 In this video, let's explore the angle between wave fronts and rays of light, the angle between these two. And why should we do that? Well, we will see in future videos, it will be really important for us to be able to reconstruct wave fronts given the rays of light or given the wave fronts draw the rays of light. It will be really easy to do that if we understand the relationship between the angle between them. And that's why we're exploring that. Now before we start, let's quickly remind ourselves what wave fronts are. Well, I always like to go back and look at the animation representing Huygens picture. Wave fronts are basically set of particles which are oscillating in sync with each other, which are in phase with each other. Over here, if you look at these ripples, these ripples are what represent the wave fronts. Since these ripples are spherical, this is in three dimension, we say the wave fronts over here are spherical in nature. And why do we say that? Well, that's because if you look at a particular sphere, notice every single particle on that sphere will be oscillating in sync with each other. Look at that. They'll be oscillating in sync. And that's why we say that this sphere represents a wave front. And so over here, every single sphere which is centered at the source will represent a wave front. And what we want to do now is find the angle between the wave fronts and the rays of light. So if I were to draw those wave fronts, those spherical wave fronts, here they are, what would be the angle between these wave fronts and the rays of light if I were to draw them? That's what you want to explore. Now, before I do that, can you pause the video and think about this from this image? If you were to draw rays of light, what would the angle be? All right, let's see. If you were to draw the rays, those rays of light would be emanating outwards from here, right? Let's draw them. All right. That's how it would look like. And notice that means these rays of light would represent radii to these spheres because these spheres have a center at the source and these rays of light are also starting from the source and therefore they are radius. They form the radius to the sphere and what's the angle between the radius and the surface of a sphere? That's always 90 degrees. So the angle between wave fronts and the rays of light over here turns out to be 90 degrees. The question is, would it always be true? Well, let's see. Let's consider another wave front. Let's consider plane wave fronts. To get plane wave fronts, we'll have to go far away. So let's do that. Let's go far away from this source. That's when we get plane wave fronts. Now notice we have plane wave fronts. Look at the direction of the rays of light. Now, the rays of light are parallel to each other. If this was like very far away, they would be perfectly parallel. But more important, look at the angle between these. It's still 90 degrees. In fact, it turns out that this is a general case. Let me close this. It turns out that if you take any wave front, wave front of any shape and the ray of light at any point on that wave front, they will always be 90 degrees. Always 90 degrees. Now I could have said that and just close the video. But what's really interesting is why. Why should this be true? It's really interesting. So let me clarify with an example what I mean. So let's say I were to draw some random wave front. Let me make it random wave front that looks like this. So now if I were to ask you to draw the rays of light due to this wave front on this wave front, the way we would draw this, if this wave front was traveling to the right, I would say that over here, the direction should be like this. This is how it should be perpendicular. Over here, it should be traveling this way. The ray of light should be this way. Over here, the ray of light should be this way. Always perpendicular. You get the point, right? It has to be. And my question to you is why should this be true in general? I mean for the sphere and the plane, we saw it made sense to us. But why is this in general? Again, can you pause the video and think a little bit about why should this be true? And I'll give you a clue. Think about what would have happened if the rays were not perpendicular. Something would break. Something in physics would break. And I want you to think a little bit about this. Alright, hopefully you have tried. So here's how I like to think about it. I know that every single particle on this wave front is oscillating in sync with each other, right? That's a definition of wave front. Another way to say that, which is going to be helpful to understand this, is that every single particle must have finished exactly the same number of oscillations. If this particle has finished three and a half oscillations, all particles must have finished the same number of oscillations because they're always oscillating in sync. So now let's think about what would have happened if the rays were not perpendicular. So let's say that this ray over here, this ray over here, was not perpendicular. Maybe it was, I don't know, maybe it was somewhat like this. What would happen in this case? Well, let me zoom in. Then the way I like to do this is I would say that this ray can now be divided into two components. Okay, bear with me. It will make sense why I'm doing this. Two components. One component which is parallel to the wave front. So let me use yellow to represent that. There'll be one component of a ray parallel to the wave front. And there will be another component which will be perpendicular to the wave front. Okay. Now let's concentrate on the ray of light that is parallel to the wave front. We agree that the ray represents the direction in which the wave is traveling, right? The direction in which the light is traveling. So since there is a parallel component, we are saying that there is some light traveling along the wave front. And that poses a problem. Why? Because if there are two particles, if I consider two particles very close to each other, consider these two particles very close to each other, we are saying there's a wave traveling from here to here. If there was a wave traveling from here to here, then these particles could not be in sync. This particle would have started oscillating first and then this particle would have started oscillating. That's what happens along the direction of a wave. I mean if you look at a string for example, if this wave is traveling this way, can these two particles ever be in sync with each other? No. Because this would always have finished, this got hit by the wave first. So it would have finished oscillations, some oscillations more than this one. And similarly, because the wave is traveling parallel, this particle would have finished more oscillations compared to this one. There would be some particle over here which would have finished more oscillations compared to this one. And that breaks our idea of wave fronts because we said in wave fronts, all particles must have finished exactly the same number of oscillations. And therefore, they can never ever be a component of the direction which is parallel to the wave front. And so the only component of the direction that can be allowed is perpendicular. And that's the reason why the rays of light must always be perpendicular. And this is super useful because if someone tells me that there are rays of light going this way and they ask me what would the wave front look like, I can tell now. If it's in three dimensions and you have parallel rays of light, I know the wave fronts need to be this way because I know this will always be perpendicular. If someone tells me that the rays of light are like this and asks me what does the wave front look like, I can try drawing that wave front. I'll try drawing the wave front in such a way that will always be perpendicular to the direction of the rays of light.