8. Give an example of interference of electromagnetic waves.
9. Figure 24.26 shows the interference pattern of two radio antennas broadcasting the same signal. Explain how this is analogous to the interference pattern for sound produced by two speakers. Could this be used to make a directional antenna system that broadcasts preferentially in certain
directions? Explain.
Figure 24.26 An overhead view of two radio broadcast antennas sending the same signal, and the interference pattern they produce.
10. Can an antenna be any length? Explain your answer.
24.3 The Electromagnetic Spectrum
11. If you live in a region that has a particular TV station, you can sometimes pick up some of its audio portion on your FM radio receiver. Explain how
this is possible. Does it imply that TV audio is broadcast as FM?
12. Explain why people who have the lens of their eye removed because of cataracts are able to see low-frequency ultraviolet.
13. How do fluorescent soap residues make clothing look “brighter and whiter” in outdoor light? Would this be effective in candlelight?
14. Give an example of resonance in the reception of electromagnetic waves.
15. Illustrate that the size of details of an object that can be detected with electromagnetic waves is related to their wavelength, by comparing details
observable with two different types (for example, radar and visible light or infrared and X-rays).
16. Why don’t buildings block radio waves as completely as they do visible light?
17. Make a list of some everyday objects and decide whether they are transparent or opaque to each of the types of electromagnetic waves.
CHAPTER 24 | ELECTROMAGNETIC WAVES 883
18. Your friend says that more patterns and colors can be seen on the wings of birds if viewed in ultraviolet light. Would you agree with your friend?
Explain your answer.
19. The rate at which information can be transmitted on an electromagnetic wave is proportional to the frequency of the wave. Is this consistent with
the fact that laser telephone transmission at visible frequencies carries far more conversations per optical fiber than conventional electronic
transmission in a wire? What is the implication for ELF radio communication with submarines?
20. Give an example of energy carried by an electromagnetic wave.
21. In an MRI scan, a higher magnetic field requires higher frequency radio waves to resonate with the nuclear type whose density and location is
being imaged. What effect does going to a larger magnetic field have on the most efficient antenna to broadcast those radio waves? Does it favor a
smaller or larger antenna?
22. Laser vision correction often uses an excimer laser that produces 193-nm electromagnetic radiation. This wavelength is extremely strongly
absorbed by the cornea and ablates it in a manner that reshapes the cornea to correct vision defects. Explain how the strong absorption helps
concentrate the energy in a thin layer and thus give greater accuracy in shaping the cornea. Also explain how this strong absorption limits damage to
the lens and retina of the eye.
884 CHAPTER 24 | ELECTROMAGNETIC WAVES
Problems & Exercises
(b) How many meters is it to Andromeda, the nearest large galaxy, given
that it is 2.00 × 106 light years away? (c) The most distant galaxy yet
24.1 Maxwell’s Equations: Electromagnetic Waves
discovered is 12.0 × 109 light years away. How far is this in meters?
19. A certain 50.0-Hz AC power line radiates an electromagnetic wave
1. Verify that the correct value for the speed of light c is obtained when
having a maximum electric field strength of 13.0 kV/m. (a) What is the
numerical values for the permeability and permittivity of free space ( µ 0
wavelength of this very low frequency electromagnetic wave? (b) What is
its maximum magnetic field strength?
and ε 0 ) are entered into the equation c =
1
µ
.
0 ε 0
20. During normal beating, the heart creates a maximum 4.00-mV
potential across 0.300 m of a person’s chest, creating a 1.00-Hz
2. Show that, when SI units for µ 0 and ε 0 are entered, the units given
electromagnetic wave. (a) What is the maximum electric field strength
by the right-hand side of the equation in the problem above are m/s.
created? (b) What is the corresponding maximum magnetic field strength
in the electromagnetic wave? (c) What is the wavelength of the
24.2 Production of Electromagnetic Waves
electromagnetic wave?
21. (a) The ideal size (most efficient) for a broadcast antenna with one
3. What is the maximum electric field strength in an electromagnetic
end on the ground is one-fourth the wavelength ( λ / 4 ) of the
wave that has a maximum magnetic field strength of 5.00×10−4 T
electromagnetic radiation being sent out. If a new radio station has such
(about 10 times the Earth’s)?
an antenna that is 50.0 m high, what frequency does it broadcast most
4. The maximum magnetic field strength of an electromagnetic field is
efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the
5×10−6 T
fundamental resonant mode of an air column closed at one end to the
. Calculate the maximum electric field strength if the wave is
resonance of currents on an antenna that is one-fourth their wavelength.
traveling in a medium in which the speed of the wave is 0.75 c .
22. (a) What is the wavelength of 100-MHz radio waves used in an MRI
5. Verify the units obtained for magnetic field strength B in Example
unit? (b) If the frequencies are swept over a ±1.00 range centered on
24.1 (using the equation B = E
100 MHz, what is the range of wavelengths broadcast?
c ) are in fact teslas (T).
23. (a) What is the frequency of the 193-nm ultraviolet radiation used in
24.3 The Electromagnetic Spectrum
laser eye surgery? (b) Assuming the accuracy with which this EM
radiation can ablate the cornea is directly proportional to wavelength,
6. (a) Two microwave frequencies are authorized for use in microwave
how much more accurate can this UV be than the shortest visible
ovens: 900 and 2560 MHz. Calculate the wavelength of each. (b) Which
wavelength of light?
frequency would produce smaller hot spots in foods due to interference
24. TV-reception antennas for VHF are constructed with cross wires
effects?
supported at their centers, as shown in Figure 24.27. The ideal length for
7. (a) Calculate the range of wavelengths for AM radio given its
the cross wires is one-half the wavelength to be received, with the more
frequency range is 540 to 1600 kHz. (b) Do the same for the FM
expensive antennas having one for each channel. Suppose you measure
frequency range of 88.0 to 108 MHz.
the lengths of the wires for particular channels and find them to be 1.94
and 0.753 m long, respectively. What are the frequencies for these
8. A radio station utilizes frequencies between commercial AM and FM.
channels?
What is the frequency of a 11.12-m-wavelength channel?
9. Find the frequency range of visible light, given that it encompasses
wavelengths from 380 to 760 nm.
10. Combing your hair leads to excess electrons on the comb. How fast
would you have to move the comb up and down to produce red light?
11. Electromagnetic radiation having a 15.0 − µ m wavelength is
classified as infrared radiation. What is its frequency?
12. Approximately what is the smallest detail observable with a
microscope that uses ultraviolet light of frequency 1.20×1015 Hz ?
13. A radar used to detect the presence of aircraft receives a pulse that
has reflected off an object 6 × 10−5 s after it was transmitted. What is
the distance from the radar station to the reflecting object?
14. Some radar systems detect the size and shape of objects such as
aircraft and geological terrain. Approximately what is the smallest
Figure 24.27 A television reception antenna has cross wires of various lengths to
observable detail utilizing 500-MHz radar?
most efficiently receive different wavelengths.
15. Determine the amount of time it takes for X-rays of frequency
25. Conversations with astronauts on lunar walks had an echo that was
3 × 1018 Hz to travel (a) 1 mm and (b) 1 cm.
used to estimate the distance to the Moon. The sound spoken by the
person on Earth was transformed into a radio signal sent to the Moon,
16. If you wish to detect details of the size of atoms (about 1 × 10−10 m
and transformed back into sound on a speaker inside the astronaut’s
) with electromagnetic radiation, it must have a wavelength of about this
space suit. This sound was picked up by the microphone in the space
size. (a) What is its frequency? (b) What type of electromagnetic radiation suit (intended for the astronaut’s voice) and sent back to Earth as a radio
might this be?
echo of sorts. If the round-trip time was 2.60 s, what was the approximate
distance to the Moon, neglecting any delays in the electronics?
17. If the Sun suddenly turned off, we would not know it until its light
stopped coming. How long would that be, given that the Sun is
26. Lunar astronauts placed a reflector on the Moon’s surface, off which a
1.50 × 1011 m
laser beam is periodically reflected. The distance to the Moon is
away?
calculated from the round-trip time. (a) To what accuracy in meters can
18. Distances in space are often quoted in units of light years, the
the distance to the Moon be determined, if this time can be measured to
distance light travels in one year. (a) How many meters is a light year?
CHAPTER 24 | ELECTROMAGNETIC WAVES 885
0.100 ns? (b) What percent accuracy is this, given the average distance
to the Moon is 3.84 × 108 m ?
27. Radar is used to determine distances to various objects by measuring
the round-trip time for an echo from the object. (a) How far away is the
planet Venus if the echo time is 1000 s? (b) What is the echo time for a
car 75.0 m from a Highway Police radar unit? (c) How accurately (in
nanoseconds) must you be able to measure the echo time to an airplane
12.0 km away to determine its distance within 10.0 m?
28. Integrated Concepts
(a) Calculate the ratio of the highest to lowest frequencies of
electromagnetic waves the eye can see, given the wavelength range of
visible light is from 380 to 760 nm. (b) Compare this with the ratio of
highest to lowest frequencies the ear can hear.
29. Integrated Concepts
(a) Calculate the rate in watts at which heat transfer through radiation
occurs (almost entirely in the infrared) from 1.0 m2 of the Earth’s
surface at night. Assume the emissivity is 0.90, the temperature of the
Figure 24.28 Satellite dishes receive TV signals sent from orbit. Although the signals
Earth is 15ºC , and that of outer space is 2.7 K. (b) Compare the
are quite weak, the receiver can detect them by being tuned to resonate at their
frequency.
intensity of this radiation with that coming to the Earth from the Sun
36. Lasers can be constructed that produce an extremely high intensity
during the day, which averages about 800 W/m2 , only half of which is
electromagnetic wave for a brief time—called pulsed lasers. They are
absorbed. (c) What is the maximum magnetic field strength in the
used to ignite nuclear fusion, for example. Such a laser may produce an
outgoing radiation, assuming it is a continuous wave?
electromagnetic wave with a maximum electric field strength of
1.00 × 1011 V / m for a time of 1.00 ns. (a) What is the maximum
24.4 Energy in Electromagnetic Waves
magnetic field strength in the wave? (b) What is the intensity of the
30. What is the intensity of an electromagnetic wave with a peak electric
beam? (c) What energy does it deliver on a 1.00-mm2 area?
field strength of 125 V/m?
37. Show that for a continuous sinusoidal electromagnetic wave, the
31. Find the intensity of an electromagnetic wave having a peak magnetic peak intensity is twice the average intensity ( I 0 = 2 I ave ), using either
field strength of 4.00 × 10−9 T .
the fact that E
32. Assume the helium-neon lasers commonly used in student physics
0 = 2 E rms , or B 0 = 2 B rms , where rms means
laboratories have power outputs of 0.500 mW. (a) If such a laser beam is
average (actually root mean square, a type of average).
projected onto a circular spot 1.00 mm in diameter, what is its intensity?
38. Suppose a source of electromagnetic waves radiates uniformly in all
(b) Find the peak magnetic field strength. (c) Find the peak electric field
directions in empty space where there are no absorption or interference
strength.
effects. (a) Show that the intensity is inversely proportional to r 2 , the
33. An AM radio transmitter broadcasts 50.0 kW of power uniformly in all
distance from the source squared. (b) Show that the magnitudes of the
directions. (a) Assuming all of the radio waves that strike the ground are
electric and magnetic fields are inversely proportional to r .
completely absorbed, and that there is no absorption by the atmosphere
or other objects, what is the intensity 30.0 km away? (Hint: Half the
39. Integrated Concepts
power will be spread over the area of a hemisphere.) (b) What is the
An LC circuit with a 5.00-pF capacitor oscillates in such a manner as to
maximum electric field strength at this distance?
radiate at a wavelength of 3.30 m. (a) What is the resonant frequency?
34. Suppose the maximum safe intensity of microwaves for human
(b) What inductance is in series with the capacitor?
exposure is taken to be 1.00 W/m2 . (a) If a radar unit leaks 10.0 W of
40. Integrated Concepts
microwaves (other than those sent by its antenna) uniformly in all
directions, how far away must you be to be exposed to an intensity
What capacitance is needed in series with an 800 − µ H inductor to
considered to be safe? Assume that the power spreads uniformly over
form a circuit that radiates a wavelength of 196 m?
the area of a sphere with no complications from absorption or reflection.
41. Integrated Concepts
(b) What is the maximum electric field strength at the safe intensity?
(Note that early radar units leaked more than modern ones do. This
Police radar determines the speed of motor vehicles using the same
caused identifiable health problems, such as cataracts, for people who
Doppler-shift technique employed for ultrasound in medical diagnostics.
worked near them.)
Beats are produced by mixing the double Doppler-shifted echo with the
35. A 2.50-m-diameter university communications satellite dish receives
original frequency. If 1.50 × 109 -Hz microwaves are used and a beat
TV signals that have a maximum electric field strength (for one channel)
frequency of 150 Hz is produced, what is the speed of the vehicle?
of 7.50 µ V/m . (See Figure 24.28.) (a) What is the intensity of this
(Assume the same Doppler-shift formulas are valid with the speed of
wave? (b) What is the power received by the antenna? (c) If the orbiting
sound replaced by the speed of light.)
satellite broadcasts uniformly over an area of 1.50 × 1013 m2 (a large
42. Integrated Concepts
fraction of North America), how much power does it radiate?
Assume the mostly infrared radiation from a heat lamp acts like a
continuous wave with wavelength 1.50 µ m . (a) If the lamp’s 200-W
output is focused on a person’s shoulder, over a circular area 25.0 cm in
diameter, what is the intensity in W/m2 ? (b) What is the peak electric
field strength? (c) Find the peak magnetic field strength. (d) How long will
it take to increase the temperature of the 4.00-kg shoulder by 2.00º C ,
886 CHAPTER 24 | ELECTROMAGNETIC WAVES
assuming no other heat transfer and given that its specific heat is
this to the power likely to be carried by the lines. An idea of how much
3.47 × 103 J/kg ⋅ ºC ?
power this is can be obtained by calculating the approximate current
responsible for µ T fields at distances of tens of meters.
43. Integrated Concepts
52. Create Your Own Problem
On its highest power setting, a microwave oven increases the
temperature of 0.400 kg of spaghetti by 45.0ºC in 120 s. (a) What was
Consider the most recent generation of residential satellite dishes that
are a little less than half a meter in diameter. Construct a problem in
the rate of power absorption by the spaghetti, given that its specific heat
which you calculate the power received by the dish and the maximum
is 3.76 × 103 J/kg ⋅ ºC ? (b) Find the average intensity of the
electric field strength of the microwave signals for a single channel
microwaves, given that they are absorbed over a circular area 20.0 cm in
received by the dish. Among the things to be considered are the power
diameter. (c) What is the peak electric field strength of the microwave?
broadcast by the satellite and the area over which the power is spread,
(d) What is its peak magnetic field strength?
as well as the area of the receiving dish.
44. Integrated Concepts
Electromagnetic radiation from a 5.00-mW laser is concentrated on a
1.00-mm2 area. (a) What is the intensity in W/m2 ? (b) Suppose a
2.00-nC static charge is in the beam. What is the maximum electric force
it experiences? (c) If the static charge moves at 400 m/s, what maximum
magnetic force can it feel?
45. Integrated Concepts
A 200-turn flat coil of wire 30.0 cm in diameter acts as an antenna for FM
radio at a frequency of 100 MHz. The magnetic field of the incoming
electromagnetic wave is perpendicular to the coil and has a maximum
strength of 1.00 × 10−12 T . (a) What power is incident on the coil? (b)
What average emf is induced in the coil over one-fourth of a cycle? (c) If
the radio receiver has an inductance of 2.50 µ H , what capacitance
must it have to resonate at 100 MHz?
46. Integrated Concepts
If electric and magnetic field strengths vary sinusoidally in time, being
zero at t = 0 , then E = E 0 sin 2π ft and B = B 0 sin 2π ft . Let
f = 1.00 GHz here. (a) When are the field strengths first zero? (b)
When do they reach their most negative value? (c) How much time is
needed for them to complete one cycle?
47. Unreasonable Results
A researcher measures the wavelength of a 1.20-GHz electromagnetic
wave to be 0.500 m. (a) Calculate the speed at which this wave
propagates. (b) What is unreasonable about this result? (c) Which
assumptions are unreasonable or inconsistent?
48. Unreasonable Results
The peak magnetic field strength in a residential microwave oven is
9.20 × 10−5 T . (a) What is the intensity of the microwave? (b) What is
unreasonable about this result? (c) What is wrong about the premise?
49. Unreasonable Results
An LC circuit containing a 2.00-H inductor oscillates at such a
frequency that it radiates at a 1.00-m wavelength. (a) What is the
capacitance of the circuit? (b) What is unreasonable about this result? (c)
Which assumptions are unreasonable or inconsistent?
50. Unreasonable Results
An LC circuit containing a 1.00-pF capacitor oscillates at such a
frequency that it radiates at a 300-nm wavelength. (a) What is the
inductance of the circuit? (b) What is unreasonable about this result? (c)
Which assumptions are unreasonable or inconsistent?
51. Create Your Own Problem
Consider electromagnetic fields produced by high voltage power lines.
Construct a problem in which you calculate the intensity of this
electromagnetic radiation in W/m2 based on the measured magnetic
field strength of the radiation in a home near the power lines. Assume
these magnetic field strengths are known to average less than a µ T .
The intensity is small enough that it is difficult to imagine mechanisms for
biological damage due to it. Discuss how much energy may be radiating
from a section of power line several hundred meters long and compare
CHAPTER 25 | GEOMETRIC OPTICS 887
25
GEOMETRIC OPTICS
Figure 25.1 Image seen as a result of reflection of light on a plane smooth surface. (credit: NASA Goddard Photo and Video, via Flickr)
888 CHAPTER 25 | GEOMETRIC OPTICS
Learning Objectives
• Define a ray.
• Define geometric optics.
• List the ways by which light travels from a source to another location.
• State the law of reflection.
• Explain reflection of light from polished and rough surfaces.
• Perform an experiment to prove the law of reflection.
• Define refraction and index of refraction and state the law of refraction.
• Discuss the various attempts at measuring the speed of light.
• Determine the index of refraction, given the speed of light in a medium.