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 --> 00:00:20,320 at ocw.mit.edu. 8 00:00:24,020 --> 00:00:26,910 PROFESSOR: OK, so happy to see you again. 9 00:00:26,910 --> 00:00:30,140 Welcome back to 8.03. 10 00:00:30,140 --> 00:00:33,360 So in the beginning of the class, 11 00:00:33,360 --> 00:00:36,720 I will give you a reminder about what we discussed last time. 12 00:00:36,720 --> 00:00:40,370 Then the main topic we are going to discuss today 13 00:00:40,370 --> 00:00:42,885 is about how to take good photos. 14 00:00:47,280 --> 00:00:48,410 So let's get started. 15 00:00:48,410 --> 00:00:52,250 So last time we have been working together 16 00:00:52,250 --> 00:00:54,440 trying to understand how to produce 17 00:00:54,440 --> 00:00:59,660 electromagnetic waves which can travel to a distance which 18 00:00:59,660 --> 00:01:02,600 is actually very, very far away, a place which is very, very 19 00:01:02,600 --> 00:01:04,519 far away from the source. 20 00:01:04,519 --> 00:01:08,480 And what we actually figured out is that in order to do that, 21 00:01:08,480 --> 00:01:15,650 you have to create a kink in the electric field line 22 00:01:15,650 --> 00:01:20,120 such that you will propagate and produce radiation. 23 00:01:20,120 --> 00:01:22,940 And this is actually what we have done last time. 24 00:01:22,940 --> 00:01:26,780 And what we concluded is that if you 25 00:01:26,780 --> 00:01:29,990 want to produce electromagnetic waves, 26 00:01:29,990 --> 00:01:33,800 you have to introduce acceleration 27 00:01:33,800 --> 00:01:37,320 of that charge, such that you will be able to produce 28 00:01:37,320 --> 00:01:38,890 electromagnetic waves. 29 00:01:38,890 --> 00:01:43,210 And also, we have derived, based on geometrical arguments, what 30 00:01:43,210 --> 00:01:48,040 would be the magnitude and the direction 31 00:01:48,040 --> 00:01:52,920 of the radiated electric field and magnetic field. 32 00:01:52,920 --> 00:01:55,910 And it's actually showing here. 33 00:01:55,910 --> 00:01:58,610 The radiated electric field is going 34 00:01:58,610 --> 00:02:03,080 to be in the opposite direction of the projection 35 00:02:03,080 --> 00:02:06,830 of the acceleration, a perp. 36 00:02:06,830 --> 00:02:09,139 The magnitude is proportional to a perp, 37 00:02:09,139 --> 00:02:14,630 but only evaluated at retarded time, t prime equal to t 38 00:02:14,630 --> 00:02:18,350 minus r/c, where r is actually the distance 39 00:02:18,350 --> 00:02:22,640 between the observer and the radiating charge. 40 00:02:22,640 --> 00:02:29,300 And the reason why we have this t prime, the retarded time, 41 00:02:29,300 --> 00:02:32,960 is because of the speed of propagation of information. 42 00:02:32,960 --> 00:02:35,810 So that's actually what we discussed last time. 43 00:02:35,810 --> 00:02:40,980 And you cannot actually instantaneously send 44 00:02:40,980 --> 00:02:44,180 the information about the acceleration of this charge 45 00:02:44,180 --> 00:02:46,760 to somewhere which is very, very far away. 46 00:02:46,760 --> 00:02:50,090 Therefore, the acceleration, a perp, 47 00:02:50,090 --> 00:02:52,760 is evaluated at t prime, which is 48 00:02:52,760 --> 00:02:57,770 t minus r/c, the amount of time for the speed 49 00:02:57,770 --> 00:03:01,960 of the information traveling at the speed of light. 50 00:03:01,960 --> 00:03:07,550 And also, the corresponding magnetic field 51 00:03:07,550 --> 00:03:12,620 can be also evaluated, whether it's through Maxwell's equation 52 00:03:12,620 --> 00:03:16,060 or using this formula, this t here. 53 00:03:16,060 --> 00:03:19,010 And finally, we will be able to evaluate 54 00:03:19,010 --> 00:03:23,300 what would be the pointing vector, the energy flux 55 00:03:23,300 --> 00:03:28,640 direction or energy flux through exactly the same equation which 56 00:03:28,640 --> 00:03:30,600 we used before. 57 00:03:30,600 --> 00:03:33,780 That's actually what we have learned last time. 58 00:03:33,780 --> 00:03:36,470 And today, what we are going to do 59 00:03:36,470 --> 00:03:40,140 is to learn how to take photos. 60 00:03:40,140 --> 00:03:41,330 So we have prepared ourself. 61 00:03:41,330 --> 00:03:45,190 We know how to produce electromagnetic waves. 62 00:03:45,190 --> 00:03:49,440 We also know about polarization of electromagnetic waves. 63 00:03:49,440 --> 00:03:52,940 And we also know how the polarizer actually works. 64 00:03:52,940 --> 00:03:58,190 So that means we will be able to make very good photos, 65 00:03:58,190 --> 00:03:59,090 theoretically. 66 00:03:59,090 --> 00:04:01,620 So yeah, that's what we all care about. 67 00:04:01,620 --> 00:04:05,480 Theoretically, we can actually make very good photos. 68 00:04:05,480 --> 00:04:08,600 So the first thing which we would like to discuss 69 00:04:08,600 --> 00:04:13,490 is how to make very good contrast when you actually 70 00:04:13,490 --> 00:04:15,890 take a photo of the sky. 71 00:04:15,890 --> 00:04:18,649 So as you can see, the left-hand side 72 00:04:18,649 --> 00:04:22,070 is a photo taken without a polarizer. 73 00:04:22,070 --> 00:04:25,850 And the right-hand side is the photo taken with a polarizer. 74 00:04:25,850 --> 00:04:28,130 You can see-- aha! -- the contrast, 75 00:04:28,130 --> 00:04:31,970 or say the sky is actually darker, 76 00:04:31,970 --> 00:04:35,740 therefore you can see the cloud much more clearly. 77 00:04:35,740 --> 00:04:38,090 And also on the same graph, you can see 78 00:04:38,090 --> 00:04:40,970 there is a photo at the beach. 79 00:04:40,970 --> 00:04:44,600 And you can see exactly the same phenomenon. 80 00:04:44,600 --> 00:04:50,130 And now we are in the position to understand what is going on. 81 00:04:50,130 --> 00:04:54,290 So this is actually why we can use polarizer 82 00:04:54,290 --> 00:04:58,720 to make such a good photo. 83 00:04:58,720 --> 00:05:03,290 So now we know what is actually happening at the sun, right? 84 00:05:03,290 --> 00:05:04,910 So at the sun, there are something 85 00:05:04,910 --> 00:05:06,680 which is oscillating-- 86 00:05:06,680 --> 00:05:14,360 OK, some kind of emission from the sun. 87 00:05:14,360 --> 00:05:17,720 And those emissions are not correlated to each other. 88 00:05:17,720 --> 00:05:21,980 And that produces unpolarized sunlight. 89 00:05:21,980 --> 00:05:24,760 So if you're looking to the sun, you 90 00:05:24,760 --> 00:05:28,300 are looking at unpolarized light. 91 00:05:28,300 --> 00:05:34,650 On the other hand, if you are looking at the sky, 92 00:05:34,650 --> 00:05:40,020 roughly like 45 or 90 degree-- 93 00:05:40,020 --> 00:05:44,110 OK, 45 degree from on the sun, what is going to happen 94 00:05:44,110 --> 00:05:47,440 is that what you are actually seeing, 95 00:05:47,440 --> 00:05:50,410 all this light from the sky, actually, 96 00:05:50,410 --> 00:05:54,850 the sunlight after scattering between sunlight 97 00:05:54,850 --> 00:05:58,930 and the dust in the air. 98 00:05:58,930 --> 00:06:03,930 So basically, on this guide, in our air, 99 00:06:03,930 --> 00:06:06,310 there are many, many little dust, right? 100 00:06:06,310 --> 00:06:11,360 And when light-- as you shine on this dust, 101 00:06:11,360 --> 00:06:15,700 they change direction, this so-called scattering. 102 00:06:15,700 --> 00:06:21,400 And those light are collected by your camera. 103 00:06:21,400 --> 00:06:25,500 The interesting thing is that if you have a molecule which 104 00:06:25,500 --> 00:06:30,040 is actually here, and you have some unpolarized sunlight 105 00:06:30,040 --> 00:06:33,760 shining on this molecule, and it changed direction 106 00:06:33,760 --> 00:06:40,640 by 90 degrees, what is going to happen is that all the things-- 107 00:06:40,640 --> 00:06:46,240 originally you have an unpolarized sunlight. 108 00:06:46,240 --> 00:06:50,080 Therefore, you have all kinds of different polarizations, 109 00:06:50,080 --> 00:06:53,210 if you look at the electromagnetic wave. 110 00:06:53,210 --> 00:06:57,130 However, if you only choose the light which are scattered 111 00:06:57,130 --> 00:07:00,870 and are going toward this direction, apparently, 112 00:07:00,870 --> 00:07:04,240 the electromagnetic wave, or the polarization, 113 00:07:04,240 --> 00:07:06,730 or let's say the direction of the electric field, 114 00:07:06,730 --> 00:07:10,960 has to be perpendicular to the direction of propagation. 115 00:07:10,960 --> 00:07:13,540 Therefore, what is going to happen 116 00:07:13,540 --> 00:07:19,450 is that only this direction, only the polarization 117 00:07:19,450 --> 00:07:22,085 in this direction, which is perpendicular to the direction 118 00:07:22,085 --> 00:07:24,820 of propagation will survive. 119 00:07:24,820 --> 00:07:27,700 All the other components, like the one 120 00:07:27,700 --> 00:07:31,810 which is pointing upward or pointing downwards, or coming 121 00:07:31,810 --> 00:07:34,510 from the original sunlight will not survive. 122 00:07:34,510 --> 00:07:38,920 Therefore, what is going to happen is that when 123 00:07:38,920 --> 00:07:41,470 you do get 45 degrees-- 124 00:07:41,470 --> 00:07:45,190 the sky 45 degrees from the sun, the sunlight 125 00:07:45,190 --> 00:07:47,950 is actually what kind of sunlight? 126 00:07:47,950 --> 00:07:51,160 Is polarized sunlight. 127 00:07:51,160 --> 00:07:54,730 Therefore, if you tune your filter 128 00:07:54,730 --> 00:07:56,770 to be aligned with the polarization, 129 00:07:56,770 --> 00:07:59,950 you will be able to filter a large amount 130 00:07:59,950 --> 00:08:02,710 of the scattered sunlight. 131 00:08:02,710 --> 00:08:05,600 And that is actually how this works. 132 00:08:05,600 --> 00:08:09,820 And then you can see that, indeed, the sky becomes darker 133 00:08:09,820 --> 00:08:14,260 after you apply this polarizer in front of your camera. 134 00:08:14,260 --> 00:08:17,950 So that's essentially the first thing which we learn. 135 00:08:17,950 --> 00:08:20,440 The second thing is that, OK, we also 136 00:08:20,440 --> 00:08:23,620 found that the polarizer is particularly 137 00:08:23,620 --> 00:08:30,820 usefull for the filtering of the reflected light on the window. 138 00:08:30,820 --> 00:08:35,080 And of course, we can also use exactly the same technique 139 00:08:35,080 --> 00:08:37,150 to filter out the reflected light 140 00:08:37,150 --> 00:08:40,240 from the water for dipole. 141 00:08:40,240 --> 00:08:41,770 And how does this work? 142 00:08:41,770 --> 00:08:45,670 And it turns out that this actually much more complicated 143 00:08:45,670 --> 00:08:47,140 than what we thought. 144 00:08:47,140 --> 00:08:50,040 And we have to actually derive this. 145 00:08:50,040 --> 00:08:54,220 And that is actually related to the electromagnetic wave 146 00:08:54,220 --> 00:08:57,730 propagation inside the material, and also 147 00:08:57,730 --> 00:08:59,740 related to Brewster's angle. 148 00:08:59,740 --> 00:09:01,790 And that is actually the main topic 149 00:09:01,790 --> 00:09:04,570 which we are going to talk about today. 150 00:09:04,570 --> 00:09:07,340 So let's immediately get started. 151 00:09:07,340 --> 00:09:10,090 So now what we are interested is, 152 00:09:10,090 --> 00:09:13,800 how does this actually work? 153 00:09:13,800 --> 00:09:15,700 And why is this happening? 154 00:09:15,700 --> 00:09:21,520 And why is the reflected light become polarized? 155 00:09:21,520 --> 00:09:25,430 So that's the question we are trying to answer. 156 00:09:25,430 --> 00:09:27,640 So we have talked about electromagnetic waves 157 00:09:27,640 --> 00:09:28,940 in vacuum. 158 00:09:28,940 --> 00:09:32,320 And we also know how to generate electromagnetic, 159 00:09:32,320 --> 00:09:35,690 and now we are interested in electromagnetic wave 160 00:09:35,690 --> 00:09:37,600 in dielectrics. 161 00:09:37,600 --> 00:09:40,910 So we have talked about two kinds of materials already. 162 00:09:40,910 --> 00:09:43,870 The first one is a perfect conductor. 163 00:09:43,870 --> 00:09:46,120 And the second one which we are going to talk about 164 00:09:46,120 --> 00:09:48,820 is a dielectrics material. 165 00:09:48,820 --> 00:09:53,680 In case of perfect conductor, it costs nothing 166 00:09:53,680 --> 00:09:58,090 to move all the charges inside this conductor around. 167 00:09:58,090 --> 00:10:02,860 And basically, that will give you a zero electric field 168 00:10:02,860 --> 00:10:05,900 inside the conductor. 169 00:10:05,900 --> 00:10:09,760 And also, we have a limited supply of charges. 170 00:10:09,760 --> 00:10:13,570 And therefore, inside this kind of perfect conductor, 171 00:10:13,570 --> 00:10:16,630 there will be no electric field. 172 00:10:16,630 --> 00:10:21,130 On the other hand, if you have a dielectrics material, what 173 00:10:21,130 --> 00:10:24,000 is going to happen is that there are 174 00:10:24,000 --> 00:10:28,380 a lot of charges inside the material, 175 00:10:28,380 --> 00:10:31,110 but all those charges are attached 176 00:10:31,110 --> 00:10:34,050 to a specific atom or molecule. 177 00:10:34,050 --> 00:10:38,250 They cannot be moving freely all over the place. 178 00:10:38,250 --> 00:10:41,550 And that is actually so-called bound 179 00:10:41,550 --> 00:10:46,050 to the atom or the location of the molecule. 180 00:10:46,050 --> 00:10:51,570 And that introduces a little bit of complication. 181 00:10:51,570 --> 00:10:54,340 So this kind of material, they also 182 00:10:54,340 --> 00:11:00,030 respond to the external magnetic field or electric field. 183 00:11:00,030 --> 00:11:04,520 For example, in this case, I have electron cloud which 184 00:11:04,520 --> 00:11:10,320 is around the nuclei in this-- 185 00:11:10,320 --> 00:11:15,890 around the positive charge nucleus in this figure. 186 00:11:15,890 --> 00:11:18,005 And you can see that before we apply 187 00:11:18,005 --> 00:11:24,900 an external electric field, it's symmetric around zero. 188 00:11:24,900 --> 00:11:30,720 After we introduce this electric field, external electric field, 189 00:11:30,720 --> 00:11:34,830 there can be some kind of polarization produced, 190 00:11:34,830 --> 00:11:40,140 because the electrons around this nucleus 191 00:11:40,140 --> 00:11:46,710 can be moved slightly, such that this material is actually 192 00:11:46,710 --> 00:11:50,100 trying to compensate a little bit the effect 193 00:11:50,100 --> 00:11:52,500 of the external force. 194 00:11:52,500 --> 00:11:56,400 So that is actually leading to a modification 195 00:11:56,400 --> 00:11:58,810 of the electric field. 196 00:11:58,810 --> 00:12:01,260 But as I mentioned, you are not going 197 00:12:01,260 --> 00:12:06,420 to cancel all the effect of the external field. 198 00:12:06,420 --> 00:12:09,340 So how do we understand this? 199 00:12:09,340 --> 00:12:11,700 The idea is the following. 200 00:12:11,700 --> 00:12:15,220 So since this system is complicated, 201 00:12:15,220 --> 00:12:17,610 you have a free charge. 202 00:12:17,610 --> 00:12:22,380 It could have free charge as we really kick out 203 00:12:22,380 --> 00:12:25,350 or add some electrons into this system. 204 00:12:25,350 --> 00:12:28,350 It can have bounce charge. 205 00:12:28,350 --> 00:12:31,551 And if becomes a rather complicated description. 206 00:12:31,551 --> 00:12:32,800 So the idea is the followiong. 207 00:12:32,800 --> 00:12:36,720 So in order to actually-- 208 00:12:36,720 --> 00:12:40,710 for our convenience, to be similar to what 209 00:12:40,710 --> 00:12:44,650 we have been doing in vacuum case, 210 00:12:44,650 --> 00:12:48,510 our goal is to define a field which 211 00:12:48,510 --> 00:12:53,520 is actually with the material itself subtracted. 212 00:12:53,520 --> 00:12:59,310 In this case, what we can do is to define a D field, 213 00:12:59,310 --> 00:13:03,300 which is related-- 214 00:13:03,300 --> 00:13:05,400 we hope that this is actually only related 215 00:13:05,400 --> 00:13:09,750 to the free charge inside the material. 216 00:13:09,750 --> 00:13:11,910 And then we can actually do all the tricks 217 00:13:11,910 --> 00:13:14,460 which is actually similar to what we have already 218 00:13:14,460 --> 00:13:17,140 learned from the vacuum-- 219 00:13:17,140 --> 00:13:21,090 Maxwell's equation to solve the problems inside material. 220 00:13:21,090 --> 00:13:24,270 So that's essentially our goal. 221 00:13:24,270 --> 00:13:29,980 In order to do that, we have to classify the total charge 222 00:13:29,980 --> 00:13:32,920 density, rho, into two components. 223 00:13:32,920 --> 00:13:35,320 The first component is the free charge, 224 00:13:35,320 --> 00:13:39,000 which essentially is the charges which can travel freely 225 00:13:39,000 --> 00:13:41,250 inside the media. 226 00:13:41,250 --> 00:13:45,990 And the bound charge, which is actually, as I mentioned, 227 00:13:45,990 --> 00:13:48,260 for example, those electrons-- 228 00:13:48,260 --> 00:13:49,440 the electron cloud. 229 00:13:49,440 --> 00:13:55,310 And essentially, they are bound around central location 230 00:13:55,310 --> 00:13:57,240 in the media. 231 00:13:57,240 --> 00:13:58,780 So the idea is the following. 232 00:13:58,780 --> 00:14:03,120 So I can now define a D field. 233 00:14:03,120 --> 00:14:10,170 This D field is a so-called electric displacement field. 234 00:14:10,170 --> 00:14:17,682 It's defined as epsilon0 times E field plus a P vector, which 235 00:14:17,682 --> 00:14:20,610 is the polarization vector. 236 00:14:20,610 --> 00:14:24,660 Where this P vector is defined in the following way-- 237 00:14:24,660 --> 00:14:31,050 minus divergence of the P factor is actually 238 00:14:31,050 --> 00:14:37,020 equal to the bound charge density, which is rho. 239 00:14:37,020 --> 00:14:41,550 If we have this definition and we actually continue 240 00:14:41,550 --> 00:14:44,190 and write down the Gauss' law-- 241 00:14:44,190 --> 00:14:48,600 and we can see that this is the Gauss' law, epsilon0-- 242 00:14:48,600 --> 00:14:54,180 divergence of E will be equal to rho. 243 00:14:54,180 --> 00:14:57,680 And we also-- under our classification, 244 00:14:57,680 --> 00:15:01,930 essentially, there are two components, rho f and rho p. 245 00:15:01,930 --> 00:15:05,130 The bound charge and the free charge. 246 00:15:05,130 --> 00:15:07,050 And according to our definition, this 247 00:15:07,050 --> 00:15:13,980 can be written as rho f minus delta P vector. 248 00:15:13,980 --> 00:15:16,200 And now what we can do is that we 249 00:15:16,200 --> 00:15:21,270 can use the definition of the electric displacement field 250 00:15:21,270 --> 00:15:24,620 and collect all the terms except the rho 251 00:15:24,620 --> 00:15:26,460 f to the left-hand side. 252 00:15:26,460 --> 00:15:30,030 Then you will conclude that the divergence of D 253 00:15:30,030 --> 00:15:33,470 will be equal to rho f. 254 00:15:33,470 --> 00:15:35,960 So after this calculation, you can 255 00:15:35,960 --> 00:15:37,790 see that what we have achieved is 256 00:15:37,790 --> 00:15:42,990 that we have defined a field, a displacement field 257 00:15:42,990 --> 00:15:51,230 D, which is totally related to the effect of the free charge. 258 00:15:51,230 --> 00:15:54,700 In this case, what we derived is actually 259 00:15:54,700 --> 00:15:57,860 delta D equal to rho f. 260 00:15:57,860 --> 00:16:02,160 And this actually looks pretty similar to the situation 261 00:16:02,160 --> 00:16:03,790 in vacuum, right? 262 00:16:03,790 --> 00:16:09,290 Because what we actually have is epsilon0 delta E equal to rho. 263 00:16:09,290 --> 00:16:12,920 And after we actually remove the contribution of the bound 264 00:16:12,920 --> 00:16:17,050 charge, this becomes this expression. 265 00:16:17,050 --> 00:16:19,710 Any questions so far? 266 00:16:19,710 --> 00:16:22,910 All right, so this is actually a purely definition. 267 00:16:22,910 --> 00:16:29,180 And we can also do a very similar thing to the current. 268 00:16:29,180 --> 00:16:33,350 So the total current, J, will have the following three 269 00:16:33,350 --> 00:16:34,680 components. 270 00:16:34,680 --> 00:16:36,830 The first component is the free current, 271 00:16:36,830 --> 00:16:43,130 which is the current related to free charge moving around 272 00:16:43,130 --> 00:16:46,190 inside this dielectrics method. 273 00:16:46,190 --> 00:16:48,950 There can be contribution from the bound current. 274 00:16:48,950 --> 00:16:50,955 The bound current is actually the current 275 00:16:50,955 --> 00:16:55,190 which is only moving around some specific location. 276 00:16:55,190 --> 00:16:58,990 And finally, the polarization contribution. 277 00:16:58,990 --> 00:17:02,900 So changing polarization also introduced a current, 278 00:17:02,900 --> 00:17:08,230 because polarization is actually defined as q dot times 279 00:17:08,230 --> 00:17:11,130 D, which is the distance between charges. 280 00:17:11,130 --> 00:17:15,349 And that if I have a changing polarization, that means also 281 00:17:15,349 --> 00:17:18,680 there are some charges floating around. 282 00:17:18,680 --> 00:17:21,569 And that actually gives you the third contribution, 283 00:17:21,569 --> 00:17:26,210 which is, let's say, P. Once we have classified 284 00:17:26,210 --> 00:17:30,290 the current into three pieces, and basically, we 285 00:17:30,290 --> 00:17:38,400 can define H field, which is actually defined as B over mu0, 286 00:17:38,400 --> 00:17:43,810 and the minus M, where M is a magnetic dipole moment. 287 00:17:43,810 --> 00:17:46,670 And the M field is actually defined 288 00:17:46,670 --> 00:17:55,120 as the curl of M defined as the magnitude and direction 289 00:17:55,120 --> 00:17:58,220 of the bound current. 290 00:17:58,220 --> 00:18:01,310 And once we have finished this definition, 291 00:18:01,310 --> 00:18:06,010 we can actually plug that into Ampere's law. 292 00:18:06,010 --> 00:18:08,560 Ampere's law, just a reminder, is 293 00:18:08,560 --> 00:18:13,670 curl B will be equal to mu0 times J. J is actually 294 00:18:13,670 --> 00:18:16,590 the total current. 295 00:18:16,590 --> 00:18:21,710 Plus a component which is actually added by Maxwell. 296 00:18:21,710 --> 00:18:26,800 So that actually results in the electromagnetic waves, 297 00:18:26,800 --> 00:18:31,460 which is epsilon0 partial E partial T. 298 00:18:31,460 --> 00:18:34,460 By using those definitions and classification, 299 00:18:34,460 --> 00:18:36,310 we can immediately write down-- 300 00:18:36,310 --> 00:18:40,570 so if we divide both sides by mu0, 301 00:18:40,570 --> 00:18:42,790 basically, you conclude that left-hand side 302 00:18:42,790 --> 00:18:49,400 is 1 over mu0 curl B. And right-hand side, 303 00:18:49,400 --> 00:18:54,290 you have the contribution of free charge current. 304 00:18:54,290 --> 00:18:58,160 And you have the contribution of bound current. 305 00:18:58,160 --> 00:18:59,960 And you have the third contribution 306 00:18:59,960 --> 00:19:04,670 which is related to a change in polarization. 307 00:19:04,670 --> 00:19:07,520 And finally, you have the term which 308 00:19:07,520 --> 00:19:11,170 is added by Maxwell epsilon 0 partial E 309 00:19:11,170 --> 00:19:15,590 partial T. By using the definition which we defined 310 00:19:15,590 --> 00:19:19,780 here for the D displacement field, 311 00:19:19,780 --> 00:19:26,390 and H, which is actually defined here, 312 00:19:26,390 --> 00:19:30,470 you can see that now we conclude that the curl of H 313 00:19:30,470 --> 00:19:37,610 will be equal to Jf plus partial D partial P. 314 00:19:37,610 --> 00:19:40,560 So remember why we are doing this. 315 00:19:40,560 --> 00:19:42,060 The reason is the following. 316 00:19:42,060 --> 00:19:47,450 We would like to classify the effect coming from free charge 317 00:19:47,450 --> 00:19:50,960 or free current inside the material 318 00:19:50,960 --> 00:19:54,830 by subtracting the effect from the bound current and the bound 319 00:19:54,830 --> 00:19:55,880 charge. 320 00:19:55,880 --> 00:19:58,700 So that's actually what we have been doing. 321 00:19:58,700 --> 00:20:05,210 Then once we define new fields, which is actually showing here, 322 00:20:05,210 --> 00:20:09,920 D, which is the displacement field, is only related to-- 323 00:20:09,920 --> 00:20:12,980 which is the field related to the free charge. 324 00:20:12,980 --> 00:20:16,400 After all, those are actually just definition. 325 00:20:16,400 --> 00:20:19,050 And the edge field, the magnetic field 326 00:20:19,050 --> 00:20:22,670 which is actually only related to the displacement field 327 00:20:22,670 --> 00:20:27,020 and the free current, we actually 328 00:20:27,020 --> 00:20:32,750 arrive something really, really similar to the vacuum 329 00:20:32,750 --> 00:20:35,540 case, Maxwell equation. 330 00:20:35,540 --> 00:20:37,580 So that's, essentially, the excitement. 331 00:20:37,580 --> 00:20:42,090 And you will see that in 8.03, we will use immediately 332 00:20:42,090 --> 00:20:43,280 those conclusions. 333 00:20:43,280 --> 00:20:44,990 And also, we would limit ourselves 334 00:20:44,990 --> 00:20:49,040 in the discussion of linear homogeneous 335 00:20:49,040 --> 00:20:51,800 and isotopic materials. 336 00:20:51,800 --> 00:20:55,070 And only work on this kind of material. 337 00:20:55,070 --> 00:20:58,240 And that is, you would need to highly simplify 338 00:20:58,240 --> 00:21:01,550 the solution for the electromagnetic field 339 00:21:01,550 --> 00:21:04,740 or waves inside material. 340 00:21:04,740 --> 00:21:07,160 Any questions so far? 341 00:21:07,160 --> 00:21:08,880 I hope those are just different issues. 342 00:21:08,880 --> 00:21:10,562 Yes. 343 00:21:10,562 --> 00:21:16,990 STUDENT: [INAUDIBLE] 344 00:21:16,990 --> 00:21:18,760 PROFESSOR: The bound current is actually 345 00:21:18,760 --> 00:21:25,070 inside the whole dielectric material. 346 00:21:25,070 --> 00:21:28,370 But of course, you can have many, many small loops 347 00:21:28,370 --> 00:21:30,320 and they will cancel. 348 00:21:30,320 --> 00:21:31,790 Because if you are looping-- 349 00:21:31,790 --> 00:21:37,090 for example, you can have many, many bound current, 350 00:21:37,090 --> 00:21:43,190 which is actually surrounding the atom. 351 00:21:43,190 --> 00:21:45,620 But you can see that all those things-- 352 00:21:45,620 --> 00:21:49,670 all the nearby little bound current will cancel each other. 353 00:21:49,670 --> 00:21:51,790 So therefore, you could do an integration, 354 00:21:51,790 --> 00:21:53,670 and it becomes a total bound current, 355 00:21:53,670 --> 00:21:56,900 which is happening around the surface 356 00:21:56,900 --> 00:22:00,780 of the dielectrics material. 357 00:22:00,780 --> 00:22:05,690 So it depends on what you mean by how this bound current 358 00:22:05,690 --> 00:22:07,560 actually moves. 359 00:22:07,560 --> 00:22:09,490 So you do have little ones. 360 00:22:09,490 --> 00:22:12,740 And then if you do integral, then it 361 00:22:12,740 --> 00:22:16,360 becomes a surface bound current. 362 00:22:16,360 --> 00:22:26,550 So if we step these two conclusions we arrive here, 363 00:22:26,550 --> 00:22:29,530 basically, we can immediately write down 364 00:22:29,530 --> 00:22:33,860 what would be the Maxwell's equation in matter. 365 00:22:33,860 --> 00:22:39,320 So now, instead of electric field and E and the-- 366 00:22:39,320 --> 00:22:43,250 E field and the magnetic field, B field, we also have-- 367 00:22:47,275 --> 00:22:52,170 oh, I think there's a typo there probably. 368 00:22:52,170 --> 00:22:55,890 Ah, there's a typo in the lower left. 369 00:22:55,890 --> 00:22:58,520 So the lower left equation, which 370 00:22:58,520 --> 00:23:01,970 is, unfortunately, propagated to many places, 371 00:23:01,970 --> 00:23:04,730 should be like this. 372 00:23:04,730 --> 00:23:07,040 So the lower left equation should 373 00:23:07,040 --> 00:23:20,260 be del cross H. That would be equal to Jf plus partial D 374 00:23:20,260 --> 00:23:23,180 partial t. 375 00:23:23,180 --> 00:23:26,290 So somehow, this is actually propagated to many places. 376 00:23:26,290 --> 00:23:30,760 So basically, we have Maxwell's equation 377 00:23:30,760 --> 00:23:34,430 in matter, which is actually really similar to what 378 00:23:34,430 --> 00:23:39,350 we have in the vacuum case. 379 00:23:39,350 --> 00:23:42,950 So very similar to a discussion we had in the vacuum case, 380 00:23:42,950 --> 00:23:49,130 what we actually do is to set the Jf and rho f equal to 0. 381 00:23:49,130 --> 00:23:52,880 If I set the Jf equal to 0, the last equation 382 00:23:52,880 --> 00:23:58,790 will become curl of H. And that would be 383 00:23:58,790 --> 00:24:01,810 equal to partial D partial t. 384 00:24:05,030 --> 00:24:10,310 So you can see that this is very similar to what we had before, 385 00:24:10,310 --> 00:24:15,230 but the problem is that we have the H field and the D field. 386 00:24:15,230 --> 00:24:17,630 The question is, how do we actually 387 00:24:17,630 --> 00:24:22,500 relate H field and the D field with the electric field? 388 00:24:25,880 --> 00:24:29,370 So as we mentioned before, the D field 389 00:24:29,370 --> 00:24:37,260 is as we defined as epsilon0 D field plus P, which is actually 390 00:24:37,260 --> 00:24:43,190 the induced polarization of the material. 391 00:24:43,190 --> 00:24:48,230 And in the case of very small electric field, 392 00:24:48,230 --> 00:24:51,470 and there is more magnetic field, and also 393 00:24:51,470 --> 00:24:57,470 a linear and homogeneous isotropic material, 394 00:24:57,470 --> 00:25:04,400 this induced polarizaiton can be proportional to the size 395 00:25:04,400 --> 00:25:06,620 of the electric field. 396 00:25:06,620 --> 00:25:11,960 So if this polarization is proportional 397 00:25:11,960 --> 00:25:15,920 to the electric field, I can immediately 398 00:25:15,920 --> 00:25:18,610 write down that the D field is actually 399 00:25:18,610 --> 00:25:27,510 some kind of constant epsilon times E. 400 00:25:27,510 --> 00:25:33,530 On the other hand, I can also discuss the H field. 401 00:25:33,530 --> 00:25:42,870 H feel is defined as 1 over mu0 B minus M, which 402 00:25:42,870 --> 00:25:46,470 is the magnetic dipole moment. 403 00:25:46,470 --> 00:25:49,000 In the case of very small magnetic field 404 00:25:49,000 --> 00:25:56,370 and the linear material, basically, this and M vector 405 00:25:56,370 --> 00:26:01,110 can be proportional to the B vector. 406 00:26:01,110 --> 00:26:04,890 Therefore, I can quickly rewrite this 407 00:26:04,890 --> 00:26:07,830 saying since the M field is also proportional to the B 408 00:26:07,830 --> 00:26:11,350 field in this linear material, therefore, 409 00:26:11,350 --> 00:26:17,300 I can rewrite that H will be equal to B divided by mu. 410 00:26:17,300 --> 00:26:19,980 What is epsilon and the mu? 411 00:26:19,980 --> 00:26:27,610 Epsilon is the permittivity inside the material. 412 00:26:27,610 --> 00:26:30,700 Permittivity is actually-- they tell you 413 00:26:30,700 --> 00:26:37,840 the resistance, the resistance of forming electric field 414 00:26:37,840 --> 00:26:39,420 in some place. 415 00:26:39,420 --> 00:26:42,600 So if you have a large epsilon, that 416 00:26:42,600 --> 00:26:46,080 means there'll be large resistance coming 417 00:26:46,080 --> 00:26:48,270 from the material. 418 00:26:48,270 --> 00:26:49,950 So this makes sense, right? 419 00:26:49,950 --> 00:26:53,820 Because that means you can easily 420 00:26:53,820 --> 00:26:59,497 introduce, or say induce, larger amount of polarization 421 00:26:59,497 --> 00:27:00,205 in your material. 422 00:27:00,205 --> 00:27:04,050 Then that means this material is not happy. 423 00:27:04,050 --> 00:27:07,050 It's going to try to cancel your electrical field. 424 00:27:07,050 --> 00:27:10,620 Therefore, if you have a large P, 425 00:27:10,620 --> 00:27:13,290 that will give you a large epsilon. 426 00:27:13,290 --> 00:27:18,810 And therefore, that means you have 427 00:27:18,810 --> 00:27:22,710 a lot of resistance of forming electric field, which is still 428 00:27:22,710 --> 00:27:26,620 the E field here. 429 00:27:26,620 --> 00:27:27,535 in matter. 430 00:27:30,300 --> 00:27:32,850 On the other hand, this mu is actually 431 00:27:32,850 --> 00:27:36,120 permeability of the, material, which 432 00:27:36,120 --> 00:27:38,910 is the resistance of forming-- 433 00:27:38,910 --> 00:27:44,820 it's related to the resistance of forming a magnetic field. 434 00:27:44,820 --> 00:27:48,470 And basically, these two quantities are actually 435 00:27:48,470 --> 00:27:52,910 telling you the D field and the H field 436 00:27:52,910 --> 00:27:58,380 are related to the electric field and the magnetic field. 437 00:27:58,380 --> 00:28:04,410 So if we assume a linear relation between D and E, 438 00:28:04,410 --> 00:28:07,380 and elsewhere at H and B, then we 439 00:28:07,380 --> 00:28:12,360 can actually immediately rewrite our Maxwell's equation 440 00:28:12,360 --> 00:28:15,550 in matter in the following way. 441 00:28:15,550 --> 00:28:19,920 So you can see that the resulting Maxwell's equation 442 00:28:19,920 --> 00:28:23,930 in matter considering a linear material 443 00:28:23,930 --> 00:28:27,300 is really remarkably similar to what 444 00:28:27,300 --> 00:28:30,740 we have in the vacuum case. 445 00:28:30,740 --> 00:28:32,040 Where is that difference? 446 00:28:32,040 --> 00:28:34,050 Can you see that? 447 00:28:34,050 --> 00:28:37,600 The only difference is in the last three equations, which 448 00:28:37,600 --> 00:28:42,870 essentially, in the lower right part of the equation, 449 00:28:42,870 --> 00:28:49,680 instead of mu0 epsilon0, you now get mu times epsilon. 450 00:28:49,680 --> 00:28:51,340 So what does that mean? 451 00:28:51,340 --> 00:28:57,760 This means that the speed of the propagation 452 00:28:57,760 --> 00:29:01,770 of the electromagnetic field is changed. 453 00:29:01,770 --> 00:29:05,260 It's now changed to-- 454 00:29:05,260 --> 00:29:07,820 instead of 1 over-- 455 00:29:07,820 --> 00:29:11,050 in the vacuum case, you have c equal to 1 456 00:29:11,050 --> 00:29:15,370 over the square root of mu0 epsilon0. 457 00:29:15,370 --> 00:29:18,010 Instead of c, we are going to get 458 00:29:18,010 --> 00:29:22,570 c over n, which n is the refractive index. 459 00:29:22,570 --> 00:29:25,965 And that is actually equal to 1 over the square root 460 00:29:25,965 --> 00:29:28,780 of mu and epsilon. 461 00:29:28,780 --> 00:29:30,890 It's as simple as that. 462 00:29:30,890 --> 00:29:32,710 So based on what we have prepared 463 00:29:32,710 --> 00:29:35,860 in the last few lectures, we can now immediately make 464 00:29:35,860 --> 00:29:38,995 sense of this equation. 465 00:29:38,995 --> 00:29:43,270 And now that we know that almost everything is the same, what 466 00:29:43,270 --> 00:29:46,645 is different is that now we have a difference 467 00:29:46,645 --> 00:29:49,390 speed of propagation, which is actually 468 00:29:49,390 --> 00:29:53,870 related to the refractive index we discussed last time. 469 00:29:53,870 --> 00:29:57,610 And now we understand what is actually this refractive index. 470 00:29:57,610 --> 00:30:03,400 n is actually equal to square root of mu epsilon divided 471 00:30:03,400 --> 00:30:06,364 by square root of mu0 epsilon0. 472 00:30:09,290 --> 00:30:12,560 Any questions? 473 00:30:12,560 --> 00:30:13,508 Yeah. 474 00:30:13,508 --> 00:30:16,352 AUDIENCE: What are the D fields and H field? 475 00:30:16,352 --> 00:30:19,200 Are they just coefficients? 476 00:30:19,200 --> 00:30:20,190 PROFESSOR: Yes. 477 00:30:20,190 --> 00:30:23,580 So the D field and the H field are 478 00:30:23,580 --> 00:30:26,580 the fields which is actually related 479 00:30:26,580 --> 00:30:29,430 to only the free charge. 480 00:30:29,430 --> 00:30:33,140 So you can see that now the gradients-- 481 00:30:33,140 --> 00:30:38,490 sorry, now the divergence of D field 482 00:30:38,490 --> 00:30:46,440 is actually only equal to the density of the free charge. 483 00:30:46,440 --> 00:30:48,990 So basically, in short, the D field 484 00:30:48,990 --> 00:30:54,870 absorbs the effect of the bound charge into this field. 485 00:30:54,870 --> 00:30:57,810 And when you try to actually understand 486 00:30:57,810 --> 00:31:00,540 what would be the field associated or induced 487 00:31:00,540 --> 00:31:03,870 by the free charge, it is actually the D field. 488 00:31:03,870 --> 00:31:09,370 But the D field is not the full story of the electric field. 489 00:31:09,370 --> 00:31:11,490 How is that related to the electric field 490 00:31:11,490 --> 00:31:12,600 in the exact form? 491 00:31:12,600 --> 00:31:15,510 This is actually defined here. 492 00:31:15,510 --> 00:31:17,670 And from the D field, you will be 493 00:31:17,670 --> 00:31:22,290 able to evaluate what would be the corresponding E field. 494 00:31:22,290 --> 00:31:26,400 And in the linear material, there's 495 00:31:26,400 --> 00:31:29,490 a spatial relation between D field and the E field, 496 00:31:29,490 --> 00:31:31,890 because the induced polarization is 497 00:31:31,890 --> 00:31:36,930 proportional to electric field in the linear material, which 498 00:31:36,930 --> 00:31:38,350 is a special case. 499 00:31:38,350 --> 00:31:44,580 And in that sense, D field is proportional to E field. 500 00:31:44,580 --> 00:31:49,210 And this factor is actually so-called epsilon, 501 00:31:49,210 --> 00:31:52,320 which is permittivity of the material. 502 00:31:52,320 --> 00:32:00,480 Describing how large is the resistance 503 00:32:00,480 --> 00:32:05,410 of forming an electric field inside a material. 504 00:32:05,410 --> 00:32:06,117 OK? 505 00:32:06,117 --> 00:32:07,430 Is that clear? 506 00:32:07,430 --> 00:32:08,610 OK, good. 507 00:32:08,610 --> 00:32:11,850 And the H field is similar argument. 508 00:32:11,850 --> 00:32:14,220 All right, so now we have made sense 509 00:32:14,220 --> 00:32:17,310 of the electric field and magnetic field 510 00:32:17,310 --> 00:32:19,950 and the Maxwell's equations in matter. 511 00:32:19,950 --> 00:32:24,900 And in 8.03, we will only discuss linear material. 512 00:32:24,900 --> 00:32:27,570 You can imagine that the relation 513 00:32:27,570 --> 00:32:32,220 between the polarization and the external electric field 514 00:32:32,220 --> 00:32:34,570 can be very complicated. 515 00:32:34,570 --> 00:32:39,720 So for example, it depends on how large 516 00:32:39,720 --> 00:32:45,390 is the wavelengths of the external force. 517 00:32:45,390 --> 00:32:51,240 So if you have very slowly varying electric field, 518 00:32:51,240 --> 00:32:53,160 you can imagine that you will not 519 00:32:53,160 --> 00:32:57,660 be able to create a polarization, which 520 00:32:57,660 --> 00:32:59,830 is the inner electron ground level. 521 00:32:59,830 --> 00:33:04,890 Because the variation of the electric field is too slow. 522 00:33:04,890 --> 00:33:09,680 But instead, you will be able to excite the polarization related 523 00:33:09,680 --> 00:33:14,460 to ions inside this material, or inside the plasma. 524 00:33:14,460 --> 00:33:18,360 So on the other hand, if you have a very, very fast 525 00:33:18,360 --> 00:33:20,580 oscillating electric field, then you 526 00:33:20,580 --> 00:33:22,650 will be able to measure the shift 527 00:33:22,650 --> 00:33:25,390 or create a polarization which is actually 528 00:33:25,390 --> 00:33:30,380 related to the displacement of the electron cloud inside that. 529 00:33:30,380 --> 00:33:33,510 So it really depends on many, many factors. 530 00:33:33,510 --> 00:33:37,680 But what we have been discussing here and elsewhere 531 00:33:37,680 --> 00:33:42,280 is highly idealized linear materials. 532 00:33:42,280 --> 00:33:46,230 And a lot of interesting things to explore in the future 533 00:33:46,230 --> 00:33:49,350 beyond 8.03. 534 00:33:49,350 --> 00:33:52,860 So once we have this relation, we 535 00:33:52,860 --> 00:33:59,040 will be able to get Maxwell's equation in matter. 536 00:33:59,040 --> 00:34:04,290 And we would like to understand, just to remind you, 537 00:34:04,290 --> 00:34:07,470 why are we doing this? 538 00:34:07,470 --> 00:34:09,600 So we have a physical question. 539 00:34:09,600 --> 00:34:12,150 We have a question about a phenomenon 540 00:34:12,150 --> 00:34:17,620 which we see in this slide. 541 00:34:17,620 --> 00:34:21,389 So when we use the polarizer, very strange, 542 00:34:21,389 --> 00:34:25,980 you will be able to filter out the reflected light 543 00:34:25,980 --> 00:34:30,750 from the sun and through-- 544 00:34:30,750 --> 00:34:34,409 when the sunlight hit the window and got reflected, 545 00:34:34,409 --> 00:34:40,110 this kind of contribution can be filtered out almost completely 546 00:34:40,110 --> 00:34:41,429 by polarizer. 547 00:34:41,429 --> 00:34:44,190 So that means the reflected light is somehow 548 00:34:44,190 --> 00:34:45,630 also polarized. 549 00:34:45,630 --> 00:34:48,690 And then why is that the case is the question 550 00:34:48,690 --> 00:34:52,520 we were trying to answer here. 551 00:34:52,520 --> 00:34:56,460 So therefore, what I am going to do is to, 552 00:34:56,460 --> 00:35:04,010 again, take a look at the boundary between two materials. 553 00:35:04,010 --> 00:35:09,560 The one side is actually in the air, which we call it number 1. 554 00:35:09,560 --> 00:35:14,060 And the other side is actually the glass, which 555 00:35:14,060 --> 00:35:17,820 I call it material number 2. 556 00:35:17,820 --> 00:35:24,960 And now I have, again, some incident prime wave, 557 00:35:24,960 --> 00:35:26,270 which is coming from the sun. 558 00:35:29,640 --> 00:35:35,660 And actually, this incident prime wave actually 559 00:35:35,660 --> 00:35:42,050 goes into this surface which is the boundary between air 560 00:35:42,050 --> 00:35:48,610 and glass in the discussion which I am trying to get into. 561 00:35:48,610 --> 00:35:53,360 I assume that this glass is actually very wide. 562 00:35:53,360 --> 00:35:56,720 It's actually fielding the whole universe. 563 00:35:56,720 --> 00:35:59,995 Therefore, this is actually just a simple plane, 564 00:35:59,995 --> 00:36:02,410 a boundary which actually divides 565 00:36:02,410 --> 00:36:04,505 our world into air and glass. 566 00:36:07,010 --> 00:36:11,760 We know from the discussion of previous lecture, 567 00:36:11,760 --> 00:36:15,470 so there must be a reflected light 568 00:36:15,470 --> 00:36:21,710 and there must be transmission into the glass. 569 00:36:21,710 --> 00:36:26,680 Because this is actually the general property of waves. 570 00:36:26,680 --> 00:36:30,860 So OK, it has nothing to do with electromagnetic wave yet, 571 00:36:30,860 --> 00:36:35,180 but if you have wave, and you have an incident wave, 572 00:36:35,180 --> 00:36:37,240 you are going to have a reflected wave 573 00:36:37,240 --> 00:36:40,760 and the transmitted wave. 574 00:36:40,760 --> 00:36:45,620 We learned from the two laws of geometrical optics, 575 00:36:45,620 --> 00:36:48,910 we know that the incident angle, theta i, 576 00:36:48,910 --> 00:36:54,200 will be equal to the reflective angle, theta r, 577 00:36:54,200 --> 00:36:58,160 with respect to the normal direction of this surface. 578 00:36:58,160 --> 00:37:04,420 And we also know that the KI, which 579 00:37:04,420 --> 00:37:10,570 is the wave number of vector of the incident wave, 580 00:37:10,570 --> 00:37:14,400 KR, which is the wave number vector, the K vector, 581 00:37:14,400 --> 00:37:19,370 of the reflected wave, and the K vector 582 00:37:19,370 --> 00:37:22,040 for the transmitting wave, KT. 583 00:37:22,040 --> 00:37:24,270 And this is actually theta T. 584 00:37:24,270 --> 00:37:28,850 These three K vectors must have a fixed relation 585 00:37:28,850 --> 00:37:35,210 so that the electromagnetic field, or say all those three 586 00:37:35,210 --> 00:37:37,570 wave equations-- the sum of these three wave 587 00:37:37,570 --> 00:37:40,790 equations in the left-hand side world and right-hand side 588 00:37:40,790 --> 00:37:43,220 world, they are connected to each other. 589 00:37:43,220 --> 00:37:44,450 They don't break. 590 00:37:44,450 --> 00:37:50,290 There's no discontinuity between the sum of the left-hand side 591 00:37:50,290 --> 00:37:55,820 plane waves and the sum of the right-hand side plane wave. 592 00:37:55,820 --> 00:38:02,060 So that means it has to obey Snell's law. 593 00:38:02,060 --> 00:38:04,580 And there will be a fixed-- 594 00:38:04,580 --> 00:38:07,180 the size of the projection of the K 595 00:38:07,180 --> 00:38:16,590 vector onto the direction of this surface will be the same. 596 00:38:16,590 --> 00:38:22,010 Otherwise, you will have different wavelengths 597 00:38:22,010 --> 00:38:24,200 in the vertical direction. 598 00:38:24,200 --> 00:38:27,320 Then the electromagnetic wave will break, right? 599 00:38:27,320 --> 00:38:31,610 Because you can-- as you always change your position 600 00:38:31,610 --> 00:38:37,600 when evaluating the total contribution of the incident, 601 00:38:37,600 --> 00:38:41,390 reflected, and transmitted waves. 602 00:38:41,390 --> 00:38:44,580 So once we have all this information in hand, 603 00:38:44,580 --> 00:38:47,120 we can actually write down the expression 604 00:38:47,120 --> 00:38:50,930 for the incident wave, EI, which is actually 605 00:38:50,930 --> 00:38:53,020 a function of r and t. 606 00:38:55,598 --> 00:39:00,230 And of course, I would like to define my coordinate system 607 00:39:00,230 --> 00:39:01,000 first. 608 00:39:01,000 --> 00:39:03,930 So this is the x direction. 609 00:39:03,930 --> 00:39:07,291 And this is y direction. 610 00:39:07,291 --> 00:39:07,790 OK, sorry. 611 00:39:07,790 --> 00:39:09,498 This is usually the z direction actually. 612 00:39:16,470 --> 00:39:19,220 The x direction is going up. 613 00:39:19,220 --> 00:39:21,330 y direction is actually pointing to you. 614 00:39:21,330 --> 00:39:24,030 And the z direction is pointing to the right-hand side 615 00:39:24,030 --> 00:39:25,410 of the board. 616 00:39:25,410 --> 00:39:29,580 And this is actually at z equal to 0, this interface 617 00:39:29,580 --> 00:39:31,340 between glass and air. 618 00:39:31,340 --> 00:39:33,000 And now I can actually write down 619 00:39:33,000 --> 00:39:37,900 what will be the incident wave electric field. 620 00:39:37,900 --> 00:39:44,040 And that is actually equal to E0I, which is a vector, 621 00:39:44,040 --> 00:39:47,960 tell you about the polarization of the incident wave. 622 00:39:47,960 --> 00:39:54,240 Cosine-- by now this should look pretty familiar to you now. 623 00:39:54,240 --> 00:39:58,710 This is actually just a KI dot r, 624 00:39:58,710 --> 00:40:03,160 which describes the direction of the propagation. 625 00:40:03,160 --> 00:40:03,860 Minus omega t. 626 00:40:07,040 --> 00:40:10,370 So this is actually describing the incident wave, 627 00:40:10,370 --> 00:40:14,540 which I call it EI. 628 00:40:14,540 --> 00:40:17,900 And I will assume that since I know 629 00:40:17,900 --> 00:40:23,540 what would be incident wave, that E0I is a known quantity. 630 00:40:23,540 --> 00:40:27,990 So I say that this is actually a known quantity. 631 00:40:30,560 --> 00:40:32,690 So now I can also do the same thing 632 00:40:32,690 --> 00:40:38,790 and write down what would be the reflected wave. 633 00:40:38,790 --> 00:40:43,910 ER as a function of r and t. 634 00:40:43,910 --> 00:40:46,460 And this will be, very similarly, 635 00:40:46,460 --> 00:40:49,460 E0R, which is actually telling you 636 00:40:49,460 --> 00:40:51,860 the magnitude and the polarization 637 00:40:51,860 --> 00:41:01,310 of the reflected wave, cosine KR dot r minus omega t. 638 00:41:01,310 --> 00:41:06,570 All right, and finally, you have the ET, which 639 00:41:06,570 --> 00:41:09,460 is a function of r and t again. 640 00:41:09,460 --> 00:41:16,520 And this will be E0T cosine KT times r minus omega t. 641 00:41:21,330 --> 00:41:25,380 Of course, they will have three electric fields, therefore, 642 00:41:25,380 --> 00:41:28,838 you must have three what field? 643 00:41:28,838 --> 00:41:29,779 AUDIENCE: Magnetic. 644 00:41:29,779 --> 00:41:31,070 PROFESSOR: Magnetic field, yes. 645 00:41:31,070 --> 00:41:34,820 So you must have the corresponding magnetic field. 646 00:41:34,820 --> 00:41:38,660 So you, for example, BI. 647 00:41:38,660 --> 00:41:41,100 It's a function of r and t. 648 00:41:41,100 --> 00:41:45,200 Will be equal to 1 over V1, which 649 00:41:45,200 --> 00:41:48,130 is the velocity in the air. 650 00:41:48,130 --> 00:41:49,940 V1 is actually equal to c. 651 00:41:54,330 --> 00:41:59,110 KI hat, the direction of the propagation cross-- 652 00:42:03,360 --> 00:42:08,640 OK, so I'm not going to write down BR and BT, 653 00:42:08,640 --> 00:42:11,730 because this is actually a very similar expression 654 00:42:11,730 --> 00:42:16,230 as the corresponding associated magnetic field for the incident 655 00:42:16,230 --> 00:42:17,870 wave. 656 00:42:17,870 --> 00:42:22,170 All right, so basically, we have actually 657 00:42:22,170 --> 00:42:24,750 translated the physical situation 658 00:42:24,750 --> 00:42:29,100 into mathematics using the coordinate system, which 659 00:42:29,100 --> 00:42:31,822 we defined here. 660 00:42:31,822 --> 00:42:33,780 So one thing which is actually very interesting 661 00:42:33,780 --> 00:42:37,980 is that we have solved half of the question 662 00:42:37,980 --> 00:42:40,000 in the previous lecture. 663 00:42:40,000 --> 00:42:42,280 So in the previous lecture when we discussed 664 00:42:42,280 --> 00:42:45,660 two-dimensional or three-dimensional waves, 665 00:42:45,660 --> 00:42:49,290 we have concluded that if you have a wave, 666 00:42:49,290 --> 00:42:55,740 and it's continuous at the surface, z equal to 0, 667 00:42:55,740 --> 00:43:05,400 you can immediately conclude that basically KI dot 668 00:43:05,400 --> 00:43:11,381 r will be equal to KR dot r. 669 00:43:11,381 --> 00:43:15,092 This will be equal to KT dot r. 670 00:43:17,880 --> 00:43:21,130 This is the first thing which we actually conclude. 671 00:43:21,130 --> 00:43:23,400 That actually leads to what? 672 00:43:23,400 --> 00:43:25,310 What law? 673 00:43:25,310 --> 00:43:26,010 Snell's law. 674 00:43:26,010 --> 00:43:27,060 Yes, very good. 675 00:43:27,060 --> 00:43:29,610 Snell's law. 676 00:43:29,610 --> 00:43:33,370 If this expression doesn't hold, then your left-hand side 677 00:43:33,370 --> 00:43:38,300 and right-hand side total electric field doesn't match 678 00:43:38,300 --> 00:43:42,720 at the surface of z equal to 0. 679 00:43:42,720 --> 00:43:44,130 The second thing which we learned 680 00:43:44,130 --> 00:43:51,270 is that theta I will be equal to theta R. OK, 681 00:43:51,270 --> 00:43:55,320 so basically, that's actually what we have learned 682 00:43:55,320 --> 00:43:57,210 from the previous lecture. 683 00:43:57,210 --> 00:44:03,840 Therefore, this match in between left-hand side electric field 684 00:44:03,840 --> 00:44:07,440 and right-hand side electric field becomes much simpler. 685 00:44:07,440 --> 00:44:14,220 Since that they are actually just some E0 times cosine, 686 00:44:14,220 --> 00:44:16,950 a functional formula showing here. 687 00:44:16,950 --> 00:44:23,460 Since these three products are identical, therefore, 688 00:44:23,460 --> 00:44:27,870 what is going to happen is that the location dependence 689 00:44:27,870 --> 00:44:34,260 and the time dependence of this relation completely cancels. 690 00:44:34,260 --> 00:44:38,700 Because you just have the individual vector, 691 00:44:38,700 --> 00:44:44,010 E0I, E0R, E0T, multiplied by a cosine function which 692 00:44:44,010 --> 00:44:48,730 is identical for those three incident 693 00:44:48,730 --> 00:44:51,030 transmitted and the reflected waves. 694 00:44:51,030 --> 00:44:54,150 Therefore, in the discussion of this kind of thing, 695 00:44:54,150 --> 00:44:58,380 you can actually ignore the location dependence 696 00:44:58,380 --> 00:45:01,650 and the time dependence, since they always cancel. 697 00:45:01,650 --> 00:45:08,400 So that's actually a pretty useful thing to have. 698 00:45:08,400 --> 00:45:11,180 And don't forget what is our goal, right? 699 00:45:11,180 --> 00:45:17,295 So our goal is to know, OK, so if I have a given incident 700 00:45:17,295 --> 00:45:22,290 prime wave, what would be the resulting reflected wave, what 701 00:45:22,290 --> 00:45:26,880 would be the resulting transmitted wave? 702 00:45:26,880 --> 00:45:31,430 And I would like to tell you the conclusion first. 703 00:45:31,430 --> 00:45:35,280 The relation between the reflected wave, 704 00:45:35,280 --> 00:45:38,370 transmitted wave, and incident wave 705 00:45:38,370 --> 00:45:42,750 depends on the Maxwell simulation. 706 00:45:42,750 --> 00:45:47,460 You can see that this relation have nothing 707 00:45:47,460 --> 00:45:51,200 to do with Maxwell's equation so far. 708 00:45:51,200 --> 00:45:56,100 It's actually really related to the wave description 709 00:45:56,100 --> 00:45:57,190 we are using. 710 00:45:57,190 --> 00:46:00,695 And also, the the match in between the left-hand side 711 00:46:00,695 --> 00:46:03,010 and right-hand side wave equation. 712 00:46:03,010 --> 00:46:07,830 So you know, now we have learned this is actually only related 713 00:46:07,830 --> 00:46:12,630 to a generic property wave equation. 714 00:46:12,630 --> 00:46:16,590 So now we need to get the help from the Maxwell's equation 715 00:46:16,590 --> 00:46:19,800 in matter so that I can make sense 716 00:46:19,800 --> 00:46:25,100 about the relation between E0R, E0T with respect 717 00:46:25,100 --> 00:46:28,650 to what is given, E0I. 718 00:46:28,650 --> 00:46:31,800 So how do we do that? 719 00:46:31,800 --> 00:46:34,530 So the first thing which I would like to do 720 00:46:34,530 --> 00:46:38,730 is that, as you see from here, I divided the whole universe 721 00:46:38,730 --> 00:46:39,660 into two parts. 722 00:46:39,660 --> 00:46:43,200 The left-hand side wall, I call it 1, is air. 723 00:46:43,200 --> 00:46:48,300 The right-hand side wall is glass, which is I call it 2. 724 00:46:48,300 --> 00:46:50,700 Therefore, I can now calculate what 725 00:46:50,700 --> 00:46:57,690 will be the sum of the electric field in the world number 1, 726 00:46:57,690 --> 00:47:05,650 which is actually defined as E0I plus E0R. 727 00:47:08,910 --> 00:47:11,670 So here you can see that I already dropped the cosine, 728 00:47:11,670 --> 00:47:13,230 because they always cancel. 729 00:47:13,230 --> 00:47:19,720 Therefore, I just write down E0I and E0R without the cosine. 730 00:47:19,720 --> 00:47:22,230 And of course, I can also calculate 731 00:47:22,230 --> 00:47:26,370 what will be the total electric field in the right-hand side 732 00:47:26,370 --> 00:47:28,110 world, inside the glass. 733 00:47:28,110 --> 00:47:30,870 And that is actually equal to E0T. 734 00:47:33,780 --> 00:47:41,090 The question is how to relate E1 and E2. 735 00:47:41,090 --> 00:47:43,640 And for that, as I mentioned, I need 736 00:47:43,640 --> 00:47:46,270 the help of Maxwell's equation in matter. 737 00:47:48,800 --> 00:47:51,780 How do we actually do that? 738 00:47:51,780 --> 00:47:54,560 So what we could do is that since we 739 00:47:54,560 --> 00:47:58,520 have Maxwell's equation, the first thing which we can do 740 00:47:58,520 --> 00:48:04,650 is that we can look at the perpendicular direction. 741 00:48:04,650 --> 00:48:09,860 The perpendicular direction means the projection 742 00:48:09,860 --> 00:48:15,530 of the electric field in the perpendicular direction 743 00:48:15,530 --> 00:48:17,420 to the surface. 744 00:48:17,420 --> 00:48:20,585 That's what I mean by perpendicular direction. 745 00:48:23,450 --> 00:48:32,195 So for that, I can make use of the first Maxwell's equation 746 00:48:32,195 --> 00:48:38,000 in matter, which is actually del dot D equal to 0. 747 00:48:38,000 --> 00:48:43,250 We are considering the case without any free charge. 748 00:48:43,250 --> 00:48:48,500 So that means I have Gauss' law, which is actually D times da. 749 00:48:51,170 --> 00:48:53,960 I do a surface integral. 750 00:48:53,960 --> 00:48:58,220 And that will be equal to 0. 751 00:48:58,220 --> 00:49:05,310 So this means that I can have a pillbox again, like this. 752 00:49:05,310 --> 00:49:07,970 I arrange this pillbox like this. 753 00:49:11,335 --> 00:49:12,710 And this is actually the surface. 754 00:49:21,040 --> 00:49:24,400 The size of one side perpendicular 755 00:49:24,400 --> 00:49:31,900 to the direction of the surface is actually called D here. 756 00:49:31,900 --> 00:49:34,150 And I have this box here. 757 00:49:36,840 --> 00:49:46,920 And the left-hand side, D1, will be equal to epsilon1 E perp-- 758 00:49:46,920 --> 00:49:53,360 D1 perp, actually, will be equal to epsilon1 E perp 1. 759 00:49:53,360 --> 00:49:59,680 Epsilon1 is the permittivity of the material number 760 00:49:59,680 --> 00:50:04,360 1, which is actually equal to epsilon0 in this case, 761 00:50:04,360 --> 00:50:07,510 because it is actually air. 762 00:50:07,510 --> 00:50:13,940 And the right-hand side, you have D perp 2. 763 00:50:13,940 --> 00:50:17,850 This is actually equal to epsilon2 E perp 2. 764 00:50:22,470 --> 00:50:24,120 So now I have figured out what would 765 00:50:24,120 --> 00:50:30,120 be the field going into this pillbox and the field 766 00:50:30,120 --> 00:50:34,180 going out of this pillbox. 767 00:50:34,180 --> 00:50:37,560 What I can do now is the following. 768 00:50:37,560 --> 00:50:46,020 I can now shrink the size of the pillbox, having d goes to 0. 769 00:50:46,020 --> 00:50:50,310 And it becomes smaller, smaller, and smaller. 770 00:50:50,310 --> 00:50:54,630 So what is going to happen is that when this d goes to 0, 771 00:50:54,630 --> 00:50:58,500 this surface integral will go to what? 772 00:50:58,500 --> 00:51:01,590 Go to also 0. 773 00:51:01,590 --> 00:51:09,900 Therefore, I can immediately conclude that epsilon1 E perp 774 00:51:09,900 --> 00:51:10,630 1-- 775 00:51:10,630 --> 00:51:14,220 OK, so when I have this d goes to 0, 776 00:51:14,220 --> 00:51:17,820 the contribution from the side goes to 0. 777 00:51:17,820 --> 00:51:22,270 I know already the sum of all the contribution of the surface 778 00:51:22,270 --> 00:51:24,930 integral will be equal to 0, but I 779 00:51:24,930 --> 00:51:27,870 don't know what is the contribution from the side. 780 00:51:27,870 --> 00:51:33,981 But if I have that d goes to 0, then the area of the four-- 781 00:51:33,981 --> 00:51:35,300 1, 2, 3, 4-- 782 00:51:35,300 --> 00:51:38,590 sides, it's actually going to go to 0. 783 00:51:38,590 --> 00:51:40,320 Therefore, you have zero contribution 784 00:51:40,320 --> 00:51:42,030 to this surface integral. 785 00:51:42,030 --> 00:51:47,250 Therefore, I can conclude that epsilon1 E perp 1 786 00:51:47,250 --> 00:51:55,244 will be equal to epsilon2 E perp 2, based on this discussion. 787 00:51:59,330 --> 00:52:00,820 All right, any questions so far? 788 00:52:03,400 --> 00:52:06,760 I hope everybody is following. 789 00:52:06,760 --> 00:52:10,660 Now, I can also use the third equation, 790 00:52:10,660 --> 00:52:16,270 which is actually del cross E-- 791 00:52:16,270 --> 00:52:22,270 the fourth equation-- which is actually the-- 792 00:52:22,270 --> 00:52:26,140 oh, sorry for that. 793 00:52:26,140 --> 00:52:28,370 I can actually use the second equation, 794 00:52:28,370 --> 00:52:34,880 which is the curl of E would be equal to minus partial B 795 00:52:34,880 --> 00:52:37,240 partial t. 796 00:52:37,240 --> 00:52:42,280 So that means I can have an integral E dot dl. 797 00:52:42,280 --> 00:52:50,672 And that is going to be equal to minus d dt, B dot dx. 798 00:52:53,440 --> 00:52:58,520 All right, so if I used the third equation 799 00:52:58,520 --> 00:53:01,840 in Maxwell's equation, basically, I 800 00:53:01,840 --> 00:53:12,020 have an integral over a loop, over some closed surface, 801 00:53:12,020 --> 00:53:17,640 and that would be equal to minus d dt, B dot-- 802 00:53:20,310 --> 00:53:25,670 this the actually the integral looking at the flux going 803 00:53:25,670 --> 00:53:29,290 through this little area. 804 00:53:29,290 --> 00:53:31,770 So what I can do is now I, again, 805 00:53:31,770 --> 00:53:38,500 zoom in to the surface which connects the two worlds. 806 00:53:38,500 --> 00:53:45,460 And I can now define a loop which is like this. 807 00:53:45,460 --> 00:53:50,160 With width equal to d. 808 00:53:50,160 --> 00:53:54,940 And there can be a contribution from the magnetic field 809 00:53:54,940 --> 00:54:01,030 in the right-hand side integral contributing to this equation. 810 00:54:03,550 --> 00:54:07,850 I can immediately write down the left-hand side 811 00:54:07,850 --> 00:54:12,680 will be E parallel 1. 812 00:54:12,680 --> 00:54:13,366 Why? 813 00:54:13,366 --> 00:54:14,740 Because right now we are actually 814 00:54:14,740 --> 00:54:20,170 looking at the component of the electric field parallel 815 00:54:20,170 --> 00:54:22,240 to the surface. 816 00:54:22,240 --> 00:54:27,632 And I can also immediately write down the right-hand side 817 00:54:27,632 --> 00:54:29,750 of this loop integral. 818 00:54:29,750 --> 00:54:35,120 You are going to have E parallel 2. 819 00:54:35,120 --> 00:54:38,500 And now I can do exactly the same trick, 820 00:54:38,500 --> 00:54:42,400 having this d goes to 0. 821 00:54:42,400 --> 00:54:45,230 The effect of this is the following. 822 00:54:45,230 --> 00:54:50,050 So if I have this rectangular loop, 823 00:54:50,050 --> 00:54:54,440 and now if I have this side d goes to 0, 824 00:54:54,440 --> 00:55:01,050 that means you will have no area to integrate the flux for the B 825 00:55:01,050 --> 00:55:01,930 field. 826 00:55:01,930 --> 00:55:06,400 So therefore, if I have d goes to 0, this will be equal-- 827 00:55:06,400 --> 00:55:08,860 this also goes to 0. 828 00:55:08,860 --> 00:55:12,220 Therefore, I can immediately conclude 829 00:55:12,220 --> 00:55:18,640 that this means that E parallel 1 will 830 00:55:18,640 --> 00:55:30,000 have to be equal to E parallel 2, based on this discussion. 831 00:55:30,000 --> 00:55:33,490 So this means that, well, we can immediately 832 00:55:33,490 --> 00:55:37,480 conclude that first of all, in order 833 00:55:37,480 --> 00:55:41,690 to figure out the relation between the incident wave, 834 00:55:41,690 --> 00:55:43,900 reflected wave, and transmitted wave, 835 00:55:43,900 --> 00:55:48,820 we need the help not only from the general property 836 00:55:48,820 --> 00:55:55,300 of the wave equation, and also some matching boundary 837 00:55:55,300 --> 00:55:58,450 conditions, we also need the help to provide 838 00:55:58,450 --> 00:56:01,490 additional boundary conditions to relay 839 00:56:01,490 --> 00:56:03,225 the electric field in the left-hand side 840 00:56:03,225 --> 00:56:04,400 and right-hand side. 841 00:56:04,400 --> 00:56:07,600 That is actually coming from our understanding of Maxwell's 842 00:56:07,600 --> 00:56:09,700 equation in matter. 843 00:56:09,700 --> 00:56:14,350 So you can see that the second two boundary 844 00:56:14,350 --> 00:56:19,770 conditions, epsilon1 E perp 1, perpendicular, 845 00:56:19,770 --> 00:56:23,780 will be equal to epsilon2 E 2 perpendicular. 846 00:56:23,780 --> 00:56:31,120 And that's essentially related to this surface integral D. 847 00:56:31,120 --> 00:56:34,960 And the second condition which tells 848 00:56:34,960 --> 00:56:41,030 us, will relate the field in the parallel direction 849 00:56:41,030 --> 00:56:45,180 is actually also coming from Maxwell's' equation. 850 00:56:45,180 --> 00:56:49,390 And we conclude that E parallel in the left-hand side 851 00:56:49,390 --> 00:56:53,410 will be equal to E parallel in the right-hand side. 852 00:56:53,410 --> 00:56:59,500 So we are almost there to solve the puzzle. 853 00:56:59,500 --> 00:57:04,550 So now what I have to do is the following. 854 00:57:04,550 --> 00:57:09,340 So now I would like to assume that we 855 00:57:09,340 --> 00:57:13,390 have some kind of polarization for the incident 856 00:57:13,390 --> 00:57:15,180 and the transmitted waves. 857 00:57:15,180 --> 00:57:19,610 So I assume that the polarization, 858 00:57:19,610 --> 00:57:22,360 it should be the following. 859 00:57:22,360 --> 00:57:27,070 The polarization is actually in parallel. 860 00:57:27,070 --> 00:57:32,030 I assume that the polarization is in parallel to the xz plane. 861 00:57:32,030 --> 00:57:34,930 So that's actually my assumption. 862 00:57:34,930 --> 00:57:37,520 So now I introduce a new assumption, 863 00:57:37,520 --> 00:57:41,800 which is that the incident wave is actually 864 00:57:41,800 --> 00:57:44,130 having a polarization. 865 00:57:44,130 --> 00:57:47,750 The electric field is actually oscillating up and down 866 00:57:47,750 --> 00:57:50,290 in this direction, which is actually parallel 867 00:57:50,290 --> 00:57:53,610 to the xz plane. 868 00:57:53,610 --> 00:57:58,940 So I can write down the corresponding E0R vector. 869 00:57:58,940 --> 00:58:02,710 And finally-- and this will be perpendicular to the direction 870 00:58:02,710 --> 00:58:04,490 of propagation. 871 00:58:04,490 --> 00:58:10,340 And finally, I will be able to also write down 872 00:58:10,340 --> 00:58:13,750 what will be the corresponding polarization 873 00:58:13,750 --> 00:58:16,690 for the transmitted wave, which is also, again, 874 00:58:16,690 --> 00:58:21,580 perpendicular to the direction of propagation 875 00:58:21,580 --> 00:58:29,020 So therefore I can now make use of the boundary condition one 876 00:58:29,020 --> 00:58:32,800 and boundary condition two to actually figure out 877 00:58:32,800 --> 00:58:39,550 what would be the relation between E0I, E0R, and E0T. 878 00:58:39,550 --> 00:58:41,500 The first thing which we consider 879 00:58:41,500 --> 00:58:45,621 is to consider the perpendicular direction. 880 00:58:49,830 --> 00:58:52,120 For equation number one, I will be 881 00:58:52,120 --> 00:59:01,750 able to conclude that epsilon1 minus E0I sine theta I. 882 00:59:01,750 --> 00:59:04,880 So basically, that is actually the contribution 883 00:59:04,880 --> 00:59:09,645 from the first vector, E0I. 884 00:59:09,645 --> 00:59:14,320 You can see that now I am trying to project everything 885 00:59:14,320 --> 00:59:19,640 to the direction perpendicular to the surface. 886 00:59:19,640 --> 00:59:22,450 And you can see that I have a minus sign. 887 00:59:22,450 --> 00:59:26,530 And that this angle is actually theta. 888 00:59:26,530 --> 00:59:27,760 So therefore, I have-- 889 00:59:32,460 --> 00:59:35,200 So this angle is actually what? 890 00:59:35,200 --> 00:59:38,350 This angle is actually I minus theta. 891 00:59:38,350 --> 00:59:42,130 So therefore, I have the cosine theta I. 892 00:59:42,130 --> 00:59:44,110 And I have this minus sign, because it's 893 00:59:44,110 --> 00:59:48,580 pointing to the left-hand side of the board. 894 00:59:48,580 --> 00:59:50,320 So that's actually the contribution 895 00:59:50,320 --> 00:59:52,540 of the incident wave. 896 00:59:52,540 --> 00:59:54,550 And I have a second term, which is 897 00:59:54,550 --> 01:00:01,000 the contribution of the reflected wave, E0R sine theta 898 01:00:01,000 --> 01:00:10,950 I. So E0I, E0R with all vector is actually just the length 899 01:00:10,950 --> 01:00:15,010 of the vector in my definition. 900 01:00:15,010 --> 01:00:19,480 And based on the first boundary condition, 901 01:00:19,480 --> 01:00:22,570 I have the left-hand side looks like this. 902 01:00:22,570 --> 01:00:29,910 On the right-hand side, I will have minus epsilon2 E0T sine 903 01:00:29,910 --> 01:00:31,630 theta T. 904 01:00:31,630 --> 01:00:38,390 Again, I am looking at the projection of this E0T vector 905 01:00:38,390 --> 01:00:40,990 in the direction which is actually 906 01:00:40,990 --> 01:00:44,260 perpendicular to the surface. 907 01:00:44,260 --> 01:00:47,220 So that's the first expression I can get. 908 01:00:47,220 --> 01:00:53,230 And then from the expression number two, I can now-- 909 01:00:53,230 --> 01:00:59,900 taking the projection, which is parallel to the surface, 910 01:00:59,900 --> 01:01:03,370 so now this is actually the parallel direction. 911 01:01:03,370 --> 01:01:05,110 And basically, what I'm going to get 912 01:01:05,110 --> 01:01:17,925 is E0I cosine theta I plus E0R cosine theta I-- 913 01:01:17,925 --> 01:01:23,120 oh sorry, theta R. Theta R is actually equal to theta I, 914 01:01:23,120 --> 01:01:27,280 so therefore, I actually replaced that by the I already. 915 01:01:27,280 --> 01:01:35,820 And that will be equal to E0T cosine theta T. Any questions 916 01:01:35,820 --> 01:01:38,650 so Far so I was pretty fast. 917 01:01:38,650 --> 01:01:42,010 So here, I already immediately write down 918 01:01:42,010 --> 01:01:45,470 this is actually theta R. Theta R is equal to theta I, 919 01:01:45,470 --> 01:01:48,250 so therefore, I already replaced that by theta I. 920 01:01:48,250 --> 01:01:51,040 Do you have any questions? 921 01:01:51,040 --> 01:01:58,360 So the painful period is going to end in like 3 minutes, OK? 922 01:01:58,360 --> 01:02:00,330 So we are almost there. 923 01:02:00,330 --> 01:02:01,930 And look what we have been doing. 924 01:02:01,930 --> 01:02:04,240 We basically figured out the boundary condition 925 01:02:04,240 --> 01:02:05,950 from Maxwell's equation. 926 01:02:05,950 --> 01:02:07,300 Then we are plugging in that. 927 01:02:07,300 --> 01:02:10,220 So we assume that the polarization is actually 928 01:02:10,220 --> 01:02:13,840 parallel on the plane of xz plane. 929 01:02:13,840 --> 01:02:16,540 And now we are actually just evaluating 930 01:02:16,540 --> 01:02:20,620 the parallel component and the perpendicular component. 931 01:02:20,620 --> 01:02:22,270 So that's actually what we've got. 932 01:02:26,380 --> 01:02:36,670 The goal, as a reminder, is to write E0R and E0T 933 01:02:36,670 --> 01:02:41,170 in terms of the known part, which is the E0I. 934 01:02:41,170 --> 01:02:44,140 I would like to relate these three-- 935 01:02:44,140 --> 01:02:47,000 the magnitude of these three vectors. 936 01:02:47,000 --> 01:02:51,640 So from equation number one, I can actually rewrite that. 937 01:02:51,640 --> 01:02:53,800 Basically, I can actually conclude 938 01:02:53,800 --> 01:03:00,080 that I can divide everything by epsilon1 sine theta I. 939 01:03:00,080 --> 01:03:06,920 So what I'm going to get is E0I minus E0R. 940 01:03:06,920 --> 01:03:15,930 And this will be equal to epsilon2 sine theta 941 01:03:15,930 --> 01:03:26,610 T divided by epsilon1 sine theta R. And this is actually E0T. 942 01:03:26,610 --> 01:03:29,010 So basically, I'm dividing everything 943 01:03:29,010 --> 01:03:33,400 by minus epsilon1 sine theta I. 944 01:03:33,400 --> 01:03:35,130 Then basically, what you are going to get 945 01:03:35,130 --> 01:03:37,073 is this expression. 946 01:03:40,140 --> 01:03:45,630 And this actually can be related to another expression, which 947 01:03:45,630 --> 01:03:52,530 is epsilon2 n1 divided by epsilon1 n2. 948 01:03:52,530 --> 01:03:55,560 Because I can use Snell's law. 949 01:03:55,560 --> 01:04:01,950 n1 sine theta I will be equal to n2 sine theta T. Therefore, 950 01:04:01,950 --> 01:04:08,070 I can replace this ratio of sine angle by refractive index. 951 01:04:08,070 --> 01:04:09,640 Everybody's following? 952 01:04:09,640 --> 01:04:11,130 OK, very good. 953 01:04:11,130 --> 01:04:16,680 And this will be multiplied by E0T. 954 01:04:16,680 --> 01:04:20,140 I can now define this. 955 01:04:20,140 --> 01:04:24,206 This is actually defined as theta E0T 956 01:04:24,206 --> 01:04:28,140 to make our life easier. 957 01:04:28,140 --> 01:04:31,480 The same thing can be done for the second expression. 958 01:04:31,480 --> 01:04:38,310 What I'm going to get is E0I plus E0R. 959 01:04:38,310 --> 01:04:41,670 Basically, what I'm doing is to divide everything 960 01:04:41,670 --> 01:04:45,810 by cosine theta I. So what is going to happen 961 01:04:45,810 --> 01:04:50,610 is that you are going to get cosine theta T divided 962 01:04:50,610 --> 01:04:56,710 by cosine theta I E0T. 963 01:04:56,710 --> 01:05:03,240 And that is actually defined as alpha E0T. 964 01:05:06,250 --> 01:05:10,410 So alpha is defined as cosine theta T divided by cosine 965 01:05:10,410 --> 01:05:12,240 theta I. 966 01:05:12,240 --> 01:05:15,040 Therefore, I can already immediately, 967 01:05:15,040 --> 01:05:21,240 based on these two expressions, to solve what would be the E0R. 968 01:05:21,240 --> 01:05:22,980 I can write down the solution. 969 01:05:22,980 --> 01:05:25,500 So basically, you can actually quickly derive 970 01:05:25,500 --> 01:05:27,210 what would be the E0R. 971 01:05:27,210 --> 01:05:30,900 And that is actually going to be my alpha minus beta divided 972 01:05:30,900 --> 01:05:34,890 by alpha plus beta, E0I. 973 01:05:37,530 --> 01:05:40,790 And you can also solve based on these two equations what 974 01:05:40,790 --> 01:05:42,960 would be the E0T. 975 01:05:42,960 --> 01:05:44,880 And now we will conclude that this 976 01:05:44,880 --> 01:05:50,330 will be equal to 2 divided by alpha plus beta, E0I. 977 01:05:53,580 --> 01:05:57,120 So this means that the refractive index will 978 01:05:57,120 --> 01:06:00,690 be equal to alpha minus beta. 979 01:06:00,690 --> 01:06:05,080 Sorry, refraction coefficient will 980 01:06:05,080 --> 01:06:10,170 be equal to alpha minus beta divided by alpha plus beta. 981 01:06:10,170 --> 01:06:13,140 And the transmission coefficient is actually 982 01:06:13,140 --> 01:06:17,190 2 over alpha plus beta. 983 01:06:17,190 --> 01:06:22,740 So we will take maybe a three minute break for you 984 01:06:22,740 --> 01:06:24,720 to be able to ask some questions. 985 01:06:24,720 --> 01:06:27,870 But you can see that we have solved the relation 986 01:06:27,870 --> 01:06:31,410 between E0I, E0R, and E0T. 987 01:06:31,410 --> 01:06:34,050 And what we have to do in the rest of the time 988 01:06:34,050 --> 01:06:36,930 is to enjoy what we actually already derived, 989 01:06:36,930 --> 01:06:39,920 and what actually that means, after the break. 990 01:06:39,920 --> 01:06:42,035 So we come back at 45. 991 01:06:48,280 --> 01:06:49,420 OK, so very good. 992 01:06:49,420 --> 01:06:51,400 So we have survived this. 993 01:06:51,400 --> 01:06:53,950 And now it's time to enjoy what we have 994 01:06:53,950 --> 01:06:56,620 learned from this equation. 995 01:06:56,620 --> 01:07:00,010 All right, so welcome back everybody. 996 01:07:00,010 --> 01:07:03,370 So we have actually saw what would 997 01:07:03,370 --> 01:07:06,960 be the refraction coefficient and the transmission 998 01:07:06,960 --> 01:07:08,740 coefficient tau. 999 01:07:08,740 --> 01:07:11,560 And those are the functions of alpha and beta. 1000 01:07:15,240 --> 01:07:18,340 So that's considered three different interesting cases. 1001 01:07:18,340 --> 01:07:22,422 So if I have normal incidence. 1002 01:07:26,790 --> 01:07:31,850 And that means alpha will be equal to cosine 1003 01:07:31,850 --> 01:07:37,950 theta T divided by cosine theta I. This is the definition. 1004 01:07:37,950 --> 01:07:40,190 If I have no more incidence, that 1005 01:07:40,190 --> 01:07:45,330 means both theta T, theta I, and theta R 1006 01:07:45,330 --> 01:07:47,400 will be equal to 90 degrees. 1007 01:07:52,230 --> 01:07:57,560 And in this case, basically, you will have the same cosine 1008 01:07:57,560 --> 01:07:58,650 theta-- 1009 01:07:58,650 --> 01:08:03,090 I mean cosine I. Theta T and cosine theta I. 1010 01:08:03,090 --> 01:08:11,820 And that means your alpha will be equal to 1 in this case. 1011 01:08:11,820 --> 01:08:17,189 So in normal material, mu1 is roughly equal to mu2 1012 01:08:17,189 --> 01:08:21,630 and roughly equal to mu0. 1013 01:08:21,630 --> 01:08:26,939 So therefore, this means that if mu1, mu2, and mu0 1014 01:08:26,939 --> 01:08:31,910 are very close to each other, then actually, 1015 01:08:31,910 --> 01:08:35,250 the refractive index based on that equation which 1016 01:08:35,250 --> 01:08:40,340 is showing there, will be basically, roughly 1017 01:08:40,340 --> 01:08:45,229 equal to square root of epsilon divided by epsilon0. 1018 01:08:48,120 --> 01:08:53,894 So therefore, beta would be equal to epsilon2 divided 1019 01:08:53,894 --> 01:08:58,540 by epsilon1, n1 divided by n2. 1020 01:08:58,540 --> 01:09:02,399 This is actually the definition. 1021 01:09:02,399 --> 01:09:07,229 And this will be basically equal to-- 1022 01:09:07,229 --> 01:09:10,180 since this is actually proportional to epsilon2 1023 01:09:10,180 --> 01:09:13,545 divided by epsilon1. 1024 01:09:13,545 --> 01:09:17,800 And n is actually proportional to epsilon, square root 1025 01:09:17,800 --> 01:09:19,080 of epsilon. 1026 01:09:19,080 --> 01:09:21,090 Therefore, you can conclude that this actually 1027 01:09:21,090 --> 01:09:26,505 will be equal to n2 squared divided by n1 squared, 1028 01:09:26,505 --> 01:09:28,569 n1 divided by n2. 1029 01:09:28,569 --> 01:09:32,275 And that will give you n2 divided by n1. 1030 01:09:32,275 --> 01:09:37,560 You can actually cancel one of the n2 and one of the n1. 1031 01:09:37,560 --> 01:09:42,540 So beta will be equal to n2 divided by n1. 1032 01:09:42,540 --> 01:09:45,420 And therefore, you can conclude that R 1033 01:09:45,420 --> 01:09:52,800 will be a function just related to the refractive index, which 1034 01:09:52,800 --> 01:09:58,410 means that you are going to get n1 minus n2 divided by n1 plus 1035 01:09:58,410 --> 01:10:00,480 n2. 1036 01:10:00,480 --> 01:10:04,230 Which means that the amount of reflected light 1037 01:10:04,230 --> 01:10:10,140 is related to the difference in the refractive index. 1038 01:10:10,140 --> 01:10:12,570 Then the amount of the transmitted light 1039 01:10:12,570 --> 01:10:20,000 will be equal two n1 divided by n1 plus n2. 1040 01:10:20,000 --> 01:10:22,110 So what does that mean? 1041 01:10:22,110 --> 01:10:26,770 This means that if you have some material which is essentially 1042 01:10:26,770 --> 01:10:35,020 like diamond, diamond have an n2 equal to something like 2.6, 1043 01:10:35,020 --> 01:10:39,770 that means a lot of light will get reflected, 1044 01:10:39,770 --> 01:10:42,230 even if you have no more incidence. 1045 01:10:42,230 --> 01:10:45,830 So that's actually why the diamonds are so beautiful, 1046 01:10:45,830 --> 01:10:49,670 because a lot of light, pretty bright, and a lot of things 1047 01:10:49,670 --> 01:10:52,380 are actually reflected. 1048 01:10:52,380 --> 01:10:57,950 The transmitted fraction is actually pretty small. 1049 01:10:57,950 --> 01:11:05,013 I can also assume that there can be a grazing incidence. 1050 01:11:09,450 --> 01:11:12,090 That happens-- this means that I am 1051 01:11:12,090 --> 01:11:22,810 going to have theta I. This theta I should be here. 1052 01:11:22,810 --> 01:11:27,420 This theta I is going to be roughly 90 degrees. 1053 01:11:27,420 --> 01:11:29,610 In the case of no more incidence, theta 1054 01:11:29,610 --> 01:11:31,680 I should be equal to zero. 1055 01:11:31,680 --> 01:11:34,950 Maybe I misspoke in the beginning. 1056 01:11:34,950 --> 01:11:36,820 So what does that mean? 1057 01:11:36,820 --> 01:11:40,730 This means that alpha will go to infinity, because theta 1058 01:11:40,730 --> 01:11:45,120 I is going to 90 degrees. 1059 01:11:45,120 --> 01:11:48,120 Therefore, you have R roughly equal 1060 01:11:48,120 --> 01:11:52,460 to 1, because R is actually alpha minus beta 1061 01:11:52,460 --> 01:11:55,300 divided by alpha plus beta. 1062 01:11:55,300 --> 01:12:00,420 If alpha goes to infinity, then R will go to 1. 1063 01:12:00,420 --> 01:12:04,265 And the tau will be-- 1064 01:12:04,265 --> 01:12:07,860 actually, roughly goes to 0. 1065 01:12:10,500 --> 01:12:13,620 So that means if you have a grazing incidence, 1066 01:12:13,620 --> 01:12:17,730 then basically, most of the light are reflected. 1067 01:12:17,730 --> 01:12:24,250 So that's actually why when we see, 1068 01:12:24,250 --> 01:12:30,600 for example, reflected light from the sun, which is actually 1069 01:12:30,600 --> 01:12:34,140 on the road, we see that a lot of light are reflected. 1070 01:12:34,140 --> 01:12:38,130 When we see a lake which is actually very far away from me 1071 01:12:38,130 --> 01:12:41,390 and the sun in front of it, you see a huge amount of lite 1072 01:12:41,390 --> 01:12:43,980 got reflected and going to your eyes. 1073 01:12:43,980 --> 01:12:48,210 Looks really bright. 1074 01:12:48,210 --> 01:12:52,800 So finally, there is a very interesting angle, 1075 01:12:52,800 --> 01:12:57,654 which is actually considering a situation when 1076 01:12:57,654 --> 01:13:01,600 alpha is equal to beta. 1077 01:13:01,600 --> 01:13:05,890 This is a very interesting angle, 1078 01:13:05,890 --> 01:13:11,050 theta B. If we choose this theta B such 1079 01:13:11,050 --> 01:13:15,770 that alpha is equal to beta, what is going to happen? 1080 01:13:15,770 --> 01:13:17,710 Somebody can tell me. 1081 01:13:17,710 --> 01:13:19,215 If I make-- 1082 01:13:19,215 --> 01:13:20,277 STUDENT: [INAUDIBLE] 1083 01:13:20,277 --> 01:13:21,360 PROFESSOR: Exactly, right? 1084 01:13:21,360 --> 01:13:25,500 So if you choose an angle such that alpha is equal to beta, 1085 01:13:25,500 --> 01:13:27,180 then R will be equal to 0. 1086 01:13:27,180 --> 01:13:28,980 There will be no reflection. 1087 01:13:28,980 --> 01:13:33,630 And everything goes through the material. 1088 01:13:33,630 --> 01:13:36,060 So that is actually so-called Brewster's angle. 1089 01:13:45,620 --> 01:13:52,325 And this happens when theta B-- 1090 01:13:56,764 --> 01:13:58,680 which theta B is actually the incident angle-- 1091 01:14:02,570 --> 01:14:09,490 plus theta T is equal to pi over 2. 1092 01:14:09,490 --> 01:14:13,310 There's a proof of this Brewster's angle in the lecture 1093 01:14:13,310 --> 01:14:13,810 notes. 1094 01:14:13,810 --> 01:14:15,700 But we are kind of running out of time. 1095 01:14:15,700 --> 01:14:19,050 But the conclusion is that you will 1096 01:14:19,050 --> 01:14:23,870 need to have theta B, which is actually equal to theta I, 1097 01:14:23,870 --> 01:14:29,870 the incident angle, plus the transmission light angle, 1098 01:14:29,870 --> 01:14:33,590 theta D. If that is equal to 90 degrees, 1099 01:14:33,590 --> 01:14:35,960 then you can make alpha equal to 0. 1100 01:14:35,960 --> 01:14:38,540 And what is going to happen is that there 1101 01:14:38,540 --> 01:14:43,820 will be no reflected light. 1102 01:14:43,820 --> 01:14:49,430 So when this happens, when the reflected light 1103 01:14:49,430 --> 01:14:54,430 and the transmitted light have an angle of 90 degrees, 1104 01:14:54,430 --> 01:14:58,760 then the amplitude goes to 0. 1105 01:14:58,760 --> 01:15:01,130 This is actually a very interesting property 1106 01:15:01,130 --> 01:15:04,010 and it only works for electromagnetic waves, 1107 01:15:04,010 --> 01:15:07,060 because this is actually coming from, really, 1108 01:15:07,060 --> 01:15:11,000 the effect of Maxwell's equation. 1109 01:15:11,000 --> 01:15:14,870 So now what I'm going to say is that basically, we 1110 01:15:14,870 --> 01:15:17,280 look at this demonstration. 1111 01:15:17,280 --> 01:15:24,330 So if we have an incident light, and the transmitted light. 1112 01:15:24,330 --> 01:15:27,920 Originally, the incident light is unpolarized. 1113 01:15:27,920 --> 01:15:30,920 So you can have all kinds of different polarization. 1114 01:15:30,920 --> 01:15:33,080 So we can become post polarization 1115 01:15:33,080 --> 01:15:36,200 into a component which is actually pointing 1116 01:15:36,200 --> 01:15:38,480 to you, which is the dot. 1117 01:15:38,480 --> 01:15:40,610 And the component, which is actually 1118 01:15:40,610 --> 01:15:45,170 the parallel to my slide, which is actually what we have been 1119 01:15:45,170 --> 01:15:47,630 working on, that situation. 1120 01:15:47,630 --> 01:15:51,110 So what is going to happen is that the component which 1121 01:15:51,110 --> 01:15:53,930 is pointing you is actually never 1122 01:15:53,930 --> 01:15:57,950 gets suppressed, because there will be no perpendicular 1123 01:15:57,950 --> 01:15:58,490 component. 1124 01:15:58,490 --> 01:16:05,990 So therefore, even if you add Brewster's angle, 1125 01:16:05,990 --> 01:16:07,880 it should get reflected. 1126 01:16:07,880 --> 01:16:10,490 On the other hand, all the components 1127 01:16:10,490 --> 01:16:12,890 which essentially heavy polarization 1128 01:16:12,890 --> 01:16:16,500 parallel to this slide is eliminated 1129 01:16:16,500 --> 01:16:19,350 because of this relation. 1130 01:16:19,350 --> 01:16:25,650 So that means the reflected light will be highly polarized. 1131 01:16:25,650 --> 01:16:27,790 Do you believe me? 1132 01:16:27,790 --> 01:16:29,200 Maybe not. 1133 01:16:29,200 --> 01:16:32,200 We can do an experiment and really show you 1134 01:16:32,200 --> 01:16:34,690 that's the case. 1135 01:16:34,690 --> 01:16:35,740 So we are almost there. 1136 01:16:38,470 --> 01:16:43,180 So now I need to turn off the light and hide the image. 1137 01:16:43,180 --> 01:16:47,140 And you can see that there is a setup here which 1138 01:16:47,140 --> 01:16:49,930 I produce unpolarized light. 1139 01:16:49,930 --> 01:16:54,360 And there's a glass here, which actually I 1140 01:16:54,360 --> 01:16:58,930 reflect the unpolarized light. 1141 01:16:58,930 --> 01:17:02,350 So now you can see that if I have some random angle, 1142 01:17:02,350 --> 01:17:04,480 and I have a polarizer here-- 1143 01:17:04,480 --> 01:17:05,860 I hope you can see it-- 1144 01:17:05,860 --> 01:17:11,590 you can see that the polarizer cannot eliminate all the light. 1145 01:17:11,590 --> 01:17:15,440 So basically, no matter what kind of direction, 1146 01:17:15,440 --> 01:17:22,030 it will not be able to eliminate all the reflected light. 1147 01:17:22,030 --> 01:17:25,600 This means there's some mixture of all kinds 1148 01:17:25,600 --> 01:17:27,635 of different polarization. 1149 01:17:27,635 --> 01:17:33,790 But now, if I change the direction to Brewster's angle, 1150 01:17:33,790 --> 01:17:39,070 it's roughly here, so you can see that now, indeed, 1151 01:17:39,070 --> 01:17:43,090 I can actually eliminate all the contribution 1152 01:17:43,090 --> 01:17:44,080 of the reflected light. 1153 01:17:44,080 --> 01:17:48,620 Because the reflected light is highly polarized. 1154 01:17:48,620 --> 01:17:56,390 As you can see from the slide, all the component 1155 01:17:56,390 --> 01:17:59,740 which is actually parallel to the slides 1156 01:17:59,740 --> 01:18:03,430 is actually eliminated due to Brewster's angle. 1157 01:18:03,430 --> 01:18:06,400 And that produces a polarized light. 1158 01:18:06,400 --> 01:18:11,640 And that can be filtered out by the polarizer. 1159 01:18:17,100 --> 01:18:20,930 So coming back to the question which we had before, 1160 01:18:20,930 --> 01:18:24,310 so why can we take such a good photo? 1161 01:18:24,310 --> 01:18:27,250 That is because of Brewster's angle. 1162 01:18:27,250 --> 01:18:30,650 So once the sunlight gets reflected by the window, 1163 01:18:30,650 --> 01:18:33,880 it becomes linearly polarized, and therefore, you can actually 1164 01:18:33,880 --> 01:18:37,170 filter out the majority of the contribution 1165 01:18:37,170 --> 01:18:41,580 by using polarization filter. 1166 01:18:41,580 --> 01:18:42,760 OK, thank you very much. 1167 01:18:42,760 --> 01:18:45,310 And I hope you enjoyed the lecture today. 1168 01:18:45,310 --> 01:18:51,220 And hope this will improve your technique, your skill, 1169 01:18:51,220 --> 01:18:53,370 for taking good photos. 1170 01:18:53,370 --> 01:18:56,045 If you have any questions, please let me know. 1171 01:19:02,700 --> 01:19:04,420 Hello, everybody. 1172 01:19:04,420 --> 01:19:09,840 So today I'm going to show you a demonstration of Brewster's 1173 01:19:09,840 --> 01:19:11,250 angle. 1174 01:19:11,250 --> 01:19:13,710 So during the class, we were discussing 1175 01:19:13,710 --> 01:19:16,260 about how to make very good photos, how 1176 01:19:16,260 --> 01:19:21,420 to use polarizer to filter out the reflected light 1177 01:19:21,420 --> 01:19:23,430 from the sun. 1178 01:19:23,430 --> 01:19:30,090 Usually, when you actually take a photo of water or a car, 1179 01:19:30,090 --> 01:19:33,060 there are refracted light from the sun on the window 1180 01:19:33,060 --> 01:19:35,190 or on the water. 1181 01:19:35,190 --> 01:19:38,970 And then you can actually use polarizer to filter them out. 1182 01:19:38,970 --> 01:19:42,030 And that has to do with the property 1183 01:19:42,030 --> 01:19:45,690 of the electromagnetic wave and the Brewster's angle. 1184 01:19:45,690 --> 01:19:49,930 So here I have an experimental setup here, 1185 01:19:49,930 --> 01:19:52,210 which consists of three components. 1186 01:19:52,210 --> 01:19:56,580 The first component is a polarized light source. 1187 01:19:56,580 --> 01:19:59,180 And it meets unpolarized light. 1188 01:19:59,180 --> 01:20:04,050 And those light are getting refracted by glass here. 1189 01:20:04,050 --> 01:20:08,670 And the unrefracted light will actually 1190 01:20:08,670 --> 01:20:12,730 be shown on the screen as a spot there. 1191 01:20:12,730 --> 01:20:21,900 So at first, if I have my glass, which essentially-- 1192 01:20:21,900 --> 01:20:24,030 the position of the glass is in a way such 1193 01:20:24,030 --> 01:20:26,880 that it's actually not on Brewster's angle. 1194 01:20:26,880 --> 01:20:29,800 And now I can actually check if this light is actually 1195 01:20:29,800 --> 01:20:33,030 polarized by using a polarizer here. 1196 01:20:33,030 --> 01:20:37,950 And if I put this polarizer between the screen 1197 01:20:37,950 --> 01:20:40,590 and the glass, you can see that, huh, 1198 01:20:40,590 --> 01:20:47,430 as a function of the angle which I am rotating this polarizer, 1199 01:20:47,430 --> 01:20:53,430 you can see that no angle can actually completely eliminate 1200 01:20:53,430 --> 01:20:55,380 the reflected light. 1201 01:20:55,380 --> 01:20:59,880 So that means that the reflected light is actually not 1202 01:20:59,880 --> 01:21:02,070 perfectly polarized. 1203 01:21:02,070 --> 01:21:05,570 But on the other hand, you can also see that in some angles, 1204 01:21:05,570 --> 01:21:09,330 you can actually significantly lower the intensity. 1205 01:21:09,330 --> 01:21:13,560 And that, essentially, is also pretty good for photo 1206 01:21:13,560 --> 01:21:18,190 taking, because that means all the reflected light, 1207 01:21:18,190 --> 01:21:20,310 although now the angle is actually not 1208 01:21:20,310 --> 01:21:25,800 at Brewster's angle, you still have the reflected light 1209 01:21:25,800 --> 01:21:27,370 slightly polarized. 1210 01:21:27,370 --> 01:21:30,770 So that actually your polarizer in front of the camera 1211 01:21:30,770 --> 01:21:33,380 will still do some work. 1212 01:21:33,380 --> 01:21:36,500 Now what I'm going to do is to change the angle so that it 1213 01:21:36,500 --> 01:21:40,590 matches with Brewster's angle. 1214 01:21:40,590 --> 01:21:43,792 So now you can see that if I insert 1215 01:21:43,792 --> 01:21:50,220 a polarizer between the glass and the screen, 1216 01:21:50,220 --> 01:21:54,060 you can see that at some angle, for example, now 1217 01:21:54,060 --> 01:21:58,366 we can actually filter out or completely eliminate 1218 01:21:58,366 --> 01:22:00,390 the spot on the screen. 1219 01:22:00,390 --> 01:22:04,440 So that means at Brewster's angle, 1220 01:22:04,440 --> 01:22:07,430 basically, the reflected light is actually 1221 01:22:07,430 --> 01:22:10,320 completely polarized, as we actually 1222 01:22:10,320 --> 01:22:13,290 predicted from the lecture. 1223 01:22:13,290 --> 01:22:14,700 And the reason is the following. 1224 01:22:14,700 --> 01:22:17,820 There's only one direction of the polarized light 1225 01:22:17,820 --> 01:22:22,050 from the unpolarized source can get reflected 1226 01:22:22,050 --> 01:22:26,940 due to the boundary condition of the electromagnetic wave. 1227 01:22:26,940 --> 01:22:30,810 And therefore, we see this very unique phenomena 1228 01:22:30,810 --> 01:22:35,050 which we can only see in electromagnetic waves.