electricity & magnetism-31

(0:00) All of you have looked at the rainbows, but very few of you have ever seen one. Looking at something is very different from seeing it. And today I will make you see the rainbow in a way that goes way beyond the beauty that we can always experience, a way that you will always remember. And I would like to start asking you fifteen, perhaps simple questions about the rainbow.

questions.pdf 

(0:34)
1. The first question then is, would any one of you remember if you see a bow, whether the red color is outside, or whether the red color is inside?

2. And then I wonder about the radius of the bow. If this is a bow in the sky, something like this. This is the horizon. (Drawing a rainbow on the blackboard) It's clearly a perfect circle, and so the perfect circle has somewhere a center. And so that means there must be a radius R. You can measure that radius in terms of how many degrees. So what is roughly that radius, you've never measured it, but is it ten degrees, is it twenty, thirty, fifty, sixty?

3. The length of the bow. This has difference. You sometimes see a very long bow, sometimes a very short one.
4. What is the width of the bow? You see colors here. How wide is that strip of color, in degrees?
5. Perhaps some of you have noticed, that there is a difference in light intensity between inside the bow and outside the bow. Maybe you've never seen it. If there is a difference, where is it brighter? Inside the bow or outside the bow? 
6. What time of the day would you see bows?
7. Would you see rainbows in the north, east, south or west? 
8. Is there perhaps a second bow in the sky?
9. And if there is a second one, where should you look for the second bow?
10. And if there is a second one, what is the color sequence of the second bow. Is the red on the outside or is the red on the inside?
11. And then you can ask the same question: what would be the radius of the second bow?
12. And what would be the width of the second bow?

All these first twelve questions, in principle, you should have been able to answer if you really have seen a rainbow. The last three is more difficult.

13. The question is: are the bows polarized?
14. In what direction are they polarized?
15. and are they weakly polarized or are the strongly polarized?

(Asking the students) Who knows the answer to twelve questions? To the first twelve questions? Who knows the answers to more than ten? Who knows the answers to nine? eight? seven? six? five? four? Do I see a hand that's four? Good, good for you. Five? Four? Three? three. Good, that's really good. Two? One? And who knows the answer to zero? Most of you, right? (3:03) I didn't see most of the hands though.

All right. So I made my points. You looked at rainbows, but you really never seen them. I'm going to make you see them today.

(3:15)
What you see here on the blackboards is one drop of water. I put the sun, for simplicity, at the horizon. Later I'll put it little bit higher on the sky. Light from the sun hits this raindrop. I've only drawn one narrow beam, which hits the raindrop right there. And you see here the angle of incidence, which with Snell's law we call theta 1. I call it i here, cause it's nicer for me, more descriptive. It means incidence angle. 

Right at that point A, some of the light will be reflected, and some of the light will go into the water, we call that refraction. And Snell's law will tell me this angle r. Whatever goes in there, reaches point B, where there is a transition back to air. And so some of the light will come out there, and some of the light will be reflected inside. And then when it reaches point C, again there is a transition from water to air. Some of the light will be reflected inside the water, and some of it will come out.

(4:32)
And as far as the geometry is concerned, if this angle is r, then this angle is also r, this is also r, and this is also r. And the angle here is i. That follows with Snell's law, and I'll leave you with that. Notice that the light came in like this, but it comes back like this. So the direction has changed over the angle delta. And the angle delta is very easy to calculate in terms of i and r. Delta is 180 degrees plus 2i minus 4r. I want you to check that at home. The four r's come in here, one, two, three, four. And the two i's come in here, and there.

(5:24)
If now, I think of all possible narrow beams of light that can strike this raindrop. One that would strike it here, would have an i of zero degrees. And then it would be 10 degrees,and 20 degrees,and 30,and 40. And the largest value of i, is when the light strikes here, would be 90 degrees.

angles.pdf

(5:46)
And so I can calculate, for all these values of i, which obviously all of them occur. Sunlight strikes this raindrop, and all these angles of i are present. So I can calculate now, for all these angles of i, what the value is for r, and then I can calculate what delta is. r follows from Snell's law, and delta follows from this geometric relationship.

(6:13)
And what you would find now, very much to your surprise, that there is a minimum value of delta, which is about 138 degrees. That means this angle phi here, has a maximum value, which is very roughly about 42 degrees. And I will show you some numbers. You can download this, by the way, this is all on the web, on the lecture supplements.

(6:47)
Here, all I have done, I've taken i to be from 0 to 90 degrees, all these angles are possible, with Snell's law, using an index of refraction of 1.336. You see it at the bottom. I calculated r, and then the last column, using that relationship, I calculated delta. And indeed you see that delta starts at 180 degrees when i is zero, and then goes to a minimum of roughly 138, after which it increases again. And this now, is crucial, is key to the understanding of the rainbow.

(7:32)
Imagine now that I have one drop of water here, and sunlight comes in at all angles of i, not just at one, at all angles of i. Whatever you see here has of course axial symmetry. This is a spherical drop, and the light comes in like this. So light can go this way, or it can also go this way. And it can also go this way, and this way. So there's completely axial symmetry, so this whole drawing, you can rotate about this line here. And everything holds same in axial symmetry.

(8:21)
So therefore, if phi maximum, if this angle phi maximum is 42 degrees, then the light, that will go back in the direction of the sun, the light that go through the journey A, B, C and then comes out of the raindrop, that's all I'm talking about now. I'm not talking about this light that sneaks out here. It's this journey, A, refraction at A, reflection at B, and then coming out at C. That light comes out in the form of a cone. And the half angle of the cone, must be roughly 42 degrees.

(9:00)
And so I'm going to draw that cone for you. Like so, and like so, and you have to think of this as a cone, completely symmetric, axial symmetric about this line. And this angle here then is roughly 42 degrees. No lights can go here because that would mean that phi will be larger than 42 degrees and that's not allowed.

(9:38)
Now, comes something very important. The index of refraction of red light for water is about 1.331, and that translates into an angle of phi max which I can calculate now, translates into an angle of phi max, which is about 42.4 degrees. But blue light has a slightly different index of refraction, therefore has a slightly different angle for phi max. And the blue light has index of refraction of something like 1.343. Notice I have blue light that I don't use violet light. Violet is much harder to see with our eyes. So I always refer to it as blue light. And that has a value for phi max, which is approximately 40.7 degrees.

(10:40)
Different index of refraction means of course that if you know i that r is slightly different. Using Snell's law, you get slightly different values for r, and so you get slightly different values for delta, so you get slightly different values for delta minimum, you get slightly different values for phi max.

(11:00)
What does this mean now? That means if you look at this cone of lights that goes back into the direction of the sun, that the outer edge, the outer surface of that cone, which has the largest possible angle, this angle is now 42.4 degrees, must be red light because blue light cannot come out in that direction. Because the maximum angle for blue is 40.7, is not 42.4. So it's only red light that can come out at the angle. The half angle of the cone is 42.4 degrees. And the blue light is not going to make it until this angle here, I put it in here, is 40.7 degrees.

(11:50)
If you look inside the cone with an half top angle of 40.7 degrees, all colors can come back. All this is saying is that for red light, phi maximum is 42.4; it can also be there at 40.7, it can be 30, it can be 20, it can be 10, it can be 0. Lights that comes in here at i equals 0, reaches point B comes straight back. That's allowed; phi would be zero then. If all the colors can make it back inside this cone that I have given here a blue color, that would mean that your brains will tell you that there is white light. Because you see red light, you see blue light, you see green light, you see yellow light, and so your brains will tell you that that is white light.

(12:46)
If I had a screen here, with a small opening to let the sunlight in, and I ask you what would you see on this screen having only one water drop, then you would see the intersection of this cone of light with this screen. And that would look as follows. The outer edge, the outer circle, which is the intersection of the cone with your screen would be red. And then there would be an inner portion, whereby all colors can come back, so there would be white light. And then as you go further in from the red until you reach that white portion, then the last column that would be added is the blue. And you can already of course sense that all the action about the rainbow occurs here. And here there is no light. It's going to be dark because there is no way that phi can be larger than 42.4 degrees. And if light would appear on the screen outside that red, it means that phi would be larger than 42.4 degrees, and that's now allowed. So it's dark here; it's white light here. It's red light here and as you go further in you will finally see the other colors. And this really is now the key to the geometry of the rainbow.

(14:20)
I'm going to put you here. So here you standing, and let the sun again be near the horizon. It's always nice when you make the picture, easy reference that you have. You standing here, sunlight is coming in in this direction, and there's rain here. If it is also raining here you won't see a rainbow because the sun will not be able to hit this raindrops. So it is essential that it is raining in the direction away from the sun but that you can still see the sun. So here are these raindrops. All right. You're looking in this direction in the sky. You're looking up in the sky like so.  And I take you one raindrop and only one, but what I'm telling you holds for all raindrops in that direction. What will that raindrop do? That raindrop will produce a cone of light which goes back in the direction of the sun whereby the edge of the cone is red light, and this angle is 42 degrees, 42.4, whatever. What do you see? Nothing, because there is no light that can come from this raindrop into your direction. There's no light because that would mean that phi is larger than 42 degrees, and that's not allowed.

(16:00)
So you look high up in the sky, you will not see any light coming back from that raindrop. And think of the whole thing, that's axial symmetric, right? Not only there, but there and there, you would not see any light. Now I'm looking, say, at some raindrops which are here. I pick one, holds for anyone in that direction. And so now, I draw a line to this point, this is what I'm looking. I'm looking this direction, I take this raindrop. I could have picked this raindrop, I could have picked this one, which make no difference. What is that raindrop doing? Well, that raindrop is throwing back at the sun lights in the form of a cone, and the cone has this angle 42 degrees, 42 degrees. What do you see now? Look! You're looking straight into that cone. You're not near the edge, so you see white lights cause green light comes back at you, red light comes back at you, everything comes back at you. So you would say, huh, I see white light, not only there, but low in the horizon because the whole thing is axial symmetric, there and there.

(17:21)
So you haven't seen the rainbow yet. Now, suppose I ask you to look up in the sky at a very specific angle and I'll make this angle 42.4 degrees. So you're looking in the sky in the directions somewhere here. I'm not very sure that I have the angle just right. So I'm looking in the sky here. Pick one raindrop, which you could pick anyone, any other one. There's nothing special about that one. So the sunlight comes in like so. What is that raindrop doing? Now, it's throwing a cone light back to the sun. And it just so happens hit you. You're only looking at the outer surface of the cone, where only red lights can go because this is the famous angle of 42.4 degrees, and so you see red lights.

(18:29)
And since the whole problem is axial symmetric, you would see red lights if you look 42.4 degrees in this direction, but also 42 degrees in this direction away from this direction to the sun. So now you see how the bow is being formed by zillions and zillions of small water drops, with each of them in their own way contributes to your rainbow.

(18:57)
And we will now put the sun higher in the sky. So I put you here now, and let's say the sun is now like so. So you look at your shadow. You're on the ground, the sun is there, there's your shadow. Here's your shadow, here's your head, the shadow. And this is the reference line that we're talking about. That is this line that was this line. So where would you see your rainbow? You have to look now 42 degrees away from that line. 42 degrees away from that line. That would be red than. A little bit lower would be your blue. And so if I sketch in there, the bow would be sort like this. And the blue would be on the inside. And here you would have white light. And this would then be the red.

(20:00)
Always relative to this reference line. And this would be roughly, I call it, 42 degrees, which according to my calculation there would be 42.4. So whenever you know where the sun is, you look at the shadow of your own heads, and you have to go 42.4 degrees away from that direction from your eye to the shadow. And so what you're seeing now is that if the sun is low in horizon, then the bow will be high above horizon, if the sun is rising, then the bow goes down, and goes down and goes down. And by the time the sun is 42 degrees above the horizon, you're not going to see a rainbow anymore, unless the water is right where you are. If the water has a distance, you won't see a rainbow. So the higher the sun in the sky, the less likely that you would ever see a rainbow.

(20:49)
Before I show you some slides, before we go into answer some of the questions, I want to mention to you that what I went through here, is refraction at A, reflection at B, and coming out at C. You can go through the same geometry, and allow for one more reflection at C, and then the lights come out. So that it would come out at point D. And if you did that, you can convince yourself that there is indeed a second rainbow, and we call that the "secondary". We call this the primary. And the secondary has a radius. This one (the primary) of course has a radius of 42.4 degrees for the red, and the blue would have a smaller radius. The secondary has a radius for the red of 50.4 degrees. So in the red, 50.4 degrees, and in the blue, it's larger, so the blue is outside, 52.5 degrees. Is that what it is? 53.5.

(22:05)
The secondary bow is fainter, and it is also wider. You could see this separation in terms of angle is larger than the separation there, that's only 1.7 degrees, and this is more than 3 degrees. So there is a secondary bow, and the secondary bow is only about roughly 10 degrees above the primary. So if you see a primary at 42 degree angle away from your shadow, there and there and there. Go another 10 degrees, and you can see a second bow. As you could see, the color's reversed. Red is there on the outside, and blue is on the inside. Red is there on the outside, and blue is on the inside. Red is there on the outside, and blue is on the inside. And here it is all reversed. Secondary bow, the red is on the inside, and blue is on the outside.

(22:54)
So let's answer some questions now. Put the questions back up on again, and you would see you can now without any difficulty already answer 12 questions, the first twelve. Red is outside, that is not negotiable, right follows immediately from what we talked about, that is not an issue anymore. The radius is about 42 degrees. And, the length of the bow. Well, that depends on if the sun is high in the sky, then the length will be very small because this whole arch will then go down and we only have a little bit left. It can also be that it is only raining here, and it is not raining there. So depending on where it's raining and how high the sun is in the sky, that will determine the length of the rainbow. The width of the bow, you will think, perhaps naively, that if you subtract the 40.7 from the 42.4, then that will give you the width of the bow, which would then be 1.7 degrees in angle. However, you would overlook then, that the sun is not a point, but that the sun in the sky has a dimension of half degree. Each point of the sun, of course, makes it own little rainbow. So you really have to add roughly half degree. So the width of the bow is more like, something like 2 degrees rather than that 1.7. Maybe 2.2 degrees. So a little wider than which you would think, which has to do with the finite size of the sun. 

(24:24)
The comparison of the light inside and outside. Clearly inside the bow there must be a lot of light which you don't see outside the bow. Big difference you'll see in the slide shortly. The time of the day, well the sun has to be a little bit low, so you don't expect it in the midday. And you want the rain as well. The late afternoon, the early morning would be ideal. And you can figure out in what direction to look. If it's early morning, sunlight is in the east, you would see the bows in the west. And late in the afternoon when the sun is the west, you would see the bows in the east.

(24:55)
Yes, there is a second bow. It is about 10 degrees higher in the sky than the primary. And the colors are reversed, the blue is outside then, the red is inside. The radius is about 52 degrees, and the width of the bow is again the difference between this two numbers, which you have to add about half a degree. So you would think on the basis of this that it is more like three degrees, and you have to add half a degree. So it's about 3 and half degrees.

(25:27)
So let's now look at some slides. And the first slide you would see is a drawing that I made, which is meant to repeat some of what I told you. So you see a person standing there, and here you see the direction to the sun. Your shadow would be this long. This would be your head of shadow. This would be your feet. And if there are water drops here, and if there's no interference between the sunlight and the water drops, then in this direction 42 degrees, you would see this water drop red. And then the water drops which were here, you would see white lights from those water drops, and maybe from this one, you would see red as well as blue. And if you have water near your feet, there's no reason why you can't see the rainbow there either, but of course  you'll have to use some garden hose to produce water there.

(26:37)
The next slide then. This is a drawing made by the master himself. Newton, who was the first to fully understand the workings of the rainbow. You see here the primary bow. And for those of you who were sitting close you can see the direction of the sun: light going in at A, reflects at B, coming out at C, which he calls E, but that's a detail. And here you see then the secondary, whereby the light comes in, one reflection, two reflections, and now comes back at you, and that gives you the secondary. So, this is red and this is blue, whereas here this is blue and this is red. The next slides.

(27:15) So if the sun is high in the sky, I've done this many times when I was watering my garden. You get this for free. You might as well do this. It's great fun. You standing there, give you immense feeling of power. And it's completely clear, there's not a cloud in the sky. And there is no rain anywhere. But you spray water around you, and you see a beautiful rainbow, and so clean your feet. And all what matters is this 42 degree angle, relative to the direction from the sun to you. And this is what the shadow of my head would be. Very easy to do, and I would recommend that you try that when you get a chance. The next slides.

(27:53)
It's a painting from the 8th century Turkey. I see a hand here. Probably makes reference to the Bible. But I think it reads in the bible, "I do set my bow in the clouds." It's very nice and very dandy. In fact the colors are wrong, the sequence of the colors as a detail. It's a nice picture, but red has to be on the outside and blue has to be on the inside. Small detail. Next slides.

(28:19)
A few years ago, actually it's more than a few, when I was first lecturing 803, I knew I was going to talk about the rainbow, ideal for 803. And so I wanted to make some rainbows myself in my backyard in Winchester, and so there is water coming out of a water hose. And the sun was behind me and I took this picture and you see all the ingredients that we just discussed. Notice the red is on the outside, and the blue is on the inside. You see all these white lights, that is that light that comes back from the water drop. But if you go here where the angle of phi would be larger than 42 degrees, which is not allowed, you don't see any light coming back at you. So you look straight through and see the forest without any white light. So you really see already here sky, so to speak, is bright here, and darker there.

(29:10)
I needed a help from my daughter, and the next slide will show you that the poor darling were suffering badly. It was January, was freezing cold. She was crying. She was really crying and I felt guilty but I said, "look, you know, I really need this slide for my 803 class. It'll only take you an hour or so, and she still remembers it. I email exchanged with her yesterday, and she said that, "I was crying. It was an awful thing which you did to me." But look! But look! You know, you've got to do something, even if your daughter of a scientist occasionally has to suffer a little bit. So she's holding here the water hose and you see the same thing, red outside, blue inside, and clearly the white light. You see this is the white light that I discussed with you earlier.

(29:56)
The next slide will show you then a real rain.. Oh no, this is an artificial one I made over my driveway. If you want to see the secondary, you need a real dark background because the secondary is quite faint. And so that's why I did it over my driveway. And you see here the primary: red on the outside, blue on the inside. And here you see the colors reversed. You can also see perhaps that it is a little whiter. But it is much fainter so it's hard to tell. 

(30:24)
And the the next slide is one of the wonderful slides made by Dr. Johnson in New Mexico. Socorro, it is where the radio telescopes are, the Very Large Array. Now look, red on the outside, blue on the inside. The sky, you got to admit, it's (a halo all brighter even?) than it is there. And you may never have noticed that. You've looked at it, but you've never seen it. Then here you see the secondary, red on the inside, blue on the outside. There is a phenomenon that we've never discussed in 802 yet. Coming up, I think, next week. And that phenomenon we call it diffraction. Snell's law cannot deal with diffraction. That occurs when water drops are very very small, say smaller than a tenth of a millimeter. What you then get over and above the bow, you get areas in the bow whereby you get as we call that destructive interference. The waves begin to kill each other and you get dark bands. And you can actually see that here. With a little bit of imagination, you see here a dark band. And when that is the case, the water is always extremely small in size, and we give this a name: we call it supernumerary bows. It's not so uncommon.

(31:42)
If the water drops become exceedingly small, let's say, smaller even than fifty microns, then its diffraction phenomenon becomes so important that in addition to the dark bands, all the colors are beginning to wash out over each other, and that creates then a white rainbow cross the dark bands. And a student of mine, who was in my 803 lecture, sent me years later the next slide, his name was Carl Wales, this picture he took a 340 miles from the North Pole. He was at (? Island) at the time. This picture was taken at 2 AM at night in July, when the sun is above the horizon. And so this must be the result of very fine water drops which somehow are there in the atmosphere. It doesn't look like it's raining, but there must have been small water drops fifty microns or less. And here you see the white, you call that rainbow, and you also see beautifully the supernumerary bows, you see the dark band in there. The next slide is a close up of that. So you see here the white rainbow, cross the dark band, which is the phenomenon, the result of diffraction.

(33:02)
Now before I will discuss the polarization of the bows, because I still owe you the answers to the last three questions, now that I'm at it, I want to show you some phenomenon which are quite common, which you may never have seen even though they are quite common, and they're rather spectacular. The first slide that comes now shows you what we call the "twenty two degree halo". It's very common around the sun, it's very common around the moon. The red is inside, it has nothing to do with water. It's the result of ice crystals way up in the atmosphere. You can see it both in the summer as well as in the winter because it's very cold way up there also in the summer. This is very common. The reason why you and I don't see it that often because who wants to look into the direction of the sun? This angle is on the 22 degrees, it's not so far. But I would advise you to at least keep an eye on the moon because the moon also has this 22 degree halo. And of course it's much easier to look in the direction of the moon. There isn't any..no problem for your eyes. This is very common. I see it at least three or four times per month, and I always look for it of course.

(34:18)
The next slide shows you both the 22 degree halo as well as the 46 degree halo, which is way less common. It's very rare to see the 46. I've seen it only a few times, it's very rare. In addition to the 22 degree halo and the 46 degree halo, you sometimes see bright spots here. You see them here, and you see them here. They're really not circles, they are arches. And they have names. They called them sun dogs, mock suns. They have various names. I see this often in Boston. The 46 degree halo is rare. All these are the results, by the way, of ice crystals, ice crystals way up in the earth atmosphere. And there is a phenomenon that then you may have seen from an airplane. If your airplane flies over clouds, and you look at the shadow of your airplane onto the clouds, you may have noticed colorful rings around the shadow of your airplane.

(35:25)
And the next slide is such an example. I took this picture several years ago. You're always right at the center of the circle, so you can see that I was sitting just behind the wings. This is the result of diffraction, had nothing to do with Snell's law. It is the result though of very very find water drops, but not in the sense of refraction and reflection that the way we discussed it was the rainbow. This is the result of diffraction. It is extremely difficult, the explanation of the glory. And even ten years ago I read an enormous article about it, I think it was in Scientific American, which made another attempt to be very quantitative about the explanation, which is not easy. The radius of this bow, if you want to call it a bow, depends on the size of the water drops. If the water drops are very small, the radius is substantially larger. So as you fly over clouds, you may fly over different clouds, with different sizes of water drops. And on a very short time scale, really you see the radius of this glory change can be very rapidly, could be on the time scale of seconds to minutes. It's very dramatic, it's very clear, and these rings can extend. This is not one ring, and you can have several rings. And so this is the result of what we call diffraction.

(36:50)
Now I want to mention to you another phenomenon, which is also the result of diffraction, and that you can see when there is fog, if you have fog. Fog consist of extremely small water drops just like the ones in clouds. I mean, fog is cloud that faces. Then, if you for instance had a headlight of a car, you could see your own shadow onto the fogs, away from the direction of the car. And that means you could see then around your head the same kind of stuff that you saw here, a glory. We call it a fog bow. That's just a name. That Germans have a much better name for that. They call it "Heiligenschein". And a Heiligen person is someone who is a saint, and so you see this around your head. So it is sort of the radiation that you would expect from a saint. 

(37:50)
Quite a few years ago I was invited to visit the Soviet Union, and they took me to the Caucasus to show me the 6-meter telescope, which was at the time the largest telescope on Earth, and I went there for a few days. And I noticed much to my surprise that every night at five o'clock, there would be fog coming out of the valley overtaking the telescope, so you could make no observations at night but that was a detail. And what I realized is that the fog was coming up, the sun was there, that was the west, and the valley was at the east. So I said to myself, "my goodness, if only at the right moment of time I'm there, then the sun is my light beam, and my shadow would fall on this wall of fog coming towards me. This is my chance to make a wonderful picture of this Heiligenschein. And I'll show you that I succeeded. I'll first show you a picture of this bizarre observatory. This is the 6-meter telescope in the Caucasus. And this is in deed you could see every night around 5:30, really like a wall, the fogs were just coming towards you, overtake you. You can see the timing of my picture was extremely crucial. If you wait a little bit too long, then the fog overtakes you, and the sun will obviously not be effective anymore, I wouldn't see my shadow on the fog. So you only had a small time window, maybe thirty seconds, maybe one minute. But I succeeded, and you would see Saint Walter coming up. There is Saint Walter. You can see here my arm, and you can see here my camera. The camera must be exactly at the center of the glory. And that's where it is. I have the camera right here. And here you see the rest of the bow. Isn't it terrific? This can be used as proof sooner or later.

(39:56)
Ok! Back to our last three questions. The last three questions deal with the polarization of the bow. Are the bows polarized? And the answer is yes. They are enormously polarized. Why are they polarized? You can answer that question, but I will answer it for you. The light that contributes to the rainbow is the light whereby the minimum angle for delta is about 138 degrees. And if you go back to the first transparency, you'll see that that's the case when the incidence angle is about 60 degrees. You can download that again. And that means that the angle of refraction is about 40 degrees. 40 degrees is very close to the Brewster's angle. The Brewster's angle, if you go from water to air, meaning where you are is water, and you bounce it off this medium, so you go from water to air, the tangent of the Brewster's angle is n2 divided by n1. n2 is where you're going, that's air, so this is one. n1 is where you are, that's the water. Let's just call that the index of refraction, 1.33. If you calculate now what the Brewster's angle is, you'll find that the Brewster's angle, theta (Brewster) is about 37 degrees. That is only three degree different from this angle, which is the crucial one that contributes to the rainbow. So you're only 3 degrees away from the Brewster's angle. So what happens as this light comes in? I'll make a separate drawing.

(42:06)
This light comes in here practically unpolarized. You should remember the meaning of the parallel and perpendicular component the way I discussed with you earlier. But right here you're so close to the Brewster's angle that almost everything that reflects is now polarized in this direction perpendicular to the blackboard. And so what comes out here is also polarized perpendicular to the blackboard. And that would mean then that if you really think through what that means the direction of the polarization, it means that the bow is polarized here in this direction, here in this direction, here in this direction, and so on. And so if you take your linear polarizers, you can hold them like this, and you can actually see that the bow is polarized. And if you put your linear polarizer at a 90 degree angle then you can kill the rainbow. It's very highly polarized, nearly 100 percent. Those of the effect that the Brewster's angle is so close to the 40 degrees.

(43:12)
I like to go to Plum Island, which is one hour drive north of here. It's beautiful beach. And the ocean in the east, sun late afternoon in the west. And then the waves come in, and the waves splash up water. And I always look for the rainbow. I look at my shadow, sun is there, look at my shadow, go 42 degrees away from the shadow wherever there is splashing water. For a split second, you see the rainbow. Red outside, blue inside, just for a split second. Of course then there's no water anywhere. You have to wait for the next wave to come in, water splashes up, rainbow, gone. And so, few years ago, I took my friend Bill (?) to the beach. And I saw the rainbow. I always look for the rainbow, right? It's clear. And I pointed it out to Bill, I didn't make a big deal out of it because he's a physicist, so he knows that stuff. That's all I said to Bill, "hey, look, you know, look for the 42 degrees away from your shadow, you see?" He looked at it, and sees nothing. He looked at it again. Beautiful rainbow. Bill, nothing. And I got extremely annoyed at Bill. I said, "man, look, the rainbow is here. It's crystal clear. And you can't have it. What's wrong with you?" He says, "I don't think there's anything wrong with me, instead maybe there's something wrong with you. Yeah, just imagine." And then I looked at Bill, and I remember what he looked like. He looked like this. And then I said to myself, "Oh! I understand why he doesn't see the rainbow. He is wearing polarized sunglasses." And the polarization of the sunglasses is in this direction. So every time that I saw this beautiful rainbow here, he had his glasses in this direction. He killed the rainbow, so I said, "Bill, would you please take your glasses off?" He took his glasses off, and he said, "there's nothing wrong with you. I see the rainbow." So whenever you want to see the rainbow at the beach, please make sure that you don't wear your polarized sunglasses.

(45:22)
Now, I will demonstrate to you the rainbow. And what I will do is use one drop of water. This is exactly what we discussed earlier. One drop of water. Light goes onto the drop, comes back. Here is that one drop of water, and here is the light, it's going to blind you. You see this water drop? You see it when you put this to make sure your eyes don't get blinded. And this light will come in your direction. And in this water drop, all these things are going to happen, all these angles of i occur, and so light will come in this direction. And I will project it on the screen, and what will you see? You'll see white is on the inside, you see red on the outside, and that light that comes out near the edge, is very strongly polarized. I will show that to you. You can't see that with your polarimeters. Because once it hits the screen, it comes back at you and it loses that direction of polarization. So you cannot use your personal polarizers. So I'll lower the screen. I'll also make it dark and give you 30 seconds for your eyes to adjust to the light. Because you can imagine I have only one drop of water, so don't expect too much. You can't do much with one drop of water. And what you're looking at is really not the rainbow itself of course, the angle of 42 degrees is only measured from this one water drop, not from where you're sitting. If you're sitting very far in the audience, the bow looks smaller than you're close. Because you'll only see the intersection of that cone with my screen. But it has all the ingredients of the rainbow. And with a little bit of imagination, you may be thinking that you see the rainbow. That would be fine with me.

(42:23)
So, we'll make it dark, so that your eyes would adjust. We'll give it a few seconds for your eyes to adjust. Mean time I can find my polarizer is here.

(47:54)
And there it is. There it is. Would you agree that it's darker here than there? You understand now why that is. Would you agree that you see the red outside and you see the blue inside? And you understand now why that is. And this light near the edge here is highly polarized. And the direction is also clear: it's polarized tangentially to the bow, polarized in this direction here, and polarized in this direction here. I have here a polarimeter. And I hold it now in the beam to allow the light through. Now I rotate it 90 degrees, and I can kill the light. You see how highly polarized it is. If I go here, I hold it parallel to the direction of the electric field, so the light goes through now. And now I rotate it 90 degrees, and I can kill that light almost for 100 percent. I kill it here and I kill it here, provided that I hold it 90 degrees relative to the tangential. Now the next time that you go to see a rainbow, you'll look at that rainbow in a very very different way. For one thing you'll check that the red is on the outside, you'll make sure that it's bright inside the bow, that it's dark outside, you'll look for the secondary, and you'll convince yourself that the color sequence of the secondary is different from the primary. It is a contagious disease which you cannot resist anymore. Your life will never be the same. And I'll be very proud of you if from now on you'll always carry a linear polarizer with you. And I gave you three, so you'd better. So that every time that you see a rainbow, you can be absolutely sure that the bows are indeed polarized. And because you know it, you would be able to see way more than just the beauty of the bows than everyone else can see. Your knowledge will add something very very special. And maybe you'll be thinking of me. Thank you. See you Friday. 

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From Wenhsin - 4/18/07 5:59 PM