearly polarized so it's 00:20:01.560 --> 00:20:06.415 in the same direction in the x-y plane. 00:20:09.400 --> 00:20:13.860 So how can we translate that into mathematics? 00:20:13.860 --> 00:20:17.990 Well, we'll use some of the knowledge 00:20:17.990 --> 00:20:20.750 we've just gained before. 00:20:20.750 --> 00:20:25.130 First of all, we know from what I discussed 00:20:25.130 --> 00:20:27.130 about the electric and magnetic field being 00:20:27.130 --> 00:20:30.190 perpendicular to each other and perpendicular 00:20:30.190 --> 00:20:35.140 to the direction of propagation that if the propagation is 00:20:35.140 --> 00:20:42.390 in this direction, then we know that the plane in which 00:20:42.390 --> 00:20:45.670 the electric and magnetic fields find themselves 00:20:45.670 --> 00:20:47.455 are perpendicular to that. 00:20:50.360 --> 00:20:52.280 Since this is propagating like this, 00:20:52.280 --> 00:21:02.100 the distance between the planes of equal phase will be lambda. 00:21:02.100 --> 00:21:04.500 That's the meaning of the wavelength. 00:21:04.500 --> 00:21:07.920 Once you've gone the distance of 1 lambda, 00:21:07.920 --> 00:21:09.750 the magnitude and direction is back 00:21:09.750 --> 00:21:14.290 to what it was before for both the electric and magnetic 00:21:14.290 --> 00:21:15.300 fields. 00:21:15.300 --> 00:21:17.240 So that's what it will look like. 00:21:20.960 --> 00:21:26.090 So the electric vector will be in this plane, 00:21:26.090 --> 00:21:28.970 but we are told furthermore it's in the xy, 00:21:28.970 --> 00:21:31.730 so it will be in this direction. 00:21:31.730 --> 00:21:34.050 If it's like this, and in this plane, 00:21:34.050 --> 00:21:37.000 so this must be the direction of the electric vector. 00:21:37.000 --> 00:21:40.150 So let's give it a magnitude E-zero. 00:21:40.150 --> 00:21:45.090 And what is this unit vector? 00:21:45.090 --> 00:21:49.550 Well, clearly that is in the x direction. 00:21:49.550 --> 00:21:56.500 It has a component like this, and in the y direction, 00:21:56.500 --> 00:21:58.430 it has a component like that. 00:21:58.430 --> 00:22:00.340 The magnitude of the components is 00:22:00.340 --> 00:22:03.380 the same, because of the 45 degrees. 00:22:03.380 --> 00:22:07.010 But for the x, it'll be negative, and for the y, 00:22:07.010 --> 00:22:08.310 positive. 00:22:08.310 --> 00:22:11.740 So the unit vector in the direction 00:22:11.740 --> 00:22:16.430 of the electric vector will be minus x-hat 00:22:16.430 --> 00:22:19.200 over root-2 plus y-hat over root-2. 00:22:19.200 --> 00:22:22.540 This is a unit vector, you can check for yourself. 00:22:22.540 --> 00:22:25.890 If you take square this, square that, take the square root, 00:22:25.890 --> 00:22:26.920 you get 1. 00:22:26.920 --> 00:22:32.240 So this is a unit vector in this direction where we wanted it. 00:22:32.240 --> 00:22:36.720 So if I write this as the amplitude 00:22:36.720 --> 00:22:39.600 and the direction of the electric field, 00:22:39.600 --> 00:22:44.940 I do have a field which is linearly polarized always 00:22:44.940 --> 00:22:47.030 in the same direction. 00:22:47.030 --> 00:22:48.840 We'll put a sine or cosine there, 00:22:48.840 --> 00:22:52.386 because we're talking about a monochromatic electromagnetic 00:22:52.386 --> 00:22:54.760 wave with the wavelengths, so it's a sinusoidal function. 00:22:58.450 --> 00:23:01.270 Where I put the sine or cosine or any other phase 00:23:01.270 --> 00:23:04.345 just determines where time equals 0. 00:23:04.345 --> 00:23:05.095 So let's put sine. 00:23:07.740 --> 00:23:13.920 It's going to be propagating in this direction, plus k, 00:23:13.920 --> 00:23:18.370 so these two will have opposite sign. 00:23:18.370 --> 00:23:21.760 This will be the frequency-- angular frequency-- 00:23:21.760 --> 00:23:24.730 of oscillations of this. 00:23:24.730 --> 00:23:31.670 And here we must describe a plane. 00:23:31.670 --> 00:23:37.200 Because along this plane, the phase has to be the same. 00:23:37.200 --> 00:23:38.720 That's what we mean by wavefront. 00:23:41.710 --> 00:23:45.720 Vectorially, how do we describe a plane? 00:23:45.720 --> 00:23:49.260 Well, we will have the plane which is perpendicular to k 00:23:49.260 --> 00:23:54.800 if we take k dot product of the vector r. 00:23:54.800 --> 00:23:56.070 r is the vector. 00:23:59.110 --> 00:24:03.790 Here is the vector r, from the origin 00:24:03.790 --> 00:24:08.300 to a point on the plane which I want to describe. 00:24:08.300 --> 00:24:10.630 So this is k dot r. 00:24:10.630 --> 00:24:17.640 So this now, we'll have k, which is the wave number, 00:24:17.640 --> 00:24:23.430 and this whole thing is called the k vector, 00:24:23.430 --> 00:24:28.300 will have a magnitude which is 2 pi over lambda. 00:24:28.300 --> 00:24:30.300 Same as in the other problem. 00:24:30.300 --> 00:24:36.300 But now it's pointing in this direction, which again, 00:24:36.300 --> 00:24:39.730 by analogy, how we calculated that is the unit vector 00:24:39.730 --> 00:24:43.460 x over root-2 plus unit vector y over root-2. 00:24:43.460 --> 00:24:45.480 So this is k. 00:24:45.480 --> 00:24:49.910 r is nothing I want to describe this point. 00:24:49.910 --> 00:24:53.560 I have x in the x direction, y in the y direction, 00:24:53.560 --> 00:24:55.210 z in the z direction. 00:24:55.210 --> 00:24:59.000 So that describes any point on that plane. 00:24:59.000 --> 00:25:01.830 If I take the dot product between them, 00:25:01.830 --> 00:25:09.160 I will get then a wave which is moving the the k direction. 00:25:09.160 --> 00:25:16.100 And this describes the position on the wavefront. 00:25:16.100 --> 00:25:20.180 So putting it all together, this electric field 00:25:20.180 --> 00:25:26.560 at every point of x, y, and t will have a magnitude 00:25:26.560 --> 00:25:32.130 is E-zero times this direction, the direction 00:25:32.130 --> 00:25:35.360 of polarization of the electric field, times sine. 00:25:40.660 --> 00:25:49.820 This is now telling me it's propagating in this direction. 00:25:49.820 --> 00:25:52.630 And with angular frequency omega. 00:25:52.630 --> 00:26:02.130 So that describes the electric part of this wave. 00:26:02.130 --> 00:26:03.730 How about the magnetic one? 00:26:03.730 --> 00:26:06.710 Well, we could do the same as before. 00:26:06.710 --> 00:26:10.420 The magnetic part is determined by this, 00:26:10.420 --> 00:26:14.480 because all Maxwell's equations have to be satisfied, 00:26:14.480 --> 00:26:16.420 including Faraday's law. 00:26:20.020 --> 00:26:23.570 But I told you, so it saves me doing it over and over again, 00:26:23.570 --> 00:26:27.510 we've learned once and for all, for a progressive wave 00:26:27.510 --> 00:26:30.130 the e and b are perpendicular to each other, 00:26:30.130 --> 00:26:33.970 and the ratio between them is c. 00:26:33.970 --> 00:26:39.860 So since I know what E is, the magnitude of the magnetic field 00:26:39.860 --> 00:26:41.880 is E-zero over c. 00:26:41.880 --> 00:26:46.320 It'll be at right angles to this direction 00:26:46.320 --> 00:26:49.470 and to the propagation, and therefore it 00:26:49.470 --> 00:26:51.830 will be out of the board. 00:26:51.830 --> 00:26:57.580 So that from E-cross-B, the vectors are in the k direction. 00:26:57.580 --> 00:27:00.280 So the b will be out of the board, which 00:27:00.280 --> 00:27:01.740 is easier this time. 00:27:01.740 --> 00:27:04.880 That's in the z direction. 00:27:04.880 --> 00:27:11.080 And it will be, as I said, exactly in phase in time 00:27:11.080 --> 00:27:14.390 and space with the electric field. 00:27:14.390 --> 00:27:16.510 The two are coupled together. 00:27:16.510 --> 00:27:21.000 So that now describes it entirely. 00:27:21.000 --> 00:27:25.720 So this is, in fact, the answer to our question. 00:27:25.720 --> 00:27:28.600 It describes an electric or magnetic field 00:27:28.600 --> 00:27:30.510 which is monochromatic. 00:27:30.510 --> 00:27:32.520 It's an electromagnetic wave. 00:27:32.520 --> 00:27:33.850 It has wavelength lambda. 00:27:33.850 --> 00:27:38.490 It propagates at 45 degrees to x and y axes, 00:27:38.490 --> 00:27:40.010 and is plane-polarized. 00:27:40.010 --> 00:27:45.080 e is always in the same direction and in the xy plane. 00:27:45.080 --> 00:27:46.400 So this is the answer. 00:27:46.400 --> 00:27:47.430 See, notice. 00:27:47.430 --> 00:27:49.840 In the past when we were doing problems, 00:27:49.840 --> 00:27:53.620 we focus more on things like what 00:27:53.620 --> 00:27:56.230 is the wave equation for this string? 00:27:56.230 --> 00:28:01.140 Or for a pipe with a gas in it? 00:28:01.140 --> 00:28:05.590 Or a transmission line, et cetera. 00:28:05.590 --> 00:28:12.460 Here, even guessing what solutions we're interested in, 00:28:12.460 --> 00:28:15.240 what kind of solution, it's already hard or even 00:28:15.240 --> 00:28:19.540 to describe the wave we're interested in. 00:28:19.540 --> 00:28:23.550 So this, for the other situations, 00:28:23.550 --> 00:28:26.030 this would have taken a few minutes. 00:28:26.030 --> 00:28:29.730 Here it needs a fair amount of analysis. 00:28:29.730 --> 00:28:34.090 And it takes much longer. 00:28:34.090 --> 00:28:35.800 Let me take one more case. 00:28:46.100 --> 00:28:51.290 The last case I'm going to exhibit is the following. 00:28:51.290 --> 00:28:58.160 Again the issue will be, there's this particular wave 00:28:58.160 --> 00:29:00.380 we want to produce. 00:29:00.380 --> 00:29:02.500 We know what we want, and we want 00:29:02.500 --> 00:29:06.000 to know how to describe it mathematically. 00:29:06.000 --> 00:29:10.470 So once again, we want to find a solution 00:29:10.470 --> 00:29:14.150 of our Maxwell's equations, which 00:29:14.150 --> 00:29:19.640 have the following property that correspond to a circularly 00:29:19.640 --> 00:29:22.260 polarized electromagnetic wave which 00:29:22.260 --> 00:29:24.560 is propagating in y direction. 00:29:24.560 --> 00:29:30.940 And it just says "any." so any, any circularly polarized 00:29:30.940 --> 00:29:35.345 electromagnetic wave which is propagating in the minus y 00:29:35.345 --> 00:29:35.845 direction. 00:29:39.600 --> 00:29:45.260 First of all, what we mean by circularly polarized wave? 00:29:45.260 --> 00:29:49.790 A circularly polarized wave is that, if I 00:29:49.790 --> 00:29:56.940 took a snapshot, if I could, at a given instant of time, 00:29:56.940 --> 00:30:02.910 one would find that the electric vector along the propagation 00:30:02.910 --> 00:30:11.890 direction is rotating like this on the spiral. 00:30:11.890 --> 00:30:17.290 If that wave is moving towards you, 00:30:17.290 --> 00:30:24.360 what you would see in any plane, a rotating electric field. 00:30:24.360 --> 00:30:29.445 And associated with a magnetic field at right angles to it. 00:30:32.360 --> 00:30:34.030 It doesn't tell us whether we want 00:30:34.030 --> 00:30:37.440 a left-handed or a right-handed rotated field. 00:30:37.440 --> 00:30:39.280 So just arbitrarily take one. 00:30:39.280 --> 00:30:42.520 And by the way, if ever you're interested in the left- 00:30:42.520 --> 00:30:46.190 and right-handed and figuring out which is which? 00:30:46.190 --> 00:30:47.310 It's a mess. 00:30:47.310 --> 00:30:50.290 Different communities use different definitions, 00:30:50.290 --> 00:30:52.230 what they mean by right- and left-handed. 00:30:52.230 --> 00:30:56.360 So I won't try to confuse you more than that. 00:30:56.360 --> 00:30:59.640 So here we want any wave, which corresponds 00:30:59.640 --> 00:31:03.920 to circular polarization, and is moving in the minus y 00:31:03.920 --> 00:31:05.310 direction. 00:31:05.310 --> 00:31:10.380 So if it's moving in plus or minus y direction, 00:31:10.380 --> 00:31:15.630 we know that the electric field will be in the xz plane 00:31:15.630 --> 00:31:18.120 at every instant of time. 00:31:18.120 --> 00:31:21.110 If it's circularly polarized, we know 00:31:21.110 --> 00:31:25.720 that the magnitude of the electric field at all locations 00:31:25.720 --> 00:31:29.630 of x, y, and z at all times will be the same. 00:31:29.630 --> 00:31:31.520 It does not change. 00:31:31.520 --> 00:31:32.320 It's a constant. 00:31:35.990 --> 00:31:38.040 Now so how do we create such a thing? 00:31:38.040 --> 00:31:40.900 Well, if we stop and think for a second, 00:31:40.900 --> 00:31:45.820 if we superimpose two solutions-- suppose 00:31:45.820 --> 00:31:50.750 we have one solution, which is a plane-polarized electromagnetic 00:31:50.750 --> 00:31:55.820 wave going towards you, and I superimpose 00:31:55.820 --> 00:32:03.190 on that another one which is out of phase with it and at 90 00:32:03.190 --> 00:32:09.040 degrees, then at every location in space, 00:32:09.040 --> 00:32:12.040 I'll have two components. 00:32:12.040 --> 00:32:16.950 If I make those components change, but in such a way 00:32:16.950 --> 00:32:19.230 that the vector addition of the two 00:32:19.230 --> 00:32:23.270 gives me a unit vector, a constant vector, 00:32:23.270 --> 00:32:26.750 I will have achieved what I wanted to do. 00:32:26.750 --> 00:32:35.110 So here is a equation which satisfies everything I've said. 00:32:35.110 --> 00:32:39.460 Let's consider an electromagnetic wave 00:32:39.460 --> 00:32:45.755 which is the same in all x and all z positions. 00:32:48.840 --> 00:32:52.640 The only variable is in the y direction. 00:32:52.640 --> 00:32:57.140 If I write that as the superposition 00:32:57.140 --> 00:33:01.520 of an electric field which is in the x direction, 00:33:01.520 --> 00:33:09.880 and propagating as a sine-- it's a sinusoidal wave-- 00:33:09.880 --> 00:33:17.760 and I add to it a cosine, which is at right angles. 00:33:17.760 --> 00:33:19.940 Furthermore I'll use the other information. 00:33:19.940 --> 00:33:22.540 It's going in the minus y direction. 00:33:22.540 --> 00:33:25.860 So I'll make these two opposite sign-- sorry, 00:33:25.860 --> 00:33:29.500 I make them the same sign, it is in minus-y direction. 00:33:29.500 --> 00:33:32.640 If it was in the plus-y, they would have opposite signs. 00:33:32.640 --> 00:33:37.590 If it's minus-y, this would have to be the same. 00:33:37.590 --> 00:33:41.690 So this is a sinusoidal wave moving 00:33:41.690 --> 00:33:43.960 in the minus-y direction. 00:33:43.960 --> 00:33:46.390 It'll have the wave number k, this is 2 pi over lambda. 00:33:46.390 --> 00:33:51.310 And this is 2 pi, the frequency or 2 pi over the period. 00:33:51.310 --> 00:33:58.650 Omega over k has to be c, the speed of electromagnetic waves. 00:33:58.650 --> 00:34:03.470 If I add to this, the resultant electric vector 00:34:03.470 --> 00:34:07.230 everywhere in space has a magnitude E-zero. 00:34:07.230 --> 00:34:08.690 I can check it. 00:34:08.690 --> 00:34:12.139 The magnitude of E is the square root 00:34:12.139 --> 00:34:14.550 of the x component of this squared 00:34:14.550 --> 00:34:17.719 plus the z component of this squared. 00:34:17.719 --> 00:34:21.770 So it's E-zero, the x component squared-- 00:34:21.770 --> 00:34:23.449 the sine squared of this. 00:34:23.449 --> 00:34:28.550 The z component is the cosine, so the squared is that. 00:34:28.550 --> 00:34:32.760 For all values of x, y, and z at all times, 00:34:32.760 --> 00:34:35.889 if I add these and take the square root, I get 1. 00:34:35.889 --> 00:34:37.580 And so this is E-zero. 00:34:37.580 --> 00:34:42.909 So this propagating wave does satisfy my requirement 00:34:42.909 --> 00:34:46.230 that everywhere is magnitude E-zero. 00:34:46.230 --> 00:34:48.110 It is a propagating wave. 00:34:48.110 --> 00:34:52.070 Each one of these are propagating 00:34:52.070 --> 00:35:02.310 with the speed of light in the direction of y. 00:35:02.310 --> 00:35:08.360 I'm sorry, forgive me, can't copy from one line to the next. 00:35:08.360 --> 00:35:11.210 This is plus, this is plus. 00:35:11.210 --> 00:35:11.830 All right. 00:35:11.830 --> 00:35:14.800 It's moving in the minus-y direction. 00:35:14.800 --> 00:35:17.420 The way I had it, it was going in the plus-y direction. 00:35:17.420 --> 00:35:18.230 I corrected it. 00:35:18.230 --> 00:35:20.670 This is in the minus-y direction. 00:35:20.670 --> 00:35:21.410 All right? 00:35:21.410 --> 00:35:24.310 And this is what was required. 00:35:24.310 --> 00:35:25.070 OK. 00:35:25.070 --> 00:35:31.960 So this mathematical description of the electric vector, 00:35:31.960 --> 00:35:35.180 how it's propagating. 00:35:35.180 --> 00:35:38.280 And now we want to know what the magnetic one is doing. 00:35:38.280 --> 00:35:44.900 Well, again, we could go back and make sure 00:35:44.900 --> 00:35:48.250 that Maxwell's equations are completely satisfied. 00:35:48.250 --> 00:35:52.880 And you'll find that here, in order for Faraday's law 00:35:52.880 --> 00:35:58.500 to hold, I have to have also a changing magnetic field. 00:35:58.500 --> 00:36:00.830 But instead of doing that, I'll make 00:36:00.830 --> 00:36:03.960 use of what we learned by the previous examples. 00:36:03.960 --> 00:36:07.860 We know that this is a superposition 00:36:07.860 --> 00:36:11.370 of two progressive waves. 00:36:11.370 --> 00:36:14.650 Each one of these is a solution of Maxwell's wave. 00:36:14.650 --> 00:36:16.140 I don't need both of them. 00:36:16.140 --> 00:36:20.620 I only needed both to get a circularly polarized wave. 00:36:20.620 --> 00:36:24.360 Each one of these has to satisfy Maxwell's equation. 00:36:24.360 --> 00:36:27.350 So associated with each of these components, 00:36:27.350 --> 00:36:32.420 I must have a magnetic field which satisfies the requirement 00:36:32.420 --> 00:36:35.570 that there is an electric vector and magnetic vector 00:36:35.570 --> 00:36:39.620 at right angle to each other moving together 00:36:39.620 --> 00:36:43.550 in the direction of propagation in phase and in time. 00:36:43.550 --> 00:36:47.440 So for each one of these, I will find 00:36:47.440 --> 00:36:49.740 the corresponding magnetic field, 00:36:49.740 --> 00:36:52.460 the magnitude will be E-zero over c, 00:36:52.460 --> 00:36:55.050 because we know that the ratio of the electric field 00:36:55.050 --> 00:36:59.990 to the magnetic field is always equal to c in vacuum. 00:36:59.990 --> 00:37:01.100 It's at right angles. 00:37:01.100 --> 00:37:05.380 This was in the x direction, this is in the z direction. 00:37:05.380 --> 00:37:09.330 And in this case, then add this one. 00:37:09.330 --> 00:37:12.040 Here, this was plus-z and this is minus-x. 00:37:12.040 --> 00:37:14.590 And you can draw yourself a little picture 00:37:14.590 --> 00:37:16.870 to make sure you get everything right. 00:37:16.870 --> 00:37:19.990 Let me just talk about, say, this one. 00:37:19.990 --> 00:37:21.930 The second component. 00:37:21.930 --> 00:37:23.460 What I have in the [INAUDIBLE], this 00:37:23.460 --> 00:37:26.860 is moving there in minus-y. 00:37:26.860 --> 00:37:30.370 This component is in the z direction, 00:37:30.370 --> 00:37:34.010 so it's over here, coming out of the board. 00:37:34.010 --> 00:37:37.830 If it's in this direction, moving down here, 00:37:37.830 --> 00:37:41.280 then the b must be in that direction. 00:37:41.280 --> 00:37:43.530 So it must be in this direction, which 00:37:43.530 --> 00:37:45.670 is minus-x, which is correct. 00:37:45.670 --> 00:37:49.190 So this is how I get this right. 00:37:49.190 --> 00:37:54.440 If I add these, I get the total magnetic field. 00:37:54.440 --> 00:38:02.000 This, now, describes one possible wave 00:38:02.000 --> 00:38:04.830 which satisfies this requirement. 00:38:04.830 --> 00:38:07.910 It's a circularly polarized electromagnetic wave 00:38:07.910 --> 00:38:11.840 propagating in the minus-y direction. 00:38:11.840 --> 00:38:17.360 OK, so let me stop at these examples of progressive waves, 00:38:17.360 --> 00:38:20.275 and I'll move over to standing waves. 00:38:24.290 --> 00:38:26.750 So let's continue in a second, thank you. 00:38:31.420 --> 00:38:33.860 So I've now erased the board, and I 00:38:33.860 --> 00:38:37.810 can continue talking about wave solutions 00:38:37.810 --> 00:38:39.300 to Maxwell's equations. 00:38:39.300 --> 00:38:40.940 But let's recap for a second. 00:38:44.640 --> 00:38:49.630 What we find is the following, that basically in vacuum 00:38:49.630 --> 00:38:53.210 at every location in space it's as if there was an oscillator. 00:38:55.950 --> 00:38:58.000 It can be displaced from equilibrium. 00:38:58.000 --> 00:39:00.750 It can be made to oscillate. 00:39:00.750 --> 00:39:04.480 Displacement from equilibrium means 00:39:04.480 --> 00:39:06.730 there is an electric field there, 00:39:06.730 --> 00:39:10.590 or there is a magnetic field there. 00:39:10.590 --> 00:39:12.940 These can oscillate. 00:39:12.940 --> 00:39:15.090 They don't have to oscillate. 00:39:15.090 --> 00:39:20.800 So for example, you could have a static field, 00:39:20.800 --> 00:39:25.150 just an electric field constant in time everywhere in space. 00:39:25.150 --> 00:39:30.660 That means every location space is displaced from equilibrium. 00:39:30.660 --> 00:39:33.570 There could be a constant magnetic field instead, 00:39:33.570 --> 00:39:34.845 or both constant. 00:39:38.770 --> 00:39:40.230 Imagine how complicated this is. 00:39:40.230 --> 00:39:45.670 At every location the direction of this displacement 00:39:45.670 --> 00:39:49.210 from equilibrium for the electric and magnetic fields, 00:39:49.210 --> 00:39:50.910 they are vectors. 00:39:50.910 --> 00:39:53.550 There are possibility of the electric field facing 00:39:53.550 --> 00:39:57.270 a different directions of the magnetic field. 00:39:57.270 --> 00:40:00.740 What we find is that whatever that combination is 00:40:00.740 --> 00:40:04.490 in space and time, that combination 00:40:04.490 --> 00:40:08.060 has to satisfy Maxwell's equations. 00:40:08.060 --> 00:40:11.940 That completely describes what happens in vacuum 00:40:11.940 --> 00:40:15.830 at every point in space and time. 00:40:15.830 --> 00:40:21.540 Now there are in particular combinations 00:40:21.540 --> 00:40:25.720 of these displacements of oscillations in space and time, 00:40:25.720 --> 00:40:31.350 which satisfy the wave equation for the electric and magnetic 00:40:31.350 --> 00:40:31.850 fields. 00:40:35.390 --> 00:40:40.580 It's a tiny subset of total, but there are such. 00:40:40.580 --> 00:40:46.980 And we are considering now for that tiny subset what kind 00:40:46.980 --> 00:40:50.850 of solutions exist, how to describe them. 00:40:50.850 --> 00:40:53.440 And even there, we're limiting ourselves 00:40:53.440 --> 00:40:58.700 to a tiny subset of a tiny subset. 00:40:58.700 --> 00:41:03.270 So far, I took the subset where this displacement 00:41:03.270 --> 00:41:08.540 from equilibrium of the electric and magnetic fields 00:41:08.540 --> 00:41:12.160 is a progressive wave. 00:41:12.160 --> 00:41:17.140 And what we found, in order to make sure 00:41:17.140 --> 00:41:21.130 that the Maxwell's equations are satisfied, 00:41:21.130 --> 00:41:23.760 you can't have any old electric field 00:41:23.760 --> 00:41:28.480 wave, or any old magnetic field wave. 00:41:28.480 --> 00:41:30.040 There's an interplay. 00:41:30.040 --> 00:41:33.960 There is, in reality, just one electromagnetic field, 00:41:33.960 --> 00:41:36.750 and that propagates. 00:41:36.750 --> 00:41:41.950 We'll now go and look for other solutions of these equations. 00:41:41.950 --> 00:41:50.980 And very interesting solutions are standing waves. 00:41:50.980 --> 00:41:56.840 So let me take a concrete example and discuss it. 00:41:56.840 --> 00:42:01.020 So here is, you could call it a problem. 00:42:01.020 --> 00:42:07.130 Suppose that I have everywhere in space an electric field 00:42:07.130 --> 00:42:10.200 which consists of a standing wave. 00:42:10.200 --> 00:42:12.140 You can recognize this when we were talking 00:42:12.140 --> 00:42:19.020 about standing waves on strings, for example. 00:42:19.020 --> 00:42:22.660 Where you have the electric field always pointing 00:42:22.660 --> 00:42:24.730 in the x direction. 00:42:24.730 --> 00:42:28.420 It's oscillating at every point in space 00:42:28.420 --> 00:42:31.840 with the same frequency and phase, cosine omega t. 00:42:31.840 --> 00:42:35.570 It's oscillating with that angular frequency. 00:42:35.570 --> 00:42:43.130 And spatially, it not change in the x and y direction, 00:42:43.130 --> 00:42:45.940 but it does in the z direction. 00:42:45.940 --> 00:42:48.490 And that is a cosine like this. 00:42:48.490 --> 00:42:54.190 So this is a standing wave of electric field. 00:42:57.080 --> 00:43:01.840 This by itself cannot be a solution. 00:43:01.840 --> 00:43:06.230 Is not a situation you can have in vacuum. 00:43:06.230 --> 00:43:10.000 It violates, by itself, Maxwell's equation. 00:43:10.000 --> 00:43:17.330 If you look at them, you find that in order for this 00:43:17.330 --> 00:43:19.250 to satisfy Maxwell's equation, the 00:43:19.250 --> 00:43:23.260 must be associated with it a magnetic field that 00:43:23.260 --> 00:43:24.980 looks like that. 00:43:24.980 --> 00:43:27.280 And so the question, the first thing is, 00:43:27.280 --> 00:43:32.610 show that if you have this, you must also have this present. 00:43:32.610 --> 00:43:35.700 The second part is some more discussion 00:43:35.700 --> 00:43:39.790 about when you have these two present, when 00:43:39.790 --> 00:43:42.270 you have a standing wave in vacuum 00:43:42.270 --> 00:43:45.080 of electromagnetic waves, for example, 00:43:45.080 --> 00:43:51.770 then what is the energy density? 00:43:51.770 --> 00:43:55.030 You know, in an electric field or a magnetic field, 00:43:55.030 --> 00:43:57.190 if you have in space, if you take 00:43:57.190 --> 00:44:03.250 any value inside the volume, there will be energy. 00:44:03.250 --> 00:44:09.830 And the energy per unit volume per cubic meter 00:44:09.830 --> 00:44:11.270 is the energy density. 00:44:11.270 --> 00:44:14.180 So we're going to calculate how much energy density there 00:44:14.180 --> 00:44:16.980 is in this standing wave. 00:44:16.980 --> 00:44:21.700 And another quantity, which is for practical reasons 00:44:21.700 --> 00:44:26.150 very important, is when you have an electric and magnetic fields 00:44:26.150 --> 00:44:33.920 present, actually energy flows through that system. 00:44:33.920 --> 00:44:39.490 And the amount of energy per unit area 00:44:39.490 --> 00:44:43.320 that flows-- per unit area perpendicular 00:44:43.320 --> 00:44:46.690 to the direction of flow-- is called the Poynting vector. 00:44:46.690 --> 00:44:48.510 And by the way, the Poynting has nothing 00:44:48.510 --> 00:44:52.150 to do with a vector that points, it's to do with a gentleman 00:44:52.150 --> 00:44:54.430 by the name of Poynting, after which this was called. 00:44:57.390 --> 00:45:00.440 So the second part of the problem 00:45:00.440 --> 00:45:07.070 is, once we found a standing wave that satisfies everything 00:45:07.070 --> 00:45:11.150 possible [INAUDIBLE] in vacuum, for this particular case 00:45:11.150 --> 00:45:14.410 what is the energy density, the magnetic and electric fields, 00:45:14.410 --> 00:45:16.380 and what's the Poynting vector? 00:45:16.380 --> 00:45:17.895 OK, so how do we do this? 00:45:23.700 --> 00:45:26.260 We know what the electric field is doing, 00:45:26.260 --> 00:45:28.370 it's the standing wave. 00:45:28.370 --> 00:45:32.890 We know that it must satisfy all Maxwell's equations, 00:45:32.890 --> 00:45:36.320 in particular Faraday's law. 00:45:36.320 --> 00:45:45.330 As before, we can calculate the curl of the electric field. 00:45:45.330 --> 00:45:51.710 Now here, the electric field is only in the x direction. 00:45:51.710 --> 00:45:54.920 And it's a function of z. 00:45:54.920 --> 00:45:59.520 And so the curl of this, to be only just one component 00:45:59.520 --> 00:46:04.100 of that, and that is given by this quantity. 00:46:04.100 --> 00:46:08.600 So this is minus the curl of this E. 00:46:08.600 --> 00:46:11.510 And we know by Faraday's law that this 00:46:11.510 --> 00:46:15.200 must equal to the rate of change of the magnetic field 00:46:15.200 --> 00:46:20.110 at that place of x, y, and z. 00:46:20.110 --> 00:46:23.250 Now I can integrate this equation, 00:46:23.250 --> 00:46:28.250 and find what B is at every point in space and every time. 00:46:28.250 --> 00:46:29.760 And that's easy enough. 00:46:29.760 --> 00:46:34.390 We just have to integrate that, which gives you the sine here, 00:46:34.390 --> 00:46:38.280 and the omega comes down, and you get this. 00:46:38.280 --> 00:46:40.910 Whenever you integrate, there is a constant. 00:46:40.910 --> 00:46:46.160 All it's telling us is that I can satisfy Maxwell's equations 00:46:46.160 --> 00:46:50.390 not only with an oscillating electric field 00:46:50.390 --> 00:46:52.760 present with an oscillating magnetic field, 00:46:52.760 --> 00:46:55.950 but I can always add a constant magnetic field 00:46:55.950 --> 00:46:56.910 throughout space. 00:46:56.910 --> 00:46:59.440 I could have also added a constant electric field. 00:46:59.440 --> 00:47:03.200 So there's an infinite number of solutions I can superimpose. 00:47:03.200 --> 00:47:04.840 I'm not interested in them. 00:47:04.840 --> 00:47:11.260 I am interested in the standing wave, the time-dependent part. 00:47:11.260 --> 00:47:15.390 So might as well make that 0. 00:47:15.390 --> 00:47:18.050 And so we are essentially home. 00:47:18.050 --> 00:47:24.880 We have found that the magnetic field is also a standing wave. 00:47:24.880 --> 00:47:27.300 And this, by the way, we look at the problem, 00:47:27.300 --> 00:47:29.360 is what we were asked to prove. 00:47:29.360 --> 00:47:32.240 So we have proven the first part, 00:47:32.240 --> 00:47:38.050 that if this is the description of the standing 00:47:38.050 --> 00:47:42.910 wave of the electric field, then there must be corresponding 00:47:42.910 --> 00:47:46.570 a standing wave magnetic field. 00:47:46.570 --> 00:47:51.910 So the two-- but notice, unlike in the case 00:47:51.910 --> 00:47:57.280 of progressive waves, where in the progressive waves, wherever 00:47:57.280 --> 00:48:01.740 you had an electric field, the magnetic field 00:48:01.740 --> 00:48:05.200 was at right angle to it and in magnitude proportional 00:48:05.200 --> 00:48:08.950 to the electric field and in phase with it, et cetera. 00:48:08.950 --> 00:48:10.970 Here, they're not. 00:48:10.970 --> 00:48:13.970 Here, the electric field, when this is cosine omega t, 00:48:13.970 --> 00:48:15.510 this is sine omega t. 00:48:15.510 --> 00:48:18.590 When this is cosine kz, this is sine kz. 00:48:18.590 --> 00:48:22.700 These two are out of phase with each other, 00:48:22.700 --> 00:48:26.680 both in time and in space. 00:48:26.680 --> 00:48:29.280 I've tried to sketch it here, it's not very good 00:48:29.280 --> 00:48:30.780 sketch, but anyway. 00:48:30.780 --> 00:48:35.010 Suppose at some instant of time, if I look at these, 00:48:35.010 --> 00:48:39.720 at some instant of time, the electric vector-- 00:48:39.720 --> 00:48:44.510 the magnitude of it-- is represented by this curve. 00:48:44.510 --> 00:48:46.950 And it is in the x direction. 00:48:46.950 --> 00:48:53.110 So the electric vector is this, like this, and like that. 00:48:53.110 --> 00:48:58.300 If this is the maximum, it is, the magnetic field at that time 00:48:58.300 --> 00:49:00.880 will be 0, if I look at these equations. 00:49:00.880 --> 00:49:03.340 So there'll be no magnetic field. 00:49:03.340 --> 00:49:06.390 Over this distance in space, there 00:49:06.390 --> 00:49:09.990 will be the electric field up here, down here, 00:49:09.990 --> 00:49:12.160 and no magnetic field. 00:49:12.160 --> 00:49:16.260 Later on, half a period later, what 00:49:16.260 --> 00:49:24.415 you find is that when this comes to 0-- it's a quarter 00:49:24.415 --> 00:49:31.030 period-- when this comes to 0, the electric field is 0, 00:49:31.030 --> 00:49:34.110 there will be a magnetic field at its maximum. 00:49:34.110 --> 00:49:35.760 But it will not be this shape. 00:49:35.760 --> 00:49:39.110 It will be, first of all, pointing in the y direction. 00:49:39.110 --> 00:49:42.450 This is in the x, it will be the y direction. 00:49:42.450 --> 00:49:44.180 It's maximum will be in the middle, 00:49:44.180 --> 00:49:47.380 well here it was always 0. 00:49:47.380 --> 00:49:49.200 And these two oscillate. 00:49:49.200 --> 00:49:50.570 It's a standing wave. 00:49:50.570 --> 00:49:58.910 The B does this, and the E does this, all in the same place. 00:49:58.910 --> 00:50:03.140 But both in space and time, the two are out of phase 00:50:03.140 --> 00:50:04.140 with each other. 00:50:04.140 --> 00:50:06.430 Completely different solution. 00:50:06.430 --> 00:50:11.720 And both progressive waves satisfy Maxwell's equations, 00:50:11.720 --> 00:50:13.880 and the standard waves. 00:50:13.880 --> 00:50:18.880 So it's important to realize there is this difference, 00:50:18.880 --> 00:50:21.180 often it's easy to get confused about it. 00:50:21.180 --> 00:50:25.610 In a progressive wave, the electric and magnetic fields 00:50:25.610 --> 00:50:27.140 are right angle. 00:50:27.140 --> 00:50:29.180 And as if they were locked together, 00:50:29.180 --> 00:50:33.140 and they move forward like this. 00:50:33.140 --> 00:50:37.590 On the other hand, in a standing situation, 00:50:37.590 --> 00:50:40.030 they're still at right angle to the other. 00:50:40.030 --> 00:50:43.090 But when one is a maximum, the other's a minimum. 00:50:43.090 --> 00:50:47.160 When this one is-- They're out of phase 00:50:47.160 --> 00:50:52.220 with each other in both space and time. 00:50:52.220 --> 00:50:53.470 So that's the first part. 00:50:53.470 --> 00:50:57.950 And the next part we were asked, now for this standing wave, 00:50:57.950 --> 00:51:00.910 imagine this could be inside your microwave oven. 00:51:00.910 --> 00:51:05.450 Inside the microwave oven, there is a standing wave. 00:51:05.450 --> 00:51:09.360 Unless they specially make it so it moves a little bit in space 00:51:09.360 --> 00:51:11.340 so you cook your meat everywhere. 00:51:11.340 --> 00:51:13.680 But then the cheapo microwave oven, 00:51:13.680 --> 00:51:17.840 you have a stationary standing wave. 00:51:17.840 --> 00:51:19.740 And suppose this is it. 00:51:19.740 --> 00:51:24.320 At every place in space, there is an energy density 00:51:24.320 --> 00:51:27.280 which actually fluctuates, goes up and down in time 00:51:27.280 --> 00:51:29.640 and is different in every location. 00:51:29.640 --> 00:51:31.340 Let's calculate that. 00:51:31.340 --> 00:51:35.550 Well, as Professor Walter Lewin showed, 00:51:35.550 --> 00:51:38.260 the energy density in an electric field, 00:51:38.260 --> 00:51:40.640 whether it's changing with time or not, 00:51:40.640 --> 00:51:45.990 if I've got in space somewhere an electric field e, 00:51:45.990 --> 00:51:49.840 at that location, I have an energy density. 00:51:49.840 --> 00:51:53.030 The amount is 1 over epsilon-zero times 00:51:53.030 --> 00:51:55.470 the magnitude of the electric field squared. 00:51:55.470 --> 00:51:59.200 That is the energy density of an electric field. 00:51:59.200 --> 00:52:01.510 It is not a vector. 00:52:01.510 --> 00:52:04.780 This is E-squared, the square of the magnitude 00:52:04.780 --> 00:52:07.890 of the electric field energy is a scalar quantity. 00:52:07.890 --> 00:52:12.340 So not surprising, this is not a vector, it's a scalar quantity. 00:52:12.340 --> 00:52:17.570 I can now immediately go over to what we know. 00:52:17.570 --> 00:52:20.820 We know the electric field, we know the magnetic field. 00:52:20.820 --> 00:52:27.880 So I can replace E-squared by what it is at every location. 00:52:27.880 --> 00:52:30.820 At every position z and every x, y. 00:52:30.820 --> 00:52:32.070 At all times. 00:52:32.070 --> 00:52:35.440 And this is the energy density. 00:52:35.440 --> 00:52:42.590 You can see it does oscillate, but there's always [INAUDIBLE]. 00:52:42.590 --> 00:52:45.070 How about the magnetic field? 00:52:45.070 --> 00:52:50.300 The magnetic field also has energy. 00:52:50.300 --> 00:52:53.700 If I take it anywhere, suppose you have a bar magnet, 00:52:53.700 --> 00:52:56.110 one of these pocket magnets, you hold it, 00:52:56.110 --> 00:52:59.450 and there's a magnetic field all around the magnet. 00:52:59.450 --> 00:53:04.490 Take any cubic meter of the volume, 00:53:04.490 --> 00:53:06.860 you'll find this amount of energy. 00:53:06.860 --> 00:53:10.590 It's 1 over 2 Mu 0 times the magnitude of the magnetic field 00:53:10.590 --> 00:53:11.410 squared. 00:53:11.410 --> 00:53:16.550 Again, I know what B is in for my standing, wave 00:53:16.550 --> 00:53:20.490 so I can calculate it, and I get this answer. 00:53:20.490 --> 00:53:25.800 So these are the two energy densities. 00:53:25.800 --> 00:53:33.160 Now what one finds, if one does-- if you plot this, 00:53:33.160 --> 00:53:38.740 or thinks about it-- that in this standing wave, 00:53:38.740 --> 00:53:45.023 you find that that energy moves backwards and forwards. 00:53:51.180 --> 00:53:57.110 At any location in space, I can calculate 00:53:57.110 --> 00:54:02.550 how much energy is moving per second 00:54:02.550 --> 00:54:07.390 per square meter-- per unit area-- perpendicular 00:54:07.390 --> 00:54:11.360 to the direction of motion of that energy. 00:54:11.360 --> 00:54:14.895 And that is what is called the Poynting vector. 00:54:17.620 --> 00:54:19.180 If you think, for example, suppose 00:54:19.180 --> 00:54:22.680 you take an electromagnetic wave like light shining 00:54:22.680 --> 00:54:24.890 that the wall. 00:54:24.890 --> 00:54:26.640 It'll warm up to the wall, I mean 00:54:26.640 --> 00:54:30.860 there's heat being transmitted, there's energy comes over. 00:54:30.860 --> 00:54:35.180 At any instant of time, how much energy per unit area 00:54:35.180 --> 00:54:36.780 is hitting the wall? 00:54:36.780 --> 00:54:42.180 It will be equal to the Poynting vector at that instant of time. 00:54:42.180 --> 00:54:46.950 And the Poynting vector s is E-cross-B over Mu 0. 00:54:52.320 --> 00:54:58.430 By the way, this applies to any electric and magnetic fields, 00:54:58.430 --> 00:55:01.840 not necessarily for progressive waves or standing 00:55:01.840 --> 00:55:03.390 waves, et cetera. 00:55:03.390 --> 00:55:06.360 It's something we want to think about 00:55:06.360 --> 00:55:09.950 and this is very surprising. 00:55:09.950 --> 00:55:13.390 Even if you have static electric and magnetic fields which 00:55:13.390 --> 00:55:17.070 are not parallel to each other, so that this is not 0, 00:55:17.070 --> 00:55:19.050 there is a flow of energy. 00:55:19.050 --> 00:55:21.140 It's something we want to think about. 00:55:21.140 --> 00:55:27.460 But in our case, E and B are perpendicular to each other. 00:55:32.260 --> 00:55:35.910 The electric field everywhere was 00:55:35.910 --> 00:55:39.930 in the x direction, the magnetic in the y direction. 00:55:39.930 --> 00:55:46.010 And so they are right angles, so it's just the x component of E 00:55:46.010 --> 00:55:48.200 and the y component of B. Well, they're 00:55:48.200 --> 00:55:49.990 the only components that are there. 00:55:49.990 --> 00:55:51.640 So it's 1 over Mu 0. 00:55:51.640 --> 00:55:55.260 E x times B y in the z direction, 00:55:55.260 --> 00:55:58.220 so this if E and B are perpendicular to each other, 00:55:58.220 --> 00:56:00.360 z is perpendicular to both of those, 00:56:00.360 --> 00:56:03.120 which is in the z direction. 00:56:03.120 --> 00:56:08.910 If I calculate this for this, I get this equation. 00:56:08.910 --> 00:56:11.550 And I can rewrite it. 00:56:11.550 --> 00:56:15.510 And I find that the energy is some constant, 00:56:15.510 --> 00:56:20.220 goes in the z direction, and this looks like sine 2 omega t 00:56:20.220 --> 00:56:22.800 times sine 2 kz. 00:56:22.800 --> 00:56:27.030 Going back to our diagram, what this 00:56:27.030 --> 00:56:34.010 looks like is that-- if you remember that E oscillates, 00:56:34.010 --> 00:56:38.740 it's a maximum here, maximum here, 00:56:38.740 --> 00:56:43.890 and it oscillates up and down, up and down, like this. 00:56:43.890 --> 00:56:50.360 B is a maximum in the middle, and that's going like this. 00:56:50.360 --> 00:56:54.970 The product of the two, it'll be 0 here, 00:56:54.970 --> 00:56:57.240 because B is always 0 here. 00:56:57.240 --> 00:57:00.530 It'll be here because E is always 0. 00:57:00.530 --> 00:57:02.720 And you cross B there for 0. 00:57:02.720 --> 00:57:05.610 So here, here, and here is going to be 0. 00:57:05.610 --> 00:57:07.180 And if you look at that function, 00:57:07.180 --> 00:57:11.260 its actually a function which has twice the frequency 00:57:11.260 --> 00:57:13.410 of the electric field oscillations 00:57:13.410 --> 00:57:15.390 or the magnetic field oscillations, 00:57:15.390 --> 00:57:21.200 and also half the wavelength. 00:57:21.200 --> 00:57:25.015 And you will find that the maximum is somewhere here. 00:57:30.800 --> 00:57:41.030 So if you look at where the maximum transfer of energy is, 00:57:41.030 --> 00:57:45.380 it's at the quarter and 3/4 location. 00:57:45.380 --> 00:57:48.110 And so it's consistent with this picture. 00:57:48.110 --> 00:57:52.990 Energy is doing this in that situation. 00:57:52.990 --> 00:57:56.490 And so that answers what they were ask. 00:57:56.490 --> 00:58:00.300 This is the Poynting vector as a function 00:58:00.300 --> 00:58:06.460 of-- for all positions in space as a function of time. 00:58:06.460 --> 00:58:09.820 This is the energy, the electric and magnetic field, 00:58:09.820 --> 00:58:14.890 and we found the magnetic field corresponding 00:58:14.890 --> 00:58:17.050 to the electric field. 00:58:17.050 --> 00:58:21.840 So this is another example of a possible solution 00:58:21.840 --> 00:58:25.700 to Maxwell's equations, this time corresponding 00:58:25.700 --> 00:58:28.410 to standing waves. 00:58:28.410 --> 00:58:32.280 As I mentioned before, I'm repeating myself , 00:58:32.280 --> 00:58:36.200 there are infinite possibilities of solutions 00:58:36.200 --> 00:58:40.090 of Maxwell's equations. 00:58:40.090 --> 00:58:43.120 So to cover them all makes no sense. 00:58:43.120 --> 00:58:48.050 What is important, that one gets a good understanding 00:58:48.050 --> 00:58:50.410 of the interesting situations. 00:58:50.410 --> 00:58:57.390 Interesting situations are some static solutions to, say, 00:58:57.390 --> 00:59:00.430 magnetic fields if you need special magnets. 00:59:00.430 --> 00:59:03.820 Or if you have a progressive wave, 00:59:03.820 --> 00:59:08.510 like light, or standing waves, like in the microwave, 00:59:08.510 --> 00:59:09.760 for example. 00:59:09.760 --> 00:59:14.280 And so I've taken two cases here. 00:59:14.280 --> 00:59:16.780 First progressive wave solution. 00:59:16.780 --> 00:59:19.970 And then standing wave solution. 00:59:19.970 --> 00:59:23.080 And from this, we will later go on to some other problems. 00:59:23.080 --> 00:59:24.630 Thank you.