 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about rainbows. Well, strictly speaking, it's dispersion when the light goes through a sphere. But yes, it's about rainbows. Now, this particular lecture is part of the course Physics 14 presented on Unisor.com. I suggest you to watch this lecture from the website because it's part of the course. So the website has a menu. All the lectures are logically connected. And every lecture has notes, very detailed notes with pictures. So it's basically like a textbook which is divided into parts. And each part belongs to one particular lecture. So you have a video and textual presentation of the same material. The website has exams. You can take it any number of times. You want it just for yourself basically. Everything is free and there are no advertising. So it's just pure knowledge. Also the website contains prerequisite course math routines. Mathematics is a mandatory knowledge which you have to possess before doing the real physics. Okay, so back to rainbows. Everybody knows what rainbow is. Everybody probably saw it above the waterfalls, for instance, or after the rain. Just recently I saw the rainbow above the Victoria Falls in Africa. Very beautiful. Now rainbows have this circular shape and they have different colors, obviously. And we will talk today about why actually it happens. Well, the short answer is a refraction. But now let's just go into the details of this refraction. Yes, one more thing. There are actually a few kind of geometrical composition between sun observer and the water droplets which are in the air after the rain or above the waterfalls. Now this composition can be different, this geometry. And obviously different geometries have different results. Some of them just don't form any rainbows. Some do. And it basically, again, depends on how it's all arranged. Now I will address first a simpler form which is basically very rarely occurring. A simpler form is when you have the sun, you have some kind of water droplet here and observer somewhere here. Now this is a simpler. And why it's actually very rarely really seen? Well, because sun is very bright and the rainbow is formed somewhere in this area where the droplets are. And in this particular case when the sun is very bright you don't really see much. Another configuration where you usually see is something like this. Now the difference is that the light should go through a more complicated path in this particular case. So let me start from the simple one and then we will go to a more complicated. So the simple one is here. Now this is the sun and let me just enlarge this particular droplet of water. It would be easier for me to draw. Now from sun the light goes in, we can consider that the light goes in parallel white rays. So the parallel one goes like this. This is white light rays from sun. Now what happens in this particular case? Well, if you have the water droplet and you have some ray of light coming on its surface there are basically two things which happen. One thing is reflection of the surface of the water. Another is when it penetrates through the water but on the border between air and water the white light diffracts into different components because different wavelengths have different refractive index. And since white light is a combination of all the wavelengths from to the whole spectrum then this spectrum actually is opening up. And now how can I really draw it? First I have to have the perpendicular to a surface. The perpendicular to a surface of the sphere is obviously the radius. So this is my angle of incident. And angle of refraction is again different for different colors. Now the red light will have a smaller deviation from the original and correspondingly bigger refraction angle. The violet, so we will use just two opposite red and violet. Everything else is in between green, blue, etc. So this would be my red light. So it's a little bit deviated. This is my continuation of the white light. Let me make it with interruptions. It's deviated. Now violet would be deviated more. That would be violet. Okay, now so the beta is as I was saying a refraction angle. And obviously we have the law of refraction. Sine A times N air in this particular case, because we are coming from air, is equal to sine of beta times refractive index of water. So this is more or less unit one. And for water this is again different for different colors, but it's somewhere around 1.33. For one color it's less, another color is greater. Average is for yellow light is 1.33, considering. Okay, now let's just in this particular case forget about reflection, because reflection can occur here and it can occur here as well. But right now we are talking about a simple case, when there is no reflection everything goes through. So right now we have another refraction, because the light comes from the water to the air. Okay, so let's again put perpendicular reduces. Okay, so this is my, now this is my incident angle, beta prime for red. I didn't really put it for the violet, because the picture would be too complicated. So we are talking about red light only. So we will have this incident angle, and the refraction would be deviated from this particular original angle. So it will be somewhere here. So it goes, well let's put observer here. This is my eye of an observer, and it goes to the eye of the observer. Now violet obviously will go somewhere here. Okay, well, am I right with this thing? I think I should really put it a little bit less inclined than this. I would probably have to put it right here. Something like this, and like here. And the eye would be here. Now this is gamma. This is a refraction after the light. And now we can say that sign of beta prime times n water is equal to sign of gamma and air. Again this is about 1, this is about 1.33. So in this particular case gamma is greater than beta prime, right? Yes, and in this case beta is less than alpha. But now, look at this very important thing. Beta and beta prime are equal to each other. Because these are two radiuses, right? So it's an equilateral triangle. And therefore this is equal to this, which means these two are equal, which means alpha equals to gamma. Well, it's just an observation, not that it's very important. But actually yes, the incident angle of the white ray which is coming onto a sphere would be equal to a refraction index as the light goes out. So that's basically what it is. Now, here is what's important. If I see the red light here, will I see the violet which comes from the same ray of sunlight? No, because they are going into different directions. And if I see red, I cannot see from the same water droplet, I cannot see the violet as well. But there is a next water droplet from which, either on the left or on the right, from which the violet will actually come to me. So again, if on a smaller scale I have somehow light from one droplet and from another droplet at different, which basically diffracts slightly differently, I will have, so if this is red, this is violet. So I think of red and violet in two different places, because how I see? I see my red light on the continuation of this thing, right? So it's somewhere here. Now, if my neighboring water droplet sends me the violet in such a way that it actually goes from there, I see the violet here. So red would be in violet and everything in between, obviously. So I will see different colors in different places. And that's why we see basically different places for the colors in the rainbow. They are not mixing together, because if I see the violet from the same droplet, they would mix together again and make the white light for me. But that's not the case. The case is that red from one droplet comes to my eye. But into the same eye, the violet would come from another droplet, because there is a slight difference, and here we have a difference in angle, so I cannot see it at the same time, too, of them. But if I have somewhere here, I don't know, another droplet which goes with the same direction and hits the eye, then I will see the violet in different places. So basically my point was that since the light is basically split into... it's like one cone and then even wider cone, I will not see the same... I will not see different colors from the same ray of light. I will see red from one and violet from another, because the result of this one, again, somehow I don't want to draw it, it will hit my eye, or in this case. Okay, so this is about how the water droplet is reflecting light and it goes into my eye. What I was saying is almost impossible to see the rainbow when the water droplet in between sun and you, too bright sun, sun is too bright. So let's just draw another picture. Another picture would be a little bit more complicated, but nevertheless it's more practical. So now we have something like... we have sun here and the water droplet here. Now what happens is sun ray goes this way. Now let's talk about one particular color, let's say red. So in this case we will have this situation. I should really put it this way. Okay, this is a white and this is red. So the first things which should happen, it should go inside the water, because if it just reflected here, it will still be white. So it's not interesting, you're not interested in this reflection. So as I was saying, there is always part of the light is reflected, part of the light is going through and deviates from the original direction. Now here, observers somewhere here. So we don't see anything very bright over there, so we see the colors. Now here we have actually a reflection, so it goes this way. And now again, partially we have probably a reflection, it goes back into the water droplet and can again reflect, refract, etc. But let's just forget about this. Now let's talk about that it goes through. It goes through and this is my perpendicular and it goes somewhere here. So this is my eye. Now this is a red. Now the violet would be, well, kind of neighboring, but again it would be at certain angle in such a way that it does not go into the same eye. So the same white ray will produce only the red component which goes to me. It's a different white ray somewhere in neighborhood going against either the same or another, most likely another rainbow water drop. It will be again partially reflected, partially refracted and it will go into the same eye but it would appear to be, now in this case I see red light appearing somewhere here. This is my red light. Now the same violet will be reflected and refracted from another droplet which would be somewhere here and I will see it eventually in this direction. So this is more practical, though a little bit more complicated path of the light, partially refracted, partially reflected. And again as I was saying, I can actually have partial reflection here as well and it goes this way and maybe another reflection, again maybe somewhere I will see it, but it will be in a different place. So I will have more than one rainbow which is again, sometimes you have double rainbow which means that the lights partially reflected, refracted once and then if you have a second, then direction would be different obviously. It will still go into your eye from some droplet but again the red will appear in still another case. So one red light you will see as single reflection from inside the water droplet but if you have double reflection some other droplet will produce from the double reflection red light and you will see it in another place. So this is very important. It explains actually why you can see a rainbow concentric to each other. We will talk about why it's concentric too. What else? I think that basically explains why you have different colors in different places on the sky. So now obviously you understand there is nothing material except the water droplets in the sky and what you see as red or white light is a result of partial reflection, partial refraction of the sunlight from the water droplets. Now let's see now why do we have this circular shape of the rainbow. Okay, this is slightly different picture and again let me start from a simpler case whenever you have this is the sun and this is observer. Now obviously we need certain angle at which I see after the refraction, refraction whatever I see this is direction where I am getting the red light from some droplet. Now where are different droplets of water which are in exactly the same position relative to parallel ray of lights and an observer which give me exactly the same angle. Well, this is a sphere, right? So my point is that if you will have a circle of the same circle of certain radius now this is basically a cone if you wish. Now all these droplets which are here on this circle they all relatively to the observer are in exactly the same position because the parallel lights from the sun comes down all of them have the same angle so this angle between my vertical well we consider sun is on the top so between the vertical and every point on this circle this angle is exactly the same. So whatever happens here with this particular water droplets happens here and if the light from here comes here then the light from here also comes here and from there comes because they are all little spheres and they are absolutely symmetrical relative to this they are circularly symmetrical relative to the axis, to the vertical axis. So it's very important to understand that every one of these water droplets is in exactly the same position relative to sun and observer as any other. They are all equivalent which means if I see the red light from here I see the red light from here. So all the places in the space in the three-dimensional space where all the droplets produce the same red light or green or violet doesn't matter which color which comes to my eye. They are all on this giant circle with this angle being whatever like 40 degree, 42 degree doesn't really matter but some kind of a relatively narrow range of degrees. So these are all red. Now we were talking that the let's say violet I should really have from a neighboring let's say from here. So this is red and violet I will get from this water droplet. But again if I see from this I'm seeing from the same kind of circle of certain radius equal to the same value as this one and I will see all these droplets which are in the second circle concentric obviously to the first because it's all symmetrical relative to the axis. So all these droplets are in exactly the same position as this one which means that all these droplets which from perspective of observer are producing the violet light are on the concentric circle. So my point is that all the water droplets which are on a circle which is in the plane perpendicular to the direction to the sun and on the same obviously radius it's a circle with a center on the axis of symmetry they all produce the same light the same color of light so that's why we see reds that's why we see violets that's why we see whatever else green in certain order the order obviously depends on the wavelengths so from red to orange to whatever I don't remember yellow, green, blue etc. Okay, but why do we see only half a circle and not even half sometimes even a quarter well that's very simple actually it all depends on the position between us and the sun so if we stand on the ground this is the ground this is observer and sun is somewhere here now these circles which I'm talking about these are circles of water droplets so first of all there must be water droplets somewhere they may be here or they may be here but in any case if I see a circle which is perpendicular to this direction between sun and observer then the circle would be like this but I'm on the ground so I cannot see this piece obviously right so that's why I see only this piece and it all depends on the direction to sun again these droplets which produce the same color are on a circle which is in the plane perpendicular to this direction in the center lying on this line so that's why from the earth we see only maximum which we can see is half but what if the cloud is not there well then you don't see this and you don't see this you might see actually one piece and another piece now if the cloud of droplets is here then again it would be something like this circle and we will see this piece of rainbow now there are some pictures of the rainbow from the airplane now from the airplane now this is the ground this is airplane and this is sun let's say and let's say this is some kind of a cloud of water droplets then this is direction so you might see something like this so you might see the rainbow in different direction instead of this you might see it this way or if you have sufficient amount of rainbow of droplets you will see the rainbow for a full circle so sometimes you can't see the full circle rainbow it all depends on where are you where is the observer etc but obviously full circle can be only from the air because if you are in the ground then horizon obviously cuts the rainbow in both sides well that's probably about it I do suggest you to read the notes for this lecture because it has nicer pictures and also if there are certain exams which will be attached to the whole pot now this whole pot actually belongs to phenomena of light so at the end I will probably put some exam it's not ready yet but I will put it there so I suggest you to take exams it's very important actually to solve problems I think it's... you see most of the knowledge let's say about mass whatever is in this particular course mass routines or any other course you probably don't need it in your life the purpose of this education is just to develop your mind and solving problems is actually the best tool to develop your mind because the problems you will have to solve in your life are numerous and the skill of analyzing the situation and producing certain solution is very important for everything else in your life so mass is good as a tool I mean as much as you go to gym to basically increase your stamina, strength and something like this you don't really... I don't know... using the track for instance in the gym you don't use exactly the same track but you have to run sometimes you don't lift weights per se I mean there are some jobs where you have to lift weights that it's not such an often thing to do but you do lift weights just to develop your muscles so that's exactly what mass for your mind is it's what lifting weights and using the track for your muscles that's what mass and physics are for your mind but physics also has another aspect of this it's always interesting to know things about why things do happen around us it helps okay, that's it thank you very much and good luck