1 00:00:00 --> 00:00:00,641 2 00:00:00,641 --> 00:00:05,285 All of you have looked at rainbows, but very few of you 3 00:00:05,285 --> 00:00:09,327 have ever seen one. Looking at something is very 4 00:00:09,327 --> 00:00:13,971 different from seeing it. And today I will make you see 5 00:00:13,971 --> 00:00:19,046 the rainbow in a way that goes way beyond the beauty that we 6 00:00:19,046 --> 00:00:22,916 can all experience, a way that you will always 7 00:00:22,916 --> 00:00:25,84 remember. And I would like to start 8 00:00:25,84 --> 00:00:30,312 asking you fifteen perhaps simple questions about the 9 00:00:30,312 --> 00:00:34,308 rainbow. The first question then is 10 00:00:34,308 --> 00:00:39,041 would any one of you remember if you see a bow whether the red 11 00:00:39,041 --> 00:00:43,541 color is outside or whether the -- the red color is inside? 12 00:00:43,541 --> 00:00:47,033 And then I wonder about the radius of the bow. 13 00:00:47,033 --> 00:00:50,757 If this is a bow in the sky, something like this, 14 00:00:50,757 --> 00:00:54,637 here is the horizon, it's clearly a perfect circle, 15 00:00:54,637 --> 00:00:58,361 and so the perfect circle has somewhere a center. 16 00:00:58,361 --> 00:01:01,62 And so that means there must be a radius R. 17 00:01:01,62 --> 00:01:06,14 You can measure that radius in terms of how many 18 00:01:06,14 --> 00:01:08,825 degrees and so what is roughly that radius. 19 00:01:08,825 --> 00:01:11,765 You've never measured it but is it ten degrees, 20 00:01:11,765 --> 00:01:13,491 is it twenty, thirty, fifty, 21 00:01:13,491 --> 00:01:15,281 sixty? The length of the bow. 22 00:01:15,281 --> 00:01:18,414 Is there a difference, do you sometimes see a very 23 00:01:18,414 --> 00:01:20,715 long bow, sometimes a very short one? 24 00:01:20,715 --> 00:01:23,848 What is the width of the bow? You see colors here. 25 00:01:23,848 --> 00:01:26,597 How wide is that strip of colors in degrees? 26 00:01:26,597 --> 00:01:29,601 Perhaps some of you have noticed that there is a 27 00:01:29,601 --> 00:01:33,117 difference in light intensity between inside the bow and 28 00:01:33,117 --> 00:01:37,337 outside the bow. Maybe you've never seen it, 29 00:01:37,337 --> 00:01:40,366 and if there is a difference where is it brighter, 30 00:01:40,366 --> 00:01:42,406 inside the bow or outside the bow? 31 00:01:42,406 --> 00:01:44,817 What time of the day would you see bows? 32 00:01:44,817 --> 00:01:48,279 Would you see rainbows in the north, east, south or west? 33 00:01:48,279 --> 00:01:50,752 Is there perhaps a second bow in the sky? 34 00:01:50,752 --> 00:01:54,338 And if there is a second one, where should you look for the 35 00:01:54,338 --> 00:01:56,749 second bow? And if there is a second one 36 00:01:56,749 --> 00:01:59,469 what is the color sequence of the second bow? 37 00:01:59,469 --> 00:02:02,746 Is the red on the outside or is the red on the inside? 38 00:02:02,746 --> 00:02:06,208 And then you can ask the same question, what would be the 39 00:02:06,208 --> 00:02:09,074 radius of the second bow? 40 00:02:09,074 --> 00:02:12,065 And what would be the width of the second bow? 41 00:02:12,065 --> 00:02:15,256 All these first twelve questions in principle you 42 00:02:15,256 --> 00:02:18,979 should have been able to answer if you really have seen a 43 00:02:18,979 --> 00:02:20,973 rainbow. The last three is more 44 00:02:20,973 --> 00:02:23,499 difficult. The question is are the bows 45 00:02:23,499 --> 00:02:25,892 polarized? In what direction are they 46 00:02:25,892 --> 00:02:28,484 polarized? And are they weakly polarized 47 00:02:28,484 --> 00:02:32,539 or are they strongly polarized? Who knows the answer to twelve 48 00:02:32,539 --> 00:02:35,198 questions, to the first twelve questions? 49 00:02:35,198 --> 00:02:38,722 Who knows the answer to more than 50 00:02:38,722 --> 00:02:42,569 ten? Who knows the answer to nine? 51 00:02:42,569 --> 00:02:43,967 Eight? Seven? 52 00:02:43,967 --> 00:02:45,017 Six? Five? 53 00:02:45,017 --> 00:02:48,397 Four? Do I see a hand at four? 54 00:02:48,397 --> 00:02:52,011 Good for you. Five, four, three? 55 00:02:52,011 --> 00:02:55,742 Three, good, that's already good. 56 00:02:55,742 --> 00:02:56,674 Two? One? 57 00:02:56,674 --> 00:03:00,405 And who knows the answer to zero? 58 00:03:00,405 --> 00:03:02,503 Most of you, right? 59 00:03:02,503 --> 00:03:09,032 I haven't seen a lot of hands though. 60 00:03:09,032 --> 00:03:11,59 All right. So I've made my point. 61 00:03:11,59 --> 00:03:16,307 You've looked at rainbows but you've really never seen them. 62 00:03:16,307 --> 00:03:19,505 And I'm going to make you see them today. 63 00:03:19,505 --> 00:03:23,983 What you see here on the blackboard is one drop of water. 64 00:03:23,983 --> 00:03:27,421 I put the sun for simplicity at the horizon. 65 00:03:27,421 --> 00:03:31,418 Later I will put it a little bit higher in the sky. 66 00:03:31,418 --> 00:03:34,377 Light from the sun hits this raindrop. 67 00:03:34,377 --> 00:03:39,174 I've only drawn one narrow beam which hits the raindrop right 68 00:03:39,174 --> 00:03:43,412 there. And you see here the angle of 69 00:03:43,412 --> 00:03:47,16 incidence, which with Snell's law we call theta one. 70 00:03:47,16 --> 00:03:50,247 I call it I here because it's nicer for me, 71 00:03:50,247 --> 00:03:53,335 more descriptive, it means incidence angle. 72 00:03:53,335 --> 00:03:57,745 Right at that point A some of the light will be reflected and 73 00:03:57,745 --> 00:04:00,685 some of the light will go into the water. 74 00:04:00,685 --> 00:04:04,508 We call that refraction. And Snell's law will tell me 75 00:04:04,508 --> 00:04:07,668 this angle R. Whatever goes in there reaches 76 00:04:07,668 --> 00:04:11,564 point B where there is a transition back to air and so 77 00:04:11,564 --> 00:04:17,591 some of that light will come out here and some of that light will 78 00:04:17,591 --> 00:04:21,161 be reflected inside. And then when it reaches point 79 00:04:21,161 --> 00:04:24,517 C again there is a transition from water to air. 80 00:04:24,517 --> 00:04:28,301 Some of that light will be reflected inside the water. 81 00:04:28,301 --> 00:04:32,443 And some of it will come out. And as far as the geometry is 82 00:04:32,443 --> 00:04:36,441 concerned, if this angle is R, then this angle is also R, 83 00:04:36,441 --> 00:04:38,868 this is also R, and this is also R. 84 00:04:38,868 --> 00:04:42,724 And the angle here is I. That follows from Snell's law, 85 00:04:42,724 --> 00:04:46,865 and I'll leave you with that. Notice that the light came in 86 00:04:46,865 --> 00:04:51,269 like this but it comes back like 87 00:04:51,269 --> 00:04:54,819 this. So the direction has changed 88 00:04:54,819 --> 00:04:59,983 over the angle delta. And the angle delta is very 89 00:04:59,983 --> 00:05:03,963 easy to calculate in terms of I and R. 90 00:05:03,963 --> 00:05:10,095 Delta is a hundred eighty degrees plus two I minus four R. 91 00:05:10,095 --> 00:05:13,537 I want you to check that at home. 92 00:05:13,537 --> 00:05:17,948 The four Rs come in here. One, two, three, 93 00:05:17,948 --> 00:05:23,541 four, and the two I's come in here and 94 00:05:23,541 --> 00:05:26,245 there. If now I think of all possible 95 00:05:26,245 --> 00:05:30,074 narrow beams of light that can strike this raindrop, 96 00:05:30,074 --> 00:05:34,654 one that would strike it here would have an I of zero degrees. 97 00:05:34,654 --> 00:05:38,859 And then here would be ten degrees and twenty degrees and 98 00:05:38,859 --> 00:05:42,388 thirty and forty. And the largest value for I is 99 00:05:42,388 --> 00:05:46,292 when the light strikes here, would be ninety degrees. 100 00:05:46,292 --> 00:05:49,896 And so I can calculate for all these values of I, 101 00:05:49,896 --> 00:05:55,339 which obviously all of them occur, sunlight strikes this 102 00:05:55,339 --> 00:05:59,264 raindrop, and all these angles for I are present. 103 00:05:59,264 --> 00:06:04,333 So I can calculate now for all these angles of I what the value 104 00:06:04,333 --> 00:06:08,176 is for R and then I can calculate what delta is. 105 00:06:08,176 --> 00:06:12,51 R follows from Snell's law and delta follows from this 106 00:06:12,51 --> 00:06:16,925 geometric relationship. And what you will find now very 107 00:06:16,925 --> 00:06:21,095 much to your surprise, that there is a minimum value 108 00:06:21,095 --> 00:06:26,983 for delta which is about hundred and thirty-eight degrees. 109 00:06:26,983 --> 00:06:31,971 That means this angle phi here has a maximum value which is 110 00:06:31,971 --> 00:06:35,068 very roughly about forty-two degrees. 111 00:06:35,068 --> 00:06:37,82 And I will show you some numbers. 112 00:06:37,82 --> 00:06:42,378 You can download this, by the way, this is on the Web, 113 00:06:42,378 --> 00:06:47,281 under lecture supplements. Here all I have done I've taken 114 00:06:47,281 --> 00:06:52,098 I to be from zero to ninety degrees, all these angles are 115 00:06:52,098 --> 00:06:57,086 possible, with Snell's law, using an index of refraction of 116 00:06:57,086 --> 00:07:00,641 one point three three six, 117 00:07:00,641 --> 00:07:05,858 that you see at the bottom, I calculate R and then in the 118 00:07:05,858 --> 00:07:10,796 last column using that relationship I calculate delta. 119 00:07:10,796 --> 00:07:15,92 And indeed you see that delta starts at a hundred eighty 120 00:07:15,92 --> 00:07:20,764 degrees when I is zero. And then goes to a minimum of 121 00:07:20,764 --> 00:07:26,54 roughly a hundred thirty-eight, after which it increases again. 122 00:07:26,54 --> 00:07:30,826 And this now is crucial, is key to 123 00:07:30,826 --> 00:07:34,655 an understanding of the rainbow. 124 00:07:34,655 --> 00:07:40,338 Imagine now that I have one drop of water here. 125 00:07:40,338 --> 00:07:47,38 And sunlight comes in at all angles of I, not just at one, 126 00:07:47,38 --> 00:07:53,311 but all angles of I. Whatever you see here has of 127 00:07:53,311 --> 00:08:00,105 course axial symmetry. It is a spherical drop. 128 00:08:00,105 --> 00:08:02,75 And the light comes in like this. 129 00:08:02,75 --> 00:08:07,047 So light can go this way but it can also go this way. 130 00:08:07,047 --> 00:08:12,005 And it can also go this way and this way, so there's complete 131 00:08:12,005 --> 00:08:15,641 axial symmetry, so this whole drawing you can 132 00:08:15,641 --> 00:08:20,268 rotate about this line here. And everything holds then in 133 00:08:20,268 --> 00:08:23,821 axial symmetry. So therefore if phi maximum, 134 00:08:23,821 --> 00:08:27,622 if this angle phi maximum is forty-two degrees, 135 00:08:27,622 --> 00:08:32,167 then the light that will go back in the direction of the 136 00:08:32,167 --> 00:08:36,394 sun, the light that goes through the 137 00:08:36,394 --> 00:08:40,251 journey A B C and then comes out of the raindrop, 138 00:08:40,251 --> 00:08:45,072 that's all I'm talking about, now, I'm not talking about this 139 00:08:45,072 --> 00:08:48,769 light that sneaks out here, it is this journey, 140 00:08:48,769 --> 00:08:51,582 A, refraction at A, reflection at B, 141 00:08:51,582 --> 00:08:55,76 and then coming out at C. That light comes out in the 142 00:08:55,76 --> 00:08:59,216 form of a cone. And the half -- top angle of 143 00:08:59,216 --> 00:09:02,591 the cone must be roughly forty-two degrees. 144 00:09:02,591 --> 00:09:08,661 And so I will go -- I'm going to draw that cone for 145 00:09:08,661 --> 00:09:09,921 you. Like so. 146 00:09:09,921 --> 00:09:14,229 And like so. And you have to think of this 147 00:09:14,229 --> 00:09:18,011 as a cone. It's completely symmetric, 148 00:09:18,011 --> 00:09:21,374 axial symmetric, about this line. 149 00:09:21,374 --> 00:09:26,942 And this angle here then is roughly forty-two degrees. 150 00:09:26,942 --> 00:09:32,09 No light can go here. Because that would mean that 151 00:09:32,09 --> 00:09:37,134 phi would be larger than forty-two 152 00:09:37,134 --> 00:09:42,017 degrees and that's not allowed. Now comes something very 153 00:09:42,017 --> 00:09:45,569 important. The index of refraction of red 154 00:09:45,569 --> 00:09:50,009 light for water is about one point three three one. 155 00:09:50,009 --> 00:09:54,981 And that translates into an angle for phi max which I can 156 00:09:54,981 --> 00:10:00,309 calculate now -- it translates into an angle of phi max which 157 00:10:00,309 --> 00:10:03,594 is about forty-two point four degrees. 158 00:10:03,594 --> 00:10:09,01 But blue light has a slightly different index of 159 00:10:09,01 --> 00:10:13,919 refraction, therefore has a slightly different angle for phi 160 00:10:13,919 --> 00:10:19,076 max, and the blue light has an index of refraction of something 161 00:10:19,076 --> 00:10:24,234 like one point three four three. Notice I have blue light and I 162 00:10:24,234 --> 00:10:28,477 don't use violet light. Violet is much harder to see 163 00:10:28,477 --> 00:10:32,221 with our eyes. So I always refer to it as blue 164 00:10:32,221 --> 00:10:35,049 light. And that has a value for phi 165 00:10:35,049 --> 00:10:40,539 max which is approximately forty point seven degrees. 166 00:10:40,539 --> 00:10:44,722 A different index of refraction means of course that if you know 167 00:10:44,722 --> 00:10:47,975 I that R is slightly different, using Snell's law, 168 00:10:47,975 --> 00:10:50,763 so you get slightly different values for R, 169 00:10:50,763 --> 00:10:54,083 and so you get slightly different values for delta, 170 00:10:54,083 --> 00:10:57,734 so you get a slightly different value for delta minimum, 171 00:10:57,734 --> 00:11:00,788 you get a slightly different value for phi max. 172 00:11:00,788 --> 00:11:04,373 What does this mean now? That means if you look at this 173 00:11:04,373 --> 00:11:08,223 cone of light that goes back into the direction of the sun, 174 00:11:08,223 --> 00:11:11,344 that the outer edge, the outer 175 00:11:11,344 --> 00:11:14,936 surface of that cone, which has the largest possible 176 00:11:14,936 --> 00:11:18,669 angle, this angle is now forty-two point four degrees, 177 00:11:18,669 --> 00:11:22,051 must be red light, because blue light cannot come 178 00:11:22,051 --> 00:11:25,643 out in that direction. Because the maximum angle for 179 00:11:25,643 --> 00:11:29,447 blue is forty point seven, is not forty-two point four. 180 00:11:29,447 --> 00:11:33,744 So it's only red light that can come out when the angle -- the 181 00:11:33,744 --> 00:11:37,548 half angle of the cone is forty-two point four degrees. 182 00:11:37,548 --> 00:11:42,337 And the blue light is not going to make it until this 183 00:11:42,337 --> 00:11:47,144 angle here -- I'll put it in here -- is forty point seven 184 00:11:47,144 --> 00:11:50,148 degrees. If you look inside the cone 185 00:11:50,148 --> 00:11:55,384 with an half top angle of forty point seven degrees all colors 186 00:11:55,384 --> 00:11:59,16 can come back. All this is saying is that for 187 00:11:59,16 --> 00:12:03,023 red light phi maximum is forty-two point four. 188 00:12:03,023 --> 00:12:06,628 It can also therefore be forty point seven. 189 00:12:06,628 --> 00:12:09,546 It can be thirty. It can be twenty. 190 00:12:09,546 --> 00:12:12,035 It can be ten. It can be zero. 191 00:12:12,035 --> 00:12:17,009 Light that comes in here at I equals zero 192 00:12:17,009 --> 00:12:20,22 reaches point P, comes straight back. 193 00:12:20,22 --> 00:12:23,61 That's allowed. Phi would be zero then. 194 00:12:23,61 --> 00:12:28,695 If all the colors can make it back inside this cone that I 195 00:12:28,695 --> 00:12:33,512 have given here a blue color, that would mean that your 196 00:12:33,512 --> 00:12:37,348 brains will tell you that it is white light. 197 00:12:37,348 --> 00:12:41,362 Because you see red light, you see blue light, 198 00:12:41,362 --> 00:12:45,927 you see green light, you see yellow light, 199 00:12:45,927 --> 00:12:50,086 and so your brains will tell you that that is white light. 200 00:12:50,086 --> 00:12:53,952 If I had a screen here with a small opening to let the 201 00:12:53,952 --> 00:12:58,329 sunlight in and I asked you what would you see on this screen 202 00:12:58,329 --> 00:13:01,903 having only one water drop, then you would see the 203 00:13:01,903 --> 00:13:05,624 intersection of this cone of light with this screen. 204 00:13:05,624 --> 00:13:08,834 And it would look as follows. The outer edge, 205 00:13:08,834 --> 00:13:12,116 the outer circle, which is the intersection of 206 00:13:12,116 --> 00:13:16,201 the cone with your screen, would be red. 207 00:13:16,201 --> 00:13:21,121 And then there would be an inner portion whereby all colors 208 00:13:21,121 --> 00:13:24,768 can come back. So that would be white light. 209 00:13:24,768 --> 00:13:28,33 And then as you go further in from the red, 210 00:13:28,33 --> 00:13:32,911 until you reach that white portion, then the last color 211 00:13:32,911 --> 00:13:38 that will be added is the blue. And you can already of course 212 00:13:38 --> 00:13:42,665 sense that all the action about the rainbow occurs here. 213 00:13:42,665 --> 00:13:46,481 And here there is no light. It's 214 00:13:46,481 --> 00:13:51,066 going to be dark because there is no way that phi can be larger 215 00:13:51,066 --> 00:13:53,506 than forty-two point four degrees. 216 00:13:53,506 --> 00:13:57,721 And if light would appear on the screen outside the red it 217 00:13:57,721 --> 00:14:01,787 means that phi would be larger than forty-two point four 218 00:14:01,787 --> 00:14:05,41 degrees and that's not allowed. So it's dark here. 219 00:14:05,41 --> 00:14:08,516 It's white light here. It's red light here. 220 00:14:08,516 --> 00:14:12,583 Then as you go further in you will finally see the other 221 00:14:12,583 --> 00:14:15,319 colors. And this really is now the key 222 00:14:15,319 --> 00:14:19,359 to the geometry of the -- of the 223 00:14:19,359 --> 00:14:22,449 rainbow. I'm going to put you here, 224 00:14:22,449 --> 00:14:27,266 so here you're standing, and let the sun again be near 225 00:14:27,266 --> 00:14:31,083 the horizon. It's always nice when you make 226 00:14:31,083 --> 00:14:34,446 a picture. The easy reference that you 227 00:14:34,446 --> 00:14:36,809 have. You're standing here. 228 00:14:36,809 --> 00:14:40,354 Sunlight is coming in in this direction. 229 00:14:40,354 --> 00:14:45,08 And there is rain here. If it's also raining here you 230 00:14:45,08 --> 00:14:48,128 won't see a rainbow. 231 00:14:48,128 --> 00:14:52,603 Because the sun will not be able to hit these raindrops. 232 00:14:52,603 --> 00:14:57,16 So it's essential that it's raining in the direction away 233 00:14:57,16 --> 00:15:00,985 from the sun but that you can still see the sun. 234 00:15:00,985 --> 00:15:04,077 So here are these raindrops. All right. 235 00:15:04,077 --> 00:15:07,576 You're looking in this direction in the sky. 236 00:15:07,576 --> 00:15:10,505 You're looking up in the sky like so. 237 00:15:10,505 --> 00:15:14,818 And I pick here one raindrop and only one but what I'm 238 00:15:14,818 --> 00:15:20,84 telling you holds for all the raindrops in that direction. 239 00:15:20,84 --> 00:15:26,351 What will that raindrop do? That raindrop will produce a 240 00:15:26,351 --> 00:15:32,765 cone of light which goes back in the direction of the sun whereby 241 00:15:32,765 --> 00:15:37,775 the edge of the cone is red light and this angle is 242 00:15:37,775 --> 00:15:41,684 forty-two degrees, forty-two point four, 243 00:15:41,684 --> 00:15:44,189 whatever. What do you see? 244 00:15:44,189 --> 00:15:47,997 Nothing. Because there is no light that 245 00:15:47,997 --> 00:15:52,185 can come from this raindrop in your 246 00:15:52,185 --> 00:15:55,2 direction. There is no light because that 247 00:15:55,2 --> 00:15:59,346 would mean that phi is larger than forty-two degrees and 248 00:15:59,346 --> 00:16:03,115 that's not allowed. So you look high up in the sky, 249 00:16:03,115 --> 00:16:07,411 you will not see any light coming back from that raindrop. 250 00:16:07,411 --> 00:16:10,954 And think of the whole thing as axial symmetric, 251 00:16:10,954 --> 00:16:12,537 right? Not only there, 252 00:16:12,537 --> 00:16:16,004 but there and there you will not see any light. 253 00:16:16,004 --> 00:16:19,999 Now I am looking say at some raindrops which are here. 254 00:16:19,999 --> 00:16:24,603 I pick one. It holds for any one in that 255 00:16:24,603 --> 00:16:27,798 direction. And so now I draw a line to 256 00:16:27,798 --> 00:16:30,907 this point. This is what I'm looking. 257 00:16:30,907 --> 00:16:34,879 I look in this direction. I take this raindrop. 258 00:16:34,879 --> 00:16:37,729 I could have picked this raindrop. 259 00:16:37,729 --> 00:16:42,824 I could have picked this one. Would have made no difference. 260 00:16:42,824 --> 00:16:47,919 What is that raindrop doing? Well, that raindrop is throwing 261 00:16:47,919 --> 00:16:53,014 back at the sun light in the form of a cone and the cone has 262 00:16:53,014 --> 00:16:58,151 this angle forty-two degrees, forty-two degrees. 263 00:16:58,151 --> 00:16:59,887 What do you see now? Look. 264 00:16:59,887 --> 00:17:02,526 You're looking straight into that cone. 265 00:17:02,526 --> 00:17:06,137 You're nowhere near the edge. So you see white light. 266 00:17:06,137 --> 00:17:10,373 Because green light comes back at you, red light comes back at 267 00:17:10,373 --> 00:17:12,665 you, everything comes back at you. 268 00:17:12,665 --> 00:17:15,235 So you will say ha, I see white light. 269 00:17:15,235 --> 00:17:17,596 Not only there, low on the horizon, 270 00:17:17,596 --> 00:17:21,763 but the whole thing is axial symmetric, also there and there. 271 00:17:21,763 --> 00:17:24,055 So you haven't seen a rainbow yet. 272 00:17:24,055 --> 00:17:27,736 But now suppose I ask you to look 273 00:17:27,736 --> 00:17:32,681 up in the sky at a very specific angle and we'll make 274 00:17:32,681 --> 00:17:36,294 the angle forty-two point four degrees. 275 00:17:36,294 --> 00:17:42 So you're looking at the sky in the direction somewhere here, 276 00:17:42 --> 00:17:46,85 I'm not quite sure that I have the angle just right. 277 00:17:46,85 --> 00:17:51,51 So I'm looking in the sky here. Pick one raindrop. 278 00:17:51,51 --> 00:17:55,979 But you could pick any one, any other other one. 279 00:17:55,979 --> 00:18:00,924 There's nothing special about that one. 280 00:18:00,924 --> 00:18:03,757 So the sunlight comes in like so. 281 00:18:03,757 --> 00:18:08,803 What is that raindrop doing? Well, it's throwing a cone of 282 00:18:08,803 --> 00:18:13,494 light back to the sun. And it just so happens that you 283 00:18:13,494 --> 00:18:18,717 are only looking at the outer surface of the cone where only 284 00:18:18,717 --> 00:18:24,117 red light can go because this is the famous angle of forty-two 285 00:18:24,117 --> 00:18:28,011 point four degrees. And so you see red light. 286 00:18:28,011 --> 00:18:33,323 And since the whole problem is axial symmetric, 287 00:18:33,323 --> 00:18:37,651 you would see red light if you look forty-two point four 288 00:18:37,651 --> 00:18:42,295 degrees in this direction but also forty-two degrees in this 289 00:18:42,295 --> 00:18:45,916 direction, away from this direction to the sun. 290 00:18:45,916 --> 00:18:50,717 And so now you see how the bow is being formed by zillions and 291 00:18:50,717 --> 00:18:55,439 zillions of small water drops which each of them in their own 292 00:18:55,439 --> 00:19:00,004 way contribute to your rainbow. And we'll now put the sun a 293 00:19:00,004 --> 00:19:03,94 little higher in the sky. So I'll put you here now. 294 00:19:03,94 --> 00:19:07,718 And let's say the sun is now like 295 00:19:07,718 --> 00:19:10,262 so. So, you look at your shadow 296 00:19:10,262 --> 00:19:13,315 here on the ground. The sun is there. 297 00:19:13,315 --> 00:19:16,623 There's your shadow. Here's your shadow. 298 00:19:16,623 --> 00:19:19,253 Here's your head -- your shadow. 299 00:19:19,253 --> 00:19:23,918 And this is the reference line that we're talking about. 300 00:19:23,918 --> 00:19:27,056 That is this line. That was this line. 301 00:19:27,056 --> 00:19:30,109 And so where would you see a rainbow? 302 00:19:30,109 --> 00:19:35,029 You have to look now forty-two degrees away from that line. 303 00:19:35,029 --> 00:19:39,693 Forty-two degrees away from that line. 304 00:19:39,693 --> 00:19:43,881 That would be red then and a little lower would be your your 305 00:19:43,881 --> 00:19:46,436 blue. And so if I sketch in there the 306 00:19:46,436 --> 00:19:50,694 bow, it would be sort of like this and then the blue would be 307 00:19:50,694 --> 00:19:55,165 on the inside and here you would have white light and this would 308 00:19:55,165 --> 00:19:58,288 then be the red. It's always relative to this 309 00:19:58,288 --> 00:20:01,553 reference line. This would be roughly -- I call 310 00:20:01,553 --> 00:20:04,391 it forty-two degrees, but according to my 311 00:20:04,391 --> 00:20:08,224 calculation, there it would be be forty-two point four. 312 00:20:08,224 --> 00:20:12,791 So whenever you know where the sun is, you look at the 313 00:20:12,791 --> 00:20:16,272 shadow of your own head and you have to go forty-two point four 314 00:20:16,272 --> 00:20:19,64 degrees away from that direction from your eye to the shadow. 315 00:20:19,64 --> 00:20:23,064 And so what you're seeing now is that if the sun is low in the 316 00:20:23,064 --> 00:20:26,376 horizon then the bow will be high above the horizon and when 317 00:20:26,376 --> 00:20:29,856 the sun is rising then the bow goes down and goes down and goes 318 00:20:29,856 --> 00:20:33,168 down and by the time that the sun is forty-two degrees above 319 00:20:33,168 --> 00:20:36,761 the horizon, you're not going to see a rainbow anymore unless the 320 00:20:36,761 --> 00:20:41,028 water is right where you are. If the water is at a distance, 321 00:20:41,028 --> 00:20:45,239 you won't see a rainbow. So the higher the sun in the 322 00:20:45,239 --> 00:20:50,018 sky, the less likely it is that you will ever see a rainbow. 323 00:20:50,018 --> 00:20:55,04 Before I show you some -- some slides and before we're going to 324 00:20:55,04 --> 00:20:59,656 answer some of the questions, uh, I want to mention to you 325 00:20:59,656 --> 00:21:03,544 that what I went through here is refraction at A, 326 00:21:03,544 --> 00:21:07,918 reflection at B -- reflection at B and coming out at C. 327 00:21:07,918 --> 00:21:13,75 You can go through the same geometry and allow for one more 328 00:21:13,75 --> 00:21:17,785 reflection at C and then let the light come out, 329 00:21:17,785 --> 00:21:22,164 so it would come out at point D and if you did that, 330 00:21:22,164 --> 00:21:27,487 you can convince yourself that there is indeed a second rainbow 331 00:21:27,487 --> 00:21:32,296 and we call that the secondary. We call this the primary. 332 00:21:32,296 --> 00:21:37,19 And the secondary has a radius -- this one of course has a 333 00:21:37,19 --> 00:21:42,513 radius of forty-two point four degrees for the red and the blue 334 00:21:42,513 --> 00:21:45,737 would have a smaller radius. 335 00:21:45,737 --> 00:21:49,833 The secondary has a radius for the red of fifty point four 336 00:21:49,833 --> 00:21:51,414 degrees. So in the red, 337 00:21:51,414 --> 00:21:54,719 it's fifty point four degrees, and in the blue, 338 00:21:54,719 --> 00:21:57,162 it's larger. So the blue is outside 339 00:21:57,162 --> 00:22:00,61 fifty-two point five degrees. Is that what it is? 340 00:22:00,61 --> 00:22:04,275 Fifty-three point five. The secondary bow is fainter 341 00:22:04,275 --> 00:22:08,801 and it is also wider because you see this separation in terms of 342 00:22:08,801 --> 00:22:11,747 angle is larger than the separation there. 343 00:22:11,747 --> 00:22:15,556 That's only one point seven degrees 344 00:22:15,556 --> 00:22:18,217 and this is, uh, more than three degrees. 345 00:22:18,217 --> 00:22:22,209 And so there is a secondary bow and the secondary bow is only 346 00:22:22,209 --> 00:22:25,07 about roughly ten degrees above the primary. 347 00:22:25,07 --> 00:22:28,862 So if you see your primary at forty-two degree angles away 348 00:22:28,862 --> 00:22:32,655 from your shadow there and there and there, go another ten 349 00:22:32,655 --> 00:22:35,249 degrees and you should see a second bow. 350 00:22:35,249 --> 00:22:37,778 And you should see the colors reversed. 351 00:22:37,778 --> 00:22:41,304 Red is there on the outside and blue is on the inside. 352 00:22:41,304 --> 00:22:45,628 Red is on the outside and blue is on the inside. 353 00:22:45,628 --> 00:22:48,788 Red is on the outside and blue is on the inside, 354 00:22:48,788 --> 00:22:51,611 but here it is reversed. The secondary bow, 355 00:22:51,611 --> 00:22:55,308 the red is on the inside and the blue is on the outside. 356 00:22:55,308 --> 00:22:57,593 So let's answer some questions now. 357 00:22:57,593 --> 00:23:01,76 We'll put the questions back up again and you will see that you 358 00:23:01,76 --> 00:23:05,928 can now without any difficulty already answer twelve questions. 359 00:23:05,928 --> 00:23:08,079 The first twelve. Red is outside. 360 00:23:08,079 --> 00:23:10,028 That is non-negotiable, right? 361 00:23:10,028 --> 00:23:13,523 That follows immediately from what we'd talked about, 362 00:23:13,523 --> 00:23:15,607 but that's not an issue anymore. 363 00:23:15,607 --> 00:23:20,696 The radius is about forty-two degrees and the length of the 364 00:23:20,696 --> 00:23:24,451 bow, well that depends on if the sun is high in the sky then the 365 00:23:24,451 --> 00:23:27,729 length will be very small because this whole s- arc will 366 00:23:27,729 --> 00:23:30,888 then go down and there will only be a little bit left. 367 00:23:30,888 --> 00:23:34,225 It could also be that it's only raining here and it's not 368 00:23:34,225 --> 00:23:36,788 raining there. So depending upon where there 369 00:23:36,788 --> 00:23:40,483 is rain and how high the sun is in the sky, that will determine 370 00:23:40,483 --> 00:23:43,046 the length of the bow. The width of the bow, 371 00:23:43,046 --> 00:23:44,953 you would think, perhaps naively, 372 00:23:44,953 --> 00:23:48,053 that if you subtract the forty-two 373 00:23:48,053 --> 00:23:51,298 from the forty -- forty point seven from the forty two point 374 00:23:51,298 --> 00:23:54,654 four that that would give you the width of the bow which would 375 00:23:54,654 --> 00:23:56,855 then be one point seven degrees in angle. 376 00:23:56,855 --> 00:23:59,771 However, you would overlook then that the sun is not a 377 00:23:59,771 --> 00:24:03,017 point, but that the sun in the sky has a dimension of half a 378 00:24:03,017 --> 00:24:05,382 degree and each point of the sun, of course, 379 00:24:05,382 --> 00:24:08,353 makes its own little rainbow, so you really have to add 380 00:24:08,353 --> 00:24:11,049 roughly half a degree. So, the width of the bow is 381 00:24:11,049 --> 00:24:14,13 more like, something like two degrees rather than the one 382 00:24:14,13 --> 00:24:16,715 point seven, maybe two -- two point two degrees. 383 00:24:16,715 --> 00:24:20,402 It's a little wider than what you would think. 384 00:24:20,402 --> 00:24:22,929 It has to do with the finite size of the sun. 385 00:24:22,929 --> 00:24:25,456 A comparison of the light inside and outside, 386 00:24:25,456 --> 00:24:28,385 clearly inside the bow, there must be a lot of white 387 00:24:28,385 --> 00:24:31,085 light which you don't see outside the bow -- big 388 00:24:31,085 --> 00:24:33,382 difference. You will see a slide shortly. 389 00:24:33,382 --> 00:24:36,943 The time of the day -- well the sun has to be a little bit low, 390 00:24:36,943 --> 00:24:39,93 so you don't expect it at midday and you want rain as 391 00:24:39,93 --> 00:24:42,917 well, so late afternoon, early morning would be ideal 392 00:24:42,917 --> 00:24:45,846 and you can figure out uh in what direction to look. 393 00:24:45,846 --> 00:24:48,603 If it's early morning, the sun rises in the east, 394 00:24:48,603 --> 00:24:52,452 you would see the bows in the west and late in 395 00:24:52,452 --> 00:24:56,81 the afternoon when the sun is in the west, you would see the 396 00:24:56,81 --> 00:24:59,987 bows in the east. Yes there is a second bow. 397 00:24:59,987 --> 00:25:04,272 It is about ten degrees higher in the sky than the primary. 398 00:25:04,272 --> 00:25:08,631 And the colors are reversed. The blue is outside and the red 399 00:25:08,631 --> 00:25:11,069 is inside. And the radius is about 400 00:25:11,069 --> 00:25:14,393 fifty-two degrees. And the width of the bow is 401 00:25:14,393 --> 00:25:18,826 again the difference between these two numbers which you have 402 00:25:18,826 --> 00:25:23,111 to add about half a degree. So you would think on the basis 403 00:25:23,111 --> 00:25:27,322 of this that it is more like three degrees and you have to 404 00:25:27,322 --> 00:25:32,642 add roughly the half a degree, so it's about three -- 405 00:25:32,642 --> 00:25:37,884 three-and-a-half degrees. So let's now look at some 406 00:25:37,884 --> 00:25:41,553 slides. And the first slide that you 407 00:25:41,553 --> 00:25:45,851 will see is a -- is a drawing that I made. 408 00:25:45,851 --> 00:25:51,198 Which uh is meant to repeat some of what I told you. 409 00:25:51,198 --> 00:25:56,86 So you see a person standing there and here you see the 410 00:25:56,86 --> 00:26:02,941 direction to the sun. Your shadow would be this 411 00:26:02,941 --> 00:26:04,984 long. This would be your head of the 412 00:26:04,984 --> 00:26:08,488 shadow, here would be your feet. And if there are water drops 413 00:26:08,488 --> 00:26:11,175 here and if there's no interference between the 414 00:26:11,175 --> 00:26:14,678 sunlight and the water drops then in this direction forty-two 415 00:26:14,678 --> 00:26:17,073 degrees you would see this water drop red. 416 00:26:17,073 --> 00:26:20,518 And then the water drops which were here you would see white 417 00:26:20,518 --> 00:26:24,08 light from those water drops and maybe from this one you would 418 00:26:24,08 --> 00:26:27,292 see red as well as blue. And if you have water near your 419 00:26:27,292 --> 00:26:30,621 feet there's no reason why you can't see the rainbow there 420 00:26:30,621 --> 00:26:34,534 either but of course you would have to lead some 421 00:26:34,534 --> 00:26:36,798 garden hose to produce water there. 422 00:26:36,798 --> 00:26:40,062 The next slide then. This is a drawing made by the 423 00:26:40,062 --> 00:26:42,993 maestro himself, Newton, who was the first to 424 00:26:42,993 --> 00:26:45,924 fully understand the workings of the rainbow. 425 00:26:45,924 --> 00:26:49,654 You see here the primary bow and for those of you who are 426 00:26:49,654 --> 00:26:53,65 sitting close you can see the direction of the sunlight going 427 00:26:53,65 --> 00:26:56,248 in at A, reflects at B, coming out at C. 428 00:26:56,248 --> 00:26:58,713 Which he calls E, but that's a detail. 429 00:26:58,713 --> 00:27:02,176 And here you see then the secondary whereby the light 430 00:27:02,176 --> 00:27:06,24 comes in, one reflection, two reflections, 431 00:27:06,24 --> 00:27:09,178 and now it comes back at you and that gives you the 432 00:27:09,178 --> 00:27:11,529 secondary. And so this is red and this is 433 00:27:11,529 --> 00:27:14,232 blue whereas here this is blue and this is red. 434 00:27:14,232 --> 00:27:16,76 The next slide. So if the sun is high in the 435 00:27:16,76 --> 00:27:19,992 sky, I've done this many times when I was uh watering my 436 00:27:19,992 --> 00:27:23,283 garden, you get this for free, you might as well do this, 437 00:27:23,283 --> 00:27:25,458 it's great fun, you're standing there, 438 00:27:25,458 --> 00:27:29,102 gives you an immense feeling of power, and is completely clear, 439 00:27:29,102 --> 00:27:32,334 there's not a -- not a cloud in the sky, there's no rain 440 00:27:32,334 --> 00:27:35,449 anywhere, but you spray water around you and you see a 441 00:27:35,449 --> 00:27:39,003 beautiful rainbow encircling your feet. 442 00:27:39,003 --> 00:27:42,903 And always what matters is this forty-two degree angle relative 443 00:27:42,903 --> 00:27:46,929 to the direction from the sun to you and this is where the shadow 444 00:27:46,929 --> 00:27:49,885 of my head would be. Very easy to do and I would 445 00:27:49,885 --> 00:27:52,967 recommend that you try that when you get a chance. 446 00:27:52,967 --> 00:27:56,238 The next slide is a painting from the eighth century, 447 00:27:56,238 --> 00:27:59,949 Turkey, I see a hand here -- probably makes reference to the 448 00:27:59,949 --> 00:28:03,849 Bible, but I think it reads in the Bible I do set my bow in the 449 00:28:03,849 --> 00:28:07,748 clouds, it's very nice and very dandy, the fact that the colors 450 00:28:07,748 --> 00:28:11,242 are wrong, the sequence of the colors, 451 00:28:11,242 --> 00:28:13,552 that's a detail, it's a nice picture, 452 00:28:13,552 --> 00:28:16,953 but red has to be on the outside and blue has to be on 453 00:28:16,953 --> 00:28:18,492 the inside. Small detail. 454 00:28:18,492 --> 00:28:19,712 Next slide. Ah yeah. 455 00:28:19,712 --> 00:28:22,984 A -- a few years ago, actually it's more than a few, 456 00:28:22,984 --> 00:28:26,385 when I was first lecturing eight oh three I knew I was 457 00:28:26,385 --> 00:28:30,042 going to talk about the rainbow, ideal for eight oh three. 458 00:28:30,042 --> 00:28:33,763 And so I wanted to make some rainbows myself in my backyard 459 00:28:33,763 --> 00:28:36,458 in Winchester. And so there is water coming 460 00:28:36,458 --> 00:28:40,821 out of a um water hose and the sun was behind me 461 00:28:40,821 --> 00:28:44,148 and I took this picture and you see indeed all the ingredients 462 00:28:44,148 --> 00:28:46,711 that we just discussed. Notice the red is on the 463 00:28:46,711 --> 00:28:48,728 outside and the blue is on the inside. 464 00:28:48,728 --> 00:28:51,891 You see all this white light. That is that light that comes 465 00:28:51,891 --> 00:28:55,217 back from the water drop but if you go here where the angle of 466 00:28:55,217 --> 00:28:57,507 phi would be larger than forty-two degrees, 467 00:28:57,507 --> 00:29:00,288 which is not allowed, you don't see any light coming 468 00:29:00,288 --> 00:29:03,124 back at you, so you look straight through and see the 469 00:29:03,124 --> 00:29:06,45 forest without any white light. So you really see already here 470 00:29:06,45 --> 00:29:10,104 sky so to speak is bright here and darker there. 471 00:29:10,104 --> 00:29:13,596 I needed help from my daughter. And the next slide will show 472 00:29:13,596 --> 00:29:15,905 you that. The poor darling was suffering 473 00:29:15,905 --> 00:29:17,148 badly. It was January, 474 00:29:17,148 --> 00:29:19,279 it was freezing cold, she was crying. 475 00:29:19,279 --> 00:29:22,535 She was really crying and I felt guilty but I said look, 476 00:29:22,535 --> 00:29:25,851 you know I really need these slides for my eight oh three 477 00:29:25,851 --> 00:29:28,515 class, it'll only take you half-an-hour or so, 478 00:29:28,515 --> 00:29:31,83 and she still remembers it, I had email exchange with her 479 00:29:31,83 --> 00:29:35,441 yesterday and she said dad I was crying, it was an awful thing 480 00:29:35,441 --> 00:29:37,691 what you did to me. But look, but look. 481 00:29:37,691 --> 00:29:40,591 You know, you got to do something, and when you're 482 00:29:40,591 --> 00:29:44,421 daughter of a scientist occasionally you have 483 00:29:44,421 --> 00:29:47,271 to suffer a little bit. And so she's holding here the 484 00:29:47,271 --> 00:29:50,559 -- the water hose and you see the same thing red outside blue 485 00:29:50,559 --> 00:29:53,628 inside, and clearly the white light, you see this is that 486 00:29:53,628 --> 00:29:56,094 white light that I discussed with you earlier. 487 00:29:56,094 --> 00:29:59,218 The next slide will show you then a real rainbow -- oh no, 488 00:29:59,218 --> 00:30:02,397 this is the um this is the artificial one I made uh over my 489 00:30:02,397 --> 00:30:05,849 driveway, if you want to see the secondary, you need a real dark 490 00:30:05,849 --> 00:30:08,48 background, because the secondary is quite faint. 491 00:30:08,48 --> 00:30:12,973 And so that's why I did it over my driveway and you see here the 492 00:30:12,973 --> 00:30:16,056 primary, red on the outside, blue on the inside, 493 00:30:16,056 --> 00:30:18,417 and here you see the colors reversed. 494 00:30:18,417 --> 00:30:21,631 You can also see perhaps that it's a little wider. 495 00:30:21,631 --> 00:30:24,32 But it's always much s- it's much fainter, 496 00:30:24,32 --> 00:30:28,19 so it's -- it's hard to tell. And then the next slide is one 497 00:30:28,19 --> 00:30:32,06 that's a wonderful slide made by Doug Johnson in New Mexico, 498 00:30:32,06 --> 00:30:35,011 Socorro, it is where the radio telescopes are, 499 00:30:35,011 --> 00:30:38,291 the very large array, now look, red on the outside, 500 00:30:38,291 --> 00:30:41,439 blue on the inside. The sky you got to admit is a 501 00:30:41,439 --> 00:30:44,885 hell of a lot brighter here than it 502 00:30:44,885 --> 00:30:47,404 is there. And you may never have noticed 503 00:30:47,404 --> 00:30:51,474 that, you've -- you've looked at it but you've never seen it and 504 00:30:51,474 --> 00:30:55,349 then here you see the secondary red on the inside blue on the 505 00:30:55,349 --> 00:30:57,739 outside. There is a phenomenon that we 506 00:30:57,739 --> 00:31:01,615 have never discussed uh in eight oh two yet, it's coming up I 507 00:31:01,615 --> 00:31:04,392 think next week. And that phenomenon we call 508 00:31:04,392 --> 00:31:06,976 diffraction. Snell's law cannot deal with 509 00:31:06,976 --> 00:31:09,56 diffraction. That occurs when water drops 510 00:31:09,56 --> 00:31:14,017 are very very small, say, smaller than a tenth of a 511 00:31:14,017 --> 00:31:16,462 millimeter. What you then get over and 512 00:31:16,462 --> 00:31:20,361 above the bow you get areas in the bow whereby you get as we 513 00:31:20,361 --> 00:31:22,609 call that destructive interference, 514 00:31:22,609 --> 00:31:26,574 the waves begin to kill each other and you get dark bands and 515 00:31:26,574 --> 00:31:29,945 you can actually see that here. With a little bit of 516 00:31:29,945 --> 00:31:32,324 imagination you see here a dark band. 517 00:31:32,324 --> 00:31:36,29 And when that is the case the water is always extremely small 518 00:31:36,29 --> 00:31:40,123 in size and we give this a name, we call this supernumerary 519 00:31:40,123 --> 00:31:41,841 bows. It's not so uncommon. 520 00:31:41,841 --> 00:31:46,043 If the water drops become exceedingly small, 521 00:31:46,043 --> 00:31:49,062 let's say, smaller even than fifty microns, 522 00:31:49,062 --> 00:31:52,799 then this diffraction phenomenon becomes so important 523 00:31:52,799 --> 00:31:56,537 that in addition to the dark bands all the colors are 524 00:31:56,537 --> 00:32:00,921 beginning to wash out over each other, and that creates then a 525 00:32:00,921 --> 00:32:03,365 white rainbow. Plus the dark bands. 526 00:32:03,365 --> 00:32:07,461 And a student of mine who was in my eight oh three lecture 527 00:32:07,461 --> 00:32:11,558 sent me years later the next slide when he was -- his name 528 00:32:11,558 --> 00:32:15,367 was Carl Wales, this picture he took, 529 00:32:15,367 --> 00:32:18,645 it's three hundred forty miles from the North Pole, 530 00:32:18,645 --> 00:32:22,579 he was at Fletcher Island at the time, this picture was taken 531 00:32:22,579 --> 00:32:26,512 at two AM at night in July when the sun is above the horizon. 532 00:32:26,512 --> 00:32:30,445 And so this must be the result of very fine water drops which 533 00:32:30,445 --> 00:32:32,74 somehow are there in the atmosphere. 534 00:32:32,74 --> 00:32:36,673 Doesn't look like it's raining but there must have been small 535 00:32:36,673 --> 00:32:38,902 water drops. Fifty microns or less. 536 00:32:38,902 --> 00:32:42,115 And here you see the white-colored rainbow and you 537 00:32:42,115 --> 00:32:46,114 also see beautifully the supernumerary bows, 538 00:32:46,114 --> 00:32:50,669 you see the -- the dark bands in there and the next slide is a 539 00:32:50,669 --> 00:32:53,806 close-up of that. So you see here the white 540 00:32:53,806 --> 00:32:57,84 rainbow plus the dark bands which is the phenomenon the 541 00:32:57,84 --> 00:33:02,097 result of uh of diffraction. Now before I will discuss the 542 00:33:02,097 --> 00:33:06,279 polarization of the bows, because I still owe you answers 543 00:33:06,279 --> 00:33:09,79 to the last three questions, now that I'm at it, 544 00:33:09,79 --> 00:33:13,449 I want to show you some phenomenon which are quite 545 00:33:13,449 --> 00:33:17,259 common which you may never have seen 546 00:33:17,259 --> 00:33:20,453 even though they're quite common and they're rather 547 00:33:20,453 --> 00:33:23,136 spectacular. The first slide that comes now 548 00:33:23,136 --> 00:33:26,266 shows you what we call the twenty-two degree halo. 549 00:33:26,266 --> 00:33:30,035 It's very common around the sun, it's very common around the 550 00:33:30,035 --> 00:33:31,504 moon. The red is inside. 551 00:33:31,504 --> 00:33:33,485 It has nothing to do with water. 552 00:33:33,485 --> 00:33:36,36 It is the result of ice crystals way up in the 553 00:33:36,36 --> 00:33:38,723 atmosphere. You can see it both in the 554 00:33:38,723 --> 00:33:42,556 summer as well as in the winter because it's very cold way up 555 00:33:42,556 --> 00:33:45,431 there also in the summer. This is very common. 556 00:33:45,431 --> 00:33:48,944 The reason why you and I don't see 557 00:33:48,944 --> 00:33:51,595 it that often, because who wants to look in 558 00:33:51,595 --> 00:33:55,003 the direction of the sun. This angle is only twenty-two 559 00:33:55,003 --> 00:33:58,348 degrees, it's not so far. But I would advise you to at 560 00:33:58,348 --> 00:34:02,135 least keep an eye on the moon. Because the moon also has this 561 00:34:02,135 --> 00:34:05,48 twenty-two degree halo. And of course it's much easier 562 00:34:05,48 --> 00:34:07,752 to look in the direction of the moon. 563 00:34:07,752 --> 00:34:10,781 That doesn't uh -- any no problems for your eyes. 564 00:34:10,781 --> 00:34:14,379 This is very common that I see this at least three or four 565 00:34:14,379 --> 00:34:18,166 times per month and I always look for it of 566 00:34:18,166 --> 00:34:20,342 course. The next um slide shows you 567 00:34:20,342 --> 00:34:24,311 both the twenty-two degree halo as well as the forty-six degree 568 00:34:24,311 --> 00:34:27,192 halo which is way more -- way way less common, 569 00:34:27,192 --> 00:34:30,84 it's very rare to see the forty-six degree -- I've seen it 570 00:34:30,84 --> 00:34:32,889 only a few times. It's very rare. 571 00:34:32,889 --> 00:34:36,602 In addition to the twenty-two degree halo and the forty-six 572 00:34:36,602 --> 00:34:39,61 degree halo you sometimes see bright spots here. 573 00:34:39,61 --> 00:34:42,107 You see them here and you see them here. 574 00:34:42,107 --> 00:34:44,732 They're really not circles. They are arcs. 575 00:34:44,732 --> 00:34:49,021 And they have names, they call them uh sun dogs, 576 00:34:49,021 --> 00:34:51,696 mock suns, they have various names. 577 00:34:51,696 --> 00:34:55,945 I've seen this often in in Boston, the forty-six degree 578 00:34:55,945 --> 00:34:59,171 halo is rare. All of this as the result by 579 00:34:59,171 --> 00:35:03,342 the way of um ice crystals. Ice crystals way up in the 580 00:35:03,342 --> 00:35:07,04 earth atmosphere. And there is a phenomenon that 581 00:35:07,04 --> 00:35:10,345 many of you may have seen from an airplane. 582 00:35:10,345 --> 00:35:14,752 Uh if your airplane flies over clouds and you look at the 583 00:35:14,752 --> 00:35:19,316 shadow of your airplane onto the cloud you may have noticed 584 00:35:19,316 --> 00:35:23,375 colorful rings around the shadow of your 585 00:35:23,375 --> 00:35:25,837 airplane. And the next slide is such an 586 00:35:25,837 --> 00:35:28,104 example. I took this picture several 587 00:35:28,104 --> 00:35:30,501 years ago. You are always right at the 588 00:35:30,501 --> 00:35:33,481 center of the circle. So you can see that I was 589 00:35:33,481 --> 00:35:36,785 sitting just behind the wings. This is the result of 590 00:35:36,785 --> 00:35:39,506 diffraction. Got nothing to do with Snell's 591 00:35:39,506 --> 00:35:41,773 law. It is the result though of very 592 00:35:41,773 --> 00:35:45,595 very fine water drops but not in the sense of refraction and 593 00:35:45,595 --> 00:35:49,223 reflection the way that we discussed it with the rainbow. 594 00:35:49,223 --> 00:35:52,722 This is the result of diffraction. 595 00:35:52,722 --> 00:35:56,146 It's extremely difficult, the explanation of the glory. 596 00:35:56,146 --> 00:35:59,253 And even ten years ago, I read an enormous article 597 00:35:59,253 --> 00:36:02,233 about it, I think it was in Scientific American. 598 00:36:02,233 --> 00:36:06,037 Which made an utter attempt to be very quantitative about the 599 00:36:06,037 --> 00:36:09,462 explanation, which is not easy. The radius of this bow, 600 00:36:09,462 --> 00:36:12,949 if you want to call it a bow, depends on the size of the 601 00:36:12,949 --> 00:36:15,422 water drops. If the water drops are very 602 00:36:15,422 --> 00:36:17,958 small the radius is substantially larger. 603 00:36:17,958 --> 00:36:22,777 So as you fly over clouds you may fly over different clouds 604 00:36:22,777 --> 00:36:27,093 with different size of water drops and on a very short time 605 00:36:27,093 --> 00:36:30,812 scale will you see the radius of this glory change. 606 00:36:30,812 --> 00:36:34,755 It can be very rapidly. It could be on a time scale of 607 00:36:34,755 --> 00:36:37,583 seconds to minutes. It's very dramatic. 608 00:36:37,583 --> 00:36:40,782 It's very clear. And these rings can extend, 609 00:36:40,782 --> 00:36:45,022 there's just not one ring, but you can have several rings. 610 00:36:45,022 --> 00:36:48,965 And so this is the result of what we call diffraction. 611 00:36:48,965 --> 00:36:54,667 Now I want to mention to you another phenomenon which is um 612 00:36:54,667 --> 00:36:59,838 also the result of diffraction. And that you can see when there 613 00:36:59,838 --> 00:37:03,008 is fog. If you have fog fog consists of 614 00:37:03,008 --> 00:37:07,762 extremely small water drops, just like the ones in clouds. 615 00:37:07,762 --> 00:37:10,765 I mean fog is a cloud, let's face it. 616 00:37:10,765 --> 00:37:15,352 Then if you for instance had the headlights of a car you 617 00:37:15,352 --> 00:37:20,44 could see your own shadow onto the fog away from the direction 618 00:37:20,44 --> 00:37:23,693 of the car. And that means you could see 619 00:37:23,693 --> 00:37:27,591 then around your head the same kind 620 00:37:27,591 --> 00:37:30,191 of stuff that you saw here. A glory. 621 00:37:30,191 --> 00:37:33,238 We call this a fogbow. That's just a name. 622 00:37:33,238 --> 00:37:36,507 The Germans have a much better name for that, 623 00:37:36,507 --> 00:37:40,742 they call it Heiligenschein. And heilige person is someone 624 00:37:40,742 --> 00:37:44,532 who is a saint and so you see this around your head, 625 00:37:44,532 --> 00:37:48,841 so it is sort of the -- the radiation that you would expect 626 00:37:48,841 --> 00:37:51,813 from a saint. Quite a few years ago I was 627 00:37:51,813 --> 00:37:55,529 invited to visit the Soviet Union. 628 00:37:55,529 --> 00:37:58,65 And they took me to the Caucasus, to show me the 629 00:37:58,65 --> 00:38:02,767 six-meter telescope which was at the time the largest telescope 630 00:38:02,767 --> 00:38:05,091 on earth. And I went there for a few 631 00:38:05,091 --> 00:38:07,017 days. And I noticed much to my 632 00:38:07,017 --> 00:38:10,935 surprise that every night at five o'clock there would be fog 633 00:38:10,935 --> 00:38:14,587 coming out of the valley overtaking the telescope so you 634 00:38:14,587 --> 00:38:18,571 could make no observations at night but that was a detail and 635 00:38:18,571 --> 00:38:21,56 what I realized is that the fog was coming up, 636 00:38:21,56 --> 00:38:23,95 the sun was there, that was the west, 637 00:38:23,95 --> 00:38:29,104 and the valley was at the east, and so I said to myself my 638 00:38:29,104 --> 00:38:33,266 goodness if only at the right moment of time I am there then 639 00:38:33,266 --> 00:38:37,639 the sun is my light beam and my shadow will fall onto this wall 640 00:38:37,639 --> 00:38:41,307 of fog coming towards me, this is my chance to make a 641 00:38:41,307 --> 00:38:44,129 wonderful picture of this Heiligenschein. 642 00:38:44,129 --> 00:38:46,527 And I'll show you that I succeeded. 643 00:38:46,527 --> 00:38:50,689 I will first show you a picture of this bizarre observatory. 644 00:38:50,689 --> 00:38:54,004 This is the six-meter telescope in the Caucasus. 645 00:38:54,004 --> 00:38:58,095 And this is indeed you could every night around five-thirty 646 00:38:58,095 --> 00:39:02,192 really like a wall the fog will just coming 647 00:39:02,192 --> 00:39:05,633 towards you overtake you. You can see the timing of my 648 00:39:05,633 --> 00:39:09,398 picture was extremely crucial. If you wait a little bit too 649 00:39:09,398 --> 00:39:13,488 long then the fog overtakes you and then the sun would obviously 650 00:39:13,488 --> 00:39:17,123 not be effective anymore. I wouldn't see my shadow on the 651 00:39:17,123 --> 00:39:19,071 fog. So I only had a small time 652 00:39:19,071 --> 00:39:22,057 window, maybe thirty seconds, maybe one minute. 653 00:39:22,057 --> 00:39:25,822 But I succeeded and you will see w- Saint Walter coming up. 654 00:39:25,822 --> 00:39:29,262 There is Saint Walter. You can see here my arm and you 655 00:39:29,262 --> 00:39:32,191 can see here my camera. 656 00:39:32,191 --> 00:39:38,146 The camera must be exactly at the center of the glory. 657 00:39:38,146 --> 00:39:43,877 And that's where it is. I have the camera right here 658 00:39:43,877 --> 00:39:50,281 and here you see the rest of my body, isn't that terrific? 659 00:39:50,281 --> 00:39:54,888 This can be used as proof sooner or later. 660 00:39:54,888 --> 00:39:57,697 OK. Back to our last three 661 00:39:57,697 --> 00:40:02,079 questions. The last three questions deal 662 00:40:02,079 --> 00:40:06,859 with the polarization of the bow. 663 00:40:06,859 --> 00:40:11,29 Are the bows polarized? And the answer is yes, 664 00:40:11,29 --> 00:40:16,509 they are enormously polarized. Why are they polarized? 665 00:40:16,509 --> 00:40:22,221 You can answer that question. But I will answer it for you. 666 00:40:22,221 --> 00:40:28,228 The light that contributes to the rainbow is the light whereby 667 00:40:28,228 --> 00:40:33,94 the minimum angle for delta is about a hundred thirty-eight 668 00:40:33,94 --> 00:40:38,469 degrees and if you go back to that 669 00:40:38,469 --> 00:40:43,81 first transparency you will see that that's the case when the 670 00:40:43,81 --> 00:40:47,459 angle of incidence is about sixty degrees. 671 00:40:47,459 --> 00:40:52,532 You can download that again. And that means that the angle 672 00:40:52,532 --> 00:40:55,736 of refraction is about forty degrees. 673 00:40:55,736 --> 00:41:00,097 Forty degrees is very close to the Brewster angle. 674 00:41:00,097 --> 00:41:05,348 The Brewster angle if you go from water to air you -- medium 675 00:41:05,348 --> 00:41:11,371 where you are is water and you bounce it off this medium, 676 00:41:11,371 --> 00:41:16,049 so you go from water to air, the tangent of the Brewster 677 00:41:16,049 --> 00:41:21,152 angle is N two divided by N one. N two is where you're going. 678 00:41:21,152 --> 00:41:23,364 That's air. So this is one. 679 00:41:23,364 --> 00:41:26,766 N one is where you are. That's the water. 680 00:41:26,766 --> 00:41:31,784 Let's just call that the index of refraction one point three 681 00:41:31,784 --> 00:41:34,761 three. If you calculate now what the 682 00:41:34,761 --> 00:41:39,864 Brewster angle is you will find that the Brewster angle theta 683 00:41:39,864 --> 00:41:44,447 Brewster is about thirty-seven degrees. 684 00:41:44,447 --> 00:41:49,275 That is only three degrees different from this angle which 685 00:41:49,275 --> 00:41:53,596 is the crucial one which contributes to the rainbow. 686 00:41:53,596 --> 00:41:58,424 So you're only three degrees away from the Brewster angle. 687 00:41:58,424 --> 00:42:02,914 So what happens as this light comes in i will make a a 688 00:42:02,914 --> 42:08 separate drawing