1 00:00:02,550 --> 00:00:04,920 The following content is provided under a Creative 2 00:00:04,920 --> 00:00:06,310 Commons license. 3 00:00:06,310 --> 00:00:08,520 Your support will help MIT OpenCourseWare 4 00:00:08,520 --> 00:00:12,610 continue to offer high quality educational resources for free. 5 00:00:12,610 --> 00:00:15,150 To make a donation or to view additional materials 6 00:00:15,150 --> 00:00:19,110 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,110 --> 00:00:20,154 at ocw.mit.edu. 8 00:00:23,880 --> 00:00:27,720 YEN-JIE LEE: Welcome back, everybody to 8.03. 9 00:00:27,720 --> 00:00:31,650 So today we are going to continue discussions 10 00:00:31,650 --> 00:00:40,170 on the examples which we started the last time, sound waves, 11 00:00:40,170 --> 00:00:42,540 and, this time EM waves, which can 12 00:00:42,540 --> 00:00:45,000 be described by wave equations. 13 00:00:47,550 --> 00:00:50,190 So far, what we have learned is that there 14 00:00:50,190 --> 00:00:52,870 are three different kinds of systems we've discussed 15 00:00:52,870 --> 00:00:55,670 in a lecture or in a textbook. 16 00:00:55,670 --> 00:00:59,140 And the first one is actually a string, a very long string, 17 00:00:59,140 --> 00:01:05,550 system with constant tension and mass on the string. 18 00:01:05,550 --> 00:01:08,640 And the behavior of the string obey wave 19 00:01:08,640 --> 00:01:11,775 equation, and can be described by a wave equation. 20 00:01:14,400 --> 00:01:19,680 We also can produce a density wave with a spring. 21 00:01:19,680 --> 00:01:22,620 And basically the density wave or spring 22 00:01:22,620 --> 00:01:26,670 can also be described by wave equations. 23 00:01:26,670 --> 00:01:29,640 So that's as you described in the textbook. 24 00:01:29,640 --> 00:01:35,400 Finally, last time we actually discussed sound waves. 25 00:01:35,400 --> 00:01:37,980 We have an open pipe, and then we 26 00:01:37,980 --> 00:01:39,450 can have air inside the pipe. 27 00:01:39,450 --> 00:01:43,830 And the behavior of the air, or the molecules 28 00:01:43,830 --> 00:01:48,310 inside the open pipe, can be described by wave equations. 29 00:01:48,310 --> 00:01:52,410 Crashes So what we are going to do today 30 00:01:52,410 --> 00:01:55,200 is to discuss with you a special kind of wave, 31 00:01:55,200 --> 00:01:57,400 which is electromagnetic waves. 32 00:01:57,400 --> 00:01:59,400 And that's actually slightly different from what 33 00:01:59,400 --> 00:02:02,580 we have learned in the last few lectures. 34 00:02:02,580 --> 00:02:08,419 And we see what this is different today in the lecture. 35 00:02:08,419 --> 00:02:09,440 All right. 36 00:02:09,440 --> 00:02:11,600 So this essentially is a reminder 37 00:02:11,600 --> 00:02:14,030 of Maxwell's equations. 38 00:02:14,030 --> 00:02:15,605 So basically what is written here 39 00:02:15,605 --> 00:02:22,080 is the differential form of Maxwell's equations. 40 00:02:22,080 --> 00:02:24,670 So the first law is Gauss law. 41 00:02:24,670 --> 00:02:29,570 It says should the divergence of e, the electric field, 42 00:02:29,570 --> 00:02:33,200 is equal to rho, which is the charge 43 00:02:33,200 --> 00:02:38,300 density as a specific point, divided by epsilon zero, 44 00:02:38,300 --> 00:02:41,240 which is actually a constant. 45 00:02:41,240 --> 00:02:45,960 We'll call it permittivity of this constant. 46 00:02:45,960 --> 00:02:46,460 OK? 47 00:02:46,460 --> 00:02:49,970 Which should relay the divergence of e 48 00:02:49,970 --> 00:02:53,910 and the density of the charge at this specific point. 49 00:02:53,910 --> 00:02:59,310 And the second law is actually Gauss law for magnetism. 50 00:02:59,310 --> 00:03:03,530 This is actually the divergence of b equal to 0. 51 00:03:03,530 --> 00:03:08,420 So divergence b is always equal to zero because we haven't yet 52 00:03:08,420 --> 00:03:13,060 discovered the magnetic monopole yet. 53 00:03:13,060 --> 00:03:13,610 Right? 54 00:03:13,610 --> 00:03:19,010 So maybe you have discovered it one time, at some time, 55 00:03:19,010 --> 00:03:20,150 in your experiment. 56 00:03:20,150 --> 00:03:22,847 Please tell me now. 57 00:03:22,847 --> 00:03:25,055 I want to be the first with who knows how to do that. 58 00:03:25,055 --> 00:03:26,960 [LAUGHS] All right? 59 00:03:26,960 --> 00:03:29,510 So promise me. 60 00:03:29,510 --> 00:03:33,350 The third one is Faraday's law. 61 00:03:33,350 --> 00:03:41,590 It's curve of e equal to minus partial e partial t and the b, 62 00:03:41,590 --> 00:03:45,970 as a reminder, is a magnetic field vector. 63 00:03:45,970 --> 00:03:49,820 And in the last law is actually Ampere's law. 64 00:03:49,820 --> 00:03:54,330 It's actually the curve of b equal to mu 0. 65 00:03:54,330 --> 00:03:58,580 Mu 0 is actually a constant, permeability. 66 00:03:58,580 --> 00:04:02,710 Which would lay the current and displacement current. 67 00:04:02,710 --> 00:04:07,640 Epsilon 0, partial e, partial t, to the curve of b. 68 00:04:07,640 --> 00:04:08,750 OK? 69 00:04:08,750 --> 00:04:13,550 And I would like to draw your attention to these term. 70 00:04:13,550 --> 00:04:18,100 This very important term is actually Maxwell's addition. 71 00:04:18,100 --> 00:04:18,620 OK? 72 00:04:18,620 --> 00:04:21,290 Without Maxwell's addition, there 73 00:04:21,290 --> 00:04:23,800 would be no electromagnetic wave. 74 00:04:23,800 --> 00:04:25,280 Then you could not see me. 75 00:04:25,280 --> 00:04:26,225 OK? 76 00:04:26,225 --> 00:04:29,250 [LAUGHS] All right. 77 00:04:29,250 --> 00:04:32,460 So, what we are going to discuss today 78 00:04:32,460 --> 00:04:36,520 is a simpler case at the beginning. 79 00:04:36,520 --> 00:04:41,250 So what will happen if we go to a vacuum? 80 00:04:41,250 --> 00:04:47,040 Going to vacuum means there will be no material charges floating 81 00:04:47,040 --> 00:04:51,470 around, and that means rho will be equal to 0. 82 00:04:51,470 --> 00:04:56,700 Therefore, the divergence of e will be equal to 0. 83 00:04:56,700 --> 00:05:02,850 And also in the last question Ampere's law, 84 00:05:02,850 --> 00:05:07,170 say which is that the current density will be equal to 0. 85 00:05:07,170 --> 00:05:11,370 Therefore, the function of curl of b 86 00:05:11,370 --> 00:05:15,870 equal to mu 0, epsilon 0, partial e, partial t. 87 00:05:15,870 --> 00:05:17,430 OK? 88 00:05:17,430 --> 00:05:21,210 So before we go into the discussion 89 00:05:21,210 --> 00:05:24,930 of Maxwell's equation's implication, 90 00:05:24,930 --> 00:05:30,630 I would like to remind you about some mathematics which 91 00:05:30,630 --> 00:05:34,270 will be used in this lecture. 92 00:05:34,270 --> 00:05:41,220 I hope you have seen this in other courses or 8.02. 93 00:05:41,220 --> 00:05:47,640 So as you can see, we use del here, which is a vector. 94 00:05:47,640 --> 00:05:51,850 This vector is defined as partial x, partial y, 95 00:05:51,850 --> 00:05:55,650 and partial z, in the x, y, and z direction. 96 00:05:55,650 --> 00:05:56,160 OK? 97 00:05:56,160 --> 00:06:00,540 So this is actually some kind of operator. 98 00:06:00,540 --> 00:06:02,470 You see that again. 99 00:06:02,470 --> 00:06:06,420 A lot more operators in 8.04. 100 00:06:06,420 --> 00:06:13,230 And we make this definition because I'm lazy. 101 00:06:13,230 --> 00:06:16,070 Because I don't want to write so many partial, partial x, 102 00:06:16,070 --> 00:06:20,170 partial, partial y, partial, partial z again and again. 103 00:06:20,170 --> 00:06:24,480 Therefore we define del, which is like this. 104 00:06:24,480 --> 00:06:26,950 Looks really crazy, but it really 105 00:06:26,950 --> 00:06:29,080 makes our lives much easier. 106 00:06:29,080 --> 00:06:29,670 OK? 107 00:06:29,670 --> 00:06:31,020 So that's the whole reason. 108 00:06:31,020 --> 00:06:33,420 As a physicist. 109 00:06:33,420 --> 00:06:39,090 And as we discussed before, we have divergence, 110 00:06:39,090 --> 00:06:44,820 which is defined here: del times a, both of them are vectors. 111 00:06:44,820 --> 00:06:46,900 Y is the operator vector, the other y 112 00:06:46,900 --> 00:06:50,790 is actually really a vector. 113 00:06:50,790 --> 00:06:54,920 You basically get partial ax, partial x, plus partial ay, 114 00:06:54,920 --> 00:06:57,870 partial y, plus partial az, partial z. 115 00:06:57,870 --> 00:07:02,740 So basically you just multiply them like a normal operation. 116 00:07:02,740 --> 00:07:06,540 And you can actually get this question. 117 00:07:06,540 --> 00:07:07,360 OK. 118 00:07:07,360 --> 00:07:10,080 Then finally there's curl. 119 00:07:10,080 --> 00:07:17,280 Curl is actually del cross a. 120 00:07:17,280 --> 00:07:20,040 So basically, maybe in the past you 121 00:07:20,040 --> 00:07:22,530 see this complicated formula. 122 00:07:22,530 --> 00:07:25,530 You know, it had maybe no meaning to you. 123 00:07:25,530 --> 00:07:28,590 And one easy way to remember this curl 124 00:07:28,590 --> 00:07:31,500 is to not care about this. 125 00:07:31,500 --> 00:07:34,350 Don't look at the right hand side part. 126 00:07:34,350 --> 00:07:38,730 But just remember that you can actually construct this curl 127 00:07:38,730 --> 00:07:41,740 by determining a matrix. 128 00:07:41,740 --> 00:07:45,090 In the matrix, I can fill the first row 129 00:07:45,090 --> 00:07:47,580 by x, y, and z unit vector. 130 00:07:47,580 --> 00:07:52,390 And the second row, I filled it with the counting of del. 131 00:07:52,390 --> 00:07:58,260 Finally I feel the content of the matrix with a vector. 132 00:07:58,260 --> 00:08:00,570 Then you will be able to calculate 133 00:08:00,570 --> 00:08:02,610 the determine of this back matrix, 134 00:08:02,610 --> 00:08:07,260 then you naturally would get this very long formula. 135 00:08:07,260 --> 00:08:09,780 So you don't really need to remember the formula, 136 00:08:09,780 --> 00:08:14,840 but you will be able to know how to calculate it really easily. 137 00:08:14,840 --> 00:08:17,260 OK? 138 00:08:17,260 --> 00:08:18,120 OK. 139 00:08:18,120 --> 00:08:20,910 So we talk about divergence. 140 00:08:20,910 --> 00:08:24,560 We talk about curl. 141 00:08:24,560 --> 00:08:27,680 What does that mean? 142 00:08:27,680 --> 00:08:30,710 Divergence, curl, what does that mean? 143 00:08:30,710 --> 00:08:35,330 So divergence is actually some kind of measure 144 00:08:35,330 --> 00:08:38,690 which measures how much the vector 145 00:08:38,690 --> 00:08:46,430 v spreads out, or diverges, from a point of interest. 146 00:08:46,430 --> 00:08:52,580 So in this example, this vector field-- 147 00:08:52,580 --> 00:08:59,960 vector field means at any point in the space which 148 00:08:59,960 --> 00:09:04,340 I am discussing, there is a vector associated with that. 149 00:09:04,340 --> 00:09:06,110 I call it vector field. 150 00:09:06,110 --> 00:09:08,360 We know scalar field very much. 151 00:09:08,360 --> 00:09:12,740 For example, the temperature as a function of position 152 00:09:12,740 --> 00:09:14,000 is a scalar field, right? 153 00:09:14,000 --> 00:09:20,390 So, every point you have scalar corresponding to that point. 154 00:09:20,390 --> 00:09:23,270 And in the case vector field, every point 155 00:09:23,270 --> 00:09:26,900 you have a vector connected to that point. 156 00:09:26,900 --> 00:09:31,160 And if I arrange the vector field like that, 157 00:09:31,160 --> 00:09:34,360 each arrow is actually a straight dimensional 158 00:09:34,360 --> 00:09:36,200 the vector. 159 00:09:36,200 --> 00:09:39,320 Then if I evaluate the divergence 160 00:09:39,320 --> 00:09:42,770 and you see the heart, it looks like something is really 161 00:09:42,770 --> 00:09:46,310 spreading out from the center of that graph. 162 00:09:46,310 --> 00:09:50,660 And that will give you positive divergence. 163 00:09:50,660 --> 00:09:51,320 OK? 164 00:09:51,320 --> 00:09:55,520 So that's the physical meaning of this formula. 165 00:09:55,520 --> 00:10:00,060 And the second formula which we discussed is the curl. 166 00:10:00,060 --> 00:10:04,470 So curl is actually del cross a. 167 00:10:04,470 --> 00:10:08,540 It's a measurement of how things are curling 168 00:10:08,540 --> 00:10:10,760 around a point of interest. 169 00:10:10,760 --> 00:10:11,720 OK? 170 00:10:11,720 --> 00:10:18,170 So you can see that if I arrange my vectors 171 00:10:18,170 --> 00:10:20,890 in a space like that, then you will 172 00:10:20,890 --> 00:10:26,480 see that something is really rotating 173 00:10:26,480 --> 00:10:28,310 around that specific point. 174 00:10:28,310 --> 00:10:30,770 Therefore, if we evaluate the curl, 175 00:10:30,770 --> 00:10:33,290 you would get the nonzero value. 176 00:10:33,290 --> 00:10:37,992 So that's actually the physics intuition which we can 177 00:10:37,992 --> 00:10:41,300 or the mathematics intuition which 178 00:10:41,300 --> 00:10:47,630 we can actually get before the discussion of Maxwell's 179 00:10:47,630 --> 00:10:49,740 equations. 180 00:10:49,740 --> 00:10:53,540 So if you accept those ideas, let's 181 00:10:53,540 --> 00:10:57,410 take a look at what we have here, especially 182 00:10:57,410 --> 00:10:59,210 in the vacuum case. 183 00:10:59,210 --> 00:11:06,530 OK, so in a vacuum case, you have a curl of e equal to minus 184 00:11:06,530 --> 00:11:09,050 partial p, partial t. 185 00:11:09,050 --> 00:11:10,560 What does that mean? 186 00:11:10,560 --> 00:11:13,640 That means, if you change the magnitude 187 00:11:13,640 --> 00:11:27,730 of the magnetic field, now we introduce a curling 188 00:11:27,730 --> 00:11:30,980 around thing in the e field. 189 00:11:30,980 --> 00:11:33,830 So if you change the size of the b field, 190 00:11:33,830 --> 00:11:39,200 then the e field will start to curl around, doing this. 191 00:11:39,200 --> 00:11:44,400 And on the other hand, if you change the electric field, 192 00:11:44,400 --> 00:11:47,410 that will do something, which is curling around in the b field. 193 00:11:50,100 --> 00:11:53,220 All right, so do you have any questions? 194 00:11:53,220 --> 00:11:56,450 I hope everybody's familiar with this notation. 195 00:11:56,450 --> 00:12:05,099 So from here actually Maxwell, see the light. 196 00:12:05,099 --> 00:12:08,220 [LAUGHS] Can you see it? 197 00:12:10,770 --> 00:12:11,540 Maybe not yet. 198 00:12:11,540 --> 00:12:17,800 Maybe we are slightly slower than Maxwell, 199 00:12:17,800 --> 00:12:20,710 but we will see that together in this lecture. 200 00:12:23,510 --> 00:12:33,970 For that I would need as usual help from the math department. 201 00:12:33,970 --> 00:12:37,510 So we are going to use this identity. 202 00:12:37,510 --> 00:12:46,595 This identity is curl of curl of a would 203 00:12:46,595 --> 00:13:02,990 be equal to del, divergence of a, minus del dot del, a. 204 00:13:02,990 --> 00:13:05,120 So this is an identity which we learned 205 00:13:05,120 --> 00:13:07,100 from the math department. 206 00:13:07,100 --> 00:13:11,000 And of course, if you are patient enough, 207 00:13:11,000 --> 00:13:13,160 you can actually expand all those terms 208 00:13:13,160 --> 00:13:16,640 and compare the left hand side of the formula and right hand 209 00:13:16,640 --> 00:13:18,020 side of the formula. 210 00:13:18,020 --> 00:13:22,880 And you will see that really this works. 211 00:13:22,880 --> 00:13:26,790 So I'm not going to do that here in front of you. 212 00:13:26,790 --> 00:13:31,760 So if you accept this is an identity, and then usually when 213 00:13:31,760 --> 00:13:35,180 we have del times del, we call it Laplace. 214 00:13:38,290 --> 00:13:41,170 And usually we write it as del squared. 215 00:13:44,310 --> 00:13:49,581 With this formula, I can now put my electric field 216 00:13:49,581 --> 00:13:50,330 into this formula. 217 00:13:55,330 --> 00:14:00,570 assuming that I am working in a situation of a vacuum 218 00:14:00,570 --> 00:14:06,900 and I plug in my electric field into that formula. 219 00:14:06,900 --> 00:14:11,470 Then this is actually what I am going to get. 220 00:14:11,470 --> 00:14:13,780 Curl of e. 221 00:14:13,780 --> 00:14:20,290 And this will be equal to del, divergence of e 222 00:14:20,290 --> 00:14:23,110 minus del squared e. 223 00:14:25,670 --> 00:14:26,170 OK? 224 00:14:28,920 --> 00:14:35,760 And based on the four Maxwell's equations, 225 00:14:35,760 --> 00:14:39,510 we can immediately recognize that divergence of e 226 00:14:39,510 --> 00:14:43,110 is equal to 0, because I don't have charges around. 227 00:14:43,110 --> 00:14:46,950 Therefore you, cannot not introduce a gradient 228 00:14:46,950 --> 00:14:50,670 or divergence. 229 00:14:50,670 --> 00:14:54,290 You can introduce positive divergence 230 00:14:54,290 --> 00:14:56,560 in the electric field. 231 00:14:56,560 --> 00:14:59,400 Therefore, when you evaluate the divergence 232 00:14:59,400 --> 00:15:03,240 of the electric field, that is equal to 0. 233 00:15:03,240 --> 00:15:06,390 According to that formula, Gauss law. 234 00:15:06,390 --> 00:15:11,010 And you can also take a look here. 235 00:15:11,010 --> 00:15:15,360 We have curl of e, according to this formula. 236 00:15:15,360 --> 00:15:18,260 Basically you can conclude that this 237 00:15:18,260 --> 00:15:23,820 will be equal to minus partial b, partial t according 238 00:15:23,820 --> 00:15:25,140 to Faraday's law. 239 00:15:27,950 --> 00:15:34,030 So if I look at the left hand side, 240 00:15:34,030 --> 00:15:45,150 that would be equal to the curl of minus partial b, partial t. 241 00:15:45,150 --> 00:15:49,160 And this will be equal to basically, 242 00:15:49,160 --> 00:15:54,870 I can take the minus sign out and take the partial partial t 243 00:15:54,870 --> 00:15:56,410 out. 244 00:15:56,410 --> 00:16:01,960 And basically you have curl of b. 245 00:16:01,960 --> 00:16:06,010 And according to Ampere's law, this 246 00:16:06,010 --> 00:16:15,618 would be equal to minus mu 0, epsilon 0, partial square e, 247 00:16:15,618 --> 00:16:17,610 partial t. 248 00:16:17,610 --> 00:16:20,980 OK, everybody is following? 249 00:16:20,980 --> 00:16:22,570 So basically what I have been doing 250 00:16:22,570 --> 00:16:28,030 is copy the left-hand side and make use of the Ampere's law. 251 00:16:28,030 --> 00:16:33,330 And basically you get minus mu, epsilon zero, partial square e, 252 00:16:33,330 --> 00:16:36,940 partial t squared 253 00:16:36,940 --> 00:16:39,520 And this thing, the left hand side, 254 00:16:39,520 --> 00:16:42,600 is equal to the right hand side. 255 00:16:42,600 --> 00:16:44,700 On the right hand side, what is left? 256 00:16:44,700 --> 00:16:45,834 This is equal to 0. 257 00:16:45,834 --> 00:16:46,500 So this is gone. 258 00:16:52,580 --> 00:17:01,059 This is equal to minus del squared, e. 259 00:17:01,059 --> 00:17:03,580 I can cancel the minus sign. 260 00:17:03,580 --> 00:17:10,550 Then basically what I am going to get is del squared e. 261 00:17:10,550 --> 00:17:15,880 And this will be equal to mu zero epsilon 0, partial square 262 00:17:15,880 --> 00:17:17,965 e, partial t squared. 263 00:17:22,380 --> 00:17:24,060 Wow, this is what? 264 00:17:24,060 --> 00:17:28,010 This looks like, what? 265 00:17:28,010 --> 00:17:29,900 Wave equation again. 266 00:17:29,900 --> 00:17:31,137 Oh my god. 267 00:17:31,137 --> 00:17:32,450 [LAUGHTER] 268 00:17:32,450 --> 00:17:35,610 But there is some difference. 269 00:17:35,610 --> 00:17:38,700 This is different from what we've seen before, right? 270 00:17:38,700 --> 00:17:45,585 Before, the wave equation only has partial squared partial x 271 00:17:45,585 --> 00:17:46,110 squared. 272 00:17:46,110 --> 00:17:49,040 This time, you have this del square. 273 00:17:49,040 --> 00:17:50,790 Very strange, right? 274 00:17:50,790 --> 00:17:52,260 So what is this? 275 00:17:52,260 --> 00:17:55,970 Del square is actually partial square, 276 00:17:55,970 --> 00:18:01,340 partial x squared cross partial square partial y square, 277 00:18:01,340 --> 00:18:05,820 plus partial square, partial z square is the operator, which 278 00:18:05,820 --> 00:18:08,790 will have three components. 279 00:18:08,790 --> 00:18:12,850 And basically, if you do this calculation, 280 00:18:12,850 --> 00:18:18,180 you are going to have how many times? 281 00:18:18,180 --> 00:18:22,320 If you do this del square e, how many times you have? 282 00:18:22,320 --> 00:18:23,210 You have how many? 283 00:18:23,210 --> 00:18:24,810 Any anybody help me? 284 00:18:27,440 --> 00:18:30,100 Yeah, you have three times in x direction, 285 00:18:30,100 --> 00:18:31,950 you have three times in y direction, 286 00:18:31,950 --> 00:18:34,270 you have three times in z direction. 287 00:18:34,270 --> 00:18:35,410 Therefore how many times? 288 00:18:35,410 --> 00:18:39,960 You have nine times, because each operator 289 00:18:39,960 --> 00:18:42,851 is acting out the vector. 290 00:18:42,851 --> 00:18:43,350 OK. 291 00:18:43,350 --> 00:18:46,620 So it's very important because this is a common mistake. 292 00:18:46,620 --> 00:18:48,780 So you have nine times. 293 00:18:48,780 --> 00:18:53,100 and it looks really like the wave equations 294 00:18:53,100 --> 00:18:55,350 it tastes like wave equation, it looks 295 00:18:55,350 --> 00:18:59,790 like equation, it feels like equation - that wave equation - 296 00:18:59,790 --> 00:19:02,827 and therefore is really the wave equation, right? 297 00:19:02,827 --> 00:19:04,170 [LAUGHTER] 298 00:19:04,170 --> 00:19:09,600 OK so this is a three-dimensional wave 299 00:19:09,600 --> 00:19:12,140 equation. 300 00:19:12,140 --> 00:19:12,810 Very cool. 301 00:19:12,810 --> 00:19:14,700 So we are increasing the dimension. 302 00:19:17,370 --> 00:19:22,020 So I can write it down more explicitly. 303 00:19:22,020 --> 00:19:26,155 So basically what I'm getting is partial square 304 00:19:26,155 --> 00:19:33,150 e partial x squared, partial square e, partial y square, 305 00:19:33,150 --> 00:19:37,830 plus partial square e, partial z square. 306 00:19:37,830 --> 00:19:45,470 And this is equal to mu zero, epsilon zero, partial square e, 307 00:19:45,470 --> 00:19:46,840 partial t squared. 308 00:19:46,840 --> 00:19:49,020 OK? 309 00:19:49,020 --> 00:19:56,600 So Maxwell sees this when he adds this additional term here. 310 00:19:56,600 --> 00:20:00,990 As you can see, if I don't have this additional term, 311 00:20:00,990 --> 00:20:05,630 the displacement of current from Maxwell, 312 00:20:05,630 --> 00:20:08,790 what is going to happen? 313 00:20:08,790 --> 00:20:14,030 This curve of b will be equal to zero. 314 00:20:14,030 --> 00:20:18,470 So what is going to happen to this identity? 315 00:20:18,470 --> 00:20:22,190 This left hand side part will be equal to zero. 316 00:20:22,190 --> 00:20:26,330 There will be no electromagnetic waves. 317 00:20:26,330 --> 00:20:28,290 OK? 318 00:20:28,290 --> 00:20:34,590 So that's really thanks to Maxwell's work. 319 00:20:34,590 --> 00:20:37,050 And this is actually really an equation 320 00:20:37,050 --> 00:20:43,020 which changed the world, because that actually gave us 321 00:20:43,020 --> 00:20:46,920 a lot of insights about how we can send energy, 322 00:20:46,920 --> 00:20:50,010 how we can actually understand the phenomenon related 323 00:20:50,010 --> 00:20:53,490 to light. 324 00:20:53,490 --> 00:20:57,600 So what is the velocity of this wave equation? 325 00:20:57,600 --> 00:21:03,600 The velocity, Vp, would be equal to what we usually call c. 326 00:21:03,600 --> 00:21:07,440 Because you have been using this constant for a long time. 327 00:21:07,440 --> 00:21:13,140 And that will be equal to 1 over square root of mu zero epsilon 328 00:21:13,140 --> 00:21:15,660 zero. 329 00:21:15,660 --> 00:21:19,260 And to measure the speed of light, 330 00:21:19,260 --> 00:21:23,870 it takes a long time to achieve that. 331 00:21:23,870 --> 00:21:26,930 Let's take a look at the history. 332 00:21:26,930 --> 00:21:36,360 So the first attempt was done by Galileo so 1638. 333 00:21:36,360 --> 00:21:39,600 He was doing an experiment, and that he was trying 334 00:21:39,600 --> 00:21:41,670 to track the speed of light. 335 00:21:41,670 --> 00:21:44,580 But he was not super successful. 336 00:21:44,580 --> 00:21:51,390 So his conclusion was that if the speed of light 337 00:21:51,390 --> 00:21:56,250 is not instantaneous, then it is super fast. 338 00:21:56,250 --> 00:22:02,190 He says it's at least 10 times faster than the speed of sound. 339 00:22:02,190 --> 00:22:06,180 He said OK, this is super awesome, very fast. 340 00:22:13,620 --> 00:22:15,180 OK. 341 00:22:15,180 --> 00:22:19,170 So that's what he found. 342 00:22:19,170 --> 00:22:33,660 And later Romer actually made use of the orbit of Jupiter. 343 00:22:33,660 --> 00:22:37,890 Basically he use Jupiter and Jupiter's satellite 344 00:22:37,890 --> 00:22:40,450 to measure the speed of light. 345 00:22:40,450 --> 00:22:43,830 So when the earth is closer to Jupiter, 346 00:22:43,830 --> 00:22:47,010 then somehow the satellite of Jupiter 347 00:22:47,010 --> 00:22:52,230 appears faster than when the earth is actually away 348 00:22:52,230 --> 00:22:53,700 from Jupiter. 349 00:22:53,700 --> 00:22:58,140 Because that light have to travel through additional time. 350 00:22:58,140 --> 00:23:02,140 Two times the radius of the orbit the Earth. 351 00:23:02,140 --> 00:23:05,230 Basically that's the math that he was using. 352 00:23:05,230 --> 00:23:12,460 He is actually making the first computative measurement 353 00:23:12,460 --> 00:23:13,860 of the speed of light. 354 00:23:13,860 --> 00:23:17,640 And what number he found is 2 times 355 00:23:17,640 --> 00:23:19,950 10 to the 9 meters per second. 356 00:23:22,520 --> 00:23:28,740 Then finally, again using the star 357 00:23:28,740 --> 00:23:35,670 the observation as a tool to actually calculate 358 00:23:35,670 --> 00:23:36,970 the speed of light. 359 00:23:36,970 --> 00:23:39,150 James actually nailed it. 360 00:23:39,150 --> 00:23:41,330 He found a value which is really close 361 00:23:41,330 --> 00:23:44,570 to the current understanding of the speed of light, 362 00:23:44,570 --> 00:23:50,540 which is 3 times 10 to 9 meters per second. 363 00:23:50,540 --> 00:23:54,680 Therefore, if you calculate that using all those constants here, 364 00:23:54,680 --> 00:23:59,120 you will be able to see that, indeed, from Maxwell's 365 00:23:59,120 --> 00:24:01,450 equation, you get-- 366 00:24:01,450 --> 00:24:04,490 oh, it should be 3 times 10 to the 8, not to the 9. 367 00:24:04,490 --> 00:24:06,090 I was saying 10 to 9. 368 00:24:06,090 --> 00:24:09,522 It should be 3 times 10 to the 8 meters per second. 369 00:24:13,330 --> 00:24:17,630 So indeed, this equation is actually 370 00:24:17,630 --> 00:24:22,430 predicting the speed of light to be 3 times 10 to the 8 371 00:24:22,430 --> 00:24:28,870 is matching the experimental result. 372 00:24:28,870 --> 00:24:31,000 So that is pretty nice. 373 00:24:31,000 --> 00:24:34,270 And you may ask a question-- 374 00:24:34,270 --> 00:24:35,780 so wait a second. 375 00:24:35,780 --> 00:24:40,060 You said this is actually an electromagnetic wave, right? 376 00:24:40,060 --> 00:24:41,950 So that's actually what I was talking about. 377 00:24:41,950 --> 00:24:48,130 But this equation only talks about electric fields. 378 00:24:48,130 --> 00:24:51,280 What is happening to the magnetic field? 379 00:24:51,280 --> 00:24:51,945 What happened? 380 00:24:54,910 --> 00:24:59,900 Can we actually choose arbitrary magnetic fields? 381 00:25:04,200 --> 00:25:07,120 Is a magnetic field also described 382 00:25:07,120 --> 00:25:10,240 by the three dimensional wave equation, right? 383 00:25:10,240 --> 00:25:14,220 The answer is that indeed you can actually 384 00:25:14,220 --> 00:25:15,640 do the same exercise. 385 00:25:15,640 --> 00:25:19,260 You can now instead of plugging in electric field, 386 00:25:19,260 --> 00:25:21,730 you can plug in a magnetic field. 387 00:25:21,730 --> 00:25:25,380 And you will extract exactly the same conclusion. 388 00:25:25,380 --> 00:25:28,180 You will conclude the del square B, 389 00:25:28,180 --> 00:25:35,390 will B equal to Mu 0, epsilon 0, partial square, 390 00:25:35,390 --> 00:25:37,920 B partial to square. 391 00:25:37,920 --> 00:25:40,440 OK, so it is actually very important 392 00:25:40,440 --> 00:25:47,450 to see that the magnetic field also obey this wave equation. 393 00:25:47,450 --> 00:25:49,760 OK? 394 00:25:49,760 --> 00:25:53,790 And also from Maxwell's equation, 395 00:25:53,790 --> 00:25:57,690 you can see that the changing electric field 396 00:25:57,690 --> 00:26:02,550 will produce a curling around a magnetic field. 397 00:26:02,550 --> 00:26:04,510 The same thing also happens here. 398 00:26:04,510 --> 00:26:08,190 A changing magnetic field also produce a curling 399 00:26:08,190 --> 00:26:10,530 around electric field. 400 00:26:10,530 --> 00:26:13,490 So what does that mean? 401 00:26:13,490 --> 00:26:18,800 That means E, electric field create magnetic field. 402 00:26:18,800 --> 00:26:21,050 Magnetic field create electric field. 403 00:26:21,050 --> 00:26:23,600 And this happens all the time. 404 00:26:23,600 --> 00:26:29,530 Therefore, one cannot live without the other. 405 00:26:29,530 --> 00:26:30,790 They are living together. 406 00:26:30,790 --> 00:26:34,600 They are all together, forever. 407 00:26:34,600 --> 00:26:40,270 All right, so what is actually oscillating 408 00:26:40,270 --> 00:26:44,320 is actually both electric field and the magnetic field, right? 409 00:26:44,320 --> 00:26:50,280 So you may ask, OK, we are talking about vacuum. 410 00:26:50,280 --> 00:26:53,910 Vacuum means there is no material, no charge, 411 00:26:53,910 --> 00:26:56,710 no whatsoever in vacuum. 412 00:26:56,710 --> 00:26:58,710 So what is actually oscillating? 413 00:26:58,710 --> 00:27:01,640 Who is oscillating? 414 00:27:01,640 --> 00:27:05,940 Is the electric field and the magnetic field. 415 00:27:05,940 --> 00:27:08,580 This so-called field, all those vectors-- 416 00:27:08,580 --> 00:27:11,310 which are actually oscillating-- it's not the material, 417 00:27:11,310 --> 00:27:15,510 but all those vectors associated with the space, 418 00:27:15,510 --> 00:27:17,710 which is actually oscillating up and down. 419 00:27:20,720 --> 00:27:23,710 All right, so originally I would like 420 00:27:23,710 --> 00:27:26,510 to show you a pulse of light in front of you. 421 00:27:26,510 --> 00:27:30,610 And show that it's moving, but it's too fast. 422 00:27:30,610 --> 00:27:32,110 So I couldn't do that. 423 00:27:32,110 --> 00:27:34,570 [LAUGHTER] 424 00:27:34,570 --> 00:27:39,070 Fortunately, we have photos. 425 00:27:39,070 --> 00:27:45,160 Photos are actually collected the recorded photons. 426 00:27:45,160 --> 00:27:50,240 Emitted from the object of the interest. 427 00:27:50,240 --> 00:27:53,930 So this is actually how we make applesauce at MIT. 428 00:27:53,930 --> 00:27:57,820 We shoot-- bullet through the apple then we have the sauce. 429 00:27:57,820 --> 00:27:59,150 [LAUGHTER] 430 00:27:59,150 --> 00:28:01,870 But not sure if that's tasty enough or not. 431 00:28:01,870 --> 00:28:04,750 But that's how we do it in MIT-- 432 00:28:04,750 --> 00:28:06,160 MIT style. 433 00:28:06,160 --> 00:28:11,070 And the good thing is that this kind of technique 434 00:28:11,070 --> 00:28:13,930 is improved dramatically in these days. 435 00:28:13,930 --> 00:28:17,980 I would like to show you a short video, which is actually 436 00:28:17,980 --> 00:28:23,145 recording a video of-- 437 00:28:23,145 --> 00:28:30,070 it's recording experiment, which you shoot some beam of light 438 00:28:30,070 --> 00:28:34,570 through some plastic container. 439 00:28:34,570 --> 00:28:38,080 And the speed of this recording corresponds 440 00:28:38,080 --> 00:28:42,310 to one trillion frame per second. 441 00:28:42,310 --> 00:28:45,040 So this is super fast recording. 442 00:28:45,040 --> 00:28:48,340 And they can actually reconstruct the propagation 443 00:28:48,340 --> 00:28:52,600 of light through this bottle. 444 00:28:52,600 --> 00:28:58,300 The credit is actually to the Media Lab Camera Culture group. 445 00:28:58,300 --> 00:29:00,852 And let's take a look at the video. 446 00:29:00,852 --> 00:29:01,910 Just one second. 447 00:29:05,110 --> 00:29:10,030 OK, so this is actually recording at one trillion frame 448 00:29:10,030 --> 00:29:11,310 per second. 449 00:29:11,310 --> 00:29:13,220 So you can see that there's a light pulse-- 450 00:29:13,220 --> 00:29:15,710 a very short pulse created. 451 00:29:15,710 --> 00:29:21,440 And is really pass through the bottle. 452 00:29:21,440 --> 00:29:27,700 And it can be recorded with the technique created by Media Lab. 453 00:29:27,700 --> 00:29:31,190 So you can see that the pulse is really propagating through it. 454 00:29:31,190 --> 00:29:33,560 And the reason why we can see the pulse 455 00:29:33,560 --> 00:29:35,390 is because there are air, there are 456 00:29:35,390 --> 00:29:37,430 material which will actually change 457 00:29:37,430 --> 00:29:39,320 the direction of the light. 458 00:29:39,320 --> 00:29:42,830 And therefore, those are recorded by the camera. 459 00:29:42,830 --> 00:29:48,620 And they take trillions of frames of this thing, 460 00:29:48,620 --> 00:29:49,760 and put them together. 461 00:29:49,760 --> 00:29:53,390 Then basically-- and they take many, many frames, 462 00:29:53,390 --> 00:29:56,700 and they put them together to reconstruct this movie. 463 00:29:56,700 --> 00:29:59,360 So as you can see that indeed you 464 00:29:59,360 --> 00:30:03,050 can see the propagation of the light 465 00:30:03,050 --> 00:30:07,100 through this kind of video. 466 00:30:07,100 --> 00:30:09,470 So I hope that we enjoyed this video. 467 00:30:09,470 --> 00:30:14,960 And let's actually take a look at some concrete example 468 00:30:14,960 --> 00:30:18,260 which make use of the wave equation, which 469 00:30:18,260 --> 00:30:20,870 we did right here. 470 00:30:20,870 --> 00:30:24,740 So let's consider a plane wave solution. 471 00:30:24,740 --> 00:30:28,100 Things we are entering a three dimensional world. 472 00:30:28,100 --> 00:30:31,816 So that's actually consider so-called a plane wave. 473 00:30:39,230 --> 00:30:45,930 So in this example, I am considering the electric field 474 00:30:45,930 --> 00:30:53,890 that's actually equal to the real part of E0 exponential i, 475 00:30:53,890 --> 00:30:57,740 kz minus omega t. 476 00:30:57,740 --> 00:31:02,930 And this electric field I actually consider here 477 00:31:02,930 --> 00:31:06,910 is in the x direction. 478 00:31:06,910 --> 00:31:10,420 And if I write all the terms from this expression 479 00:31:10,420 --> 00:31:14,790 expressively, that's actually what I'm getting is-- 480 00:31:14,790 --> 00:31:26,650 x component will be E0, cosine kz minus omega t, 0 and 0. 481 00:31:26,650 --> 00:31:28,960 So what does this mean? 482 00:31:28,960 --> 00:31:31,710 What is actually a plane wave? 483 00:31:31,710 --> 00:31:36,010 The plane wave basically is actually 484 00:31:36,010 --> 00:31:37,840 fielding the whole space. 485 00:31:37,840 --> 00:31:41,530 What I mean by plane wave is I feel the whole space 486 00:31:41,530 --> 00:31:43,690 with electric field. 487 00:31:43,690 --> 00:31:47,200 This electric field only have one-- 488 00:31:47,200 --> 00:31:50,800 only one direction have non-zero value, which 489 00:31:50,800 --> 00:31:54,010 is x direction in this example. 490 00:31:54,010 --> 00:31:58,696 And then the other direction, there's no-- 491 00:31:58,696 --> 00:32:02,460 the magnitude is actually equal to 0. 492 00:32:02,460 --> 00:32:05,420 So that's actually what I mean by plane wave. 493 00:32:05,420 --> 00:32:11,310 And also the electric field is filling a whole space 494 00:32:11,310 --> 00:32:17,140 in the discussion-- in the example which I discussed here. 495 00:32:17,140 --> 00:32:22,800 And if I define my coordinate system like this, 496 00:32:22,800 --> 00:32:24,220 x is in the horizontal direction. 497 00:32:24,220 --> 00:32:28,240 Then that means everything is actually-- 498 00:32:28,240 --> 00:32:30,220 all the electric field is actually 499 00:32:30,220 --> 00:32:36,830 pointing toward the x direction in this coordinate system. 500 00:32:39,600 --> 00:32:45,840 So we have discussed progressing wave in the past few lectures. 501 00:32:45,840 --> 00:32:50,640 Can somebody actually tell me the direction of propagation 502 00:32:50,640 --> 00:32:52,750 of this plane wave? 503 00:32:52,750 --> 00:32:56,200 So the hint is that this is actually equal to E0, 504 00:32:56,200 --> 00:32:58,860 that the magnitude of the x component 505 00:32:58,860 --> 00:33:03,390 is equal to E0, cosine kz minus omega t. 506 00:33:03,390 --> 00:33:07,290 What is actually the direction of propagation 507 00:33:07,290 --> 00:33:09,516 of this electric field? 508 00:33:09,516 --> 00:33:10,016 AUDIENCE: z. 509 00:33:10,016 --> 00:33:11,557 YEN-JIE LEE: It's in the z direction. 510 00:33:11,557 --> 00:33:12,690 Yeah, very good. 511 00:33:12,690 --> 00:33:14,670 Because we know that this is actually 512 00:33:14,670 --> 00:33:17,090 going in the positive z direction. 513 00:33:17,090 --> 00:33:20,480 Because this is actually kz minus-- 514 00:33:20,480 --> 00:33:22,330 there's a minus sign-- omega t. 515 00:33:22,330 --> 00:33:27,140 So therefore it's going toward the positive z direction. 516 00:33:27,140 --> 00:33:32,130 Not x direction. x direction is where the electric field is 517 00:33:32,130 --> 00:33:33,930 pointing to. 518 00:33:33,930 --> 00:33:40,980 And the direction of propagation is toward the z direction. 519 00:33:40,980 --> 00:33:45,330 So there's a difference. 520 00:33:45,330 --> 00:33:49,170 So first thing which I would like to do is to check if this 521 00:33:49,170 --> 00:33:55,300 so-called plane wave solution actually satisfy the equation-- 522 00:33:55,300 --> 00:33:58,710 the wave equation which we derive here. 523 00:33:58,710 --> 00:34:03,660 Del square E equal to Mu 0 epsilon 0, partial square E, 524 00:34:03,660 --> 00:34:04,650 partial t square. 525 00:34:04,650 --> 00:34:09,909 So I can now plug that in to that equation. 526 00:34:09,909 --> 00:34:13,830 I can now plug in to this equation. 527 00:34:13,830 --> 00:34:17,580 If I plug in the wave-- the plane wave solution, which 528 00:34:17,580 --> 00:34:21,239 I have here to that equation-- basically, 529 00:34:21,239 --> 00:34:23,429 I can get the left-hand side. 530 00:34:23,429 --> 00:34:25,020 The left-hand side of the equation, 531 00:34:25,020 --> 00:34:30,060 you will get minus E0. 532 00:34:30,060 --> 00:34:33,090 Only one term which contribute is 533 00:34:33,090 --> 00:34:37,159 the partial square E partial z square term which contribute. 534 00:34:37,159 --> 00:34:38,040 Right? 535 00:34:38,040 --> 00:34:42,340 Because the magnitude of the electric field 536 00:34:42,340 --> 00:34:46,110 only depends on z and t. 537 00:34:46,110 --> 00:34:53,730 Therefore, you get minus E0, k square, cosine, kz minus omega 538 00:34:53,730 --> 00:34:57,600 t in the left-hand side of the wave equation. 539 00:35:00,480 --> 00:35:03,510 How about the right-hand side? 540 00:35:03,510 --> 00:35:07,140 Right-hand side actually you are taking partial derivative, 541 00:35:07,140 --> 00:35:09,360 which is fed to t. 542 00:35:09,360 --> 00:35:13,035 Basically, you get minus Mu 0-- 543 00:35:13,035 --> 00:35:14,280 Mu 0, epsilon 0-- 544 00:35:14,280 --> 00:35:18,540 I copied from that formula there. 545 00:35:18,540 --> 00:35:23,110 And you basically get omega square out 546 00:35:23,110 --> 00:35:25,720 of it because of the partial square, partial t 547 00:35:25,720 --> 00:35:28,220 square operator. 548 00:35:28,220 --> 00:35:33,964 And then you basically get cosine kz minus omega t. 549 00:35:37,170 --> 00:35:39,970 And of course I missed the E0 term. 550 00:35:39,970 --> 00:35:42,100 E0 should be copied from-- on there. 551 00:35:45,030 --> 00:35:46,730 So now I can show that-- 552 00:35:46,730 --> 00:35:48,450 OK, this cancel. 553 00:35:48,450 --> 00:35:53,954 Basically, this is the same cosine kz minus omega t. 554 00:35:53,954 --> 00:35:57,235 And E0 also cancel. 555 00:35:57,235 --> 00:36:01,010 And I can cancel the minus sign. 556 00:36:01,010 --> 00:36:05,080 What I'm going to get is k square 557 00:36:05,080 --> 00:36:10,860 is equal to Mu 0, epsilon 0, omega square. 558 00:36:10,860 --> 00:36:15,720 So that means there should be a fixed relation between k 559 00:36:15,720 --> 00:36:20,850 and omega, which is actually omega over k 560 00:36:20,850 --> 00:36:26,130 will be equal to 1 over square to the Mu 0, epsilon 0. 561 00:36:26,130 --> 00:36:27,900 And this is equal to c. 562 00:36:27,900 --> 00:36:33,030 If this is satisfied, then the plane wave 563 00:36:33,030 --> 00:36:36,630 is a solution to the wave equation-- 564 00:36:36,630 --> 00:36:40,470 only when this is actually satisfied. 565 00:36:40,470 --> 00:36:44,910 Otherwise we can write arbitrary plane wave equation, 566 00:36:44,910 --> 00:36:48,480 but they are not the solution of that equation 567 00:36:48,480 --> 00:36:49,480 from Maxwell's equation. 568 00:36:52,630 --> 00:36:57,820 So now, I have derived the electric field 569 00:36:57,820 --> 00:37:02,440 and also know the relation between omega, the angle 570 00:37:02,440 --> 00:37:05,830 frequency, and the k, the wave number. 571 00:37:05,830 --> 00:37:10,540 And now, what about magnetic field? 572 00:37:10,540 --> 00:37:15,100 So I just mentioned before, magnetic field cannot live 573 00:37:15,100 --> 00:37:16,220 without electric field. 574 00:37:16,220 --> 00:37:20,620 And electric field cannot live without magnetic field. 575 00:37:20,620 --> 00:37:23,740 So what is actually responding magnetic field? 576 00:37:23,740 --> 00:37:26,470 We can actually evaluate that. 577 00:37:26,470 --> 00:37:31,830 So now, the question is what is actually the magnetic field? 578 00:37:31,830 --> 00:37:35,170 And how is that vary as a function of time 579 00:37:35,170 --> 00:37:39,990 and as a function of position in the space? 580 00:37:39,990 --> 00:37:45,610 So we are facing a choice. 581 00:37:45,610 --> 00:37:48,830 So there are two equations, which 582 00:37:48,830 --> 00:37:54,690 relate electric field and the magnetic field. 583 00:37:54,690 --> 00:37:58,380 It is actually very important you make the right choice when 584 00:37:58,380 --> 00:38:02,340 you start your calculation. 585 00:38:02,340 --> 00:38:04,920 So we can use Faraday's law. 586 00:38:04,920 --> 00:38:07,770 We can also use Ampere's law. 587 00:38:07,770 --> 00:38:11,050 But there's only one, which is actually much easier 588 00:38:11,050 --> 00:38:17,280 to derive a solution, which is the choice of Faraday's law. 589 00:38:17,280 --> 00:38:23,310 If you choose to use Ampere's law to evaluate B, 590 00:38:23,310 --> 00:38:24,780 then you are going to get a really 591 00:38:24,780 --> 00:38:27,960 super complicated problem to solve. 592 00:38:27,960 --> 00:38:30,570 But on the other hand, if you choose 593 00:38:30,570 --> 00:38:33,570 to use Faraday's law to solve this problem, 594 00:38:33,570 --> 00:38:41,160 then you can see that the unknown is the magnetic field-- 595 00:38:41,160 --> 00:38:43,530 the field which I would like to evaluate. 596 00:38:43,530 --> 00:38:47,830 And the expression for the B is actually rather simple. 597 00:38:47,830 --> 00:38:50,640 It's actually just a partial derivative, partial B, 598 00:38:50,640 --> 00:38:52,140 partial t. 599 00:38:52,140 --> 00:38:54,480 So it's pretty simple and you can actually 600 00:38:54,480 --> 00:38:56,360 evaluate the known part. 601 00:38:56,360 --> 00:38:59,460 This curl looks pretty complicated. 602 00:38:59,460 --> 00:39:02,130 So you can actually evaluate that because you know 603 00:39:02,130 --> 00:39:04,380 what is the electric field. 604 00:39:04,380 --> 00:39:07,650 On the other hand, if you will use Ampere's Law 605 00:39:07,650 --> 00:39:11,820 then you will be in trouble because you don't 606 00:39:11,820 --> 00:39:14,610 know what is a B, xBy and Bz. 607 00:39:14,610 --> 00:39:16,350 And you have to evaluate curl. 608 00:39:16,350 --> 00:39:18,740 And you get a lot of terms, and that is actually 609 00:39:18,740 --> 00:39:21,290 equal to something from-- the information 610 00:39:21,290 --> 00:39:22,290 from the electric field. 611 00:39:22,290 --> 00:39:25,680 And that would be very difficult to evaluate. 612 00:39:25,680 --> 00:39:31,510 So therefore, what we are going to do is to use Faraday's law, 613 00:39:31,510 --> 00:39:38,070 curl of E will be equal to minus partial B, partial t. 614 00:39:38,070 --> 00:39:44,200 So basically, as I mentioned in the beginning, 615 00:39:44,200 --> 00:39:51,270 we can make use of the equations the determinant of matrix 616 00:39:51,270 --> 00:39:53,760 to evaluate the curl. 617 00:39:53,760 --> 00:39:56,310 So therefore, I am going to use that. 618 00:39:56,310 --> 00:40:03,040 And then what I'm going to get is x, y, z unit vector for fill 619 00:40:03,040 --> 00:40:05,460 the first row. 620 00:40:05,460 --> 00:40:09,680 And the partial partial x, partial partial y, 621 00:40:09,680 --> 00:40:14,820 partial partial z, which fill the second row of the matrix. 622 00:40:14,820 --> 00:40:20,610 Then I get Ex, 0, 0 because the electric field is only 623 00:40:20,610 --> 00:40:23,490 in the x direction. 624 00:40:23,490 --> 00:40:25,980 And this will be equal to-- 625 00:40:25,980 --> 00:40:30,880 only two terms survive because of these two 0's. 626 00:40:30,880 --> 00:40:35,430 So all other terms are killed, and only two terms are now 0. 627 00:40:35,430 --> 00:40:39,924 The first term is actually partial Ex, 628 00:40:39,924 --> 00:40:44,770 partial z in the y direction. 629 00:40:44,770 --> 00:40:50,390 And the second term is actually minus partial Ex, 630 00:40:50,390 --> 00:40:55,700 partial y in the z direction. 631 00:40:55,700 --> 00:40:56,840 Any questions? 632 00:40:56,840 --> 00:40:57,870 Am I going too fast? 633 00:41:01,540 --> 00:41:04,850 All right, so you can see that the electric field only 634 00:41:04,850 --> 00:41:08,310 depends on the position z. 635 00:41:08,310 --> 00:41:11,710 It's independent of y. 636 00:41:11,710 --> 00:41:17,020 Therefore, partial Ex, partial y, is actually equal to 0. 637 00:41:17,020 --> 00:41:19,900 Wow, this become much, much easier 638 00:41:19,900 --> 00:41:23,710 because there's only one term which is surviving. 639 00:41:23,710 --> 00:41:26,670 This is a operator. 640 00:41:26,670 --> 00:41:28,690 Then basically what we're going to get 641 00:41:28,690 --> 00:41:34,120 is I can now calculate partial Ex, partial z based 642 00:41:34,120 --> 00:41:38,440 on that equation, E0 cosine kz minus omega t. 643 00:41:38,440 --> 00:41:41,460 Then basically, what I can get is minus-- 644 00:41:41,460 --> 00:41:49,504 I get a K out of it, E0 sine kz, and it's omega t. 645 00:41:55,130 --> 00:41:59,590 So this is actually the result of the left-hand side. 646 00:41:59,590 --> 00:42:04,460 The right-hand side of that equation of the Faraday's law 647 00:42:04,460 --> 00:42:07,160 is minus partial B, partial t. 648 00:42:07,160 --> 00:42:12,550 So this will give you equal to minus partial B, partial t. 649 00:42:17,530 --> 00:42:22,410 So very important-- I don't want to drop the y direction. 650 00:42:22,410 --> 00:42:25,270 So this is just y direction. 651 00:42:25,270 --> 00:42:29,980 And this is actually a vector and this is also a vector. 652 00:42:29,980 --> 00:42:37,600 So what I could do is to do a integration over t. 653 00:42:37,600 --> 00:42:40,720 And those will cancel the minus sign. 654 00:42:40,720 --> 00:42:45,730 So if I integrate over t, then basically what I'm going to get 655 00:42:45,730 --> 00:42:57,100 is K over omega is 0, cosine kz minus omega t 656 00:42:57,100 --> 00:42:58,476 in the y direction only. 657 00:43:01,020 --> 00:43:04,740 So I'm doing a integration of t, cancel the minus sign, 658 00:43:04,740 --> 00:43:07,420 then this is what you want to get. 659 00:43:07,420 --> 00:43:14,350 And of course, k/omega is actually 1/c. 660 00:43:14,350 --> 00:43:17,320 So therefore, you have E0-- 661 00:43:17,320 --> 00:43:19,450 you can actually simplify this fraction-- 662 00:43:19,450 --> 00:43:26,160 and this is actually equal to E0 over c, cosine, kz minus omega 663 00:43:26,160 --> 00:43:30,960 t in the y direction. 664 00:43:30,960 --> 00:43:35,500 OK, look at what we have learned from here. 665 00:43:35,500 --> 00:43:38,650 What we have learned from here is that-- 666 00:43:38,650 --> 00:43:44,020 I got started with a plane wave solution of the electric field. 667 00:43:44,020 --> 00:43:48,430 And I can show that only when omega over k 668 00:43:48,430 --> 00:43:51,880 is equal to the speed of light this is actually 669 00:43:51,880 --> 00:43:56,260 a solution to my wave equation. 670 00:43:56,260 --> 00:44:00,700 And also because the electric field and the magnetic field 671 00:44:00,700 --> 00:44:05,350 have to satisfy the Maxwell's equation all the time-- 672 00:44:05,350 --> 00:44:07,310 because that's the fundamental law-- 673 00:44:07,310 --> 00:44:10,690 therefore, I can use those equations 674 00:44:10,690 --> 00:44:13,680 to evaluate and to find what is actually 675 00:44:13,680 --> 00:44:16,090 the corresponding magnetic field. 676 00:44:16,090 --> 00:44:19,480 And using Faraday's law and plugging in 677 00:44:19,480 --> 00:44:21,280 and the solving the question, I will 678 00:44:21,280 --> 00:44:27,120 be able to figure out that B is also what kind of wave? 679 00:44:27,120 --> 00:44:30,710 B is also what kind of wave I was talking about-- 680 00:44:30,710 --> 00:44:32,730 also? 681 00:44:32,730 --> 00:44:33,620 AUDIENCE: Plane wave. 682 00:44:33,620 --> 00:44:34,911 YEN-JIE LEE: Plane wave, right? 683 00:44:34,911 --> 00:44:36,270 It's also plane wave. 684 00:44:36,270 --> 00:44:37,020 You see? 685 00:44:37,020 --> 00:44:39,950 So if I got started with a plane wave 686 00:44:39,950 --> 00:44:42,360 in the electric field side, and I also 687 00:44:42,360 --> 00:44:46,560 get the plane wave in the magnetic field side. 688 00:44:46,560 --> 00:44:49,630 They are proportional to each other. 689 00:44:49,630 --> 00:44:54,120 Originally, the magnitude of the electric field is E0. 690 00:44:54,120 --> 00:44:57,450 The corresponding magnetic field-- 691 00:44:57,450 --> 00:45:00,880 the magnitude is proportional to E0. 692 00:45:00,880 --> 00:45:04,530 But there's a factor of 1/c difference 693 00:45:04,530 --> 00:45:07,240 between the magnetic field amplitude 694 00:45:07,240 --> 00:45:11,010 and the electric field amplitude. 695 00:45:11,010 --> 00:45:13,380 The third thing which we learned from here 696 00:45:13,380 --> 00:45:19,830 is that electric field is actually in the x direction. 697 00:45:19,830 --> 00:45:24,250 B field is actually not in the x direction, 698 00:45:24,250 --> 00:45:25,835 it's in the y direction. 699 00:45:29,320 --> 00:45:36,610 What we learn from here is that the direction of the B field 700 00:45:36,610 --> 00:45:42,860 can be determined by a simple calculation. 701 00:45:42,860 --> 00:45:46,980 So basically, the B is proportional-- 702 00:45:46,980 --> 00:45:48,940 the magnitude of B is proportional 703 00:45:48,940 --> 00:45:50,060 to the electric field. 704 00:45:52,770 --> 00:45:55,920 But you have to multiply the magnitude by 1/c. 705 00:45:58,750 --> 00:46:00,760 And also this is actually not correct 706 00:46:00,760 --> 00:46:06,550 because the B is actually in the y direction. 707 00:46:06,550 --> 00:46:10,410 So the original direction of the electric field 708 00:46:10,410 --> 00:46:12,190 is in the x direction. 709 00:46:12,190 --> 00:46:16,950 Also we know the direction of a propagation 710 00:46:16,950 --> 00:46:21,010 is in the z direction. 711 00:46:21,010 --> 00:46:26,690 Therefore, if I take unit vector K-- 712 00:46:26,690 --> 00:46:31,000 K is actually the wave number, but now I make it a vector 713 00:46:31,000 --> 00:46:35,000 and I take the unit vector is equal to z-- 714 00:46:35,000 --> 00:46:42,550 so direction of propagation. 715 00:46:42,550 --> 00:46:47,410 If I make this definition then I can now rewrite this relation. 716 00:46:47,410 --> 00:46:51,250 Basically, I can express the magnitude of B 717 00:46:51,250 --> 00:46:59,290 by K hat, which is the direction of propagation cross the E 718 00:46:59,290 --> 00:47:01,090 field. 719 00:47:01,090 --> 00:47:02,560 And we can check this. 720 00:47:02,560 --> 00:47:05,830 And then basically what you are going to get is z cross E-- 721 00:47:05,830 --> 00:47:10,780 then actually really z cross x, you 722 00:47:10,780 --> 00:47:14,380 are going to get y direction. 723 00:47:14,380 --> 00:47:18,430 And that is actually telling you that B and the E 724 00:47:18,430 --> 00:47:21,940 have a rather simple relation. 725 00:47:21,940 --> 00:47:25,510 And also you don't really need to go through 726 00:47:25,510 --> 00:47:28,240 all those calculation again because now you 727 00:47:28,240 --> 00:47:32,590 can see that if you know the direction of propagation 728 00:47:32,590 --> 00:47:35,010 and you know the direction of the electric field, 729 00:47:35,010 --> 00:47:41,230 then you can already evaluate what will be in the B field. 730 00:47:41,230 --> 00:47:44,050 So we will take a five minute break. 731 00:47:44,050 --> 00:47:48,002 We'll come back in 29, and we will continue the discussion 732 00:47:48,002 --> 00:47:48,710 of this solution. 733 00:47:52,860 --> 00:47:56,590 Let me know if you have any questions about the content we 734 00:47:56,590 --> 00:47:57,200 discussed. 735 00:48:05,160 --> 00:48:06,810 Welcome back, everybody. 736 00:48:06,810 --> 00:48:08,670 So we will continue the discussion 737 00:48:08,670 --> 00:48:12,180 of what we have learned from the wave equation. 738 00:48:12,180 --> 00:48:17,360 So basically we start with plane wave in the electric field. 739 00:48:17,360 --> 00:48:20,100 And this electric field is in the x direction. 740 00:48:20,100 --> 00:48:23,430 And we evaluated the corresponding B field 741 00:48:23,430 --> 00:48:25,890 which is in the y direction. 742 00:48:25,890 --> 00:48:30,690 And what we found is that actually we can find a pretty 743 00:48:30,690 --> 00:48:34,930 simple relation between electric field and the magnetic field, 744 00:48:34,930 --> 00:48:40,055 which is actually magnetic field vector is equal to 1/c, 745 00:48:40,055 --> 00:48:46,170 K hat cross E. And the K hat now which is-- you find here-- 746 00:48:46,170 --> 00:48:48,700 is actually the direction of propagation. 747 00:48:48,700 --> 00:48:53,070 So basically, in this case in the discussion we had before, 748 00:48:53,070 --> 00:48:58,590 the direction of propagation is in the positive z direction. 749 00:48:58,590 --> 00:49:02,540 So if I go ahead and visualize the whole-- 750 00:49:02,540 --> 00:49:06,840 solution-- plot the magnetic field and the electric field is 751 00:49:06,840 --> 00:49:11,200 a function of z, x and the y-- 752 00:49:11,200 --> 00:49:12,850 it's a function of z actually here. 753 00:49:12,850 --> 00:49:16,680 And I only evaluate the value at x equal to 0, 754 00:49:16,680 --> 00:49:18,480 and the y equal to 0. 755 00:49:18,480 --> 00:49:20,490 And basically, this is actually what you have. 756 00:49:20,490 --> 00:49:22,770 So basically, you have two sine wave. 757 00:49:22,770 --> 00:49:26,250 One is actually pointing to the x direction. 758 00:49:26,250 --> 00:49:28,950 And the other one is actually pointing to the y direction, 759 00:49:28,950 --> 00:49:31,260 which is the B field. 760 00:49:31,260 --> 00:49:33,690 And those lines doesn't mean a lot 761 00:49:33,690 --> 00:49:37,650 because those lines are just connecting the end 762 00:49:37,650 --> 00:49:39,810 point of all those vectors. 763 00:49:39,810 --> 00:49:47,550 So you can see that they are cosine wave structure when 764 00:49:47,550 --> 00:49:50,040 you connect all those vectors. 765 00:49:50,040 --> 00:49:53,970 And keep in mind that those are evaluated 766 00:49:53,970 --> 00:49:57,270 at x and y equal to 0. 767 00:49:57,270 --> 00:50:01,170 Therefore, what we actually get is actually a lot of vectors. 768 00:50:01,170 --> 00:50:03,930 So those individual arrows are vectors. 769 00:50:03,930 --> 00:50:09,390 And this whole thing-- this whole electromagnetic wave 770 00:50:09,390 --> 00:50:14,010 is propagating to the positive z direction. 771 00:50:14,010 --> 00:50:18,480 And those electric field and the magnetic field 772 00:50:18,480 --> 00:50:22,430 are propagating at the speed of light, which you see. 773 00:50:22,430 --> 00:50:25,370 And also you can see that the magnitude-- 774 00:50:25,370 --> 00:50:26,790 also I plotted here-- 775 00:50:26,790 --> 00:50:31,020 the magnitude, there's no phase difference 776 00:50:31,020 --> 00:50:37,369 between electric field and the magnetic field. 777 00:50:37,369 --> 00:50:38,910 This is actually not always the case. 778 00:50:38,910 --> 00:50:45,670 In which we will show a example probably later in the lecture. 779 00:50:45,670 --> 00:50:49,380 So in general, what we can actually do 780 00:50:49,380 --> 00:50:56,800 is to write down a general expression for the plane wave. 781 00:50:56,800 --> 00:50:59,490 So for example, I can have a plane wave, which is actually 782 00:50:59,490 --> 00:51:02,160 propagating in some direction. 783 00:51:02,160 --> 00:51:05,990 Which is actually given by this K vector. 784 00:51:05,990 --> 00:51:09,540 K vector is actually giving you information 785 00:51:09,540 --> 00:51:12,930 about the wave number. 786 00:51:12,930 --> 00:51:18,390 And also the direction of propagation. 787 00:51:18,390 --> 00:51:21,660 And in this case, what I am trying to construct 788 00:51:21,660 --> 00:51:25,820 is a solution, which is actually propagating along 789 00:51:25,820 --> 00:51:28,920 in the direction of the K vector. 790 00:51:28,920 --> 00:51:32,990 And the electric field is actually 791 00:51:32,990 --> 00:51:36,540 going to be pointing to a direction perpendicular 792 00:51:36,540 --> 00:51:40,680 to the direction of the K vector. 793 00:51:40,680 --> 00:51:44,490 So basically, what I can do is I can write this plane 794 00:51:44,490 --> 00:51:46,605 wave in this functional form. 795 00:51:46,605 --> 00:51:49,380 E0 is actually a vector, which is actually 796 00:51:49,380 --> 00:51:54,250 telling you the direction of the electric field-- 797 00:51:54,250 --> 00:52:00,210 E0 vector-- is actually have this function of form. 798 00:52:03,110 --> 00:52:06,490 And the K vector is actually placed 799 00:52:06,490 --> 00:52:08,670 in the exponential function-- inside 800 00:52:08,670 --> 00:52:10,650 the exponential function. 801 00:52:10,650 --> 00:52:16,518 Exponential i, k dot r minus omega t. 802 00:52:16,518 --> 00:52:18,870 And what is actually r? 803 00:52:18,870 --> 00:52:25,080 r is actually x x hat, plus y y hat, plus z z hat. 804 00:52:25,080 --> 00:52:29,700 And omega is actually the angular frequency 805 00:52:29,700 --> 00:52:33,330 which we are familiar with and that's actually 806 00:52:33,330 --> 00:52:39,870 equal to c times the magnitude of K, 807 00:52:39,870 --> 00:52:43,470 which is actually the wave number. 808 00:52:43,470 --> 00:52:45,660 And you can actually show that-- 809 00:52:45,660 --> 00:52:51,150 OK, indeed this expression can satisfy 810 00:52:51,150 --> 00:52:55,980 the wave equation, which we did right for the electric field. 811 00:52:55,980 --> 00:52:59,370 And of course there are some requirements, 812 00:52:59,370 --> 00:53:06,040 which is actually that the direction of the electric field 813 00:53:06,040 --> 00:53:10,740 have to be perpendicular to the direction of propagation. 814 00:53:10,740 --> 00:53:12,770 Which you can actually derive that. 815 00:53:12,770 --> 00:53:19,710 And finally, this expression B field equal to 1/c. 816 00:53:19,710 --> 00:53:21,860 K, which is the direction of propagation 817 00:53:21,860 --> 00:53:26,080 cross E field is still valid because basically we 818 00:53:26,080 --> 00:53:30,730 have shown that it works for the plane wave pointing 819 00:53:30,730 --> 00:53:33,940 to the x direction propagating to the z direction. 820 00:53:33,940 --> 00:53:37,220 We can always redefine the coordinate system 821 00:53:37,220 --> 00:53:41,610 because we can actually rotate this coordinate system 822 00:53:41,610 --> 00:53:44,650 and the physics should not change. 823 00:53:44,650 --> 00:53:48,610 Therefore, you must see that this expression 824 00:53:48,610 --> 00:53:50,370 must be still valid. 825 00:53:50,370 --> 00:53:55,990 And also that the direction of the electric field, 826 00:53:55,990 --> 00:53:58,570 which is actually proportional to E0, 827 00:53:58,570 --> 00:54:03,760 must be perpendicular to the direction of propagation. 828 00:54:03,760 --> 00:54:09,750 So that is actually what we can actually 829 00:54:09,750 --> 00:54:14,440 learn a general description of electric field pointing 830 00:54:14,440 --> 00:54:17,950 to some random direction. 831 00:54:17,950 --> 00:54:21,640 So we have talked about the progressing wave 832 00:54:21,640 --> 00:54:26,020 solution and also the plane wave and also 833 00:54:26,020 --> 00:54:30,590 the corresponding magnetic field. 834 00:54:30,590 --> 00:54:32,770 I hope that you can actually apply this-- 835 00:54:32,770 --> 00:54:34,990 the technique which we learned here-- 836 00:54:34,990 --> 00:54:37,700 if you are given a magnetic field, 837 00:54:37,700 --> 00:54:41,890 you must know that there must be a corresponding electric field 838 00:54:41,890 --> 00:54:45,280 because they cannot be separated from each other. 839 00:54:45,280 --> 00:54:49,510 And you can actually obtain the corresponding electric field 840 00:54:49,510 --> 00:54:56,150 if you are given magnetic field by using Maxwell's equations. 841 00:54:56,150 --> 00:54:59,650 So what is going to happen is that 842 00:54:59,650 --> 00:55:02,680 now if I emit this photon-- 843 00:55:02,680 --> 00:55:06,790 or say this electromagnetic wave from the light source, 844 00:55:06,790 --> 00:55:08,580 for example, that one-- 845 00:55:08,580 --> 00:55:12,490 the one of which is pointing at my face. 846 00:55:12,490 --> 00:55:14,440 Basically, my face is going to bounce 847 00:55:14,440 --> 00:55:18,370 some of the electromagnetic field around. 848 00:55:18,370 --> 00:55:21,910 And some that actually go out of the window. 849 00:55:21,910 --> 00:55:24,220 And then when they go out the window, 850 00:55:24,220 --> 00:55:25,940 maybe they are lucky they are not hitting 851 00:55:25,940 --> 00:55:27,650 any building in the MIT. 852 00:55:27,650 --> 00:55:30,040 Then what is going to happen is that they're 853 00:55:30,040 --> 00:55:35,030 going to propagate forever toward the end of the universe. 854 00:55:35,030 --> 00:55:38,890 Really, they are going straight forever as you 855 00:55:38,890 --> 00:55:40,580 can see from this solution. 856 00:55:40,580 --> 00:55:43,570 It's like some kind of wave propagating forever 857 00:55:43,570 --> 00:55:44,710 at the speed of light. 858 00:55:44,710 --> 00:55:51,100 If they don't encounter anything before the end of life 859 00:55:51,100 --> 00:55:53,020 of the electromagnetic wave, it's 860 00:55:53,020 --> 00:55:56,920 going to be propagating forever toward that direction-- 861 00:55:56,920 --> 00:55:59,380 escaping from that window. 862 00:55:59,380 --> 00:56:04,930 So that is actually fascinating and-- 863 00:56:04,930 --> 00:56:07,870 but we would like to introduce some more excitement 864 00:56:07,870 --> 00:56:09,460 to see what is going to happen. 865 00:56:09,460 --> 00:56:14,340 So what I'm going to do is now instead of only discussing 866 00:56:14,340 --> 00:56:16,560 about the plane wave-- 867 00:56:16,560 --> 00:56:18,940 what I'm going to do is that I would 868 00:56:18,940 --> 00:56:23,170 like to add a perfect conductor into the game 869 00:56:23,170 --> 00:56:26,500 and see what is going to happen. 870 00:56:26,500 --> 00:56:31,056 So what do I mean by a perfect conductor? 871 00:56:31,056 --> 00:56:34,531 A perfect conductor can be seen in a musical, 872 00:56:34,531 --> 00:56:35,280 like in a concert. 873 00:56:35,280 --> 00:56:36,030 [LAUGHTER] 874 00:56:36,030 --> 00:56:38,280 But the one which I am talking about 875 00:56:38,280 --> 00:56:41,920 is not that one, which is also fascinating, 876 00:56:41,920 --> 00:56:45,090 but this is a different system. 877 00:56:45,090 --> 00:56:48,870 The interesting thing is that both the conductors 878 00:56:48,870 --> 00:56:52,670 in the concert and this one is very busy. 879 00:56:52,670 --> 00:56:54,960 It's a very busy system. 880 00:56:54,960 --> 00:56:58,480 What do I mean by perfect conductor? 881 00:56:58,480 --> 00:57:02,820 That means all the little charges inside the conductor 882 00:57:02,820 --> 00:57:04,650 can move freely. 883 00:57:04,650 --> 00:57:08,820 So if they move they don't actually cause any energy. 884 00:57:08,820 --> 00:57:10,290 They can move around-- 885 00:57:10,290 --> 00:57:13,410 all the electrons inside the conductor 886 00:57:13,410 --> 00:57:19,240 can be moved freely without costing anything, without any 887 00:57:19,240 --> 00:57:21,520 of this energy dissipation. 888 00:57:21,520 --> 00:57:26,040 So that's actually what I mean by perfect conductor. 889 00:57:26,040 --> 00:57:30,090 What do I mean by a very busy system? 890 00:57:30,090 --> 00:57:34,690 That means whenever there are any distortion 891 00:57:34,690 --> 00:57:36,540 on the electric field-- 892 00:57:36,540 --> 00:57:40,980 any electric field approaching to this conductor-- 893 00:57:40,980 --> 00:57:43,360 what is going to happen is that this conductor will, 894 00:57:43,360 --> 00:57:45,134 oh, this is electric field, so I have 895 00:57:45,134 --> 00:57:46,550 to move from some of my electrons. 896 00:57:46,550 --> 00:57:50,280 Then it's going to cancel all the electric field 897 00:57:50,280 --> 00:57:53,220 inside the conductor because it cost nothing. 898 00:57:53,220 --> 00:57:55,290 So you have fast-- 899 00:57:55,290 --> 00:57:59,340 really fast the react to this change in the electric field 900 00:57:59,340 --> 00:58:02,130 and they really carefully arrange all the electrons. 901 00:58:02,130 --> 00:58:05,730 And so that the electric field is canceled. 902 00:58:05,730 --> 00:58:09,150 Otherwise, all those electrons will continue to move around 903 00:58:09,150 --> 00:58:10,910 until this happens-- 904 00:58:10,910 --> 00:58:13,290 this cancellation happens. 905 00:58:13,290 --> 00:58:16,830 So that's actually what I mean by a busy world 906 00:58:16,830 --> 00:58:22,220 and what I mean by a perfect conductor. 907 00:58:22,220 --> 00:58:27,490 If I put this conductor into game, what is going to happen? 908 00:58:27,490 --> 00:58:33,170 What is going to happen is that if I consider a situation-- 909 00:58:33,170 --> 00:58:39,050 if I have my x's defined here pointing up to be the x-axis, 910 00:58:39,050 --> 00:58:42,890 pointing to the right to the z-axis, pointing to the-- 911 00:58:42,890 --> 00:58:46,010 pointing toward you is actually the y-axis. 912 00:58:46,010 --> 00:58:50,500 So I can now again take the plane wave 913 00:58:50,500 --> 00:58:53,210 which I started with. 914 00:58:53,210 --> 00:58:55,700 There will be a plane wave like this. 915 00:58:55,700 --> 00:59:03,280 And it's going toward a piece of perfect conductor. 916 00:59:03,280 --> 00:59:05,590 What is going to happen is that as I actually 917 00:59:05,590 --> 00:59:11,320 mentioned before there are many charges all over the place. 918 00:59:11,320 --> 00:59:15,370 They are going to quickly rearrange-- 919 00:59:15,370 --> 00:59:18,920 all those charges to cancel the electric field. 920 00:59:18,920 --> 00:59:22,090 So if you have a plane wave going 921 00:59:22,090 --> 00:59:26,110 toward the perfect conductor at the surface 922 00:59:26,110 --> 00:59:27,900 of the perfect conductor-- 923 00:59:27,900 --> 00:59:30,880 the electric field will become 0. 924 00:59:33,630 --> 00:59:37,980 But if you have only one plane wave it cannot be-- 925 00:59:37,980 --> 00:59:41,680 the magnitude cannot be equal to 0 because I know the functional 926 00:59:41,680 --> 00:59:42,180 form. 927 00:59:42,180 --> 00:59:45,420 I know that the functional form of that electric field 928 00:59:45,420 --> 00:59:49,260 is E0 cosine kz minus omega t. 929 00:59:49,260 --> 00:59:55,760 If I place this perfect conductor at Z equal to 0, 930 00:59:55,760 --> 00:59:58,470 then I can evaluate the electric field 931 00:59:58,470 --> 01:00:01,740 is not equal to 0 because it is actually equal to E0 932 01:00:01,740 --> 01:00:04,890 cosine minus omega t. 933 01:00:04,890 --> 01:00:09,480 So what can I do to cancel the electric field? 934 01:00:09,480 --> 01:00:13,170 This is actually very similar to the situation 935 01:00:13,170 --> 01:00:22,640 when you have a progressing wave on this string hitting a wall. 936 01:00:22,640 --> 01:00:27,910 Because the magnitude of the string 937 01:00:27,910 --> 01:00:30,800 which is fed to the equilibrium position 938 01:00:30,800 --> 01:00:33,650 is actually equal to 0. 939 01:00:33,650 --> 01:00:35,260 That's actually what we have learned 940 01:00:35,260 --> 01:00:37,100 in the last few lectures. 941 01:00:37,100 --> 01:00:40,400 And this is actually exactly the same situation, right? 942 01:00:40,400 --> 01:00:42,430 You have a progressing wave. 943 01:00:42,430 --> 01:00:46,670 And there is some kind of boundary, which is actually 944 01:00:46,670 --> 01:00:51,380 when this progressing plane wave encounter 945 01:00:51,380 --> 01:00:53,300 this perfect conductor. 946 01:00:53,300 --> 01:00:55,640 There-- the electric field-- 947 01:00:55,640 --> 01:01:11,170 the boundary condition-- has to be E is actually-- 948 01:01:17,070 --> 01:01:22,180 E, x, y, 0, which is actually the position of the z 949 01:01:22,180 --> 01:01:24,510 of the perfect conductor. 950 01:01:24,510 --> 01:01:29,130 As a function of time will be equal to 0. 951 01:01:29,130 --> 01:01:35,100 The whole plane will have 0 electric field. 952 01:01:35,100 --> 01:01:39,660 So that means there must be what kind of wave? 953 01:01:39,660 --> 01:01:45,770 There must be a reflective wave because of the presence 954 01:01:45,770 --> 01:01:48,170 of the perfect conductor. 955 01:01:48,170 --> 01:01:50,480 It's actually similar to the situation 956 01:01:50,480 --> 01:01:54,170 which we discussed there's a progressing wave hitting 957 01:01:54,170 --> 01:01:55,130 the wall. 958 01:01:55,130 --> 01:01:58,130 And this string wall system-- there 959 01:01:58,130 --> 01:02:01,320 will be a reflecting wave coming out of it. 960 01:02:01,320 --> 01:02:05,030 So therefore, what we are expecting is some kind of-- 961 01:02:08,940 --> 01:02:11,910 refracting wave which actually cancel 962 01:02:11,910 --> 01:02:16,980 the magnitude of the electric field at Z equal to 0. 963 01:02:16,980 --> 01:02:20,600 And then this progressing wave is going to the left-hand side 964 01:02:20,600 --> 01:02:22,140 direction. 965 01:02:22,140 --> 01:02:25,460 So now, I can actually write down the incident wave-- 966 01:02:29,170 --> 01:02:29,950 expression. 967 01:02:29,950 --> 01:02:32,440 The incident wave-- 968 01:02:32,440 --> 01:02:38,202 I call it Ei, this is Ei-- 969 01:02:38,202 --> 01:02:48,810 is expressed as E0 over 2, cosine kz minus omega t. 970 01:02:48,810 --> 01:02:52,280 This is actually what I putting to the system. 971 01:02:52,280 --> 01:02:57,030 The magnitude is E0 over 2, and it's actually 972 01:02:57,030 --> 01:03:00,440 propagating toward the z direction, 973 01:03:00,440 --> 01:03:02,010 as you can see from here. 974 01:03:02,010 --> 01:03:05,760 And that the direction of the electric field 975 01:03:05,760 --> 01:03:08,730 is in the x direction. 976 01:03:08,730 --> 01:03:11,490 And of course the E field will have 977 01:03:11,490 --> 01:03:15,840 a corresponding magnetic field, which is actually-- 978 01:03:15,840 --> 01:03:20,880 you can actually write it down directly using this formula-- 979 01:03:20,880 --> 01:03:26,040 B equal to 1 over c, K cross E. K here is z, 980 01:03:26,040 --> 01:03:28,500 therefore you can quickly evaluate 981 01:03:28,500 --> 01:03:32,490 and then conclude that the magnetic field 982 01:03:32,490 --> 01:03:35,900 must be in the y direction. 983 01:03:35,900 --> 01:03:38,910 And that the magnitude of the magnetic field 984 01:03:38,910 --> 01:03:43,270 would be E0 divided by 2 c. 985 01:03:43,270 --> 01:03:48,960 Cosine Kz and this omega t. 986 01:03:48,960 --> 01:03:51,960 So that is actually the incident wave. 987 01:03:51,960 --> 01:03:55,200 And of course I also need, as I discussed, 988 01:03:55,200 --> 01:04:00,900 there must be a reflective wave, Er, 989 01:04:00,900 --> 01:04:07,020 which you actually cancel the electric field at z equal to 0. 990 01:04:07,020 --> 01:04:09,810 If that cancels the incident wave, 991 01:04:09,810 --> 01:04:14,180 that means the magnitude must be in the opposite direction 992 01:04:14,180 --> 01:04:15,870 of the incident wave. 993 01:04:15,870 --> 01:04:17,850 Therefore, I can quickly write down 994 01:04:17,850 --> 01:04:22,140 what would be the resulting reflective wave that 995 01:04:22,140 --> 01:04:27,130 would be equal to minus E0 over 2, 996 01:04:27,130 --> 01:04:35,760 cosine minus Kz, minus omega t in the x direction. 997 01:04:35,760 --> 01:04:38,340 And then the corresponding B field, 998 01:04:38,340 --> 01:04:43,500 I can also write it down using exactly the same formula. 999 01:04:43,500 --> 01:04:46,110 And basically what I conclude is that this 1000 01:04:46,110 --> 01:04:53,830 will be equal to E0 over 2 c, cosine minus kz, 1001 01:04:53,830 --> 01:04:58,500 minus omega t in the y direction. 1002 01:04:58,500 --> 01:05:05,922 So you can actually check this expression after the direction. 1003 01:05:08,630 --> 01:05:11,840 So now, I would like to check what 1004 01:05:11,840 --> 01:05:16,310 would be the magnitude of the electric field at z equal to 0. 1005 01:05:16,310 --> 01:05:18,740 So basically, at z equal to 0, you 1006 01:05:18,740 --> 01:05:21,350 have something which is proportional to cosine 1007 01:05:21,350 --> 01:05:24,140 minus omega t for the incident wave. 1008 01:05:24,140 --> 01:05:27,890 And then the magnitude is E0 over 2. 1009 01:05:27,890 --> 01:05:33,900 And if you evaluate z equal to 0, basically you get minus E0 1010 01:05:33,900 --> 01:05:37,550 over 2 cosine minus omega t. 1011 01:05:37,550 --> 01:05:40,640 Therefore, they really cancel and give you 1012 01:05:40,640 --> 01:05:43,820 the desired boundary condition, which 1013 01:05:43,820 --> 01:05:46,390 is actually E equal to 0 and the surface 1014 01:05:46,390 --> 01:05:49,010 of the perfect conductor. 1015 01:05:49,010 --> 01:05:51,530 So that's very nice. 1016 01:05:51,530 --> 01:05:54,920 And this is actually the physics of which we already learned 1017 01:05:54,920 --> 01:05:58,070 from this string wall system. 1018 01:05:58,070 --> 01:06:01,130 So what I can do now is to calculate 1019 01:06:01,130 --> 01:06:04,900 the total electric field if I add them together. 1020 01:06:04,900 --> 01:06:07,190 Basically, I would get E-- 1021 01:06:07,190 --> 01:06:11,040 total electric field, which is actually 1022 01:06:11,040 --> 01:06:14,180 overlapping the incident and the reflective wave. 1023 01:06:14,180 --> 01:06:21,920 What I am going to get is Ei plus Er. 1024 01:06:21,920 --> 01:06:28,230 And basically, what I get is E0 over 2 because the incident 1025 01:06:28,230 --> 01:06:31,110 wave and the reflective wave of the electric field 1026 01:06:31,110 --> 01:06:33,320 is always in the x direction. 1027 01:06:33,320 --> 01:06:37,070 Therefore, I only need to take care of the x direction. 1028 01:06:37,070 --> 01:06:45,620 So basically, I have cosine kz minus omega t, minus-- 1029 01:06:45,620 --> 01:06:47,480 right, because there's a minus sign here-- 1030 01:06:47,480 --> 01:06:55,597 minus cosine minus kz, minus omega t in the x direction. 1031 01:06:55,597 --> 01:06:58,520 And there should be-- 1032 01:06:58,520 --> 01:07:02,570 And of course this is a cosine minus cosine. 1033 01:07:02,570 --> 01:07:04,220 So we have all the formulas-- one 1034 01:07:04,220 --> 01:07:07,820 from, for example, Wikipedia, or from your textbook. 1035 01:07:07,820 --> 01:07:10,610 So you can actually calculate this-- 1036 01:07:10,610 --> 01:07:22,280 rewrite this expression to be E0 sine omega t, sine kz. 1037 01:07:22,280 --> 01:07:25,551 And then this is actually in the x direction. 1038 01:07:28,150 --> 01:07:29,950 Everybody's following? 1039 01:07:29,950 --> 01:07:32,860 I hope it's not too fast. 1040 01:07:32,860 --> 01:07:34,570 All right, and of course, I can also 1041 01:07:34,570 --> 01:07:37,690 calculate the corresponding B field. 1042 01:07:37,690 --> 01:07:41,725 So it's actually again, exactly the same thing-- 1043 01:07:41,725 --> 01:07:45,340 Bi plus Br. 1044 01:07:45,340 --> 01:07:48,230 And basically, I will skip the step. 1045 01:07:48,230 --> 01:07:52,990 Basically, you can add this term and that term. 1046 01:07:52,990 --> 01:07:56,140 And you will be able to conclude that the B field will 1047 01:07:56,140 --> 01:08:02,280 be equal to E0 over c cosine omega t, 1048 01:08:02,280 --> 01:08:07,388 cosine kz in the y direction. 1049 01:08:11,750 --> 01:08:15,800 This is actually pretty interesting. 1050 01:08:15,800 --> 01:08:20,779 If you look at this result, I have a electric field, 1051 01:08:20,779 --> 01:08:24,920 which is proportional to E0, the magnitude, sine omega 1052 01:08:24,920 --> 01:08:28,720 t, and sine kz. 1053 01:08:28,720 --> 01:08:30,470 What does that I mean? 1054 01:08:30,470 --> 01:08:34,160 This is a special kind of wave which we learned before. 1055 01:08:34,160 --> 01:08:36,416 What kind of wave is this? 1056 01:08:36,416 --> 01:08:37,490 AUDIENCE: Standing. 1057 01:08:37,490 --> 01:08:39,740 YEN-JIE LEE: It's a standing wave 1058 01:08:39,740 --> 01:08:44,510 because the shape is actually fixed, the sine kz. 1059 01:08:44,510 --> 01:08:48,279 And the magnitude is actually changing up and down 1060 01:08:48,279 --> 01:08:50,246 at the angle frequency omega t. 1061 01:08:50,246 --> 01:08:51,120 It's a standing wave. 1062 01:08:58,300 --> 01:09:03,840 Another thing which is really interesting is that if we look 1063 01:09:03,840 --> 01:09:08,430 at the expression of a electric field and the magnetic field-- 1064 01:09:08,430 --> 01:09:09,460 if we compare that-- 1065 01:09:09,460 --> 01:09:10,950 one is actually sine, sine. 1066 01:09:10,950 --> 01:09:12,870 The other one is cosine, cosine. 1067 01:09:15,770 --> 01:09:19,240 That's kind of interesting because this is actually 1068 01:09:19,240 --> 01:09:22,910 different from what we actually usually learn 1069 01:09:22,910 --> 01:09:25,850 from the progressing wave solution, 1070 01:09:25,850 --> 01:09:27,350 or traveling wave solution. 1071 01:09:27,350 --> 01:09:34,160 Where the electric field and the magnetic field are in phase. 1072 01:09:34,160 --> 01:09:35,540 There's no phase difference. 1073 01:09:35,540 --> 01:09:39,529 In the case of the superposition of the incident 1074 01:09:39,529 --> 01:09:41,540 and the reflective wave-- 1075 01:09:41,540 --> 01:09:44,960 the solution of a standing wave-- 1076 01:09:44,960 --> 01:09:50,959 actually you can see that the phase of the B field and the E 1077 01:09:50,959 --> 01:09:55,090 field are different. 1078 01:09:55,090 --> 01:10:00,220 Finally, very important-- you will see that-- 1079 01:10:00,220 --> 01:10:01,780 look at this expression-- 1080 01:10:01,780 --> 01:10:04,420 B equal to 1/c, K cross E-- 1081 01:10:07,020 --> 01:10:11,860 that means this only work for traveling wave. 1082 01:10:11,860 --> 01:10:17,860 Clearly, this doesn't work for standing waves. 1083 01:10:17,860 --> 01:10:19,010 So very important. 1084 01:10:19,010 --> 01:10:23,180 So don't blindly apply this expression. 1085 01:10:23,180 --> 01:10:26,500 This is only useful for the traveling wave solution. 1086 01:10:26,500 --> 01:10:31,470 And you can see a very concrete example here. 1087 01:10:31,470 --> 01:10:35,980 This doesn't work for standing waves. 1088 01:10:35,980 --> 01:10:38,080 That's kind of interesting. 1089 01:10:38,080 --> 01:10:41,760 And if you look at this result, you 1090 01:10:41,760 --> 01:10:46,190 will see that if I don't have magnetic field-- 1091 01:10:46,190 --> 01:10:48,810 if I only have the electric field-- 1092 01:10:48,810 --> 01:10:53,590 there will be a instant of time, for example, t equal to 0. 1093 01:10:53,590 --> 01:10:57,160 When t is equal to 0, sine is equal to 0. 1094 01:10:57,160 --> 01:10:58,660 What is going to happen? 1095 01:10:58,660 --> 01:11:00,810 You will have no electric field. 1096 01:11:03,640 --> 01:11:06,220 That means electric field completely 1097 01:11:06,220 --> 01:11:08,170 disappear because we are operating 1098 01:11:08,170 --> 01:11:10,540 this system in vacuum. 1099 01:11:10,540 --> 01:11:12,180 There's nowhere to hide. 1100 01:11:12,180 --> 01:11:14,950 Where is the energy? 1101 01:11:14,950 --> 01:11:19,630 The energy, fortunately-- electric field 1102 01:11:19,630 --> 01:11:23,840 have a very good partner, which is actually B field. 1103 01:11:23,840 --> 01:11:33,320 All the energy's actually stored in the form of magnetic field. 1104 01:11:33,320 --> 01:11:39,220 You can see that now magnetic field is reaching the maximum. 1105 01:11:39,220 --> 01:11:43,428 So of course I can now calculate the Poynting vector. 1106 01:11:50,680 --> 01:11:57,070 Poynting vector is E cross B divided by mu zero. 1107 01:11:57,070 --> 01:12:02,860 And these will be equal to one over Mu 0, Ex, By, 1108 01:12:02,860 --> 01:12:04,300 and the z direction. 1109 01:12:04,300 --> 01:12:06,580 There's only one term which survive. 1110 01:12:06,580 --> 01:12:09,700 So Poynting vector is not pointing vector. 1111 01:12:09,700 --> 01:12:12,100 It's not pointing around. 1112 01:12:12,100 --> 01:12:14,500 There's a gentleman who is called Poynting 1113 01:12:14,500 --> 01:12:15,970 and he has a vector. 1114 01:12:15,970 --> 01:12:21,530 And this vector is a directional energy flux. 1115 01:12:21,530 --> 01:12:26,415 It's a directional energy flux, or the rate of energy transfer 1116 01:12:26,415 --> 01:12:31,390 per unit area. 1117 01:12:31,390 --> 01:12:35,460 So that is actually the meaning of Poynting vector. 1118 01:12:35,460 --> 01:12:39,050 And then each magnitude is proportional to E cross 1119 01:12:39,050 --> 01:12:42,280 B divided by Mu 0. 1120 01:12:42,280 --> 01:12:45,100 So I can calculate that. 1121 01:12:45,100 --> 01:12:47,330 Basically, I have the E and the B-- 1122 01:12:47,330 --> 01:12:50,080 Ex and By, then I can calculate. 1123 01:12:50,080 --> 01:12:54,620 That would be equal to E0 square, over Mu 0, 1124 01:12:54,620 --> 01:13:02,080 sine omega t, cosine omega t, cosine Kz, 1125 01:13:02,080 --> 01:13:09,540 sine Kz in the z direction because I have x cross y. 1126 01:13:09,540 --> 01:13:12,400 And I'm going to get the z direction. 1127 01:13:12,400 --> 01:13:14,230 And I can simplify this. 1128 01:13:14,230 --> 01:13:17,540 I have the sine, cosine. 1129 01:13:17,540 --> 01:13:20,290 And also all have cosine, and sine. 1130 01:13:20,290 --> 01:13:23,320 Basically, you can simplify this expression 1131 01:13:23,320 --> 01:13:31,870 and get E0 squared divided by 4, Mu 0c sine 2 omega t, 1132 01:13:31,870 --> 01:13:37,270 sine 2kz in the z direction. 1133 01:13:37,270 --> 01:13:43,090 So you can see that the directional energy 1134 01:13:43,090 --> 01:13:46,050 flux is in the z direction. 1135 01:13:50,090 --> 01:13:53,520 It has a vector-- 1136 01:13:53,520 --> 01:13:58,310 it has a wave number 2 times of the original wave number. 1137 01:13:58,310 --> 01:14:02,855 And it's actually going up and down 2 times 1138 01:14:02,855 --> 01:14:07,730 of the speed of the oscillation of the original electromagnetic 1139 01:14:07,730 --> 01:14:09,770 wave. 1140 01:14:09,770 --> 01:14:14,720 And this energy is actually vibrating up and down. 1141 01:14:14,720 --> 01:14:20,080 And the shape of this energy transfer Poynting vector 1142 01:14:20,080 --> 01:14:23,600 is actually a sine wave. 1143 01:14:23,600 --> 01:14:28,710 So that this is actually how the microwave actually works. 1144 01:14:28,710 --> 01:14:30,530 So basically, what we are doing is 1145 01:14:30,530 --> 01:14:37,360 to have generate microwave inside your device. 1146 01:14:37,360 --> 01:14:39,680 And in the oven this microwave is actually 1147 01:14:39,680 --> 01:14:42,080 bouncing back and forth because you 1148 01:14:42,080 --> 01:14:45,680 have metal walls, which actually bounce 1149 01:14:45,680 --> 01:14:48,380 the electromagnetic field back and forth. 1150 01:14:48,380 --> 01:14:55,400 And it really can cook the food by vibrating the molecules 1151 01:14:55,400 --> 01:14:58,330 inside the food back and forth. 1152 01:14:58,330 --> 01:15:02,810 So as you can see the magnitude of the Poynting vector 1153 01:15:02,810 --> 01:15:06,340 is actually isolating up and down. 1154 01:15:06,340 --> 01:15:09,380 That actually cause additional vibration and that heat up 1155 01:15:09,380 --> 01:15:10,290 the food. 1156 01:15:10,290 --> 01:15:12,350 So after this lecture, you will be 1157 01:15:12,350 --> 01:15:15,590 able to say proudly that you understand 1158 01:15:15,590 --> 01:15:17,450 the physics of microwave oven. 1159 01:15:17,450 --> 01:15:19,970 [LAUGHTER] 1160 01:15:19,970 --> 01:15:21,000 Thank you very much. 1161 01:15:21,000 --> 01:15:23,030 I hope you enjoyed the lecture today. 1162 01:15:23,030 --> 01:15:26,230 And you have any questions, I will be here.