Snell's laws proof using Huygen's principle

hygiene says light is a wave and gives us a way to figure out how the wavefronts evolve a wavefront can be thought of as a set of particles which are oscillating in sync with each other so if this was say for example light then according to hygienes every point on this wavefront acts as a source for secondary waves and the envelope of these secondary waves a common tangent to them represents a new wavefront but can we use this to prove laws of refraction let's say we have light moving from one medium to another say air to water and here is our incident wavefront the moment it hits the surface of the water every point on the water behaves as a hygiene source and starts producing secondary waves in the water we're of course neglecting the reflected waves over here and notice the waves are slower in water compared to that of air and now when we draw a common tangent to these secondary waves we get the refracted wave front and indeed we see because the waves are traveling slower the refracted wavefront is bent towards the normal so now let's see how to draw this step by step on a piece of paper and see if we can prove snell's law so here's our air and water so let me draw a couple of incident rays of light so i'm gonna put my ruler over here and here's my first incident ray and i'm going to draw two incident rays of light and here's my secondary i'm going to draw them parallel to each other and so basically i'm imagining that the source is far away you might recall that when the source is far away the rays of light become parallel to each other the source is at infinity you can say and when that happens what happens to our wavefront remember when the source is at infinity the wavefronts are plane wavefronts and when i'm drawing over here they're going to be straight lines and how do i draw my wavefront over here well wavefronts are always perpendicular to the direction of the rays and so i can now bring in my set square and draw my incident wavefront so let me use dotted lines or dashed lines so that we can differentiate between reason wavefront so here's my incident wavefront all right so let me write that as well this is incident wave front and this is going to be perpendicular to our perpendicular to our incident rays and now to prove snell's law we have to draw the refracted wave front and we can do that using heightens principle hygiene says that every point on this can be thought of as a source and when the incident wave front hits that point it's going to start it you can imagine it gets activated and it starts producing waves secondary views spherical secondary views now the waves produced in the same medium are the reflected waves and we're going to ignore that because we are only interested in the waves in the water and the important thing as you will see now is that the waves in the water are slower than the waves in air and so now notice as the wavefront goes forward as the eigen sources get activated more and more secondary waves are formed but notice the secondary waves are slower than the incident waves and now a common tangent to all these circles will represent our refracted wave front and you can immediately see that the refracted wavefront is bent in fact it is bent towards the normal and i hope you can imagine if you had some different media when which the wave was faster than the first media then the refracted waveform would have been bent the other way around away from the normal okay but how do i construct this now how do i draw this in a piece of paper well i don't need all the circles to draw the tangent i just need two so i'll only consider two eigen sources one eigen source over here the first one that got activated i'll draw one circle over here and another eigen source which i'll consider is over here the last one which just got activated when the wavefront hit over here and so it has just started producing the wave so that wave is a point and so all i have to do then is draw a tangent from this point onto that circle so let's go ahead and draw that okay but how big should i draw that circle well remember that that the radius of that cell curve represents the distance traveled by the secondary wave in the time the incident wavefront went from here to here so if we are in a denser medium then the distant star would be smaller because the speed is slower in that case that radius should be smaller than this distance but if the second medium was a rarer medium then that radius should be bigger than this distance does that make sense so in this case since we are considering a denser medium water is denser than the air let's draw a circle which is which has a radius smaller than this distance and to do that i'm going to bring in my compass and i'm going to set the length of the compass to be a little smaller than this distance and i'll bring that compost over here and i'm going to make an arc so here goes okay so this is the secondary wave from this eigen source and at this moment the secondary wave from this eigen source has just started so it's just a point and so my refracted wave will be a common tangent which is basically a you know a tangent from here to here so to draw that tangent let me bring in my ruler i'm going to move that so that becomes a tangent it's going to look somewhat like this okay and here we go refracted wavefront and again let me go and write that this is our refracted wavefront but remember i have to prove snell's law which means i have to draw incident uh incident ray and refracted ray because i want to bring in angle of incidence and angle of refraction so how do i draw the refracted rays now well rays will be perpendicular to the wavefront so i'm going to bring in my set square again so here's my set square i've already set it perpendicular to the wavefront and so i'm going to draw one ray from here to here so here is our first refracted ray and let me draw a similar one over here as well this is going to be our second refracted ray there we go and again this is perpendicular so we're done with the construction now let's go ahead and prove snell's law to do that we have to prove sin i by sine r is a constant so we need to draw angle of incidence and angle of refraction so for that i'm going to drop a normal we can drop a normal over here so let me oops let's all right so here's the normal and let's write down our angles of incidences and refraction so this is going to be our angle of incidence between the incident ray and the normal and between the refracted ray and the normal that's going to be our angle of refraction and i need to prove that sine i by sine r is a constant how do i do that well now we're in the geometry world so let's think mathematically we're talking about sine i and sine are trigonometry so we require right angle triangles and i can see two right angle triangles over here so maybe if i can somehow bring these angles into the triangle if you know what i mean then i can use sinai i can calculate sinai and sine r from the triangle and then i can take the ratio and somehow prove that that's really a constant i know it's a little big but i really want you to give this a shot so why don't you pause the video and see if you can prove this okay if you have tried let's see let's try to first bring in the angles into the triangle let's concentrate on the incident right angle triangles let me dim this so if you look over here because this is perpendicular i know that if this is i this has to be 90 minus i this has to be 90 i'm going to write that over here 90 minus i okay but if i look at this whole thing should also be 90 degrees because normal is perpendicular to our surface and so if this is 90 minus i then that means that this should be this should be i so let me write that over here this should be i so i brought in the angle let's do the same for our refracted triangle okay now if we concentrate here because again normal is perpendicular to the surface if this is r this has to be 90 minus r but wait i don't want 90 minus r let me write that this is 90 minus r 90 minus r but i want r i want to bring in r well look at this right angle triangle since it's a right angle triangle these two angles should add up to become 90 because it's already 90 total should be 180. and so if these two should add up to become 90 if this is 90 minus r this should be r there we have it we have brought in i's and rs into our triangle now let's calculate sine i by sine r and see what happens so if i do sine i sine i divided by sine r what do i get what do i get so if i look at sine i in this triangle it's going to be the opposite side divided by the hypotenuse for sine r it's going to be the opposite side divided by the hypotenuse same hypotenuse so the hypotenuse cancels out so that's nice but the big question now is how much is this opposite side how much is this opposite side how do i figure that out well again think about it what does this length represent that's the distance traveled by the incident wave front in some time from here to here and that distance traveled i can write in terms of speed because i know that in this equation speed is going to come i know that we've already seen that before therefore let me write this in terms of speed speed equals distance into time so distance is speed into time so if the wave is traveling at a speed v1 let's say then i can say that oh this distance has to be v1 times the time the time it took to go from here to here does that make sense the distance is speed into time similarly over here the wave has a different speed in this case a slower speed let's call that as v2 and the time taken is exactly the same as the time taken to end from here to here so time taken will be t itself and so over here it's going to be this distance over here is going to be v2 times t so our sine i is going to be v 1 times t the opposite side divided by the hypotenuse i'm just going to call that as hypotenuse because i know it's going to cancel out and divided by sine r is going to be v 2 times t divided by the hypotenuse and so notice the hypotenuse cancels out because there's the same hypotenuse the t cancels out and i end up with v1 over v2 which means that sine i by sine r is equal to the ratio of the speeds which is a constant for a given media for air and water this will be one number so it doesn't matter what i what angle of incidence i choose sine i by sine r will be that same number so we have indeed proved snell's law sine i by sine r is a constant and hygiene also helps us understand what that constant is it's the ratio of the speeds in the two media victory for hygienes to check your understanding i highly encourage you to reconstruct and prove snell's law all by yourself do it right now when everything is fresh and if at any point you get stuck no worries just go back and revisit that part of the video this is the best way to learn new concepts trying to recall stuff rather than just summarizing