Essential University Physics Global Edition

Richard Wolfson

Chapter 29

Maxwell's Equations and Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Why is Maxwell’s modification of Ampère’s law essential to the
existence of electromagnetic waves?

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Problem 2

The presence of magnetic monopoles would require a modification of Gauss’s law for magnetism. Which other Maxwell equation
would need modification?

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Problem 3

What is the nature of the magnetic field vector in an electromagnetic wave?

Carlos Henrique De Lima

Carlos Henrique De Lima

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Problem 4

The speed of an electromagnetic wave is given by $c=\lambda f$. How does the speed depend on frequency? On wavelength?

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Problem 5

When astronomers observe a supernova explosion in a distant galaxy, they see a sudden, simultaneous rise in visible light and other forms of electromagnetic radiation. How is this evidence that the speed of light is independent of frequency?

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Problem 6

The Sun emits about half of its electromagnetic-wave energy in the visible region of the spectrum. Where do you think it emits most of the remainder?

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Problem 7

An $L C$ circuit is made entirely from superconducting materials, yet its oscillations eventually damp out. Why?

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Problem 8

If you double the field strength in an electromagnetic wave, what happens to the intensity?

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Problem 9

The intensity of light drops as the inverse square of the distance from the source. Does this mean that electromagnetic energy is lost? Explain.

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Problem 10

Electromagnetic waves don’t readily penetrate metals. Why not?

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Problem 11

A uniform electric field is increasing at $1.0(\mathrm{~V} / \mathrm{m}) / \mu \mathrm{s}$. Find the displacement current through a $1.6-\mathrm{cm}^{2}$ area perpendicular to the field.

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Problem 12

A parallel-plate capacitor has square plates $13 \mathrm{~cm}$ on a side and $0.50 \mathrm{~cm}$ apart. The voltage across the plates is increasing at $220 \mathrm{~V} / \mathrm{ms}$. What's the displacement current in the capacitor?

Ankur S

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Problem 13

The fields of an electromagnetic wave are $\vec{E}=E_{\mathrm{p}} \sin (k z+\omega t) \hat{\jmath}$ and $\vec{B}=B_{\mathrm{p}} \sin (k z+\omega t) \hat{l}$. Give a unit vector in the wave's propagation direction.

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Problem 14

A radio wave's electric field is given by the expression $\vec{E}=E \sin (k z-\omega t) \times(\hat{\imath}+\hat{\jmath})$. (a) Find the peak electric field.
(b) Give a unit vector in the direction of the magnetic field at a place and time where $\sin (k z-\omega t)$ is positive.

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Problem 15

A light-minute is the distance light travels in 1 minute. Show that the Sun is about 8 light-minutes from Earth.

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Problem 16

Your intercontinental telephone call is carried by electromagnetic waves routed via a satellite in geostationary orbit at $36,000 \mathrm{~km}$ altitude. Approximately how long does it take before your voice is heard at the other end?

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Problem 17

An airplane's radar altimeter works by bouncing radio waves off the ground and measuring the round-trip travel time. If that time is $74.7 \mu \mathrm{s}$, what should the pilot report to the passengers as the current altitude?

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Problem 18

Roughly how long does it take light to travel 1 foot?

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Problem 19

If you speak via radio from Earth to an astronaut on the Moon, how long is it before you can receive a reply?

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Problem 20

What are the wavelengths of (a) a $108-\mathrm{MHz} \mathrm{FM}$ radio wave, (b) a 2.4-GHz WiFi signal, (c) a 620-THz light wave, and (d) a $2.0-\mathrm{EHz} \mathrm{X}$ ray?

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Problem 21

A $50-\mathrm{Hz}$ power line emits electromagnetic radiation. What's the wavelength?

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Problem 22

Microwave ovens for consumers' use operate at $2.45 \mathrm{GHz}$. What's the distance between wave crests in such a microwave?

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Problem 23

An electromagnetic wave is propagating in the z-direction. What's its polarization direction if its magnetic field is in the $y$-direction?

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Problem 24

Polarized light is incident on a sheet of polarizing material, and only $20 \%$ of the light gets through. Find the angle between the electric field and the material's transmission axis.

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Problem 25

Vertically polarized light passes through a polarizer with its axis at $69^{\circ}$ to the vertical. What fraction of the incident intensity emerges from the polarizer?

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Problem 26

A typical laboratory electric field is $1500 \mathrm{~V} / \mathrm{m}$. Find the average intensity of an electromagnetic wave with this value for its peak field.

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Problem 27

What would be the average intensity of a laser beam so strong that its electric field produced dielectric breakdown of air (which requires $\left.E_{\mathrm{p}}=3 \mathrm{MV} / \mathrm{m}\right)$ ?

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Problem 28

Estimate the peak electric field inside a $1.1-\mathrm{kW}$ microwave oven under the simplifying approximation that the microwaves propagate as a plane wave through the oven's $750-\mathrm{cm}^{2}$ cross-sectional area.

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Problem 29

Your new radio says it can pick up signals with peak electric fields as weak as $450 \mu \mathrm{V} / \mathrm{m}$. Will it work if you take it to your remote cabin, where the intensity of your favorite radio station is $0.35 \mathrm{nW} / \mathrm{m}^{2}$ ?

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Problem 30

A laser pointer delivers $0.10-\mathrm{mW}$ average power in a beam $0.90 \mathrm{~mm}$ in diameter. Find (a) the average intensity, (b) the peak electric field, and (c) the peak magnetic field.

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Problem 31

Your university radio station has a 4.6-kW radio transmitter that broadcasts uniformly in all directions; listeners within $15 \mathrm{~km}$ have reliable reception. You want to increase the power to double that range. What should be the new power?

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Problem 32

A green laser pointer produces 532-nm light that's propagating in the $+y$-direction. Its electric field is parallel to the $z$-axis and has amplitude $2.19 \mathrm{kV} / \mathrm{m}$. Find (a) the wave frequency, (b) the amplitude of the wave's magnetic field, and (c) the direction of the magnetic field.

Laszlo Zalavari

Laszlo Zalavari

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Problem 33

An infrared laser that sends signals along optical fibers that comprise the Internet operates at a frequency of 194 $\mathrm{THz}$ and produces electromagnetic waves with magnetic-field amplitude of $328 \mu \mathrm{T}$. The magnetic field is parallel to the $y$-axis. Find (a) the wavelength of the laser light if it's propagating in air, (b) the wave's electric-field amplitude, and (c) the electric-field direction when the wave is propagating in the $+x$-direction.

Laszlo Zalavari

Laszlo Zalavari

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Problem 34

An AM radio station broadcasts with a wavelength of $484 \mathrm{~m}$. Its transmitting antenna is vertical, resulting in vertically polarized radio waves (i.e., the electric field of the waves is vertical). You observe the station's signal at a point where it's propagating due east, and where the amplitude of the wave's electric field is $347 \mathrm{mV} / \mathrm{m}$. Find (a) the wave frequency, (b) the amplitude of the wave's magnetic field, and (c) the direction of the magnetic field.

Prabhu Ramji

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Problem 35

A public FM radio station broadcasts at $88.7 \mathrm{MHz}$. Its transmitting antenna produces waves whose magnetic field is horizontal. You observe the station's signal at a point where it's propagating due south, and where the amplitude of the magnetic field is $28.5 \mathrm{nT}$. Find (a) the wavelength, (b) the amplitude of the wave's electric field, and (c) the direction of the electric field.

Prabhu Ramji

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Problem 36

When the cellphone of Example $29.4$ finds itself in a rural area, it automatically raises its transmitter power to $3.0$ W. At this power, how far can it be from the cell tower described in the example?

Shoukat Ali

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Problem 37

What transmitter power would be needed for a
cellphone to communicate reliably with the cell tower described
in Example 29.4, if the phone is 8.7 km from the tower?

Allison Knapp

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Problem 38

The Voyager 1 spacecraft, now in interstellar space, radios data back to Earth using a 22.4-W transmitter. A high-gain antenna focuses a narrow beam of radio waves toward Earth, making the transmitter equivalent to one that, if it were broadcasting in all directions, would have 50,000 times the power of the actual transmitter. In the current epoch, the Deep Space Network antennas that receive Voyager's broadcasts detect Voyager signals with electric-field amplitudes of $3.9 \mathrm{pV} / \mathrm{m}$. How far away is Voyager?

Justin Hameline

Justin Hameline

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Problem 39

Rovers on the surface of Mars communicate data to the Mars Reconnaisance Orbiter (MRO), which orbits Mars and relays the data to Earth. $M R O$ uses a 100 -W radio transmitter, and its high-gain antenna (HGA) has a 3-m parabolic dish that focuses the radio signal into a narrow band, effectively increasing the transmitter power by a factor of 50,000 . What sensitivity must Earth-based receivers have to receive signals from $M R O ?$ Express it as a minimum value for the electric-field amplitude in the radio waves at the receiver, and base your answer on the greatest possible Earth-Mars distance, about 400 million $\mathrm{km}$.

Mirza Aslam Beig

Mirza Aslam Beig

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Problem 40

A parallel-plate capacitor has circular plates with radius $50.0 \mathrm{~cm}$ and spacing $1.0 \mathrm{~mm}$. A uniform electric field between the plates is changing at the rate of $1.0 \mathrm{MV} / \mathrm{m} \cdot \mathrm{s}$. Find the magnetic field strength between the plates (a) on the symmetry axis, (b) $15 \mathrm{~cm}$ from the axis, and (c) $150 \mathrm{~cm}$ from the axis.

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Problem 41

An electric field points into the page and occupies a circular region of radius $1.0 \mathrm{~m}$, as shown in Fig. 29.14. There are no electric charges in the region, but there is a magnetic field forming closed loops pointing clockwise, as shown. The magnetic-field strength $80 \mathrm{~cm}$ from the center of the region is $1.9 \mu T$. (a) What's the rate of change of the electric field? (b) Is the electric field increasing or decreasing?

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Problem 42

The medical profession divides the ultraviolet region of the electromagnetic spectrum into three bands: UVA $(320 \mathrm{~nm}-420 \mathrm{~nm})$, UVB $(290 \mathrm{~nm}-320 \mathrm{~nm})$, and UVC (100 nm-290 nm). UVA and UVB promote skin cancer and premature skin aging; UVB also causes sunburn, but helpfully fosters production of vitamin D. Ozone in Earth's atmosphere blocks most of the more dangerous UVC. Find the frequency range associated with UVB radiation.

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Problem 43

Dielectric breakdown in air occurs when the electric field is approximately $3 \mathrm{MV} / \mathrm{m}$. What would be the peak magnetic field in an electromagnetic wave with this peak electric field?

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Problem 44

A polarizer blocks $75 \%$ of a polarized light beam. What's the angle between the beam's polarization and the polarizer's axis?

Guilherme Barros

Guilherme Barros

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Problem 45

An electro-optic modulator is a device that switches a laser beam rapidly from off to on by switching the polarization direction through $90^{\circ}$ when a voltage is applied. But a brownout results in only enough voltage for a $78^{\circ}$ rotation. What fraction of the light is transmitted during the brownout when the beam is supposed to be fully on?

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Problem 46

Unpolarized light of intensity $S_{0}$ passes first through a polarizer with its axis vertical and then through one with its axis at $35^{\circ}$ to the vertical. Find the intensity after the second polarizer.

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Problem 47

Vertically polarized light passes through two polarizers, the first at $60^{\circ}$ to the vertical and the second at $90^{\circ}$ to the vertical. What fraction of the light gets through?

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Problem 48

A magnetic field (not necessarily uniform) lies in the plane of this page, and a uniform electric field is perpendicular to the page and points out of the page. You evaluate $\oint \vec{B} \cdot \overrightarrow{d l}$ going clockwise around the edge of the page, and the result is $5.12 \times 10^{-8} \mathrm{~T} \cdot \mathrm{m}$. (a) At what rate is the electric field changing? (b) Is the electric field increasing or decreasing? Hint: You'll need to do an experimental measurement.

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Problem 49

High microwave intensities can cause biological damage through heating of tissue; a particular concern is cataract formation. The U.S. Food and Drug Administration limits microwave radiation near the door of a microwave oven to $5.0 \mathrm{~mW} / \mathrm{m}^{2}$. The window in a particular oven door measures $40 \mathrm{~cm}$ by $17 \mathrm{~cm}$ and is covered with a metal screen to block microwaves. Assuming power leaks uniformly through the window area, what percent of the oven's $900-\mathrm{W}$ microwave power can leak without exceeding the FDA standards?

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Problem 50

Use the fact that sunlight intensity at Earth's orbit is $1360 \mathrm{~W} / \mathrm{m}^{2}$ to determine the Sun's total power output.

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Problem 51

A quasar 10 billion light-years from Earth appears the same brightness as a star 57,000 light-years away. How do the power outputs of the quasar and the star compare?

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Problem 52

Lasers are classified according to the eye-damage danger they pose. Class-2 lasers, including many laser pointers, produce visible light with no greater than $1 \mathrm{~mW}$ total power. They're relatively safe because the eye's blink reflex limits exposure time to $250 \mathrm{~ms}$. Find (a) the intensity of a 1-mW class-2 laser with beam diameter $1.8 \mathrm{~mm}$, (b) the total energy delivered before the blink reflex shuts the eye, and (c) the peak electric field in the laser beam.

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Problem 53

At $2.0 \mathrm{~km}$ from a radio transmitter, the peak electric field is $350 \mathrm{mV} / \mathrm{m}$. Assuming the transmitter broadcasts equally in all directions, find (a) the transmitted power and (b) the peak electric field $10 \mathrm{~km}$ from the transmitter.

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Problem 54

Find the peak electric and magnetic fields $1.7 \mathrm{~m}$ from a 60-W lightbulb that radiates equally in all directions.

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Problem 55

Note the situation shown in Fig. $29.14$, but now consider that the electric field strength is increasing at the rate of $4.88 \times 10^{10} \mathrm{~V} / \mathrm{m} \cdot \mathrm{s}$, and that you don't know the magnetic field at the $50.0-\mathrm{cm}$ distance from the center where the figure shows a field line. Find (a) the magnetic field strength at the $50 \mathrm{~cm}$ distance shown and (b) also at $1.50 \mathrm{~m}$ from the center of the circular region (e.g., outside the electric-field region).

Keshav Singh

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Problem 56

A camera flash delivers $4.5 \mathrm{~kW}$ of light power for $1.0 \mathrm{~ms}$. Find (a) the total energy and (b) the total momentum carried by the flash.

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Problem 57

A laser produces an average power of $6.0 \mathrm{~W}$ in a $1.0-\mathrm{mm}$-diameter beam. Find (a) the average intensity and (b) the peak electric field of the laser light.

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Problem 58

A $180-\mathrm{W} / \mathrm{cm}^{2}$ laser beam shines on a light-absorbing surface. What's the radiation pressure on the surface?

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Problem 59

A 79-kg astronaut is floating in empty space. If she shines a 1.0-W flashlight in a fixed direction, how long will it take her to accelerate to $10 \mathrm{~m} / \mathrm{s}$ ?

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Problem 60

Read the Application: Starshot on page 597 and assume that the mass of each StarChip spacecraft is $2.4$ grams. Consider that $100 \mathrm{GW}$ of laser power is directed toward a StarChip's sail for 10 minutes. The sail is so thin that some of the light goes right through it, and the laser beam is also wider than the sail. Assume, therefore, that only $34 \mathrm{GW}$ of laser power actually reflects from the sail. Find (a) the total energy associated with the $10-\min$ burst of this 34-GW pulse, (b) the associated momentum, and (c) the spacecraft's final speed, expressed as a decimal fraction of the speed of light, assuming it's accelerated from rest. For (c), don't forget that the sail reflects the incident light rather than absorbing it.

Suzanne W.

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Problem 61

A white dwarf star is approximately the size of Earth but radiates about as much power as the Sun. Estimate the radiation pressure on a light-absorbing object at the white dwarf's surface.

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Problem 62

A cylindrical resistor of length $L$, radius $a$, and resistance $R$ carries current $I$. Calculate the electric and magnetic fields at the surface of the resistor, assuming the electric field is uniform over the surface. Calculate the Poynting vector and show that it points into the resistor. Calculate the flux of the Poynting vector (that is, $\int \vec{S} \cdot d \vec{A}$ ) over the resistor's surface to get the rate of electromagnetic energy flow into the resistor, and show that the result is $I^{2} R$. Your result shows that the energy heating the resistor comes from the fields surrounding it. These fields are sustained by the source of electric energy that drives the current.

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Problem 63

In a stack of polarizing sheets, each sheet has its transmission axis rotated $14^{\circ}$ with respect to the preceding sheet. If the stack passes $37 \%$ of the incident unpolarized light, how many sheets does it contain?

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Problem 64

You're an astronomer studying the origin of the solar system, and you're evaluating a hypothesis that sufficiently small particles were blown out of the solar system by the force of sunlight. To see how small such particles must be, compare the force of sunlight with the force of solar gravity, and solve for the particle radius at which the two are equal. Assume spherical particles with density $2 \mathrm{~g} / \mathrm{cm}^{3}$. (Note: Distance from the Sun doesn't matter. Why not?)

Dominador Tan

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Problem 65

Differentiate Equation $29.12$ with respect to $x$ and Equation $29.13$ with respect to $t$. Then, using the fact that mixed derivatives are equal (e.g., $\frac{\partial}{\partial t}\left(\frac{\partial B}{d x}\right)=\frac{\partial}{\partial x}\left(\frac{\partial B}{d t}\right)$, combine the resulting equations and show that the result is the wave equation (Equation 14.5) for waves with speed $c=1 / \sqrt{\epsilon_{0} \mu_{0}}$.

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Problem 66

Maxwell's equations in a dielectric resemble those in vacuum (Equations 29.6-29.9) but with $\epsilon_{0}$ replaced by $\kappa \epsilon_{0}$, where $\kappa$ is the dielectric constant introduced in Chapter 23 . Show that the speed of electromagnetic waves in a dielectric is $c / \sqrt{\kappa}$.

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Problem 67

You're designing a WiFi router that operates at both $2.40 \mathrm{GHz}$ and $5.00 \mathrm{GHz}$. The design calls for antennas that are half a wavelength high. How high should you make the two antennas?

Narayan Hari

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Problem 68

Your roommate's father is CEO of a coal company, so your roommate is understandably skeptical of alternative energy proposals. He claims that there's no future for solar energy, because the power in sunlight is insufficient to meet humankind's energy demand. Is he right? To find out, compare the solar power incident on Earth with our human energy consumption rate of about 18 TW.

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Problem 69

Earth emits infrared radiation at very nearly the rate at which it absorbs sunlight energy - namely, $239 \mathrm{~W}$ for every $\mathrm{m}^{2}$ of Earth's surface area. Estimate the intensity of that infrared radiation at the Sun's distance from Earth. Your answer shows why the energy flow between Sun and Earth is essentially one way, all from Sun to Earth.

Supratim Pal

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Problem 70

Your friend who works for the college radio station must make electric-field measurements for a report to be filed with the station's application for license renewal. The measurement is made $4.6 \mathrm{~km}$ from the antenna, where your friend measures the electric field at $380 \mathrm{~V} / \mathrm{m}$. The station is allowed to broadcast at no more than $55-\mathrm{kW}$ power. Assuming power spreads equally in all directions, is the station in compliance with its license?

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Problem 71

The National Ignition Facility at Lawrence Livermore National Laboratory initiates nuclear fusion by converging 192 laser beams on a deuterium-tritium target. Each beam has a square cross section $38 \mathrm{~cm}$ on a side, and each beam delivers $10.0 \mathrm{~kJ}$ of energy in $20.0 \mathrm{~ns}$. Find (a) the peak electric field and (b) the peak magnetic field in each laser beam. (c) Find the combined power of all 192 laser beams while they're firing, and compare with humankind's energy consumption rate of about $18 \mathrm{TW}$.

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Problem 72

The table below shows the intensity of the radio signal received at Earth from a spacecraft on its way to the outer solar system, as a function of its distance from Earth. Distances are in astronomical units (AU, with $1 \mathrm{AU}$ being the mean Earth-Sun distance; see Appendix E). Determine a quantity that, when you plot $\bar{S}$ versus that quantity, should give a straight line. Make your plot, establish a bestfit line, and use it to determine the spacecraft's transmitter power.
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Distance (AU) } & 1.56 & 1.81 & 2.14 & 2.78 & 3.17 & 4.25 \\
\hline \begin{array}{l}
\text { Intensity, } \\
\bar{S}\left(10^{-23} \mathrm{~W} / \mathrm{m}^{2}\right)
\end{array} & 22.5 & 17.8 & 11.6 & 7.10 & 5.63 & 3.01 \\
\hline
\end{array}
$$

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Problem 73

If a sunlight-powered sailing spacecraft accelerated at $1 \mathrm{~m} / \mathrm{s}^{2}$ in the vicinity of Earth's orbit, what would be its acceleration a Mars, about $1.5$ times as far from the Sun as Earth?
a. about $0.25 \mathrm{~m} / \mathrm{s}^{2}$
b. a little less than $0.5 \mathrm{~m} / \mathrm{s}^{2}$
c. a little more than $0.5 \mathrm{~m} / \mathrm{s}^{2}$
d. about $0.66 \mathrm{~m} / \mathrm{s}^{2}$

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Problem 74

One spacecraft has a sail that absorbs all light incident on it; the other has a perfectly reflective sail. How do their accelerations compare in light with the same intensity?
a. The absorptive sail gives twice the acceleration.
b. The reflective sail gives twice the acceleration.
c. The absorptive sail gives greater acceleration, but not twice as much.
d. The reflective sail gives greater acceleration, but not twice as much.

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Problem 75

A sail capable of propelling a spacecraft to the outer solar system must be able to overcome the Sun's gravity. Suppose a spacecraft is designed so the force of sunlight on its sail is 20 times that of solar gravity in the vicinity of Earth's orbit. If the spacecraft reaches Jupiter, some 5 times as far from the Sun as Earth,
a. the sail force will still exceed solar gravity, now by a factor of 4 .
b. the sail force will be slightly less than solar gravity.
c. the sail force will now be 25 times solar gravity.
d. the sail force will still be 20 times solar gravity.

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Problem 76

The intensity of sunlight at Earth's orbit is about $1.4 \mathrm{~kW} / \mathrm{m}^{2}$. A $100-\mathrm{kg}$ sailing spacecraft with 1- $\mathrm{km}^{2}$ sail area would experience an acceleration of about
a. $5 \mathrm{~mm} / \mathrm{s}^{2}$.
b. $5 \mathrm{~cm} / \mathrm{s}^{2}$.
c. $5 \mathrm{~m} / \mathrm{s}^{2}$.
d. $5 \mathrm{~km} / \mathrm{s}^{2}$.

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