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A light ray traveling in air is incident on one face of a right- angle prism with index of refraction $n=1.50,$ as shown in Figure P 22.54 , and the ray follows the path shown in the figure. Assuming $\theta=60.0^{\circ}$ and the base of the prism is mirrored, determine the angle $\phi$ made by the outgoing ray with the normal to the right face of the prism.

$\begin{array}{l}{\text { The angle } \phi \text { made by the outgoing ray with the normal to the right face of }} \\ {\text { the prism is } 7.90^{\circ} .}\end{array}$

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we were told that we have a light ray traveling an heir with an index of refraction. I call hints of A that is incident on a prism, which has an index of refraction. It's a p of 1.5. Okay, and we're told that this initial angle of incident data is equal to 60 degrees. Okay. Were asked to look at figure P 22.54 to follow. The ray path is shown in the figure. Um, and it wants us to find angle find made by the outgoing ray with the prism State Prison face there. Okay, So using smells long, we're going to start with the index of refraction of air times the sign of the incident angle Fada is equal to index of refraction of prism, timesthe sign of our, which is, uh, the angle of refraction there. And we call this one Little are in that photo. Okay, so then solving for are we find that this is equal to the inner sign of in sub a divided by inside p multiplied by the sign of the angles data, which we told the 60 degrees. So something for this we find that our is equal to 35.26 degrees. That's the angle of that first re fraction there. Okay, so from the triangle, the sum of the angles should be 100 and 80 degrees. Um, therefore, we can conclude on let's get a new page going here that veda plus 90 which is, uh, 90 degrees minus that angle. We just found our plus 90 degrees minus. Uh, the angle I, which we're going to find which we need here, is equal to 180 degrees. Therefore, solving for the angle I we find this is equal to locate a minus R. So 60 minus 35.26 is 24 0.74 degrees. Okay, so now using that angle I and using again the fact that the inside of a triangle has to add up to 100 and 80 degrees, we realized that in the 90 minus data plus 90 minus R Mrs Big R, which is what we need here to finally find Fi. The outgoing angle plus 90 minus I, which we just found it all has to add up to 180 degrees or in other words are is equal to 90 degrees minus data minus I Looking those values into this expression, we find that our is equal to 5.26 degrees. Okay, so now using this as our angle of incident on the prison toe air interface, we can once again and finally use smells law to say that in sub three, right, that little nicer in stumpy supposed to be a sub script, uh, times this sign of that angle of incident, which is our that we just found is equal to in some a cause. That's the medium is traveling into times the sign of five which were asked to find here. Or, in other words, fi is equal to the inverse sine of the ratio of in some P Thio two. Inside a multiplied by the sign of angle are that we just found to be 5.5 point 26 So plugging those values into this expression, we find that fei is equal to 7.9 degrees weaken box that in as their solution to our question