1 00:00:00,000 --> 00:00:24,000 2 00:00:24,000 --> 00:00:29,330 Today's lecture is going to be about electromagnetic radiation 3 00:00:29,330 --> 00:00:34,398 which is one of the highlights in 8.03 and one of the great 4 00:00:34,398 --> 00:00:37,281 victories of 19th century physics. 5 00:00:37,281 --> 00:00:41,912 So far we have dealt with mechanical waves and we have 6 00:00:41,912 --> 00:00:47,504 dealt with sound waves but today we are going to enter the domain 7 00:00:47,504 --> 00:00:50,737 of electromagnetic waves. Radio waves, 8 00:00:50,737 --> 00:00:53,533 radar, infrareds, visible lights, 9 00:00:53,533 --> 00:00:56,330 ultraviolets, gamma rays, x-rays, 10 00:00:56,330 --> 00:01:03,377 electromagnetic radiation. And electric field, 11 00:01:03,377 --> 00:01:10,666 which is associated with a magnetic field, 12 00:01:10,666 --> 00:01:18,133 one cannot be thought of without the other, 13 00:01:18,133 --> 00:01:26,133 propagates through space, even in empty space, 14 00:01:26,133 --> 00:01:34,311 even in vacuum. And I will start with Maxwell's 15 00:01:34,311 --> 00:01:45,333 equations, and then you will see a little bit more math which I 16 00:01:45,333 --> 00:01:53,333 do to brush up on what you may have forgotten. 17 00:01:53,333 --> 00:02:00,266 So let's start with Maxwell's equations. 18 00:02:00,266 --> 00:02:16,071 01:30 And then we have the curl of E 19 00:02:16,071 --> 00:02:28,946 equal minus dB/dt, Faraday's Law. 20 00:02:28,946 --> 00:02:50,272 And then we have the curl of B equals mu zero times J, 21 00:02:50,272 --> 00:03:06,769 J being current density that is in vacuum. 22 00:03:06,769 --> 00:03:16,769 02:30 If you apply the dell vector on 23 00:03:16,769 --> 00:03:23,692 a function phi, this is called the gradient of 24 00:03:23,692 --> 00:03:29,692 phi, that is a vector. That is d phi/dx, 25 00:03:29,692 --> 00:03:38,000 x roof plus d phi/dy y roof plus d phi/dz z roof. 26 00:03:38,000 --> 00:03:48,565 And then we have the divergence of a vector A. 27 00:03:48,565 --> 00:04:01,243 That itself is a scalar. And that is dAx/dx plus dAy/dy 28 00:04:01,243 --> 00:04:05,000 plus dAz/dz. 29 00:04:05,000 --> 00:04:12,000 30 00:04:12,000 --> 00:04:21,578 I will raise this again later but I want you to see this. 31 00:04:21,578 --> 00:04:27,736 The curl of A, the way I remember it, 32 00:04:27,736 --> 00:04:35,092 and you probably do the same, x, y, z, d/dx, 33 00:04:35,092 --> 00:04:43,555 d/dy, d/dz. And then you get your Ax, 34 00:04:43,555 --> 00:04:51,333 Ay, Az. And that then becomes dAz/dy 35 00:04:51,333 --> 00:05:02,000 minus dAy/dz. And that is in the x direction. 36 00:05:02,000 --> 00:05:13,793 And then you get dAx/dz minus dAz/dx, and that is in the y 37 00:05:13,793 --> 00:05:21,862 direction. And then you would get dAy/dx 38 00:05:21,862 --> 00:05:32,000 minus dAx/dy, and that is in the z direction. 39 00:05:32,000 --> 00:05:37,261 And let me check that because it is so easy to make a little 40 00:05:37,261 --> 00:05:42,611 slip, and that is very awkward for me and for you also later. 41 00:05:42,611 --> 00:05:46,267 That is fine. And then we get dAy/dx minus 42 00:05:46,267 --> 00:05:49,299 dAx/dy, that is fine. There is one, 43 00:05:49,299 --> 00:05:52,955 and only one, vector manipulation with the 44 00:05:52,955 --> 00:05:56,076 dell vector that I want you to know. 45 00:05:56,076 --> 00:06:00,000 I don't want you to remember it. 46 00:06:00,000 --> 00:06:07,688 I certainly don't remember it, but we need it today. 47 00:06:07,688 --> 00:06:12,512 And that is the curl of the curl. 48 00:06:12,512 --> 00:06:20,954 The curl of the curl of A, and I will show you the result 49 00:06:20,954 --> 00:06:26,381 without proof, is the gradient of the 50 00:06:26,381 --> 00:06:35,723 divergence of A. Minus dell dot dell vector A. 51 00:06:35,723 --> 00:06:44,802 And this we often write as simply dell squared. 52 00:06:44,802 --> 00:06:51,513 For this we write dell square of A. 53 00:06:51,513 --> 00:07:00,000 And this is also called the Laplacian. 54 00:07:00,000 --> 00:07:08,010 And so the dell square of a vector is d2A/dx2, 55 00:07:08,010 --> 00:07:15,486 the whole vector plus d2A/dy2 plus d2A/dz2. 56 00:07:15,486 --> 00:07:24,743 And this has nine terms because this has three terms, 57 00:07:24,743 --> 00:07:34,000 this has three terms and this has three terms. 58 00:07:34,000 --> 00:07:42,654 I will do the first one, and then you have to do the 59 00:07:42,654 --> 00:07:49,103 others. I will only do the component in 60 00:07:49,103 --> 00:07:58,606 the x direction which then becomes d2Ax/dx2 plus d2Ax/dy2 61 00:07:58,606 --> 00:08:05,028 plus d2Ax/dz2. And that is then the component 62 00:08:05,028 --> 00:08:09,974 in the x direction. And there is also one in the y 63 00:08:09,974 --> 00:08:14,517 direction and there is one in the z direction. 64 00:08:14,517 --> 00:08:20,776 This first term comes from this one, the second term comes from 65 00:08:20,776 --> 00:08:25,621 this one, and the third term comes from this one. 66 00:08:25,621 --> 00:08:30,870 Now we are in a good position to start to use this on 67 00:08:30,870 --> 00:08:36,258 Maxwell's equations. I will raise this again, 68 00:08:36,258 --> 00:08:41,341 I will lower this so you can see both, and I will start 69 00:08:41,341 --> 00:08:46,611 working on the center board. What I am going to do now is 70 00:08:46,611 --> 00:08:50,000 take the curl of the curl of E. 71 00:08:50,000 --> 00:09:00,000 72 00:09:00,000 --> 00:09:07,587 And I know that the curl of the curl of E is minus d/dt because 73 00:09:07,587 --> 00:09:14,685 the curl of E is minus dB/dt. And so it is minus d/dt times 74 00:09:14,685 --> 00:09:19,947 the curl of B. Now I use the only thing that 75 00:09:19,947 --> 00:09:25,454 we should remember, and that is this identity, 76 00:09:25,454 --> 00:09:31,818 that the curl of the curl of A is the gradient of the 77 00:09:31,818 --> 00:09:38,316 divergence of A. But the divergence of E is zero 78 00:09:38,316 --> 00:09:41,953 in empty space, you see that there, 79 00:09:41,953 --> 00:09:48,051 so this first term is zero. I only have this term which is 80 00:09:48,051 --> 00:09:53,293 then minus the Laplacian operator on the vector A. 81 00:09:53,293 --> 00:09:58,000 So this is also minus dell squared E. 82 00:09:58,000 --> 00:10:06,000 83 00:10:06,000 --> 00:10:13,559 Now, the curl of B is epsilon zero mu zero dE/dt. 84 00:10:13,559 --> 00:10:22,062 So when I take the time derivative, I get minus epsilon 85 00:10:22,062 --> 00:10:26,000 zero mu zero times -- 86 00:10:26,000 --> 00:10:35,000 87 00:10:35,000 --> 00:10:43,212 -- d2E/dt2, so that is this part. 88 00:10:43,212 --> 00:10:57,584 And now I get on the right side, I get minus d2E/dx2 plus 89 00:10:57,584 --> 00:11:06,609 d2E/dy2 plus d2E/dz2. And this minus sign and this 90 00:11:06,609 --> 00:11:12,200 minus sign eats each other up. And this equation that you are 91 00:11:12,200 --> 00:11:17,326 looking now at is a milestone in the history of mankind. 92 00:11:17,326 --> 00:11:23,103 This equation changed our whole way of thinking about the world 93 00:11:23,103 --> 00:11:28,415 and even about the universe. This was the great victory of 94 00:11:28,415 --> 00:11:33,025 Maxwell. This is a wave equation in an 95 00:11:33,025 --> 00:11:38,866 electric field in vacuum. So this tells you that you must 96 00:11:38,866 --> 00:11:44,812 be able to create electric fields which move with speed v, 97 00:11:44,812 --> 00:11:50,654 for which we always write c, which is one over the square 98 00:11:50,654 --> 00:11:55,557 root of epsilon zero mu zero. Because, remember, 99 00:11:55,557 --> 00:12:01,183 this is a wave equation. And what you see here is always 100 00:12:01,183 --> 00:12:03,859 one over v squared. And it was Maxwell, 101 00:12:03,859 --> 00:12:07,098 of course, who was the first to recognize that, 102 00:12:07,098 --> 00:12:11,253 because this was only possible because he added this term to 103 00:12:11,253 --> 00:12:14,211 [UNINTELLIGIBLE] equation. He was a genius. 104 00:12:14,211 --> 00:12:17,239 Now, you can go through a similar reasoning. 105 00:12:17,239 --> 00:12:20,267 Instead of taking the curl of the curl of E, 106 00:12:20,267 --> 00:12:24,000 you could take the curl of the curl of B 107 00:12:24,000 --> 00:12:30,473 And then you will find that there must be an associated 108 00:12:30,473 --> 00:12:38,145 magnetic field for which you get that the dell square B equals mu 109 00:12:38,145 --> 00:12:45,337 zero epsilon zero times d2b/dt2. One cannot exist without the 110 00:12:45,337 --> 00:12:49,413 other. To put it even more bluntly, 111 00:12:49,413 --> 00:12:54,687 one is the other. You cannot think of them as 112 00:12:54,687 --> 00:12:59,842 being separate. One cannot exist without the 113 00:12:59,842 --> 00:13:04,175 other. So we are going to have 114 00:13:04,175 --> 00:13:07,959 three-dimensional wave equations, and the 115 00:13:07,959 --> 00:13:12,689 electromagnetic waves are then characterized by two 116 00:13:12,689 --> 00:13:18,270 interdependent oscillations, one in the E field and one in B 117 00:13:18,270 --> 00:13:21,675 fields. And the speed with which they 118 00:13:21,675 --> 00:13:27,067 propagate in vacuum is this. That follows immediately from 119 00:13:27,067 --> 00:13:32,119 Maxwell's equations. Untouched by human hands, 120 00:13:32,119 --> 00:13:35,596 so to speak. And, if you use the value for 121 00:13:35,596 --> 00:13:39,921 epsilon zero and mu zero, you will find that this is 122 00:13:39,921 --> 00:13:44,923 extremely close to 3.00 times 10 to the 8 meters per second. 123 00:13:44,923 --> 00:13:49,757 And the amazing thing is that epsilon zero can be measured 124 00:13:49,757 --> 00:13:54,336 without any time variability. Epsilon zero follows from 125 00:13:54,336 --> 00:13:59,000 Coulomb's Law, as static as you can have it. 126 00:13:59,000 --> 00:14:04,002 And mu zero can be measured without any time variability. 127 00:14:04,002 --> 00:14:07,397 Mu zero follows [UNINTELLIGIBLE]'s Law. 128 00:14:07,397 --> 00:14:12,311 And so it was these two, which I call static quantities, 129 00:14:12,311 --> 00:14:17,760 Maxwell was able to demonstrate that they dictate the speed of 130 00:14:17,760 --> 00:14:21,066 light in vacuum. Mu zero is called the 131 00:14:21,066 --> 00:14:25,890 permeability of free space. You can look up what it is. 132 00:14:25,890 --> 00:14:30,000 It is 4pi times 10 to the minus 7. 133 00:14:30,000 --> 00:14:36,195 And epsilon zero is called the permittivity of free space, 134 00:14:36,195 --> 00:14:42,065 8.8 times 10 to the minus 12. Maxwell knew the speed of 135 00:14:42,065 --> 00:14:45,217 light. It was known way before 136 00:14:45,217 --> 00:14:47,717 Maxwell. In 1676, Romer, 137 00:14:47,717 --> 00:14:54,130 in a brilliant way using the eclipsing times of the moons of 138 00:14:54,130 --> 00:14:57,826 Jupiter derived the speed of light. 139 00:14:57,826 --> 00:15:02,233 In 1676. And he came up with 214,000 140 00:15:02,233 --> 00:15:06,535 kilometers per second. And the only reason why he was 141 00:15:06,535 --> 00:15:11,250 on the low side is that it wasn't well-known in those days 142 00:15:11,250 --> 00:15:14,476 what the distances were between planets. 143 00:15:14,476 --> 00:15:18,778 And then in 1728 James Bradley used another brilliant 144 00:15:18,778 --> 00:15:21,922 technique. I won't go into the details, 145 00:15:21,922 --> 00:15:24,817 but it is called stellar aberration. 146 00:15:24,817 --> 00:15:29,450 And he used that technique to determine that the speed of 147 00:15:29,450 --> 00:15:34,000 light was 301,000 kilometers per second. 148 00:15:34,000 --> 00:15:37,299 And then in the late 19th century, in fact, 149 00:15:37,299 --> 00:15:41,462 in 1849 both Foucault and Fizeau measured the speed of 150 00:15:41,462 --> 00:15:44,211 light in their laboratory in France. 151 00:15:44,211 --> 00:15:48,610 One used the rotating mirror. The other used the rotating 152 00:15:48,610 --> 00:15:51,280 disk. And so they were even able to 153 00:15:51,280 --> 00:15:55,287 do it in the laboratory. And they found values which 154 00:15:55,287 --> 00:16:00,000 were within 5% of 300,000 kilometers per second. 155 00:16:00,000 --> 00:16:05,645 So Maxwell knew the answer. And so when he saw this, 156 00:16:05,645 --> 00:16:11,623 he immediately realized, he postulated that light is an 157 00:16:11,623 --> 00:16:17,158 electromagnetic phenomenon. And in 1865 he laid the 158 00:16:17,158 --> 00:16:23,800 foundation of the very famous Electromagnetic Theory of Light 159 00:16:23,800 --> 00:16:30,000 which changed the way that we look at the world. 160 00:16:30,000 --> 00:16:36,369 So now comes the question what are the solutions to this wave 161 00:16:36,369 --> 00:16:39,979 equation. Well, I will start with a 162 00:16:39,979 --> 00:16:45,924 simple one, and the simple one actually holds all the key 163 00:16:45,924 --> 00:16:51,657 information that you need. But then we will build it up 164 00:16:51,657 --> 00:16:56,010 and make it a little bit more complicated. 165 00:16:56,010 --> 00:17:01,000 I start with a xyz coordinate system. 166 00:17:01,000 --> 00:17:05,547 And whenever in physics you deal with cross products. 167 00:17:05,547 --> 00:17:09,657 Always make your coordinate system right-handed. 168 00:17:09,657 --> 00:17:14,030 Don't even think of left-handed coordinate systems. 169 00:17:14,030 --> 00:17:19,102 And a right-handed coordinate system is a coordinate system 170 00:17:19,102 --> 00:17:24,000 whereby x roof crossed with y roof equal z roof. 171 00:17:24,000 --> 00:17:34,000 172 00:17:34,000 --> 00:17:41,330 I assume a simple case that the E vector only exists in the x 173 00:17:41,330 --> 00:17:46,217 direction, so it has a certain amplitude. 174 00:17:46,217 --> 00:17:52,325 I call it E zero x. And I make it a traveling wave, 175 00:17:52,325 --> 00:17:56,479 of course, cosine omega t minus kz. 176 00:17:56,479 --> 00:18:03,467 So Ey is zero and Ez is zero. Only E vector in the x 177 00:18:03,467 --> 00:18:06,271 direction. The complicated 178 00:18:06,271 --> 00:18:11,766 three-dimensional wave equation collapses now to a 179 00:18:11,766 --> 00:18:18,607 one-dimensional wave equation. Of the nine terms that you have 180 00:18:18,607 --> 00:18:25,000 her, only one term survives, and that is d2Ex/dz2. 181 00:18:25,000 --> 00:18:31,000 182 00:18:31,000 --> 00:18:36,775 And that then becomes mu zero times epsilon zero. 183 00:18:36,775 --> 00:18:43,995 And, when you take the d2E/dt2 only one term survives because 184 00:18:43,995 --> 00:18:47,605 there is no Ey, there is no Ez, 185 00:18:47,605 --> 00:18:51,456 and so this now becomes d2Ex/dt2. 186 00:18:51,456 --> 00:18:56,991 So this is now a one-dimensional wave equation. 187 00:18:56,991 --> 00:19:03,643 I started easy. Now, remember that the curl of 188 00:19:03,643 --> 00:19:07,773 E is minus dB/dt. So the curl of E, 189 00:19:07,773 --> 00:19:12,995 in this case, and you can check that because 190 00:19:12,995 --> 00:19:18,218 you know what the curl is, here is the curl. 191 00:19:18,218 --> 00:19:23,198 The curl of E, there is only one term that 192 00:19:23,198 --> 00:19:28,056 survives. I will write down here the curl 193 00:19:28,056 --> 00:19:33,033 of E. There is only one term that 194 00:19:33,033 --> 00:19:38,314 survives, and that is dEx/dz in the y direction. 195 00:19:38,314 --> 00:19:44,606 And that now must be equal to minus dB/dt because that is 196 00:19:44,606 --> 00:19:48,202 Maxwell. I will lower this later. 197 00:19:48,202 --> 00:19:54,943 I will raise it so that you can see this, but it is important 198 00:19:54,943 --> 00:20:00,000 that you can see this above my head. 199 00:20:00,000 --> 00:20:05,858 My question now is what is B? This is the traveling wave in 200 00:20:05,858 --> 00:20:12,020 the z direction which has an E vector only in the x direction, 201 00:20:12,020 --> 00:20:17,979 as of now, and I want to know what the associated B field is 202 00:20:17,979 --> 00:20:23,030 Well, for one thing, you can already see that the B 203 00:20:23,030 --> 00:20:26,969 field is going to be in the y direction. 204 00:20:26,969 --> 00:20:32,378 If we take dEx/dz -- So this is the Ex. 205 00:20:32,378 --> 00:20:39,515 And we take the derivative against z, we get a minus k, 206 00:20:39,515 --> 00:20:45,726 then you get to E0x. And then the cosine becomes 207 00:20:45,726 --> 00:20:53,524 minus sine, so this becomes a plus times the sine of omega t 208 00:20:53,524 --> 00:20:58,017 minus kz. And that now equals minus 209 00:20:58,017 --> 00:21:00,000 dB/dt. 210 00:21:00,000 --> 00:21:05,000 211 00:21:05,000 --> 00:21:08,191 We are almost there. All we have to do now is do an 212 00:21:08,191 --> 00:21:11,446 integration in time. It is important that we get the 213 00:21:11,446 --> 00:21:14,000 y direction. That is very important. 214 00:21:14,000 --> 00:21:21,000 215 00:21:21,000 --> 00:21:23,343 Remember this y is very important. 216 00:21:23,343 --> 00:21:26,751 That is going to be the direction of the B field. 217 00:21:26,751 --> 00:21:30,159 Don't forget that y. Now you bring the [dt?] here 218 00:21:30,159 --> 00:21:33,000 and you do an integration in time. 219 00:21:33,000 --> 00:21:38,958 And so now you get the B field that is going to be associated 220 00:21:38,958 --> 00:21:42,832 with that traveling wave electric field. 221 00:21:42,832 --> 00:21:47,996 If you do an integral, the omega pops out below here, 222 00:21:47,996 --> 00:21:51,174 so you get kE0x divided by omega. 223 00:21:51,174 --> 00:21:56,835 The sign becomes minus the cosine, but I also have a minus 224 00:21:56,835 --> 00:22:02,000 sign here so those two minus signs cancel. 225 00:22:02,000 --> 00:22:08,056 And so I get the cosine of omega t minus kz. 226 00:22:08,056 --> 00:22:15,802 But omega divided by k is c. That is the speed of light. 227 00:22:15,802 --> 00:22:23,549 And so I can also write this now B in the y direction to 228 00:22:23,549 --> 00:22:31,436 indicate that it is in the y direction, E0x divided by c, 229 00:22:31,436 --> 00:22:39,323 because omega divided by k is c, times the cosine omega t 230 00:22:39,323 --> 00:22:44,556 minus kz. And, just to remind you, 231 00:22:44,556 --> 00:22:48,390 here is your y again. But, of course, 232 00:22:48,390 --> 00:22:54,887 the By already indicates that. And so now you have found the B 233 00:22:54,887 --> 00:23:00,000 field that is associated with this E field. 234 00:23:00,000 --> 00:23:03,972 As I used to say, and I said earlier, 235 00:23:03,972 --> 00:23:08,937 one is the other. One cannot exist without the 236 00:23:08,937 --> 00:23:12,468 other. If we compare the two now, 237 00:23:12,468 --> 00:23:17,434 you see several things which holds in general. 238 00:23:17,434 --> 00:23:22,731 And that is you see the magnitude of the B field, 239 00:23:22,731 --> 00:23:27,917 you can call this B0y, is c times lower than the 240 00:23:27,917 --> 00:23:33,196 magnitude of the E field. See the c here? 241 00:23:33,196 --> 00:23:37,663 Notice also that the E field and the B field are in phase 242 00:23:37,663 --> 00:23:41,173 with each other. You have here cosine omega t 243 00:23:41,173 --> 00:23:43,965 minus kz and you have the same here. 244 00:23:43,965 --> 00:23:47,236 There is no phase difference between them. 245 00:23:47,236 --> 00:23:51,544 That means if one reaches a maximum the other reaches a 246 00:23:51,544 --> 00:23:54,655 maximum. And if one is zero the other is 247 00:23:54,655 --> 00:24:00,000 zero because they oscillate with that frequency omega. 248 00:24:00,000 --> 00:24:04,984 Notice also that B is perpendicular to E because E is 249 00:24:04,984 --> 00:24:09,392 in the x direction and B is in the y direction. 250 00:24:09,392 --> 00:24:14,760 Notice also that each one of them is perpendicular to the 251 00:24:14,760 --> 00:24:19,744 direction of propagation. E is perpendicular to the z 252 00:24:19,744 --> 00:24:24,632 direction and B is perpendicular to the z direction. 253 00:24:24,632 --> 00:24:30,000 And all this follows from Maxwell's equations. 254 00:24:30,000 --> 00:24:35,232 Now I would like to make a sketch of what this 255 00:24:35,232 --> 00:24:42,325 electromagnetic wave looks like, and I will make an attempt to 256 00:24:42,325 --> 00:24:46,279 make you see it. It is not so easy, 257 00:24:46,279 --> 00:24:52,558 but I will make an attempt. Here is the x direction and 258 00:24:52,558 --> 00:25:00,000 here is the y direction and let this be the z direction. 259 00:25:00,000 --> 00:25:10,000 260 00:25:10,000 --> 00:25:14,619 I pick a particular moment in time. 261 00:25:14,619 --> 00:25:22,635 I know that the E vector is a cosine function in z because I 262 00:25:22,635 --> 00:25:26,983 pick a particular moment in time. 263 00:25:26,983 --> 00:25:35,000 And so I will try to put in here the cosine function. 264 00:25:35,000 --> 00:25:39,588 If you think it is a sine function that, 265 00:25:39,588 --> 00:25:46,647 of course, is the same thing. And so at this frozen moment in 266 00:25:46,647 --> 00:25:52,176 time the E vector would be like this, like this, 267 00:25:52,176 --> 00:25:56,176 like this, like this and like this. 268 00:25:56,176 --> 00:26:01,000 And this value here would be E0x. 269 00:26:01,000 --> 00:26:06,000 270 00:26:06,000 --> 00:26:14,609 Associated with that E field is a B field that is in the y 271 00:26:14,609 --> 00:26:18,536 direction. It is like this. 272 00:26:18,536 --> 00:26:26,541 It is in the yz plane. And so the B vector is like so, 273 00:26:26,541 --> 00:26:31,223 like so, like so, like so, here, 274 00:26:31,223 --> 00:26:39,500 here, here, here. And this value then is B0y. 275 00:26:39,500 --> 00:26:48,000 And this whole pattern moves with velocity c in this 276 00:26:48,000 --> 00:26:55,166 direction, and the two are married together. 277 00:26:55,166 --> 00:27:01,921 They are stuck together. And, if you take any plane 278 00:27:01,921 --> 00:27:05,911 perpendicular to the z axis, a plane that is infinitely 279 00:27:05,911 --> 00:27:09,310 large in this direction, infinity large in this 280 00:27:09,310 --> 00:27:13,596 direction, infinitely large in that directly and infinitely 281 00:27:13,596 --> 00:27:17,438 large in that direction, at that moment in time the E 282 00:27:17,438 --> 00:27:21,650 vector is everywhere in that plane exactly the same value. 283 00:27:21,650 --> 00:27:25,418 Look at the equation. There is no dependent on y and 284 00:27:25,418 --> 00:27:27,783 x. That is why we call them plane 285 00:27:27,783 --> 00:27:31,932 wave solutions. How realistic they are is 286 00:27:31,932 --> 00:27:35,257 another matter, but they are consistent with 287 00:27:35,257 --> 00:27:39,123 Maxwell's equations. And the same is true for the B 288 00:27:39,123 --> 00:27:41,365 field. Any plane that you take 289 00:27:41,365 --> 00:27:44,922 perpendicular to z, someone is sitting 25 miles 290 00:27:44,922 --> 00:27:48,247 away from you, but on that same plane at any 291 00:27:48,247 --> 00:27:51,726 moment in time you will see the same E vector, 292 00:27:51,726 --> 00:27:54,278 all of you, and the same B vector. 293 00:27:54,278 --> 00:28:00,000 And that whole pattern then moves with velocity c in space. 294 00:28:00,000 --> 00:28:03,906 And so if I am standing here in one of the many planes 295 00:28:03,906 --> 00:28:08,402 perpendicular to the z axis and the electromagnetic wave comes 296 00:28:08,402 --> 00:28:12,383 to me and there is another person standing there in the 297 00:28:12,383 --> 00:28:16,732 same plane, we will see the E vector go like this and we see 298 00:28:16,732 --> 00:28:20,270 the B vector go like this, but they go in unison. 299 00:28:20,270 --> 00:28:24,398 When the E vector is maximum here the B vector is maximum 300 00:28:24,398 --> 00:28:26,977 there. And then they reach both zero 301 00:28:26,977 --> 00:28:31,418 at the same moment in time. And then the E vector goes 302 00:28:31,418 --> 00:28:33,599 negative and then the B vector does that. 303 00:28:33,599 --> 00:28:36,000 So that is the way the oscillation works. 304 00:28:36,000 --> 00:28:42,000 305 00:28:42,000 --> 00:28:47,046 I would like to summarize for you what is actually the bottom 306 00:28:47,046 --> 00:28:50,663 line of all this. And when I have to solve B 307 00:28:50,663 --> 00:28:55,373 fields for given E fields or E fields for given B fields, 308 00:28:55,373 --> 00:29:00,000 that is really the only way that I used to think. 309 00:29:00,000 --> 00:29:04,831 So we have a traveling electromagnetic wave. 310 00:29:04,831 --> 00:29:11,460 What follows only holds for traveling electromagnetic waves, 311 00:29:11,460 --> 00:29:16,292 not for standing electromagnetic waves which 312 00:29:16,292 --> 00:29:21,348 comes in the future. Traveling electromagnetic 313 00:29:21,348 --> 00:29:25,280 waves. The E vector is perpendicular 314 00:29:25,280 --> 00:29:33,692 to the direction of propagation. The B vector is perpendicular 315 00:29:33,692 --> 00:29:40,440 to the direction of propagation. E and B are in phase. 316 00:29:40,440 --> 00:29:45,788 If one reaches zero the other reaches zero. 317 00:29:45,788 --> 00:29:52,409 E is perpendicular to B. And, as a result of the fact 318 00:29:52,409 --> 00:30:00,559 that they are both perpendicular to the direction of propagation, 319 00:30:00,559 --> 00:30:08,071 it follows immediately that E cross B is in the direction of 320 00:30:08,071 --> 00:30:12,272 propagation. And then last, 321 00:30:12,272 --> 00:30:15,545 but not least, I have no room for it, 322 00:30:15,545 --> 00:30:20,000 I will put it here, that the magnitude of B at any 323 00:30:20,000 --> 00:30:25,000 moment in time is the magnitude of E divided by c. 324 00:30:25,000 --> 00:30:32,000 325 00:30:32,000 --> 00:30:37,098 If the E vector is only in one direction, as it is here in the 326 00:30:37,098 --> 00:30:41,611 x direction, we call that linearly polarized radiation. 327 00:30:41,611 --> 00:30:44,453 The word speaks for itself, linear. 328 00:30:44,453 --> 00:30:48,967 Now, of course it is entirely possible, which is also a 329 00:30:48,967 --> 00:30:54,065 perfect solution to Maxwell's equations and to wave equations, 330 00:30:54,065 --> 00:30:58,161 you could easily have an E field which is in the y 331 00:30:58,161 --> 00:31:02,000 direction. At this moment in time, 332 00:31:02,000 --> 00:31:06,782 it would be coupled to a B field which is in the minus x 333 00:31:06,782 --> 00:31:11,652 direction, so that E cross B is still in the direction of 334 00:31:11,652 --> 00:31:14,782 propagation. And that would also be a 335 00:31:14,782 --> 00:31:18,000 perfect solution to the wave equation. 336 00:31:18,000 --> 00:31:22,956 And so the sum of the two or linear combination of the two 337 00:31:22,956 --> 00:31:28,000 should also be a solution to our wave equation. 338 00:31:28,000 --> 00:31:33,791 Therefore, I can write now in somewhat more complicated form 339 00:31:33,791 --> 00:31:39,680 for traveling wave in which I give it both a component in the 340 00:31:39,680 --> 00:31:44,981 x direction, as well as a component in the y direction. 341 00:31:44,981 --> 00:31:49,595 And I can even change the phase between the two. 342 00:31:49,595 --> 00:31:53,521 In other words, the following E vector is 343 00:31:53,521 --> 00:31:57,938 perfectly kosher. E0x times the cosine omega t 344 00:31:57,938 --> 00:32:02,709 minus kz. That would then be in the x 345 00:32:02,709 --> 00:32:06,622 direction. And then I would have another 346 00:32:06,622 --> 00:32:10,334 E0y times the cosine omega t minus kz. 347 00:32:10,334 --> 00:32:14,949 And I can give it any random phase angle delta. 348 00:32:14,949 --> 00:32:19,063 And that would then be in the y direction. 349 00:32:19,063 --> 00:32:24,983 And so clearly this satisfies the wave equation because each 350 00:32:24,983 --> 00:32:30,000 separately satisfies the wave equation. 351 00:32:30,000 --> 00:32:36,265 Now we make delta equal zero. You still have linearly 352 00:32:36,265 --> 00:32:41,204 polarized radiation. Look at the xy plane. 353 00:32:41,204 --> 00:32:47,349 This is x and this is y, and the radiation is coming 354 00:32:47,349 --> 00:32:52,650 straight to you. At a certain moment in time, 355 00:32:52,650 --> 00:33:00,000 the E vector in this direction reaches a maximum E0x. 356 00:33:00,000 --> 00:33:05,719 And, if delta is zero, that is the moment in time that 357 00:33:05,719 --> 00:33:12,302 the E vector in the y direction also reaches a maximum because 358 00:33:12,302 --> 00:33:15,755 delta is zero. So this one is 0y, 359 00:33:15,755 --> 00:33:19,748 of course, at the same moment as that. 360 00:33:19,748 --> 00:33:26,115 So what is the net E vector that is the factorial sum of the 361 00:33:26,115 --> 00:33:30,000 two? The E vector is this. 362 00:33:30,000 --> 00:33:36,909 This is E total which is the square root of E0x2 plus 0y2. 363 00:33:36,909 --> 00:33:44,060 And, if you see this coming to you, you will see an electric 364 00:33:44,060 --> 00:33:49,030 field going like this, linearly polarized. 365 00:33:49,030 --> 00:33:54,727 No longer linearly polarized in the x direction, 366 00:33:54,727 --> 00:34:02,000 no longer in the y direction, but in this direction. 367 00:34:02,000 --> 00:34:09,091 I can also make the phase angle between the two 90 degrees. 368 00:34:09,091 --> 00:34:16,061 I could make delta pi over two. Now you get something very 369 00:34:16,061 --> 00:34:21,318 interesting. This is now x and this is now y 370 00:34:21,318 --> 00:34:25,475 so that the radiation comes to you. 371 00:34:25,475 --> 00:34:33,163 X cross y is a new direction. And now I pick a moment in time 372 00:34:33,163 --> 00:34:37,959 that E0x is maximum here, so this is the vector, 373 00:34:37,959 --> 00:34:42,346 but now E0y, the E vector in the y direction 374 00:34:42,346 --> 00:34:47,653 is now zero because they are 90 degrees out of phase. 375 00:34:47,653 --> 00:34:52,551 And so, therefore, if this one reaches a maximum, 376 00:34:52,551 --> 00:34:56,632 this one is zero. A quarter period later, 377 00:34:56,632 --> 00:35:02,561 this one becomes zero. But now you have here E0y. 378 00:35:02,561 --> 00:35:07,585 A quarter period later, this one is back to zero and 379 00:35:07,585 --> 00:35:12,313 now this one is here. And a quarter period later, 380 00:35:12,313 --> 00:35:16,549 this one again is zero and this one is here. 381 00:35:16,549 --> 00:35:22,262 Now what you are going to see, there is never a moment that 382 00:35:22,262 --> 00:35:27,286 the E vector is zero, but E vector rotates around in 383 00:35:27,286 --> 00:35:29,946 an ellipse. It goes like so, 384 00:35:29,946 --> 00:35:35,372 like so and so on. And so it rotates around like 385 00:35:35,372 --> 00:35:38,972 this, and we call that elliptically polarized 386 00:35:38,972 --> 00:35:42,163 radiation. There is nothing very special 387 00:35:42,163 --> 00:35:45,109 about it. It is a perfect solution to 388 00:35:45,109 --> 00:35:49,281 Maxwell's equations. You have one component in the x 389 00:35:49,281 --> 00:35:53,945 direction, another in the y direction, and you offset them 390 00:35:53,945 --> 00:35:57,463 by 90 degrees. You can choose this angle any 391 00:35:57,463 --> 00:36:02,562 value you want to. If E0x is the same as E0y, 392 00:36:02,562 --> 00:36:06,997 it is a circle. And then we call it circularly 393 00:36:06,997 --> 00:36:10,249 polarized radiation. In this case, 394 00:36:10,249 --> 00:36:13,895 it is going clockwise. But, of course, 395 00:36:13,895 --> 00:36:20,104 if you make delta minus pi over two it will go counterclockwise. 396 00:36:20,104 --> 00:36:25,623 Now, suppose you were asked to calculate the associated B 397 00:36:25,623 --> 00:36:31,437 field, that is a piece of cake because you just follow these 398 00:36:31,437 --> 00:36:37,395 simple rules. You take the component in the x 399 00:36:37,395 --> 00:36:44,187 direction and you calculate the associated traveling wave in B. 400 00:36:44,187 --> 00:36:50,321 And then you do for the y direction and you calculate the 401 00:36:50,321 --> 00:36:55,579 associated traveling B wave. And you add them up. 402 00:36:55,579 --> 00:36:59,632 That gives you then the solution in B. 403 00:36:59,632 --> 00:37:05,000 This situation is nice and simple in 2D. 404 00:37:05,000 --> 00:37:15,232 But I think I owe you a more general description to widen 405 00:37:15,232 --> 00:37:20,166 your insight. And I want to, 406 00:37:20,166 --> 00:37:27,292 at least in terms of the math, go in 3D. 407 00:37:27,292 --> 00:37:34,418 So we now have an xyz coordinate system. 408 00:37:34,418 --> 00:37:43,372 And now we want the option of having the E vector, 409 00:37:43,372 --> 00:37:54,700 not just in the xy plane or not in the xz plane but in a random 410 00:37:54,700 --> 00:38:00,000 direction. 37:33 411 00:38:00,000 --> 00:38:04,816 It is now a vector. It is not in x or y or in z. 412 00:38:04,816 --> 00:38:10,453 It is in three-dimension. So this E0 has an x component, 413 00:38:10,453 --> 00:38:13,630 a y component and a z component. 414 00:38:13,630 --> 00:38:19,574 And now I can write down here the cosine of omega t minus k 415 00:38:19,574 --> 00:38:25,313 dot r, which is now the most general way that I can write 416 00:38:25,313 --> 00:38:30,540 this electromagnetic wave. Whereby, k is kx in the x 417 00:38:30,540 --> 00:38:37,109 direction -- -- plus ky in the y direction 418 00:38:37,109 --> 00:38:45,327 plus kz in the z direction. And k, as you will see shortly, 419 00:38:45,327 --> 00:38:49,720 is the direction of propagation. 420 00:38:49,720 --> 00:38:57,372 And the magnitude of k is always 2pi divided by lambda. 421 00:38:57,372 --> 00:39:06,157 And that magnitude of k is the square root of kx2 plus ky2 plus 422 00:39:06,157 --> 00:39:10,604 kz2. And now I would like to give 423 00:39:10,604 --> 00:39:14,976 you some insight in the meaning of this k dot r. 424 00:39:14,976 --> 00:39:18,976 And for that I need a little bit more space. 425 00:39:18,976 --> 00:39:24,558 And now we have to make a touch decision what we are going to 426 00:39:24,558 --> 00:39:28,000 kill. I am going to kill this. 427 00:39:28,000 --> 00:39:40,000 428 00:39:40,000 --> 00:39:45,233 To make you see what a geometric meaning is of k dot r, 429 00:39:45,233 --> 00:39:48,237 I will go first two-dimensional, 430 00:39:48,237 --> 00:39:53,859 and then you will immediately get the picture what it is in 431 00:39:53,859 --> 00:39:58,511 three-dimensions. I have here the x direction and 432 00:39:58,511 --> 00:40:10,992 I have here the y direction. Here I have the k vector. 433 00:40:10,992 --> 00:40:22,503 And so this is kx and this is ky. 434 00:40:22,503 --> 00:40:36,892 It is a vector. I am going to draw a line 435 00:40:36,892 --> 00:40:52,000 perpendicular to the k vector. 40:30 436 00:40:52,000 --> 00:41:00,000 437 00:41:00,000 --> 00:41:05,379 You can see that immediately. If this is my vector r, 438 00:41:05,379 --> 00:41:11,896 which is the position vector in space given by the relation that 439 00:41:11,896 --> 00:41:16,448 you see there, then this angle here between k 440 00:41:16,448 --> 00:41:21,620 and r, I call that theta. Then k dot r is a scalar, 441 00:41:21,620 --> 00:41:28,034 is the magnitude of k times the magnitude of r times the cosine 442 00:41:28,034 --> 00:41:32,742 of theta. And what is r cosine theta? 443 00:41:32,742 --> 00:41:36,298 That is this here is r cosine theta. 444 00:41:36,298 --> 00:41:42,495 And any vector r that ends on this red line will have the same 445 00:41:42,495 --> 00:41:46,457 value for r cosine theta. So, therefore, 446 00:41:46,457 --> 00:41:51,739 on the entire red line, which in this two-dimensional 447 00:41:51,739 --> 00:41:56,920 plot is just a line, anywhere on this line the value 448 00:41:56,920 --> 00:42:02,000 for k dot r is a constant, is the same. 449 00:42:02,000 --> 00:42:06,720 Now, to give a moment in time, t equals zero, 450 00:42:06,720 --> 00:42:13,478 we have an electromagnetic wave traveling in the direction of k. 451 00:42:13,478 --> 00:42:19,272 And this line happens to be the line where the E field, 452 00:42:19,272 --> 00:42:24,957 at this moment in time, is the maximum and pointing in 453 00:42:24,957 --> 00:42:30,000 your direction say. It is a crest in E. 454 00:42:30,000 --> 00:42:38,860 It is a mountain coming out of the board in maximum value. 455 00:42:38,860 --> 00:42:46,165 And here I draw another line perpendicular to r. 456 00:42:46,165 --> 00:42:53,160 Here k dot r is zero. The dot product is zero. 457 00:42:53,160 --> 00:43:00,000 R is perpendicular to k to this point. 458 00:43:00,000 --> 00:43:04,982 I will also assume that the E vector here is a maximum 459 00:43:04,982 --> 00:43:10,059 pointing in your direction. And then there is one here. 460 00:43:10,059 --> 00:43:12,974 K dot r here is, therefore, 2pi. 461 00:43:12,974 --> 00:43:18,427 And k dot r here is 4pi because this now represents a wave. 462 00:43:18,427 --> 00:43:24,256 This is the fool wavelength of the wave where E is a maximum in 463 00:43:24,256 --> 00:43:27,735 your direction, E is a maximum in your 464 00:43:27,735 --> 00:43:33,000 direction, E is a maximum in your direction. 465 00:43:33,000 --> 00:43:38,800 In other words, this here is by definition of 466 00:43:38,800 --> 00:43:43,677 lambda. And this whole thing moves out 467 00:43:43,677 --> 00:43:50,531 in space with speed c. And so you see the lines k dot 468 00:43:50,531 --> 00:43:58,309 r perpendicular to k represent the maxima of the E vector at 469 00:43:58,309 --> 00:44:04,086 this moment in time. And all that starts to move 470 00:44:04,086 --> 00:44:06,173 out. Now you go to the third 471 00:44:06,173 --> 00:44:09,032 dimension. Then k dot r is a constant, 472 00:44:09,032 --> 00:44:12,355 are now no longer lines, but they are planes 473 00:44:12,355 --> 00:44:16,220 perpendicular to the vector k. And this whole plane 474 00:44:16,220 --> 00:44:20,548 perpendicular to the vector k, in that whole plane at the 475 00:44:20,548 --> 00:44:24,025 moment in time, the E vector is everywhere the 476 00:44:24,025 --> 00:44:26,112 same. Whether it is linearly 477 00:44:26,112 --> 00:44:30,362 polarized, whether it is circularly polarized or whether 478 00:44:30,362 --> 00:44:35,000 it is elliptically polarized is irrelevant. 479 00:44:35,000 --> 00:44:40,010 It is everywhere the same. And that whole plane then moves 480 00:44:40,010 --> 00:44:44,318 out with the speed of light in the direction of k. 481 00:44:44,318 --> 00:44:47,395 And so k dot r, in three dimensions, 482 00:44:47,395 --> 00:44:51,263 are all planes perpendicular to the vector k. 483 00:44:51,263 --> 00:44:56,450 If you stand anywhere in space and you look in the direction 484 00:44:56,450 --> 00:45:01,637 where the radiation is coming from then in any plane that is 485 00:45:01,637 --> 00:45:05,329 perpendicular to k going to infinity there, 486 00:45:05,329 --> 00:45:09,373 to there, to there, at any moment in time the E 487 00:45:09,373 --> 00:45:15,000 vector is the same and the B vector is the same. 488 00:45:15,000 --> 00:45:19,859 And I said whether it is linearly polarized radiation or 489 00:45:19,859 --> 00:45:24,188 circular or elliptical that is a different matter. 490 00:45:24,188 --> 00:45:26,927 It could be either one of those. 491 00:45:26,927 --> 00:45:31,522 And so this is the best way that you can think of the 492 00:45:31,522 --> 00:45:37,000 general form of an E vector going in three-dimensions. 493 00:45:37,000 --> 00:45:42,750 These are then planes perpendicular to the direction 494 00:45:42,750 --> 00:45:46,470 of propagation. In our first case, 495 00:45:46,470 --> 00:45:52,671 which was so very simple because I wanted to warm you up 496 00:45:52,671 --> 00:45:56,279 slowly, k dot r became simply kz. 497 00:45:56,279 --> 00:46:00,000 Kx was zero. Ky was zero. 498 00:46:00,000 --> 00:46:03,628 Because it was only going in the z direction, 499 00:46:03,628 --> 00:46:07,835 kz was the only one which was not zero, which was k. 500 00:46:07,835 --> 00:46:11,958 And so my dot product, k dot r collapses into a kz. 501 00:46:11,958 --> 00:46:16,329 And so the wavelength lambda is 2pi divided by this k. 502 00:46:16,329 --> 00:46:20,123 But, of course, if you have a three-dimensional 503 00:46:20,123 --> 00:46:24,659 case then the situation is a little bit more complicated 504 00:46:24,659 --> 00:46:28,783 because then k is the square root, as you see here, 505 00:46:28,783 --> 00:46:34,610 of kx2 plus ky2 plus kz2. Our first case was to make it 506 00:46:34,610 --> 00:46:38,392 simple for you. This is the right moment to 507 00:46:38,392 --> 00:46:41,543 stop. After the break we can look at 508 00:46:41,543 --> 00:46:46,315 some demonstrations also. We have to relax a little to 509 00:46:46,315 --> 00:46:49,286 digest all this. This is not easy. 510 00:46:49,286 --> 00:46:53,337 And so let's start handing out this mini-quiz. 511 00:46:53,337 --> 00:46:58,379 To make you feel good about yourself, I made it easy this 512 00:46:58,379 --> 00:47:02,086 time. Don't start yet. 513 00:47:02,086 --> 00:47:06,000 Can you start handing it out? 514 00:47:06,000 --> 00:47:12,000 515 00:47:12,000 --> 00:47:17,386 I put here on the blackboard the xy plane. 516 00:47:17,386 --> 00:47:24,744 This is your electromagnetic wave going in this direction 517 00:47:24,744 --> 00:47:30,000 perpendicular to the direction of k. 518 00:47:30,000 --> 00:47:33,439 This is, per definition, a wavelength. 519 00:47:33,439 --> 00:47:39,295 This is where E is a maximum in your direction and this is where 520 00:47:39,295 --> 00:47:44,129 it is a maximum in your direction, and so this is the 521 00:47:44,129 --> 00:47:47,847 wavelength. Now, look at the intersection 522 00:47:47,847 --> 00:47:53,239 of this wave in the y-axis. This wave interacts here and it 523 00:47:53,239 --> 00:47:57,515 intersects there. And so the distance from here 524 00:47:57,515 --> 00:48:01,047 to here is ly, which is way larger than 525 00:48:01,047 --> 00:48:05,879 lambda. And the same is true for the 526 00:48:05,879 --> 00:48:11,843 distance in the x direction. It is also larger than lambda. 527 00:48:11,843 --> 00:48:17,088 And in the z direction, in general, it would also be 528 00:48:17,088 --> 00:48:22,024 larger than lambda. Now, this wave moves with the 529 00:48:22,024 --> 00:48:25,521 speed of c. And when it has moved a 530 00:48:25,521 --> 00:48:32,000 distance lambda in the k direction this crest is here. 531 00:48:32,000 --> 00:48:36,987 How far has it moved in the y direction then? 532 00:48:36,987 --> 00:48:44,016 All the way from here to there. So its speed in the y direction 533 00:48:44,016 --> 00:48:47,983 is larger than c. And so that speed, 534 00:48:47,983 --> 00:48:53,991 which we call the phase velocity in the y direction is 535 00:48:53,991 --> 00:49:00,000 very simply, is ly divided by lambda times c. 536 00:49:00,000 --> 00:49:07,015 And that is larger than c. It follows immediately from the 537 00:49:07,015 --> 00:49:14,892 geometry because this angle here is the same as this angle there. 538 00:49:14,892 --> 00:49:19,323 This is also k divided by ky times c. 539 00:49:19,323 --> 00:49:26,584 And that is not only larger than c, but it can be way larger 540 00:49:26,584 --> 00:49:32,000 than c. Ky is 2pi divided by l of y. 541 00:49:32,000 --> 00:49:35,299 I refuse to call l of y lambda of y. 542 00:49:35,299 --> 00:49:41,144 I do not want to have to think in terms of a wavelength in this 543 00:49:41,144 --> 00:49:47,084 direction, a wavelength in this direction and another wavelength 544 00:49:47,084 --> 00:49:51,043 in that direction. For me there is only one 545 00:49:51,043 --> 00:49:55,851 wavelength, and that wavelength is 2pi divided by k. 546 00:49:55,851 --> 00:50:00,000 I refuse to call this lambda of y. 547 00:50:00,000 --> 00:50:02,981 Ky is 2pi divided by that l of y. 548 00:50:02,981 --> 00:50:08,012 And you can do the same, of course, in the x and in the 549 00:50:08,012 --> 00:50:12,018 z direction. And you will find then that the 550 00:50:12,018 --> 00:50:17,142 phase velocity in the c direction equals k divided by kx 551 00:50:17,142 --> 00:50:20,683 times c. And the phase direction in the 552 00:50:20,683 --> 00:50:24,223 z direction is k divided by kz times c. 553 00:50:24,223 --> 00:50:30,000 And what is the phase velocity in the direction k? 554 00:50:30,000 --> 00:50:35,602 That is k divided by k times c. That is c, of course. 555 00:50:35,602 --> 00:50:41,313 In the direction of k, it propagates with the speed of 556 00:50:41,313 --> 00:50:44,760 light. But this pattern moves way 557 00:50:44,760 --> 00:50:50,579 faster than the speed of light. And it can actually get 558 00:50:50,579 --> 00:50:54,673 completely out of hand. It can be very, 559 00:50:54,673 --> 00:50:58,983 very large. Suppose these waves travel in 560 00:50:58,983 --> 00:51:04,493 the x direction. That means these red lines will 561 00:51:04,493 --> 00:51:07,710 be vertical. That means l of y will go to 562 00:51:07,710 --> 00:51:10,686 infinity. It means that k of y goes to 563 00:51:10,686 --> 00:51:12,938 zero. It means that the phase 564 00:51:12,938 --> 00:51:16,477 velocity goes to infinity. This pattern here, 565 00:51:16,477 --> 00:51:20,739 the intersection here will then go as a speed which is 566 00:51:20,739 --> 00:51:24,198 infinitely high, and that is no violation of 567 00:51:24,198 --> 00:51:28,702 Einstein's theory of special relativity because no energy 568 00:51:28,702 --> 00:51:34,888 will flow with that speed. And I can best convince you of 569 00:51:34,888 --> 00:51:40,664 that by showing that something similar can happen with water. 570 00:51:40,664 --> 00:51:46,633 Suppose we have a shoreline and we have some water waves coming 571 00:51:46,633 --> 00:51:50,773 in like this. Maybe I should put them in red 572 00:51:50,773 --> 00:51:55,201 so you begin to make the connection between the 573 00:51:55,201 --> 00:51:58,763 electromagnetic waves and water waves. 574 00:51:58,763 --> 00:52:03,000 Here is a wave rolling in nicely. 575 00:52:03,000 --> 00:52:06,853 This is, by definition, the wavelength lambda. 576 00:52:06,853 --> 00:52:10,021 And all of that moves with velocity v. 577 00:52:10,021 --> 00:52:13,788 But the intersection of these two waves here, 578 00:52:13,788 --> 00:52:18,840 at point A and here at point B, I call this distance l of x. 579 00:52:18,840 --> 00:52:22,608 I could have called it l of y. As I did here, 580 00:52:22,608 --> 00:52:27,660 it is the intersection with this axis that I gave the symbol 581 00:52:27,660 --> 00:52:31,000 l of y. I called it l of x. 582 00:52:31,000 --> 00:52:36,000 583 00:52:36,000 --> 00:52:43,630 The difference in arrival time between wave one and wave two at 584 00:52:43,630 --> 00:52:50,523 point B is the period of the wave, which is simply lambda 585 00:52:50,523 --> 00:52:54,092 divided by B. That is trivial. 586 00:52:54,092 --> 00:53:00,000 But that is not only the case for point B. 587 00:53:00,000 --> 00:53:03,288 That is also the case for point A. 588 00:53:03,288 --> 00:53:08,768 This wave one reaches A before wave two reaches point A. 589 00:53:08,768 --> 00:53:15,046 And this is the time in between the arrival time of wave one and 590 00:53:15,046 --> 00:53:21,224 wave two, which is a completely different question from what is 591 00:53:21,224 --> 00:53:28,000 the difference in arrival time of wave one alone at A and B. 592 00:53:28,000 --> 00:53:31,766 That difference in arrival time between one wave, 593 00:53:31,766 --> 00:53:35,612 between point A and B depends on this angle theta. 594 00:53:35,612 --> 00:53:40,320 And, when theta goes to zero, that difference in arrival time 595 00:53:40,320 --> 00:53:42,753 goes to zero. Both A and B will, 596 00:53:42,753 --> 00:53:45,970 at the same moment in time, see that wave. 597 00:53:45,970 --> 00:53:49,345 That means in that case, when theta is zero, 598 00:53:49,345 --> 00:53:52,170 this l of x becomes infinitely large. 599 00:53:52,170 --> 00:53:56,721 And so, if you express that in terms of a phase velocity in 600 00:53:56,721 --> 00:54:00,331 this direction, the phase velocity then becomes 601 00:54:00,331 --> 00:54:05,187 infinitely high. It is this pattern that moves 602 00:54:05,187 --> 00:54:08,731 with the velocity that can be way larger than c, 603 00:54:08,731 --> 00:54:11,824 but no water will move with that velocity. 604 00:54:11,824 --> 00:54:16,273 It is very clear there is no water going from here to there. 605 00:54:16,273 --> 00:54:20,874 And so this is not a violation of Einstein's theory of special 606 00:54:20,874 --> 00:54:23,891 relativity. And so several very dedicated 607 00:54:23,891 --> 00:54:28,265 students wrote me emails that after last lecture they could 608 00:54:28,265 --> 00:54:32,897 not sleep. And I did not even feel guilty. 609 00:54:32,897 --> 00:54:38,317 And the reason why they could not sleep is that we had this 610 00:54:38,317 --> 00:54:43,925 wonderful demonstration whereby I have here an aluminum place 611 00:54:43,925 --> 00:54:49,906 and I had another aluminum plate and this was the z direction and 612 00:54:49,906 --> 00:54:54,485 the separation between the plates was A and we had 613 00:54:54,485 --> 00:55:00,000 electromagnetic radiation going in that direction. 614 00:55:00,000 --> 00:55:03,421 And we concluded, I will not go over the 615 00:55:03,421 --> 00:55:07,456 reasoning again, that the phase velocity in the 616 00:55:07,456 --> 00:55:12,982 z direction was omega divided by k of z and that was larger than 617 00:55:12,982 --> 00:55:15,701 c. That is why you guys couldn't 618 00:55:15,701 --> 00:55:17,368 sleep. And, in fact, 619 00:55:17,368 --> 00:55:21,754 I even demonstrated that if you make the radiation, 620 00:55:21,754 --> 00:55:26,842 the frequency close to the cutoff, that this phase velocity 621 00:55:26,842 --> 00:55:32,753 even goes to infinity. Now you know that there is no 622 00:55:32,753 --> 00:55:36,455 problem. There is no energy flowing with 623 00:55:36,455 --> 00:55:40,063 that speed. It is no different from the 624 00:55:40,063 --> 00:55:43,196 water. In fact, the energy flow is 625 00:55:43,196 --> 00:55:47,848 with the group velocity which is d omega over dkz, 626 00:55:47,848 --> 00:55:53,164 and that was always less than c, as you perhaps remember. 627 00:55:53,164 --> 00:55:56,297 Not zero. My enthusiasm is getting 628 00:55:56,297 --> 00:56:00,000 carried away. Less than c. 629 00:56:00,000 --> 00:56:03,853 And remember at the cutoff frequency where no longer 630 00:56:03,853 --> 00:56:07,934 radiation would go through, this phase velocity went to 631 00:56:07,934 --> 00:56:11,259 infinity and the group velocity went to zero. 632 00:56:11,259 --> 00:56:15,869 I have fulfilled a promise to those of you who could not sleep 633 00:56:15,869 --> 00:56:20,629 to tell you that the meaning of phase velocity is larger than c. 634 00:56:20,629 --> 00:56:24,030 It is natural. There is nothing wrong with it. 635 00:56:24,030 --> 00:56:26,977 You have it with water. You have it with 636 00:56:26,977 --> 00:56:32,788 electromagnetic radiation. You cannot transport any mass 637 00:56:32,788 --> 00:56:37,307 with that speed. You cannot transport any energy 638 00:56:37,307 --> 00:56:41,730 with that speed. There is nothing obscene about 639 00:56:41,730 --> 00:56:43,365 it. Very straight. 640 00:56:43,365 --> 00:56:47,980 I now want to do a demonstration to show you that 641 00:56:47,980 --> 00:56:52,500 electromagnetic waves can be linearly polarized. 642 00:56:52,500 --> 00:56:58,173 Next lecture we will discuss how we generate electromagnetic 643 00:56:58,173 --> 00:57:03,058 waves. We do that by accelerating 644 00:57:03,058 --> 00:57:05,529 charges. In this case, 645 00:57:05,529 --> 00:57:10,235 electrons. And I have here a transmitter. 646 00:57:10,235 --> 00:57:16,941 This is an antenna through which we are going to oscillate 647 00:57:16,941 --> 00:57:21,647 a current at a frequency of 80 megahertz. 648 00:57:21,647 --> 00:57:27,294 F is 80 megahertz. The wavelength lambda is about 649 00:57:27,294 --> 00:57:32,628 3.75 meters. The wavelength is the speed of 650 00:57:32,628 --> 00:57:37,709 light divided by the frequency, which comes out to be about 651 00:57:37,709 --> 00:57:41,477 3.75 meters. And, as we oscillate 80 million 652 00:57:41,477 --> 00:57:46,471 times per second back and forth electrons in this antenna, 653 00:57:46,471 --> 00:57:50,326 that means a current is going back and forth. 654 00:57:50,326 --> 00:57:53,567 Electromagnetic radiation is produced. 655 00:57:53,567 --> 00:57:59,000 Next lecture you will exactly see how much and why. 656 00:57:59,000 --> 00:58:03,102 And that electromagnetic radiation is polarized in this 657 00:58:03,102 --> 00:58:06,750 direction linearly. That should not surprise you. 658 00:58:06,750 --> 00:58:11,308 If the antenna is like this and the electrons move like this, 659 00:58:11,308 --> 00:58:15,563 it should not surprise you, but you will see next Tuesday 660 00:58:15,563 --> 00:58:19,058 why it is linearly polarized in this direction. 661 00:58:19,058 --> 00:58:21,414 Here I have a receiving antenna. 662 00:58:21,414 --> 00:58:25,745 It is a copper wire that is cut in half, and where the two 663 00:58:25,745 --> 00:58:30,000 copper rods connect there is a light bulb. 664 00:58:30,000 --> 00:58:34,402 Any current that flows in here must go through the light bulb. 665 00:58:34,402 --> 00:58:38,010 When I turn on this transmitter, I want to show you 666 00:58:38,010 --> 00:58:42,268 that as long as I hold this receiving antenna like this that 667 00:58:42,268 --> 00:58:46,237 I will see this light go on, but when I do this I won't. 668 00:58:46,237 --> 00:58:50,494 Because the electromagnetic radiation, the E field goes like 669 00:58:50,494 --> 00:58:54,608 this, is going to slosh a current in here with a frequency 670 00:58:54,608 --> 00:58:58,000 80 megahertz and the light will go. 671 00:58:58,000 --> 00:59:02,734 But when I do it like this the electric field is like this. 672 00:59:02,734 --> 00:59:06,408 There is no current flowing in this direction, 673 00:59:06,408 --> 00:59:11,224 and so this is a dramatic way of demonstrating that there is 674 00:59:11,224 --> 00:59:14,816 such a thing as linearly polarized radiation. 675 00:59:14,816 --> 00:59:17,591 What have we decided on the lights? 676 00:59:17,591 --> 00:59:21,183 We were going to TV. We made it as dark as we 677 00:59:21,183 --> 00:59:24,612 possibly could. Here is the transmitter and 678 00:59:24,612 --> 00:59:30,000 here is the receiver. And I am holding it like this. 679 00:59:30,000 --> 00:59:33,109 And you see that I am receiving a signal. 680 00:59:33,109 --> 00:59:37,462 And so there is a current sloshing back and forth and the 681 00:59:37,462 --> 00:59:42,203 light is on, and now it is off. The E field comes in like this 682 00:59:42,203 --> 00:59:46,712 and the light says tough luck, I do not see a current going 683 00:59:46,712 --> 00:59:50,754 through me, but now it does. This is sensitive to the 684 00:59:50,754 --> 00:59:53,474 polarization of the incoming signal. 685 00:59:53,474 --> 00:59:56,894 And when I go a little bit to this side then, 686 00:59:56,894 --> 1:00:00,081 as we just saw, electromagnetic radiation, 687 1:00:00,081 --> 1:00:04,434 the E field must be always perpendicular to the direction 688 1:00:04,434 --> 1:00:09,063 of propagation. The direction of propagation 689 1:00:09,063 --> 1:00:13,189 now is in this direction. That is why I hold it like this 690 1:00:13,189 --> 1:00:16,873 for maximum effect, because now the E field is like 691 1:00:16,873 --> 1:00:19,305 this. Whereas, here the E field is 692 1:00:19,305 --> 1:00:22,178 like this. Now the E field is like this. 693 1:00:22,178 --> 1:00:26,599 And the light is not as bright. And you will see Tuesday why. 694 1:00:26,599 --> 1:00:30,652 But clearly when I do this nothing because of the linear 695 1:00:30,652 --> 1:00:35,000 polarization of the electromagnetic radiation. 696 1:00:35,000 --> 1:00:41,428 If I come too close to this then I will burn the light. 697 1:00:41,428 --> 1:00:46,904 Do you want to see that? Who wants to see that? 698 1:00:46,904 --> 1:00:50,000 Oh, God, you children. 699 1:00:50,000 --> 1:01:00,000 700 1:01:00,000 --> 1:01:04,455 There is another way that I can show you this, 701 1:01:04,455 --> 1:01:09,702 and that we do with radar, with the same setup that we 702 1:01:09,702 --> 1:01:14,158 used last time, except we did a different kind 703 1:01:14,158 --> 1:01:19,306 of experiment last time. This is 10 gigahertz so that 704 1:01:19,306 --> 1:01:25,148 produces three centimeter radar. And this is the transmitter 705 1:01:25,148 --> 1:01:30,000 where the antenna is in this direction. 706 1:01:30,000 --> 1:01:34,352 It is only a very small antenna of only three centimeters 707 1:01:34,352 --> 1:01:37,538 wavelength. The E field goes like this and 708 1:01:37,538 --> 1:01:41,424 here is the receiver. And, if we put the antenna of 709 1:01:41,424 --> 1:01:45,310 the receiver like this, it will receive the signal. 710 1:01:45,310 --> 1:01:48,808 If we put the antenna like this it says sorry, 711 1:01:48,808 --> 1:01:52,772 I cannot receive it. And this 10 gigahertz signal we 712 1:01:52,772 --> 1:01:57,668 modulate with 550 hertz which is a triangular modulation so that 713 1:01:57,668 --> 1:02:02,401 you can hear it. And triangular audio signal are 714 1:02:02,401 --> 1:02:06,430 always not very pleasant because, if you do a Fourier 715 1:02:06,430 --> 1:02:08,832 analysis of this, you have many, 716 1:02:08,832 --> 1:02:12,707 many high harmonics. It is not just a beautiful 550 717 1:02:12,707 --> 1:02:16,116 hertz sinusoidal. You hear a very sharp tone. 718 1:02:16,116 --> 1:02:19,525 But, in any case, the period is such that the 719 1:02:19,525 --> 1:02:23,554 frequency is 550 hertz. I will make you hear it and I 720 1:02:23,554 --> 1:02:26,808 will make you see it. You can see it there. 721 1:02:26,808 --> 1:02:32,000 That is the signal as it is sent by the transmitter. 722 1:02:32,000 --> 1:02:36,502 That is the triangular shape. And now I will turn on the 723 1:02:36,502 --> 1:02:39,532 receiver and you will hear the sounds. 724 1:02:39,532 --> 1:02:43,953 This is the loudspeaker. And you will see the signal of 725 1:02:43,953 --> 1:02:46,000 the receiver there. 726 1:02:46,000 --> 1:02:50,000 727 1:02:50,000 --> 1:02:53,298 The radiation comes in like this. 728 1:02:53,298 --> 1:02:58,247 And now I am going to rotate this one 90 degrees, 729 1:02:58,247 --> 1:03:01,578 the receiver. And it is gone. 730 1:03:01,578 --> 1:03:07,157 The same phenomenon that I just showed you with the 80 731 1:03:07,157 --> 1:03:10,526 megahertz. How can I get it back? 732 1:03:10,526 --> 1:03:15,473 One way I can get it back is rotating this back. 733 1:03:15,473 --> 1:03:21,368 But I can do something better, something more convincing. 734 1:03:21,368 --> 1:03:24,631 I can rotate this by 90 degrees. 735 1:03:24,631 --> 1:03:30,000 Now the transmitter, now I have it back. 736 1:03:30,000 --> 1:03:35,000 You see here two dramatic cases of linear polarization. 737 1:03:35,000 --> 1:03:42,000 738 1:03:42,000 --> 1:03:46,000 My hand absorbs it. A nice feeling. 739 1:03:46,000 --> 1:03:51,000 740 1:03:51,000 --> 1:03:54,138 Make sure we have some light back. 741 1:03:54,138 --> 1:03:58,419 We will discuss extensively in 8.03, actually, 742 1:03:58,419 --> 1:04:01,462 one of my favorite parts of 8.03. 743 1:04:01,462 --> 1:04:06,884 How you can turn unpolarized light into linearly polarized 744 1:04:06,884 --> 1:04:10,308 light. There are various ways that we 745 1:04:10,308 --> 1:04:12,686 can do that. The cheapest, 746 1:04:12,686 --> 1:04:17,822 which is really a copout, is to buy a linear polarizer. 747 1:04:17,822 --> 1:04:22,197 You have three of them in your little envelope. 748 1:04:22,197 --> 1:04:28,000 Don't take them out yet. This is one of them. 749 1:04:28,000 --> 1:04:30,876 They were invented by Edwin Land. 750 1:04:30,876 --> 1:04:36,089 And what they do is they change unpolarized light into 100% 751 1:04:36,089 --> 1:04:41,662 linearly polarized light at the expense of the light intensity. 752 1:04:41,662 --> 1:04:47,235 An ideal linear polarizer would reduce the light intensity by a 753 1:04:47,235 --> 1:04:51,011 factor of two. Let us first discuss what is 754 1:04:51,011 --> 1:04:55,146 unpolarized light? The light from the desk lamp 755 1:04:55,146 --> 1:05:00,000 and the light from the sun is unpolarized. 756 1:05:00,000 --> 1:05:03,646 Imagine that there is a desk lamp here and light comes 757 1:05:03,646 --> 1:05:06,399 straight to you. And here is a plane wave 758 1:05:06,399 --> 1:05:10,045 solution by the E vector oscillate linearly polarized. 759 1:05:10,045 --> 1:05:13,830 It is coming straight at you in this direction and it is 760 1:05:13,830 --> 1:05:17,270 linearly polarized. But a little later there is one 761 1:05:17,270 --> 1:05:20,573 that comes in like this, also linearly polarized. 762 1:05:20,573 --> 1:05:24,839 And then one like this and then one like this and then one like 763 1:05:24,839 --> 1:05:25,596 that. Chaos. 764 1:05:25,596 --> 1:05:30,000 All possible angles of linearly polarized light. 765 1:05:30,000 --> 1:05:33,427 And we call that unpolarized radiation. 766 1:05:33,427 --> 1:05:38,930 Now, what is this magic piece of plastic doing that Edwin Land 767 1:05:38,930 --> 1:05:42,087 invented? It is doing the following. 768 1:05:42,087 --> 1:05:46,597 There is one direction, not always indicated on the 769 1:05:46,597 --> 1:05:50,476 plastic what that is, and I will just put it 770 1:05:50,476 --> 1:05:55,167 arbitrarily vertical, but I can put it any direction. 771 1:05:55,167 --> 1:06:00,579 And that is the direction that the E vector will have when it 772 1:06:00,579 --> 1:06:05,000 emerges from this linear polarizer. 773 1:06:05,000 --> 1:06:10,517 Now let us suppose that linearly polarized light comes 774 1:06:10,517 --> 1:06:15,722 in, which is one of the many unpolarized radiation. 775 1:06:15,722 --> 1:06:21,760 And this has an amplitude E zero and it is oscillating like 776 1:06:21,760 --> 1:06:26,965 this, cosine omega t, and that this angle be theta. 777 1:06:26,965 --> 1:06:33,107 Now, one way to explain the reduction in light is to project 778 1:06:33,107 --> 1:06:38,000 this E onto this preferred direction. 779 1:06:38,000 --> 1:06:42,445 It is the only direction in which the E vector will emerge. 780 1:06:42,445 --> 1:06:46,355 And so you see there is a reduction of the E vector, 781 1:06:46,355 --> 1:06:49,421 and the reduction is the cosine of theta. 782 1:06:49,421 --> 1:06:53,484 Now comes the question, if there is a reduction of the 783 1:06:53,484 --> 1:06:58,006 cosine of the theta in the E field, what is the reduction in 784 1:06:58,006 --> 1:07:02,098 light intensity? Well, light intensity means 785 1:07:02,098 --> 1:07:04,196 energy. And energy is always 786 1:07:04,196 --> 1:07:07,694 proportional with the square of the amplitude. 787 1:07:07,694 --> 1:07:12,435 If that does not convince you, next Tuesday we will talk about 788 1:07:12,435 --> 1:07:15,777 the pointing vector. You remember from 8.02, 789 1:07:15,777 --> 1:07:20,362 the pointing vector that is responsible for energy transport 790 1:07:20,362 --> 1:07:25,181 which is proportional to E cross B, but B is proportional to E. 791 1:07:25,181 --> 1:07:30,000 So E cross B is always proportional to E squared. 792 1:07:30,000 --> 1:07:33,974 And so, therefore, the light intensity that 793 1:07:33,974 --> 1:07:39,179 emerges from this plate is proportional to cosine square 794 1:07:39,179 --> 1:07:42,397 theta. And that is the reduction of 795 1:07:42,397 --> 1:07:46,750 the light intensity. And that is referred to as 796 1:07:46,750 --> 1:07:50,725 Malus' Law. Now, if all angles are present, 797 1:07:50,725 --> 1:07:56,025 randomly angles of theta then the average value of cosine 798 1:07:56,025 --> 1:08:01,938 square theta becomes one-half. For an ideal polarizer, 799 1:08:01,938 --> 1:08:05,814 which don't exist, but an ideal polarizer would 800 1:08:05,814 --> 1:08:10,786 turn unpolarized light into 100% polarized light all in this 801 1:08:10,786 --> 1:08:13,735 direction, if that is the direction. 802 1:08:13,735 --> 1:08:18,707 And then the intensity of the light would be twice as low as 803 1:08:18,707 --> 1:08:21,151 the incoming one. In practice, 804 1:08:21,151 --> 1:08:25,280 however, you may not get 50%, but you may get 40%. 805 1:08:25,280 --> 1:08:30,000 There is always some additional absorption. 806 1:08:30,000 --> 1:08:33,981 And, in problem set number seven, you will be able to 807 1:08:33,981 --> 1:08:38,193 answer some interesting questions and use the polarizers 808 1:08:38,193 --> 1:08:42,022 that you have in your envelope, in your optics kit. 809 1:08:42,022 --> 1:08:45,774 Clearly, if I have one linear polarizer like this, 810 1:08:45,774 --> 1:08:49,756 this is the preferred direction, and the other one is 811 1:08:49,756 --> 1:08:53,279 like this, for which we have a name in physics, 812 1:08:53,279 --> 1:08:56,801 we call them cross-polarizers, then, of course, 813 1:08:56,801 --> 1:09:01,243 independent of this absorption phenomenon that I mentioned, 814 1:09:01,243 --> 1:09:07,209 no light can emerge. Because now the cosine of the 815 1:09:07,209 --> 1:09:11,635 angle is zero. Cross-polarizers will turn 816 1:09:11,635 --> 1:09:18,274 unpolarized light into darkness, and that is something that I 817 1:09:18,274 --> 1:09:24,471 would like you to see now. Let's make it a little darker. 818 1:09:24,471 --> 1:09:31,000 I have here several sheets of that linear polarizer. 819 1:09:31,000 --> 1:09:34,948 And the first thing I want you to see is this absorption 820 1:09:34,948 --> 1:09:37,245 phenomenon that I just mentioned. 821 1:09:37,245 --> 1:09:41,193 Here is a linear polarizer. The light that comes through 822 1:09:41,193 --> 1:09:45,643 here now is linearly polarized. The light that reflects off the 823 1:09:45,643 --> 1:09:47,940 screen is not linearly polarized. 824 1:09:47,940 --> 1:09:51,529 That is a different thing because it goes through a 825 1:09:51,529 --> 1:09:54,975 reflection process. This light that comes up here 826 1:09:54,975 --> 1:09:58,277 is linearly polarized. You and I have no way of 827 1:09:58,277 --> 1:10:01,956 telling. There are animals that can see 828 1:10:01,956 --> 1:10:04,760 polarized light, who can distinguish it from 829 1:10:04,760 --> 1:10:07,565 unpolarized light, and they can even see the 830 1:10:07,565 --> 1:10:11,152 direction of polarization. Bees can see the direction of 831 1:10:11,152 --> 1:10:13,956 polarization. And I have learned also to see 832 1:10:13,956 --> 1:10:17,478 it under ideal conditions. If any of you is interested, 833 1:10:17,478 --> 1:10:19,760 I can teach you, but it is not easy. 834 1:10:19,760 --> 1:10:23,673 You have to come to my office. There is a way that humans can 835 1:10:23,673 --> 1:10:27,195 actually recognize linearly polarized light and see the 836 1:10:27,195 --> 1:10:31,339 direction of polarization. But, apart from that, 837 1:10:31,339 --> 1:10:34,911 we have no way of knowing that this is linearly polarized 838 1:10:34,911 --> 1:10:36,697 light. And, if I rotate this, 839 1:10:36,697 --> 1:10:39,759 the direction of linear polarization will change, 840 1:10:39,759 --> 1:10:43,076 but we have no way of knowing. In fact, the preferred 841 1:10:43,076 --> 1:10:46,329 direction, as indicated on this sheet, is like this. 842 1:10:46,329 --> 1:10:49,072 Now I take a second one, which is identical. 843 1:10:49,072 --> 1:10:53,090 And I am going to put it on top of it so that the two directions 844 1:10:53,090 --> 1:10:55,705 are aligned. If they were ideal then there 845 1:10:55,705 --> 1:10:59,150 would be no light reduction anymore, but there is light 846 1:10:59,150 --> 1:11:03,007 reduction. Look at this middle portion 847 1:11:03,007 --> 1:11:06,205 where they overlap. There is still a further 848 1:11:06,205 --> 1:11:10,443 reduction in light intensity. Even though the direction of 849 1:11:10,443 --> 1:11:14,830 polarization has not changed, it is not the result of Malus' 850 1:11:14,830 --> 1:11:18,920 Law that there is a reduction but there is an absorption 851 1:11:18,920 --> 1:11:23,084 phenomenon that I mentioned. Only here are two plates and 852 1:11:23,084 --> 1:11:26,728 here is one plate. Now I have here a black stripe. 853 1:11:26,728 --> 1:11:31,762 That is on top of the overhead. Here now is one linear 854 1:11:31,762 --> 1:11:34,485 polarizer and here comes the other. 855 1:11:34,485 --> 1:11:38,569 They are now aligned. And I am going to rotate them. 856 1:11:38,569 --> 1:11:42,574 The angle of theta is increasing and increasing and 857 1:11:42,574 --> 1:11:46,578 increasing, and I am approaching slowly 90 degrees. 858 1:11:46,578 --> 1:11:50,263 And there you are. And it is as dark as you can 859 1:11:50,263 --> 1:11:52,826 have it. Two cross-polarizers now 860 1:11:52,826 --> 1:11:57,471 through an unpolarized light, first into linearly polarized 861 1:11:57,471 --> 1:12:02,517 light, that is sheet number one, and then sheet number two kills 862 1:12:02,517 --> 1:12:06,780 it altogether. Nothing can get through. 863 1:12:06,780 --> 1:12:10,934 Cosine of the angle is zero. Now comes the great miracle, 864 1:12:10,934 --> 1:12:15,237 which is something you can also do at home because you have 865 1:12:15,237 --> 1:12:18,353 three linear polarizers in your optics kit. 866 1:12:18,353 --> 1:12:22,878 Suppose I stick a third linear polarizer in between the two at 867 1:12:22,878 --> 1:12:26,290 a random angle, will I see light coming through 868 1:12:26,290 --> 1:12:30,000 or will I not see light coming through? 869 1:12:30,000 --> 1:12:33,431 Very good. If I do it in front of number 870 1:12:33,431 --> 1:12:36,422 one, darkness will remain darkness. 871 1:12:36,422 --> 1:12:41,524 If I do it above number two, darkness will remain darkness. 872 1:12:41,524 --> 1:12:46,891 But if I do it in between one and two, there is my line again, 873 1:12:46,891 --> 1:12:52,082 light comes through and there is an immediately consequence, 874 1:12:52,082 --> 1:12:57,272 of course, of the fact that you have to apply now the cosine 875 1:12:57,272 --> 1:13:02,583 square theta twice. But you never will see it go 876 1:13:02,583 --> 1:13:06,273 down to zero. Now I want you to open your 877 1:13:06,273 --> 1:13:09,779 envelope. Don't put your fingers on all 878 1:13:09,779 --> 1:13:15,500 the beautiful pieces that are in there that we will have to use 879 1:13:15,500 --> 1:13:19,467 in the future. Open it and get out of it one 880 1:13:19,467 --> 1:13:23,619 linear polarizer. And you will recognize them. 881 1:13:23,619 --> 1:13:27,863 They are green plates and they have this shape. 882 1:13:27,863 --> 1:13:31,000 And you just get one. 883 1:13:31,000 --> 1:13:37,000 884 1:13:37,000 --> 1:13:42,728 Don't drop them on the floor and don't make them too dirty. 885 1:13:42,728 --> 1:13:45,987 Can you see me? Who cannot see me? 886 1:13:45,987 --> 1:13:49,444 Good. You can rotate your polarizers 887 1:13:49,444 --> 1:13:54,975 until you are purple in your face, and you won't see much 888 1:13:54,975 --> 1:13:58,827 difference. You will see Walter Lewin no 889 1:13:58,827 --> 1:14:04,309 matter what. Use your polarizers and rotate 890 1:14:04,309 --> 1:14:11,149 them around and look at my face. It is not going to change very 891 1:14:11,149 --> 1:14:14,569 much. Now, I am going to hold in 892 1:14:14,569 --> 1:14:18,761 front of my face this linear polarizer. 893 1:14:18,761 --> 1:14:24,718 Now you are looking at Walter Lewin in polarized light. 894 1:14:24,718 --> 1:14:32,000 And that now gives you the option to make me come and go. 895 1:14:32,000 --> 1:14:36,858 You can rotate in such a direction that you say there he 896 1:14:36,858 --> 1:14:42,246 is, but if you rotate it at 90 degrees you say thank goodness, 897 1:14:42,246 --> 1:14:45,779 he is gone. Now, keep in mind that if you 898 1:14:45,779 --> 1:14:50,902 rotate your linear polarizers such that you can see me that 899 1:14:50,902 --> 1:14:55,583 the reverse is also true. I can then look through your 900 1:14:55,583 --> 1:15:00,000 linear polarizer and I can see your eye. 901 1:15:00,000 --> 1:15:02,675 And if somehow, in an evil way, 902 1:15:02,675 --> 1:15:08,116 you prefer 90 degrees so that you don't see me then you have a 903 1:15:08,116 --> 1:15:11,326 black eye. And who wants a black eye? 904 1:15:11,326 See you next Tuesday.