College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 24

Electromagnetic Waves

Educators


Problem 1

The coils of a horizontal Slinky vibrate vertically up and down; the wave propagates in the horizontal direction. The amplitude of vibrations is 20 ${cm} .$ You thread the Slinky through a board with a $50-{cm}$ slot at an angle of $60^{\circ}$ relative to the vertical. What is the amplitude of the wave that moves past the slot? In what direction do the coils now vibrate?

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Problem 2

A 40-W lightbulb is 2.0 m from a screen. What is the intensity of light incident on the screen? What assumptions did you make?

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Problem 3

Assume that the bulb in Problem 2 radiates unpolarized light. You place a tourmaline crystal in front of the screen. (a) What is the intensity of light hitting the screen after passing through the crystal? (b) You place a second tourmaline crystal between the first crystal and the screen so that it is oriented at
an angle of 50° relative to the axis of the first. What is the in- tensity of light hitting the screen? What assumptions did you make?

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Problem 4

Investigate the properties of Iceland spar and explain how it contributed to the understanding of light as a wave.

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Problem 5

Describe an experiment you can design to find out whether sound can be polarized. Discuss what outcome would help you make a conclusive judgment.

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Problem 6

Place two small electrically charged objects at a distance d from each other. The charge of the first object is +Q; the change of the second object is $-2 Q$ . (a) Draw $\vec{E}$ field lines for the electric field due to these two objects. (b) Calculate the magnitude of the $\vec{E}$ field at a point that is exactly between the two objects. What assumptions did you make?

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Problem 7

A current in a straight wire is 0.20 A. What is the magnitude of the $\vec{B}$ field at a point that is 30 ${cm}$ from the wire? What assumptions did you make?

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Problem 8

Draw a bar magnet with marked poles. (a) Show the direction of the $\vec{B}$ field lines. Where do they start? Where do they end? (b) Imagine this magnet falling as a result of the pull of Earth with its north pole pointing down. Draw the $\vec{E}$ field lines for the induced electric field in front of the magnet and behind the magnet.

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Problem 9

A uniform $\vec{B}$ field of $10^{-3}$ T decreases to zero in 30 s. What is the magnitude of the induced current in a $20-\Omega$ closed loop of wire that is placed perpendicular to the magnetic field lines? The area of the loop is $2.00 \times 10^{-3} {m}^{2}$

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Problem 10

(a) A uniform $\vec{B}$ field whose lines are oriented in the S-N direction increases steadily. Draw the $\vec{E}$ field lines of the induced electric field. (b) A uniform magnetic field whose lines are oriented E-W decreases steadily. Draw the $\vec{E}$ field lines of the induced electric field. (c) Repeat part (b) for a case when the $\vec{B}$ field changes at twice the rate.

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Problem 11

Investigate in detail how Hertz’s apparatus worked and describe how it was used to produce and detect electromagnetic waves.

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Problem 12

Describe experiments that allow you to observe reflection and refraction of electromagnetic waves.

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Problem 13

Radar parameters Radar is characterized by a pulse width and pulse repetition period. (a) Explain what these words mean and how you can use them to estimate the range of distances that radar can measure. (b) If you wish to be able to detect objects at a close distance, what parameter of the radar should you change and how? If you wish to be able to detect objects at a long distance, what parameter of the radar should you change and how?

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Problem 14

EST Radar distance Estimate the range of distances at which you can detect an object using radar with a pulse width of 12 ?s and a pulse repetition of 15 kHz.

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Problem 15

EST Radar antenna height Why do you need to raise a radar emission antenna above the ground to be able to detect objects near the surface of Earth at the farthest range of the radar? Support your answer with a sketch. Estimate how high you need to raise the radar.

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Problem 16

More radar detection Radar uses radio waves of a wavelength of 2.0 m. The time interval for one radiation pulse is 100 times larger than the time of one oscillation; the time between pulses is 10 times larger than the time of one pulse. What is the shortest distance to an object that this radar can detect?

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Problem 17

Communicating with Mars Imagine that you have a vehicle traveling on Mars. Can you use radio signals to give commands to the vehicle? The shortest distance between Earth and Mars is $56 \times 10^{6} {km} ;$ the longest is $400 \times 10^{6} {km} .$ What is the delay time for the signal that you send to Mars from Earth?

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Problem 18

TV tower transmission distance The height of a TV tower is 500 m and the radius of Earth is 6371 km. What is the maximum distance along the ground at which signals from the tower can be received?

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Problem 19

(a) The amplitude of the $\vec{E}$ field oscillations in an electromagnetic wave traveling in air is 20.00 ${N} / {C}$ . What is the amplitude of the $\vec{B}$ field oscillations? (b) The amplitude of the $\vec{B}$ field oscillations in an electromagnetic wave traveling in a vacuum is $3.00 \times 10^{-3} {T}$ . What is the amplitude of the oscillations of the $\vec{E}$ field?

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Problem 20

Milky Way The Sun is a star in the Milky Way galaxy. When viewed from the side, the galaxy looks like a disk that is approximately 100,000 light-years in diameter (a light- year is the distance light travels in one year) and about 1000 light-years thick (Figure P24.20). What is the diameter and thickness of the Milky Way in meters? In kilometers? In miles?

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Problem 21

We observe an increase in brightness of a star that is $5.96 \times 10^{19} {m}$ away. When did the actual increase in brightness take place? Is the star in the Milky Way galaxy? What assumptions did you make?

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Problem 22

Milky Way neighbors Two neighboring galaxies to the Milky Way, the Large and Small Magellanic Clouds, are about 180,000 light-years away. How far away are they in meters? In miles?

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Problem 23

Limits of human vision The wavelength limits of human vision are 400 nm to 700 nm. What range of frequencies of light can we see? How do the wavelength and frequency ranges change when we are underwater? The speed of light in water is 1.33 times less than in the air.

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Problem 24

AM radio An AM radio station has a carrier frequency of 600 kHz. What is the wavelength of the broadcast?

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Problem 25

The electric field in a sinusoidal wave changes as
$$E=(25 {N} / {C}) \cos \left[\left(1.2 \times 10^{11} {rad} / {s}\right) t\right.$$
$$+\left(4.2 \times 10^{2} {rad} / {m}\right) x ]$$
(a) In what direction does the wave propagate? (b) What is the amplitude of the electric field oscillations? (c) What is the amplitude of the magnetic field oscillations? (d) What is the frequency of the wave? (e) What is the wavelength? (f) What is the speed? (g) What other information can you infer from the equation?

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Problem 26

A sinusoidal electromagnetic wave propagates in a vacuum in the positive $x$ -direction. The $\vec{B}$ field oscillates in the $z$ -direction. The wavelength of the wave is 30 ${nm}$ and the amplitude of the $\vec{B}$ field oscillations is $1.0 \times 10^{-2} {T} .$ What is the amplitude of the $E$ field oscillations? (b) Write an equation that describes the $\vec{B}$ field of the wave as a function of time and location. (c) Write an equation that describes the $\vec{E}$ field in the wave as a function of time and location.

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Problem 27

For the previous problem determine (a) the frequency with which the electric energy in the wave oscillates; (b) the frequency at which magnetic field energy oscillates; (c) the maximum energy density; (d) the minimal energy density; (e) the average energy density; and (f) the intensity of the wave.

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Problem 28

BIO Ultraviolet A and B UV-A rays are important for the skin’s production of vitamin D; however, they tan and damage the skin. UV-B rays can cause skin cancer. The wavelength range of UV-A rays is 320 nm to 400 nm and that of UV-B rays is 280 to 320 nm. What is the range of frequencies corresponding to the two types of rays?

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Problem 29

BIO X-rays used in medicine The wavelengths of X-rays use in medicine range from about $8.3 \times 10^{-11} {m}$ for mammography to shorter than $6.2 \times 10^{-14} {m}$ for radiation therapy. What are the frequencies of the corresponding waves? What assumption did you make in your answer?

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Problem 30

Power of sunlight on Earth The Sun emits about $3.9 \times 10^{26} {J}$ of electromagnetic radiation each second. (a) Estimate the power that each square meter of the Sun's surface radiates. (b) Estimate the power that 1 ${m}^{2}$ of Earth's surface receives. (c) What assumptions did you make in part
(b)? The distance from Earth to the Sun is about $1.5 \times 10^{11} {m}$ and the diameter of the Sun is about $1.4 \times 10^{9} {m} .$

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Problem 31

Light from an incandescent bulb Only about 10% of the electromagnetic energy from an incandescent lightbulb is visible light. The bulb radiates most of its energy in the infrared part of the electromagnetic spectrum. If you place a 100-W lightbulb 2.0 m away from you, (a) what is the intensity of the infrared radiation at your location? (b) What is the infrared energy density? (c) What are the approximate

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Problem 32

Explain how the information about energy radiated by a lightbulb in Problem 31 can be used to compare the magnitudes of $\vec{E}$ fields oscillating at different frequencies. Pose a problem that requires the use of this information.

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Problem 33

EST Estimate the amplitude of oscillations of the $\vec{E}$ and $\vec{B}$ fields close to the surface of the Sun. List all of the assumptions that you made.

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Problem 34

BIO New laser to treat cancer The HERCULES pulsed laser has the potential to help in the treatment of cancer, as it focuses its power on a tiny area and essentially burns individual cancer cells. The HERCULES can be focused on a surface that is $0.8 \times 10^{-6} {m}$ across. This pulsed laser provides 1.2 ${J}$ of energy during a $27 \times 10^{-15}$ -s time interval at a wavelength of approximately 800 ${nm}$ . (a) Determine the power provided during the pulse. (b) Determine the magnitude of the maximum $\vec{E}$ field produced by the pulse.

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Problem 35

Equation jeopardy 1 Tell everything you can about the wave described by the equations below.
$E_{y}=(2.4 \times 10^{5} {N} / {C}) \cos [2 \pi(\frac{t}{1.5 \times 10^{-15} {s}}-\frac{x}{4.5 \times 10^{-7} {m}})]$
$B_{z}=(8.0 \times 10^{-4} {T}) \cos [2 \pi(\frac{t}{1.5 \times 10^{-15} {s}}-\frac{x}{4.5 \times 10^{-7} {m}})]$

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Problem 36

Equation jeopardy 2 Tell everything you can about the wave described by the equations below.
$E_{y}=(9.0 \times 10^{5} {N} / {C}) \cos [2 \pi(\frac{t}{5.3 \times 10^{-15} {s}}-\frac{x}{1.6 \times 10^{-6} {m}})]$
$B_{z}=B_{\max } \cos [2 \pi(\frac{t}{5.3 \times 10^{-15} {s}}-\frac{x}{1.6 \times 10^{-6} {m}})]$

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Problem 37

Equation jeopardy 3 Tell everything you can about the situation described below.
$E_{\max }=\sqrt{\frac{2 \cdot 640 {kW} / {m}^{2}}{\left(3.0 \times 10^{8} {m} / {s}\right)\left(8.85 \times 10^{-12} {C}^{2} / {N} \cdot {m}^{2}\right)}}$

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Problem 38

Red filter A color filter is a transparent material that permits only light of a certain color to pass through. When you place a red filter between a lightbulb and a grating, the pattern on the screen beyond the grating consists of rather wide m = 0, 1, 2, ... red bands. Why aren’t the bright bands on the screen narrow, as they are when you shine a red laser on the grating?

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Problem 39

Describe an experiment that you can perform to determine whether the light from a particular source is unpolarized or linearly polarized. If the latter, then how can you determine the polarization direction?

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Problem 40

An unpolarized beam of light passes through two polarizing sheets that are initially aligned so that the transmitted beam is maximal. By what angle should the second polarized sheet be rotated relative to the first to reduce the transmitted intensity to (a) one-half and (b) one-tenth the intensity that was transmitted through both polarizing sheets when aligned?

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Problem 41

BIO Spider polarized light navigation The gnaphosid spider Drassodes cupreus has evolved a pair of lensless eyes for detecting polarized light. Each eye is sensitive to polarized light in perpendicular directions. Near sunset, the spider leaves its nest in search of prey. Light from overhead is linearly polarized and indicates the direction the spider is moving. After the hunt, the spider uses the polarized light to return to it nest. Suppose that the spider orients its head so that one of these two eyes detects light of intensity 800 ${W} / {m}^{2}$ and the other eye detects zero light intensity. What intensities do the two eyes detect if the spider now rotates its head $20^{\circ}$ from the previous orientation?

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Problem 42

Two polarizing sheets are oriented at an angle of 60° relative to each other. (a) Determine the factor by which the intensity of an unpolarized light beam is reduced after passing through both sheets. (b) Determine the factor by which the intensity of a po1arized beam oriented at 30° relative to each polarizing sheet is reduced after passing through both sheets.

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Problem 43

Light reflected from a pond At what angle of incidence (and reflection) does light reflected from a smooth pond become completely polarized parallel to the pond’s surface? How do you know? In which direction does the $E$ field vector oscillate in this reflected light wave?

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Problem 44

Light reflected from water in a cake pan At what angle is reflected light from a water-glass interface at the bottom of a cake pan holding water completely polarized parallel to the surface of the glass whose refractive index is 1.65?

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Problem 45

*Unpolarized light passes through three polarizers. The second makes an angle of 25° relative to the first, and the third makes an angle of 45° relative to the first. The intensity of light measured after the third polarizer is 40 ${W} / {m}^{2}$ . Determine the intensity of the unpolarized light incident on the
first polarizer.

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Problem 46

You have two pairs of polarized glasses. Make a list of experimental questions you can answer using one of them or both. Describe the experiments you will design to answer them and discuss how you will ensure that the solutions (answers) you find make sense.

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Problem 47

EST Density of Milky Way What is the average density of the Milky Way assuming that it contains about 200 million stars with masses comparable to the mass of the Sun? What additional information do you need to know to answer the question? What assumptions should you make?

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Problem 48

Supernova in neighboring galaxy On February 23, 1987 astronomers noticed that a relatively faint star in the Tarantula Nebula in the Large Magellanic Cloud suddenly became so bright that it could be seen with the naked eye. Astronomers suspected that the star exploded as a supernova. Late-stage stars eject material that forms a ring before they explode as supernovas. The ring is at first invisible. However, following the explosion the light from the supernova reaches the ring. Astronomers observed the ring exactly one year after the phenomenon itself occurred (Figure P24.48). (a) Use geometry and the speed of light to estimate the distance to the supernova. The angular size of the ring is 0.81 arcseconds (b) How do you know whether your answer makes sense? (c) Use the width of the ring to estimate the uncertainty in the value of the distance.

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Problem 49

BIO EST Human vision power sensitivity A rod in the eye’s retina can detect light of energy $4 \times 10^{-19} {J}$ . Estimate the power of this light that the rod can detect. Indicate any assumptions you made. You will need more information than what is provided.

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Problem 50

Effect of weather on radio transmission Weather affects the transmission of AM stations but does not affect FM stations. FM stations do not broadcast in remote areas. Suggest possible reasons for these phenomena. [Hint: Find information about the wavelengths of the waves and decide how they might propagate in Earth’s atmosphere.]

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Problem 51

Measuring speed of light In 1849 A. Fizeau conducted an experiment to determine the speed of light in a laboratory (before that time, all methods involved astronomical distances). He used an apparatus described in Figure P24.51. Light from the source S went through an interrupter K and
after reflecting from the mirror M returned to the rotating wheel again. If light passed between the teeth of the wheel, Fizeau could see it; if light on the way back hit the tooth of the wheel, Fizeau would not see it. He could measure the speed of light by relating the time interval between teeth crossing the beam and the distance light traveled during those time intervals. The information that Fizeau had about the system was: L = the distance between the wheel and the mirror; T = the
period of rotation of the wheel; and N = the number of teeth in the wheel (the width of one tooth was equal to the width of the gap between the teeth). Using these parameters Fizeau derived a formula for calculating the speed of light: $c=(4 L N / T)$. Explain how he arrived at this equation.

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Problem 52

Speed of light In Fizeau’s experiment (described in Problem 51) the distance between the wheel and the mirror was 3.733 km, the wheel had 720 teeth, and the wheel made 29 rotations per second. What was the speed of light determined by Fizeau? What were the uncertainties in this value?

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Problem 53

A sinusoidal electromagnetic wave has a $20-{N} / {C} \vec{E}$ field amplitude. Determine everything you can about this wave.

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Problem 54

Bees know the direction to the Sun because
(a) direct sunlight is linearly polarized.
(b) direct sunlight is unpolarized.
(c) they detect infrared radiation from the sun.
(d) a and c.
(e) b and c.

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Problem 55

Suppose that the scout bee in a waggle dance indicates that a good food source is at a 90° angle from the direction to the Sun. Bees who have never been to the food source when leaving the hive head toward a region of the sky that
(a) has a flowery odor.
(b) has completely unpolarized light coming from it.
(c) has completely linearly polarized light coming from it.
(d) has partially unpolarized light coming from it.
(e) None of these answers

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Problem 56

If the direction of the middle line of the scout’s waggle dance figure eight points 53° to the left side of the upward direction in the hive, in what direction to the left should bees leaving the nest for the food source head relative to light coming from the Sun?
(a) Where the light is 36% linearly polarized to the left of the Sun’s direction
(b) Where the light is 60% linearly polarized to the left of the Sun’s direction
(c) Where the light is 64% linearly polarized to the left of the Sun’s direction
(d) Where the light is 80% linearly polarized to the left of the Sun’s direction
(e) Where the light is 100% linearly polarized to the left of the Sun’s direction

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Problem 57

Polar molecules are caused to vibrate in all directions perpendicular to the direction of travel of sunlight. When the Sun is rising in the east and you look at the molecules directly overhead, the light from these molecules is
(a) unpolarized.
(b) linearly polarized along an axis between you and the molecules overhead.
(c) linearly polarized parallel to a plane with you, the Sun, and the molecules.
(d) linearly polarized perpendicular to a plane with you, the Sun, and the molecules.
(e) All of these are correct.

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Problem 58

In what directions is the light from the sky completely unpolarized?
(a) Looking directly at the Sun
(b) Looking directly away from the Sun
(c) Looking at 90° relative to the Sun
(d) a and b
(e) b and c

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Problem 59

Why are the United States and other countries banning the use of incandescent lightbulbs?
(a) The bulbs get too hot.
(b) The bulbs have tungsten filaments.
(c) Ninety percent of the electric energy used is converted to thermal energy.
(d) The bulbs are only 10% efficient in converting electric energy to light.
(e) c and

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Problem 60

Which answer below is closest to the rate of visible light emission from a 100-W incandescent lightbulb?
(a) 10 W
(b) 20 W
(c) 50 W
(d) 100 W
(e) Not enough information to make a determination.

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Problem 61

What does the surface of the body at about 35 °C primarily emit?
(a) No electromagnetic radiation
(b) Long-wavelength infrared radiation
(c) Short-wavelength infrared radiation
(d) Light
(e) Invisible ultraviolet light

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Problem 62

Suppose you changed all incandescent lightbulbs to more energy-efficient bulbs that used one-fourth the amount of energy to get the same light. About how many $3.3 \times 10^{9}-{kW} \cdot {h} /$ year electric power plants could be removed from the power grid?
(a) 3
(b) 10
(c) 20
(d) 50
(e) 100

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Problem 63

How much money will you save on your electric bill each year if you replace five $100-{W}$ incandescent bulbs with five CFL or LED $25-{W}$ bulbs that produce the same amount of light? Assume that the bulbs are on 3.0 ${h} / {day}$ and that electric energy costs $\$ 0.12 / {kW} \cdot {h}$ .
(a) $5
(b) $10
(c) $20
(d) $50 (e) $100

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