### Video Transcript

So, in my video with Steven
Strogatz about the brachistochrone, we referenced this thing called “Snell’s
law.” It’s the principle in physics that
tells you how light bends as it travels from one medium into another, where its
speed changes. Our conversation did talk about
this in detail, but it was a little bit too much detail. So I ended up cutting it out of the
video. So what I wanna do here is just
show you a condensed version of that because it references a pretty clever argument
by Mark Levi, and it also gives a sense of completion to the brachistochrone
solution as a whole.

Consider when light travels from
air into water. The speed of light is a little bit
slower in water than it is in air. And this results in a beam of light
bending as it enters the water. Why? There are many ways that you can
think about this, but a pretty neat one is to use Fermat’s principle. We talked about this in detail in
the brachistochrone video. But in short, it tells you that if
light goes from some point to another, it will always do it in the fastest way
possible.

Consider some point 𝐴 in its
trajectory in the air and some point 𝐵 on its trajectory in the water. First, you might think that the
straight line between them is the fastest path. Strogatz: The only problem with
that strategy though, even though it’s the shortest path, is that you may be
spending a long time in the water. Sanderson: Light is slower in the
water, so the path can become faster if we shift things to favour spending more time
in the air. You might even try to minimize the
time spent in the water by shifting it all the way to the right. Strogatz: However, it’s not not
actually the best thing to do either. Sanderson: As with the
brachistochrone problem, we find ourselves trying to balance these two competing
factors. Strogatz: It’s a problem that you
can write down with geometry. Sanderson: And if this was a
calculus class, we would set up the appropriate equation with a single variable 𝑥
and find where its derivative is zero.

But we’ve got something better than
calculus: a Mark Levi solution. He recognized that optics is not
the only time that nature seeks out a minimum. It does so with energy as well. Any mechanical setup will stabilize
when the potential energy is at a minimum.

So for this “light into a media”
problem, he imagines putting a rod on the border between the air and the water and
placing a ring on the rod, which is free to slide left and right. Now, attach a spring from the point
𝐴 to the ring and a second spring between the ring and point 𝐵. You can think of the layout of the
springs as a potential path that light could take between 𝐴 and 𝐵. To finagle things so that the
potential energy in the springs equals the amount of time that light would take on
that path, you just need to make sure that each spring has a constant tension, which
is inversely proportional to the speed of light in its medium.

Sanderson: The only problem with
this is that constant tension springs don’t actually exist. Strogatz: That’s right; they’re
unphysical springs, but there’s still the aspect of the system wanting to minimize
its total energy. That physical principle will hold
even though these springs don’t exist in the world as we know it. Sanderson: The reason that springs
make the problem simpler though is that we can find the stable state just by
balancing forces. The leftward component of the force
in the top spring has to cancel out with the rightward component in the force of the
bottom spring. In this case, the horizontal
component in each spring is just the total force times the sine of the angle that
that spring makes with the vertical. Strogatz: And from that out pops
this thing called “Snell’s law,” which many of us learned in our first physics
class.

Sanderson: Snell’s law says that
sine of 𝜃 divided by the speed of light stays constant when light travels from one
medium to another, where 𝜃 is the angle that that beam of light makes with a line
perpendicular to the interface between the two media. So there you go! No calculus necessary.