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The index of refraction for violet light in silica flint glass is 1.66 and that for red light is 1.62. What is the angular dispersion of visible light passing through an equilateral prism of apex angle $60.0^{\circ}$ if the angle of incidence is $50.0^{\circ}$? (See Fig. P22.62.)

$\text { Dispersion angle }=4.6 \text { deg }$

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University of Michigan - Ann Arbor

Numerade Educator

University of Washington

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Okay. So for our question was told that the index of refraction for violet Light, which I on this glass, which I call in G V, is 1.66 And for red light, I call n g. R is 1.62 and it wants us to find the Anglo angular dispersion of visible light passing through an equilibrium prism of ankle 60 degrees if the angle of incidence is 50 degrees. So we're gonna first solve the, uh, angle of refraction for violent. Then we're going to solve it for red. And we're gonna compare the two to get that answer here. Okay, So I, uh, wrote out what all the index of refraction czar and this a refraction of air is one. So any time we would normally write that, I'm just gonna ignore it, because anything times one or divided by one is itself. So starting out we have industry refraction of air times a sign of data of the incident angle, which is 50 degrees. So, like I said, I'm just not gonna write the index refraction of their cause. It's one is equal to index of refraction of the violent Glad. Uh, glass for violet times The sign of the angle of refraction are so solving for are we find that our is equal to the inverse sine of the sign of 50 degrees divided by the index of refraction of violent in glass. Gov. So we find that our is equal to 57.5 degrees. Okay, so now just using the, uh, fact that the internal angle of a triangle has to add up to 180 degrees. We have 180 degrees is equal to 60 degrees the apex of the prism, plus 90 degrees minus r, which we just found, plus data. So therefore solving for data, we find that data was equal to 57 point ah, five degrees. Oh, excuse me. I wrote 57.5 earlier for our but if you actually carry out that expression, it is not equal to 57.5 degrees. It is equal to 27.5 degrees. So let's go ahead and fix that really quick. This is 27.5. My apologies. I just wrote the wrong number there. I was reading a little bit ahead, So it's 27.5 degrees. So then plugging 27.5 in for our and that expression we confined data and data is equal to 57.5 degrees. Okay, so now that we found data, we can find the angle of incident data One V for the angle of incident for violet, um, as follows. So fatal One for violet is equal to 90 degrees, minus that value fay that we just found. So it's equal to 32.5 degrees. Comes to write that five. Here we go. Okay, then. Using Snell's Law, we have n g V now times The sign of that angle is equal to now it's entering into the air and leaving the glass. So index of refraction of air again Not gonna write X is just equal to one times the sign we'll call this state of two V. Okay, so therefore, state a two V's solving for that is equal to the inverse sine of the index of refraction of glass for violet times, sign data one V. So we find that data to for violet is equal to 50 or 63.2 degrees. Units are degrees. Here Okay, So then fate of E in general is equal to 50 degrees minus R plus data too. The minus they don't wanna be. So we find that beta V is equal to 53.2 degrees. Okay. And now we simply have to do the same thing for red, and then we can find the difference between the two. So now for red light, if you go back to finding our we found this relationship for our here. But we're simply gonna be placing replacing in G ve within G R. Okay, So we find that our is equal to the inverse sine of the re re ratio between the sign of 50 degrees, divided by a knicks of reflect refraction of glass for the red light. So we find that our is equal to 28.22 degrees. And just as before, we can now go ahead and find data because the internals of a triangle must add up to 100 degrees. So therefore 180 degrees is equal to 60 degrees, same apex angle plus 90 minus are just it before, But we just have a new value for our for the angle of refraction coming into the prison, plus data. So solving for data you find that data is equal to 58.22 degrees. Okay. And data one are just as before is going to be equal to 90 degrees minus ada. Who's so this is equal to 31.78 degrees. Okay. And then again, using Snell's lives before we confined data to our is going to be equal to the inverse sine of the index of refraction of glass for a red light. Time stayed alone are which we just found. They don't want our Okay, so we find data to our to be equal to 58.56 degrees. And just as before, data are then is equal to 50 degrees minus R which this is the Are that we just found to be 28.22 not the first Ark. This is for the red light. Um, plus data to our This is a plus data to our minus. They don't want our So we find data are is equal to 48.56 degrees, 38.56 degrees. Okay, so now we have the angle for violet and light on the angle for a red light. And it wants us to find the angular dispersion here. Okay, so the end of the dispersion Omega is going to be equal to the difference between the two. This would be the ankle for Violet. Now we found minus the angle for our that we found written about his dangle for Are we could go back here and see the angle for Violet is 53.2 plugging those values and we see that this is equal to 4.64 degrees. Plus, that is the solution to our question, and we can go ahead and box it in.