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Huygen's principle - reflection laws proof

Huygen's principle states that every point on a wavefront behaves as a source for secondary waves, whose common tangent (envelop) becomes the new wavefront. Using this principle, let's prove the laws of reflection. Created by Mahesh Shenoy.

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Video transcript

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 this is great but can it prove the laws of reflection say this is a mirror and here is an incoming plane wavefront when the wavefront hits the mirror the point at which it does that becomes a source for reflected waves and now a common tangent to all of these reflected waves represent the reflected wavefront so let's see if we can really prove that the reflected wavefront obeys the laws of reflection so here's our mirror let me draw a couple of incident rays of light let me bring in my pencil and my ruler as well okay so here is going to be my incident light let's say this is one ray and let me take this somewhere over here to draw a second ray as well because i want to draw a wave front so i need to draw at least two rays of light so here's my second ray of light and these are my incident rays now because i'm using heightens principle i have to draw a wavefront and the way to draw a wavefront remember waveforms are always perpendicular to the incident ray so to draw a perpendicular line let me bring in my set square now so here's my set square i've already set it to make sure it's perpendicular i can draw the waveforms wherever i want but as you will see the reason because for reasons as you will see i'm going to draw a wavefront over here this is going to be my incident wavefront and so this is our incident wavefront incident wavefront i've drawn it dashed so that we don't confuse it with incident rays and remember this is perpendicular to our incident rays that's the property of our wavefront all right so how do we draw now our reflected wave front to do that we need to draw reflected secondary waves and from fighting's principle then we can draw a common tangent and since these reflected secondary waves come from the mirror every single point on the mirror become our hygiene sources and when the incident wavefront hits the hygiene source it gets activated and starts giving out secondary waves for example right now this particular hygiene source has been activated and it's going to start giving out secondary waves and as the wavefront moves forward more and more secondary waves start getting activated as you can see and they start giving out secondary waves and a common tangent to all these circles represent our reflected wavefront now this is great but how do you draw this on a piece of paper well what's interesting is that we don't need all the circles to draw a tangent since we're drawing a tangent from this point one circle would be enough and so what we'll do is we'll ignore all the other eigen sources and we'll only consider this eigen source and draw one circle and then we'll draw a tangent from here to there and that's going to represent our reflected wave front so let's go back all right now comes the question how big should i draw that circle if i had a compass in my hand how big that should that radius be because that radius represents the distance traveled by this wave the reflected wave in the time the incident wavefront went from here to here now would be a good time to actually pause the video and see if you can answer this question yourself all right hopefully you've given this a shot the way i'm thinking about it is i know the time for which the secondary wave the reflected wave was traveling at the same time it took this incident wavefront to go from here to here and it also has the same speed the reflected wave has exactly the same speed as the incident wave so the radius or the distance traveled by this wave this reflected wave should be the same as the distance travelled by the incident wave and we know the distance child by the incident wave is this length it comes from here to here so if i had my compass i would take this much length and then put over here and take an arc in fact i do have my compass here's my compass i'm going to take this much radius and i'm going to bring that compass over here and i'll draw an arc somewhere over here so here goes okay and what does that arc represent this represents the reflected wave from this eigen source and at the same time this eigen source it just got activated because now that incident wavefront is over here and so the wave produced by this source is still a point which is convenient because now to draw the reflected wave front i have to draw tangent from this point or this wave point wave to this and i can directly do that using my ruler so again if i bring in my ruler i'm going to point this over here i'm going to try and make a tangent so yeah this is the tangent and i'm going to draw dashed line from here to here and that would represent my reflected wavefront there you go so what next well remember we're trying to prove angle of incidence equals angle of reflection and we draw those angles with respect to rays of light so we have the incident ray which means we have drawn out the reflected ray how do we draw the reflected ray well i have the wavefront and rays are parallel perpendicular to the front and so i have to draw perpendicular to this now bring in my set square again here i've already said it perpendicular so i'm gonna draw a ray from here to here so here we go this is one ray and let me draw a second reflector tray this is going to be my second reflected ray and there we have it these are our reflected rays so we are done with our construction this is also perpendicular and so finally now it's time to see if i equals r and you can pretty much look at the figure and guess that it has to be true but let's go ahead and prove it anyways to do that we have to first drop a normal so let me go ahead and drop a normal over here this is the point of incidence and this angle over here is the angle of incidence i while this angle becomes the angle of reflection r so how do we prove now we are in the geometry world and in geometry we have to always look for some familiar shapes and see what relationship we can find between them i can see two right angle triangles and we need to prove i equals r so maybe you can guess that somehow if we can figure out what is the relationship between the triangles and these angles then maybe somehow we can do it using some laws of geometry i know it's vague but i really want you to give this a shot this is the last piece of the puzzle and i don't want to steal that away from you so go crazy pause the video and give this a shot okay let's do this let's first concentrate on the incident triangles let me dim this all right all right i'm trying to bring this angle into the triangle if you know what i mean okay and do that i'm going to look at this this is right angle because the wavefront is always perpendicular to the incident ray and therefore this angle becomes this this total becomes 90 this becomes 90 minus i i'm not going to write that this becomes 90 minus i but now if you concentrate on this angle these two should also be 90 because this is normal it's perpendicular to the mirror and therefore if this is 90 minus i this should be i have brought the angle into the triangle okay similarly let's now look at the reflected triangle and again feel free to pause in between and see if you can try this since this angle is r i know this angle is 90 minus r and therefore this angle must be r that's great so let me write that this angle must be r so i've brought in the angles into the triangle and they look congruent to me and so maybe they are so let's see if we can go ahead and prove them since they are right angle triangles we have to prove we can use rhs postulate so first of all i see the hypotenuse to be common that's great then i have one right angle that is also common nice and now i need one more side well look at this side this side has to be equal to this side can you see why this is the most important thing well remember when we took that arc that compass we took the same distance as from here to here because the distance tower by this wave was exactly equal to the distance traveled by this wave which is from here to here and so because the reflected wave has the same speed as the incident wave these two sides are this have to be equal therefore these two triangles are happening to be equal and because they are congruent so they're congruent these corresponding angles should also be equal to each other this is the angle opposite to this side and therefore correspondingly this is the angle opposite to this side and so i has to be equal to r booyah victory for huygens as a long story short from heigen's principle because the reflected wave has the same speed as the incident wave that's the reason why we found i the two angles to be equal to each other and now maybe you can stretch this and you can guess that in the refraction that's not true because the waves don't have the same speed in this different media and that's why in refraction the angle of incidence is not equal to the angle of refraction but that's something we will talk about in a in a separate video