OK, well everyone knows that rainbows are made when light shines on water droplets in the air, but why do we see all the different colors and why is a rainbow a curved shape?
First of all we need to think about light. White light is actually made up of a mixture of all the different colors Light takes the form of waves and all the different colors have a different wavelength. Red light has the longest wavelength (720 nm) while violet has the shortest (380 nm).
These different wavelengths can be used to explain why you see all the different colors in the rainbow. When light hits water droplets in the air, it is bent slightly - this bending is called diffraction. The amount of diffraction depends on the wavelength - Red light is bent less than blue light so when white light meets the water droplets, it is separated into all the different colors
Now onto the complicated bit - why is a rainbow an arch? Well, to start with, it isn't actually an arch at all - its a complete circle. We just don't see the whole thing because the horizon gets in the way. Sometimes people on planes will see the entire circle with the shadow of the plane in the center.
Let's think about how exactly light is reflected by a raindrop. As can be seen in the picture below, light enters the raindrop, is bent slightly (how much depends on the wavelength), hits the curved surface at the back of the raindrop, is reflected and finally exits the raindrop (again being bent slightly depending on its wavelength). Consider the difference in angle between the light going in and the light coming out of the droplet - this angle is around 42 degrees. (Because all the wavelengths are bent slightly differently by the raindrop, this angle will vary everso slightly for each color).
So to be able to see the light reflected by the rain we must be standing directly in the path of this light. (Remember that we see things as colored depending on what light they reflect into our eyes - its the same in a rainbow. We see raindrops as red when they reflect red light into our eyes etc.) Imagine that the light from the sun is hitting a flat wall of raindrops (I know its not really like this in real life!). Only raindrops that reflect light into our eyes will appear as the rainbow to us, all the other droplets are still reflecting light but this light is missing our eyes so we don't see them. Another way to illustrate this differing reflection of light is to hold a shiny drinking mug on its side and look at the light reflected from the curved surface. You'll be able to see a band of light that will move depending on where your head is. The rest of the surface is also reflecting light but its heading off in the wrong direction to be detected by your eyes.
The picture below shows all those droplets that will reflect light into our eyes. Imagine rotating the gray triangle as shown (with the triangles short side making a motion like a windscreen wiper) to find all the raindrops that will reflect light with the correct angle. That's where the arch of the rainbow comes from - all these raindrops will reflect light at the correct angle to hit our eyes.
The picture below illustrates what happens to the light that hits raindrops at different points outside of the arch. The angle with which the light is bent is the same, and this means that none of this light will reach the eyes.
Just to confuse you now that we've sorted that out! A rainbow isn't actually in one plane (flat surface) like we've described above. All the raindrops between your eye and the rainbow are also reflecting light and the rainbow is actually a cone shape with the pointy bit held against your eye. However, if you were to look through a cone like this, you will see the round edge at the end, not the rest of the cone closer to you. You can try this with one of those little drinking fountain paper cups if you don't believe me....
You may also have noticed that often you don't just see one rainbow,
but two. The second is always much fainter than the first and
the colors are in the opposite order. This rainbow occurs from
light that has been reflected within the raindrop twice and has
escaped at a different angle.
