OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Page 0 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Conceptual Questions
1. Under what conditions can light be modeled like a ray? Like a wave?
Solution
Light can be modeled as a ray when devices are large compared to wavelength, and as a wave
when devices are comparable or small compared to wavelength.
2. Why is the index of refraction always greater than or equal to 1?
Solution
The index of refraction is the ratio of the speed of light in a vacuum to the speed of light in a
medium. Because the former is the fastest that the speed of light can be, this ratio is, by
definition, greater than or equal to 1.
3. Does the fact that the light flash from lightning reaches you before its sound prove that the
speed of light is extremely large or simply that it is greater than the speed of sound? Discuss how
you could use this effect to get an estimate of the speed of light.
Solution
This fact simply proves that the speed of light is greater than that of sound. If one knows the
distance to the location of the lightning and the speed of sound, one could, in principle,
determine the speed of light from the data. In practice, because the speed of light is so great, the
data would have to be known to impractically high precision.
4. Speculate as to what physical process might be responsible for light traveling more slowly in a
medium than in a vacuum.
Solution
answers may vary
5. Using the law of reflection, explain how powder takes the shine off of a person’s nose. What
is the name of the optical effect?
Solution
Powder consists of many small particles with randomly oriented surfaces. This leads to diffuse
reflection, reducing shine.
6. Diffusion by reflection from a rough surface is described in this chapter. Light can also be
diffused by refraction. Describe how this occurs in a specific situation, such as light interacting
with crushed ice.
Solution
Crushed ice has surface segments with a variety of orientations, resulting in refracted rays in
many directions, creating a diffuse effect.
7. Will light change direction toward or away from the perpendicular when it goes from air to
water? Water to glass? Glass to air?
Solution
“toward” when increasing n (air to water, water to glass); “away” when decreasing n (glass to
air)
8. Explain why an object in water always appears to be at a depth shallower than it actually is?
Solution
Rays from underwater objects emerge from water subject to refraction. The observer in air
perceives an apparent location for the source, which is different from the physical location.
Page 1 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
9. Explain why a person’s legs appear very short when wading in a pool. Justify your explanation
with a ray diagram showing the path of rays from the feet to the eye of an observer who is out of
the water.
Solution
A ray from a leg emerges from water after refraction. The observer in air perceives an apparent
location for the source, as if a ray traveled in a straight line. See the dashed ray below.
10. Explain why an oar that is partially submerged in water appears bent.
Solution
Although rays from the parts of the oar above water travel straight to the observer’s eye, rays
from the underwater parts of the oar reach the eye after refraction, making them appear to come
from another direction. This difference in direction appears as a bend in the oar.
11. A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible
when submerged. Can it be a diamond? Explain.
Solution
The gemstone becomes invisible when its index of refraction is the same, or at least similar to,
the water surrounding it. Because diamond has a particularly high index of refraction, it can still
sparkle as a result of total internal reflection, not invisible.
12. The most common type of mirage is an illusion that light from faraway objects is reflected by
a pool of water that is not really there. Mirages are generally observed in deserts, when there is a
hot layer of air near the ground. Given that the refractive index of air is lower for air at higher
temperatures, explain how mirages can be formed.
Solution
A mirage can be regarded as a result of two layers of air: a hot layer near the ground (low n) and
a cool layer above (high n). Rays originating in the cool layer can reflect back up due to an effect
analogous to total internal reflection.
13. How can you use total internal reflection to estimate the index of refraction of a medium?
Solution
One can measure the critical angle by looking for the onset of total internal reflection as the
angle of incidence is varied. can then be applied to compute the
index of refraction.
Page 2 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
14. Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices
of refraction and angles, perhaps referring to that shown below. Some of us have seen the
formation of a double rainbow; is it physically possible to observe a triple rainbow?
Solution
Total internal reflection is a feature at the back surface of the water droplets that form rainbows
(see the following figure). When the figure is inverted, it illustrates the upper bow in a double
rainbow. Third and higher-order rainbows can be caused by multiple internal reflections; see
http://www.atoptics.co.uk/rainbows/ord34.htm
15. A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths
with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated
by white light.
Solution
In addition to total internal reflection, rays that refract into and out of diamond crystals are
subject to dispersion due to varying values of n across the spectrum, resulting in a sparkling
display of colors.
16. How do wave effects depend on the size of the object with which the wave interacts? For
example, why does sound bend around the corner of a building while light does not?
Solution
Compare wavelength with the size of the object. Sound wavelengths are much larger than light
wavelengths, making diffraction around objects the size of buildings observable.
17. Does Huygens’s principle apply to all types of waves?
Solution
yes
Page 3 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
18. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a
wave. Does the reverse hold true? That is, if diffraction is not observed, does that mean the
phenomenon is not a wave?
Solution
It could simply mean that diffraction exists but is too small to be observed. In any case, light that
does not display diffraction can be reasonably modeled as rays.
19. Can a sound wave in air be polarized? Explain.
Solution
No. Sound waves are not transverse waves.
20. No light passes through two perfect polarizing filters with perpendicular axes. However, if a
third polarizing filter is placed between the original two, some light can pass. Why is this? Under
what circumstances does most of the light pass?
Solution
The third polarizer rotates the polarization axis such that it is no longer perpendicular to the
second polarizer. Maximum transmission occurs at .
21. Explain what happens to the energy carried by light that it is dimmed by passing it through
two crossed polarizing filters.
Solution
Energy is absorbed into the filters.
22. When particles scattering light are much smaller than its wavelength, the amount of
scattering is proportional to . Does this mean there is more scattering for small than large
How does this relate to the fact that the sky is blue?
Solution
Small is indeed preferentially scattered. Within the visible spectrum, this results in more blue
(small than red (large being scattered into the direction of our eyes, resulting in an overall
blue color for the sky.
23. Using the information given in the preceding question, explain why sunsets are red.
Solution
Sunsets are viewed with light traveling straight from the Sun toward us. When blue light is
scattered out of this path, the remaining red light dominates the overall appearance of the setting
Sun.
24. When light is reflected at Brewster’s angle from a smooth surface, it is polarized
parallel to the surface. Part of the light will be refracted into the surface. Describe how you
would do an experiment to determine the polarization of the refracted light. What direction
would you expect the polarization to have and would you expect it to be ?
Solution
The refracted light can be put through a polarizing filter to measure for polarization. A
polarization direction perpendicular to that of the reflected light can be expected if the original
source is unpolarized.
25. If you lie on a beach looking at the water with your head tipped slightly sideways, your
polarized sunglasses do not work very well. Why not?
Solution
The axis of polarization for the sunglasses has been rotated .
Page 4 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Problems
26. What is the speed of light in water? In glycerine?
Solution
;
27. What is the speed of light in air? In crown glass?
Solution
;
Use the following table for problems 28 and 29.
Medium n
Gases at , 1 atm
Air 1.000293
Carbon dioxide 1.00045
Hydrogen 1.000139
Oxygen 1.000271
Liquids at
Benzene 1.501
Carbon disulfide 1.628
Carbon tetrachloride 1.461
Ethanol 1.361
Glycerine 1.473
Water, fresh 1.333
Solids at
Diamond 2.419
Fluorite 1.434
Glass, crown 1.52
Glass, flint 1.66
Ice (at ) 1.309
Polystyrene 1.49
Plexiglas 1.51
Quartz, crystalline 1.544
Quartz, fused 1.458
Sodium chloride 1.544
Zircon 1.923
28. Calculate the index of refraction for a medium in which the speed of light is
and identify the most likely substance based on the table above.
Solution
1.490; polystyrene
29. In what substance in the table above is the speed of light
Solution
ice at
30. There was a major collision of an asteroid with the Moon in medieval times. It was described
by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long
after the asteroid hit the Moon, which is away, would the light first arrive on
Earth?
Page 5 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Solution
1.28 s
31. Components of some computers communicate with each other through optical fibers having
an index of refraction What time in nanoseconds is required for a signal to travel 0.200
m through such a fiber?
Solution
1.03 ns
32. Compare the time it takes for light to travel 1000 m on the surface of Earth and in outer
space.
Solution
0.977 ns slower on Earth’s surface
33. How far does light travel underwater during a time interval of ?
Solution
337 m
34. Suppose a man stands in front of a mirror as shown below. His eyes are 1.65 m above the
floor and the top of his head is 0.13 m higher. Find the height above the floor of the top and
bottom of the smallest mirror in which he can see both the top of his head and his feet. How is
this distance related to the man’s height?
Solution
The bottom is from the floor and the top is from the floor.
35. Show that when light reflects from two mirrors that meet each other at a right angle, the
outgoing ray is parallel to the incoming ray, as illustrated below.
Page 6 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Solution
proof
36. On the Moon’s surface, lunar astronauts placed a corner reflector, off which a laser beam is
periodically reflected. The distance to the Moon is calculated from the round-trip time. What
percent correction is needed to account for the delay in time due to the slowing of light in Earth’s
atmosphere? Assume the distance to the Moon is precisely and Earth’s atmosphere
(which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant
index of refraction
Solution
37. A flat mirror is neither converging nor diverging. To prove this, consider two rays originating
from the same point and diverging at an angle (see below). Show that after striking a plane
mirror, the angle between their directions remains
Solution
proof
Unless otherwise specified, for problems 38 through 45, the indices of refraction of glass and
water should be taken to be 1.50 and 1.333, respectively.
38. A light beam in air has an angle of incidence of at the surface of a glass plate. What are
the angles of reflection and refraction?
Solution
and
Page 7 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
39. A light beam in air is incident on the surface of a pond, making an angle of with respect
to the surface. What are the angles of reflection and refraction?
Solution
reflection, ; refraction,
40. When a light ray crosses from water into glass, it emerges at an angle of with respect to
the normal of the interface. What is its angle of incidence?
Solution
41. A pencil flashlight submerged in water sends a light beam toward the surface at an angle of
incidence of . What is the angle of refraction in air?
Solution
42. Light rays from the Sun make a angle to the vertical when seen from below the surface
of a body of water. At what angle above the horizon is the Sun?
Solution
43. The path of a light beam in air goes from an angle of incidence of to an angle of
refraction of when it enters a rectangular block of plastic. What is the index of refraction of
the plastic?
Solution
1.53
44. A scuba diver training in a pool looks at his instructor as shown below. What angle does the
ray from the instructor’s face make with the perpendicular to the water at the point where the ray
enters? The angle between the ray in the water and the perpendicular to the water is .
Solution
45. (a) Using information in the preceding problem, find the height of the instructor’s head above
the water, noting that you will first have to calculate the angle of incidence. (b) Find the apparent
depth of the diver’s head below water as seen by the instructor.
Solution
Page 8 of 18
, OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
a. 2.9 m; b. 1.4 m
46. Verify that the critical angle for light going from water to air is , as discussed at the
end of Determining a Critical Angle, regarding the critical angle for light traveling in a
polystyrene (a type of plastic) pipe surrounded by air.
Solution
47. (a) At the end of Determining the Critical Angle, it was stated that the critical angle for light
going from diamond to air is . Verify this. (b) What is the critical angle for light going
from zircon to air?
Solution
a. ; b.
48. An optical fiber uses flint glass clad with crown glass. What is the critical angle?
Solution
49. At what minimum angle will you get total internal reflection of light traveling in water and
reflected from ice?
Solution
50. Suppose you are using total internal reflection to make an efficient corner reflector. If there is
air outside and the incident angle is , what must be the minimum index of refraction of the
material from which the reflector is made?
Solution
1.414
51. You can determine the index of refraction of a substance by determining its critical angle. (a)
What is the index of refraction of a substance that has a critical angle of when submerged
in water? What is the substance, based on the following table? (b) What would the critical angle
be for this substance in air?
Medium n
Gases at , 1 atm
Air 1.000293
Carbon dioxide 1.00045
Hydrogen 1.000139
Oxygen 1.000271
Liquids at
Benzene 1.501
Carbon disulfide 1.628
Carbon tetrachloride 1.461
Ethanol 1.361
Glycerine 1.473
Water, fresh 1.333
Solids at
Diamond 2.419
Fluorite 1.434
Glass, crown 1.52
Glass, flint 1.66
Page 9 of 18
Unit 1: Optics
Chapter 1: The Nature of Light
Page 0 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Conceptual Questions
1. Under what conditions can light be modeled like a ray? Like a wave?
Solution
Light can be modeled as a ray when devices are large compared to wavelength, and as a wave
when devices are comparable or small compared to wavelength.
2. Why is the index of refraction always greater than or equal to 1?
Solution
The index of refraction is the ratio of the speed of light in a vacuum to the speed of light in a
medium. Because the former is the fastest that the speed of light can be, this ratio is, by
definition, greater than or equal to 1.
3. Does the fact that the light flash from lightning reaches you before its sound prove that the
speed of light is extremely large or simply that it is greater than the speed of sound? Discuss how
you could use this effect to get an estimate of the speed of light.
Solution
This fact simply proves that the speed of light is greater than that of sound. If one knows the
distance to the location of the lightning and the speed of sound, one could, in principle,
determine the speed of light from the data. In practice, because the speed of light is so great, the
data would have to be known to impractically high precision.
4. Speculate as to what physical process might be responsible for light traveling more slowly in a
medium than in a vacuum.
Solution
answers may vary
5. Using the law of reflection, explain how powder takes the shine off of a person’s nose. What
is the name of the optical effect?
Solution
Powder consists of many small particles with randomly oriented surfaces. This leads to diffuse
reflection, reducing shine.
6. Diffusion by reflection from a rough surface is described in this chapter. Light can also be
diffused by refraction. Describe how this occurs in a specific situation, such as light interacting
with crushed ice.
Solution
Crushed ice has surface segments with a variety of orientations, resulting in refracted rays in
many directions, creating a diffuse effect.
7. Will light change direction toward or away from the perpendicular when it goes from air to
water? Water to glass? Glass to air?
Solution
“toward” when increasing n (air to water, water to glass); “away” when decreasing n (glass to
air)
8. Explain why an object in water always appears to be at a depth shallower than it actually is?
Solution
Rays from underwater objects emerge from water subject to refraction. The observer in air
perceives an apparent location for the source, which is different from the physical location.
Page 1 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
9. Explain why a person’s legs appear very short when wading in a pool. Justify your explanation
with a ray diagram showing the path of rays from the feet to the eye of an observer who is out of
the water.
Solution
A ray from a leg emerges from water after refraction. The observer in air perceives an apparent
location for the source, as if a ray traveled in a straight line. See the dashed ray below.
10. Explain why an oar that is partially submerged in water appears bent.
Solution
Although rays from the parts of the oar above water travel straight to the observer’s eye, rays
from the underwater parts of the oar reach the eye after refraction, making them appear to come
from another direction. This difference in direction appears as a bend in the oar.
11. A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible
when submerged. Can it be a diamond? Explain.
Solution
The gemstone becomes invisible when its index of refraction is the same, or at least similar to,
the water surrounding it. Because diamond has a particularly high index of refraction, it can still
sparkle as a result of total internal reflection, not invisible.
12. The most common type of mirage is an illusion that light from faraway objects is reflected by
a pool of water that is not really there. Mirages are generally observed in deserts, when there is a
hot layer of air near the ground. Given that the refractive index of air is lower for air at higher
temperatures, explain how mirages can be formed.
Solution
A mirage can be regarded as a result of two layers of air: a hot layer near the ground (low n) and
a cool layer above (high n). Rays originating in the cool layer can reflect back up due to an effect
analogous to total internal reflection.
13. How can you use total internal reflection to estimate the index of refraction of a medium?
Solution
One can measure the critical angle by looking for the onset of total internal reflection as the
angle of incidence is varied. can then be applied to compute the
index of refraction.
Page 2 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
14. Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices
of refraction and angles, perhaps referring to that shown below. Some of us have seen the
formation of a double rainbow; is it physically possible to observe a triple rainbow?
Solution
Total internal reflection is a feature at the back surface of the water droplets that form rainbows
(see the following figure). When the figure is inverted, it illustrates the upper bow in a double
rainbow. Third and higher-order rainbows can be caused by multiple internal reflections; see
http://www.atoptics.co.uk/rainbows/ord34.htm
15. A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths
with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated
by white light.
Solution
In addition to total internal reflection, rays that refract into and out of diamond crystals are
subject to dispersion due to varying values of n across the spectrum, resulting in a sparkling
display of colors.
16. How do wave effects depend on the size of the object with which the wave interacts? For
example, why does sound bend around the corner of a building while light does not?
Solution
Compare wavelength with the size of the object. Sound wavelengths are much larger than light
wavelengths, making diffraction around objects the size of buildings observable.
17. Does Huygens’s principle apply to all types of waves?
Solution
yes
Page 3 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
18. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a
wave. Does the reverse hold true? That is, if diffraction is not observed, does that mean the
phenomenon is not a wave?
Solution
It could simply mean that diffraction exists but is too small to be observed. In any case, light that
does not display diffraction can be reasonably modeled as rays.
19. Can a sound wave in air be polarized? Explain.
Solution
No. Sound waves are not transverse waves.
20. No light passes through two perfect polarizing filters with perpendicular axes. However, if a
third polarizing filter is placed between the original two, some light can pass. Why is this? Under
what circumstances does most of the light pass?
Solution
The third polarizer rotates the polarization axis such that it is no longer perpendicular to the
second polarizer. Maximum transmission occurs at .
21. Explain what happens to the energy carried by light that it is dimmed by passing it through
two crossed polarizing filters.
Solution
Energy is absorbed into the filters.
22. When particles scattering light are much smaller than its wavelength, the amount of
scattering is proportional to . Does this mean there is more scattering for small than large
How does this relate to the fact that the sky is blue?
Solution
Small is indeed preferentially scattered. Within the visible spectrum, this results in more blue
(small than red (large being scattered into the direction of our eyes, resulting in an overall
blue color for the sky.
23. Using the information given in the preceding question, explain why sunsets are red.
Solution
Sunsets are viewed with light traveling straight from the Sun toward us. When blue light is
scattered out of this path, the remaining red light dominates the overall appearance of the setting
Sun.
24. When light is reflected at Brewster’s angle from a smooth surface, it is polarized
parallel to the surface. Part of the light will be refracted into the surface. Describe how you
would do an experiment to determine the polarization of the refracted light. What direction
would you expect the polarization to have and would you expect it to be ?
Solution
The refracted light can be put through a polarizing filter to measure for polarization. A
polarization direction perpendicular to that of the reflected light can be expected if the original
source is unpolarized.
25. If you lie on a beach looking at the water with your head tipped slightly sideways, your
polarized sunglasses do not work very well. Why not?
Solution
The axis of polarization for the sunglasses has been rotated .
Page 4 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Problems
26. What is the speed of light in water? In glycerine?
Solution
;
27. What is the speed of light in air? In crown glass?
Solution
;
Use the following table for problems 28 and 29.
Medium n
Gases at , 1 atm
Air 1.000293
Carbon dioxide 1.00045
Hydrogen 1.000139
Oxygen 1.000271
Liquids at
Benzene 1.501
Carbon disulfide 1.628
Carbon tetrachloride 1.461
Ethanol 1.361
Glycerine 1.473
Water, fresh 1.333
Solids at
Diamond 2.419
Fluorite 1.434
Glass, crown 1.52
Glass, flint 1.66
Ice (at ) 1.309
Polystyrene 1.49
Plexiglas 1.51
Quartz, crystalline 1.544
Quartz, fused 1.458
Sodium chloride 1.544
Zircon 1.923
28. Calculate the index of refraction for a medium in which the speed of light is
and identify the most likely substance based on the table above.
Solution
1.490; polystyrene
29. In what substance in the table above is the speed of light
Solution
ice at
30. There was a major collision of an asteroid with the Moon in medieval times. It was described
by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long
after the asteroid hit the Moon, which is away, would the light first arrive on
Earth?
Page 5 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Solution
1.28 s
31. Components of some computers communicate with each other through optical fibers having
an index of refraction What time in nanoseconds is required for a signal to travel 0.200
m through such a fiber?
Solution
1.03 ns
32. Compare the time it takes for light to travel 1000 m on the surface of Earth and in outer
space.
Solution
0.977 ns slower on Earth’s surface
33. How far does light travel underwater during a time interval of ?
Solution
337 m
34. Suppose a man stands in front of a mirror as shown below. His eyes are 1.65 m above the
floor and the top of his head is 0.13 m higher. Find the height above the floor of the top and
bottom of the smallest mirror in which he can see both the top of his head and his feet. How is
this distance related to the man’s height?
Solution
The bottom is from the floor and the top is from the floor.
35. Show that when light reflects from two mirrors that meet each other at a right angle, the
outgoing ray is parallel to the incoming ray, as illustrated below.
Page 6 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
Solution
proof
36. On the Moon’s surface, lunar astronauts placed a corner reflector, off which a laser beam is
periodically reflected. The distance to the Moon is calculated from the round-trip time. What
percent correction is needed to account for the delay in time due to the slowing of light in Earth’s
atmosphere? Assume the distance to the Moon is precisely and Earth’s atmosphere
(which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant
index of refraction
Solution
37. A flat mirror is neither converging nor diverging. To prove this, consider two rays originating
from the same point and diverging at an angle (see below). Show that after striking a plane
mirror, the angle between their directions remains
Solution
proof
Unless otherwise specified, for problems 38 through 45, the indices of refraction of glass and
water should be taken to be 1.50 and 1.333, respectively.
38. A light beam in air has an angle of incidence of at the surface of a glass plate. What are
the angles of reflection and refraction?
Solution
and
Page 7 of 18
,OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
39. A light beam in air is incident on the surface of a pond, making an angle of with respect
to the surface. What are the angles of reflection and refraction?
Solution
reflection, ; refraction,
40. When a light ray crosses from water into glass, it emerges at an angle of with respect to
the normal of the interface. What is its angle of incidence?
Solution
41. A pencil flashlight submerged in water sends a light beam toward the surface at an angle of
incidence of . What is the angle of refraction in air?
Solution
42. Light rays from the Sun make a angle to the vertical when seen from below the surface
of a body of water. At what angle above the horizon is the Sun?
Solution
43. The path of a light beam in air goes from an angle of incidence of to an angle of
refraction of when it enters a rectangular block of plastic. What is the index of refraction of
the plastic?
Solution
1.53
44. A scuba diver training in a pool looks at his instructor as shown below. What angle does the
ray from the instructor’s face make with the perpendicular to the water at the point where the ray
enters? The angle between the ray in the water and the perpendicular to the water is .
Solution
45. (a) Using information in the preceding problem, find the height of the instructor’s head above
the water, noting that you will first have to calculate the angle of incidence. (b) Find the apparent
depth of the diver’s head below water as seen by the instructor.
Solution
Page 8 of 18
, OpenStax University Physics Volume III
Unit 1: Optics
Chapter 1: The Nature of Light
a. 2.9 m; b. 1.4 m
46. Verify that the critical angle for light going from water to air is , as discussed at the
end of Determining a Critical Angle, regarding the critical angle for light traveling in a
polystyrene (a type of plastic) pipe surrounded by air.
Solution
47. (a) At the end of Determining the Critical Angle, it was stated that the critical angle for light
going from diamond to air is . Verify this. (b) What is the critical angle for light going
from zircon to air?
Solution
a. ; b.
48. An optical fiber uses flint glass clad with crown glass. What is the critical angle?
Solution
49. At what minimum angle will you get total internal reflection of light traveling in water and
reflected from ice?
Solution
50. Suppose you are using total internal reflection to make an efficient corner reflector. If there is
air outside and the incident angle is , what must be the minimum index of refraction of the
material from which the reflector is made?
Solution
1.414
51. You can determine the index of refraction of a substance by determining its critical angle. (a)
What is the index of refraction of a substance that has a critical angle of when submerged
in water? What is the substance, based on the following table? (b) What would the critical angle
be for this substance in air?
Medium n
Gases at , 1 atm
Air 1.000293
Carbon dioxide 1.00045
Hydrogen 1.000139
Oxygen 1.000271
Liquids at
Benzene 1.501
Carbon disulfide 1.628
Carbon tetrachloride 1.461
Ethanol 1.361
Glycerine 1.473
Water, fresh 1.333
Solids at
Diamond 2.419
Fluorite 1.434
Glass, crown 1.52
Glass, flint 1.66
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