FIGURE EX17.27 shows the circular wave fronts emitted by two wave...

Question

FIGURE EX17.27 shows the circular wave fronts emitted by two wave sources. b. Make a table with rows labeled P, Q, and R and columns labeled r1 ,r2 , Δr , and C/D. Fill in the table for points P, Q, and R, giving the distances as multiples of λ and indicating, with a C or a D, whether the interference at that point is constructive or destructive.

Accepted Answer

Hey, everyone in this problem, two stones are dropped and create a ripple into a calm pond. As illustrated in the diagram we're given and we're asked to determine the type of interference that would be observed at points NNO. So we have four answer choices for this problem K A through D, each of them just containing a different combination of constructive or destructive interference for each point. And we're gonna come back to those as we work through this problem. So let's get back to our diagram. We have stone A that was dropped. Stone B that was dropped and we can see these ripples from each stone. OK. So what we wanna do is look at each point of interest and try to figure out what's going on here. OK. Now, we can see that there's no phase difference between the sources. The wave fronts have moved the same distance. So if we have no phase difference, hey, what we want to look at is the path difference. OK. The path difference which we're gonna call delta X, we know it's gonna be equal to N multiplied by Lambda or N is equal to 012 et cetera. This is true if we have constructive interference and the path difference delta X is gonna be equal to N lambda divided by two. OK? For odd values of NN equals 135, et cetera for destructive interference. All right. So we have no phase difference. We know that means we have to look at the path difference. OK? We have these two conditions for the path difference to give us either constructive or destructive interference. Let's go ahead and see what we have. OK. So let's start with point M. So at point at what do we have? OK. So looking at point M, if we look at the ripples from A, OK. The diagram tells us that each one is one wavelength. OK. So from point A two point M is one wavelength. OK. So our path difference delta X which is gonna be the difference, the absolute value of the difference from stone A to stone B. And when I say from stone A to stone B, I don't mean the distance between A and B, what I mean is a distance from our point M to stone and then the distance from our point M to stone B. OK. What's the difference between those two distance? Uh So we said from A, we have, we, OK, one wave we've gone from A to M one wavelength. Now, if we look at B and we can see that for B, we have to travel one wavelength. And then another way they were going to ripples outside of B to get to point it. So we have Linda minus two. And now the absolute value of Lambda minus two. Lambda is just going to be Lambda. OK. Now Linda, this is an integer multiple of the OK. And if N is equal to one, then this satisfies the first equation. So this is constructive interference. All right. So let's take a look at our answer choices before we go further. Option B has that point M is destructive. So we can eliminate that AC and D all have constructive interference. So those could be correct. So let's keep going now and see which one it's going to be. Now we're gonna move to point. OK. Now, for point N, we're gonna do the exact same thing. OK? What is the path difference? What is the difference in the distances between our point and stone A or where stone A was dropped our point and where stone B was strong? OK. We're just using D A and DB to represent those distances to either stone A or stone B respectively. All right. Now, in this case, if we look at A and point from A, we have to go one wavelength, two wavelengths and then half a wavelength more to get two points. And so the distance D A is gonna be 2. multiplied by LA. Now, from point B, we're gonna do the same thing, we're gonna go one wavelength, two wavelengths and we have to go half a wavelength more 2. wavelengths. OK. So what we end up here is 2.5 lambda minus 2.5 lambda. This just gives us a path difference of zero. Now, if we have a path difference of zero, that means we have constructive interference, right? So again, constructive interference, all right, taking a look at our answer choices. Option A is the only one that has constructive interference for both point F and point A. OK. So we're expecting this to be answered. A, let's take a look at 0.0 though and just double check that we haven't made any mistakes, just be sure that we know how to do this process. All right. So from 0.0 let's go from a, a the distance from a one wavelength, two wave blanks, three wavelengths four weeks. All right. So let's write in our equation A for 0. The path difference delta X is again the absolute value of D A minus DB. And we've just said that D A is going to be four now for DB. Well, we start A B, we go out one wavelength and then we have to go half a wavelength more to get to 0.0 Can you just always double check, start back at stone A or stone B? So that you can see which ripples kind of correspond to which stone. All right. So from B again, 1.5 wavelength, yeah, four wavelengths minus 1.5 that's gonna give us 2.5 multiplied by length. Now, 2.5 multiplied wavelength, that's an odd multiple of half a wavelength that corresponds to destructive interference. OK. So at 0.0 we have destructive interference, which is exactly what we had in option A and so the correct answer here is going to be option A point M and N have constructive interference and 0.0 we will have destructive interference. Thanks everyone for watching. I hope this video helped see you in the next one.

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