Content check — College Physics (AP), Chapter 24: Electromagnetic Waves

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Chapter 24: Electromagnetic Waves

31 problems · 2 figures

What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of

5.00 \times 10^{- 4} T

(about 10 times the Earth’s)?

Answer

150 kV/m

24-6 §24.3 source ↗

(a) Two microwave frequencies are authorized for use in microwave ovens: 900 and 2560 MHz. Calculate the wavelength of each. (b) Which frequency would produce smaller hot spots in foods due to interference effects?

Answer

(a) 33.3 cm (900 MHz) 11.7 cm (2560 MHz)

(b) The microwave oven with the smaller wavelength would produce smaller hot spots in foods, corresponding to the one with the frequency 2560 MHz.

24-8 §24.3 source ↗

A radio station utilizes frequencies between commercial AM and FM. What is the frequency of a 11.12-m-wavelength channel?

Answer

26.96 MHz

24-10 §24.3 source ↗

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?

Answer

5.0 \times 10^{14}

24-12 §24.3 source ↗

Approximately what is the smallest detail observable with a microscope that uses ultraviolet light of frequency

1.20 \times 10^{15} Hz

Answer

λ = \frac{c}{f} = \frac{3.00 \times 10^{8} m/s}{1.20 \times 10^{15} Hz} = 2.50 \times 10^{- 7} m

24-14 §24.3 source ↗

Some radar systems detect the size and shape of objects such as aircraft and geological terrain. Approximately what is the smallest observable detail utilizing 500-MHz radar?

Answer

0.600 m

24-16 §24.3 source ↗

If you wish to detect details of the size of atoms (about

1 \times 10^{- 10} m

) with electromagnetic radiation, it must have a wavelength of about this size. (a) What is its frequency? (b) What type of electromagnetic radiation might this be?

Answer

(a) $f =$ $\frac{c}{λ} =$ $\frac{3.00 \times 10^{8} m/s}{1 \times 10^{- 10} m} =$ $3 \times 10^{18} Hz$

(b) X-rays

24-19 §24.3 source ↗

A certain 50.0-Hz AC power line radiates an electromagnetic wave having a maximum electric field strength of 13.0 kV/m. (a) What is the wavelength of this very low frequency electromagnetic wave? (b) What is its maximum magnetic field strength?

Answer

(a) $6.00 \times 10^{6} m$

(b) $4 . 33 \times 10^{- 5} T$

24-21 §24.3 source ↗

(a) The ideal size (most efficient) for a broadcast antenna with one end on the ground is one-fourth the wavelength (

λ / 4

) of the electromagnetic radiation being sent out. If a new radio station has such an antenna that is 50.0 m high, what frequency does it broadcast most efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the fundamental resonant mode of an air column closed at one end to the resonance of currents on an antenna that is one-fourth their wavelength.

Answer

(a) 1.50 × 10 ⁶ Hz, AM band
(b) The resonance of currents on an antenna that is 1/4 their wavelength is analogous to the fundamental resonant mode of an air column closed at one end, since the tube also has a length equal to 1/4 the wavelength of the fundamental oscillation.

24-23 §24.3 source ↗

(a) What is the frequency of the 193-nm ultraviolet radiation used in laser eye surgery? (b) Assuming the accuracy with which this EM radiation can ablate the cornea is directly proportional to wavelength, how much more accurate can this UV be than the shortest visible wavelength of light?

Answer

(a) $1 . 55 \times 10^{15} Hz$

(b) The shortest wavelength of visible light is 380 nm, so that

\begin{array}{l} \frac{λ_{visible}}{λ_{UV}} \\ = \frac{380 nm}{193 nm} \\ = 1.97 . \end{array}

In other words, the UV radiation is 97% more accurate than the shortest wavelength of visible light, or almost twice as accurate!

24-25 §24.3 source ↗

Conversations with astronauts on lunar walks had an echo that was 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, and transformed back into sound on a speaker inside the astronaut’s space suit. This sound was picked up by the microphone in the space suit (intended for the astronaut’s voice) and sent back to Earth as a radio 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?

Answer

3 . 90 \times 10^{8} m

24-27 §24.3 source ↗

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?

Answer

(a) $1 . 50 \times 10^{11} m$

(b) $0.500 μ s$

24-29 §24.3 source ↗

Integrated Concepts

(a) Calculate the rate in watts at which heat transfer through radiation occurs (almost entirely in the infrared) from $1.0 m^{2}$ of the Earth’s surface at night. Assume the emissivity is 0.90, the temperature of the Earth is $15º C$ , and that of outer space is 2.7 K. (b) Compare the intensity of this radiation with that coming to the Earth from the Sun during the day, which averages about $800 {W/m}^{2}$ , only half of which is absorbed. (c) What is the maximum magnetic field strength in the outgoing radiation, assuming it is a continuous wave?

Answer

(a) $-$ $3.5 \times 10^{2} {W/m}^{2}$

(b) 88%

24-30 §24.4 source ↗

What is the intensity of an electromagnetic wave with a peak electric field strength of 125 V/m?

Answer

\begin{array}{l} I & = & \frac{{cε}_{0} E_{0}^{2}}{2} \\ = & \frac{3.00 \times 10^{8} m/s 8.85 \times 10^{–12} C^{2} /N \cdot m^{2} {1 25 V/m}^{2}}{2} \\ = & 20. {7 W/m}^{2} \end{array}

24-32 §24.4 source ↗

Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of 0.250 mW. (a) If such a laser beam is projected onto a circular spot 1.00 mm in diameter, what is its intensity? (b) Find the peak magnetic field strength. (c) Find the peak electric field strength.

Answer

(a) $I =$ $\frac{P}{A} =$ $\frac{P}{π r^{2}} =$ $\frac{0.250 \times 10^{- 3} W}{π {0.500 \times 10^{- 3} m}^{2}} =$ ${318 W/m}^{2}$

(b) $\begin{array}{l} I_{ave} & = & \frac{{cB}_{0}^{2}}{{2μ}_{0}} \Rightarrow B_{0} = {\frac{{2μ}_{0} I}{c}}^{1 / 2} \\ = & {\frac{2 4 π \times 10^{- 7} T \cdot m/A 318 . {3 W/m}^{2}}{3.00 \times 10^{8} m/s}}^{1 / 2} \\ = & 1.63 \times 10^{- 6} T \end{array}$

(c) $\begin{array}{l} E_{0} & = & {cB}_{0} = 3 .00 \times 10^{8} m/s 1.633 \times 10^{- 6} T \\ = & 4.90 \times 10^{2} V/m \end{array}$

24-34 §24.4 source ↗

Suppose the maximum safe intensity of microwaves for human exposure is taken to be

1 . 00 W {/m}^{2}

. (a) If a radar unit leaks 10.0 W of microwaves (other than those sent by its antenna) uniformly in all directions, how far away must you be to be exposed to an intensity considered to be safe? Assume that the power spreads uniformly over the area of a sphere with no complications from absorption or reflection. (b) What is the maximum electric field strength at the safe intensity? (Note that early radar units leaked more than modern ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)

Answer

(a) 89.2 cm

(b) 27.4 V/m

24-36 §24.4 source ↗

Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to ignite nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of

1.00 \times 10^{11} V / m

for a time of 1.00 ns. (a) What is the maximum magnetic field strength in the wave? (b) What is the intensity of the beam? (c) What energy does it deliver on a

1.00 {-mm}^{2}

area?

Answer

(a) 333 T

(b) $1.33 \times 10^{19} {W/m}^{2}$

24-38 §24.4 source ↗

Suppose a source of electromagnetic waves radiates uniformly in all directions in empty space where there are no absorption or interference effects. (a) Show that the intensity is inversely proportional to

r^{2}

, the distance from the source squared. (b) Show that the magnitudes of the electric and magnetic fields are inversely proportional to

r

Answer

(a) $I =$ $\frac{P}{A} =$ $\frac{P}{4π r^{2}} \propto \frac{1}{r^{2}}$

(b) ${I∝E}_{0}^{2}, B_{0}^{2} \Rightarrow E_{0}^{2}, B_{0}^{2} \propto \frac{1}{r^{2}} \Rightarrow E_{0}, B_{0} \propto \frac{1}{r}$

24-40 §24.4 source ↗

Integrated Concepts

What capacitance is needed in series with an $800 - μ H$ inductor to form a circuit that radiates a wavelength of 196 m?

Answer

13.5 pF

24-42 §24.4 source ↗

Integrated Concepts

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 ${W/m}^{2}$ ? (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 $2.00º C$ , assuming no other heat transfer and given that its specific heat is $3.47 \times 10^{3} J/kg \cdot ºC$ ?

Answer

(a) $4 . 07 k {W/m}^{2}$

(b) 1.75 kV/m

(d) 2 min 19 s

24-44 §24.4 source ↗

Integrated Concepts

Electromagnetic radiation from a 5.00-mW laser is concentrated on a $1.00 {-mm}^{2}$ area. (a) What is the intensity in ${W/m}^{2}$ ? (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?

Answer

(a) $5.00 \times 10^{3} {W/m}^{2}$

(b) $3 . 88 \times 10^{- 6} N$

24-46 §24.4 source ↗

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?

Answer

(a) $t = 0$

(b) $7 . 50 \times 10^{- 10} s$

24-48 §24.4 source ↗

Unreasonable Results

The peak magnetic field strength in a residential microwave oven is $9.20 \times 10^{- 5} T$ . (a) What is the intensity of the microwave? (b) What is unreasonable about this result? (c) What is wrong about the premise?

Answer

(a) $1 . 01 \times 10^{6} {W/m}^{2}$

(b) Much too great for an oven.

24-50 §24.4 source ↗

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?

Answer

(a) $2 . 53 \times 10^{- 20} H$

(b) L is much too small.

24-53 §24.4 source ↗

Critical Thinking (a) Find the magnetic field strength when the electric field strength is 1,250 V/m. (b) If E is increased, would B be increased, decreased, or remain the same? (c) By what factor would the change in part b be, if any?

Answer

(a) $\begin{array}{l} B = \frac{E}{c} \\ B = \frac{1250}{3.00 \times 10^{8}} T = 4.17 \times 10^{- 6} T \end{array}$

(b) E would increase.

AP Test Prep 24.1 §24.2 source ↗

If an electromagnetic wave is described as having a frequency of 3 GHz, what are its period and wavelength (in a vacuum)?

a.
3.0 × 10⁹s, 10 cm
b.
3.3 × 10⁻¹⁰s, 10 cm
c.
3.3 × 10⁻¹⁰s, 10 m
d.
3.0 × 10⁹s, 10 m

Answer

B. 3.3 × 10⁻¹⁰s, 10 cm

AP Test Prep 24.3 §24.2 source ↗

A wave is travelling through a medium until it hits the end of the medium and there is nothing but vacuum beyond. What happens to a mechanical wave? Electromagnetic wave?

a.
reflects backward, continues on
b.
reflects backward, reflects backward
c.
continues on, continues on
d.
stops, continues on

Answer

A. reflects backward, continues on

AP Test Prep 24.5 §24.2 source ↗

Given the waveform in Figure 24.5(d), if

T = 3.0 \times 10^{- 9} s

and

E =

5.0 \times 10^{5} N / C

,
which of the following is the correct equation
for the wave at the antenna?

a.
$(5 .0 \times 10^{5} N / C) sin (2 π (3 .0 \times 10^{- 9} s) t)$
b.
$(5 .0 \times 10^{5} N / C) sin (\frac{2 π t}{(3 .0 \times 10^{- 9} s)})$
c.
$(5 .0 \times 10^{5} N / C) cos (2 π (3 .0 \times 10^{- 9} s) t)$
d.
$(5 .0 \times 10^{5} N / C) cos (\frac{2 π t}{(3 .0 \times 10^{- 9} s)})$

A long straight gray wire with an A C generator at its center, functioning as a broadcast antenna for electromagnetic waves, is shown. The wave distributions at four different times are shown in four different parts. Part a of the diagram shows a long straight gray wire with an A C generator at its center. The time is marked t equals zero. The bottom part of the antenna is positive and the upper end of the antenna is negative. An electric field E acting upward is shown by an upward arrow. Part b of the diagram shows a long straight gray wire with an A C generator at its center. The time is marked t equals capital T divided by four. The antenna has no polarity marked and a wave is shown to emerge from the A C source. An electric field E acting upward as shown by an upward arrow. The electric field E propagates away from the antenna at the speed of light, forming part of the electromagnetic wave from the A C source. A quarter portion of the wave is shown above the horizontal axis. Part c of the diagram shows a long straight gray wire with an A C generator at its center. The time is marked t equals capital T divided by two. The bottom part of the antenna is negative and the upper end of the antenna is positive and a wave is shown to emerge from the A C source. The electric field E propagates away from the antenna at the speed of light, forming part of the electromagnetic wave from the A C source. A quarter portion of the wave is shown below the horizontal axis and a quarter portion of the wave is above the horizontal axis. Part d of the diagram shows a long straight gray wire with an AC generator at its center. The time is marked t equals capital T. The bottom part of the antenna is positive and the upper end of the antenna is negative. A wave is shown to emerge from the A C source. The electric field E propagates away from the antenna at the speed of light, forming part of the electromagnetic wave from the A C source. A quarter portion of the wave is shown above the horizontal axis followed by a half wave below the horizontal axis and then again a quarter of a wave above the horizontal axis. — Figure 24.5 in the book ↗

Answer

(5 .0 \times 10^{5} N / C) cos (\frac{2 π t}{(3 .0 \times 10^{- 9} s)})

AP Test Prep 24.7 §24.2 source ↗

In Heinrich Hertz’s spark gap experiment (Figure 24.4), how will the induced sparks in Loop 2 compare to those created in Loop 1?

a.
Stronger
b.
Weaker
c.
Need to know the tuner settings to tell
d.
Weaker, but how much depends on the tuner settings.

The circuit diagram shows a simple circuit containing an alternating voltage source, a resistor R, capacitor C and a transformer, which provides the impedance. The transformer is shown to consist of two coils separated by a core. In parallel with the transformer is connected a wire loop labeled as Loop one Transmitter with a small gap that creates sparks across the gap. The sparks create electromagnetic waves, which are transmitted through the air to a similar loop next to it labeled as Loop two Receiver. These waves induce sparks in Loop two, and are detected by the tuner shown as a rectangular box connected to it. — Figure 24.4 in the book ↗

Answer

D. Weaker, but how much depends on the tuner settings.

AP Test Prep 24.9 §24.3 source ↗

The correct ordering from least to greatest wavelength is:

a.
ELF, FM radio, microwaves, infrared, red, green, ultraviolet, x-ray, gamma ray
b.
ELF, FM radio, microwaves, infrared, green, red, ultraviolet, x-ray, gamma ray
c.
gamma ray, x-ray, ultraviolet, red, green, infrared, microwaves, FM radio, ELF
d.
gamma ray, x-ray, ultraviolet, green, red, infrared, microwaves, FM radio, ELF

Answer

D. gamma ray, x-ray, ultraviolet, green, red, infrared, microwaves, FM radio, ELF

AP Test Prep 24.11 §24.4 source ↗

An old microwave oven outputs only half the electric field it used to. How much longer does it take to cook things in this microwave oven?

a.
Four times as long
b.
Twice as long
c.
Half the time
d.
One fourth the time

Answer

A. Four times as long

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Problems and answers are adapted from College Physics for AP® Courses 2e by Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson and others, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-ap-courses-2e/pages/1-connection-for-ap-r-courses. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.