College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 21

Reflection and Refraction

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Chapter Questions

Problem 1

You are 165 cm tall and stand under a tree. The tree shadow is 34 m long and your shadow is about twice your height. How tall is the tree?

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Problem 2

A shadow from a surgeon’s hand obstructs her view while operating. Make suggestions for an alternative light source that avoids this difficulty. Include one or more sketches for your proposed plan.

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Problem 3

A lunar eclipse happens when the Moon, Earth, and Sun are aligned in that order (the Moon is in the
shadow of Earth). Aristotle used this phenomenon to determine the shape of Earth. He proposed that Earth has a round shape. Draw a picture to describe his reasoning.

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Problem 4

You and a friend are having a romantic candlelight dinner. You notice that the light shadows of your hands on the wall look fuzzy. However, the shadow of a glass is very sharp and crisp. Where are you and your friend sitting with respect to the candle and the wall? Where is the glass? To answer these questions, draw ray diagrams assuming that the candle is an extended light source.

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Problem 5

You want to make a pinhole camera with a blank wall as the screen and you (or a friend) as the object of interest. Draw a sketch showing the wall, the pinhole, the place you or your friend will sit, the best location for the Sun or some other light source, and the location of people viewing the wall. Will the image appear upright or inverted? Explain.

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Problem 6

Only observers in a very narrow region on Earth (about 200 km diameter) can see a total solar eclipse. In the region of such an eclipse, there is no sunlight and a person can see stars during daytime. Draw the arrangement of the Sun, the Moon, and Earth during a total solar eclipse.

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Problem 7

Your summer ecology research job involves documenting the growth of trees at an experimental site. One day you forget your tree-height-measuring instrument. How can you determine the height of trees without it? Provide a sketch for your method.

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Problem 8

The same day that you forgot the height-measuring instrument (see Problem 7), you also forgot your watch. You left your house at 7:00 a.m. and drove to the site in 2.0 hours. How can you build a sundial so that you can leave the site at 6:00 p.m. to make it to a concert? What assumptions did you make?

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Problem 9

You have a small mirror. While holding the mirror, you see a light spot on a wall at the same height as the mirror. At what angle are you holding the mirror if the Sun is 50° above the horizontal? Draw a ray diagram to answer the question. What assumptions did you make?

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Problem 10

You see a well and wonder if there is water in it. How should you hold a small mirror to see the well’s bottom? The Sun’s rays are hitting the ground at an angle of 60° above the horizontal. Draw a ray diagram to answer the question. What assumptions did you make?

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Problem 11

Design an experiment that you can perform to test the law of reflection. Describe the instruments that you are going to use and their experimental uncertainty (half of the smallest division). How certain will you be of your results?

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Problem 12

Draw four separate mirrors: one horizontal, the next 30° above the horizontal, the third 90° above the horizontal (vertical), and the fourth at an angle of 120° relative to the first. For each mirror draw an incident ray hitting the mirror at its middle (the ray should not be perpendicular to the mirror) and then draw a reflected ray.

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Problem 13

Design a mirror arrangement so that light from a laser pointer will travel in exactly the opposite direction after it reflects off the mirror(s), even if you change the direction the laser pointer is pointing.

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Problem 14

Two mirrors are oriented at right angles. A narrow light beam strikes the horizontal mirror at an incident angle of 65°, reflects from it, and then hits the vertical mirror. Determine the angle of incidence at the vertical mirror and the direction of the light after leaving the vertical mirror. Include a sketch with your explanation.

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Problem 15

Draw rays (maybe a side view) to help explain how the Sun shines on the back of the inverted bird in

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Problem 16

You are driving along a street on a sunny day. Only a few apartment building windows appear bright; the
rest are pitch black. Explain this difference and include a ray diagram to help with your explanation.

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Problem 17

A light bulb is placed somewhere on the radial axis of the bowl mirror shown in Figure P 21.17. Carefully draw three different narrow light beams leaving the bulb that reflect from the bowl mirror and move away. Would this arrangement be useful for a headlight? Explain.

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Problem 18

A flat mirror is rotated 17° about an axis in the plane of the mirror. What is the angle change of a reflected light beam if the direction of the incident beam does not change?

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Problem 19

(a) A laser beam passes from air into a 25% glucose solution at an incident angle of 35°. In what direction does light travel in the glucose solution? (b) The beam travels from ethyl alcohol to air at an incident angle of 12°. Determine the angle of the refracted beam in the air. (c) Draw pictures for (a) and (b) showing the interface between the media, the normal line, the incident rays, the reflected rays, the refracted rays, and the angles of these rays relative to the normal line.

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Problem 20

A beam of light passes from glass with refractive index 1.58 into water with a refractive index 1.33. The angle of the refracted ray in water is 58.0°. Draw a sketch of the situation showing the interface between the media, the normal line, the incident ray, the reflected ray, the refracted ray, and the angles of these rays relative to the normal line.

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Problem 21

A beam of light passes from air into a transparent petroleum product, cyclohexane, at an incident angle of 48°. The angle of refraction is 31°. What is the index of refraction of the cyclohexane?

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Problem 22

An aquarium open at the top has 30-cm-deep water in it. You shine a laser pointer into the top opening so it is incident on the air-water interface at a 45° angle relative to the vertical. You see a bright spot where the beam hits the bottom of the aquarium. How much water (in terms of height) should you add to the tank so the bright spot on the bottom moves 5.0 cm?

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Problem 23

You have a V-shaped transparent empty container such as shown in Figure P 21.23. When you shine a laser pointer horizontally through the empty container, the beam goes straight through and makes a spot on the wall. (a) What happens to this spot if you fill the container with water just a little above the level at which the laser beam passes through the container? (b) What happens if you fill the container to the very top? Indicate any assumptions used and draw a ray diagram for each situation.

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Problem 24

A light beam hits the interface between air and an unknown material at an angle of 43° relative to the normal. The reflected ray and the refracted ray make an angle of 108° with respect to each other. What is the index of refraction of the material?A light beam hits the interface between air and an unknown material at an angle of 43° relative to the normal. The reflected ray and the refracted ray make an angle of 108° with respect to each other. What is the index of refraction of the material?

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Problem 25

A light ray passes from air through a glass plate with refractive index 1.60 into water. The angle of the refracted ray in the water is 42.0°. Determine the angle of the incident ray at the air-glass interface.

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Problem 26

Behind the lens of the eye is the vitreous humor, a jellylike substance that occupies most of the eyeball. The refractive index of the vitreous humor is 1.35 and that of the lens is 1.44. A narrow beam of light traveling in the lens comes to the interface with the vitreous humor at a 23° angle. What is its direction relative to the interface when in the vitreous humor?

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Problem 27

You have a block of glass with an equilateral prism- shaped opening filled with air inside. Draw the path of a light ray that strikes the glass plate parallel to one of the sides of the prism and then passes through the prism.

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Problem 28

You watch a crab in an aquarium. Light traveling in air enters a sheet of glass at the side of the aquarium and then passes into the water. If the angle of incidence at the air-glass interface is 22°, what are the angles at which the light wave travels in the glass and in the water? Indicate any assumptions you made.

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Problem 29

Light moving up and toward the right in air enters the side of a cube of gelatin of refractive index 1.30 at an incident angle of 80°, Determine the angle at which the light leaves the top surface of the cube. How does the angle change if the refractive index of the gelatin is slightly greater? Explain.

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Problem 30

Can your light be seen? You swim under water at night and shine a laser pointer so that it hits the water-air interface at an incident angle of 52°. Will a friend see the light above the water? Explain.

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Problem 31

Light is incident on the boundary between two media at an angle of 30°. If the refracted light makes an angle of 42°, what is the critical angle for light incident on the same boundary?

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Problem 32

Determine the critical angle for light inside a diamond incident on an interface with air.

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Problem 33

Determine the refractive index of a glucose solution for which the critical angle for light traveling in the solution incident on an interface with air is 42.5°. How would the critical angle change if the glucose concentration were slightly greater? Explain.

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Problem 34

You wish to use a prism to change the direction of a beam of light 90 with respect to its original direction. Describe the shape of the prism and its orientation with respect to the original beam to achieve this goal.

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Problem 35

Light is incident on the boundary between two media at an angle of $34^{\circ}$. If the refracted light makes an angle of $37^{\circ}$, what is the critical angle for light incident on the same boundary?

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Problem 36

What must be the minimum value of the refractive index of the prism shown in Figure P 21.36 in order that light is totally reflected where indicated? Will some of the light make it out of the top surface? Explain.

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Problem 37

In gemology, two of the most useful pieces of information concerning an unknown gem are the refractive index of the stone and its mass density. The refractive index is often determined using a critical angle measurement. Determine the refractive index of the gemstone shown in Figure P 21.37 a. The critical angle for the total reflection of light at the gem-air interface is $37.28^{\circ}$. A simple sketch of the experiment combined with the ray diagram is shown in Figure P 21.37 b. (The apparatus used to make these measurements is a complicated optical instrument.)

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Problem 38

(a) The refractive index for the gem aquamarine is 1.57. Determine the critical angle for light traveling inside aqua-marine when reaching an air interface. (b) You have a tourmaline gem and find that a laser beam in air incident on the air-tourmaline interface at a $50^{\circ}$ angle has a refracted angle in the tourmaline of $28.2^{\circ}$. Determine the refractive index and critical angle of tourmaline.

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Problem 39

While you are sitting on a chair at the edge of a 1.2-m-deep swimming pool, your eyes are 1.5 m
above the surface of the water. You see an acorn on the bottom of the pool at a $45^{\circ}$ angle below the horizontal. What is the horizontal distance between you and the acorn?

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Problem 40

A swimming pool is 1.4 m deep and 12 m long. Is it possible for you to dive to the very bottom of the pool so people standing on the deck at the end of the pool do not see you? Explain.

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Problem 41

(a) Rays of light are incident on a glass-air interface. Determine the critical angle for total internal reflection $\left(n_{\mathrm{glass}}=1.58\right) .(\mathrm{b})$ If there is a thin, horizontal layer of water $\left(n_{\text { water }}=1.33\right)$ on the glass, will a ray incident on the glass-water interface at the critical angle determined in part (a) be able to leave the water (Figure P 21.41)? Justify your answer.

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Problem 42

Tell all that you can about a process described by the following equation: $1.33 \sin 30^{\circ}=1.00 \sin \theta_{2}$ .

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Problem 43

Tell all that you can about a process described by the following equation: $1.33 \sin \theta_{c}=1.00 \sin 90^{\circ} .$

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Problem 44

Tell all that you can about a process described by the following equation: $1.00 \sin 53^{\circ}=1.60 \sin \theta_{2}=1.33 \sin \theta_{3}$

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Problem 45

When reaching a boundary between two media, an incident ray is partially reflected and partially refracted. At what angle of incidence is the reflected ray perpendicular to the incident ray? The indexes of refraction for the two media are known.

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Problem 46

A laser beam travels from air $(n=1.00)$ into glass $(n=1.52)$ and then into gelatin. The incident ray makes a $58.0^{\circ}$ angle with the normal as it enters the glass and a $36.4^{\circ}$ angle with the normal in the gelatin. (a) Determine the angle of the refracted ray in the glass. (b) Determine the index of refraction of the gelatin.

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Problem 47

The height of the Sun above the horizon is $25^{\circ} .$ You sit on a raft and want to orient a mirror so that sunlight reflects off the mirror and travels at an angle of $45^{\circ}$ in the water of refractive index $1.33 .$ How should you orient the mirror?

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Problem 48

The prism shown in Figure P 21.36 is immersed in water of refractive index 1.33. Determine the minimum value of the refractive index for the prism so that the light is totally internally reflected where shown. Will any light leave the top surface of the prism? Explain.

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Problem 49

Rays of light enter the end of a light pipe from air at an angle $\theta_{1}$ (Figure $P 21.49 )$ . The refractive index of the pipe is 1.64 . Determine the greatest angle $\theta_{1}$ for which the ray is totally reflected at the top surface of the glass-air interface inside the pipe.

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Problem 50

A light ray is incident on a flat piece of glass. The angle of incidence is $\theta_{1}$ and the thickness of the glass is $d$ . Determine the distance that the ray travels in the glass. At what angle will the ray emerge from the glass on the other side? Draw a ray diagram to help explain your solution.

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Problem 51

You have a triangular prism made of glass of refractive index 1.60, with angles of $30^{\circ}-90^{\circ}-60^{\circ}.$ The short side is oriented vertically. A horizontal ray hits the middle of the slanted side of the prism. Draw the path of a ray as it passes into and through the prism. Determine all angles for its trip through the prism.

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Problem 52

In the third century B.C. Euclid per- formed the following experiment. He put a coin at the bottom of a mug with opaque sides. He placed his head above the mug at a position where he could not see the coin. Then without changing the position of his eyes, he added water to the mug and could see the coin. Repeat the experiment and explain how it works.

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Problem 53

You have a candle and a large piece of paper with a triangular hole slightly larger than the candle flame cut in it. You place the paper between the candle and the wall. Draw ray diagrams to show what you see on the wall when the paper is placed (a) near the candle, (b) halfway between the candle and wall, and (c) near the wall.

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Problem 54

You wish to investigate the properties of different transparent materials and decide to design a refractometer, a device used to measure the refractive index of an unknown liquid. A light beam in the liquid is refracted into a material whose refractive index is known and is less than that of the unknown liquid. The incident beam is adjusted for total internal reflection, and the equation for the angle of total internal reflection is used to determine the unknown index. Provide suggestions for a design for such a device and perform a sample calculation to show how it might work.

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Problem 55

You place a point-like source of light at the bottom of a container filled with vegetable oil of refractive index 1.60. At what height from the light source do you need to place a 0.30-cm circular cover so no light emerges from the liquid?

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Problem 56

There is a light pole on one bank of a small pond. You are standing up while fishing on the other bank. After reflection from the surface of the water, part of the light from the bulb at the top of the pole reaches your eyes. Use a ray diagram to help find a point on the surface of the water from where the reflected ray reaches your eyes. Determine an expression for the distance from this point on the water to the bottom of the light pole if the height of the pole is $H,$ your height is h, and the distance between you and the light pole is $l.$

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Problem 57

Imagine that while in Rome you are admiring the famous Fontana dei Quattro Fiumi, or the Fountain of the Four Rivers, in the center of Piazza Navona. You think about retrieving a coin at the bottom of the fountain. Light from the coin reaches your eye at an angle of $37^{\circ}$ below the horizontal (the angle between your line of sight and the vertical normal line to the water is $53^{\circ}$ ). The depth of the fountain is 0.40 m and your height is 1.6 m (1.2 m above the water level). Where is the coin?

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Problem 58

An optic fiber of refractive index 1.72 is coated with a protective covering of glass of refractive index 1.50. (a) Determine the critical angle for the fiber-glass interface. (b) Determine the critical angle for the glass-air interface. (c) Determine the critical angle for a fiber-air interface (no glass covering). (d) Suppose a ray hits the fiber-glass interface at the angle calculated in (c). What is the angle of
refraction of that ray when it reaches the glass-air interface? Will it leave the optic fiber? Explain.

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Problem 59

You put a mirror at the bottom of a 1.4-m-deep pool. A laser beam enters the water at $30^{\circ}$ relative to the normal, hits the mirror, reflects, and comes back out of the water. How far from the water entry point will the beam come out of the water? Draw a ray diagram.

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Problem 60

A scuba diver stands at the bottom of a lake that is 12 m deep. What is the distance to the closest points at the bottom of the lake that the diver can see due to light from these points being totally reflected by the water surface? The height of the diver is 1.8 m and $n_{\text { water }}=1.33.$

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Problem 61

Why do you see rainbows only when you are between the Sun and the location of the rainbow?

(a) The sunlight must be refracted and reflected from rain- drops and needs to move back and downward from its original direction.
(b) You need to intercept the light coming from the Sun.
(c) If you are looking toward the Sun, rainbows would block the sunlight.
(d) The sunlight is refracted by the raindrops and changes direction.

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Problem 62

Suppose that light entered one side of a square water droplet of refractive index 1.33 at an incident angle of $20^{\circ} .$ Some of the light reflects off the back surface and then goes back out on the same side as it entered. What is the closest angle to the angle of refraction of light leaving the square droplet?

$\begin{array}{ll}{\text { (a) } 15^{\circ}} & {\text { (b) } 18^{\circ}} & {\text { (c) } 20^{\circ}} \\ {\text { (d) } 24^{\circ}} & {\text { (e) } 35^{\circ}}\end{array}$

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Problem 63

Raindrops reflect different colors of light at different angles. Why do we see the parts of the rainbow as different colors of light rather than all colors coming from each?

(a) The net deflection of light seen from one drop depends on its refractive index.
(b) All of the raindrops reflecting a particular color have the same angular deflection relative to the direction of the Sun.
(c) Your eye sees only one color coming from a particular raindrop.
(d) The different colors are reflected and refracted at different angles.
(e) All of the above

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Problem 64

Which choice is closest to the minimum angle of incidence of red light against the back wall of the water drop in Figure 21.24 so that it is totally reflected at that wall?

$\begin{array}{ll}{\text { (a) } 16^{\circ}} & {\text { (b) } 21^{\circ}} & {\text { (b) } 41^{\circ}} \\ {\text { (d) } 45^{\circ}} & {\text { (e) } 49^{\circ}}\end{array}$

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Problem 65

Repeat the previous problem, only for the violet light.

$\begin{array}{ll}{\text { (a) } 15^{\circ}} & {\text { (b) } 20^{\circ}} & {\text { (c) } 41^{\circ}} \\ {\text { (d) } 45^{\circ}} & {\text { (e) } 49^{\circ}}\end{array}$

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Problem 66

Why is violet light refracted more than red light?

(a) The violet light travels a shorter distance in the drop than the red light.
(b) The red light travels more slowly than the violet light.
(c) The refractive index of water for violet light is greater than that for red light.
(d) All of the above

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Problem 67

Why is there essentially zero greenhouse effect on the Moon?

(a) There is no photosynthesis on the Moon.
(b) There is no carbon dioxide on the Moon.
(c) There is no gaseous atmosphere on the Moon.
(d) Only b and c are correct.
(e) All of a, b, and c are correct.

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Problem 68

The average Earth surface temperature without its atmosphere would be 255 K. Why?

(a) At that temperature, the emission rate of radiation from Earth would just balance the absorption rate of radiation from the Sun.
(b) The emission of Earth radiation is from a sphere of area $4 \pi R_{\text { Earth }},$ whereas the absorption of radiation from the Sun is from an area $\pi R_{\text { Earth }}$ .
(c) Earth’s cross section would be significantly reduced be- cause of the lack of the atmosphere.
(d) Answers a and b are correct.
(e) Answers a, b, and c contribute about equally to this temperature calculation.

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Problem 69

The Sun irradiates Earth’s outer atmosphere at a rate that is closest to which of the following?

(a) $1 \times 10^{16} \mathrm{J} / \mathrm{s}$
(b) $4 \times 10^{16} \mathrm{J} / \mathrm{s}$
(c) $2 \times 10^{17} \mathrm{J} / \mathrm{s}$
(d) $7 \times 10^{17} \mathrm{J} / \mathrm{s}$
(e) 1400 $\mathrm{J} / \mathrm{s}$

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Problem 70

Because of the greenhouse effect, Earth’s average surface temperature is 288 K instead of 255 K. Because of this higher temperature, Earth’s surface emits radiation at a rate that is higher by a factor of approximately which of the following?

$\begin{array}{ll}{\text { (a) } 1.13} & {\text { (b) } 1.28} & {\text { (b) } 1.63} \\ {\text { (d) } 1.87} & {\text { (e) } 2.21}\end{array}$

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Problem 71

Because of the increased temperature of Earth’s surface (288 K compared to 255 K) due to the greenhouse effect, its energy emission rate has increased significantly. Why hasn’t Earth cooled down as a result?

(a) It is cooling down.
(b) The Sun absorbs much of Earth’s extra radiation, and the Sun is warming and emitting more radiation.
(c) There is a long delay between the change in temperature and the increased rate of radiation by Earth.
(d) Earth’s atmosphere absorbs some of the outgoing radiation and emits it back to Earth’s surface.
(e) All of the above are correct.

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Problem 72

What will the increasing concentration of greenhouse gases do?

(a) Increase the reflection rate of sunlight.
(b) Increase the reflection of Earth’s radiation back to the surface.
(c) Increase the cross-sectional area of the Earth, causing more sunlight to be absorbed.
(d) Shield Earth from dangerous ultraviolet radiation.
(e) Provide extra nourishment for plants on Earth.

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