Snell's Law

In the previous blog post I talked briefly about Snell's law, however, I couldn't cover this topic due to the fact that I did not know the maths required. Guess what! I know them now. Well, at least at a basic level. Let's get into it.

The thing I needed to learn was the "sine" of an angle. The cool thing is that in a right-angle triangle, like the one below, if you take one side of the right-angle ("a" below) and divide it by the opposite side of the right-angle ("h" below) then you get same number no matter what size the triangle is. 

As you can see a1/h1 is 0.708 and so is a2/h2 almost perfectly. This number is called the sine of the angle O1. By the way "h" is called the hypotenuse ... who came up with that name!

I have learnt how to work with a table of sines to calculate the sine of an angle. I use this one from NASA.

So, now we can understand Snell's law. It predicts how much the light will bend when it changes media based on the refraction indexes.

n1 and n2 are the indexes of refraction of both media as we explained in the previous post.

If we know both indexes of refraction and we know the incoming angle we can calculate the outgoing angle. For example, I pointed the laser to a water tank at 40 degrees. From the previous post we know air has n1=1.000277 and water has n2=1.333333. So, if I put these numbers in Snell's formula I get an angle of approx 29 degrees.

And that's what you can see in this photo, 40 degrees in and 29 degrees out.

A cool thing is that when we point the laser perpendicular, the light doesn't bend because the angle is 0 degrees, and the sine of 0 is 0 as you can see in the table of sines. 

Also, when light goes from a slow medium to a fast medium, if you point the laser at a big enough angle the outgoing angle will be 90 degrees. This is called the "critical angle". From this angle on all the light reflects back into the slow medium. Check the photo below. You can even see the laser hitting back at the table after bouncing 3 times in the tank.

Finally, here are my tips if you want to do this experiment at home:

• Lasers are dangerous. Don't point them at your eyes or anyone else's and do this experiment with an adult
• It's very hard to see the laser in clear water. You can add a little drop of milk and then you will be able to see the laser because it is crashing against the tiny milk particles and getting scattered. Plus, if you add only a drop or two it won't change the index of refraction
• The laser is also very hard to see in the air. In the action movies they use smoke to uncover the laser trip wires, but if don't have smoke you can spray water as in the photo below
• The green laser device I used in the previous photos is from an optics kit from Aliexpress and it produces a wall of laser instead of a single point which helps a lot

In the next post we will use Snell's law to understand lenses.

And that's it! I hope this post doesn't give you nightmares of hypotenuses chasing you around a triangle.