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A triangular glass prism with apex angle $60.0^{\circ}$ has an index of refraction of $1.50 .$ (a) Show that if its angle of incidence on the first surface is $\theta_{1}=48.6^{\circ},$ light will pass symmetrically through the prism as shown in Figure $35.17 .$ (b) Find the angle of deviation $\delta_{\min }$ for $\theta_{1}=48.6^{\circ} .$ (c) What If? Find the angle of deviation if the angle of incidence on the first surface is $45.6^{\circ} .$ (d) Find the angle of deviation if $\theta_{1}=51.6^{\circ} .$

a. $48.6^{\circ}$

b. $37.2^{\circ}$

c. $=\left|37.3^{\circ}\right|$

d. $37.32^{\circ}$

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Cornell University

University of Washington

Simon Fraser University

University of Sheffield

we have been given a prism within a pax angle of 60 degrees and a refractive index off and to which is equal to 1.5. Now we have an angle of entry of array into this president, 48.6 degrees. And we're being asked to prove that the, um the rays symmetrical, has shown and GM is that they have provided for us. So this is a 60 degree angle. This is our Ray. And of course, Ah, this is going to be symmetric on normal to D to the But, um, so we have an angle entering at, what, 48.6 degrees. So this is state of one. And this is the normal? No, Sorry. This is the normal to the surface, and this will be exiting at the exact same angle. So we want to prove that, um, this angle right here, we want to prove that this angle is the same as this one. So to do that, we're going to have to do some magic. So we're going to say that and one sign data one equal to end to sign there. Two. And since we know the outside medium is air. We know that's Ah one. So we're going to want to find angle to which is located, right? So are are fated to is located right here. So we want to determine what that is. So to do that we're going to take sign of State award divided by N to and then find the inverse sign of that whole situation devastated to and I will give us sorry. In a 48.6 over 1.5 equal to dated to the two will don't be equal to 30 degrees. We have no determined that the A two is 30 degrees. We know that this is 60 degrees. So if this guy is 30 degrees and this is a normal, that means that, uh, this angle which we're gonna call Alfa right Alfa plus Stena to is going to have to equal 90 degrees, which means that this angle right here, Alfa, since this is 30 degrees, Alfa is going to be 60 degrees. So that means if we have to 60 degree angles, this one also have to be 60 degrees. Um, that's just basic geometry. All the angles have to add up to 180 degrees. So now we determined that this angle is also 60 degrees. So I'm gonna make this a little more clear Since we know that this angle of 60 degrees, that means that this angle and this angle which we're gonna call, see, the three have to equal. You guessed it. They have to equal 90 degrees because this is a normal right. This is normal to the service, so they have to equal 90 which means the fate of three is equal to 30 degrees. So if they three is equal to 30 degrees, Um, we'll call this instead of calling the state and one week all day before um, we can say that N three of Okay, I'm gonna have to do this over a little. But the, uh, n three times sign of Day three is equal to end for, ah, sign data four. And we know this is equal to want to get the outside medium is air on three is 1.5. So to find the sign of to find a uniform and I'd find the inverse sign off 1.5 times sign of 30 what they before and that of course will give us 48.6 degrees, which is equal to fatal one. So that means that they'd affords equal of fate a one. So that's it. We have successfully proved that the, um that they raise symmetrical on both Theo entry and exit. So next thing we want to do is part B in which we have to find what is known as the angle of deviation. So to find the angle of deviation, we need to use a special equation. And that equation is given to you in your textbook. So I'm gonna write that out for you. I e an equal sign off the, uh, apex angle, plus the, uh, angle deviation all over, too. And that is over sine of the angle deviation by the way, to seize my handwriting there so it will multiply sine of the angle of deviation over to buy end. Find the inverse ein of that, and then we're gonna subtract the apex angle, and that will give us our angle deviation. So in this case, for me, right for being, we're gonna have signed negative one off sign of 60 divided by two times 1.5 minus 60 and that is going to give us, um, an answer off 37.2 degrees. So for C, we're going to do the same thing. We're going to use the same equation, but this time we're plugging in a state of value. Hold up. So what you'll notice is that in no part of this equation to the angle of incidence, play a factor. So for B, C and D, the answer is going to be 37.2 degrees because the angle incident does not play a role in the angle of deviation. All that matters is the apex angle, the the apex angle, the n value of the prison, and that's it. So this is your answer. We have successfully proved the symmetry of the prison at that angle, and we have determined that these three are the angles of deviation for this prison