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A light ray of wavelength 589 nm is incident at an angle $\theta$ on the top surface of a block of polystyrene surrounded by air, as shown in Figure P22.48. (a) Find the maximum value of $\theta$ for which the refracted ray will undergo total internal reflection at the left vertical face of the block. (b) Repeat the calculation for the case in which the polystyrene block is immersed in water. (c) What happens if the block is immersed in carbon disulfide?

a. $90.0^{\circ}$

b.$30.0^{\circ}$

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were us to consider the situation in which ah, light Ray is incident. Yeah, on Polly staring from surrounding air. And then we're gonna do it when it's surrounded by water. And then when also, that Polly staring. It's surrounded by carbon die sulfide. So I wrote the index of refraction for ERAS and survey Polly staring as in soapy water, is in sub w and carbon, uh, die sulfide as in subsea. Okay, so we want to find that critical angle for all of these and using Snell's law. The critical angle happens when the angle of refraction is 90 degrees. So sign of 90 is one. So we can easily drive the critical angle from that so sign of fate A. C. Then the critical angle would be equal to incident to reverence of one. But for part, a were asked to consider air. So in sub two is the index of refraction of air Annan. Someone is the index of refraction of Polly steering and okay, so now solving for data see where the index are the critical angle, we find that it's equal to the inverse sine of that ratio. Insa bait and stumpy, so plugging those values in for those index of re fractions. We find that this angle is approximately 90 degrees for part B were asked to do the exact same thing. Except for this time we're gonna surrounded by water instead of air. So solving for that critical angle, same formula, we find that it's equal to the inverse sine of this time the ratio of the index of refraction of water police. Terry. We find that this is approximately 30 degrees, which can be boxing is their solution to be. And then for Parsi asked us to do it for carbon die sulfide. But if you go back to this first page here, we see that carbon die sulfide has a greater index of refraction than Polly staring. So we can conclude then that since ah, in subsea is greater than and, said P, it's not possible for there to be told any total internal refraction. You can't take the sign of something there, the inverse sine of something that's larger than one. So, um, not possible for internal reflection, Lincoln box it in as their solution for part C.