Problem

A number of factors play a role in determining th…

02:34

Problem 106 Easy Difficulty

Predict/Calculate A beam of light enters the sloping side of a $45^{\circ}-90^{\circ}-45^{\circ}$ glass prism with an index of refraction $n=1.66$
The situation is similar to that shown in Figure $26-68$ , except that
the angle of incidence of the incoming beam can be varied. (a) Find
the angle of incidence for which the reflection on the vertical side of the prism exactly satisfies the condition for total internal reflection. (b) If the angle of incidence is increased, is the reflection at the
vertical surface still total? Explain. (c) What is the minimum value
of $n$ such that a horizontal beam like that in Figure 26-68 undergoes
total internal reflection at the vertical side of the prism?

Answer

(a) 13.3 $\mathrm{deg}$
(b) see explanation
(c) see work

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Video Transcript

in this question, being off light enters the sloping side off. I saw this wrangle triangle then, um, that's the you want. Oh, actually, parts in this question in part A. We want to find the angle of incidence published. Reflection on the vertical side is, uh, undergoes total internal reflection. Okay, so to labor some angles. Okay, So this is our angle of incidence. This is and go refraction. And then I'm going to call this data, see, for now and then this angle would be 135 degrees. Mhm. Yeah. So, um, we're going to use snails slow in the condition for total internal reflection to buy to solve for part a. Okay, so, uh, using. So this is the critical angle conditions for total internal reflection. So And we know that man is given to be 1.66 Mhm. Okay, So scientists see is to go to 1/1 0.66 And so did you see is calculated to be 37.0 degrees. Okay. And from the diagram, you know that data are plastic. Asi is 45 degrees, So data are if you go to 45 minus 37 degrees and you get 7.96 degrees. Okay. And from now it's law. Okay, we have scientist, I request to end. Sign there are. Okay, So, uh, side I is equal to 1.66 sign. 7.96 degrees. Right. It's just, uh this gives the I to be equal to 13 country degrees. Yeah. So this is the answer for party in part B. I went off. If the angle of incidents increases are, do we still get total internal reflection? Okay. Okay. So in part B, the answer is no. Okay, who's in? Uh, if data, I increase us. Okay. There there are increases in from sales law. Okay, then, did you see? Ah is, you know, uh, critical anger. So, uh, total internal reflection. Okay. Mhm. Yeah, because, uh, data are plus data, See, is 45 degree so data increases the labor data, See? Is going to be go smaller. Okay, So then then we are not going to get total internal reflection. Okay, Keeton Proxy, we want to find a minimum value off the total Internet. The minimum value off reflective index such that when we show very shy and horizontal bean. Ah, like that. Oh, yes. Grants. Yes. So, in part A, the being is not shining at horizontally. This is for policy. Okay, so, for a, uh, open in the beautiful party, this is what the being is coming. It looks like this. Okay, so this is for part A. Yeah, And then this game is for Patsy. Yeah. Sorry about that, Pete. So we want to find a minimum value of and such that when you have a horizontal beam and undergoes total internal reflection at the particular side of the prison. Okay, So wish horizontal bean he did. I would be 45 degrees so we can calculate are we're fact there the angle of refraction if using snows law. Okay, so we have sign whether I equals two and, uh, sign they are. Okay, we are supposed to find. And so and it's unknown. Okay, so, um, and we'll just be equal to, uh, sign 45 degrees. Bye bye. Sign there are so yeah given as such. And then from the total internal reflection condition. Yeah, you have sign. They see it goes through a wonderful and okay then from the diagram from diagram of both. Ah, they are us. They see becasue 45 degrees. Okay, so, um, is this what I'm going to do? I'm going to say okay. Scientists are is good, too. Sorry. 45 degrees minus status e. Okay. And then this is equal to one over n sign. 45 degrees. Okay. And, uh, this equation we expand, use the ah, you start trigonometry identity. So we have sai 45 degrees because I did a C minus co sign. 45 degrees site C is equal to when you were in Sorry. 45 degrees, right? Mm. 7. 45 degrees And cause I 45 degrees of the same at the same value. Right. Then, uh, you can just cancel them out. And because I didn't see minus one, they will end goes through when they will end. Okay. And then, uh, besides, jealousy is equal toe end. And this is, uh, one minus side square. They see square root. It goes to two over end and one minus one will end. Square is you goto for end square. So you're going to get, um, five. Who and square is equal to one and square it was five. Okay, so and it goes to 2.24 Okay, so this is the minimum. Ah, value of n you find it. Want to have total internal reflection on the vertical site off the prism refunding horizontal incident being okay, and that's all.

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