p34_068 - 68. (a) The first contribution to the overall...

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Unformatted text preview: 68. (a) The first contribution to the overall deviation is at the first refraction: δθ1 = θi − θr . The next contribution(s) to the overall deviation is (are) the reflection(s). Noting that the angle between the ray right before reflection and the axis normal to the back surface of the sphere is equal to θr , and recalling the law of reflection, we conclude that the angle by which the ray turns (comparing the direction of propagation before and after [each] reflection) is δθr = 180◦ − 2θr . Thus, for k reflections, we have δθ2 = kθr to account for these contributions. The final contribution is the refraction suffered by the ray upon leaving the sphere: δθ3 = θi − θr again. Therefore, θ dev = δθ1 + δθ2 + δθ3 = 2 (θi − θr ) + k (180◦ − 2θr ) = k (180◦ ) + 2θi − 2(k + 1)θr . (b) For k = 2 and n = 1.331 (given in problem 67), we search for the second-order rainbow angle numerically. We find that the θ dev minimum for red light is 230.37◦ , and this occurs at θi = 71.90◦ . (c) Similarly, we find that the second-order θ dev minimum for blue light (for which n = 1.343) is 233.48◦ , and this occurs at θi = 71.52◦ . (d) The difference in θ dev in the previous two parts is 3.11◦ . (e) Setting k = 3, we search for the third-order rainbow angle numerically. We find that the θ dev minimum for red light is 317.53◦ , and this occurs at θi = 76.88◦ . (f) Similarly, we find that the third-order θ dev minimum for blue light is 321.89◦ , and this occurs at θi = 76.62◦ . (g) The difference in θ dev in the previous two parts is 4.37◦ . ...
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