1. Work your way through an argument using dimensional analysis to
establish the λ-4 dependence of the percentage of light scattered in
Ilayleigh Scattering. Let E0i and E0s be the incident and scattered
amplitudes, the latter at a distance r from the scatterer. Assume E0s cc
E0i and E0s α 1/r. Furthermore, plausibly assume that the scattered
amplitude is proportional to the volume, V, of the scatterer; within
limits this is reasonable. Determine the units of the constant of
proportionality. Get solution
2. A white floodlight beam crosses a large volume containing a tenuous molecular gas mixture of mostly oxygen and nitrogen. Compare the relative amount of scattering occurring for the yellow (580 nm) component with that of the violet (400 nm) component. Get solution
3. Figure P.4.3 depicts light emerging from a point source. It shows three different representations of radiant energy streaming outward. Identify each one and discuss its relationship to the others.Figure P.4.3 A segment of a spherical wave... Get solution
4. The equation for a driven damped oscillator is...(a) Explain the significance of each term.(b) Let ... where E0 and x0 are real quantities. Substitute into the above expression and show that...(c) Derive an expression for the phase lag, α, and discuss how α varies as to goes from ... Get solution
5. Imagine that we have a non absorbing glass plate of index n and thickness Δy, which stands between a source S and an observer P.(a) If the unobstructed wave (without the plate present) is Eu = E0 exp iω(t – y/c) show that with the plate in place the observer sees a wave...(b) Show that if either n ≈ 1 or Δy is very small, then...The second term on the right may be envisioned as the field arising from the oscillators in the plate. Get solution
6. A very narrow laserbeam is incident at an angle of 58° on a horizontal mirror. The reflected beam strikes a wall at a spot 5.0 m away from the point of incidence where the beam hit the mirror. How far horizontally is the wall from that point of incidence? Get solution
7. On entering the tomb of FRED the Hero of Nod, you find your¬self in a dark closed chamber with a small hole in a wall 3.0 m up from the floor. Once a year, on FRED's birthday, a beam of sunlight enters via the hole, strikes a small polished gold disk on the floor 4.0 m from the wall and reflects off it, lighting up a great diamond embedded in the forehead of a glorious statue of FRED, 20 m from the wall. Roughly how tall is the statue? Get solution
8. Figure P.4.8 shows what's called a corner mirror. Determine the direction of the exiting ray with respect to the incident ray.Figure P.4.8... Get solution
9. Get solution
10. Get solution
11. Calculate the transmission angle for a ray incident in air at 30° on a block of crown glass (ng = 1.52). Get solution
12. The construction in Fig. P.4.10 corresponds to. Descartes! s derivation of the Law of Refraction. Light moves from S to O in the same time it travels from O to P. Moreover, its transverse momentum is unchanged on traversing the interface. Use all of this to "derive" Snell's Law.Figure P.4.10... Get solution
13. Get solution
14. Figure P.4.11 is a plot of the sine of the angle-of-incidence versus the sine of the transmission angle measured as light passed from air into a more optically dense medium. Discuss the curve. What is the significance of the slope of the line? Guess at what the dense medium might be.Figure P.4.11... Get solution
15. A ray of yellow light from a sodium discharge lamp falls on the surface of a diamond in air at 45°. If at that frequency nd = 2.42, compute the angular deviation suffered upon transmission. Get solution
16. Given an interface between water ... and glass ... compute the transmission angle for a beam incident in the water at 45°. If the transmitted beam is reversed so that it impinges on the interface, show that θt, — 45°. Get solution
17. A beam of 12-cm planar microwaves strikes the surface of a dielectric at 45°. If ... compute (a) the wavelength in the transmitting medium, and (b) the angle θt. Get solution
18. Light of wavelength 600 nm in vacuum enters a block of glass where ng = 1.5. Compute its wavelength in the glass. What color would it appear to someone embedded in the glass (see Table 3.4)? Get solution
19. A laserbeam impinges on an air-liquid interface at an angle of 55°. The refracted ray is observed to be transmitted at 40°. What is the refractive index of the liquid? Get solution
20. An underwater swimmer shines a beam of light up toward the surface. It strikes the air-water interface at 35°. At what angle will it emerge into the air? Get solution
21. Make a plot of θi- versus θt, for an air-glass boundary where nga — 1.5. Discuss the shape of the curve. Get solution
22. A laserbeam having a diameter D in air strikes a piece of glass (ng) at an angle θi. What is the diameter of the beam in the glass? Get solution
23. An exceedingly narrow beam of white light is incident at 60.0° on a sheet of glass 10.0 cm thick in air. The index of refraction for red light is 1.505 and for violet light it's 1.545. Determine the approximate diameter of the emerging beam. Get solution
24. Get solution
25. Get solution
26. A laserbeam impinges on the top surface of a 2.00-cm-thiek parallel glass (n = 1.50) plate at an angle of 35°. How long is the actual path through the glass? Get solution
27. Light is incident in the air on an air-glass interface. If the index of refraction of the glass is 1.70, find the incident angle such that the transmission angle is to equal ... Get solution
28. Suppose that you focus a camera with a close-up bellows attachment directly down on a letter printed on this page. The letter is then covered with a 1.00-mm-thick microscope slide (n = 1.55). How high must the camera be raised in order to keep the letter in focus? Get solution
29. A coin is resting on the bottom of a tank of water (nw = 1.33) LOO m deep. On top of the water floats a layer of benzene (nb = 1.50), which is 20.0 cm thick. Looking down nearly perpendicularly, how far beneath the topmost surface does the coin appear? Draw a ray diagram. Get solution
30. In Fig. P.4.26 the wavefronts in the incident medium match the fronts in the transmitting medium everywhere on the interface—a concept known as wavefront continuity. Write expressions for the number of waves per unit length along the interface in terms of θi; and λi- in one case and θt, and λt in the other. Use these to derive Snell's Law. Do you think Snell's Law applies to sound waves? Explain.Figure P.4.26... Get solution
31. With the previous problem in mind, return to Eq. (4.19) and take the origin of the coordinate system in the plane-of-incidence and on the interface (Fig. 4.39). Show that that equation is then equivalent to equating the x-components of the various propagation vectors. Show that it is also equivalent to the notion of wavefront continuity. Get solution
32. Making use of the ideas of equal transit times between corresponding points and the orthogonality of rays and wavefronts, derive the law of reflection and Snell's Law. The ray diagram of Fig. P.4.28 should be helpful.Figure P.4.28... Get solution
33. Starting with Snell's Law, prove that the vector refraction equation has the form... Get solution
34. Derive a vector expression equivalent to the Law of Reflection. As before, let the normal go from the incident to the transmitting medium, even though it obviously doesn't really matter. Get solution
35. In the case of reflection from a planar surface, use Fermat's Principle to prove that the incident and reflected rays share a common plane with the normal ûm namely, the plane-of-incidence. Get solution
36. Derive the Law of Reflection, θt- = θr by using the calculus to minimize the transit time, as required by Fermat's Principle. Get solution
37. According to the mathematician Hermann Schwarz, there is one triangle that can be inscribed within an acute triangle such that it has a minimal perimeter. Using two planar mirrors, a laserbeam, and Fermat's Principle, explain how you can show that this inscribed triangle has its vertices at the points where the altitudes of the acute triangle intersect its corresponding sides Get solution
38. Show analytically that a beam entering a planar transparent plate, as in Fig. P.4.34, emerges parallel to its initial direction. Derive an expression for the lateral displacement of the beam. Incidentally, the incoming and outgoing rays would be parallel even for a stack of plates of different material.Figure P.4.34... Get solution
39. Show that the two rays that enter the system in Fig. P.4.35 parallel to each other emerge from it being parallel.Figure. P.4.35... Get solution
40. Discuss the results of Problem 4.34 in the light of Fermat's Principle; that is, how does the relative index n21 affect things? To see the lateral displacement, look at a broad source through a thick piece of glass (≈... inch) or a stack (four will do) of microscope slides held at an angle. There will be an obvious shift between the region of the source seen directly and the region viewed through the glass. Get solution
41. Get solution
42. Suppose a lightwave that is linearly polarized in the plane-of-incidence impinges at 30° on a crown-glass (ng = 1.52) plate in air. Compute the appropriate amplitude reflection and transmission coefficients at the interface. Compare your results with Fig. 4.39. Get solution
43. Derive Eqs. (4.42) through (4.45) for r┴, r║, t┴, and r║. Get solution
44. A beam of light in air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 20.0°. The incident light has component E-field amplitudes parallel and perpendicular to the plane-of-incidence of 10.0 V/m and 20.0 V/m, respectively. Determine the corresponding reflected field amplitudes. Get solution
45. A laserbeam is incident on the interface between air and some dielectric of index n. For small values of θi- show that θt = θi/n. Use this and Eq. (4.42) to establish that at near-normal incidence [-r┴]θi ≈ o = (n-1)/(n+ 1). Get solution
46. Get solution
47. Get solution
48. Get solution
49. Light is incident in ah* perpendicularly on a sheet of crown glass having an index of refraction of 1.522. Determine both the reflectance and the transmittance. Get solution
50. A beam of quasimonochromalic light having an irradiance of 500 W/m2 is incident in air perpendicularly on the surface of a tank of water (nw = 1.333). Determine the transmitted irradiance. Get solution
51. Get solution
52. Get solution
53. Get solution
55. Get solution
56. Get solution
57. Quasimonochromatic light having an irradiance of 400 W/m2 is incident normally on the cornea (nc = 1.376) of the human eye. If the person is swimming under the water (nw = 1.33), determine the transmitted irradiance into the cornea. Get solution
58. Compare the amplitude reflection coefficients for an air-water (nw = 4/3) interface with that of an air-crown glass (ng = 3/2) interface, both at near-normal incidence. What are the corresponding ratios of the reflected to the incident irradiances? Get solution
59. Use Eq. (4.42) and the power series expansion of the sine function to establish that at near-normal incidence we can obtain a better approximation than the one in Problem 4.41, which is [-r┴]θi ≈ o = (n-1)/(n+ 1), namely... Get solution
60. Establish that at near-normal incidence the equation...is a good approximation. [Hint: Use the results of the previous problem, Eq. (4.43), and the power scries expansions of the sine and cosine functions.] Get solution
61. Prove that for a vacuum-dielectric interface at glancing incidence r┴ → - 1, as in Fig. 4.41. Get solution
62. In Fig. 4.41 the curve of r┴ approaches -1.0 as the angle-of-incidence approaches 90°. Prove that if α┴ is the angle the curve makes with the vertical at θi-, = 90°, then...[Hint: First show that dθt/dθi = 0.] Get solution
63. Prove that...for all θt first from the boundary conditions and then from the Fresnel Equations. Get solution
64. Verify that...for θi = 30° at a crown-glass and air interface (nti= 1.52). Get solution
65. Use the Fresnel Equations to prove that light incident at θp ½π-θt results in a reflected beam that is indeed polarized. Get solution
66. Show that tan θp = nt/ni and calculate the polarization angle for external incidence on a plate of crown glass (ng = 1.52) in air. Get solution
67. Beginning with Eq. (4.38), show that for two dielectric media, in general tan ... Get solution
68. Show that the polarization angles for internal and external reflection at a given interface are complementary, that is, θp + θ’p = 90° (see Problem 4.64). Get solution
69. It is often useful to work with the azimutkal angle γ, which isFigure P.4.61 (Photo and diagram courtesy S. Reich, The Weizmann Institute of Science, Israel....defined as the angle between the plane-of-vibration and the plane-of-incidence. Thus for linearly polarized light,...Figure P.4.67 is a plot of γr versus θi, for internal and external reflection at an air-glass interface (nga = 1.51), where γt = 45°. Verify a few of the points on the curves and in addition show that...Figure P.4.67... Get solution
70. Making use of the definitions of the azimuthal angles in Problem 4.67, show that... Get solution
71. Make a sketch of R┴ and R║ and nt = 1.5 and n, = 1 (i.e., internal reflection). Get solution
72. Show that... Get solution
73. Using the results of Problem 4.70, that is, Eqs. (4.98) and (4.99), show that...... Get solution
74. Suppose that we look at a source perpendicularly through a slack of N microscope slides. The source seen through even a dozen slides will be noticeably darker. Assuming negligible absorption, show that the total iransmillance of the stack is given by...and evaluate Tt, for three slides in air. Get solution
75. Making use of the expression...for an absorbing medium, we define a quantity called the unit trans-mittance. T1. At normal incidence, Eq. (4.55), T = It/Ii and thus when y = 1, T1 ≡ I(1)/Io. If the total thickness of the slides in the previous problem is d and if they now have a transmittanee per unit length T1, show that... Get solution
76. Show that at normal incidence on the boundary between two dielectrics, as ... Moreover, prove that as nti ... Thus as the two media take on more similar indices of refraction, less and less energy is carried off in the reflected wave. It should be obvious that when nti = 1 there will be no interface and no reflection. Get solution
77. Derive the expressions for r┴ and r║ given by Eqs. (4.70) and (4.71). Get solution
78. Show that when θi>θe at a dielectric interface, r┴ and r║ are complex and ... Get solution
79. Calculate the critical angle beyond which there is total internal reflection at an air-glass (ng = 1.5) interface. Compare this result with that of Problem 4.15. Get solution
80. Referring back to Problem 4.18, note that as θi,. increases θt increases. Prove that the maximum value θt ( may have is θc. Get solution
81. What is the critical angle for total internal reflection for diamond? What, if anything, does the critical angle have to do with the luster of a well-cut diamond? Get solution
82. Using a block of a transparent, unknown material, it is found that a beam of light inside the material is totally internally reflected at the air-block interface at an angle of 48.0°. What is its index of refraetion? Get solution
83. A prism, ABC, is configured such that angle BCA = 90° and angle CBA = 45°. What is the minimum value of its index of refraction if, while immersed in air, a beam traversing face AC is to be totally internally reflected from face BC. Get solution
84. A fish looking straight up toward the smooth surface of a pond receives a cone of rays and sees a circle of light filled with the images of sky and birds and whatever else is up there. This bright circular field is surrounded by darkness. Explain what is happening and compute the cone-angle. Get solution
85. A glass block having an index of 1.55 is covered with a layer of water of index 1.33. For light traveling in the glass, what is the critical angle at the interface? Get solution
86. Derive an expression for the speed of the evanescent wave in the case of internal reflection. Write it in terms of c, ni and θt Get solution
87. Light having a vacuum wavelength of 600 nm, traveling in a...glass (ng = 1.50) block, is incident at 45° on a glass-air interface. It is then totally internally reflected. Determine the distance into the air at which the amplitude of the evanescent wave has dropped to a value of 1/e of its maximum value at the interface. Get solution
88. Get solution
89. Get solution
90. Get solution
91. Figure P.4.61 shows a laserbeam incident on a wet piece of filter paper atop a sheet of glass whose index of refraction is to be measured—the photograph shows the resulting light pattern. Explain what is happening and derive an expression for ni in terms of R and d. Get solution
92. Consider the common mirage associated with an inhomoge-neous distribution of air situated above a warm roadway. Envision the bending of the rays as if it were instead a problem in total intet nal-reflection. If an observer, at whose head na = 1.000 29, sees an apparent wet spot at θi,≥ 88.7° down the road, find the index of the air immediately above the road. Get solution
93. Figure P.4.80 depicts a glass cube surrounded by four glass prisms in very close proximity to its sides. Sketch in the paths that will be taken by the two rays shown and discuss a possible application for the device.Figure P.4.80... Get solution
94. Figure P.4.82 shows a prism-coupler arrangement developed al the Bell Telephone Laboratories. Its function is to feed a laserbeam into a thin (0.00001-inch) transparent film, which then serves as a sort of waveguide. One application is that of thin-film laserbeam circuitry—a kind of integrated optics. How do you think it works?Figure P.4.82... Get solution
95. Figure P.4.81 is a plot of nt and nR versus A for a common am al. Identify the metal by comparing its characteristics with those con sidered in the chapter and discuss its optical properties.Figure P.4.81... Get solution
96. Get solution
97. Get solution
98. Get solution
99. Figure P.4.77 depicts a ray being multiply reflected by a transparent dielectric plate (the amplitudes of the resulting fragments arc indicated). As in Section 4.5, we use the primed coefficient notation because the angles are related by Snell's Law.(a) Finish labeling the amplitudes of the last four rays.(b) Show, using the Fresnel Equations, that... Get solution
100. A wave, linearly polarized in the plane-of-incidence, impinges on the interface between two dielectric media. If ... there is no reflected wave, that is, ... Using Stokes's techFigure P.4.77...nique, start from scratch to show that ... and θt = θp (Problem 4.66). How docs this compare with Eq. (4.100)? Get solution
101. Making use of the Fresnel Equations, show that ...as in the previous problem. Get solution
2. A white floodlight beam crosses a large volume containing a tenuous molecular gas mixture of mostly oxygen and nitrogen. Compare the relative amount of scattering occurring for the yellow (580 nm) component with that of the violet (400 nm) component. Get solution
3. Figure P.4.3 depicts light emerging from a point source. It shows three different representations of radiant energy streaming outward. Identify each one and discuss its relationship to the others.Figure P.4.3 A segment of a spherical wave... Get solution
4. The equation for a driven damped oscillator is...(a) Explain the significance of each term.(b) Let ... where E0 and x0 are real quantities. Substitute into the above expression and show that...(c) Derive an expression for the phase lag, α, and discuss how α varies as to goes from ... Get solution
5. Imagine that we have a non absorbing glass plate of index n and thickness Δy, which stands between a source S and an observer P.(a) If the unobstructed wave (without the plate present) is Eu = E0 exp iω(t – y/c) show that with the plate in place the observer sees a wave...(b) Show that if either n ≈ 1 or Δy is very small, then...The second term on the right may be envisioned as the field arising from the oscillators in the plate. Get solution
6. A very narrow laserbeam is incident at an angle of 58° on a horizontal mirror. The reflected beam strikes a wall at a spot 5.0 m away from the point of incidence where the beam hit the mirror. How far horizontally is the wall from that point of incidence? Get solution
7. On entering the tomb of FRED the Hero of Nod, you find your¬self in a dark closed chamber with a small hole in a wall 3.0 m up from the floor. Once a year, on FRED's birthday, a beam of sunlight enters via the hole, strikes a small polished gold disk on the floor 4.0 m from the wall and reflects off it, lighting up a great diamond embedded in the forehead of a glorious statue of FRED, 20 m from the wall. Roughly how tall is the statue? Get solution
8. Figure P.4.8 shows what's called a corner mirror. Determine the direction of the exiting ray with respect to the incident ray.Figure P.4.8... Get solution
9. Get solution
10. Get solution
11. Calculate the transmission angle for a ray incident in air at 30° on a block of crown glass (ng = 1.52). Get solution
12. The construction in Fig. P.4.10 corresponds to. Descartes! s derivation of the Law of Refraction. Light moves from S to O in the same time it travels from O to P. Moreover, its transverse momentum is unchanged on traversing the interface. Use all of this to "derive" Snell's Law.Figure P.4.10... Get solution
13. Get solution
14. Figure P.4.11 is a plot of the sine of the angle-of-incidence versus the sine of the transmission angle measured as light passed from air into a more optically dense medium. Discuss the curve. What is the significance of the slope of the line? Guess at what the dense medium might be.Figure P.4.11... Get solution
15. A ray of yellow light from a sodium discharge lamp falls on the surface of a diamond in air at 45°. If at that frequency nd = 2.42, compute the angular deviation suffered upon transmission. Get solution
16. Given an interface between water ... and glass ... compute the transmission angle for a beam incident in the water at 45°. If the transmitted beam is reversed so that it impinges on the interface, show that θt, — 45°. Get solution
17. A beam of 12-cm planar microwaves strikes the surface of a dielectric at 45°. If ... compute (a) the wavelength in the transmitting medium, and (b) the angle θt. Get solution
18. Light of wavelength 600 nm in vacuum enters a block of glass where ng = 1.5. Compute its wavelength in the glass. What color would it appear to someone embedded in the glass (see Table 3.4)? Get solution
19. A laserbeam impinges on an air-liquid interface at an angle of 55°. The refracted ray is observed to be transmitted at 40°. What is the refractive index of the liquid? Get solution
20. An underwater swimmer shines a beam of light up toward the surface. It strikes the air-water interface at 35°. At what angle will it emerge into the air? Get solution
21. Make a plot of θi- versus θt, for an air-glass boundary where nga — 1.5. Discuss the shape of the curve. Get solution
22. A laserbeam having a diameter D in air strikes a piece of glass (ng) at an angle θi. What is the diameter of the beam in the glass? Get solution
23. An exceedingly narrow beam of white light is incident at 60.0° on a sheet of glass 10.0 cm thick in air. The index of refraction for red light is 1.505 and for violet light it's 1.545. Determine the approximate diameter of the emerging beam. Get solution
24. Get solution
25. Get solution
26. A laserbeam impinges on the top surface of a 2.00-cm-thiek parallel glass (n = 1.50) plate at an angle of 35°. How long is the actual path through the glass? Get solution
27. Light is incident in the air on an air-glass interface. If the index of refraction of the glass is 1.70, find the incident angle such that the transmission angle is to equal ... Get solution
28. Suppose that you focus a camera with a close-up bellows attachment directly down on a letter printed on this page. The letter is then covered with a 1.00-mm-thick microscope slide (n = 1.55). How high must the camera be raised in order to keep the letter in focus? Get solution
29. A coin is resting on the bottom of a tank of water (nw = 1.33) LOO m deep. On top of the water floats a layer of benzene (nb = 1.50), which is 20.0 cm thick. Looking down nearly perpendicularly, how far beneath the topmost surface does the coin appear? Draw a ray diagram. Get solution
30. In Fig. P.4.26 the wavefronts in the incident medium match the fronts in the transmitting medium everywhere on the interface—a concept known as wavefront continuity. Write expressions for the number of waves per unit length along the interface in terms of θi; and λi- in one case and θt, and λt in the other. Use these to derive Snell's Law. Do you think Snell's Law applies to sound waves? Explain.Figure P.4.26... Get solution
31. With the previous problem in mind, return to Eq. (4.19) and take the origin of the coordinate system in the plane-of-incidence and on the interface (Fig. 4.39). Show that that equation is then equivalent to equating the x-components of the various propagation vectors. Show that it is also equivalent to the notion of wavefront continuity. Get solution
32. Making use of the ideas of equal transit times between corresponding points and the orthogonality of rays and wavefronts, derive the law of reflection and Snell's Law. The ray diagram of Fig. P.4.28 should be helpful.Figure P.4.28... Get solution
33. Starting with Snell's Law, prove that the vector refraction equation has the form... Get solution
34. Derive a vector expression equivalent to the Law of Reflection. As before, let the normal go from the incident to the transmitting medium, even though it obviously doesn't really matter. Get solution
35. In the case of reflection from a planar surface, use Fermat's Principle to prove that the incident and reflected rays share a common plane with the normal ûm namely, the plane-of-incidence. Get solution
36. Derive the Law of Reflection, θt- = θr by using the calculus to minimize the transit time, as required by Fermat's Principle. Get solution
37. According to the mathematician Hermann Schwarz, there is one triangle that can be inscribed within an acute triangle such that it has a minimal perimeter. Using two planar mirrors, a laserbeam, and Fermat's Principle, explain how you can show that this inscribed triangle has its vertices at the points where the altitudes of the acute triangle intersect its corresponding sides Get solution
38. Show analytically that a beam entering a planar transparent plate, as in Fig. P.4.34, emerges parallel to its initial direction. Derive an expression for the lateral displacement of the beam. Incidentally, the incoming and outgoing rays would be parallel even for a stack of plates of different material.Figure P.4.34... Get solution
39. Show that the two rays that enter the system in Fig. P.4.35 parallel to each other emerge from it being parallel.Figure. P.4.35... Get solution
40. Discuss the results of Problem 4.34 in the light of Fermat's Principle; that is, how does the relative index n21 affect things? To see the lateral displacement, look at a broad source through a thick piece of glass (≈... inch) or a stack (four will do) of microscope slides held at an angle. There will be an obvious shift between the region of the source seen directly and the region viewed through the glass. Get solution
41. Get solution
42. Suppose a lightwave that is linearly polarized in the plane-of-incidence impinges at 30° on a crown-glass (ng = 1.52) plate in air. Compute the appropriate amplitude reflection and transmission coefficients at the interface. Compare your results with Fig. 4.39. Get solution
43. Derive Eqs. (4.42) through (4.45) for r┴, r║, t┴, and r║. Get solution
44. A beam of light in air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 20.0°. The incident light has component E-field amplitudes parallel and perpendicular to the plane-of-incidence of 10.0 V/m and 20.0 V/m, respectively. Determine the corresponding reflected field amplitudes. Get solution
45. A laserbeam is incident on the interface between air and some dielectric of index n. For small values of θi- show that θt = θi/n. Use this and Eq. (4.42) to establish that at near-normal incidence [-r┴]θi ≈ o = (n-1)/(n+ 1). Get solution
46. Get solution
47. Get solution
48. Get solution
49. Light is incident in ah* perpendicularly on a sheet of crown glass having an index of refraction of 1.522. Determine both the reflectance and the transmittance. Get solution
50. A beam of quasimonochromalic light having an irradiance of 500 W/m2 is incident in air perpendicularly on the surface of a tank of water (nw = 1.333). Determine the transmitted irradiance. Get solution
51. Get solution
52. Get solution
53. Get solution
55. Get solution
56. Get solution
57. Quasimonochromatic light having an irradiance of 400 W/m2 is incident normally on the cornea (nc = 1.376) of the human eye. If the person is swimming under the water (nw = 1.33), determine the transmitted irradiance into the cornea. Get solution
58. Compare the amplitude reflection coefficients for an air-water (nw = 4/3) interface with that of an air-crown glass (ng = 3/2) interface, both at near-normal incidence. What are the corresponding ratios of the reflected to the incident irradiances? Get solution
59. Use Eq. (4.42) and the power series expansion of the sine function to establish that at near-normal incidence we can obtain a better approximation than the one in Problem 4.41, which is [-r┴]θi ≈ o = (n-1)/(n+ 1), namely... Get solution
60. Establish that at near-normal incidence the equation...is a good approximation. [Hint: Use the results of the previous problem, Eq. (4.43), and the power scries expansions of the sine and cosine functions.] Get solution
61. Prove that for a vacuum-dielectric interface at glancing incidence r┴ → - 1, as in Fig. 4.41. Get solution
62. In Fig. 4.41 the curve of r┴ approaches -1.0 as the angle-of-incidence approaches 90°. Prove that if α┴ is the angle the curve makes with the vertical at θi-, = 90°, then...[Hint: First show that dθt/dθi = 0.] Get solution
63. Prove that...for all θt first from the boundary conditions and then from the Fresnel Equations. Get solution
64. Verify that...for θi = 30° at a crown-glass and air interface (nti= 1.52). Get solution
65. Use the Fresnel Equations to prove that light incident at θp ½π-θt results in a reflected beam that is indeed polarized. Get solution
66. Show that tan θp = nt/ni and calculate the polarization angle for external incidence on a plate of crown glass (ng = 1.52) in air. Get solution
67. Beginning with Eq. (4.38), show that for two dielectric media, in general tan ... Get solution
68. Show that the polarization angles for internal and external reflection at a given interface are complementary, that is, θp + θ’p = 90° (see Problem 4.64). Get solution
69. It is often useful to work with the azimutkal angle γ, which isFigure P.4.61 (Photo and diagram courtesy S. Reich, The Weizmann Institute of Science, Israel....defined as the angle between the plane-of-vibration and the plane-of-incidence. Thus for linearly polarized light,...Figure P.4.67 is a plot of γr versus θi, for internal and external reflection at an air-glass interface (nga = 1.51), where γt = 45°. Verify a few of the points on the curves and in addition show that...Figure P.4.67... Get solution
70. Making use of the definitions of the azimuthal angles in Problem 4.67, show that... Get solution
71. Make a sketch of R┴ and R║ and nt = 1.5 and n, = 1 (i.e., internal reflection). Get solution
72. Show that... Get solution
73. Using the results of Problem 4.70, that is, Eqs. (4.98) and (4.99), show that...... Get solution
74. Suppose that we look at a source perpendicularly through a slack of N microscope slides. The source seen through even a dozen slides will be noticeably darker. Assuming negligible absorption, show that the total iransmillance of the stack is given by...and evaluate Tt, for three slides in air. Get solution
75. Making use of the expression...for an absorbing medium, we define a quantity called the unit trans-mittance. T1. At normal incidence, Eq. (4.55), T = It/Ii and thus when y = 1, T1 ≡ I(1)/Io. If the total thickness of the slides in the previous problem is d and if they now have a transmittanee per unit length T1, show that... Get solution
76. Show that at normal incidence on the boundary between two dielectrics, as ... Moreover, prove that as nti ... Thus as the two media take on more similar indices of refraction, less and less energy is carried off in the reflected wave. It should be obvious that when nti = 1 there will be no interface and no reflection. Get solution
77. Derive the expressions for r┴ and r║ given by Eqs. (4.70) and (4.71). Get solution
78. Show that when θi>θe at a dielectric interface, r┴ and r║ are complex and ... Get solution
79. Calculate the critical angle beyond which there is total internal reflection at an air-glass (ng = 1.5) interface. Compare this result with that of Problem 4.15. Get solution
80. Referring back to Problem 4.18, note that as θi,. increases θt increases. Prove that the maximum value θt ( may have is θc. Get solution
81. What is the critical angle for total internal reflection for diamond? What, if anything, does the critical angle have to do with the luster of a well-cut diamond? Get solution
82. Using a block of a transparent, unknown material, it is found that a beam of light inside the material is totally internally reflected at the air-block interface at an angle of 48.0°. What is its index of refraetion? Get solution
83. A prism, ABC, is configured such that angle BCA = 90° and angle CBA = 45°. What is the minimum value of its index of refraction if, while immersed in air, a beam traversing face AC is to be totally internally reflected from face BC. Get solution
84. A fish looking straight up toward the smooth surface of a pond receives a cone of rays and sees a circle of light filled with the images of sky and birds and whatever else is up there. This bright circular field is surrounded by darkness. Explain what is happening and compute the cone-angle. Get solution
85. A glass block having an index of 1.55 is covered with a layer of water of index 1.33. For light traveling in the glass, what is the critical angle at the interface? Get solution
86. Derive an expression for the speed of the evanescent wave in the case of internal reflection. Write it in terms of c, ni and θt Get solution
87. Light having a vacuum wavelength of 600 nm, traveling in a...glass (ng = 1.50) block, is incident at 45° on a glass-air interface. It is then totally internally reflected. Determine the distance into the air at which the amplitude of the evanescent wave has dropped to a value of 1/e of its maximum value at the interface. Get solution
88. Get solution
89. Get solution
90. Get solution
91. Figure P.4.61 shows a laserbeam incident on a wet piece of filter paper atop a sheet of glass whose index of refraction is to be measured—the photograph shows the resulting light pattern. Explain what is happening and derive an expression for ni in terms of R and d. Get solution
92. Consider the common mirage associated with an inhomoge-neous distribution of air situated above a warm roadway. Envision the bending of the rays as if it were instead a problem in total intet nal-reflection. If an observer, at whose head na = 1.000 29, sees an apparent wet spot at θi,≥ 88.7° down the road, find the index of the air immediately above the road. Get solution
93. Figure P.4.80 depicts a glass cube surrounded by four glass prisms in very close proximity to its sides. Sketch in the paths that will be taken by the two rays shown and discuss a possible application for the device.Figure P.4.80... Get solution
94. Figure P.4.82 shows a prism-coupler arrangement developed al the Bell Telephone Laboratories. Its function is to feed a laserbeam into a thin (0.00001-inch) transparent film, which then serves as a sort of waveguide. One application is that of thin-film laserbeam circuitry—a kind of integrated optics. How do you think it works?Figure P.4.82... Get solution
95. Figure P.4.81 is a plot of nt and nR versus A for a common am al. Identify the metal by comparing its characteristics with those con sidered in the chapter and discuss its optical properties.Figure P.4.81... Get solution
96. Get solution
97. Get solution
98. Get solution
99. Figure P.4.77 depicts a ray being multiply reflected by a transparent dielectric plate (the amplitudes of the resulting fragments arc indicated). As in Section 4.5, we use the primed coefficient notation because the angles are related by Snell's Law.(a) Finish labeling the amplitudes of the last four rays.(b) Show, using the Fresnel Equations, that... Get solution
100. A wave, linearly polarized in the plane-of-incidence, impinges on the interface between two dielectric media. If ... there is no reflected wave, that is, ... Using Stokes's techFigure P.4.77...nique, start from scratch to show that ... and θt = θp (Problem 4.66). How docs this compare with Eq. (4.100)? Get solution
101. Making use of the Fresnel Equations, show that ...as in the previous problem. Get solution
