 Huygens says light is a wave and gives us a way to figure out how the wave fronts evolve. A wave front 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 Huygens, every point on this wave front acts as a source for secondary waves and the envelope of these secondary waves, a common tangent to them, represents a new wave front. This is great, but can it prove the laws of reflection? Say this is a mirror and here is an incoming plane wave front. When the wave front 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 wave front. So let's see if we can really prove that the reflected wave front 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's 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 Huygens principle, I have to draw a wave front. And the way to draw a wave front, remember wave fronts 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 said it to make sure it's perpendicular. I can draw the wave fronts wherever I want, but as you will see, the reason, because for reasons as you will see, I'm going to draw a wave front over here. This is going to be my incident wave front. And so this is our incident wave front. Incident wave front. I've drawn it dashed so that we don't confuse it with the incident rays. And remember this is perpendicular to our incident rays. That's the property of our wave front. All right. So how do we draw now our reflected wave front to do that? We need to draw reflected secondary waves from Huygens 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 Huygens sources. And when the incident wave front hits the Huygens source, it gets activated and starts giving out secondary waves. For example, right now this particular Huygens source has been activated and it's going to start giving out secondary waves. And as the wave front 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 wave front. 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 Huygens sources and we'll only consider this Huygens 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. Alright, now comes the question, how big should I draw that circle? If I had a compass in my hand, how big should that radius be? Because that radius represents the distance traveled by this wave, the reflected wave, in the time the incident wave front 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. Alright, hopefully you're 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. It's the same time it took this incident wave front 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 traveled by the incident wave. And we know the distance traveled 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 it 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 it goes. Okay. And what does that arc represent? This represents the reflected wave from this Huygens source. And at the same time, this Huygens source just got activated because now that incident wave front 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 a dash line from here to here and that would represent my reflected wave front. 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 you have drawn out the reflected ray. How do we draw the reflected ray? We'll have the wave front. Raise a parallel, a perpendicular to the wave front and so I have to draw perpendicular to this. I'll bring in my set square again. Yeah, I've already said it perpendicular. So I'm going to draw a ray from here to here. So here we go. This is one ray. And let me draw a second reflected ray. This is going to be my second reflected ray. And there we have it. These are our reflected rays. So we are done with the 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. Let's do this. Let's first concentrate on the incident triangles. Let me dim this. I'm trying to bring this angle into the triangle if you know what I mean. And you know that I'm going to look at this. This is the right angle because the wave front is always perpendicular to the incident ray. And therefore 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. I've brought the angle into the triangle. Similarly let's now look at the reflected triangle. And again feel free to pause in between and see if you can try this. 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 traveled 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 have to be equal. Therefore these two triangles are happening to be equal. And because they are 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. And so long story short from Huygens 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 separate video.