White light is incident onto a 30° prism at the 40° angle shown i...

Question

White light is incident onto a 30° prism at the 40° angle shown in FIGURE P35.41. Violet light emerges perpendicular to the rear face of the prism. The index of refraction of violet light in this glass is 2.0% larger than the index of refraction of red light. At what angle Φ does red light emerge from the rear face?< >

Accepted Answer

Hey, everyone in this problem, a beam of sunlight strikes a right triangle prism as shown in the figure below. We're gonna take a look at that picture in just a minute. The incident angle of the beam on the side BC of the prism is 60 degrees. So we take a look at our picture side BC. That is our hypotenuse. Yeah, we can see this dotted line that's gonna be normal to that side and we have our sunlight coming in. So the angle that incident angle between the sunlight and that normal we're told is 60 degrees. And we're told that the blue light exits side A B parallel to the normal of side A B. OK. So parallel to the normal, we can see that means that we are perpendicular to side A B. Given that the refractive index for yellow in this type of glass is 3.5% higher than that of blue. We are asked to calculate the angular deviation between the yellow light and the blue light exiting the prison. OK. So what we are looking for is this angle between the blue and yellow light at the exit. We have four answer choices all in degrees. Option A 1.1 option B 1.6 option C 2.1 and option D two point A. All right. So we're gonna have to work through this kind of step by step. So if we take a look at our triangle, blue light is exiting side A B perpendicular to side A. OK. So that means it's actually not getting refracted, it's not changing directions when it goes through that side because it's meeting it at a 90 degree angle. And so we can imagine that that blue light is getting refracted coming in from BC at the exact same angle that it's leaving at. And if we look at our diagram, we have a 35 degree angle here, that means that we have a 35 degree angle between the refraction of blue light inside of the triangle and that initial normal to side BC. OK. So that's what we have about the blue light. So let's start and figure out the refractive index for blue light. OK. Now, by Snell's law, we know that the refractive index or blue light in this triangle hm multiplied by sign of the angle, theta the refraction angle. So the angle that it makes inside of that triangle is gonna be equal to the refractive index of air multiplied by sign of the incident angle. So sign of the, and we'll call it in air hm filling in what we know So we wanna find this refractive index of the blue light. I remember that the yellow light we're told has this relationship for its refractive index to blue light. So if we can figure out refractive index for blue light, then we can figure it out for yellow light and then we can figure out what's going on with these yellow light to finally get that angle, we want, we have N BT multiplied by sine of 35 degrees is equal to the refractive index in air is just one, a multiplied by sign of that incident angle 60 degrees. And this tells us that a refractive index or the blue light in this triangle is gonna be equal to sine of 60 degrees divided by sine of 35 degrees, which is gonna be approximately equal to 1.5099. We have a refractive index for the blue light. We wanna figure out what's going on with this yellow light. So for the yellow light, we're gonna do this in green. So it's easier to see than in yellow. OK. And we know that the refractive index for yellow, so we'll call that NYT. So the refractive index for yellow in this triangle is 3.5% higher, then the one we just found for the blue light. So for then N BT, what that tells us is that NYT is just gonna be equal to 1.035. OK. So 3.5% higher multiplied by N BT 1.5099 which gives a refractive index for this yellow light. In this triangle of 1.5627 is we have both of our refractive indices. Now, let's figure out what happens to the yellow light. Then when we enter the prism from side BC, we already knew what happened to the blue light, but we don't know what happened to the yellow light. So we're gonna again use Snell's Law. We're gonna do the exact same thing we did for the blue light. But now we have the refractive index, we need to find the angle. And with the same equation, we're just looking for something different. Yeah. So we have that the refractive index of yellow through this prism multiplied by a sign of that angle of refraction. It's gonna be equal to the angle of oh sorry, the refractive index in error multiplied by sign of that incident angle go again, the air filling in what we know 1.56275 multiplied by sign of this angle. The that we are looking for is gonna be equal to one multiplied by s of 60 degrees. We can divide both sides by 1.562759 of theta YT will be equal to sine of 60 degrees divided by 1.56275. If we work that out and we take the inverse sign on both sides. We're gonna get that theta YT is equal to 33.65 degrees. Right now. We have to be a little bit careful here. And let's go to our diagram and take a look at what we just found. So we found this angle of refraction. So the yellow light, once it comes into the prism, it's gonna kind of move at an angle of 33 degrees, 33.65 degrees. But once it leaves side A B, it's actually gonna be refracted again, it's going to change direction, exiting that side because it's not meeting that side at a 90 degree angle like the blue light did. Hm. So in order to figure out the angle that the yellow light is leaving, exiting side A B, we have to figure out the incidence angle of it coming into side A B right. Now, what we're gonna do is we're gonna have to subtract two angles. OK. So we know that the angle we are shown is 35 degrees and we know that the angle that this yellow light leaves side BC from is 33.65 degrees. Now, the angle of incidence for yellow light exiting side A B is gonna be measured from that perpendicular line we have. So it's actually gonna be 35 degrees minus 33.65 degrees. It's gonna be that difference in this little angle up here, right So if we think about the IY, OK. So this is gonna be the incidence for that yellow light leaving side A B it's gonna be equal to that angle that, that blue light makes coming into side BC minus the angle the yellow light made, hey, that's what we showed in that diagram. So it's gonna be 35 degrees minus 33.65 degrees, which gives us an incident angle of 1.35 degrees. Hm. So we have this incident angle. Now we want to figure out the angle after it exits that side. So again, we go back to Snell's Law. So now this time we have that the refractive index or the yellow light in this triangular prism multiplied by sine of this incidence angle, theta IY is gonna be equal to the refractive index of air multiplied by sign. And we'll just call this theta Y. OK. So the angle of that yellow light coming out of side A B exiting that person filling in what we know 1.56275 multiplied by s of 1.35 degrees is equal to one multiplied by s of theta. What you can take the inverse sign on both sides. And we get that theta Y is approximately equal to 2.11 degrees. Let's go back to our diagram and take a look at what we just found and we found the angle of yellow light leaving side A B and that is measured from the normal. And so if that's measured from the normal, that's the angle between the blue light and the yellow light, that's exactly the angle we were trying to find that is theta one. And so the answer that we found is that this angle we were looking for is equal to 2.11 degrees approximately. If we go to our answer choices and we round to two significant digits, you can see that this corresponds with answer choice. C Thanks everyone for watching. I hope this video helped see you in the next one.

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