Refraction: Behind the Rainbow

So in my last post we saw how rainbows are the product of refraction, but this is just one aspect of light interactions, and one specific case of refraction. When light travels from one medium to another, there is always some sort of interaction. Sometimes this is reflection, like when the light hits a really hard surface like a mirror, or that floor your mum insists you keep shiny, although this means you can’t walk on it in your socks.

But if light can travel through the second medium (like water, plastic or glass) then we can get refraction, which means a change in direction of the light beam, although doesn't always have to lead to a rainbow. It’s easiest to see in our experiments when we use a really thin beam, so we can get accurate measurements, but to start with we are going to imagine our wave as rows and rows of people to see what really happens to make this bend.

Light is a wave (a transverse wave, more on those at another time) so we can imagine rows of people, holding hands, marking off sections of this beam, marching forward in rows. When light travels through air, like when the row of people walk on good ground, good progress is made, it can travel quickly and in a straight line. But imagine hitting some really horrible sticky mud, the kind that tries to rob your feet of its wellies. If you hit that straight on, then everyone will slow down at the same time, but if one end of the row meets the mud (the light beam meets the water) at an angle, then this slowing down will happen from one end first.

This has two effects, the first is that the angle your row is walking at changes direction (remember, there’s no map to follow here, these are mindless walking robots), and secondly, the rows get closer together as everyone slows down. The gaps between the rows are called wavelengths, as they mark sections along the wave, se we can see that the wavelength shortens.

Well that’s all very well, but how do we know how much our light beam will change?  Surely that would be useful for someone? In fact it’s really important to opticians, who use exactly these sorts of calculations to make sure that the refraction of light through glasses is just right so that you can see perfectly. Time for a little bit of maths.

To find the angle of the resultant ray r (the one that is in the water), the important pieces of information we need to know are the incident ray (incoming ray) angle i, and the refractive index n of the air and the water. This is a special number relating to how hard it is for light to travel through something, very similar to a scale that you could make up for our poor rows of people struggling in the mud, to let them know if it was ok mud or really really bad bog.

So, we either find out these refractive indices, or if we come across them in an exam we should be given them as part of the question. Alternatively, if you have both angles i and r, and the refractive index of one of the materials, you can work out the other.

This equation (Snell's Law, thank you Mr. Snell) can be rearranged to find any of the variables once you know the other three.

Let’s finish this off by wondering why we see the different colours in the rainbow, now we know what happens when refraction occurs. So, we know that white light is made up of all the colours of the rainbow. These are all mixed up together, so we can imagine this as our rows of people wearing different coloured outfits. Now, the wavelength of the different colours is ever so slightly different, so the red people will be walking slightly faster than the yellow people and so on. Now, when they all hit the mud in this big tangled mass these different walking speeds have the effect that the angles the rows alter by are slightly different. So instead of all walking in a jumble, they have been separated into the rainbow colours.