A beam of red light is incident on an equilateral prism as shown below (a) If th

Question

A beam of red light is incident on an equilateral prism as shown below

(a) If the index of refraction of red light of the prism is 1.400, at what angle (theta) does the beam emerge from the other face of the prism?

(b) Suppose the incident beam were white light. What would be the angular separation of the red and blue components in the emergent beam if the index of refraction of blue light were 1.403?

(c) What would it be if the index of refraction of blue light were 1.405?

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Explanation / Answer

The first angle is 80 (I am going to leave out the degree sign for simplicity). Call the second angle, the refraction of the first beam 1; go across the prism and let 2 be the angle to the normal internal to the prism; call 3 the outside angle on the right side of the prism. 3 is what we seek.

Simple trigonmetry using complementary angles and the top 60 angle of the equilateral triangle tells us that 1 + 2 = 60 so 2 = 60 - 1.

From Snell's law, sin 80 = n sin 1 and sin 3 = n sin 2.

Substituting for 2 we get:

3 = arcsin [n sin(60 - arcsin(sin 80/n))]

If we plug in n = 1.40, 3 = 21.67538 which is the answer to part (a).

If we plug in n = 1.403, 3 = 21.90069 so the answer to part (b) is the difference between that and part (a): 0.225313

If we plug in n = 1.405, 3 = 22.05083 so the answer to part (c) is the difference between that and part (a): 0.375459

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