So, in my video with Steve Strogatz about the brachistochrone, we reference 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 wanted to 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 the 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 brachistochrome 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.
—3B1B: Light is slower in the water, so the path can become faster if we shift things to favor spending more time in the air.
You might even 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.
—3B1B: As with the brachistochrone problem, we find ourselves trying to balance these two competing factors.
—Strogatz: It's a problem you can write down using geometry.
—3B1B: And, if this was a calculus class, we would set up the appropriate equation with a single variable x and find 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 set-up will stabilize when the potential energy is at a minimum.
So for this "light in two 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.
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.
—3B1B: The reason 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.
—3B1B: Snell's law says that sine of theta divided by the speed of light stays
constant when light travels from one medium to another, where theta 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.