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Single Slit Diffraciton

Patrick Ford
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Hey, guys, in this video, we're gonna talk about single slit diffraction. So what happens to the light as it passes through a single slip, as opposed to what we saw before with the double slit? All right, let's get to it now. Light shone through a double slit had unexpected results, as we talked about. If you don't consider diffraction okay, if you do not consider diffraction, then it's an un obvious result. And obviously, back before they understood diffraction, they had certain expectations for the experiment, and the experiment turned out differently. Likewise, like Shone through, a single Slip also displays this same sort of unexpected result, which we call a diffraction pattern, which is alternating peaks. Sorry, alternating spots of brightness and darkness, right? The big difference between the double slit experiment and the single slit experiment is concerned with the central bright spot. Okay, in a double slit, the central bright spot is just as wide as all of the others. It's the same with as all the others, so every single bright spot across the entire screen is gonna be of uniform with. But in a single slit. The central bright spot is actually twice as large as all of the other ones. It's also considerably brighter, so that central one is definitely going to be larger than any of the other bright ones. But all the other bright spots, all the other bright fringes are gonna have the same width. Okay, that only applies to the central bright fringe. All of the dark fringes have the same width in the single slip, just as they did in the double slit experiment. Okay, let me remind myself, so we can see this figure like in a double slit. The diffraction pattern is produced due to interference. Okay, The big difference between the double slit and the single slit is that in the double slit, you actually have two sources of light that air interfering in the single slip. You have one source of light. It's just that light leaving at the top part of the slit and light leaving at the bottom part of the slit does not leave at the same angle. Okay. Light leaving different parts of the slit leave at different angles. Okay, so you have all of these different angles that the light can travel at leaving both slits. Okay, sometimes two beams of light will arrange themselves so that when they arrive they're both at a peak, right when they arrive on the screen there, both at a peak. This produces constructive interference and like that constructively interferes produces bright fringes. The amplitude of the light increases under constructive interference. Other times you can have a wave arrive at a peak. One wave arrived at a peak and another wave arrived at a trough. And when you have a peak and a trough meeting, you have destructive interference. And like that, destructively interferes, produces a dark spot or a dark fringe. Okay, because with destructive interference comes a smaller amplitude for the interfered wave. Smaller amplitude means it's darker. Okay, just like we did for the double slit experiment. We talked about the single slip conceptually, and now we want to actually talk about the mathematics of solving single slit problems. Where are these fringe is actually located. OK, now, the key difference in the math between the single slit and the double slit experiment is that in the single slit experiment, you Onley have an equation for bright friend. Sorry, dark fringes. Okay. In the double slit, we had two equations, One for the bright fringes. One for the dark fringes. For the single slit. We Onley have one for the dark fringes. Okay, dark fringes air located at angles given by sign if they t m equals m lambda over de. Okay, where m is our indexing number this time. Okay, but because em index is the dark fringes and not the bright fringes, it turns out that a requirement is we cannot have m equals zero. Okay, this is crucial to remember there's no m equals zero index for dark fringes due to a single slip. Okay, so what we have here is we have the first dark fringe or the M equals one dark fringe. We have a corresponding M equals one dark fringe on the other side. And then we would have the M equals two dark fringe on the top side and the bottom side, right? And then we could say some arbitrary M. The Mpath dark fringe is given by Fada Sub m where Fada follows this equation. Okay, Now, solving problems with a single slip is going to be exactly the same as solving problems with a double slit. Okay, The first thing that we're gonna do is we're gonna draw the figure that I have above me in the green box for a single slip, and it's gonna look identical to the figure for a double slit. Okay, let me minimize myself so I could draw this off to the side. All right, so here's our single slit. Right. This drawing is going toe look absolutely identical to that of a double slit. The Onley differences that I physically drawn one slip instead of two. Now I'm gonna draw that central axis, okay? And let me read the problem before continuing. Ah, 450 nanometer laser is shone through a single slip of with 4500. millimeters. If the screen is at a distance of 140 centimeters away from the slit, how wide is the central bright spot? Okay, so the screen is a distance of 140 centimeters away, which is equivalent to 1.4 m. Okay. And what we're looking for is the width of the central bright spot. So the central bright spot looks like this, right, and it will continue and there'll be a second bright spot and a third bright spot, etcetera. The equation that we have for the single slip tells us the locations off the dark fringes. So we know this is the M equals one dark fringe. This is the equivalent M equals one dark French on the bottom side. And this is a fatal one, that first dark fringe angle. And this is fatal one as well. And notice this distance is just that distance that we're looking for. This is the width, which I'll call w. That's the width of the central bright spot. Okay, now, to make solving this problem easier, we can notice that the top triangle triangle above the horizontal axis and the triangle below the horizontal axis are identical. So this height, why one and this height, Why one are the same. Okay, now all we need to do is find the angle fate a one using our equation so that we confined. Why one? So first, remember that our equation for the location of dark fringes in a single slit experiment is M lambda over D, where M starts at one. Remember, this is different then the location for the bright spots on a single slit experiment where M equals sorry. M started at zero. We're looking for theta one, so it is just gonna be one. What's Lambda? It's a 450 nanometer laser. Nano's 10 to the negative nine. What is the single slit width D O K. D was also different in the the single slit than it is in the double slit in the double slit. D represented the separation between the two slits for a single slit. D represents the width of the single slit. Okay, we're told that the single slides a width of 0.1 nanometers. So this is 0.1. Sorry millimeters and millions 10 to the negative three. Okay. Plugging this into a calculator, you're gonna find that this is 45 or that fate a one 0.26 degrees minus itself again. Now we can see we've just found fate. A one with this triangle right here. We can then find why one Okay, so I'm gonna redraw that triangle. The angle is data one which we know is 0.26 degrees. The base is 14 m and the height we called. Why one Okay. And once again, the width is not gonna be. Why won the width is actually going to be two times. Why one? Okay, so let's solve this triangle. Notice that we have the opposite edge and the adjacent edge to that angle. So we can say that the tangent of the angle is equal to the opposite edge, divided by de adjacent edge. And so all I have to do is multiply 1.4 up to the other side of the equation, Plug this into a calculator and we get an answer of 0.64 m, which is equivalent to 6.4 millimeters. Don't forget, this is not the final answer. Okay, This is simply why one the width w, which is what we're looking for, is two times why one It's twice this height or the some of those two heights, but they're the same. So this is to time 6. millimeters, which is 12.8 millimeters. Okay, Now, don't forget that the central bright fringe is twice as wide as all the other bright fringes. So what would the width of all the other bright fringes be? It would just be Why one which was 6.4 millimeters. Alright, guys, that wraps up this video on the single slit diffraction. Thanks for watching