 In this video, let's derive the expression for intensity in the Young's double slit experiment. So we'll derive expression for intensity of the resulting wave. And we'll do that as a function of its phase difference. Phase difference is usually denoted by phi. Now just a quick heads up, this most of this video is gonna be visual. So a way for you to logically visualize what's going on. All right, so let's begin. You may be by now familiar with the setup. We have two holes and there's a light source behind it. And the light hits the two holes. They act like coherent sources of light. And then the light meets on the screen at different different places to produce an interference pattern. And what we want to do now is figure out when they meet at every point, what will be the intensity or the brightness of the light? That's what we need to figure out. How do we do that? Well, intensity of any oscillation, intensity of any oscillation, I'm just gonna write I, turns out to be proportional to the amplitude of the oscillation squared. To the square of the amplitude of the oscillation. And don't worry, we'll visualize all these complex terms. We'll do that in a second. But because I want to calculate, because intensity is proportional to amplitude squared. That means to figure out the intensity of the light here, I need to figure out what will be the resulting amplitude when the light meets. And so let's just assume that the amplitude of the waves produced by these is A. So then we'll figure out what the resulting amplitude is going to be when they meet. And we'll be able to figure out when you square that, we'll get the resulting intensity. All right, now before we begin, a lot of complex terms. Let's start with something. Let's start with coherence. What do we mean by these sources are coherent? Here's how I like to visualize it. Coherent sources means that their phase relationship is locked in time. Here's what I mean. So if you were to look at these oscillations, notice they are in sync with each other. But what's important is that is locked in time, meaning forever they will be in sync with each other. That's coherent sources. And this is another example of coherent sources. Over here, notice when one thing is, they're oscillating in the opposite direction. Why are they coherent? Because, again, that relationship stays locked in time. They will always oscillate in the opposite direction. Get that? So that's the meaning of coherence. Phase relationship is locked. All right, let's consider the simplest example of coherence. The one in which they are oscillating in sync with each other. And if these are sources of light, they will start producing light waves. And these light waves are in three dimension, but it's easier to visualize in one dimension. And now, this distance, the distance of the peak from the mean position is what we mean by amplitude. And if you square that, that represents, or it's proportional to the intensity of the oscillation or the intensity of the wave, okay? And if these waves are going to hit a point on the screen, imagine this is a point on the screen, then the waves are gonna make that point oscillate as well. And notice, that point will oscillate with the same frequency as this one, and it'll have the same amplitude. Can you see that? Same amplitude. Now, what we are interested in is what happens when they meet at a point on the screen. Then what happens to the resulting amplitude? Well, over here, notice their individual contributions, their individual oscillations are in sync with each other. Can you see that? Because this distance and this distance is exactly same, these two pictures are identical, these two oscillations end up becoming in sync. And so, if these two were to meet at one point, let me show you that now, then the resulting, the resulting oscillation would have twice the amplitude. Does that make sense? Look, each one has amplitude A and they're in sync. It's like two forces which are in the same direction, the resulting would be twice that. And so, the resulting amplitude would just be two A and the square of that would be the intensity, right? But here's the thing, in this example, these two waves travel the same distance to meet, but that's not always going to be true and that's what makes things interesting. So, let me show you what I mean. If you consider a point over here where the two light waves are meeting, then notice this one has to travel a shorter distance and this one has to travel a longer distance. Now, individually, they will not be oscillating this particle in sync with each other. Again, let me show you that. Again, here we go. This time I've kept the yellow closer to the particle compared to the blue because that's the case we're dealing with, yellow closer to the particle compared to the blue. Because of this, you will see that the wave from the yellow will reach the particle first. And as a result, before this particle starts oscillating, this would have finished some number of oscillations. And so, we say this has a phase difference compared to this. And how do we calculate that? Well, if this has finished one extra oscillation compared to this, we say it has two pi extra phase. One oscillation corresponds to two pi extra phase. If it has half an oscillation extra compared to this one, then we would say it has pi extra phase and so on and so forth. Every extra oscillation corresponds to two pi extra phase. And because there is a phase difference, notice they are not oscillating in sync with each other. So in general, when the two waves meet on the screen, the two particles may not oscillate in sync, which is their contributions may not be in sync. That's what I want to mean. And because their contributions are not in sync, now comes the question, what happens when they meet? Well, when they meet, because their contributions are not in sync with each other, the resulting oscillation will not have an amplitude twice. It can have any amplitude between zero and two way, actually. And so, our goal now is to figure out what that resulting amplitude is in general, because the square of that is going to be the intensity. So how do we figure that out? There are a couple of ways to do that. One is algebraic, another one is vectorial. Vectorial is visual, so let's do that. There are a couple of ways to do that. One is algebraic, another one is vectors. Now I like vectors because again, we can visualize it. So let's do that, one more, one last visualization. So the idea is you treat oscillations as a rotating vector. It's an imaginary vector. If you've studied alternating currents, same idea, we use phasors over there, it's the same thing. The whole idea is you draw a vector which has the length equal to the amplitude of the oscillation and it's spinning at the same rate of the vibration, so same frequency, okay? And you set that up in such a way that if you were to shine light from the side horizontally, whatever shadow gets casted on a wall, imagine there's a vertical wall over here, the tip of that shadow represents the position of this particle. That's how you represent a rotating vector. At any point, notice, right now, if you cast a shadow, sorry, shine light, you cast a shadow, again, the tip of that represents the position of this. Now why this is important because now I don't have to look at the position, I don't have to look at the vibration oscillations anymore. This vector, if I know what the vector is doing, I automatically know what the position of my particle is. And vectors are easier to add, at least compared to oscillations when you're adding oscillations. So what we're gonna do now is we're gonna take another vector for this one. And when these two waves meet, we're just gonna keep these two vectors together. So they're gonna meet now, there we go. And all we have to do now is figure out when you add these two vectors, what happens to the resultant? That resultant will represent our resulting amplitude. Does that make sense? That's what we need to do now. And what does the angle between them represent? The angle between them represents the phase difference. If the two particles were oscillating in phase, these two vectors would be in line. If the two particles were oscillating in the opposite direction, then these two would be opposite to each other, giving a phase of 180 degrees. So can you see that the angle directly represents the phase? And so I know how to add up vectors using the parallelogram law. This now represents the amplitude of the resultant. And just to be clear, just that we are on the same page, what this means now is that if I run this animation, then the shadow of this one on the vertical wall that represents the oscillation or the position of the resultant oscillation. This is our resultant oscillation. So for that, all I have to do is calculate what the amplitude, what this length is, that represents the amplitude of this. And the square of that will give me the intensity. So let's set that up. Here is the first vector that represents the amplitude of the first wave, first source. This is the second vector that represents the amplitude of the second wave. The angle between them is phi. That's the phase difference when they reach here. And the resultant of that represents the resulting amplitude. So we know how to add vectors. We know the formula to calculate the resultant of two vectors. So why don't you pause the video now and figure out what the resulting amplitude is. All right, if you're given this a shot, let's see. The resulting amplitude is, how do you calculate? Well, the formula is a squared plus b squared plus two a b cos theta. So the square of the amplitude is going to be a squared plus b squared plus two a b. So a b becomes a square cos phi. Okay, now we just have to simplify this. I get a two a square here. There's a two a square here. I'll take that common. And what I end up with inside the bracket is one plus cos phi. Can I simplify this further? Yes, I can. Using trigonometric relation, one plus cos phi is two cos squared phi over two. And so if I put them all together, if I multiply them, I get the resulting amplitude square equals four a squared cos squared phi over two. All right, I have amplitudes, but I want intensity. Notice I already have amplitudes squared. So these should be proportional to their intensities. So this is proportional to the resulting intensity. So I'll write as K times IR. IR is the resulting intensity. I'm writing K because this is not equal to, right? There is some proportionality constant. Similarly, this is proportional to the intensity of the incoming wave. So it'll be K times I naught times cos squared phi over two and the K cancels. And I get my relationship. I now have the intensity. The resulting intensity becomes four times I naught times cos squared phi over two. And notice I found what I wanted. I now have the expression for the intensity as a function of phase difference. In another video, we will play with this equation more and we'll get a deeper understanding of what really happens to the intensity when the waves interfere. But for now, just reflect on how we derive this. We spent most of the time visualizing and the math was just last few minutes. So what I like about this method is if you can visualize properly what's going on, you can derive most of that stuff with practice in your head, which means I don't have to remember any of these equations.