502 CHAPTER 7 Analytic Trigonometry 117. Ptolemy, who lived in the city of Alexandria in Egypt during the second century ad, gave the measured values in the following table for the angle of incidence θ1 and the angle of refraction θ2 for a light beam passing from air into water. Do these values agree with Snell’s Law? If so, what index of refraction results? (These data are of interest as the oldest recorded physical measurements.) 1θ 2θ 1θ 2θ ° 10 °8 ° 50 ° ′ 35 0 ° 20 ° ′ 15 30 ° 60 ° ′ 40 30 ° 30 ° ′ 22 30 ° 70 ° ′ 45 30 ° 40 ° ′ 29 0 ° 80 ° ′ 50 0 118. Bending Light The speed of yellow sodium light (wavelength, 589 nanometers) in a certain liquid is measured to be × 1.92 108 meters per second.What is the index of refraction of this liquid, with respect to air, for sodium light?‡ [Hint: The speed of light in air is approximately × 2.998 108 meters per second.] 119. Bending Light A beam of light with a wavelength of 589 nanometers traveling in air makes an angle of incidence of ° 40 on a slab of transparent material, and the refracted beam makes an angle of refraction of ° 26 . Find the index of refraction of the material.‡ 120. Bending Light A light ray with a wavelength of 589 nanometers (produced by a sodium lamp) traveling through air makes an angle of incidence of ° 30 on a smooth, flat slab of crown glass. Find the angle of refraction.‡ 121. Brewster’s Law If the angle of incidence and the angle of refraction are complementary angles, the angle of incidence is referred to as the Brewster angle θ .B The Brewster angle is related to the indices of refraction of the two media, n1 and n ,2 by the equation θ θ = n n sin cos , B B 1 2 where n1 is the index of refraction of the incident medium and n2 is the index of refraction of the refractive medium. Determine the Brewster angle for a light beam traveling through water (at ° 20 C) that makes an angle of incidence with a smooth, flat slab of crown glass. 122. Challenge Problem A light beam passes through a thick slab of material whose index of refraction is n .2 Show that the emerging beam is parallel to the incident beam.‡ 123. Challenge Problem Give the general formula for the solutions of the equation. θ θ + = 3sin 3 cos 0 124. Challenge Problem If θ θ ( ) + + + = x x tan cot 1 0 2 has two real solutions, { } − + 2 3, 2 3 , find θ θ sin cos . The following discussion of Snell’s Law of Refraction* (named after Willebrord Snell, 1580–1626) is needed for Problems 115–122. Light, sound, and other waves travel at different speeds, depending on the medium (air, water, wood, and so on) through which they pass. Suppose that light travels from a point A in one medium, where its speed is v ,1 to a point B in another medium, where its speed is v .2 Refer to the figure, where the angle θ1 is called the angle of incidence and the angle θ2 is the angle of refraction. Snell’s Law, which can be proved using calculus, states that θ θ = v v sin sin 1 2 1 2 The ratio v v 1 2 is called the index of refraction. Some values are given in the table shown below. A B Angle of refraction Angle of incidence Incident ray, speed v1 Refracted ray, speed v2 u1 u2 Some Indexes of Refraction Medium Index of Refraction† Water 1.33 Ethyl alcohol ( ) ° 20 C 1.36 Carbon disulfide 1.63 Air (1 atm and °0 C) 1.00029 Diamond 2.42 Fused quartz 1.46 Glass, crown 1.52 Glass, dense flint 1.66 Sodium chloride 1.54 115. The index of refraction of light in passing from a vacuum into water is 1.33. If the angle of incidence is ° 40 , determine the angle of refraction. 116. The index of refraction of light in passing from a vacuum into dense flint glass is 1.66. If the angle of incidence is ° 50 , determine the angle of refraction. *Because this law was also deduced by René Descartes in France, it is also known as Descartes’ Law. †For light of wavelength 589 nanometers, measured with respect to a vacuum. The index with respect to air is negligibly different in most cases. ‡Adapted from Halliday, Resnick, and Walker, Fundamentals of Physics, 10th ed., 2014, John Wiley & Sons.
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