Interference of electromagnetic waves
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Young's double slit problem solving
- [Voiceover] I think we should look at an example of Young's Double Slit. Let's consider the light of wavelength 700 nanometers. That would mean this distance right here between pix is 700 nanometers apart shines through a double slit whose holes are 200 nanometers wide. That means from here to there is 200 nanometers and they're spaced 1300 nanometers apart. That means from the center of one to the center of the other is 1300 nanometers. If the screen is placed three meters away, here's our screen, and it is three meters away. Then what would be the distance from the central bright spot on the screen to the next bright spot? The central bright spot is going to be, well, it's in the center. You can follow this line. Look at it, it's kind of like a shadowy line. Right there, there's our bright spot constructive point. How far will it be from that point vertically to our next one? Our next one is right here. You can see these lines of constructive interference. This one's about right here. So the question is, how far is this distance right here? How do we figure this out? Well, we're going to use the equation we found which is to say d sin theta. Remember, this is the formula right here. D sin theta is the path link difference. That's supposed to equal m lambda. Sometimes you'll see this as n lambda but n reminds me of index over fraction which confuses me so I'm going to write that as m. All right, so what do we do? D, what is the d? We got all these numbers in here. D is defined to be the distance between the holes. So d in this case is this 1300 nanometers. I got 1300 nanometers times the sin of an angle. What angle are we going to talk about? Well, what we want to know is this distance here. So I'm going to worry about this angle. I'm going to worry about the angle from, here's my center line, from there to the point I'm concerned with is this first bright spot pass the center point. That's the angle I'm concerned with. This angle right here. Equals m, what should m be? Well, this is zero. Should I put zero? No, because I don't want the angle to the center. I want the angle to the first one over here. So this is m equals one. The first order bright spot constructive point. Times the wavelength, what's the wavelength? The wavelength of the light we said was 700 nanometers. Now you might be wondering, "Wait a minute. "What about this 200 nanometers wide piece of information? "Do we have to use that?" No, we don't. In fact, that does not play in here. The only time that this spacing is important. It's not going to change your calculation. You just need the spacing to be small, small enough that you get enough diffraction, that you get a wide enough angle of diffraction that these two waves will overlap significantly enough that they'll create the interference pattern that you want to see over here to a degree that's visible. Okay, but we don't need it. We only know how to use that one in our calculation. All right, so we calculate the angle. Here we go. I'm going to find sin of theta. I'd get the sin of theta equals, one is just once, so I'll divide both sides by 1300. I get 700 over 1300. The nanometers cancels nanometers. As long as I'm in the same units, it doesn't matter. I'll solve this for theta. How do I get theta? I got to use inverse sin of both sides. So the inverse sin of sin theta is just theta. The inverse sin of this side gives me 32.6 degrees. That's what this angle is right here. 32.6 degrees but that's not what I was trying to find. What I'm trying to find is this distance, not this angle. How do we do that? Well, this side, this side right here, I'll call it delta y because it looks like a vertical distance. This is the opposite side to that angle. That's the opposite side. We know the adjacent side. This adjacent side we were told. This three meters away from the screen. The screen was three meters away from the double slit. How do we relate the opposite side to the adjacent side? Sure, I know how to do that tangent theta equals opposite over adjacent and our opposite is delta y over three meters in this case. If I solve this for delta y, I'm going to get delta y equals, multiply both sides by three meters times the tangent of theta. Theta we solved for right here, 32.6 degrees. If you multiply all that out, you get 1.92 meters. That's how big this would be from here, center point, to the next bright spot is 1.92 meters. That's how you solve this problem. You got to use a little trigonometry. Once you get your angle, you got to relate it to a distance vertically on the screen. This is a common problem using Young's Double Slit. I will say one more thing. Oftentimes, a popular question, a follow-up question is, what would happen if we reduce the distance between the slits? What would happen if we take this distance between slits and we make it smaller? We scrunch these together. Would this angle get bigger or smaller? Well mathematically, let's just look at it. If the distance over here goes down, in our formula, if d goes down. Notice I'm not changing the wavelength. That's the term by the laser I fire in here. This wavelength staying the same. So this whole side has got to stay the same because m is still one, this point. What's going to happen to theta if the d goes down and the whole thing has to remain the same? The angle's got to go up because sin of a bigger angle will give me a bigger number. Sin of zero is zero. Sin of something bigger than zero gives me something bigger than zero. The bigger the theta, the bigger sin theta. So as d decreases, sin theta has got to go up. That's mathematically why that I can show you in here. Check this out. Let's just take this. Let's take this here and I'm going to move this whole thing down and watch what happens. Can you see the shadowy lines spread out? See how they're spreading out? Then we come back together and those shadowy lines have constructive. It's kind of ironic. They look like shadows but they should be bright. It's just the way this visually looks. We get more and more lines. This way, they get squashed together. As you push d closer together, they get smaller. They spread apart. The bright spots spread apart. So in other words, if I were to move these distances and the slits closer together, you would see these bright spots get farther and farther away from each other on the screen. So that's an application of Young's Double Slit. Good luck.
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