 In this video, we will talk about where and how are the maximas formed in a single slit interference and also why does the intensity of the maxima decrease as you go further away from the center. We know that the condition for destructive interference is given by this relation right here. This is a sin theta equals to n lambda, where a is the width of the slit, lambda is the wavelength of the light and theta is the angular position of the minimas formed. Here this theta is the angular position of the first minima. So the first minima, this is formed when n is equals to 1. So theta, which is considered as a very small angle, so we can write theta in place of sin theta. The angular position over here, this is lambda divided by a. Similarly, for second minima, for this minima right here, n would be 2 and the angular position of the second minima, this would be 2 lambda by a. Now the central maxima is formed when there is no part difference among all the light sources from the single slit, theta is zero in that case. The first minima is formed at theta equals to lambda by a. Second is formed at 2 lambda by a. And we can see that there is one maxima between these two minimas. So it is safe to assume that the angular position of this maxima right here, this could be 3 lambda by 2a. This is lambda by a, this is 1.5 lambda by a and this is 2 lambda by a. Let's check if this is right, if this is the angular position of the first maxima. So if we make for the first maxima, this is where the light rays could be meeting at the first maxima. And the part difference between the top source, between the top most source and the bottom most source, this part difference right here, this is, this is 3 lambda, 3 lambda by 2. And we know that the part difference between the top source and the bottom most source was lambda for the first minima, for the second minima it is 2 lambda. So for the first maxima, it should be 3 lambda by 2, we are checking if that really makes sense, if that's right. And I'm calling these sources because in a single slit, all of the points, all of the points in the slit, they act as secondary sources as per Huygens principle, they then interfere and create this pattern. Okay, so the part difference between the top most source and the bottom most source is 3 lambda by 2. So they should destructively interfere, but the top most source will also destructively interfere with the source with which its part difference is lambda by 2. Here the part difference is 3 lambda by 2, but the top most source will destructively interfere even before, even with a source that is in between, because the part difference with that source will be lambda by 2. You will come across 0.5 lambda right, lambda by 2 before reaching 1.5 lambda. So let's try and find where that source could be. We know that for this case, we can write this as a sin theta, the part difference a sin theta is 3 lambda by 2, a sin theta is 3 lambda by 2. Now if we divide both the sides by 3, if we do that, if we do that, this becomes a by 3 sin theta and this will be then equal to, 3 will get cancelled, this will be just equal to lambda by 2. So what this tells us really is that all the sources that are a by 3 distance apart from each other, all the sources that are a by 3 distance apart from each other, the vertical distance of a by 3 from each other, they will have a path difference of lambda by 2 and if they have a path difference of lambda by 2, they should destructively interfere. So let's divide this, let's divide this whole a, this big a into 3 smaller a's. When we do that, this is let's say, like this, so this right here is a by 3, this is a by 3, this is a by 3. So the top most point, fill destructively interfere at a point source over here because a path difference between them, the path difference between them, that will be lambda by 2. This distance right here, this distance, this will be lambda by 2. But this is not true for just these two sources which are a by 3 apart, even a source which is let's say over here, this source and some source that is in the middle section maybe over here, this source is let's say in the middle of the top section and this source is in the middle of the middle section. Even the distance between them will be a by 3, this distance is also a by 3. So that means that even these two sources, even these two sources, the path difference between them, the path difference between them will be lambda by 2. So this distance, the shaded white part, this distance it is lambda by 2. It turns out for all the points in the top section, in the top one-third part of the slate, all the sources in the top one-third part will have a source in the middle one-third part with which they will destructively interfere because the path difference between them will be lambda by 2. If we pick let's say this point and this point right where the middle section ends. Even the distance between them is a by 3 and the path difference would be lambda by 2. Because of the top section, the top section cancels with the middle section. They destructively interfere with each other. So the top two-thirds of the slate, the top two sections, they do not contribute anything. So if there is a maxima over here, it's not coming from these two parts of the slate. But no two sources are destructively interfering within the last one-third part of the slate. Only this part of the slate, only this one-third of the slate contributes to the intensity at a point between the two minimas. So only this part of the slate leads to a maxima between the two minimas. And clearly this is much weaker than the central maximum. Because at the central maximum, the entire slate contributed. Nothing got cancelled from each other. They all constructively interfered. Here only one-third of the slate contributes in phase. So the intensity is much less. And we can say, we can say that maxima, that maximas are formed when the part difference, when the part difference is, this is equal to n plus 1 by 2 into lambda. So when n is equals to 1, A sin theta, this equals 3 lambda by 2. And there will be a maxima that will be the first maxima. When n is equals to 2, you will get, you will get this maxima right here. But why does it peak, why does it peak decrease? What is the intensity of the maxima decrease as you go further away from the center? When we were looking at the first maxima, that was the angular position was 3 lambda by 2A. We divided the slate into three parts. If you look at the second maxima, the angular position here would be, this would be 5 lambda by 2A. So here we will actually divide the slate into five parts. Let's see how that would look like. So let's say here we have, here we have a slate and we divide it in five parts. We divide it, we divide it in five parts. This is one, this is two, three, four and five. So in this case, what happens is all the sources in the top, in the top section of the slate, destructively interfere with all the sources in this part of the slate. Because the part difference between them remains to be as lambda by 2. And we can also see that mathematically, initially, in the top most source and the bottom most source, we could have written A sin theta equal to 5 lambda by 2. But now when we divide the slate into five parts, this becomes A by 5. A by 5 and this 5 goes away. So any two sources which are A by 5 distance apart, they will destructively interfere. And that happens with any source in the top part and any source in this part in the yellow part of the slate. And similarly, that happens with all the sources in this part of the slate and any source, any source in let's say, let's say this part of the slate. Even between them, the distance remains to be as A by 5, which comes out to a part difference of lambda by 2 and a destructively interference. That only leaves the bottom most part of the slate, only this part of the slate. And here, we see only one-fifth of the entire slate contributes or leads to a formation of a maxima on the screen. Only one-fifth of the slate contributes in this case. So the maxima's intensity is much less.