1 00:00:00 --> 00:00:00,35 2 00:00:00,35 --> 00:00:05,147 I'm very proud of you. You did very well on the last 3 00:00:05,147 --> 00:00:08,345 exam. Class average is a little bit 4 00:00:08,345 --> 00:00:11,167 above seventy. Congratulations. 5 00:00:11,167 --> 00:00:15,964 There were twenty-two students who scored a hundred. 6 00:00:15,964 --> 00:00:21,138 Many of you are interested in where the dividing line is 7 00:00:21,138 --> 00:00:25,465 between C and D. If I take only the three exams 8 00:00:25,465 --> 00:00:28,475 into account, forget the quizzes, 9 00:00:28,475 --> 00:00:32,802 forget the homework, forget the 10 00:00:32,802 --> 00:00:37,689 motor, and you add up the three grades of your three exams, 11 00:00:37,689 --> 00:00:42,493 the dividing line between C and D will be somewhere in the 12 00:00:42,493 --> 00:00:47,044 region a hundred thirty-five to a hundred thirty-eight. 13 00:00:47,044 --> 00:00:51,764 So you can use that for your calibration where you stand. 14 00:00:51,764 --> 00:00:56,904 The controversy between Newton and Huygens about the nature of 15 00:00:56,904 --> 00:01:01,877 light was settled in eighteen oh one when Young demonstrated 16 00:01:01,877 --> 00:01:06,28 convincingly that light shows all the 17 00:01:06,28 --> 00:01:11,063 characteristic of waves. Now in the early twentieth 18 00:01:11,063 --> 00:01:17,091 century, the particle character of light surfaced again and this 19 00:01:17,091 --> 00:01:21,875 um mysterious and very fascinating duality of being 20 00:01:21,875 --> 00:01:27,711 waves and particles at the same time is now beautifully merged 21 00:01:27,711 --> 00:01:32,495 in quantum mechanics. But today I will focus on the 22 00:01:32,495 --> 00:01:35,562 wave character only. 23 00:01:35,562 --> 00:01:41,71 Very characteristic for waves are interference patterns which 24 00:01:41,71 --> 00:01:47,447 are produced by two sources, which simultaneously produce 25 00:01:47,447 --> 00:01:52,058 traveling waves at exactly the same frequency. 26 00:01:52,058 --> 00:01:58,001 Let this be source number one and let this be source number 27 00:01:58,001 --> 00:02:01,177 two. And they each produce waves 28 00:02:01,177 --> 00:02:08,042 with the same frequency, therefore the same wavelength, 29 00:02:08,042 --> 00:02:12,573 and they go out let's say in all directions. 30 00:02:12,573 --> 00:02:18,158 They could be spherical, in the case of water surface, 31 00:02:18,158 --> 00:02:23,321 going out like rings. And suppose you were here at 32 00:02:23,321 --> 00:02:28,59 position P in space at a distance R one from source 33 00:02:28,59 --> 00:02:34,596 number one and at a distance R two from source number two. 34 00:02:34,596 --> 00:02:40,603 Then it is possible that at the point P the two waves that 35 00:02:40,603 --> 00:02:45,05 arrive are in phase with each other. 36 00:02:45,05 --> 00:02:49,628 That means the mountain from two arrives at the same time as 37 00:02:49,628 --> 00:02:53,585 a mountain from one, and the valley from two arrives 38 00:02:53,585 --> 00:02:56,611 at the same time as the valley from one. 39 00:02:56,611 --> 00:03:01,189 So the mountains become higher and the valleys become lower. 40 00:03:01,189 --> 00:03:04,138 We call that constructive interference. 41 00:03:04,138 --> 00:03:08,716 It is also possible that the waves as they arrive at point P 42 00:03:08,716 --> 00:03:12,518 are exactly a hundred eighty degrees out of phase, 43 00:03:12,518 --> 00:03:17,647 so that means that the mountain from two arrives at 44 00:03:17,647 --> 00:03:20,45 the same time as the valley from one. 45 00:03:20,45 --> 00:03:24,732 In which case they can kill each other, and that we call 46 00:03:24,732 --> 00:03:28,859 destructive interference. You can have this with water 47 00:03:28,859 --> 00:03:32,207 waves, so it's on a two-dimensional surface. 48 00:03:32,207 --> 00:03:35,789 You can also have it with sound, which would be 49 00:03:35,789 --> 00:03:39,059 three-dimensional. So the waves go out on a 50 00:03:39,059 --> 00:03:41,473 sphere. And you can have it with 51 00:03:41,473 --> 00:03:45,521 electromagnetic radiation as we will 52 00:03:45,521 --> 00:03:49,18 also see today, which is of course also three 53 00:03:49,18 --> 00:03:52,34 dimensions. If particles oscillate then 54 00:03:52,34 --> 00:03:57,496 their energy is proportional to the square of their amplitudes. 55 00:03:57,496 --> 00:04:01,072 So therefore since energy must be conserved, 56 00:04:01,072 --> 00:04:05,23 the amplitude of sound oscillations and also of the 57 00:04:05,23 --> 00:04:09,887 electric vector in the case of electromagnetic radiation, 58 00:04:09,887 --> 00:04:14,211 the amplitude must fall off as one over the distance, 59 00:04:14,211 --> 00:04:18,286 one over R. Because you're talking 60 00:04:18,286 --> 00:04:22,687 about three-D waves. You're talking about spherical 61 00:04:22,687 --> 00:04:25,415 waves. And the surface area of a 62 00:04:25,415 --> 00:04:30,52 sphere grows with R squared. And so the amplitude must fall 63 00:04:30,52 --> 00:04:33,953 off as one over R. Now if we look at the 64 00:04:33,953 --> 00:04:39,322 superposition of two waves in this case at point P and we make 65 00:04:39,322 --> 00:04:44,78 the distance large so that R one and R two are much much larger 66 00:04:44,78 --> 00:04:48,652 than the separation between these two points, 67 00:04:48,652 --> 00:04:53,141 then this fact that the amplitude of 68 00:04:53,141 --> 00:04:58,366 the wave from two is slightly smaller than the amplitude from 69 00:04:58,366 --> 00:05:02,632 the wave from one can then be pretty much ignored. 70 00:05:02,632 --> 00:05:07,421 Imagine that the path from here to here is one-half of a 71 00:05:07,421 --> 00:05:11,687 wavelength longer than the path from here to here. 72 00:05:11,687 --> 00:05:16,998 That means that this wave from here to here will have traveled 73 00:05:16,998 --> 00:05:21,526 half a period of an oscillation longer than this one. 74 00:05:21,526 --> 00:05:27,001 And that means they are exactly a hundred eighty 75 00:05:27,001 --> 00:05:32,363 degrees out of phase and so the two can kill each other. 76 00:05:32,363 --> 00:05:36,361 And we call that destructive interference. 77 00:05:36,361 --> 00:05:41,918 And so we're going to have destructive interference when R 78 00:05:41,918 --> 00:05:47,768 two minus R one is for instance plus or minus one-half lambda 79 00:05:47,768 --> 00:05:53,033 but it could also be plus or minus three-halves lambda, 80 00:05:53,033 --> 00:05:57,129 five-halves lambda, and so on. 81 00:05:57,129 --> 00:06:03,522 And so in general you would have destructive interference if 82 00:06:03,522 --> 00:06:10,132 the difference between R two and R one is two N plus one times 83 00:06:10,132 --> 00:06:15,008 lambda divided by two whereby N is an integer, 84 00:06:15,008 --> 00:06:18,801 could be zero, or plus or minus one, 85 00:06:18,801 --> 00:06:22,16 or plus or minus two, and so on. 86 00:06:22,16 --> 00:06:27,686 That's when you would have destructive interference. 87 00:06:27,686 --> 00:06:35,313 We would have constructive interference if R two minus R 88 00:06:35,313 --> 00:06:41,997 one is simply N times lambda. So then the waves at point P 89 00:06:41,997 --> 00:06:46,805 are in phase and N is again could be zero, 90 00:06:46,805 --> 00:06:51,027 plus or minus one, plus or minus two, 91 00:06:51,027 --> 00:06:55,6 and so on. If the sum of the distance to 92 00:06:55,6 --> 00:07:03,692 two points is a constant you get an ellipse in mathematics. 93 00:07:03,692 --> 00:07:07,881 If the difference is a constant, which is the case 94 00:07:07,881 --> 00:07:12,498 here, the difference to two points is a constant value, 95 00:07:12,498 --> 00:07:17,543 for instance one-half lambda, then the curve is a hyperbola. 96 00:07:17,543 --> 00:07:22,331 It would be a hyperbola if we deal with a two-dimensional 97 00:07:22,331 --> 00:07:25,239 surface. But if we think of this as 98 00:07:25,239 --> 00:07:29,086 three-dimensional, so you can rotate the whole 99 00:07:29,086 --> 00:07:34,387 thing about this axis, then you get hyperboloids, 100 00:07:34,387 --> 00:07:40,428 you get bowl-shaped surfaces. And so if I'm now trying to 101 00:07:40,428 --> 00:07:46,468 tighten the nuts a little bit, suppose I have here two of 102 00:07:46,468 --> 00:07:53,263 these sources that produce waves and the separation between them 103 00:07:53,263 --> 00:07:59,195 is D, then it is obvious that the line right through the 104 00:07:59,195 --> 00:08:05,882 middle of them and perpendicular to them is always a maximum if 105 00:08:05,882 --> 00:08:10,841 the two sources are oscillating in phase. 106 00:08:10,841 --> 00:08:15,838 So this line is immediately clear that R two minus R one is 107 00:08:15,838 --> 00:08:18,768 zero here. If the two are in phase. 108 00:08:18,768 --> 00:08:23,335 And they always have to generate the same frequency of 109 00:08:23,335 --> 00:08:26,523 course. So this line would be always a 110 00:08:26,523 --> 00:08:29,452 maximum. Constructive interference. 111 00:08:29,452 --> 00:08:32,124 It's this zero substitute there. 112 00:08:32,124 --> 00:08:36,777 And in case that we're talking about three-dimensional, 113 00:08:36,777 --> 00:08:43,009 this is of course a plane. Going perpendicular to the 114 00:08:43,009 --> 00:08:46,527 blackboard right through the middle. 115 00:08:46,527 --> 00:08:52,257 The difference R two minus R one equals lambda would again 116 00:08:52,257 --> 00:08:55,575 give me constructive interference. 117 00:08:55,575 --> 00:09:01,104 That would be a hyperbola then, R two minus R one equals 118 00:09:01,104 --> 00:09:06,934 lambda, that would again be a maximum, and you can draw the 119 00:09:06,934 --> 00:09:13,167 same line on this side, and then R two minus R one 120 00:09:13,167 --> 00:09:17,113 being two lambda again would be a maximum. 121 00:09:17,113 --> 00:09:22,119 And again if this is three-dimensional you can rotate 122 00:09:22,119 --> 00:09:25,584 it about this line and you get bowls. 123 00:09:25,584 --> 00:09:30,397 And so in between you're obviously going to get the 124 00:09:30,397 --> 00:09:33,863 minima, the destructive interference, 125 00:09:33,863 --> 00:09:38,868 lambda divided by two, and then here you would have R 126 00:09:38,868 --> 00:09:42,526 two minus R one is three-halves lambda. 127 00:09:42,526 --> 00:09:49,246 We call these lines where you kill each other destructive 128 00:09:49,246 --> 00:09:53,226 interference, we call them nodal lines or in 129 00:09:53,226 --> 00:09:57,299 case you have a surface it's a nodal surface. 130 00:09:57,299 --> 00:10:01,928 And the maxima are sometimes also called antinodes, 131 00:10:01,928 --> 00:10:06,093 but I may also refer to them simply as maxima. 132 00:10:06,093 --> 00:10:10,814 And so this is what we call an interference pattern. 133 00:10:10,814 --> 00:10:16,09 If you look right here between on the line between the two 134 00:10:16,09 --> 00:10:21,054 points then you should be able to convince 135 00:10:21,054 --> 00:10:25,557 yourself that the linear separation here between two 136 00:10:25,557 --> 00:10:28,56 lines of maxima is one-half lambda. 137 00:10:28,56 --> 00:10:32,181 Figure that out at home. That's very easy. 138 00:10:32,181 --> 00:10:37,567 Also the distance between these two yellow lines here right in 139 00:10:37,567 --> 00:10:42,689 between is one-half lambda. And so that tells you then that 140 00:10:42,689 --> 00:10:48,253 the number of lines or surfaces which are maxima is very roughly 141 00:10:48,253 --> 00:10:52,014 two D divided by one-half lambda. 142 00:10:52,014 --> 00:10:56,181 So this is the number of maxima, which is also the same 143 00:10:56,181 --> 00:11:00,733 roughly as the number of minima, is then approximately two D 144 00:11:00,733 --> 00:11:04,515 divided by lambda. And so if you want more maxima, 145 00:11:04,515 --> 00:11:08,604 if you want more of these surfaces, you have a choice, 146 00:11:08,604 --> 00:11:13,234 you can make D larger or you can make the wavelength shorter. 147 00:11:13,234 --> 00:11:17,555 And if you make the wavelength shorter you can do that by 148 00:11:17,555 --> 00:11:23,393 increasing the frequency if you had that control. 149 00:11:23,393 --> 00:11:30,658 The first thing that I'm going to do is to make you see these 150 00:11:30,658 --> 00:11:35,622 nodal lines with a demonstration of water. 151 00:11:35,622 --> 00:11:42,887 We have here two sources that we can tap on the water and the 152 00:11:42,887 --> 00:11:49,788 distance between those two tappers, D, is ten centimeters, 153 00:11:49,788 --> 00:11:54,094 so we're talking about water here. 154 00:11:54,094 --> 00:11:59,196 Uh we will tap with a frequency of about seven hertz and what 155 00:11:59,196 --> 00:12:03,108 you're going to see are very clear nodal lines, 156 00:12:03,108 --> 00:12:07,956 this is a two-dimensional surface, where the water doesn't 157 00:12:07,956 --> 00:12:11,442 move at all. The mountains and the valleys 158 00:12:11,442 --> 00:12:15,865 arrive at the same time. The water is never moving at 159 00:12:15,865 --> 00:12:18,586 all. So let me make sure that you 160 00:12:18,586 --> 00:12:23,988 can see that well. And so I have to change my -- 161 00:12:23,988 --> 00:12:26,946 my lights. I'll first turn it on, 162 00:12:26,946 --> 00:12:31,291 that may be the easiest. Starts tapping already. 163 00:12:31,291 --> 00:12:34,527 I can see the nodal lines very well. 164 00:12:34,527 --> 00:12:40,258 So here you see the two tappers and here you see a line whereby 165 00:12:40,258 --> 00:12:45,62 the water is not moving at all. At all moments in time it's 166 00:12:45,62 --> 00:12:48,023 standing still. Here's one. 167 00:12:48,023 --> 00:12:51,906 Here is one. And you even with a little bit 168 00:12:51,906 --> 00:12:55,596 of imagination can see that they 169 00:12:55,596 --> 00:12:59,3 are really not straight lines but they are hyperbolas. 170 00:12:59,3 --> 00:13:03,213 If you're very close to one tapper, the zero can never be 171 00:13:03,213 --> 00:13:06,078 exactly zero, because the amplitude of the 172 00:13:06,078 --> 00:13:09,782 wave from this one then will always be larger than the 173 00:13:09,782 --> 00:13:13,625 amplitude from that one, because as you go away from the 174 00:13:13,625 --> 00:13:17,958 source the amplitude must fall off on a two-dimensional surface 175 00:13:17,958 --> 00:13:20,194 as one over the square root of R. 176 00:13:20,194 --> 00:13:25,155 In a three-dimensional wave must fall of as one over R. 177 00:13:25,155 --> 00:13:28,045 But if you're far enough away then the distance is 178 00:13:28,045 --> 00:13:31,642 approximately the same and so the amplitudes of the individual 179 00:13:31,642 --> 00:13:35,18 waves are very closely the same and you can then like you see 180 00:13:35,18 --> 00:13:37,716 here the water is absolutely standing still. 181 00:13:37,716 --> 00:13:41,195 And here are then the areas whereby you see traveling waves, 182 00:13:41,195 --> 00:13:44,261 they are traveling waves, they're not standing waves, 183 00:13:44,261 --> 00:13:47,799 that here you see if you were sitting here in space the water 184 00:13:47,799 --> 00:13:50,217 would be up and down, bobbing up and down, 185 00:13:50,217 --> 00:13:53,106 and the amplitude that you would have is twice the 186 00:13:53,106 --> 00:13:57,647 amplitude that you get from one, because the mountains add 187 00:13:57,647 --> 00:14:02,335 to the mountains and the valleys add to the valleys. 188 00:14:02,335 --> 00:14:07,484 But if you were here in space you would be sitting still. 189 00:14:07,484 --> 00:14:11,438 You would not be bobbing up and down at all. 190 00:14:11,438 --> 00:14:15,207 And that is very characteristic for waves. 191 00:14:15,207 --> 00:14:20,632 If I were to tap them a hundred eighty degrees out of phase, 192 00:14:20,632 --> 00:14:26,056 which I didn't -- they were in phase -- then all nodal lines 193 00:14:26,056 --> 00:14:30,5 would become maxima and all maximum lines 194 00:14:30,5 --> 00:14:34,024 would become nodes, that goes without saying of 195 00:14:34,024 --> 00:14:36,629 course. It is essential that you -- 196 00:14:36,629 --> 00:14:41,149 that the frequencies are the same, that is an absolute must. 197 00:14:41,149 --> 00:14:44,75 They don't have to be in phase, the two tappers, 198 00:14:44,75 --> 00:14:49,347 if they're not in phase then the positions in space where you 199 00:14:49,347 --> 00:14:53,638 have maxima and minima will change but a must is that the 200 00:14:53,638 --> 00:14:57,545 frequency is the same. Now I was hiking last year in 201 00:14:57,545 --> 00:15:01,682 Utah when I noticed a butterfly in the 202 00:15:01,682 --> 00:15:05,176 water of a pond which was fighting for its life. 203 00:15:05,176 --> 00:15:07,481 And you see that butterfly here. 204 00:15:07,481 --> 00:15:10,901 Tom, perhaps you can turn off that uh overhead. 205 00:15:10,901 --> 00:15:15,064 You see the butterfly here, and you see here projected on 206 00:15:15,064 --> 00:15:18,483 the bottom the beautiful rings dark and bright, 207 00:15:18,483 --> 00:15:22,051 because these rings on the water act like lenses, 208 00:15:22,051 --> 00:15:25,768 and what you see very dramatically is indeed what I 209 00:15:25,768 --> 00:15:30,452 said, that the amplitude of the wave must go down with distance, 210 00:15:30,452 --> 00:15:34,317 because energy must be conserved of 211 00:15:34,317 --> 00:15:37,567 course in the wave, and since the circumference 212 00:15:37,567 --> 00:15:41,171 grows linearly with R, the amplitude must go down as 213 00:15:41,171 --> 00:15:45,339 one over the square root of R because the energy in the wave 214 00:15:45,339 --> 00:15:48,165 is proportional to the amplitude squared. 215 00:15:48,165 --> 00:15:52,334 So when I saw this it occurred to me that it would be a good 216 00:15:52,334 --> 00:15:55,866 idea to catch another butterfly, put it next to it, 217 00:15:55,866 --> 00:15:59,752 and then photograph -- make a fantastic photograph of an 218 00:15:59,752 --> 00:16:02,931 interference pattern. But I realized of course 219 00:16:02,931 --> 00:16:07,383 immediately having taken eight oh two that the 220 00:16:07,383 --> 00:16:13,634 frequencies of the two butterflies would have to be 221 00:16:13,634 --> 00:16:21,385 exactly the same and so I gave up the idea and I decided not to 222 00:16:21,385 --> 00:16:25,635 be cruel. So no other butterfly was 223 00:16:25,635 --> 00:16:30,511 sacrificed. If we look at the directions 224 00:16:30,511 --> 00:16:37,762 where we expect the maxima as seen from the location of the 225 00:16:37,762 --> 00:16:44,639 sources, then I want to remind you of what a 226 00:16:44,639 --> 00:16:50,222 hyperbola looks like. If here are these two sources 227 00:16:50,222 --> 00:16:55,248 and here is the center I can draw a line here, 228 00:16:55,248 --> 00:16:59,38 then a hyperbola would look like this. 229 00:16:59,38 --> 00:17:06,192 Let me re- remove the part on the left, doesn't look too good, 230 00:17:06,192 --> 00:17:10,547 but it's the same on the left of course. 231 00:17:10,547 --> 00:17:17,248 And what you remember from your high school math, 232 00:17:17,248 --> 00:17:23,072 that it approaches that line. And therefore you can define 233 00:17:23,072 --> 00:17:28,998 angle theta as seen from the center between these two which 234 00:17:28,998 --> 00:17:34,925 are the directions where you have maxima and where you have 235 00:17:34,925 --> 00:17:38,603 minima. And that's what I am going to 236 00:17:38,603 --> 00:17:43,099 work out for you now on this blackboard here. 237 00:17:43,099 --> 00:17:47,799 So here are now the two sources that oscillate, 238 00:17:47,799 --> 00:17:52,397 there's one here and there's one 239 00:17:52,397 --> 00:17:57,222 here and here is the center in between them, 240 00:17:57,222 --> 00:18:03,842 and let this separation be D. And I am looking very far away 241 00:18:03,842 --> 00:18:10,462 so that I'm approaching this line where the hyperbolas merge 242 00:18:10,462 --> 00:18:14,277 so to speak with the straight line. 243 00:18:14,277 --> 00:18:20,335 And so I look very far away without being -- committing 244 00:18:20,335 --> 00:18:25,734 myself how far, I'm looking in the direction 245 00:18:25,734 --> 00:18:27,95 theta away. This is theta. 246 00:18:27,95 --> 00:18:32,205 And so this is theta. And I want to know in which 247 00:18:32,205 --> 00:18:37,08 directions of theta I expect to see maxima, and in which 248 00:18:37,08 --> 00:18:39,916 direction I expect to see minima. 249 00:18:39,916 --> 00:18:44,968 So this is what we called earlier R one and we called this 250 00:18:44,968 --> 00:18:48,514 earlier R two, it is the distance to that 251 00:18:48,514 --> 00:18:52,768 point very far away. If I want to know what R two 252 00:18:52,768 --> 00:18:57,667 minus R one is that's very easy now. 253 00:18:57,667 --> 00:19:03,032 I draw a line from here perpendicular to this line and 254 00:19:03,032 --> 00:19:09,005 you see immediately that this distance here is R two minus R 255 00:19:09,005 --> 00:19:12,346 one. But that distance is also you 256 00:19:12,346 --> 00:19:18,623 realize that this angle is theta it's the same one as that one, 257 00:19:18,623 --> 00:19:22,875 so that distance here is also D sine theta. 258 00:19:22,875 --> 00:19:27,431 And so now I'm in business, I can 259 00:19:27,431 --> 00:19:32,364 predict in what directions we will see constructive 260 00:19:32,364 --> 00:19:36,409 interference. Because all we are demanding 261 00:19:36,409 --> 00:19:40,651 now, requesting, that R two minus R one is N 262 00:19:40,651 --> 00:19:44,499 times lambda. And so we need that D sine 263 00:19:44,499 --> 00:19:49,235 theta and I'll give it a subindex N, as in Nancy, 264 00:19:49,235 --> 00:19:54,267 equals N times lambda. In others words that the sine 265 00:19:54,267 --> 00:19:59,553 of theta N is simply N lambda divided by D. 266 00:19:59,553 --> 00:20:03,598 And that uniquely defines all those directions, 267 00:20:03,598 --> 00:20:08,873 the whole zoo of directions N equals zero, that is the center 268 00:20:08,873 --> 00:20:11,686 line, N equals one, N equals two, 269 00:20:11,686 --> 00:20:13,884 N equals three, and so on. 270 00:20:13,884 --> 00:20:17,928 And then I have the whole family of destructive 271 00:20:17,928 --> 00:20:21,796 interference. Which would require that lambda 272 00:20:21,796 --> 00:20:26,72 R two minus R one which is D sine theta must now be two N 273 00:20:26,72 --> 00:20:32,33 plus one times lambda divided by two. 274 00:20:32,33 --> 00:20:37,457 Just as we had it on the blackboard there. 275 00:20:37,457 --> 00:20:44,459 We discussed that earlier. And so that requires then that 276 00:20:44,459 --> 00:20:51,211 the sine of theta N for the destructive interference is 277 00:20:51,211 --> 00:20:58,213 going to be two N plus one times lambda divided by two D. 278 00:20:58,213 --> 00:21:03,204 So this indicates the directions where 279 00:21:03,204 --> 00:21:08,839 we expect maxima and where we expect minima as seen from the 280 00:21:08,839 --> 00:21:14,475 center between the two sources. But now I would like to know 281 00:21:14,475 --> 00:21:20,588 what the linear distance is if I project this onto a screen which 282 00:21:20,588 --> 00:21:25,077 is very far away. And so let us have a screen at 283 00:21:25,077 --> 00:21:29,853 a distance capital L which has to be very far away, 284 00:21:29,853 --> 00:21:34,056 so here are now the two sources. 285 00:21:34,056 --> 00:21:38,74 It's a different scale. And here is a screen. 286 00:21:38,74 --> 00:21:44,702 And the distance b- from the two sources to the screen is 287 00:21:44,702 --> 00:21:48,321 capital L. And here is one of those 288 00:21:48,321 --> 00:21:53,325 direction theta. And you see immediately that if 289 00:21:53,325 --> 00:21:58,222 I call this the direction X, X being zero here, 290 00:21:58,222 --> 00:22:02,8 that the tangent of theta is X divided by L. 291 00:22:02,8 --> 00:22:06,952 If but only if I deal with small angles, 292 00:22:06,952 --> 00:22:13,983 the tangent of theta is the same as the sine of theta. 293 00:22:13,983 --> 00:22:19,889 And therefore I can now tell you where the maxima will lie on 294 00:22:19,889 --> 00:22:25,402 that screen away from the center line, which I call zero, 295 00:22:25,402 --> 00:22:30,521 that is now when X of N is L times the sine of theta, 296 00:22:30,521 --> 00:22:35,935 in small angle approximation. So this is approximately L 297 00:22:35,935 --> 00:22:41,35 times N lambda divided by D, and for the same reason you 298 00:22:41,35 --> 00:22:44,838 will get here c- destructive 299 00:22:44,838 --> 00:22:49,395 interference when X of N is going to be L times two N plus 300 00:22:49,395 --> 00:22:52,033 one times lambda divided by two D. 301 00:22:52,033 --> 00:22:55,711 That is simple geometry. So now we have all the 302 00:22:55,711 --> 00:22:59,708 ingredients here on the blackboard and I'm going to 303 00:22:59,708 --> 00:23:03,066 leave it there for the rest of the lecture. 304 00:23:03,066 --> 00:23:07,543 Whenever we're going to do an experiment with two sources 305 00:23:07,543 --> 00:23:12,34 which are in phase at the same frequency, you can predict the 306 00:23:12,34 --> 00:23:17,287 directions of maxima and minima and you can 307 00:23:17,287 --> 00:23:21,82 even predict the separation, the linear separation, 308 00:23:21,82 --> 00:23:26,443 if you know how far away you are from these sources. 309 00:23:26,443 --> 00:23:31,429 And the first demonstration that I'm going to do is with 310 00:23:31,429 --> 00:23:34,692 sound. We have here two loudspeakers. 311 00:23:34,692 --> 00:23:38,953 And the distance between those two loudspeakers, 312 00:23:38,953 --> 00:23:43,666 we're going to do it with sound, D, is one point five 313 00:23:43,666 --> 00:23:46,204 meters. That's 314 00:23:46,204 --> 00:23:49,733 a given. And the frequency is three 315 00:23:49,733 --> 00:23:53,884 thousand hertz. The wavelength therefore, 316 00:23:53,884 --> 00:24:00,421 lambda, equals V divided by the frequency, the speed of sound is 317 00:24:00,421 --> 00:24:06,648 about three hundred forty meters per second, divided by three 318 00:24:06,648 --> 00:24:11,629 thousand, is about oh point one one three meters. 319 00:24:11,629 --> 00:24:17,544 So the wavelength is about eleven point three centimeters. 320 00:24:17,544 --> 00:24:22,795 I can now calculate everyone who is sitting here 321 00:24:22,795 --> 00:24:26,518 fweet right in the middle through this whole plane will 322 00:24:26,518 --> 00:24:30,035 have a maximum of sound, and then when we go away at 323 00:24:30,035 --> 00:24:34,034 angle theta, some will again have maxima, and we go further 324 00:24:34,034 --> 00:24:37,551 away theta, again maxima, and in between will be the 325 00:24:37,551 --> 00:24:39,826 minima. And I'm going to calculate 326 00:24:39,826 --> 00:24:42,239 where they fall in the lecture hall. 327 00:24:42,239 --> 00:24:46,514 The first thing that I'm going to do is I'm going to give you N 328 00:24:46,514 --> 00:24:50,513 as in Nancy and calculate that angle theta 329 00:24:50,513 --> 00:24:53,509 of N and I will do it for the maxima. 330 00:24:53,509 --> 00:24:57,171 In other words, I'm going to use constructive 331 00:24:57,171 --> 00:25:01,998 interference and you see that the sine of theta N is lambda 332 00:25:01,998 --> 00:25:05,244 divided by D. That's the equation I use. 333 00:25:05,244 --> 00:25:07,907 When N is zero the angle is zero. 334 00:25:07,907 --> 00:25:11,569 That is zero angle. Everyone here will hear a 335 00:25:11,569 --> 00:25:13,4 maximum. When N is one, 336 00:25:13,4 --> 00:25:18,559 and you may want to check that at home, I find an angle of four 337 00:25:18,559 --> 00:25:23,072 point three degrees, and when N is two, 338 00:25:23,072 --> 00:25:27,611 the angle is about double that, is about eight point seven 339 00:25:27,611 --> 00:25:32,31 degrees, and when N is three, it should be close to thirteen 340 00:25:32,31 --> 00:25:36,611 degrees, thirteen point one. In case you take N is ten, 341 00:25:36,611 --> 00:25:39,796 so I skip a few, you get about forty-nine 342 00:25:39,796 --> 00:25:42,902 degrees. This is where the maximum fall. 343 00:25:42,902 --> 00:25:47,68 And so there's going to be a maximum here and then four point 344 00:25:47,68 --> 00:25:50,627 three degrees away is again a maximum. 345 00:25:50,627 --> 00:25:56,108 But surely we would like to know how far you in the 346 00:25:56,108 --> 00:26:01,405 audience will have to move in order to go from a maximum to a 347 00:26:01,405 --> 00:26:04,406 minimum. And so the way you have to 348 00:26:04,406 --> 00:26:09,703 think of this is that if I make here a picture of the lecture 349 00:26:09,703 --> 00:26:14,912 hall, if here are these two sources, you are at a distance L 350 00:26:14,912 --> 00:26:18,62 away from here. Some of you are five meters 351 00:26:18,62 --> 00:26:21,268 away. Some are ten meters away. 352 00:26:21,268 --> 00:26:27,529 Some are fifteen meters away, all the way in the back of the 353 00:26:27,529 --> 00:26:30,403 audience. And you want to know where 354 00:26:30,403 --> 00:26:34,51 you're going to hear the maxima. I call this X one, 355 00:26:34,51 --> 00:26:37,96 I call this X two, and I call this X three, 356 00:26:37,96 --> 00:26:41,903 and this is zero. So this is the meaning of theta 357 00:26:41,903 --> 00:26:44,367 one. And this is the meaning of 358 00:26:44,367 --> 00:26:48,556 theta three, and this angle here would be theta two. 359 00:26:48,556 --> 00:26:51,349 That's the meaning of these angles. 360 00:26:51,349 --> 00:26:55,455 And so I can calculate now how far 361 00:26:55,455 --> 00:27:02,495 you have to move depending upon what capital L is to hear to go 362 00:27:02,495 --> 00:27:07,491 from one maximum in sound to another maximum. 363 00:27:07,491 --> 00:27:14,303 And we raise the a little more. And so I will show you now s- 364 00:27:14,303 --> 00:27:20,094 some of the results for maxima. So I only go now for 365 00:27:20,094 --> 00:27:27,587 constructive interference. And I have done this for three 366 00:27:27,587 --> 00:27:31,323 different distances. Those of you who are five 367 00:27:31,323 --> 00:27:34,975 meters away from me, ten meters away from me, 368 00:27:34,975 --> 00:27:37,548 and fifteen meters away from me. 369 00:27:37,548 --> 00:27:41,699 And what you see on the left side is going to be X, 370 00:27:41,699 --> 00:27:46,43 that is the linear separation, and these, so these were in 371 00:27:46,43 --> 00:27:50,83 meters, forgive me but I will do these in centimeters. 372 00:27:50,83 --> 00:27:54,565 And this is X one, if you are five meters away 373 00:27:54,565 --> 00:27:59,462 from me, you will have -- I will put X one a 374 00:27:59,462 --> 00:28:04,339 little lower than I have it now, you will see shortly why I 375 00:28:04,339 --> 00:28:09,048 put it a little lower -- X one this is about thirty-eight 376 00:28:09,048 --> 00:28:12,496 centimeters. So the linear separation from 377 00:28:12,496 --> 00:28:16,111 one to the next is thirty-eight centimeters. 378 00:28:16,111 --> 00:28:19,811 And you're ten meters away, it's double that, 379 00:28:19,811 --> 00:28:23,427 that's no surprise, seventy-six centimeters. 380 00:28:23,427 --> 00:28:27,799 And if you're fifteen meters away it is a hundred and 381 00:28:27,799 --> 00:28:33,063 thirteen centimeters. And then X two which is the 382 00:28:33,063 --> 00:28:38,198 position where you have another maximum would be at seventy-six 383 00:28:38,198 --> 00:28:42,588 centimeters and it would be at a hundred and fifty-two 384 00:28:42,588 --> 00:28:47,392 centimeters and it would be at two hundred and twenty-eight 385 00:28:47,392 --> 00:28:51,451 centimeters if you're fifteen meters away from me. 386 00:28:51,451 --> 00:28:56,338 So the minima will fall almost exactly in between and so the 387 00:28:56,338 --> 00:29:01,887 minima where in an ideal case there is no sound at all, 388 00:29:01,887 --> 00:29:05,095 sound plus sound gives silence, think about it, 389 00:29:05,095 --> 00:29:08,931 sound plus sound will give silence, will be when you are 390 00:29:08,931 --> 00:29:12,068 roughly at nineteen centimeters, half of this, 391 00:29:12,068 --> 00:29:14,649 this will be thirty-eight centimeters. 392 00:29:14,649 --> 00:29:17,717 Half of this, and here will be something like 393 00:29:17,717 --> 00:29:21,203 fifty-seven centimeters. And you can calculate what 394 00:29:21,203 --> 00:29:24,341 these values are, they are exactly in between. 395 00:29:24,341 --> 00:29:28,525 And so the conclusion is that if you're five meters away from 396 00:29:28,525 --> 00:29:32,152 me and you're near the center line, 397 00:29:32,152 --> 00:29:35,654 but you can also be a little bit in this direction, 398 00:29:35,654 --> 00:29:38,457 that the separation between bright sound, 399 00:29:38,457 --> 00:29:42,59 loud sound, which is always at zero of course in the middle, 400 00:29:42,59 --> 00:29:44,972 to silence is nineteen centimeters. 401 00:29:44,972 --> 00:29:48,755 And then you move another nineteen centimeters and then 402 00:29:48,755 --> 00:29:52,188 you hear loud sound. If you are however ten meters 403 00:29:52,188 --> 00:29:54,64 away from me, just past the cameras, 404 00:29:54,64 --> 00:29:58,213 then you have to move thirty-eight centimeters to go 405 00:29:58,213 --> 00:30:02,067 from loud sound to silence. And if you're all the way in 406 00:30:02,067 --> 00:30:05,64 the audience, in the back of the 407 00:30:05,64 --> 00:30:09,41 audience, it's more like sixty centimeters. 408 00:30:09,41 --> 00:30:13,629 And this is what we're going to do now together. 409 00:30:13,629 --> 00:30:19,104 I want you all to stand up and I'm going to make you listen to 410 00:30:19,104 --> 00:30:23,682 three thousand hertz. And what I want you to do when 411 00:30:23,682 --> 00:30:28,978 I turn on the two loudspeakers, I want you to move your head 412 00:30:28,978 --> 00:30:33,556 very slowly and try to find locations where you hear 413 00:30:33,556 --> 00:30:37,777 silence. The position of silence is 414 00:30:37,777 --> 00:30:41,73 extremely well-defined, so don't go too fast, 415 00:30:41,73 --> 00:30:45,324 you miss it, also keep in mind that there 416 00:30:45,324 --> 00:30:49,187 are reflections of the sound from the walls, 417 00:30:49,187 --> 00:30:54,038 and from the blackboard, and so the pattern that I have 418 00:30:54,038 --> 00:30:59,249 calculated here is not perfect. But you will see that there 419 00:30:59,249 --> 00:31:04,01 will be locations where sound plus sound will give you 420 00:31:04,01 --> 00:31:06,615 silence. [tone] 421 00:31:06,615 --> 00:31:12,125 [tone] You are a couple of lousy scientists. 422 00:31:12,125 --> 00:31:17,635 [tone] You are a couple of lousy scientists. 423 00:31:17,635 --> 00:31:25,707 If the separation between a lot of sound and silence is nineteen 424 00:31:25,707 --> 00:31:32,498 centimeters, that's about the separation of your ears, 425 00:31:32,498 --> 00:31:37,111 you dummies, so one ear could be at a 426 00:31:37,111 --> 00:31:42,749 maximum, the other ear could be at a minimum, 427 00:31:42,749 --> 00:31:49,284 so at least close one ear. [laughter] [tone] Go very 428 00:31:49,284 --> 00:31:53 slowly. [tone] 429 00:31:53 --> 00:31:57,404 Who has found clear location where [tone] the sound is nearly 430 00:31:57,404 --> 00:32:01,148 zero [tone] or practically zero? [tone] Most of you. 431 00:32:01,148 --> 00:32:05,038 And you certainly can hear if you move that there's an 432 00:32:05,038 --> 00:32:08,342 enormous difference [tone] in sound intensity. 433 00:32:08,342 --> 00:32:12,232 [tone] [tone] So again, who has found locations [tone] 434 00:32:12,232 --> 00:32:15,976 whereby you clearly say this is practically silence? 435 00:32:15,976 --> 00:32:19,426 [tone] Ah, you see them all the way in the back, 436 00:32:19,426 --> 00:32:23,243 [tone] [tone] and the separation, how far you have to 437 00:32:23,243 --> 00:32:28,434 move, depends on how far you are away from me. 438 00:32:28,434 --> 00:32:32,539 Sit down again. [audience noise] Young was a 439 00:32:32,539 --> 00:32:38,076 sound engineer and as a sound engineer he was very familiar 440 00:32:38,076 --> 00:32:43,709 with the interference of sound. He knew that sound and sound 441 00:32:43,709 --> 00:32:48,005 can make silence. And so in eighteen oh one he 442 00:32:48,005 --> 00:32:54,115 demonstrated in a convincing way that light plus light can create 443 00:32:54,115 --> 00:32:57,743 darkness. That would be the nail in the 444 00:32:57,743 --> 00:33:01,944 coffin that would demonstrate uniquely 445 00:33:01,944 --> 00:33:05,787 that light are indeed waves, and there was still this 446 00:33:05,787 --> 00:33:10,369 controversy between Huygens and Newton as you perhaps remember. 447 00:33:10,369 --> 00:33:14,286 Newton wanted light to be particles but Huygens wanted 448 00:33:14,286 --> 00:33:17,76 them to be waves. And the way that Young did his 449 00:33:17,76 --> 00:33:20,79 experiment is as follows. He had a screen, 450 00:33:20,79 --> 00:33:24,781 don't think of it as this big, you're talking now about 451 00:33:24,781 --> 00:33:28,624 extraordinarily small dimensions, you will understand 452 00:33:28,624 --> 00:33:32,888 shortly how small, and in this screen are 453 00:33:32,888 --> 00:33:34,923 two openings, two pinholes, 454 00:33:34,923 --> 00:33:39,617 and light is coming from the left and think of light as being 455 00:33:39,617 --> 00:33:42,825 plain waves. They reach these two openings 456 00:33:42,825 --> 00:33:47,832 and these two openings according to Huygens will produce circular 457 00:33:47,832 --> 00:33:51,979 waves, spherical waves of course, three-dimensionally. 458 00:33:51,979 --> 00:33:56,751 These openings become Huygens sources and spherical waves will 459 00:33:56,751 --> 00:34:02,776 propagate out in this direction. And so now we have exactly the 460 00:34:02,776 --> 00:34:05,483 situation that we had with our sound. 461 00:34:05,483 --> 00:34:09,845 Now if all works well there should be directions theta away 462 00:34:09,845 --> 00:34:14,131 from this line where you see darkness and other directions 463 00:34:14,131 --> 00:34:18,418 where you see bright light. And we are going to do it in a 464 00:34:18,418 --> 00:34:23,005 way, we have the luxury of laser beams, so we have very strong 465 00:34:23,005 --> 00:34:25,938 light sources, which Young did not have. 466 00:34:25,938 --> 00:34:29,773 The way we are going to do it, we have a -- a slide, 467 00:34:29,773 --> 00:34:34,987 which is completely black, but with a razor blade two 468 00:34:34,987 --> 00:34:39,739 lines have been drawn on it. And so I will draw these lines 469 00:34:39,739 --> 00:34:43,262 as white lines. But they're really openings. 470 00:34:43,262 --> 00:34:47,85 And there is another one here. And the separation between 471 00:34:47,85 --> 00:34:52,111 these lines D is oh point oh eight eight millimeters, 472 00:34:52,111 --> 00:34:54,815 less than a tenth of a millimeter. 473 00:34:54,815 --> 00:34:59,239 When you look at them you cannot even see that they are 474 00:34:59,239 --> 00:35:02,434 two lines. Our laser beam has a diameter 475 00:35:02,434 --> 00:35:07,911 of about three millimeters, which is thirty times larger 476 00:35:07,911 --> 00:35:10,631 than this distance, thirty times larger. 477 00:35:10,631 --> 00:35:14,886 So what I'm going to show you now that this is our laser beam, 478 00:35:14,886 --> 00:35:18,025 is not to scale, the laser beam is much larger 479 00:35:18,025 --> 00:35:20,397 than that. And so the light will go 480 00:35:20,397 --> 00:35:24,443 through some parts of these slots, as far as our laser beam 481 00:35:24,443 --> 00:35:28,698 reaches, and we are now capable of predicting when we're going 482 00:35:28,698 --> 00:35:32,046 to project it there, here are the two slots which 483 00:35:32,046 --> 00:35:35,882 are like so, and so you're going to get 484 00:35:35,882 --> 00:35:39,998 interference patterns in these directions theta, 485 00:35:39,998 --> 00:35:45,252 and we can calculate what the position X is going to be there 486 00:35:45,252 --> 00:35:49,367 between the maxima. And so if that is the screen 487 00:35:49,367 --> 00:35:54,446 and if this is X equals zero, and if this is X one and this 488 00:35:54,446 --> 00:35:58,999 is X two, and of course the whole thing is symmetric, 489 00:35:58,999 --> 00:36:02,764 you can always go in the opposite direction, 490 00:36:02,764 --> 00:36:08,28 you can now calculate, and you have all the tools, 491 00:36:08,28 --> 00:36:11,543 I did it for you in great detail using sound, 492 00:36:11,543 --> 00:36:15,325 but you have all the tools to do it now, you know D, 493 00:36:15,325 --> 00:36:19,329 I'm going to tell you what lambda is, it's six thousand 494 00:36:19,329 --> 00:36:23,778 three hundred and twenty-eight Angstroms, and one Angstrom is 495 00:36:23,778 --> 00:36:27,93 ten to the minus ten meters, so you can calculate all the 496 00:36:27,93 --> 00:36:32,453 direction theta for which there are maxima and for which there 497 00:36:32,453 --> 00:36:35,419 are minima. Minima means light plus light 498 00:36:35,419 --> 00:36:37,941 gives darkness, an amazing concept. 499 00:36:37,941 --> 00:36:42,619 And you can then if you know the distance from 500 00:36:42,619 --> 00:36:45,544 here to the screen, which is capital L, 501 00:36:45,544 --> 00:36:49,777 you can calculate what the separation is as we see it on 502 00:36:49,777 --> 00:36:53,856 the screen, and L is roughly ten meters, maybe eleven, 503 00:36:53,856 --> 00:36:57,551 but that's not so important. And so I calculated, 504 00:36:57,551 --> 00:37:02,399 and you can confirm that -- and you should confirm that that the 505 00:37:02,399 --> 00:37:05,709 angle theta one, I will only calculate theta 506 00:37:05,709 --> 00:37:10,25 one, which is the angle then to this point, theta zero is of 507 00:37:10,25 --> 00:37:14,888 course always zero, right, that's the easiest, 508 00:37:14,888 --> 00:37:18,627 I find that theta one is oh point four one degrees, 509 00:37:18,627 --> 00:37:22,441 that is for maximum, and that means that X one given 510 00:37:22,441 --> 00:37:26,479 the distance of ten meters then becomes seven point two 511 00:37:26,479 --> 00:37:29,395 centimeters. So from here to here on the 512 00:37:29,395 --> 00:37:33,957 screen, from maximum to maximum, will be about seven point two 513 00:37:33,957 --> 00:37:38,518 centimeters, and from here to here will then of course also be 514 00:37:38,518 --> 00:37:42,631 about seven two centimeters, and in between you will see 515 00:37:42,631 --> 00:37:48,883 darkness. The light from the two sources, 516 00:37:48,883 --> 00:37:57,365 a hundred eighty degrees out of phase, and that will give you 517 00:37:57,365 --> 00:38:02,172 darkness. Let me turn on the laser. 518 00:38:02,172 --> 00:38:09,24 And turn off the lights. Make sure I have my -- OK. 519 00:38:09,24 --> 00:38:15,602 And there you see it. There you see a maximum, 520 00:38:15,602 --> 00:38:21,257 darkness, a maximum, darkness, 521 00:38:21,257 --> 00:38:23,698 a maximum, darkness, and so on. 522 00:38:23,698 --> 00:38:28,255 And the separation if I didn't make a mistake between the 523 00:38:28,255 --> 00:38:31,51 maxima is indeed about seven centimeters. 524 00:38:31,51 --> 00:38:35,904 Imagine what an incredible moment this is in your life, 525 00:38:35,904 --> 00:38:40,054 that you actually see that light plus light can make 526 00:38:40,054 --> 00:38:43,227 darkness. So the waves go simultaneously 527 00:38:43,227 --> 00:38:47,296 through both openings, and each opening acts like a 528 00:38:47,296 --> 00:38:51,934 Huygens source, and the net result is that 529 00:38:51,934 --> 00:38:55,86 these two waves arrive there on the screen a hundred eighty 530 00:38:55,86 --> 00:38:59,176 degrees out of phase at the locations of darkness. 531 00:38:59,176 --> 00:39:02,83 The censor is of course that they have exactly the same 532 00:39:02,83 --> 00:39:06,959 frequency which is what they do, because we have one laser gun 533 00:39:06,959 --> 00:39:10,207 going in and so the wave goes through both slots. 534 00:39:10,207 --> 00:39:13,185 So we're guaranteed, and that was the secret, 535 00:39:13,185 --> 00:39:17,042 that Young understood you're guaranteed that the waves are 536 00:39:17,042 --> 00:39:21,644 not only the same frequency but they're even in phase 537 00:39:21,644 --> 00:39:24,692 because they both go through to both slots. 538 00:39:24,692 --> 00:39:29,191 Now if you look very carefully here you will see of course that 539 00:39:29,191 --> 00:39:32,166 these maxima don't have the same strength. 540 00:39:32,166 --> 00:39:35,866 We will understand next lecture why that's the case. 541 00:39:35,866 --> 00:39:40,22 They would have very closely the same strength if the opening 542 00:39:40,22 --> 00:39:44,428 where we scratched out the black on the slide was much much 543 00:39:44,428 --> 00:39:47,984 smaller than the separation between the two slots, 544 00:39:47,984 --> 00:39:51,612 so to speak. And that separation is 545 00:39:51,612 --> 00:39:53,787 point oh eight eight millimeter. 546 00:39:53,787 --> 00:39:56,944 If we make the openings much narrower, indeed, 547 00:39:56,944 --> 00:39:59,961 the light intensities would be more uniform, 548 00:39:59,961 --> 00:40:03,68 each maximum would be approximately the same strength, 549 00:40:03,68 --> 00:40:06,627 but then very little light will go through. 550 00:40:06,627 --> 00:40:10,275 And so it's a tradeoff. And the moment you make these 551 00:40:10,275 --> 00:40:12,31 two openings, these two slots, 552 00:40:12,31 --> 00:40:15,678 larger and larger, you will understand Friday why 553 00:40:15,678 --> 00:40:18,695 then the light intensities are not the same, 554 00:40:18,695 --> 00:40:22,765 why the light intensity is a maximum at the center and then 555 00:40:22,765 --> 00:40:26,283 falls off near the edge. 556 00:40:26,283 --> 00:40:29,412 As you see. It's a maximum here, 557 00:40:29,412 --> 00:40:33,954 and then the light intensities become smaller. 558 00:40:33,954 --> 00:40:39,303 I've shown you now the interference pattern with sound 559 00:40:39,303 --> 00:40:44,754 and for red laser light, but imagine now that I did the 560 00:40:44,754 --> 00:40:49,699 same with white light. The situation would be very 561 00:40:49,699 --> 00:40:55,755 different, and maybe even disappointing for you. 562 00:40:55,755 --> 00:40:59,978 Let this be the location on the screen. 563 00:40:59,978 --> 00:41:04,757 So we -- we have X here, and here X is zero. 564 00:41:04,757 --> 00:41:10,313 And I want to know where the maxima are in the red, 565 00:41:10,313 --> 00:41:15,869 well that's very easy, there will be a maximum here 566 00:41:15,869 --> 00:41:21,314 when this position is L times lambda divided by D. 567 00:41:21,314 --> 00:41:28,034 This is when Nancy is one. And there will also be a 568 00:41:28,034 --> 00:41:34,231 maximum here when we have two L times lambda divided by D. 569 00:41:34,231 --> 00:41:40,536 And of course there will be one on this side same distance. 570 00:41:40,536 --> 00:41:45,971 And there will be one here, this is when N is zero. 571 00:41:45,971 --> 00:41:47,928 N is one. N is two. 572 00:41:47,928 --> 00:41:54,015 The red light will have maxima. How about the blue light? 573 00:41:54,015 --> 00:42:00,972 The blue light will have maxima here, where L lambda 574 00:42:00,972 --> 00:42:03,633 divided by D, but lambda is different, 575 00:42:03,633 --> 00:42:05,934 lambda for blue light is smaller. 576 00:42:05,934 --> 00:42:10,104 Substantially smaller than red light, so the maximum of the 577 00:42:10,104 --> 00:42:13,555 blue will fall here, the maximum of the blue will 578 00:42:13,555 --> 00:42:16,863 always fall at N equals zero together with red, 579 00:42:16,863 --> 00:42:20,386 and then Nancy equals two the blue will fall here, 580 00:42:20,386 --> 00:42:23,478 so this is Nancy two, Nancy one, Nancy zero. 581 00:42:23,478 --> 00:42:26,426 And here Nancy zero, Nancy one, Nancy two. 582 00:42:26,426 --> 00:42:30,812 And so the red and the blue and therefore all the other colors 583 00:42:30,812 --> 00:42:36,252 live a life of their own. They don't talk to each other. 584 00:42:36,252 --> 00:42:40,501 They come in with their own separation in terms of angles 585 00:42:40,501 --> 00:42:44,827 and in terms of locations X. That's the reason why I chose 586 00:42:44,827 --> 00:42:47,938 one and only one frequency with the sound. 587 00:42:47,938 --> 00:42:52,339 Because if I had exposed you to many different frequencies, 588 00:42:52,339 --> 00:42:56,513 many different wavelengths, then the location of silence 589 00:42:56,513 --> 00:43:00,762 for one wavelength is not the location of silence for the 590 00:43:00,762 --> 00:43:05,627 other wavelength. And so the experiment would not 591 00:43:05,627 --> 00:43:08,788 have worked. And that's why it worked so 592 00:43:08,788 --> 00:43:11,544 well with the laser, the red laser, 593 00:43:11,544 --> 00:43:15,84 which is practically one wavelength, and so the minima 594 00:43:15,84 --> 00:43:19,164 and the maxima are extremely well-defined. 595 00:43:19,164 --> 00:43:22,892 If we had done the experiment with white light, 596 00:43:22,892 --> 00:43:27,026 it wouldn't have been so impressive, and on the next 597 00:43:27,026 --> 00:43:30,998 slide I show you what you w- would have seen then. 598 00:43:30,998 --> 00:43:35,132 This is what white light would have 599 00:43:35,132 --> 00:43:38,295 done, this is a two-slit interference pattern. 600 00:43:38,295 --> 00:43:40,967 This is what red light would have done. 601 00:43:40,967 --> 00:43:44,201 Red light is a narrow bandwidth of wavelengths, 602 00:43:44,201 --> 00:43:48,138 well-defined black lines, light plus light give darkness, 603 00:43:48,138 --> 00:43:51,513 well-defined maxima, and the blue notice that the 604 00:43:51,513 --> 00:43:55,379 separation between the dark lines and therefore also the 605 00:43:55,379 --> 00:43:59,598 separation between the bright lines is substantially smaller. 606 00:43:59,598 --> 00:44:03,394 Because blue light has a wavelength of about forty-five 607 00:44:03,394 --> 00:44:07,737 hundred Angstroms and red light roughly 608 00:44:07,737 --> 00:44:11,764 sixty-five hundred. So there's a big difference. 609 00:44:11,764 --> 00:44:16,732 And so white light would then give you the superposition of 610 00:44:16,732 --> 00:44:20,673 all these colors, and so you don't really get a 611 00:44:20,673 --> 00:44:25,899 very nice interference pattern of dark areas and bright areas, 612 00:44:25,899 --> 00:44:31,21 because all the colors begin to overlap and each live a life of 613 00:44:31,21 --> 00:44:34,466 their own. What I can do with sound and 614 00:44:34,466 --> 00:44:40,719 what I did with water and what I have done with laser light 615 00:44:40,719 --> 00:44:44,649 I can also do with radio electromagnetic waves. 616 00:44:44,649 --> 00:44:48,921 With radar we have a ten-gigahertz transmitter here 617 00:44:48,921 --> 00:44:52,338 that we have used earlier in this course. 618 00:44:52,338 --> 00:44:57,379 And so I will now show you that with radar you can also show 619 00:44:57,379 --> 00:45:02,419 interference patterns and the calculation that you see there 620 00:45:02,419 --> 00:45:07,203 are absolutely identical. The only thing I want to remind 621 00:45:07,203 --> 00:45:11,646 you of, that the approximation when you 622 00:45:11,646 --> 00:45:15,841 know capital L that the tangent theta is roughly the same as the 623 00:45:15,841 --> 00:45:18,705 sine of theta is only true for small angles. 624 00:45:18,705 --> 00:45:21,435 Five degrees is fine, ten degrees is fine, 625 00:45:21,435 --> 00:45:25,497 but by the time that you reach fifty, sixty or seventy degrees 626 00:45:25,497 --> 00:45:29,626 that approximation is not true. So then you really have to take 627 00:45:29,626 --> 00:45:32,956 the tangent of theta. That's no problem because you 628 00:45:32,956 --> 00:45:36,552 first calculate what theta is, because that equation is 629 00:45:36,552 --> 00:45:41,148 correct, and then you can calculate always where X is, 630 00:45:41,148 --> 00:45:45,435 but then you use the tangent and not the sine. 631 00:45:45,435 --> 00:45:50,676 So these are approximations which hold for small angles. 632 00:45:50,676 --> 00:45:55,63 And so if now we look at a ten-gigahertz transmitter, 633 00:45:55,63 --> 00:45:58,965 that means we have two transmitters, 634 00:45:58,965 --> 00:46:03,443 one here and one here. And their separation D is 635 00:46:03,443 --> 00:46:07,54 twenty-three centimeters. You see them here. 636 00:46:07,54 --> 00:46:13,447 This is where they are. Here's one and here's the 637 00:46:13,447 --> 00:46:16,77 other, twenty-three centimeters apart. 638 00:46:16,77 --> 00:46:21,438 At ten gigahertz the wavelength is three centimeters. 639 00:46:21,438 --> 00:46:25,838 You can confirm that. The speed is speed of light. 640 00:46:25,838 --> 00:46:30,237 Lambda is the speed of light divided by frequency. 641 00:46:30,237 --> 00:46:35,624 That gives you the wavelength. And we have here at a distance 642 00:46:35,624 --> 00:46:40,832 L which is a hundred and twenty centimeters, we have here a 643 00:46:40,832 --> 00:46:46,237 receiver and a track, so this is X equals zero and 644 00:46:46,237 --> 00:46:51,531 here we can move it along X and so you can calculate now at what 645 00:46:51,531 --> 00:46:56,32 angles seen from this point there will be a maximum there. 646 00:46:56,32 --> 00:47:00,353 Theta zero is obvious. Right here there will be a 647 00:47:00,353 --> 00:47:02,202 maximum. The two waves, 648 00:47:02,202 --> 00:47:07,243 the distance between them is zero, R two minus R one is zero. 649 00:47:07,243 --> 00:47:10,352 So they will constructively interfere. 650 00:47:10,352 --> 00:47:14,805 But there is another angle, theta one, for which again 651 00:47:14,805 --> 00:47:18,196 there will be constructive 652 00:47:18,196 --> 00:47:21,367 interference. And you can confirm that I 653 00:47:21,367 --> 00:47:25,107 found for these numbers that theta one is about 654 00:47:25,107 --> 00:47:29,01 seven-and-a-half degrees. This is now for maxima. 655 00:47:29,01 --> 00:47:33,401 And so roughly at an angle which is half that value you 656 00:47:33,401 --> 00:47:36,979 will find silence. Silence means that the two 657 00:47:36,979 --> 00:47:39,581 radio waves will kill each other. 658 00:47:39,581 --> 00:47:44,541 Essential for the maximum to be here is of course that the two 659 00:47:44,541 --> 00:47:49,984 transmitters are in phase. We could have rigged it up so 660 00:47:49,984 --> 00:47:54,177 that they were a hundred eighty degrees out of phase in which 661 00:47:54,177 --> 00:47:56,414 case there would be silence here. 662 00:47:56,414 --> 00:48:00,748 Silence in this case means that the radar would kill the radar. 663 00:48:00,748 --> 00:48:04,033 But I do use the word silence for a good reason, 664 00:48:04,033 --> 00:48:08,087 because the way we rigged this up is the same way we did it 665 00:48:08,087 --> 00:48:10,673 before. We modulate this ten-gigahertz 666 00:48:10,673 --> 00:48:13,539 signal with a thousand-hertz audio signal. 667 00:48:13,539 --> 00:48:15,846 We call that amplitude modulation. 668 00:48:15,846 --> 00:48:20,261 And the receiver which is here receives the 669 00:48:20,261 --> 00:48:25,121 ten-gigahertz radiation which is modulated at a thousand hertz. 670 00:48:25,121 --> 00:48:29,51 We feed it to an amplifier. We demodulate it and you will 671 00:48:29,51 --> 00:48:33,43 hear the thousand hertz. And so we can also move it 672 00:48:33,43 --> 00:48:38,055 along this track here and find the location X one whereby we 673 00:48:38,055 --> 00:48:41,347 have our first maximum apart from the zero. 674 00:48:41,347 --> 00:48:45,893 And I found that that is very roughly at fifteen point five 675 00:48:45,893 --> 00:48:48,95 centimeters. And you should confirm that 676 00:48:48,95 --> 00:48:52,102 using those equations. 677 00:48:52,102 --> 00:48:57,091 Equations are the same. Whether you deal with sound or 678 00:48:57,091 --> 00:49:02,55 with red laser light or with gigahertz makes no difference. 679 00:49:02,55 --> 00:49:06,597 And so let's turn now to this demonstration. 680 00:49:06,597 --> 00:49:11,962 I will turn on the -- the two transmitters and here is the 681 00:49:11,962 --> 00:49:16,198 receiver which is exactly at angle theta zero. 682 00:49:16,198 --> 00:49:20,621 So there's a maximum. I'm now going to close one 683 00:49:20,621 --> 00:49:25,001 transmitter, [tone] put my hand in front. 684 00:49:25,001 --> 00:49:28,798 [tone] And you think about what will be the reduction of 685 00:49:28,798 --> 00:49:32,249 intensity of the sound here. [tone] If I close one, 686 00:49:32,249 --> 00:49:35,907 it's substantially down. You may think that it is down 687 00:49:35,907 --> 00:49:38,875 by a factor of two. Because we have only one 688 00:49:38,875 --> 00:49:41,636 transmitter instead of two. You're wrong. 689 00:49:41,636 --> 00:49:44,328 If you think that, you do not understand 690 00:49:44,328 --> 00:49:47,227 interference. It is four times lower when I 691 00:49:47,227 --> 00:49:50,678 hold my hands over one. Figure it out for yourself. 692 00:49:50,678 --> 00:49:54,199 I'll test you on the final to see 693 00:49:54,199 --> 00:49:56,646 whether you really understood that. 694 00:49:56,646 --> 00:50:01,109 So now the sound is four times larger than when I cover one up. 695 00:50:01,109 --> 00:50:05,284 I'll cover the other one up. It's down by a factor of four. 696 00:50:05,284 --> 00:50:09,459 [tone] I can cover this one up and then you hear nothing of 697 00:50:09,459 --> 00:50:12,194 course. [tone] Now I will move this one 698 00:50:12,194 --> 00:50:16,153 [tone] to the location [tone] where there is destructive 699 00:50:16,153 --> 00:50:19,105 interference, [tone] which should be about 700 00:50:19,105 --> 00:50:21,768 half of fifteen point six centimeters. 701 00:50:21,768 --> 00:50:26,238 [tone] [cough] Maybe you have good ears but I 702 00:50:26,238 --> 00:50:29,893 hear nothing anymore. Now I go through it and find 703 00:50:29,893 --> 00:50:34,146 the maximum, which is about fifteen point six centimeters. 704 00:50:34,146 --> 00:50:37,428 [tone] Here it is. [tone] And the other side, 705 00:50:37,428 --> 00:50:41,532 [tone] here's the maximum at center, so here should be a 706 00:50:41,532 --> 00:50:45,113 minimum, there it is, and I go to the other side, 707 00:50:45,113 --> 00:50:48,768 there should be a maximum. [tone] And there it is. 708 00:50:48,768 --> 00:50:54,215 [tone] So what I have shown you today is I've shown you the 709 00:50:54,215 --> 00:50:58,619 interference pattern of sound, of water, of red laser light, 710 00:50:58,619 --> 00:51:03,024 of radar, and at the very least I hope I've convinced you as 711 00:51:03,024 --> 00:51:07,354 Young convinced the world in eighteen oh one that light are 712 00:51:07,354 --> 00:51:10,116 waves. And that means that Huygens was 713 00:51:10,116 --> 00:51:14,147 right and Newton was wrong. Now that should perhaps not 714 00:51:14,147 --> 51:19 surprise you because Huygens was Dutch.