### Light Takes the Quickest Path

Can we actually prove this..? The angle of incidence is equal to the angle of reflection… why? Why not something else..? So if you were to be curious enough to ask that question… let’s see what we can do…

It’s a very beautiful thing… Right..? Let’s start with a story… let’s say there is a pole here… and there’s an other pole here, a taller one… and let’s say there’s a crow sitting on this one… and there’s a mouse running between these two poles continuously… right..? And the crow’s nest is on the other pole… now what does the crow want to do..? It wants to take this mouse and carry it to its nest… and the crow flies in straight lines… So you might be beginning to guess what we’re getting to with this… and the crow wants to do it as quickly as possible… let’s say… So you have a very simple situation here… So the question is: where must the crow catch the mouse? Should it catch when it is right close to that point..? …pretty close to this pole when it is right in the middle..? …where exactly..? How do we answer such a question..? Think about it…

Now, if you were imagine… the second pole… and reflect the second pole on that rod so that it comes down… you might be wondering why we’re doing this… but you watch in a couple of seconds and look at what happens… What is the shortest part between that pole and this end of the pole now..? It’s straight line that connects them… that’s very clear, because… between any two points the shortest distance is the straight line… Now why did we do this..? Because these two triangles now are similar… which means that… if that’s the shortest path between these two points… then the exact reflection path must be the shortest path between these two poles… which is what we’re trying to find… because in this case, the shortest path is the quickest path as well… So where will they intersect..? Where will the crow catch the mouse..? The part where that line intersects the rod… like you can see…

Now, what if these two poles are of equal height..? What happens..? Watch what happens if these two poles are of equal height… you reflect it…and draw a line, where does it pass..? right through the center of that rod… you reflect it back… now what do you see..? These two triangles are congruent..?

You already know what congruent triangles are… which means that these two angles are going to be equal… and those two angles are also going to be equal… and what have we just proved..? that the shortest path between this point and that point, touching the mirror, is going to be the point, such that it’s the center of that mirror… right..? Why am I talking about mirror all of a sudden..? Because, right now our story was… a crow taking a mouse and carrying it to the other pole, right..? Now when we do that… we realize that if the crow was light… then what does it become..? … light also wants to take the quickest path..? So, light will go through the center… and reach the other side, if this was a mirror…

Of course, the shortest path to take in general will be the straight line, connecting them… but we’re talking about the ray of light that gets reflected… So the question is… The ray that has to touch the mirror and still go there… and then we shall see the quickest path… and what’s it…? just to reach the center point… and by proving that, what have we proved..? The angle of incidence equals the angle of reflection. And, we have not told it as a law, but we’ve told that does light want the angle of incidence to be equal to the angle of reflection..? not really..! All the light is trying to do, is to make it so that it takes the quickest path… and it so happens that… that quickest path is such that the angle of incidence equal to the angle of reflection.. So, if we notice something… till now we proved already, that rectilinear propagation of light… or that light travels in straight lines can be proved using just one assumption… light is in a hurry! And, angle of incidence equals angle of reflection, can also be proved using the same thing… Light is in a hurry. Light wants to take the quickest path between any two points… Now, we have told it in a very general manner… or a very funny manner that light is in a hurry… there’s a name for this… it’s called ‘Fermat’s principle of least time’. In other words, Fermat’s principle: Fermat stated… that… between two points, light will always take the quickest path… that’s called Fermat’s principle of least time… and that’s going to be our focus, because, with that one statement we’re going to prove everything that we can do in this entire chapter … called ‘Ray optics’.