1.! Questions about reflected intensity. [Use the formulas on p. 8 of Light.] , no matter

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1 Reading: Light Key concepts: Huygens s principle; reflection; refraction; reflectivity; total reflection; Brewster angle; polarization by absorption, reflection and Rayleigh scattering. 1.! Questions about reflected intensity. [Use the formulas on p. 8 of Light.] Show that at normal incidence the reflectivity is R = (n 1 n 2 )2, no matter 2 (n 1 + n 2 ) which index is larger or how the light is polarized. [Use small angle approximations.] Evaluate R for light incident from air on water ( n 1 = 1; n 2 = 4/3 ). Show that the sum of the Brewster angles for incidence from medium 1 and from medium 2 is 90. Show that if there is a critical angle it is larger than the Brewster angle. d.! Show that R > R! except at normal incidence and when both are 1. 2.! A beam of light in air has a square cross section of width d 1. It is shown from the side, incident on a glass plate. The refracted beam is widened in the dimension shown, to d 2, while the dimension perpendicular to the page is unchanged. Express d 2 /d 1 in terms of the angles of incidence and refraction, θ 1 and θ 2. d 1 θ 1 d 2 Suppose this beam consists of light with E in the plane of incidence, and that θ 1 is the Brewster angle, so there is no reflected beam. Express d 2 /d 1 in terms of n, the refractive index of the glass. With no reflected beam, the total power in the refracted beam must be the same as in the incident beam. What is the ratio of the intensities I 2 /I 1? 3.! You are handed a pair of Polaroid sunglasses on a clear day at the beach. Give two methods by which you can determine the direction of the transmission axis of the lenses.

2 4.! A rod made of transparent plastic is immersed in a liquid in a glass beaker, and is being viewed from the side as shown. Despite both the plastic and water being transparent, one can easily see the rod in the water. How? What light rays allow us to distinguish the plastic from the water? Suppose instead of water the liquid is an oil with the same index of refraction as that of the plastic. How would this affect our ability to see the plastic rod? Explain. 5.! Explain three things about sunset on a day with only a few clouds. The sun is still visible for several minutes after it has really set. That is, without the atmosphere the sun would disappear earlier than it does with the atmosphere. The sun s color at sunset is orange-yellow and the intensity of its light is low enough that we can look directly at it. The sky away from the sun has a blue color, perhaps with orange and pink colored clouds. 6.! A beam of unpolarized light is incident at angle θ from a medium with index n 1 onto the surface of a medium with index n 2. Comment on the validity of each statement. If n 1 < n 2, then at all values of θ other than 0 or 90 the reflected light is at least partially polarized. If n 1 > n 2, then there is no value of θ for which the reflected light is totally polarized. If n 1 < n 2, then reflection is total only for θ = 90. d.! There is no value of θ for which the reflection is total and the reflected light is also totally polarized.

3 7.! Three questions about mirrors. The two mirrors used in a clothing store for customers to see themselves are arranged so that (as seen from above) they make a right angle as shown. Draw rays from the right and left shoulders of the person to show that the image seen is not left-right reversed. Draw rays to prove that light entering a corner cube mirror is reflected straight back. Your overall height is h and your eyes are distance d below the top of your head. Prove that a plane mirror of height 1 h will allow you to see your 2 entire body in reflection, and determine how far above the floor its bottom should be placed. 8.! Light is incident as shown on a triangular prism made of glass with refractive index n. It is totally reflected internally. When the prism is immersed in water (refractive index = 4/3), the light is no longer totally reflected. Use this information to put upper and lower limits on the value of n ! Optical fibers are often protected by a transparent coating with a different index of coating refraction. Shown is the surface of such a fiber. θ 1 The fiber has refractive index n and the fiber coating has index n. Above the coating is air. We wish to show that the coating has no effect on total reflection by the system. Find the angle θ 2 at which the ray strikes the upper surface of the coating.! If θ 1 is the critical angle for the fiber-air interface, show that θ 2 is also the critical angle for the coating-air interface. What happens if θ 1 is greater than the critical angle for the fiber-air interface?

4 10.! An object viewed through a plate glass window appears to be slightly closer than it is. Shown is the situation, with the object at distance x from the window. The ray indicated emerges from the window displaced as shown, appearing to have come from the position at distance x from the window. You are to find the distance x x. The glass has thickness t and refractive index n. [Throughout the problem, use small angle approximations.] θ 1 x x y y At what vertical distance y from the horizontal axis does the ray enter the glass, in terms of x and the angle of incidence θ 1? t At what vertical distance y does it emerge from the glass? [Draw an enlarged picture of the region where the ray passes through the glass so the refracted ray can be clearly shown, and find y y.] Use the similar triangles to find the answer. 11.! A beam of light of intensity I 0 impinges at nearly normal incidence on a film of a transparent material as shown. The medium above and below the film is air, and the reflectivity at the surfaces is R. We are interested in the intensities of the reflected beams a, b and c shown. In terms of I 0 and R, what are the intensities of beams a, b and c? [Recall that R + T = 1, and that (at normal incidence) R for a given pair of media is the same no matter which medium the light comes from.] Let the film be water, for which n 4/3. (You found R in Question #1.) Find the intensities of the three beams. What can you conclude about the importance of beam c and those made by more reflections? Suppose the film is made highly reflecting (R 1) by evaporating a thin coating of a metal onto both its surfaces. What can you say about the relative importance of the three beams in this case? I 0 a b c

5 12.! Now we consider the transmitted intensities for light that passes through the film. Shown are three transmitted beams. I 0 Find the intensities of the beams in terms of I 0 and R. If the film is water, what are these intensities? What can you conclude about the importance of beams b and c? As before, let the film be made highly reflecting. What can you say about the relative importance of these three beams? a b c 13.! Incident light of intensity I 0 moving in the z-direction is polarized along the x- axis. It passes through N filters, at angles π /2N, 2π /2N, 3π /2N,..., π /2 relative to the x-axis, so the light emerges polarized along the y-axis. Express the final intensity I in terms of I 0 and the angle π /2N. For large N this angle is small and we can use the small angle approximation for the cosine: cosθ 1 θ 2 /2. Write the formula for I using this approximation. Finally, use the binomial formula (1 + ε) n 1 + nε (for small ε ) to show that as N, I I 0.