p34_068

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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◦ . ...
View Full Document

Ask a homework question - tutors are online