You know how frustrating it feels when a tune or half muddled lyrics is stuck in your head and you just can't put a finger on the song... I had one such tune for years. After lot of indexing ans searching through my memories, the only information my brain could come up with was , it ( perhaps ) starts with the words - rainbows and roses. And that it is something to do with my childhood memory.
One normal day, amongst the things in our weekday routine , I went to drop my son in his music class. I heard a song and I shrieked involuntarily ," THATS IT!!! That's THE song". And before the teacher could throw me and my son out of the class for disturbing her, I gave a sheepish grin and dropped my son and slid out with a huge content grin writ all over my face. I solved the puzzle finally. It was a song from the movie sound of music and it starts like this - raindrops and roses. Not bad. I was close. Maybe I associated happiness with rainbows. Understandable. I'm sure most of us do that. we all love rainbows don't we.. The colourful hue, the lovely bow shape.. My face still lights up if I see one, and I still make it a point to stop and look at it when it appears.
Anyhoooo... Let's try and forget about the dream, and get on to rainbows. What do we know already about them... rainbow is an arc that magically appears across the sky when it is raining during the day, keywords here being - arc, rain, day. It comprises of 7 colours - violet, indigo, blue, green, yellow, orange and red, in that particular order with red on top of the arc and violet towards the bottom. The colours are abbreviated or acronym-ed "VIBGYOR". Did you know, newton was the one who conducted a prism experiment and found out that white light comprised of these colours. And apparently, he first categorised them as 5 colours, and later went on to make it 7 colours, in line with the notes in music and with the days of the weeks.
And this phenomenon of splitting of the white light into 7 colours is called dispersion. Before getting into hunky dory details, let's talk motivation . Who amongst you has heard of the pot of gold at the end of the rainbow. How can we get there ? Maybe if we understand how rainbow is formed, we might be able to do some mighty calculations as to where the end of the rainbow might fall.. super cool isn't it? Wouldn't you want to try? Let's buckle up then....
Imagine a group of people, walking along each other, side by side, in a straight line. (This is called the marching soldiers analogy) Draw a line in front of them, just to simulate a different medium situation. ( The line is not parallel to the line of people) And tell them that they have to take long steps when they are in medium one , and as soon as they cross the line, they have to take short steps. If you carefully observe, the line of people, when they are crossing the line, change their direction. That is, their path gets bent. And once they cross the medium, they will go straight in the bent path. Why does that happen? When person1 reaches the line, he starts taking small steps. But, when he reaches the line, the others still haven't. They still take long steps. So the line bends and takes a different direction. Do you get it? That's exactly what happens to the ray of light too when it enters a denser medium. It gets bent while entering the medium. This phenomenon is called refraction.
Remember the concept of alternate angles? I remember the alphabet 'z' when someone says
alternate angles. My son, when in lower kg, used to say this when I said 'z'. " go to the right, slant right down, and go to the right again". Let's refer to the angle between the first go-the-right and the slant-right-down as angle one and the angle between slant-right-down and the second go-to-the-right as angle two. Apparently, angle one and angle two are alternate angles and are equal ( the prerequisite being, the two go-to-the-right lines are parallel to each other, and the slant-right-down line is a straight line)
That's it, no more new physics concepts, or no trigonometry concepts. Just imagine this. You are standing facing the rain, with the sun behind you. The sun rays reach the raindrops in front of you, get refracted, reflected and again refracted out as a band of colours. Each colour gets refracted at a different angle. So whichever colour reaches your eye, you see that corresponding colour on top. Isn't it? Look at this illustration. Red deviates the most, so I see red on top. Remember, Violet deviates the least. Raindrop1 , has split the light it received into 7 colours, and only the red reached my eye.. the rest of the colours went above my eye. As long as they didn't reach my eye, I can't see them. Raindrop 2, which was a little below raindrop1, sends orange to my eye and the rest go above and below my eye level. So , I see orange under red. You get the logic.. so on... till violet . That explains the order of the colours.
https://www.scienceabc.com/nature/rainbows-always-curved.html
https://www.scientificamerican.com/article/why-are-rainbows-curved-i/
One normal day, amongst the things in our weekday routine , I went to drop my son in his music class. I heard a song and I shrieked involuntarily ," THATS IT!!! That's THE song". And before the teacher could throw me and my son out of the class for disturbing her, I gave a sheepish grin and dropped my son and slid out with a huge content grin writ all over my face. I solved the puzzle finally. It was a song from the movie sound of music and it starts like this - raindrops and roses. Not bad. I was close. Maybe I associated happiness with rainbows. Understandable. I'm sure most of us do that. we all love rainbows don't we.. The colourful hue, the lovely bow shape.. My face still lights up if I see one, and I still make it a point to stop and look at it when it appears.
That night I had a dream. Lovely garden, full of roses. I love it there. There's thunder, it begins to drizzle .. A lovely rainbow appears ... And suddenly, out of nowhere, the music teacher comes sliding down the rainbow!! ( I'm going to give you a minute to process that..) Apparently, I woke up screaming , "She broke the rainbow. She broke it"... Hilarious isn't it. And partly scary too. ( absolutely no offence meant to the teacher.. Just the fact that the rainbow was broken scared me )
Anyhoooo... Let's try and forget about the dream, and get on to rainbows. What do we know already about them... rainbow is an arc that magically appears across the sky when it is raining during the day, keywords here being - arc, rain, day. It comprises of 7 colours - violet, indigo, blue, green, yellow, orange and red, in that particular order with red on top of the arc and violet towards the bottom. The colours are abbreviated or acronym-ed "VIBGYOR". Did you know, newton was the one who conducted a prism experiment and found out that white light comprised of these colours. And apparently, he first categorised them as 5 colours, and later went on to make it 7 colours, in line with the notes in music and with the days of the weeks.
Where were we... yes, dispersion. Dispersion is the splitting up of white light into its constituent colours. Before we get to the why, we need to know the 'when'. When does it split into 7 colours? It splits only when it enters into another medium of different density. Isn't it ? In newton's case, the prism and in the case of the rainbow, it splits when it enters water droplets. What happens to light when it enters another medium?
Let's think. Let's imagine a kid running, running on ground. He runs so fast, that he reaches the
sea shore ( we are assuming ideal conditions, like no traffic and that it's safe for a kid to run that far ) now he is running on the sea shore. He will involuntarily slow down isn't it... because of the beach sand. Now he runs to the waves and wants to play in the waves . His pace will slow down even further because of the water isn't it ? Now imagine another kid ( say his brother ) running along with him. He will also slow down when he reaches the sea shore and the sea, but the amount of speed he loses will be different from that of his brother. Both of them slow down in a different media, but the first kid slows down a lot and the second slows down a little . Different people react differently to similar situations isn't it . Remember this for now. Let's discuss another phenomenon.
sea shore ( we are assuming ideal conditions, like no traffic and that it's safe for a kid to run that far ) now he is running on the sea shore. He will involuntarily slow down isn't it... because of the beach sand. Now he runs to the waves and wants to play in the waves . His pace will slow down even further because of the water isn't it ? Now imagine another kid ( say his brother ) running along with him. He will also slow down when he reaches the sea shore and the sea, but the amount of speed he loses will be different from that of his brother. Both of them slow down in a different media, but the first kid slows down a lot and the second slows down a little . Different people react differently to similar situations isn't it . Remember this for now. Let's discuss another phenomenon. 
Now, along with the bending, we know that the ray slows down. ( like we discussed in the para, a little above ) what happens is, each of the seven colours, have their own way of reacting to the denser medium. All of the colours slow down, but they slow down at different paces. Hence they split and you get to see them distinctly. Lovely isn't it. I love it when the 'aaha' feeling settles on my brain.
Applying all this knowledge to a light entering a raindrop, this is what happens. Light from the sun, when it reaches the water drop, first, gets refracted, that is bent and moves tad bit slower. In the process, it splits into 7 colours. When this band of colours reach the back of the drop, they bounce from it like light would bounce from a mirror or a reflective surface. This is called total internal reflection.
To understand this, let's look a little deeper into refraction, that is, the Bending of light when it
enters a different medium. The different medium either can be denser or less denser than the first medium. And the angle of bending and he direction of bending depends on whether medium 2 is denser or less denser than medium 1. For example, when light enters from air to water, that is from less dense to more dense medium, the ray bends towards an imaginary perpendicular line drawn at the point of incidence of the Ray of light on medium2.
The inverse case, when light moves from water to air, it bends away from the imaginary
perpendicular line drawn at the point of incidence of the light on air. The amount of bending depends on the refractive index of the medium 2, let's say density of medium 2 for all practical purposes and to ease our understanding. And it also depends on the angle of incidence, that is, The angle with which the light strikes the medium 2. The more the angle of incidence, the more the Ray will bend away from the perpendicular line. At one point of time, it will reflect back into medium 1 itself instead of coming through medium 2. That is total internal reflection. And that incident angle after which the rays will reflect back is called critical angle.
This is not a new concept you know, even if you felt that the names and the terms are new. I'm sure all of would've seen a mirage. If not in a desert, for sure when travelling by road on a real sunny day. You might see puddles of water on the road that disappeared when you approached them. That is a classic case of total internal reflection.
Just to recap, white light enters the water droplet, that is from a less dense medium air to the more dense medium water, it refracts, and splits into 7 colours. Now when it reaches the back of the drop, it's time to come out. That is, from the more dense water medium to the less dense
air medium. It so happens, that at a particular angle of incidence, this ray gets total internal reflected back to the water medium and reaches the front of the water drop where it tries coming out of the water drop. Here, it gets refracted again and comes out as a band of colours. ( second refraction, so again speed of the individual colours and the angle of refraction differs and they separate out ) phew !!!
What would you do, if I told you that, that, was the easy part so far. We have accounted for the colours in the rainbow.. good job !! ( like how my son would put it ) So when sunlight enters the water drops in the rain, it splits into seven colours. If u remember the prism experiment from school, the rays split into 7 colours like a band . That is, straight line. There is no curve. So how does the bow in a rainbow appear.. this is the most crucial part with regard to the pot of gold. So pay attention!
What if I told you , the rainbow is actually not a bow but a circle. A 'raincircle'. Doesn't sound nice na. "Raincircle". I personally prefer the bow. The rainbow. It's got a nice ring to it! However, sadly so, it's a circle it seems. Sometimes, most of the times, we just don't get to the see the bottom part of the circle. And what's visible, becomes a bow! I'm sure you are already upset that this article is seeming like a science lesson, especially a physics one.. oh boy!! I hear you. I was scared of physics too from back then as a kid. So, just to bring in your trust back and more so to give you a break, try this at home. Go to your garden, where there's ample space. Turn your back to the sun, take a hose pipe in your hand and spray water in front of you. And when you see the 'raincircle' , come back here to continue. You can also alternatively search for 'make your own rainbows' in youtube or the internet. Or maybe watch this link from youtube.
For those of you who (correctly so) wondered how trigonometry would help us understand the world and why all of us were asked to search for theta in a scary looking polygon with many angles, I'm going to give you a good news. One concept of trigonometry here is going to help you understand the bow in a rainbow. Wow!! Finally, I get to use my trigonometry after all . This one is real simple, trust me.
alternate angles. My son, when in lower kg, used to say this when I said 'z'. " go to the right, slant right down, and go to the right again". Let's refer to the angle between the first go-the-right and the slant-right-down as angle one and the angle between slant-right-down and the second go-to-the-right as angle two. Apparently, angle one and angle two are alternate angles and are equal ( the prerequisite being, the two go-to-the-right lines are parallel to each other, and the slant-right-down line is a straight line) 
Now for the circle or the bow if you may, the angle in which the rays enter the water drop is key to the total internal reflection to happen and hence for the colour to reach my eye. Isn't it. Now let's think, raindrop1 has sent red to my eye. Which other raindrop can do that.. that is, which other drop can send red to me. The raindrops where the angle of incidence is the same. Right ? Those drops would send red colour to my eye. For easy visualisation, imagine a z here. That is, imagine the ray from the sun to the drop as the first go-to-the-right, and the Ray that eventually comes out of the drop and enters my eye as the slant-right-down. Imagine the Second go-to-the-right as an imaginary line parallel to the first go-to-the-right line. That's a complete z.
Applying the alternate angle theory, the angle the ray reaching my eye makes with this new imaginary line is all we need to worry about. Keeping this angle constant, I can make a
complete circle of water drops. Isn't it? That is, imagine holding a compass. Where the compass-hand holding a pencil is the Ray that reaches my eye, and the imaginary line is the other hand of the compass that remains fixed. Rotating the compass-hand u can draw a circle isn't it. And the angle of the compass is still fixed. So all the drops that fall in the circle, will send the same red colour to my eye. Because the angle of entering the water drop will be the same for all the drops and hence will produce the same beam of light where red reaches my eye. That explains the red part of the circle. And similarly the same logic will explain the remaining colours. Why don't I see the circle and why do I see the bow.... cause the below part of the circle is obstructed by the earth I'm standing on. Hence a rainbow.. and not a rain circle. Yay!! I told you I liked the ring to the rainbow!
Oh no, now does this mean, the beauty of the rainbow also lies in the eye of the beholder. It's an illusion is it ? If the angle the ray makes with the eye matters, then will it move when I move? Oh oh... will I ever be able to get to the end of the rainbow.. our ancestors were real clever people isn't it.. more than clever, I think the word wise would be more apt. They've left us stories about pot of gold at the end of the rainbow, only to make us go berserk about it and let us learn a lesson that we shouldn't pin our hopes on the elusive pot of gold which is hidden but instead spend the time earning and making our own pot of gold maybe. Anyways, don't worry. Illusion or not, we still have the rainbow, and magical or logical, it is a wow, a wonder of the world around us!
References
https://en.wikipedia.org/wiki/Rainbowhttps://www.scienceabc.com/nature/rainbows-always-curved.html
https://www.scientificamerican.com/article/why-are-rainbows-curved-i/
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