Snell's law proof using springs

Key Takeaways

so in my video with Steve stroat about the Briston 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 want to do here is just show you a condensed version of that because it references a pretty clever argument by Mark Levy and it also gives a sense of completion to the broyone solution as a [Music] 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 Fair principle we talked about this in detail in the brochis 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 his trajectory in the water first you might think that the straight line between them is the fastest path 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 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 try to minimize the time spent in the Water by shifting it all the way to the right however it's it's not actually the best thing to do either as with the Briston problem we find ourselves trying to balance these two competing factors it's a problem that you can write down with geometry 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 Levy solution he recognized that op 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 media problem he imagines putting a rod on the border between the air in 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 spring as a potential path that light could take between A and B to fin angle 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 that's right they're they're unphysical Springs but they're still the aspect of the system wanting to minimize it its total energy that that that physical principle will hold even though these Springs don't exist in the world as we know it 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 sign of the angle that that spring makes with the vertical and from that out pops this thing called Snell's law which many of us learned in our first physics class Snell's law says that sign 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 [Music]