WEBVTT 00:00:02.550 --> 00:00:04.920 The following content is provided under a Creative 00:00:04.920 --> 00:00:06.310 Commons license. 00:00:06.310 --> 00:00:08.520 Your support will help MIT OpenCourseWare 00:00:08.520 --> 00:00:12.610 continue to offer high quality educational resources for free. 00:00:12.610 --> 00:00:15.150 To make a donation or to view additional materials 00:00:15.150 --> 00:00:19.110 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.110 --> 00:00:20.154 at ocw.mit.edu. 00:00:23.880 --> 00:00:27.720 YEN-JIE LEE: Welcome back, everybody to 8.03. 00:00:27.720 --> 00:00:31.650 So today we are going to continue discussions 00:00:31.650 --> 00:00:40.170 on the examples which we started the last time, sound waves, 00:00:40.170 --> 00:00:42.540 and, this time EM waves, which can 00:00:42.540 --> 00:00:45.000 be described by wave equations. 00:00:47.550 --> 00:00:50.190 So far, what we have learned is that there 00:00:50.190 --> 00:00:52.870 are three different kinds of systems we've discussed 00:00:52.870 --> 00:00:55.670 in a lecture or in a textbook. 00:00:55.670 --> 00:00:59.140 And the first one is actually a string, a very long string, 00:00:59.140 --> 00:01:05.550 system with constant tension and mass on the string. 00:01:05.550 --> 00:01:08.640 And the behavior of the string obey wave 00:01:08.640 --> 00:01:11.775 equation, and can be described by a wave equation. 00:01:14.400 --> 00:01:19.680 We also can produce a density wave with a spring. 00:01:19.680 --> 00:01:22.620 And basically the density wave or spring 00:01:22.620 --> 00:01:26.670 can also be described by wave equations. 00:01:26.670 --> 00:01:29.640 So that's as you described in the textbook. 00:01:29.640 --> 00:01:35.400 Finally, last time we actually discussed sound waves. 00:01:35.400 --> 00:01:37.980 We have an open pipe, and then we 00:01:37.980 --> 00:01:39.450 can have air inside the pipe. 00:01:39.450 --> 00:01:43.830 And the behavior of the air, or the molecules 00:01:43.830 --> 00:01:48.310 inside the open pipe, can be described by wave equations. 00:01:48.310 --> 00:01:52.410 Crashes So what we are going to do today 00:01:52.410 --> 00:01:55.200 is to discuss with you a special kind of wave, 00:01:55.200 --> 00:01:57.400 which is electromagnetic waves. 00:01:57.400 --> 00:01:59.400 And that's actually slightly different from what 00:01:59.400 --> 00:02:02.580 we have learned in the last few lectures. 00:02:02.580 --> 00:02:08.419 And we see what this is different today in the lecture. 00:02:08.419 --> 00:02:09.440 All right. 00:02:09.440 --> 00:02:11.600 So this essentially is a reminder 00:02:11.600 --> 00:02:14.030 of Maxwell's equations. 00:02:14.030 --> 00:02:15.605 So basically what is written here 00:02:15.605 --> 00:02:22.080 is the differential form of Maxwell's equations. 00:02:22.080 --> 00:02:24.670 So the first law is Gauss law. 00:02:24.670 --> 00:02:29.570 It says should the divergence of e, the electric field, 00:02:29.570 --> 00:02:33.200 is equal to rho, which is the charge 00:02:33.200 --> 00:02:38.300 density as a specific point, divided by epsilon zero, 00:02:38.300 --> 00:02:41.240 which is actually a constant. 00:02:41.240 --> 00:02:45.960 We'll call it permittivity of this constant. 00:02:45.960 --> 00:02:46.460 OK? 00:02:46.460 --> 00:02:49.970 Which should relay the divergence of e 00:02:49.970 --> 00:02:53.910 and the density of the charge at this specific point. 00:02:53.910 --> 00:02:59.310 And the second law is actually Gauss law for magnetism. 00:02:59.310 --> 00:03:03.530 This is actually the divergence of b equal to 0. 00:03:03.530 --> 00:03:08.420 So divergence b is always equal to zero because we haven't yet 00:03:08.420 --> 00:03:13.060 discovered the magnetic monopole yet. 00:03:13.060 --> 00:03:13.610 Right? 00:03:13.610 --> 00:03:19.010 So maybe you have discovered it one time, at some time, 00:03:19.010 --> 00:03:20.150 in your experiment. 00:03:20.150 --> 00:03:22.847 Please tell me now. 00:03:22.847 --> 00:03:25.055 I want to be the first with who knows how to do that. 00:03:25.055 --> 00:03:26.960 [LAUGHS] All right? 00:03:26.960 --> 00:03:29.510 So promise me. 00:03:29.510 --> 00:03:33.350 The third one is Faraday's law. 00:03:33.350 --> 00:03:41.590 It's curve of e equal to minus partial e partial t and the b, 00:03:41.590 --> 00:03:45.970 as a reminder, is a magnetic field vector. 00:03:45.970 --> 00:03:49.820 And in the last law is actually Ampere's law. 00:03:49.820 --> 00:03:54.330 It's actually the curve of b equal to mu 0. 00:03:54.330 --> 00:03:58.580 Mu 0 is actually a constant, permeability. 00:03:58.580 --> 00:04:02.710 Which would lay the current and displacement current. 00:04:02.710 --> 00:04:07.640 Epsilon 0, partial e, partial t, to the curve of b. 00:04:07.640 --> 00:04:08.750 OK? 00:04:08.750 --> 00:04:13.550 And I would like to draw your attention to these term. 00:04:13.550 --> 00:04:18.100 This very important term is actually Maxwell's addition. 00:04:18.100 --> 00:04:18.620 OK? 00:04:18.620 --> 00:04:21.290 Without Maxwell's addition, there 00:04:21.290 --> 00:04:23.800 would be no electromagnetic wave. 00:04:23.800 --> 00:04:25.280 Then you could not see me. 00:04:25.280 --> 00:04:26.225 OK? 00:04:26.225 --> 00:04:29.250 [LAUGHS] All right. 00:04:29.250 --> 00:04:32.460 So, what we are going to discuss today 00:04:32.460 --> 00:04:36.520 is a simpler case at the beginning. 00:04:36.520 --> 00:04:41.250 So what will happen if we go to a vacuum? 00:04:41.250 --> 00:04:47.040 Going to vacuum means there will be no material charges floating 00:04:47.040 --> 00:04:51.470 around, and that means rho will be equal to 0. 00:04:51.470 --> 00:04:56.700 Therefore, the divergence of e will be equal to 0. 00:04:56.700 --> 00:05:02.850 And also in the last question Ampere's law, 00:05:02.850 --> 00:05:07.170 say which is that the current density will be equal to 0. 00:05:07.170 --> 00:05:11.370 Therefore, the function of curl of b 00:05:11.370 --> 00:05:15.870 equal to mu 0, epsilon 0, partial e, partial t. 00:05:15.870 --> 00:05:17.430 OK? 00:05:17.430 --> 00:05:21.210 So before we go into the discussion 00:05:21.210 --> 00:05:24.930 of Maxwell's equation's implication, 00:05:24.930 --> 00:05:30.630 I would like to remind you about some mathematics which 00:05:30.630 --> 00:05:34.270 will be used in this lecture. 00:05:34.270 --> 00:05:41.220 I hope you have seen this in other courses or 8.02. 00:05:41.220 --> 00:05:47.640 So as you can see, we use del here, which is a vector. 00:05:47.640 --> 00:05:51.850 This vector is defined as partial x, partial y, 00:05:51.850 --> 00:05:55.650 and partial z, in the x, y, and z direction. 00:05:55.650 --> 00:05:56.160 OK? 00:05:56.160 --> 00:06:00.540 So this is actually some kind of operator. 00:06:00.540 --> 00:06:02.470 You see that again. 00:06:02.470 --> 00:06:06.420 A lot more operators in 8.04. 00:06:06.420 --> 00:06:13.230 And we make this definition because I'm lazy. 00:06:13.230 --> 00:06:16.070 Because I don't want to write so many partial, partial x, 00:06:16.070 --> 00:06:20.170 partial, partial y, partial, partial z again and again. 00:06:20.170 --> 00:06:24.480 Therefore we define del, which is like this. 00:06:24.480 --> 00:06:26.950 Looks really crazy, but it really 00:06:26.950 --> 00:06:29.080 makes our lives much easier. 00:06:29.080 --> 00:06:29.670 OK? 00:06:29.670 --> 00:06:31.020 So that's the whole reason. 00:06:31.020 --> 00:06:33.420 As a physicist. 00:06:33.420 --> 00:06:39.090 And as we discussed before, we have divergence, 00:06:39.090 --> 00:06:44.820 which is defined here: del times a, both of them are vectors. 00:06:44.820 --> 00:06:46.900 Y is the operator vector, the other y 00:06:46.900 --> 00:06:50.790 is actually really a vector. 00:06:50.790 --> 00:06:54.920 You basically get partial ax, partial x, plus partial ay, 00:06:54.920 --> 00:06:57.870 partial y, plus partial az, partial z. 00:06:57.870 --> 00:07:02.740 So basically you just multiply them like a normal operation. 00:07:02.740 --> 00:07:06.540 And you can actually get this question. 00:07:06.540 --> 00:07:07.360 OK. 00:07:07.360 --> 00:07:10.080 Then finally there's curl. 00:07:10.080 --> 00:07:17.280 Curl is actually del cross a. 00:07:17.280 --> 00:07:20.040 So basically, maybe in the past you 00:07:20.040 --> 00:07:22.530 see this complicated formula. 00:07:22.530 --> 00:07:25.530 You know, it had maybe no meaning to you. 00:07:25.530 --> 00:07:28.590 And one easy way to remember this curl 00:07:28.590 --> 00:07:31.500 is to not care about this. 00:07:31.500 --> 00:07:34.350 Don't look at the right hand side part. 00:07:34.350 --> 00:07:38.730 But just remember that you can actually construct this curl 00:07:38.730 --> 00:07:41.740 by determining a matrix. 00:07:41.740 --> 00:07:45.090 In the matrix, I can fill the first row 00:07:45.090 --> 00:07:47.580 by x, y, and z unit vector. 00:07:47.580 --> 00:07:52.390 And the second row, I filled it with the counting of del. 00:07:52.390 --> 00:07:58.260 Finally I feel the content of the matrix with a vector. 00:07:58.260 --> 00:08:00.570 Then you will be able to calculate 00:08:00.570 --> 00:08:02.610 the determine of this back matrix, 00:08:02.610 --> 00:08:07.260 then you naturally would get this very long formula. 00:08:07.260 --> 00:08:09.780 So you don't really need to remember the formula, 00:08:09.780 --> 00:08:14.840 but you will be able to know how to calculate it really easily. 00:08:14.840 --> 00:08:17.260 OK? 00:08:17.260 --> 00:08:18.120 OK. 00:08:18.120 --> 00:08:20.910 So we talk about divergence. 00:08:20.910 --> 00:08:24.560 We talk about curl. 00:08:24.560 --> 00:08:27.680 What does that mean? 00:08:27.680 --> 00:08:30.710 Divergence, curl, what does that mean? 00:08:30.710 --> 00:08:35.330 So divergence is actually some kind of measure 00:08:35.330 --> 00:08:38.690 which measures how much the vector 00:08:38.690 --> 00:08:46.430 v spreads out, or diverges, from a point of interest. 00:08:46.430 --> 00:08:52.580 So in this example, this vector field-- 00:08:52.580 --> 00:08:59.960 vector field means at any point in the space which 00:08:59.960 --> 00:09:04.340 I am discussing, there is a vector associated with that. 00:09:04.340 --> 00:09:06.110 I call it vector field. 00:09:06.110 --> 00:09:08.360 We know scalar field very much. 00:09:08.360 --> 00:09:12.740 For example, the temperature as a function of position 00:09:12.740 --> 00:09:14.000 is a scalar field, right? 00:09:14.000 --> 00:09:20.390 So, every point you have scalar corresponding to that point. 00:09:20.390 --> 00:09:23.270 And in the case vector field, every point 00:09:23.270 --> 00:09:26.900 you have a vector connected to that point. 00:09:26.900 --> 00:09:31.160 And if I arrange the vector field like that, 00:09:31.160 --> 00:09:34.360 each arrow is actually a straight dimensional 00:09:34.360 --> 00:09:36.200 the vector. 00:09:36.200 --> 00:09:39.320 Then if I evaluate the divergence 00:09:39.320 --> 00:09:42.770 and you see the heart, it looks like something is really 00:09:42.770 --> 00:09:46.310 spreading out from the center of that graph. 00:09:46.310 --> 00:09:50.660 And that will give you positive divergence. 00:09:50.660 --> 00:09:51.320 OK? 00:09:51.320 --> 00:09:55.520 So that's the physical meaning of this formula. 00:09:55.520 --> 00:10:00.060 And the second formula which we discussed is the curl. 00:10:00.060 --> 00:10:04.470 So curl is actually del cross a. 00:10:04.470 --> 00:10:08.540 It's a measurement of how things are curling 00:10:08.540 --> 00:10:10.760 around a point of interest. 00:10:10.760 --> 00:10:11.720 OK? 00:10:11.720 --> 00:10:18.170 So you can see that if I arrange my vectors 00:10:18.170 --> 00:10:20.890 in a space like that, then you will 00:10:20.890 --> 00:10:26.480 see that something is really rotating 00:10:26.480 --> 00:10:28.310 around that specific point. 00:10:28.310 --> 00:10:30.770 Therefore, if we evaluate the curl, 00:10:30.770 --> 00:10:33.290 you would get the nonzero value. 00:10:33.290 --> 00:10:37.992 So that's actually the physics intuition which we can 00:10:37.992 --> 00:10:41.300 or the mathematics intuition which 00:10:41.300 --> 00:10:47.630 we can actually get before the discussion of Maxwell's 00:10:47.630 --> 00:10:49.740 equations. 00:10:49.740 --> 00:10:53.540 So if you accept those ideas, let's 00:10:53.540 --> 00:10:57.410 take a look at what we have here, especially 00:10:57.410 --> 00:10:59.210 in the vacuum case. 00:10:59.210 --> 00:11:06.530 OK, so in a vacuum case, you have a curl of e equal to minus 00:11:06.530 --> 00:11:09.050 partial p, partial t. 00:11:09.050 --> 00:11:10.560 What does that mean? 00:11:10.560 --> 00:11:13.640 That means, if you change the magnitude 00:11:13.640 --> 00:11:27.730 of the magnetic field, now we introduce a curling 00:11:27.730 --> 00:11:30.980 around thing in the e field. 00:11:30.980 --> 00:11:33.830 So if you change the size of the b field, 00:11:33.830 --> 00:11:39.200 then the e field will start to curl around, doing this. 00:11:39.200 --> 00:11:44.400 And on the other hand, if you change the electric field, 00:11:44.400 --> 00:11:47.410 that will do something, which is curling around in the b field. 00:11:50.100 --> 00:11:53.220 All right, so do you have any questions? 00:11:53.220 --> 00:11:56.450 I hope everybody's familiar with this notation. 00:11:56.450 --> 00:12:05.099 So from here actually Maxwell, see the light. 00:12:05.099 --> 00:12:08.220 [LAUGHS] Can you see it? 00:12:10.770 --> 00:12:11.540 Maybe not yet. 00:12:11.540 --> 00:12:17.800 Maybe we are slightly slower than Maxwell, 00:12:17.800 --> 00:12:20.710 but we will see that together in this lecture. 00:12:23.510 --> 00:12:33.970 For that I would need as usual help from the math department. 00:12:33.970 --> 00:12:37.510 So we are going to use this identity. 00:12:37.510 --> 00:12:46.595 This identity is curl of curl of a would 00:12:46.595 --> 00:13:02.990 be equal to del, divergence of a, minus del dot del, a. 00:13:02.990 --> 00:13:05.120 So this is an identity which we learned 00:13:05.120 --> 00:13:07.100 from the math department. 00:13:07.100 --> 00:13:11.000 And of course, if you are patient enough, 00:13:11.000 --> 00:13:13.160 you can actually expand all those terms 00:13:13.160 --> 00:13:16.640 and compare the left hand side of the formula and right hand 00:13:16.640 --> 00:13:18.020 side of the formula. 00:13:18.020 --> 00:13:22.880 And you will see that really this works. 00:13:22.880 --> 00:13:26.790 So I'm not going to do that here in front of you. 00:13:26.790 --> 00:13:31.760 So if you accept this is an identity, and then usually when 00:13:31.760 --> 00:13:35.180 we have del times del, we call it Laplace. 00:13:38.290 --> 00:13:41.170 And usually we write it as del squared. 00:13:44.310 --> 00:13:49.581 With this formula, I can now put my electric field 00:13:49.581 --> 00:13:50.330 into this formula. 00:13:55.330 --> 00:14:00.570 assuming that I am working in a situation of a vacuum 00:14:00.570 --> 00:14:06.900 and I plug in my electric field into that formula. 00:14:06.900 --> 00:14:11.470 Then this is actually what I am going to get. 00:14:11.470 --> 00:14:13.780 Curl of e. 00:14:13.780 --> 00:14:20.290 And this will be equal to del, divergence of e 00:14:20.290 --> 00:14:23.110 minus del squared e. 00:14:25.670 --> 00:14:26.170 OK? 00:14:28.920 --> 00:14:35.760 And based on the four Maxwell's equations, 00:14:35.760 --> 00:14:39.510 we can immediately recognize that divergence of e 00:14:39.510 --> 00:14:43.110 is equal to 0, because I don't have charges around. 00:14:43.110 --> 00:14:46.950 Therefore you, cannot not introduce a gradient 00:14:46.950 --> 00:14:50.670 or divergence. 00:14:50.670 --> 00:14:54.290 You can introduce positive divergence 00:14:54.290 --> 00:14:56.560 in the electric field. 00:14:56.560 --> 00:14:59.400 Therefore, when you evaluate the divergence 00:14:59.400 --> 00:15:03.240 of the electric field, that is equal to 0. 00:15:03.240 --> 00:15:06.390 According to that formula, Gauss law. 00:15:06.390 --> 00:15:11.010 And you can also take a look here. 00:15:11.010 --> 00:15:15.360 We have curl of e, according to this formula. 00:15:15.360 --> 00:15:18.260 Basically you can conclude that this 00:15:18.260 --> 00:15:23.820 will be equal to minus partial b, partial t according 00:15:23.820 --> 00:15:25.140 to Faraday's law. 00:15:27.950 --> 00:15:34.030 So if I look at the left hand side, 00:15:34.030 --> 00:15:45.150 that would be equal to the curl of minus partial b, partial t. 00:15:45.150 --> 00:15:49.160 And this will be equal to basically, 00:15:49.160 --> 00:15:54.870 I can take the minus sign out and take the partial partial t 00:15:54.870 --> 00:15:56.410 out. 00:15:56.410 --> 00:16:01.960 And basically you have curl of b. 00:16:01.960 --> 00:16:06.010 And according to Ampere's law, this 00:16:06.010 --> 00:16:15.618 would be equal to minus mu 0, epsilon 0, partial square e, 00:16:15.618 --> 00:16:17.610 partial t. 00:16:17.610 --> 00:16:20.980 OK, everybody is following? 00:16:20.980 --> 00:16:22.570 So basically what I have been doing 00:16:22.570 --> 00:16:28.030 is copy the left-hand side and make use of the Ampere's law. 00:16:28.030 --> 00:16:33.330 And basically you get minus mu, epsilon zero, partial square e, 00:16:33.330 --> 00:16:36.940 partial t squared 00:16:36.940 --> 00:16:39.520 And this thing, the left hand side, 00:16:39.520 --> 00:16:42.600 is equal to the right hand side. 00:16:42.600 --> 00:16:44.700 On the right hand side, what is left? 00:16:44.700 --> 00:16:45.834 This is equal to 0. 00:16:45.834 --> 00:16:46.500 So this is gone. 00:16:52.580 --> 00:17:01.059 This is equal to minus del squared, e. 00:17:01.059 --> 00:17:03.580 I can cancel the minus sign. 00:17:03.580 --> 00:17:10.550 Then basically what I am going to get is del squared e. 00:17:10.550 --> 00:17:15.880 And this will be equal to mu zero epsilon 0, partial square 00:17:15.880 --> 00:17:17.965 e, partial t squared. 00:17:22.380 --> 00:17:24.060 Wow, this is what? 00:17:24.060 --> 00:17:28.010 This looks like, what? 00:17:28.010 --> 00:17:29.900 Wave equation again. 00:17:29.900 --> 00:17:31.137 Oh my god. 00:17:31.137 --> 00:17:32.450 [LAUGHTER] 00:17:32.450 --> 00:17:35.610 But there is some difference. 00:17:35.610 --> 00:17:38.700 This is different from what we've seen before, right? 00:17:38.700 --> 00:17:45.585 Before, the wave equation only has partial squared partial x 00:17:45.585 --> 00:17:46.110 squared. 00:17:46.110 --> 00:17:49.040 This time, you have this del square. 00:17:49.040 --> 00:17:50.790 Very strange, right? 00:17:50.790 --> 00:17:52.260 So what is this? 00:17:52.260 --> 00:17:55.970 Del square is actually partial square, 00:17:55.970 --> 00:18:01.340 partial x squared cross partial square partial y square, 00:18:01.340 --> 00:18:05.820 plus partial square, partial z square is the operator, which 00:18:05.820 --> 00:18:08.790 will have three components. 00:18:08.790 --> 00:18:12.850 And basically, if you do this calculation, 00:18:12.850 --> 00:18:18.180 you are going to have how many times? 00:18:18.180 --> 00:18:22.320 If you do this del square e, how many times you have? 00:18:22.320 --> 00:18:23.210 You have how many? 00:18:23.210 --> 00:18:24.810 Any anybody help me? 00:18:27.440 --> 00:18:30.100 Yeah, you have three times in x direction, 00:18:30.100 --> 00:18:31.950 you have three times in y direction, 00:18:31.950 --> 00:18:34.270 you have three times in z direction. 00:18:34.270 --> 00:18:35.410 Therefore how many times? 00:18:35.410 --> 00:18:39.960 You have nine times, because each operator 00:18:39.960 --> 00:18:42.851 is acting out the vector. 00:18:42.851 --> 00:18:43.350 OK. 00:18:43.350 --> 00:18:46.620 So it's very important because this is a common mistake. 00:18:46.620 --> 00:18:48.780 So you have nine times. 00:18:48.780 --> 00:18:53.100 and it looks really like the wave equations 00:18:53.100 --> 00:18:55.350 it tastes like wave equation, it looks 00:18:55.350 --> 00:18:59.790 like equation, it feels like equation - that wave equation - 00:18:59.790 --> 00:19:02.827 and therefore is really the wave equation, right? 00:19:02.827 --> 00:19:04.170 [LAUGHTER] 00:19:04.170 --> 00:19:09.600 OK so this is a three-dimensional wave 00:19:09.600 --> 00:19:12.140 equation. 00:19:12.140 --> 00:19:12.810 Very cool. 00:19:12.810 --> 00:19:14.700 So we are increasing the dimension. 00:19:17.370 --> 00:19:22.020 So I can write it down more explicitly. 00:19:22.020 --> 00:19:26.155 So basically what I'm getting is partial square 00:19:26.155 --> 00:19:33.150 e partial x squared, partial square e, partial y square, 00:19:33.150 --> 00:19:37.830 plus partial square e, partial z square. 00:19:37.830 --> 00:19:45.470 And this is equal to mu zero, epsilon zero, partial square e, 00:19:45.470 --> 00:19:46.840 partial t squared. 00:19:46.840 --> 00:19:49.020 OK? 00:19:49.020 --> 00:19:56.600 So Maxwell sees this when he adds this additional term here. 00:19:56.600 --> 00:20:00.990 As you can see, if I don't have this additional term, 00:20:00.990 --> 00:20:05.630 the displacement of current from Maxwell, 00:20:05.630 --> 00:20:08.790 what is going to happen? 00:20:08.790 --> 00:20:14.030 This curve of b will be equal to zero. 00:20:14.030 --> 00:20:18.470 So what is going to happen to this identity? 00:20:18.470 --> 00:20:22.190 This left hand side part will be equal to zero. 00:20:22.190 --> 00:20:26.330 There will be no electromagnetic waves. 00:20:26.330 --> 00:20:28.290 OK? 00:20:28.290 --> 00:20:34.590 So that's really thanks to Maxwell's work. 00:20:34.590 --> 00:20:37.050 And this is actually really an equation 00:20:37.050 --> 00:20:43.020 which changed the world, because that actually gave us 00:20:43.020 --> 00:20:46.920 a lot of insights about how we can send energy, 00:20:46.920 --> 00:20:50.010 how we can actually understand the phenomenon related 00:20:50.010 --> 00:20:53.490 to light. 00:20:53.490 --> 00:20:57.600 So what is the velocity of this wave equation? 00:20:57.600 --> 00:21:03.600 The velocity, Vp, would be equal to what we usually call c. 00:21:03.600 --> 00:21:07.440 Because you have been using this constant for a long time. 00:21:07.440 --> 00:21:13.140 And that will be equal to 1 over square root of mu zero epsilon 00:21:13.140 --> 00:21:15.660 zero. 00:21:15.660 --> 00:21:19.260 And to measure the speed of light, 00:21:19.260 --> 00:21:23.870 it takes a long time to achieve that. 00:21:23.870 --> 00:21:26.930 Let's take a look at the history. 00:21:26.930 --> 00:21:36.360 So the first attempt was done by Galileo so 1638. 00:21:36.360 --> 00:21:39.600 He was doing an experiment, and that he was trying 00:21:39.600 --> 00:21:41.670 to track the speed of light. 00:21:41.670 --> 00:21:44.580 But he was not super successful. 00:21:44.580 --> 00:21:51.390 So his conclusion was that if the speed of light 00:21:51.390 --> 00:21:56.250 is not instantaneous, then it is super fast. 00:21:56.250 --> 00:22:02.190 He says it's at least 10 times faster than the speed of sound. 00:22:02.190 --> 00:22:06.180 He said OK, this is super awesome, very fast. 00:22:13.620 --> 00:22:15.180 OK. 00:22:15.180 --> 00:22:19.170 So that's what he found. 00:22:19.170 --> 00:22:33.660 And later Romer actually made use of the orbit of Jupiter. 00:22:33.660 --> 00:22:37.890 Basically he use Jupiter and Jupiter's satellite 00:22:37.890 --> 00:22:40.450 to measure the speed of light. 00:22:40.450 --> 00:22:43.830 So when the earth is closer to Jupiter, 00:22:43.830 --> 00:22:47.010 then somehow the satellite of Jupiter 00:22:47.010 --> 00:22:52.230 appears faster than when the earth is actually away 00:22:52.230 --> 00:22:53.700 from Jupiter. 00:22:53.700 --> 00:22:58.140 Because that light have to travel through additional time. 00:22:58.140 --> 00:23:02.140 Two times the radius of the orbit the Earth. 00:23:02.140 --> 00:23:05.230 Basically that's the math that he was using. 00:23:05.230 --> 00:23:12.460 He is actually making the first computative measurement 00:23:12.460 --> 00:23:13.860 of the speed of light. 00:23:13.860 --> 00:23:17.640 And what number he found is 2 times 00:23:17.640 --> 00:23:19.950 10 to the 9 meters per second. 00:23:22.520 --> 00:23:28.740 Then finally, again using the star 00:23:28.740 --> 00:23:35.670 the observation as a tool to actually calculate 00:23:35.670 --> 00:23:36.970 the speed of light. 00:23:36.970 --> 00:23:39.150 James actually nailed it. 00:23:39.150 --> 00:23:41.330 He found a value which is really close 00:23:41.330 --> 00:23:44.570 to the current understanding of the speed of light, 00:23:44.570 --> 00:23:50.540 which is 3 times 10 to 9 meters per second. 00:23:50.540 --> 00:23:54.680 Therefore, if you calculate that using all those constants here, 00:23:54.680 --> 00:23:59.120 you will be able to see that, indeed, from Maxwell's 00:23:59.120 --> 00:24:01.450 equation, you get-- 00:24:01.450 --> 00:24:04.490 oh, it should be 3 times 10 to the 8, not to the 9. 00:24:04.490 --> 00:24:06.090 I was saying 10 to 9. 00:24:06.090 --> 00:24:09.522 It should be 3 times 10 to the 8 meters per second. 00:24:13.330 --> 00:24:17.630 So indeed, this equation is actually 00:24:17.630 --> 00:24:22.430 predicting the speed of light to be 3 times 10 to the 8 00:24:22.430 --> 00:24:28.870 is matching the experimental result. 00:24:28.870 --> 00:24:31.000 So that is pretty nice. 00:24:31.000 --> 00:24:34.270 And you may ask a question-- 00:24:34.270 --> 00:24:35.780 so wait a second. 00:24:35.780 --> 00:24:40.060 You said this is actually an electromagnetic wave, right? 00:24:40.060 --> 00:24:41.950 So that's actually what I was talking about. 00:24:41.950 --> 00:24:48.130 But this equation only talks about electric fields. 00:24:48.130 --> 00:24:51.280 What is happening to the magnetic field? 00:24:51.280 --> 00:24:51.945 What happened? 00:24:54.910 --> 00:24:59.900 Can we actually choose arbitrary magnetic fields? 00:25:04.200 --> 00:25:07.120 Is a magnetic field also described 00:25:07.120 --> 00:25:10.240 by the three dimensional wave equation, right? 00:25:10.240 --> 00:25:14.220 The answer is that indeed you can actually 00:25:14.220 --> 00:25:15.640 do the same exercise. 00:25:15.640 --> 00:25:19.260 You can now instead of plugging in electric field, 00:25:19.260 --> 00:25:21.730 you can plug in a magnetic field. 00:25:21.730 --> 00:25:25.380 And you will extract exactly the same conclusion. 00:25:25.380 --> 00:25:28.180 You will conclude the del square B, 00:25:28.180 --> 00:25:35.390 will B equal to Mu 0, epsilon 0, partial square, 00:25:35.390 --> 00:25:37.920 B partial to square. 00:25:37.920 --> 00:25:40.440 OK, so it is actually very important 00:25:40.440 --> 00:25:47.450 to see that the magnetic field also obey this wave equation. 00:25:47.450 --> 00:25:49.760 OK? 00:25:49.760 --> 00:25:53.790 And also from Maxwell's equation, 00:25:53.790 --> 00:25:57.690 you can see that the changing electric field 00:25:57.690 --> 00:26:02.550 will produce a curling around a magnetic field. 00:26:02.550 --> 00:26:04.510 The same thing also happens here. 00:26:04.510 --> 00:26:08.190 A changing magnetic field also produce a curling 00:26:08.190 --> 00:26:10.530 around electric field. 00:26:10.530 --> 00:26:13.490 So what does that mean? 00:26:13.490 --> 00:26:18.800 That means E, electric field create magnetic field. 00:26:18.800 --> 00:26:21.050 Magnetic field create electric field. 00:26:21.050 --> 00:26:23.600 And this happens all the time. 00:26:23.600 --> 00:26:29.530 Therefore, one cannot live without the other. 00:26:29.530 --> 00:26:30.790 They are living together. 00:26:30.790 --> 00:26:34.600 They are all together, forever. 00:26:34.600 --> 00:26:40.270 All right, so what is actually oscillating 00:26:40.270 --> 00:26:44.320 is actually both electric field and the magnetic field, right? 00:26:44.320 --> 00:26:50.280 So you may ask, OK, we are talking about vacuum. 00:26:50.280 --> 00:26:53.910 Vacuum means there is no material, no charge, 00:26:53.910 --> 00:26:56.710 no whatsoever in vacuum. 00:26:56.710 --> 00:26:58.710 So what is actually oscillating? 00:26:58.710 --> 00:27:01.640 Who is oscillating? 00:27:01.640 --> 00:27:05.940 Is the electric field and the magnetic field. 00:27:05.940 --> 00:27:08.580 This so-called field, all those vectors-- 00:27:08.580 --> 00:27:11.310 which are actually oscillating-- it's not the material, 00:27:11.310 --> 00:27:15.510 but all those vectors associated with the space, 00:27:15.510 --> 00:27:17.710 which is actually oscillating up and down. 00:27:20.720 --> 00:27:23.710 All right, so originally I would like 00:27:23.710 --> 00:27:26.510 to show you a pulse of light in front of you. 00:27:26.510 --> 00:27:30.610 And show that it's moving, but it's too fast. 00:27:30.610 --> 00:27:32.110 So I couldn't do that. 00:27:32.110 --> 00:27:34.570 [LAUGHTER] 00:27:34.570 --> 00:27:39.070 Fortunately, we have photos. 00:27:39.070 --> 00:27:45.160 Photos are actually collected the recorded photons. 00:27:45.160 --> 00:27:50.240 Emitted from the object of the interest. 00:27:50.240 --> 00:27:53.930 So this is actually how we make applesauce at MIT. 00:27:53.930 --> 00:27:57.820 We shoot-- bullet through the apple then we have the sauce. 00:27:57.820 --> 00:27:59.150 [LAUGHTER] 00:27:59.150 --> 00:28:01.870 But not sure if that's tasty enough or not. 00:28:01.870 --> 00:28:04.750 But that's how we do it in MIT-- 00:28:04.750 --> 00:28:06.160 MIT style. 00:28:06.160 --> 00:28:11.070 And the good thing is that this kind of technique 00:28:11.070 --> 00:28:13.930 is improved dramatically in these days. 00:28:13.930 --> 00:28:17.980 I would like to show you a short video, which is actually 00:28:17.980 --> 00:28:23.145 recording a video of-- 00:28:23.145 --> 00:28:30.070 it's recording experiment, which you shoot some beam of light 00:28:30.070 --> 00:28:34.570 through some plastic container. 00:28:34.570 --> 00:28:38.080 And the speed of this recording corresponds 00:28:38.080 --> 00:28:42.310 to one trillion frame per second. 00:28:42.310 --> 00:28:45.040 So this is super fast recording. 00:28:45.040 --> 00:28:48.340 And they can actually reconstruct the propagation 00:28:48.340 --> 00:28:52.600 of light through this bottle. 00:28:52.600 --> 00:28:58.300 The credit is actually to the Media Lab Camera Culture group. 00:28:58.300 --> 00:29:00.852 And let's take a look at the video. 00:29:00.852 --> 00:29:01.910 Just one second. 00:29:05.110 --> 00:29:10.030 OK, so this is actually recording at one trillion frame 00:29:10.030 --> 00:29:11.310 per second. 00:29:11.310 --> 00:29:13.220 So you can see that there's a light pulse-- 00:29:13.220 --> 00:29:15.710 a very short pulse created. 00:29:15.710 --> 00:29:21.440 And is really pass through the bottle. 00:29:21.440 --> 00:29:27.700 And it can be recorded with the technique created by Media Lab. 00:29:27.700 --> 00:29:31.190 So you can see that the pulse is really propagating through it. 00:29:31.190 --> 00:29:33.560 And the reason why we can see the pulse 00:29:33.560 --> 00:29:35.390 is because there are air, there are 00:29:35.390 --> 00:29:37.430 material which will actually change 00:29:37.430 --> 00:29:39.320 the direction of the light. 00:29:39.320 --> 00:29:42.830 And therefore, those are recorded by the camera. 00:29:42.830 --> 00:29:48.620 And they take trillions of frames of this thing, 00:29:48.620 --> 00:29:49.760 and put them together. 00:29:49.760 --> 00:29:53.390 Then basically-- and they take many, many frames, 00:29:53.390 --> 00:29:56.700 and they put them together to reconstruct this movie. 00:29:56.700 --> 00:29:59.360 So as you can see that indeed you 00:29:59.360 --> 00:30:03.050 can see the propagation of the light 00:30:03.050 --> 00:30:07.100 through this kind of video. 00:30:07.100 --> 00:30:09.470 So I hope that we enjoyed this video. 00:30:09.470 --> 00:30:14.960 And let's actually take a look at some concrete example 00:30:14.960 --> 00:30:18.260 which make use of the wave equation, which 00:30:18.260 --> 00:30:20.870 we did right here. 00:30:20.870 --> 00:30:24.740 So let's consider a plane wave solution. 00:30:24.740 --> 00:30:28.100 Things we are entering a three dimensional world. 00:30:28.100 --> 00:30:31.816 So that's actually consider so-called a plane wave. 00:30:39.230 --> 00:30:45.930 So in this example, I am considering the electric field 00:30:45.930 --> 00:30:53.890 that's actually equal to the real part of E0 exponential i, 00:30:53.890 --> 00:30:57.740 kz minus omega t. 00:30:57.740 --> 00:31:02.930 And this electric field I actually consider here 00:31:02.930 --> 00:31:06.910 is in the x direction. 00:31:06.910 --> 00:31:10.420 And if I write all the terms from this expression 00:31:10.420 --> 00:31:14.790 expressively, that's actually what I'm getting is-- 00:31:14.790 --> 00:31:26.650 x component will be E0, cosine kz minus omega t, 0 and 0. 00:31:26.650 --> 00:31:28.960 So what does this mean? 00:31:28.960 --> 00:31:31.710 What is actually a plane wave? 00:31:31.710 --> 00:31:36.010 The plane wave basically is actually 00:31:36.010 --> 00:31:37.840 fielding the whole space. 00:31:37.840 --> 00:31:41.530 What I mean by plane wave is I feel the whole space 00:31:41.530 --> 00:31:43.690 with electric field. 00:31:43.690 --> 00:31:47.200 This electric field only have one-- 00:31:47.200 --> 00:31:50.800 only one direction have non-zero value, which 00:31:50.800 --> 00:31:54.010 is x direction in this example. 00:31:54.010 --> 00:31:58.696 And then the other direction, there's no-- 00:31:58.696 --> 00:32:02.460 the magnitude is actually equal to 0. 00:32:02.460 --> 00:32:05.420 So that's actually what I mean by plane wave. 00:32:05.420 --> 00:32:11.310 And also the electric field is filling a whole space 00:32:11.310 --> 00:32:17.140 in the discussion-- in the example which I discussed here. 00:32:17.140 --> 00:32:22.800 And if I define my coordinate system like this, 00:32:22.800 --> 00:32:24.220 x is in the horizontal direction. 00:32:24.220 --> 00:32:28.240 Then that means everything is actually-- 00:32:28.240 --> 00:32:30.220 all the electric field is actually 00:32:30.220 --> 00:32:36.830 pointing toward the x direction in this coordinate system. 00:32:39.600 --> 00:32:45.840 So we have discussed progressing wave in the past few lectures. 00:32:45.840 --> 00:32:50.640 Can somebody actually tell me the direction of propagation 00:32:50.640 --> 00:32:52.750 of this plane wave? 00:32:52.750 --> 00:32:56.200 So the hint is that this is actually equal to E0, 00:32:56.200 --> 00:32:58.860 that the magnitude of the x component 00:32:58.860 --> 00:33:03.390 is equal to E0, cosine kz minus omega t. 00:33:03.390 --> 00:33:07.290 What is actually the direction of propagation 00:33:07.290 --> 00:33:09.516 of this electric field? 00:33:09.516 --> 00:33:10.016 AUDIENCE: z. 00:33:10.016 --> 00:33:11.557 YEN-JIE LEE: It's in the z direction. 00:33:11.557 --> 00:33:12.690 Yeah, very good. 00:33:12.690 --> 00:33:14.670 Because we know that this is actually 00:33:14.670 --> 00:33:17.090 going in the positive z direction. 00:33:17.090 --> 00:33:20.480 Because this is actually kz minus-- 00:33:20.480 --> 00:33:22.330 there's a minus sign-- omega t. 00:33:22.330 --> 00:33:27.140 So therefore it's going toward the positive z direction. 00:33:27.140 --> 00:33:32.130 Not x direction. x direction is where the electric field is 00:33:32.130 --> 00:33:33.930 pointing to. 00:33:33.930 --> 00:33:40.980 And the direction of propagation is toward the z direction. 00:33:40.980 --> 00:33:45.330 So there's a difference. 00:33:45.330 --> 00:33:49.170 So first thing which I would like to do is to check if this 00:33:49.170 --> 00:33:55.300 so-called plane wave solution actually satisfy the equation-- 00:33:55.300 --> 00:33:58.710 the wave equation which we derive here. 00:33:58.710 --> 00:34:03.660 Del square E equal to Mu 0 epsilon 0, partial square E, 00:34:03.660 --> 00:34:04.650 partial t square. 00:34:04.650 --> 00:34:09.909 So I can now plug that in to that equation. 00:34:09.909 --> 00:34:13.830 I can now plug in to this equation. 00:34:13.830 --> 00:34:17.580 If I plug in the wave-- the plane wave solution, which 00:34:17.580 --> 00:34:21.239 I have here to that equation-- basically, 00:34:21.239 --> 00:34:23.429 I can get the left-hand side. 00:34:23.429 --> 00:34:25.020 The left-hand side of the equation, 00:34:25.020 --> 00:34:30.060 you will get minus E0. 00:34:30.060 --> 00:34:33.090 Only one term which contribute is 00:34:33.090 --> 00:34:37.159 the partial square E partial z square term which contribute. 00:34:37.159 --> 00:34:38.040 Right? 00:34:38.040 --> 00:34:42.340 Because the magnitude of the electric field 00:34:42.340 --> 00:34:46.110 only depends on z and t. 00:34:46.110 --> 00:34:53.730 Therefore, you get minus E0, k square, cosine, kz minus omega 00:34:53.730 --> 00:34:57.600 t in the left-hand side of the wave equation. 00:35:00.480 --> 00:35:03.510 How about the right-hand side? 00:35:03.510 --> 00:35:07.140 Right-hand side actually you are taking partial derivative, 00:35:07.140 --> 00:35:09.360 which is fed to t. 00:35:09.360 --> 00:35:13.035 Basically, you get minus Mu 0-- 00:35:13.035 --> 00:35:14.280 Mu 0, epsilon 0-- 00:35:14.280 --> 00:35:18.540 I copied from that formula there. 00:35:18.540 --> 00:35:23.110 And you basically get omega square out 00:35:23.110 --> 00:35:25.720 of it because of the partial square, partial t 00:35:25.720 --> 00:35:28.220 square operator. 00:35:28.220 --> 00:35:33.964 And then you basically get cosine kz minus omega t. 00:35:37.170 --> 00:35:39.970 And of course I missed the E0 term. 00:35:39.970 --> 00:35:42.100 E0 should be copied from-- on there. 00:35:45.030 --> 00:35:46.730 So now I can show that-- 00:35:46.730 --> 00:35:48.450 OK, this cancel. 00:35:48.450 --> 00:35:53.954 Basically, this is the same cosine kz minus omega t. 00:35:53.954 --> 00:35:57.235 And E0 also cancel. 00:35:57.235 --> 00:36:01.010 And I can cancel the minus sign. 00:36:01.010 --> 00:36:05.080 What I'm going to get is k square 00:36:05.080 --> 00:36:10.860 is equal to Mu 0, epsilon 0, omega square. 00:36:10.860 --> 00:36:15.720 So that means there should be a fixed relation between k 00:36:15.720 --> 00:36:20.850 and omega, which is actually omega over k 00:36:20.850 --> 00:36:26.130 will be equal to 1 over square to the Mu 0, epsilon 0. 00:36:26.130 --> 00:36:27.900 And this is equal to c. 00:36:27.900 --> 00:36:33.030 If this is satisfied, then the plane wave 00:36:33.030 --> 00:36:36.630 is a solution to the wave equation-- 00:36:36.630 --> 00:36:40.470 only when this is actually satisfied. 00:36:40.470 --> 00:36:44.910 Otherwise we can write arbitrary plane wave equation, 00:36:44.910 --> 00:36:48.480 but they are not the solution of that equation 00:36:48.480 --> 00:36:49.480 from Maxwell's equation. 00:36:52.630 --> 00:36:57.820 So now, I have derived the electric field 00:36:57.820 --> 00:37:02.440 and also know the relation between omega, the angle 00:37:02.440 --> 00:37:05.830 frequency, and the k, the wave number. 00:37:05.830 --> 00:37:10.540 And now, what about magnetic field? 00:37:10.540 --> 00:37:15.100 So I just mentioned before, magnetic field cannot live 00:37:15.100 --> 00:37:16.220 without electric field. 00:37:16.220 --> 00:37:20.620 And electric field cannot live without magnetic field. 00:37:20.620 --> 00:37:23.740 So what is actually responding magnetic field? 00:37:23.740 --> 00:37:26.470 We can actually evaluate that. 00:37:26.470 --> 00:37:31.830 So now, the question is what is actually the magnetic field? 00:37:31.830 --> 00:37:35.170 And how is that vary as a function of time 00:37:35.170 --> 00:37:39.990 and as a function of position in the space? 00:37:39.990 --> 00:37:45.610 So we are facing a choice. 00:37:45.610 --> 00:37:48.830 So there are two equations, which 00:37:48.830 --> 00:37:54.690 relate electric field and the magnetic field. 00:37:54.690 --> 00:37:58.380 It is actually very important you make the right choice when 00:37:58.380 --> 00:38:02.340 you start your calculation. 00:38:02.340 --> 00:38:04.920 So we can use Faraday's law. 00:38:04.920 --> 00:38:07.770 We can also use Ampere's law. 00:38:07.770 --> 00:38:11.050 But there's only one, which is actually much easier 00:38:11.050 --> 00:38:17.280 to derive a solution, which is the choice of Faraday's law. 00:38:17.280 --> 00:38:23.310 If you choose to use Ampere's law to evaluate B, 00:38:23.310 --> 00:38:24.780 then you are going to get a really 00:38:24.780 --> 00:38:27.960 super complicated problem to solve. 00:38:27.960 --> 00:38:30.570 But on the other hand, if you choose 00:38:30.570 --> 00:38:33.570 to use Faraday's law to solve this problem, 00:38:33.570 --> 00:38:41.160 then you can see that the unknown is the magnetic field-- 00:38:41.160 --> 00:38:43.530 the field which I would like to evaluate. 00:38:43.530 --> 00:38:47.830 And the expression for the B is actually rather simple. 00:38:47.830 --> 00:38:50.640 It's actually just a partial derivative, partial B, 00:38:50.640 --> 00:38:52.140 partial t. 00:38:52.140 --> 00:38:54.480 So it's pretty simple and you can actually 00:38:54.480 --> 00:38:56.360 evaluate the known part. 00:38:56.360 --> 00:38:59.460 This curl looks pretty complicated. 00:38:59.460 --> 00:39:02.130 So you can actually evaluate that because you know 00:39:02.130 --> 00:39:04.380 what is the electric field. 00:39:04.380 --> 00:39:07.650 On the other hand, if you will use Ampere's Law 00:39:07.650 --> 00:39:11.820 then you will be in trouble because you don't 00:39:11.820 --> 00:39:14.610 know what is a B, xBy and Bz. 00:39:14.610 --> 00:39:16.350 And you have to evaluate curl. 00:39:16.350 --> 00:39:18.740 And you get a lot of terms, and that is actually 00:39:18.740 --> 00:39:21.290 equal to something from-- the information 00:39:21.290 --> 00:39:22.290 from the electric field. 00:39:22.290 --> 00:39:25.680 And that would be very difficult to evaluate. 00:39:25.680 --> 00:39:31.510 So therefore, what we are going to do is to use Faraday's law, 00:39:31.510 --> 00:39:38.070 curl of E will be equal to minus partial B, partial t. 00:39:38.070 --> 00:39:44.200 So basically, as I mentioned in the beginning, 00:39:44.200 --> 00:39:51.270 we can make use of the equations the determinant of matrix 00:39:51.270 --> 00:39:53.760 to evaluate the curl. 00:39:53.760 --> 00:39:56.310 So therefore, I am going to use that. 00:39:56.310 --> 00:40:03.040 And then what I'm going to get is x, y, z unit vector for fill 00:40:03.040 --> 00:40:05.460 the first row. 00:40:05.460 --> 00:40:09.680 And the partial partial x, partial partial y, 00:40:09.680 --> 00:40:14.820 partial partial z, which fill the second row of the matrix. 00:40:14.820 --> 00:40:20.610 Then I get Ex, 0, 0 because the electric field is only 00:40:20.610 --> 00:40:23.490 in the x direction. 00:40:23.490 --> 00:40:25.980 And this will be equal to-- 00:40:25.980 --> 00:40:30.880 only two terms survive because of these two 0's. 00:40:30.880 --> 00:40:35.430 So all other terms are killed, and only two terms are now 0. 00:40:35.430 --> 00:40:39.924 The first term is actually partial Ex, 00:40:39.924 --> 00:40:44.770 partial z in the y direction. 00:40:44.770 --> 00:40:50.390 And the second term is actually minus partial Ex, 00:40:50.390 --> 00:40:55.700 partial y in the z direction. 00:40:55.700 --> 00:40:56.840 Any questions? 00:40:56.840 --> 00:40:57.870 Am I going too fast? 00:41:01.540 --> 00:41:04.850 All right, so you can see that the electric field only 00:41:04.850 --> 00:41:08.310 depends on the position z. 00:41:08.310 --> 00:41:11.710 It's independent of y. 00:41:11.710 --> 00:41:17.020 Therefore, partial Ex, partial y, is actually equal to 0. 00:41:17.020 --> 00:41:19.900 Wow, this become much, much easier 00:41:19.900 --> 00:41:23.710 because there's only one term which is surviving. 00:41:23.710 --> 00:41:26.670 This is a operator. 00:41:26.670 --> 00:41:28.690 Then basically what we're going to get 00:41:28.690 --> 00:41:34.120 is I can now calculate partial Ex, partial z based 00:41:34.120 --> 00:41:38.440 on that equation, E0 cosine kz minus omega t. 00:41:38.440 --> 00:41:41.460 Then basically, what I can get is minus-- 00:41:41.460 --> 00:41:49.504 I get a K out of it, E0 sine kz, and it's omega t. 00:41:55.130 --> 00:41:59.590 So this is actually the result of the left-hand side. 00:41:59.590 --> 00:42:04.460 The right-hand side of that equation of the Faraday's law 00:42:04.460 --> 00:42:07.160 is minus partial B, partial t. 00:42:07.160 --> 00:42:12.550 So this will give you equal to minus partial B, partial t. 00:42:17.530 --> 00:42:22.410 So very important-- I don't want to drop the y direction. 00:42:22.410 --> 00:42:25.270 So this is just y direction. 00:42:25.270 --> 00:42:29.980 And this is actually a vector and this is also a vector. 00:42:29.980 --> 00:42:37.600 So what I could do is to do a integration over t. 00:42:37.600 --> 00:42:40.720 And those will cancel the minus sign. 00:42:40.720 --> 00:42:45.730 So if I integrate over t, then basically what I'm going to get 00:42:45.730 --> 00:42:57.100 is K over omega is 0, cosine kz minus omega t 00:42:57.100 --> 00:42:58.476 in the y direction only. 00:43:01.020 --> 00:43:04.740 So I'm doing a integration of t, cancel the minus sign, 00:43:04.740 --> 00:43:07.420 then this is what you want to get. 00:43:07.420 --> 00:43:14.350 And of course, k/omega is actually 1/c. 00:43:14.350 --> 00:43:17.320 So therefore, you have E0-- 00:43:17.320 --> 00:43:19.450 you can actually simplify this fraction-- 00:43:19.450 --> 00:43:26.160 and this is actually equal to E0 over c, cosine, kz minus omega 00:43:26.160 --> 00:43:30.960 t in the y direction. 00:43:30.960 --> 00:43:35.500 OK, look at what we have learned from here. 00:43:35.500 --> 00:43:38.650 What we have learned from here is that-- 00:43:38.650 --> 00:43:44.020 I got started with a plane wave solution of the electric field. 00:43:44.020 --> 00:43:48.430 And I can show that only when omega over k 00:43:48.430 --> 00:43:51.880 is equal to the speed of light this is actually 00:43:51.880 --> 00:43:56.260 a solution to my wave equation. 00:43:56.260 --> 00:44:00.700 And also because the electric field and the magnetic field 00:44:00.700 --> 00:44:05.350 have to satisfy the Maxwell's equation all the time-- 00:44:05.350 --> 00:44:07.310 because that's the fundamental law-- 00:44:07.310 --> 00:44:10.690 therefore, I can use those equations 00:44:10.690 --> 00:44:13.680 to evaluate and to find what is actually 00:44:13.680 --> 00:44:16.090 the corresponding magnetic field. 00:44:16.090 --> 00:44:19.480 And using Faraday's law and plugging in 00:44:19.480 --> 00:44:21.280 and the solving the question, I will 00:44:21.280 --> 00:44:27.120 be able to figure out that B is also what kind of wave? 00:44:27.120 --> 00:44:30.710 B is also what kind of wave I was talking about-- 00:44:30.710 --> 00:44:32.730 also? 00:44:32.730 --> 00:44:33.620 AUDIENCE: Plane wave. 00:44:33.620 --> 00:44:34.911 YEN-JIE LEE: Plane wave, right? 00:44:34.911 --> 00:44:36.270 It's also plane wave. 00:44:36.270 --> 00:44:37.020 You see? 00:44:37.020 --> 00:44:39.950 So if I got started with a plane wave 00:44:39.950 --> 00:44:42.360 in the electric field side, and I also 00:44:42.360 --> 00:44:46.560 get the plane wave in the magnetic field side. 00:44:46.560 --> 00:44:49.630 They are proportional to each other. 00:44:49.630 --> 00:44:54.120 Originally, the magnitude of the electric field is E0. 00:44:54.120 --> 00:44:57.450 The corresponding magnetic field-- 00:44:57.450 --> 00:45:00.880 the magnitude is proportional to E0. 00:45:00.880 --> 00:45:04.530 But there's a factor of 1/c difference 00:45:04.530 --> 00:45:07.240 between the magnetic field amplitude 00:45:07.240 --> 00:45:11.010 and the electric field amplitude. 00:45:11.010 --> 00:45:13.380 The third thing which we learned from here 00:45:13.380 --> 00:45:19.830 is that electric field is actually in the x direction. 00:45:19.830 --> 00:45:24.250 B field is actually not in the x direction, 00:45:24.250 --> 00:45:25.835 it's in the y direction. 00:45:29.320 --> 00:45:36.610 What we learn from here is that the direction of the B field 00:45:36.610 --> 00:45:42.860 can be determined by a simple calculation. 00:45:42.860 --> 00:45:46.980 So basically, the B is proportional-- 00:45:46.980 --> 00:45:48.940 the magnitude of B is proportional 00:45:48.940 --> 00:45:50.060 to the electric field. 00:45:52.770 --> 00:45:55.920 But you have to multiply the magnitude by 1/c. 00:45:58.750 --> 00:46:00.760 And also this is actually not correct 00:46:00.760 --> 00:46:06.550 because the B is actually in the y direction. 00:46:06.550 --> 00:46:10.410 So the original direction of the electric field 00:46:10.410 --> 00:46:12.190 is in the x direction. 00:46:12.190 --> 00:46:16.950 Also we know the direction of a propagation 00:46:16.950 --> 00:46:21.010 is in the z direction. 00:46:21.010 --> 00:46:26.690 Therefore, if I take unit vector K-- 00:46:26.690 --> 00:46:31.000 K is actually the wave number, but now I make it a vector 00:46:31.000 --> 00:46:35.000 and I take the unit vector is equal to z-- 00:46:35.000 --> 00:46:42.550 so direction of propagation. 00:46:42.550 --> 00:46:47.410 If I make this definition then I can now rewrite this relation. 00:46:47.410 --> 00:46:51.250 Basically, I can express the magnitude of B 00:46:51.250 --> 00:46:59.290 by K hat, which is the direction of propagation cross the E 00:46:59.290 --> 00:47:01.090 field. 00:47:01.090 --> 00:47:02.560 And we can check this. 00:47:02.560 --> 00:47:05.830 And then basically what you are going to get is z cross E-- 00:47:05.830 --> 00:47:10.780 then actually really z cross x, you 00:47:10.780 --> 00:47:14.380 are going to get y direction. 00:47:14.380 --> 00:47:18.430 And that is actually telling you that B and the E 00:47:18.430 --> 00:47:21.940 have a rather simple relation. 00:47:21.940 --> 00:47:25.510 And also you don't really need to go through 00:47:25.510 --> 00:47:28.240 all those calculation again because now you 00:47:28.240 --> 00:47:32.590 can see that if you know the direction of propagation 00:47:32.590 --> 00:47:35.010 and you know the direction of the electric field, 00:47:35.010 --> 00:47:41.230 then you can already evaluate what will be in the B field. 00:47:41.230 --> 00:47:44.050 So we will take a five minute break. 00:47:44.050 --> 00:47:48.002 We'll come back in 29, and we will continue the discussion 00:47:48.002 --> 00:47:48.710 of this solution. 00:47:52.860 --> 00:47:56.590 Let me know if you have any questions about the content we 00:47:56.590 --> 00:47:57.200 discussed. 00:48:05.160 --> 00:48:06.810 Welcome back, everybody. 00:48:06.810 --> 00:48:08.670 So we will continue the discussion 00:48:08.670 --> 00:48:12.180 of what we have learned from the wave equation. 00:48:12.180 --> 00:48:17.360 So basically we start with plane wave in the electric field. 00:48:17.360 --> 00:48:20.100 And this electric field is in the x direction. 00:48:20.100 --> 00:48:23.430 And we evaluated the corresponding B field 00:48:23.430 --> 00:48:25.890 which is in the y direction. 00:48:25.890 --> 00:48:30.690 And what we found is that actually we can find a pretty 00:48:30.690 --> 00:48:34.930 simple relation between electric field and the magnetic field, 00:48:34.930 --> 00:48:40.055 which is actually magnetic field vector is equal to 1/c, 00:48:40.055 --> 00:48:46.170 K hat cross E. And the K hat now which is-- you find here-- 00:48:46.170 --> 00:48:48.700 is actually the direction of propagation. 00:48:48.700 --> 00:48:53.070 So basically, in this case in the discussion we had before, 00:48:53.070 --> 00:48:58.590 the direction of propagation is in the positive z direction. 00:48:58.590 --> 00:49:02.540 So if I go ahead and visualize the whole-- 00:49:02.540 --> 00:49:06.840 solution-- plot the magnetic field and the electric field is 00:49:06.840 --> 00:49:11.200 a function of z, x and the y-- 00:49:11.200 --> 00:49:12.850 it's a function of z actually here. 00:49:12.850 --> 00:49:16.680 And I only evaluate the value at x equal to 0, 00:49:16.680 --> 00:49:18.480 and the y equal to 0. 00:49:18.480 --> 00:49:20.490 And basically, this is actually what you have. 00:49:20.490 --> 00:49:22.770 So basically, you have two sine wave. 00:49:22.770 --> 00:49:26.250 One is actually pointing to the x direction. 00:49:26.250 --> 00:49:28.950 And the other one is actually pointing to the y direction, 00:49:28.950 --> 00:49:31.260 which is the B field. 00:49:31.260 --> 00:49:33.690 And those lines doesn't mean a lot 00:49:33.690 --> 00:49:37.650 because those lines are just connecting the end 00:49:37.650 --> 00:49:39.810 point of all those vectors. 00:49:39.810 --> 00:49:47.550 So you can see that they are cosine wave structure when 00:49:47.550 --> 00:49:50.040 you connect all those vectors. 00:49:50.040 --> 00:49:53.970 And keep in mind that those are evaluated 00:49:53.970 --> 00:49:57.270 at x and y equal to 0. 00:49:57.270 --> 00:50:01.170 Therefore, what we actually get is actually a lot of vectors. 00:50:01.170 --> 00:50:03.930 So those individual arrows are vectors. 00:50:03.930 --> 00:50:09.390 And this whole thing-- this whole electromagnetic wave 00:50:09.390 --> 00:50:14.010 is propagating to the positive z direction. 00:50:14.010 --> 00:50:18.480 And those electric field and the magnetic field 00:50:18.480 --> 00:50:22.430 are propagating at the speed of light, which you see. 00:50:22.430 --> 00:50:25.370 And also you can see that the magnitude-- 00:50:25.370 --> 00:50:26.790 also I plotted here-- 00:50:26.790 --> 00:50:31.020 the magnitude, there's no phase difference 00:50:31.020 --> 00:50:37.369 between electric field and the magnetic field. 00:50:37.369 --> 00:50:38.910 This is actually not always the case. 00:50:38.910 --> 00:50:45.670 In which we will show a example probably later in the lecture. 00:50:45.670 --> 00:50:49.380 So in general, what we can actually do 00:50:49.380 --> 00:50:56.800 is to write down a general expression for the plane wave. 00:50:56.800 --> 00:50:59.490 So for example, I can have a plane wave, which is actually 00:50:59.490 --> 00:51:02.160 propagating in some direction. 00:51:02.160 --> 00:51:05.990 Which is actually given by this K vector. 00:51:05.990 --> 00:51:09.540 K vector is actually giving you information 00:51:09.540 --> 00:51:12.930 about the wave number. 00:51:12.930 --> 00:51:18.390 And also the direction of propagation. 00:51:18.390 --> 00:51:21.660 And in this case, what I am trying to construct 00:51:21.660 --> 00:51:25.820 is a solution, which is actually propagating along 00:51:25.820 --> 00:51:28.920 in the direction of the K vector. 00:51:28.920 --> 00:51:32.990 And the electric field is actually 00:51:32.990 --> 00:51:36.540 going to be pointing to a direction perpendicular 00:51:36.540 --> 00:51:40.680 to the direction of the K vector. 00:51:40.680 --> 00:51:44.490 So basically, what I can do is I can write this plane 00:51:44.490 --> 00:51:46.605 wave in this functional form. 00:51:46.605 --> 00:51:49.380 E0 is actually a vector, which is actually 00:51:49.380 --> 00:51:54.250 telling you the direction of the electric field-- 00:51:54.250 --> 00:52:00.210 E0 vector-- is actually have this function of form. 00:52:03.110 --> 00:52:06.490 And the K vector is actually placed 00:52:06.490 --> 00:52:08.670 in the exponential function-- inside 00:52:08.670 --> 00:52:10.650 the exponential function. 00:52:10.650 --> 00:52:16.518 Exponential i, k dot r minus omega t. 00:52:16.518 --> 00:52:18.870 And what is actually r? 00:52:18.870 --> 00:52:25.080 r is actually x x hat, plus y y hat, plus z z hat. 00:52:25.080 --> 00:52:29.700 And omega is actually the angular frequency 00:52:29.700 --> 00:52:33.330 which we are familiar with and that's actually 00:52:33.330 --> 00:52:39.870 equal to c times the magnitude of K, 00:52:39.870 --> 00:52:43.470 which is actually the wave number. 00:52:43.470 --> 00:52:45.660 And you can actually show that-- 00:52:45.660 --> 00:52:51.150 OK, indeed this expression can satisfy 00:52:51.150 --> 00:52:55.980 the wave equation, which we did right for the electric field. 00:52:55.980 --> 00:52:59.370 And of course there are some requirements, 00:52:59.370 --> 00:53:06.040 which is actually that the direction of the electric field 00:53:06.040 --> 00:53:10.740 have to be perpendicular to the direction of propagation. 00:53:10.740 --> 00:53:12.770 Which you can actually derive that. 00:53:12.770 --> 00:53:19.710 And finally, this expression B field equal to 1/c. 00:53:19.710 --> 00:53:21.860 K, which is the direction of propagation 00:53:21.860 --> 00:53:26.080 cross E field is still valid because basically we 00:53:26.080 --> 00:53:30.730 have shown that it works for the plane wave pointing 00:53:30.730 --> 00:53:33.940 to the x direction propagating to the z direction. 00:53:33.940 --> 00:53:37.220 We can always redefine the coordinate system 00:53:37.220 --> 00:53:41.610 because we can actually rotate this coordinate system 00:53:41.610 --> 00:53:44.650 and the physics should not change. 00:53:44.650 --> 00:53:48.610 Therefore, you must see that this expression 00:53:48.610 --> 00:53:50.370 must be still valid. 00:53:50.370 --> 00:53:55.990 And also that the direction of the electric field, 00:53:55.990 --> 00:53:58.570 which is actually proportional to E0, 00:53:58.570 --> 00:54:03.760 must be perpendicular to the direction of propagation. 00:54:03.760 --> 00:54:09.750 So that is actually what we can actually 00:54:09.750 --> 00:54:14.440 learn a general description of electric field pointing 00:54:14.440 --> 00:54:17.950 to some random direction. 00:54:17.950 --> 00:54:21.640 So we have talked about the progressing wave 00:54:21.640 --> 00:54:26.020 solution and also the plane wave and also 00:54:26.020 --> 00:54:30.590 the corresponding magnetic field. 00:54:30.590 --> 00:54:32.770 I hope that you can actually apply this-- 00:54:32.770 --> 00:54:34.990 the technique which we learned here-- 00:54:34.990 --> 00:54:37.700 if you are given a magnetic field, 00:54:37.700 --> 00:54:41.890 you must know that there must be a corresponding electric field 00:54:41.890 --> 00:54:45.280 because they cannot be separated from each other. 00:54:45.280 --> 00:54:49.510 And you can actually obtain the corresponding electric field 00:54:49.510 --> 00:54:56.150 if you are given magnetic field by using Maxwell's equations. 00:54:56.150 --> 00:54:59.650 So what is going to happen is that 00:54:59.650 --> 00:55:02.680 now if I emit this photon-- 00:55:02.680 --> 00:55:06.790 or say this electromagnetic wave from the light source, 00:55:06.790 --> 00:55:08.580 for example, that one-- 00:55:08.580 --> 00:55:12.490 the one of which is pointing at my face. 00:55:12.490 --> 00:55:14.440 Basically, my face is going to bounce 00:55:14.440 --> 00:55:18.370 some of the electromagnetic field around. 00:55:18.370 --> 00:55:21.910 And some that actually go out of the window. 00:55:21.910 --> 00:55:24.220 And then when they go out the window, 00:55:24.220 --> 00:55:25.940 maybe they are lucky they are not hitting 00:55:25.940 --> 00:55:27.650 any building in the MIT. 00:55:27.650 --> 00:55:30.040 Then what is going to happen is that they're 00:55:30.040 --> 00:55:35.030 going to propagate forever toward the end of the universe. 00:55:35.030 --> 00:55:38.890 Really, they are going straight forever as you 00:55:38.890 --> 00:55:40.580 can see from this solution. 00:55:40.580 --> 00:55:43.570 It's like some kind of wave propagating forever 00:55:43.570 --> 00:55:44.710 at the speed of light. 00:55:44.710 --> 00:55:51.100 If they don't encounter anything before the end of life 00:55:51.100 --> 00:55:53.020 of the electromagnetic wave, it's 00:55:53.020 --> 00:55:56.920 going to be propagating forever toward that direction-- 00:55:56.920 --> 00:55:59.380 escaping from that window. 00:55:59.380 --> 00:56:04.930 So that is actually fascinating and-- 00:56:04.930 --> 00:56:07.870 but we would like to introduce some more excitement 00:56:07.870 --> 00:56:09.460 to see what is going to happen. 00:56:09.460 --> 00:56:14.340 So what I'm going to do is now instead of only discussing 00:56:14.340 --> 00:56:16.560 about the plane wave-- 00:56:16.560 --> 00:56:18.940 what I'm going to do is that I would 00:56:18.940 --> 00:56:23.170 like to add a perfect conductor into the game 00:56:23.170 --> 00:56:26.500 and see what is going to happen. 00:56:26.500 --> 00:56:31.056 So what do I mean by a perfect conductor? 00:56:31.056 --> 00:56:34.531 A perfect conductor can be seen in a musical, 00:56:34.531 --> 00:56:35.280 like in a concert. 00:56:35.280 --> 00:56:36.030 [LAUGHTER] 00:56:36.030 --> 00:56:38.280 But the one which I am talking about 00:56:38.280 --> 00:56:41.920 is not that one, which is also fascinating, 00:56:41.920 --> 00:56:45.090 but this is a different system. 00:56:45.090 --> 00:56:48.870 The interesting thing is that both the conductors 00:56:48.870 --> 00:56:52.670 in the concert and this one is very busy. 00:56:52.670 --> 00:56:54.960 It's a very busy system. 00:56:54.960 --> 00:56:58.480 What do I mean by perfect conductor? 00:56:58.480 --> 00:57:02.820 That means all the little charges inside the conductor 00:57:02.820 --> 00:57:04.650 can move freely. 00:57:04.650 --> 00:57:08.820 So if they move they don't actually cause any energy. 00:57:08.820 --> 00:57:10.290 They can move around-- 00:57:10.290 --> 00:57:13.410 all the electrons inside the conductor 00:57:13.410 --> 00:57:19.240 can be moved freely without costing anything, without any 00:57:19.240 --> 00:57:21.520 of this energy dissipation. 00:57:21.520 --> 00:57:26.040 So that's actually what I mean by perfect conductor. 00:57:26.040 --> 00:57:30.090 What do I mean by a very busy system? 00:57:30.090 --> 00:57:34.690 That means whenever there are any distortion 00:57:34.690 --> 00:57:36.540 on the electric field-- 00:57:36.540 --> 00:57:40.980 any electric field approaching to this conductor-- 00:57:40.980 --> 00:57:43.360 what is going to happen is that this conductor will, 00:57:43.360 --> 00:57:45.134 oh, this is electric field, so I have 00:57:45.134 --> 00:57:46.550 to move from some of my electrons. 00:57:46.550 --> 00:57:50.280 Then it's going to cancel all the electric field 00:57:50.280 --> 00:57:53.220 inside the conductor because it cost nothing. 00:57:53.220 --> 00:57:55.290 So you have fast-- 00:57:55.290 --> 00:57:59.340 really fast the react to this change in the electric field 00:57:59.340 --> 00:58:02.130 and they really carefully arrange all the electrons. 00:58:02.130 --> 00:58:05.730 And so that the electric field is canceled. 00:58:05.730 --> 00:58:09.150 Otherwise, all those electrons will continue to move around 00:58:09.150 --> 00:58:10.910 until this happens-- 00:58:10.910 --> 00:58:13.290 this cancellation happens. 00:58:13.290 --> 00:58:16.830 So that's actually what I mean by a busy world 00:58:16.830 --> 00:58:22.220 and what I mean by a perfect conductor. 00:58:22.220 --> 00:58:27.490 If I put this conductor into game, what is going to happen? 00:58:27.490 --> 00:58:33.170 What is going to happen is that if I consider a situation-- 00:58:33.170 --> 00:58:39.050 if I have my x's defined here pointing up to be the x-axis, 00:58:39.050 --> 00:58:42.890 pointing to the right to the z-axis, pointing to the-- 00:58:42.890 --> 00:58:46.010 pointing toward you is actually the y-axis. 00:58:46.010 --> 00:58:50.500 So I can now again take the plane wave 00:58:50.500 --> 00:58:53.210 which I started with. 00:58:53.210 --> 00:58:55.700 There will be a plane wave like this. 00:58:55.700 --> 00:59:03.280 And it's going toward a piece of perfect conductor. 00:59:03.280 --> 00:59:05.590 What is going to happen is that as I actually 00:59:05.590 --> 00:59:11.320 mentioned before there are many charges all over the place. 00:59:11.320 --> 00:59:15.370 They are going to quickly rearrange-- 00:59:15.370 --> 00:59:18.920 all those charges to cancel the electric field. 00:59:18.920 --> 00:59:22.090 So if you have a plane wave going 00:59:22.090 --> 00:59:26.110 toward the perfect conductor at the surface 00:59:26.110 --> 00:59:27.900 of the perfect conductor-- 00:59:27.900 --> 00:59:30.880 the electric field will become 0. 00:59:33.630 --> 00:59:37.980 But if you have only one plane wave it cannot be-- 00:59:37.980 --> 00:59:41.680 the magnitude cannot be equal to 0 because I know the functional 00:59:41.680 --> 00:59:42.180 form. 00:59:42.180 --> 00:59:45.420 I know that the functional form of that electric field 00:59:45.420 --> 00:59:49.260 is E0 cosine kz minus omega t. 00:59:49.260 --> 00:59:55.760 If I place this perfect conductor at Z equal to 0, 00:59:55.760 --> 00:59:58.470 then I can evaluate the electric field 00:59:58.470 --> 01:00:01.740 is not equal to 0 because it is actually equal to E0 01:00:01.740 --> 01:00:04.890 cosine minus omega t. 01:00:04.890 --> 01:00:09.480 So what can I do to cancel the electric field? 01:00:09.480 --> 01:00:13.170 This is actually very similar to the situation 01:00:13.170 --> 01:00:22.640 when you have a progressing wave on this string hitting a wall. 01:00:22.640 --> 01:00:27.910 Because the magnitude of the string 01:00:27.910 --> 01:00:30.800 which is fed to the equilibrium position 01:00:30.800 --> 01:00:33.650 is actually equal to 0. 01:00:33.650 --> 01:00:35.260 That's actually what we have learned 01:00:35.260 --> 01:00:37.100 in the last few lectures. 01:00:37.100 --> 01:00:40.400 And this is actually exactly the same situation, right? 01:00:40.400 --> 01:00:42.430 You have a progressing wave. 01:00:42.430 --> 01:00:46.670 And there is some kind of boundary, which is actually 01:00:46.670 --> 01:00:51.380 when this progressing plane wave encounter 01:00:51.380 --> 01:00:53.300 this perfect conductor. 01:00:53.300 --> 01:00:55.640 There-- the electric field-- 01:00:55.640 --> 01:01:11.170 the boundary condition-- has to be E is actually-- 01:01:17.070 --> 01:01:22.180 E, x, y, 0, which is actually the position of the z 01:01:22.180 --> 01:01:24.510 of the perfect conductor. 01:01:24.510 --> 01:01:29.130 As a function of time will be equal to 0. 01:01:29.130 --> 01:01:35.100 The whole plane will have 0 electric field. 01:01:35.100 --> 01:01:39.660 So that means there must be what kind of wave? 01:01:39.660 --> 01:01:45.770 There must be a reflective wave because of the presence 01:01:45.770 --> 01:01:48.170 of the perfect conductor. 01:01:48.170 --> 01:01:50.480 It's actually similar to the situation 01:01:50.480 --> 01:01:54.170 which we discussed there's a progressing wave hitting 01:01:54.170 --> 01:01:55.130 the wall. 01:01:55.130 --> 01:01:58.130 And this string wall system-- there 01:01:58.130 --> 01:02:01.320 will be a reflecting wave coming out of it. 01:02:01.320 --> 01:02:05.030 So therefore, what we are expecting is some kind of-- 01:02:08.940 --> 01:02:11.910 refracting wave which actually cancel 01:02:11.910 --> 01:02:16.980 the magnitude of the electric field at Z equal to 0. 01:02:16.980 --> 01:02:20.600 And then this progressing wave is going to the left-hand side 01:02:20.600 --> 01:02:22.140 direction. 01:02:22.140 --> 01:02:25.460 So now, I can actually write down the incident wave-- 01:02:29.170 --> 01:02:29.950 expression. 01:02:29.950 --> 01:02:32.440 The incident wave-- 01:02:32.440 --> 01:02:38.202 I call it Ei, this is Ei-- 01:02:38.202 --> 01:02:48.810 is expressed as E0 over 2, cosine kz minus omega t. 01:02:48.810 --> 01:02:52.280 This is actually what I putting to the system. 01:02:52.280 --> 01:02:57.030 The magnitude is E0 over 2, and it's actually 01:02:57.030 --> 01:03:00.440 propagating toward the z direction, 01:03:00.440 --> 01:03:02.010 as you can see from here. 01:03:02.010 --> 01:03:05.760 And that the direction of the electric field 01:03:05.760 --> 01:03:08.730 is in the x direction. 01:03:08.730 --> 01:03:11.490 And of course the E field will have 01:03:11.490 --> 01:03:15.840 a corresponding magnetic field, which is actually-- 01:03:15.840 --> 01:03:20.880 you can actually write it down directly using this formula-- 01:03:20.880 --> 01:03:26.040 B equal to 1 over c, K cross E. K here is z, 01:03:26.040 --> 01:03:28.500 therefore you can quickly evaluate 01:03:28.500 --> 01:03:32.490 and then conclude that the magnetic field 01:03:32.490 --> 01:03:35.900 must be in the y direction. 01:03:35.900 --> 01:03:38.910 And that the magnitude of the magnetic field 01:03:38.910 --> 01:03:43.270 would be E0 divided by 2 c. 01:03:43.270 --> 01:03:48.960 Cosine Kz and this omega t. 01:03:48.960 --> 01:03:51.960 So that is actually the incident wave. 01:03:51.960 --> 01:03:55.200 And of course I also need, as I discussed, 01:03:55.200 --> 01:04:00.900 there must be a reflective wave, Er, 01:04:00.900 --> 01:04:07.020 which you actually cancel the electric field at z equal to 0. 01:04:07.020 --> 01:04:09.810 If that cancels the incident wave, 01:04:09.810 --> 01:04:14.180 that means the magnitude must be in the opposite direction 01:04:14.180 --> 01:04:15.870 of the incident wave. 01:04:15.870 --> 01:04:17.850 Therefore, I can quickly write down 01:04:17.850 --> 01:04:22.140 what would be the resulting reflective wave that 01:04:22.140 --> 01:04:27.130 would be equal to minus E0 over 2, 01:04:27.130 --> 01:04:35.760 cosine minus Kz, minus omega t in the x direction. 01:04:35.760 --> 01:04:38.340 And then the corresponding B field, 01:04:38.340 --> 01:04:43.500 I can also write it down using exactly the same formula. 01:04:43.500 --> 01:04:46.110 And basically what I conclude is that this 01:04:46.110 --> 01:04:53.830 will be equal to E0 over 2 c, cosine minus kz, 01:04:53.830 --> 01:04:58.500 minus omega t in the y direction. 01:04:58.500 --> 01:05:05.922 So you can actually check this expression after the direction. 01:05:08.630 --> 01:05:11.840 So now, I would like to check what 01:05:11.840 --> 01:05:16.310 would be the magnitude of the electric field at z equal to 0. 01:05:16.310 --> 01:05:18.740 So basically, at z equal to 0, you 01:05:18.740 --> 01:05:21.350 have something which is proportional to cosine 01:05:21.350 --> 01:05:24.140 minus omega t for the incident wave. 01:05:24.140 --> 01:05:27.890 And then the magnitude is E0 over 2. 01:05:27.890 --> 01:05:33.900 And if you evaluate z equal to 0, basically you get minus E0 01:05:33.900 --> 01:05:37.550 over 2 cosine minus omega t. 01:05:37.550 --> 01:05:40.640 Therefore, they really cancel and give you 01:05:40.640 --> 01:05:43.820 the desired boundary condition, which 01:05:43.820 --> 01:05:46.390 is actually E equal to 0 and the surface 01:05:46.390 --> 01:05:49.010 of the perfect conductor. 01:05:49.010 --> 01:05:51.530 So that's very nice. 01:05:51.530 --> 01:05:54.920 And this is actually the physics of which we already learned 01:05:54.920 --> 01:05:58.070 from this string wall system. 01:05:58.070 --> 01:06:01.130 So what I can do now is to calculate 01:06:01.130 --> 01:06:04.900 the total electric field if I add them together. 01:06:04.900 --> 01:06:07.190 Basically, I would get E-- 01:06:07.190 --> 01:06:11.040 total electric field, which is actually 01:06:11.040 --> 01:06:14.180 overlapping the incident and the reflective wave. 01:06:14.180 --> 01:06:21.920 What I am going to get is Ei plus Er. 01:06:21.920 --> 01:06:28.230 And basically, what I get is E0 over 2 because the incident 01:06:28.230 --> 01:06:31.110 wave and the reflective wave of the electric field 01:06:31.110 --> 01:06:33.320 is always in the x direction. 01:06:33.320 --> 01:06:37.070 Therefore, I only need to take care of the x direction. 01:06:37.070 --> 01:06:45.620 So basically, I have cosine kz minus omega t, minus-- 01:06:45.620 --> 01:06:47.480 right, because there's a minus sign here-- 01:06:47.480 --> 01:06:55.597 minus cosine minus kz, minus omega t in the x direction. 01:06:55.597 --> 01:06:58.520 And there should be-- 01:06:58.520 --> 01:07:02.570 And of course this is a cosine minus cosine. 01:07:02.570 --> 01:07:04.220 So we have all the formulas-- one 01:07:04.220 --> 01:07:07.820 from, for example, Wikipedia, or from your textbook. 01:07:07.820 --> 01:07:10.610 So you can actually calculate this-- 01:07:10.610 --> 01:07:22.280 rewrite this expression to be E0 sine omega t, sine kz. 01:07:22.280 --> 01:07:25.551 And then this is actually in the x direction. 01:07:28.150 --> 01:07:29.950 Everybody's following? 01:07:29.950 --> 01:07:32.860 I hope it's not too fast. 01:07:32.860 --> 01:07:34.570 All right, and of course, I can also 01:07:34.570 --> 01:07:37.690 calculate the corresponding B field. 01:07:37.690 --> 01:07:41.725 So it's actually again, exactly the same thing-- 01:07:41.725 --> 01:07:45.340 Bi plus Br. 01:07:45.340 --> 01:07:48.230 And basically, I will skip the step. 01:07:48.230 --> 01:07:52.990 Basically, you can add this term and that term. 01:07:52.990 --> 01:07:56.140 And you will be able to conclude that the B field will 01:07:56.140 --> 01:08:02.280 be equal to E0 over c cosine omega t, 01:08:02.280 --> 01:08:07.388 cosine kz in the y direction. 01:08:11.750 --> 01:08:15.800 This is actually pretty interesting. 01:08:15.800 --> 01:08:20.779 If you look at this result, I have a electric field, 01:08:20.779 --> 01:08:24.920 which is proportional to E0, the magnitude, sine omega 01:08:24.920 --> 01:08:28.720 t, and sine kz. 01:08:28.720 --> 01:08:30.470 What does that I mean? 01:08:30.470 --> 01:08:34.160 This is a special kind of wave which we learned before. 01:08:34.160 --> 01:08:36.416 What kind of wave is this? 01:08:36.416 --> 01:08:37.490 AUDIENCE: Standing. 01:08:37.490 --> 01:08:39.740 YEN-JIE LEE: It's a standing wave 01:08:39.740 --> 01:08:44.510 because the shape is actually fixed, the sine kz. 01:08:44.510 --> 01:08:48.279 And the magnitude is actually changing up and down 01:08:48.279 --> 01:08:50.246 at the angle frequency omega t. 01:08:50.246 --> 01:08:51.120 It's a standing wave. 01:08:58.300 --> 01:09:03.840 Another thing which is really interesting is that if we look 01:09:03.840 --> 01:09:08.430 at the expression of a electric field and the magnetic field-- 01:09:08.430 --> 01:09:09.460 if we compare that-- 01:09:09.460 --> 01:09:10.950 one is actually sine, sine. 01:09:10.950 --> 01:09:12.870 The other one is cosine, cosine. 01:09:15.770 --> 01:09:19.240 That's kind of interesting because this is actually 01:09:19.240 --> 01:09:22.910 different from what we actually usually learn 01:09:22.910 --> 01:09:25.850 from the progressing wave solution, 01:09:25.850 --> 01:09:27.350 or traveling wave solution. 01:09:27.350 --> 01:09:34.160 Where the electric field and the magnetic field are in phase. 01:09:34.160 --> 01:09:35.540 There's no phase difference. 01:09:35.540 --> 01:09:39.529 In the case of the superposition of the incident 01:09:39.529 --> 01:09:41.540 and the reflective wave-- 01:09:41.540 --> 01:09:44.960 the solution of a standing wave-- 01:09:44.960 --> 01:09:50.959 actually you can see that the phase of the B field and the E 01:09:50.959 --> 01:09:55.090 field are different. 01:09:55.090 --> 01:10:00.220 Finally, very important-- you will see that-- 01:10:00.220 --> 01:10:01.780 look at this expression-- 01:10:01.780 --> 01:10:04.420 B equal to 1/c, K cross E-- 01:10:07.020 --> 01:10:11.860 that means this only work for traveling wave. 01:10:11.860 --> 01:10:17.860 Clearly, this doesn't work for standing waves. 01:10:17.860 --> 01:10:19.010 So very important. 01:10:19.010 --> 01:10:23.180 So don't blindly apply this expression. 01:10:23.180 --> 01:10:26.500 This is only useful for the traveling wave solution. 01:10:26.500 --> 01:10:31.470 And you can see a very concrete example here. 01:10:31.470 --> 01:10:35.980 This doesn't work for standing waves. 01:10:35.980 --> 01:10:38.080 That's kind of interesting. 01:10:38.080 --> 01:10:41.760 And if you look at this result, you 01:10:41.760 --> 01:10:46.190 will see that if I don't have magnetic field-- 01:10:46.190 --> 01:10:48.810 if I only have the electric field-- 01:10:48.810 --> 01:10:53.590 there will be a instant of time, for example, t equal to 0. 01:10:53.590 --> 01:10:57.160 When t is equal to 0, sine is equal to 0. 01:10:57.160 --> 01:10:58.660 What is going to happen? 01:10:58.660 --> 01:11:00.810 You will have no electric field. 01:11:03.640 --> 01:11:06.220 That means electric field completely 01:11:06.220 --> 01:11:08.170 disappear because we are operating 01:11:08.170 --> 01:11:10.540 this system in vacuum. 01:11:10.540 --> 01:11:12.180 There's nowhere to hide. 01:11:12.180 --> 01:11:14.950 Where is the energy? 01:11:14.950 --> 01:11:19.630 The energy, fortunately-- electric field 01:11:19.630 --> 01:11:23.840 have a very good partner, which is actually B field. 01:11:23.840 --> 01:11:33.320 All the energy's actually stored in the form of magnetic field. 01:11:33.320 --> 01:11:39.220 You can see that now magnetic field is reaching the maximum. 01:11:39.220 --> 01:11:43.428 So of course I can now calculate the Poynting vector. 01:11:50.680 --> 01:11:57.070 Poynting vector is E cross B divided by mu zero. 01:11:57.070 --> 01:12:02.860 And these will be equal to one over Mu 0, Ex, By, 01:12:02.860 --> 01:12:04.300 and the z direction. 01:12:04.300 --> 01:12:06.580 There's only one term which survive. 01:12:06.580 --> 01:12:09.700 So Poynting vector is not pointing vector. 01:12:09.700 --> 01:12:12.100 It's not pointing around. 01:12:12.100 --> 01:12:14.500 There's a gentleman who is called Poynting 01:12:14.500 --> 01:12:15.970 and he has a vector. 01:12:15.970 --> 01:12:21.530 And this vector is a directional energy flux. 01:12:21.530 --> 01:12:26.415 It's a directional energy flux, or the rate of energy transfer 01:12:26.415 --> 01:12:31.390 per unit area. 01:12:31.390 --> 01:12:35.460 So that is actually the meaning of Poynting vector. 01:12:35.460 --> 01:12:39.050 And then each magnitude is proportional to E cross 01:12:39.050 --> 01:12:42.280 B divided by Mu 0. 01:12:42.280 --> 01:12:45.100 So I can calculate that. 01:12:45.100 --> 01:12:47.330 Basically, I have the E and the B-- 01:12:47.330 --> 01:12:50.080 Ex and By, then I can calculate. 01:12:50.080 --> 01:12:54.620 That would be equal to E0 square, over Mu 0, 01:12:54.620 --> 01:13:02.080 sine omega t, cosine omega t, cosine Kz, 01:13:02.080 --> 01:13:09.540 sine Kz in the z direction because I have x cross y. 01:13:09.540 --> 01:13:12.400 And I'm going to get the z direction. 01:13:12.400 --> 01:13:14.230 And I can simplify this. 01:13:14.230 --> 01:13:17.540 I have the sine, cosine. 01:13:17.540 --> 01:13:20.290 And also all have cosine, and sine. 01:13:20.290 --> 01:13:23.320 Basically, you can simplify this expression 01:13:23.320 --> 01:13:31.870 and get E0 squared divided by 4, Mu 0c sine 2 omega t, 01:13:31.870 --> 01:13:37.270 sine 2kz in the z direction. 01:13:37.270 --> 01:13:43.090 So you can see that the directional energy 01:13:43.090 --> 01:13:46.050 flux is in the z direction. 01:13:50.090 --> 01:13:53.520 It has a vector-- 01:13:53.520 --> 01:13:58.310 it has a wave number 2 times of the original wave number. 01:13:58.310 --> 01:14:02.855 And it's actually going up and down 2 times 01:14:02.855 --> 01:14:07.730 of the speed of the oscillation of the original electromagnetic 01:14:07.730 --> 01:14:09.770 wave. 01:14:09.770 --> 01:14:14.720 And this energy is actually vibrating up and down. 01:14:14.720 --> 01:14:20.080 And the shape of this energy transfer Poynting vector 01:14:20.080 --> 01:14:23.600 is actually a sine wave. 01:14:23.600 --> 01:14:28.710 So that this is actually how the microwave actually works. 01:14:28.710 --> 01:14:30.530 So basically, what we are doing is 01:14:30.530 --> 01:14:37.360 to have generate microwave inside your device. 01:14:37.360 --> 01:14:39.680 And in the oven this microwave is actually 01:14:39.680 --> 01:14:42.080 bouncing back and forth because you 01:14:42.080 --> 01:14:45.680 have metal walls, which actually bounce 01:14:45.680 --> 01:14:48.380 the electromagnetic field back and forth. 01:14:48.380 --> 01:14:55.400 And it really can cook the food by vibrating the molecules 01:14:55.400 --> 01:14:58.330 inside the food back and forth. 01:14:58.330 --> 01:15:02.810 So as you can see the magnitude of the Poynting vector 01:15:02.810 --> 01:15:06.340 is actually isolating up and down. 01:15:06.340 --> 01:15:09.380 That actually cause additional vibration and that heat up 01:15:09.380 --> 01:15:10.290 the food. 01:15:10.290 --> 01:15:12.350 So after this lecture, you will be 01:15:12.350 --> 01:15:15.590 able to say proudly that you understand 01:15:15.590 --> 01:15:17.450 the physics of microwave oven. 01:15:17.450 --> 01:15:19.970 [LAUGHTER] 01:15:19.970 --> 01:15:21.000 Thank you very much. 01:15:21.000 --> 01:15:23.030 I hope you enjoyed the lecture today. 01:15:23.030 --> 01:15:26.230 And you have any questions, I will be here.