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University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 32

Electromagnetic Waves - all with Video Answers

Educators


Chapter Questions

03:40

Problem 1

(a) How much time does it take light to travel from the moon to the earth, a distance of $384,000 \mathrm{km}$ ? (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance from earth to Sirius in kilometers?

Artemisa Mazón
Artemisa Mazón
Numerade Educator
01:41

Problem 2

In a TV picture, ghost images are formed when the signal from the transmitter travels to the receiver both directly and indirectly after reflection from a building or other large metallic mass. In a 25 -inch set, the ghost is about 1.0 $\mathrm{cm}$ to the right of the principal image if the reflected signal arrives 0.60$\mu$ s after the principal signal. In this case, what is the difference in path lengths for the two signals?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:33

Problem 3

For an electromagnetic wave propagating in air, determine the frequency of a wave with a wavelength of (a) $5.0 \mathrm{km} ;$ (b) 5.0 $\mathrm{m}$ (c) $5.0 \mu \mathrm{m} ;$ (d) 5.0 $\mathrm{nm}$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
04:08

Problem 4

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 $\mathrm{nm}$ to 400 $\mathrm{nm}$ . It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 $\mathrm{nm}$ and 320 $\mathrm{mm}$ , is much more dangerous because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:32

Problem 5

A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25$\mu \mathrm{T}$ and a wavelength of 432 $\mathrm{nm}$ is traveling in the $+x$ -direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of $x$ and $t$ in the form of Eqs. $(32.17) .$

Ze-Han Lee
Ze-Han Lee
Numerade Educator
07:05

Problem 6

An electromagnetic wave of wavelength $435 \mathrm{~nm}$ is traveling in vacuum in the $-z$ -direction. The electric field has amplitude $2.70 \times 10^{-3} \mathrm{~V} / \mathrm{m}$ and is parallel to the $x$ -axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations for $\overrightarrow{\boldsymbol{E}}(z, t)$ and $\overrightarrow{\boldsymbol{B}}(z, t)$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
10:43

Problem 7

A sinusoidal electromagnetic wave of frequency $6.10 \times 10^{14} \mathrm{Hz}$ travels in vacuum in the $+z$ -direction. The $\overrightarrow{\boldsymbol{B}}$ -field is parallel to the $y$ -axis and has amplitude $5.80 \times 10^{-4}$ T. Write the vector equations for $\overrightarrow{\boldsymbol{E}}(z, t)$ and $\overrightarrow{\boldsymbol{B}}(z, t) .$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:17

Problem 8

The electric field of a sinusoidal electromagnetic wave obeys the equation $E=-(375 \mathrm{V} / \mathrm{m}) \sin \left[\left(5.97 \times 10^{15} \mathrm{rad} / \mathrm{s}\right) t+\right.$ $\left(1.99 \times 10^{7} \mathrm{rad} / \mathrm{m}\right) x ]$ . (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans?(c) What is the speed of the wave?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
08:17

Problem 9

An electromagnetic wave has an electric field given by $\vec{E}(y, t)=-\left(3.10 \times 10^{5} \mathrm{~V} / \mathrm{m}\right) \hat{k} \sin \left[\mathrm{k} y-\left(12.65 \times 10^{12} \mathrm{rad} / \mathrm{s}\right) t\right]$
(a) In which direction is the wave traveling? (b) What is the wavelength of the wave? (c) Write the vector equation for $\vec{R}(y, t)$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
07:46

Problem 10

An electromagnetic wave has a magnetic field given by $\overrightarrow{\boldsymbol{B}}(x, t)=\left(8.25 \times 10^{-9} \mathrm{T}\right) \hat{j} \sin \left[\left(1.38 \times 10^{4} \mathrm{rad} / \mathrm{m}\right) x+\omega t\right]$ (a) In which direction is the wave traveling? (b) What is the frequency $f$ of the wave? (c) Write the vector equation for $\vec{E}(x, t)$ .

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:14

Problem 11

Radio station WCCO in Minneapolis broadcasts at a frequency of 830 $\mathrm{kHz}$ . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from $\mathrm{WCCO}$ is $4.82 \times 10^{-11}$ T. Calculate (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

Ryan Hood
Ryan Hood
Numerade Educator
02:23

Problem 12

The electric-field amplitude near a certain radio transmitter is $3.85 \times 10^{-3} \mathrm{V} / \mathrm{m}$ What is the amplitude of $\overrightarrow{\boldsymbol{B}} ?$ How does this compare in magnitude with the earth's field?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:44

Problem 13

An electromagnetic wave with frequency $5.70 \times 10^{14} \mathrm{Hz}$ propagates with a speed of $2.17 \times 10^{8} \mathrm{m} / \mathrm{s}$ in a certain piece of glass. Find (a) the wavelength of the wave in the glass; (b) the wavelength of a wave of the same frequency propagating in air; (c) the index of refraction $n$ of the glass for an electromagnetic wave with this frequency; (d) the dielectric constant for glass at this frequency, assuming that the relative permeability is unity.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:00

Problem 14

An electromagnetic wave with frequency 65.0 $\mathrm{Hz}$ travels in an insulating magnetic material that has dielectric constant 3.64 and relative permeability 5.18 at this frequency. The electric field has amplitude $7.20 \times 10^{-3} \mathrm{V} / \mathrm{m}$ . (a) What is the speed of propagation of the wave? (b) what is the wavelength of the wave? (c) What is the amplitude of the magnetic field? (d) What is the intensity of the wave?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:07

Problem 15

We can reasonably model a 75-W incandescent light-bulb as a sphere 6.0 $\mathrm{cm}$ in diameter. Typically, only about 5$\%$ of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible-light intensity (in $\mathrm{W} / \mathrm{m}^{2} )$ at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:47

Problem 16

Consider each of the following electric and magnetic-ficld orientations. In each case, what is the direction of propagation of the wave? $-E \hat{k}, \vec{B}=-B i,(\mathrm{~d}) \vec{E}=E \hat{\imath}, \vec{B}=-B \hat{k}$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
04:19

Problem 17

A sinusoidal electromagnetic wave is propagating in a vacunm in the $+z$ -direction. If at a particular instant and at a certain point in space the electric field is in the $+x$ -direction and has magnitude 4.00 $\mathrm{V} / \mathrm{m}$ , what are the magnitude and direction of the magnetic field of the wave at this same point in space and instant in time?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:34

Problem 18

A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area $0.500 \mathrm{m}^{2} .$ At the window, the electric field of the wave has rms value 0.0200 $\mathrm{V} / \mathrm{m}$ . How much energy docs this wave carry through the window during a 30.0 -s commercial?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:11

Problem 19

You are a NASA mission specialist on your first flight aboard the space shuttle. Thanks to your extensive training in pliysics, you have been assigned to evaluate the performance of a new radio transmitter on board the International Space Station (ISS). Perched on the shuttle's movable arm, you aim a sensitive detector at the ISS, which is 2.5 $\mathrm{km}$ away. You find that the electric-field amplitude of the radio waves coming from the ISS transmitter is 0.090 $\mathrm{V} / \mathrm{m}$ and that the frequency of the waves is 244 $\mathrm{MHz}$ . Find the following: (a) the intensity of the radio wave at your location; (b) the magnetic-field amplitude of the wave at your location; (c) the total power output of the ISS radio transmitter. (d) What assumptions, if any, did you make in your calculations?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:58

Problem 20

The intensity of a cylindrical laser beam is 0.800 $\mathrm{W} / \mathrm{m}^{2}$ The cross-sectional area of the beam is $3.0 \times 10^{-4} \mathrm{m}^{2}$ and the intensity is uniform across the cross section of the beam. (a) What is the average power output of the laser? (b) What is the rms value of the electric field in the beam?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:21

Problem 21

A space probe $2.0 \times 10^{10} \mathrm{m}$ from a star measures the total intensity of electrumagnetic radiation from the star to be $5.0 \times 10^{3} \mathrm{W} / \mathrm{m}^{2} .$ If the star radiates uniformly in all directions, what is its total average power output?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:56

Problem 22

A simusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 $\mathrm{cm}$ and an electric-field amplitude of $5.40 \times 10^{-2} \mathrm{V} / \mathrm{m}$ at a distance of 250 $\mathrm{m}$ from the antenna. Calculate (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:14

Problem 23

A monochromatic light source with power output 60.0 $\mathrm{W}$ radiates light of wavelength 700 $\mathrm{nm}$ uniformly in all directions. Calculate $E_{\max }$ and $B_{\max }$ for the 700 -nm light at a distance of 5.00 $\mathrm{m}$ from the source.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:45

Problem 24

For the electromagnetic wave represented by Eq. $(32.19)$ show that the Poynting vector $(a)$ is in the same direction as the propagation of the wave and $(b)$ has average magnitude given by Eqs. $(32.29) .$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:50

Problem 25

An intense light source radiates uniformly in all directions. At a distance of 5.0 $\mathrm{m}$ from the source, the radiation pressure on a perfectly absorbing surface is $9.0 \times 10^{-6} \mathrm{Pa}$ . What is the total average power output of the source?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:30

Problem 26

Public television station KQED in San Francisco broadcasts a sinusoidal radio signal at a power of 316 $\mathrm{kW}$ . Assume that the wave spreads out uniformly into a hemisphere above the ground. At a home 5.00 $\mathrm{km}$ away from the antenna, (a) what average pressure does this wave exert on a totally reflecting surface, $(b)$ what are the amplitudes of the electric and magnetic fields of the wave, and (c) what is the aver- age density of the energy this wave carries? (d) For the energy density in part (c), what percentage is due to the electric field and what percentage is due to the magnetic field?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:07

Problem 27

If the intensity of direct sunlight at a point on the earth's surface is $0.78 \mathrm{kW} / \mathrm{m}^{2},$ find $(\mathrm{a})$ the average momentum density (momentum per unit volume) in the sunlight and (b) the average momentum flow rate in the sunlight.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:55

Problem 28

In the $25-$ fil Space Simulator facility at NASA's Jet Propulsion Laboratory, a bank of overhead arc lamps can produce light of intensity 2500 $\mathrm{W} / \mathrm{m}^{2}$ at the floor of the facility. (This simulates the intensity of sunlight near the planet Venus.) Find the average radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing section of the floor and (b) a totally reflecting section of the floor. (c) Find the average momentum density (momentum per unit volume) in the light at the floor.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:45

Problem 29

Verify that all the expressions in Eqs. $(32.27)$ are equivalent to $\mathrm{Eq} .(32.26)$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
04:16

Problem 30

An electromagnetic standing wave in air of frequency 750 MHz is set up between two conducting planes 80.0 $\mathrm{cm}$ apart. At which positions between the planes could a point charge be placed at rest so that it would remain at rest? Explain.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:17

Problem 31

A standing electromagnetic wave in a certain material has frequency $2.20 \times 10^{10} \mathrm{Hz}$ . The nodal planes of $\overrightarrow{\boldsymbol{B}}$ are 3.55 $\mathrm{mm}$ apart. Find (a) the wavelength of the wave in this material; (b) the distance between adjacent nodal planes of the $\overrightarrow{\boldsymbol{E}}$ field: (c) the speed of propagation of the wave.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
04:57

Problem 32

An electromagnetic standing wave in air has frequency 75.0 MHz. (a) What is the distance between nodal planes of the $\overrightarrow{\boldsymbol{E}} $ field? (b) What is the distance between a nodal plane of $\vec{E}$ and the closest nodal plane of $\overrightarrow{\boldsymbol{B}} ?$

Mohit Khurana
Mohit Khurana
Texas A&M University
05:33

Problem 33

An electromagnetic standing wave in a certain material has frequency $1.20 \times 10^{10} \mathrm{Hz}$ and speed of propagation $210 \times 10^{8} \mathrm{m} / \mathrm{s}$ . (a) What is the distance between a nodal plane of $\overrightarrow{\boldsymbol{B}}$ and the closest antinodal plane of $\overrightarrow{\boldsymbol{B}} ?$ (b) What is the distance between an antinodal plane of $\overrightarrow{\boldsymbol{E}}$ and the closest antinodal plane of $\overrightarrow{\boldsymbol{B}}$ ? (c) What is the distance between a nodal plane of $\overrightarrow{\boldsymbol{E}}$ and the closest nodal plane of $\overrightarrow{\boldsymbol{B}} ?$

Nathan Nowack
Nathan Nowack
Numerade Educator
09:09

Problem 34

Show that the electric and magnetic fields for standing waves given by Eqs. $(32.34)$ and $(32.35)$ (a) satisfy the wave equation, Eq. $(32.15),$ and $(b)$ satisfy $\mathrm{Eqs} .(32.12)$ and $(32.14).$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:41

Problem 35

The microwaves in a certain microwave oven have a wavelength of $12.2 \mathrm{cm} .$ (a) How wide must this oven be so that it will contain five antinodal planes of the electric field along its width in the standing wave pattern? (b) What is the frequency of these microwaves? (c) Suppose a manufacturing error occurred and the oven was made 5.0 $\mathrm{cm}$ longer than specificd in part (a). In this case, what would have to be the frequency of the microwaves for there still to be five antinodal planes of the electric field along the width of the oven?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
06:40

Problem 36

Consider a sinusoidal electromagnetic wave with fields $\overrightarrow{\boldsymbol{E}}=\boldsymbol{E}_{\max } \hat{\boldsymbol{j}} \sin (k x-\omega t)$ and $\overrightarrow{\boldsymbol{B}}=\boldsymbol{B}_{\max } \boldsymbol{k} \sin (k x-\omega t+\phi),$ with $-\pi \leq \phi \leq \pi .$ Show that if $\overrightarrow{\boldsymbol{E}}$ and $\overrightarrow{\boldsymbol{B}}$ are to satisfy Eqs. $(32.12)$ and $(32.14),$ then $E_{\text { max }}=c B_{\text { max }}$ and $\phi=0 .$ (The result $\phi=0$ means the $\vec{E}$ and $\vec{B}$ fields oscillate in phase.)

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:44

Problem 37

Show that the magnetic field $B_{z}(x, t)$ in aplane electromagnetic wave propagating in the $+x$ -direction must satisfy Eq. $(32.15) .$ (Hint . Take the partial derivative of Eq. $(32.12)$ with respect to $t$ and the partial derivative of Eq. $(32.14)$ with respect to $x .$ Then combine the results.)

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
06:54

Problem 38

For a sinusoidal electromagnetic wave in vacuum, such as that described by Eq. $(32.16),$ show that the average energy density in the electric field is the same as that in the magnetic field.

Mohit Khurana
Mohit Khurana
Texas A&M University
05:09

Problem 39

A satellite 575 $\mathrm{km}$ above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 $\mathrm{MHz}$ uniformly in all directions, with a power of 25.0 $\mathrm{kW}$ . (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0 $\mathrm{cm}$ by 40.0 $\mathrm{cm}$ oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
03:30

Problem 40

A plane sinusoidal clectromagnetic wave in air has a wavelength of 3.84 $\mathrm{cm}$ and an $\overrightarrow{\boldsymbol{E}}$ -field amplitude of 1.35 $\mathrm{V} / \mathrm{m}$ . (a) What is the frequency? (b) What is the $\overrightarrow{\boldsymbol{B}}$ -field amplitude? (c) What is the intensity?(d) What average force does this radiation exert on a totally absorbing surface with area 0.240 $\mathrm{m}^{2}$ perpendicular to the direction of propagation?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
04:58

Problem 41

A small helium-neon laser emits red visible light with a power of 3.20 $\mathrm{mW}$ in a beam that has a diameter of 2.50 $\mathrm{mm}$ . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a $1.00-\mathrm{m}$ length of the beam?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
08:52

Problem 42

Consider aplane electromagnetic wave such as that shown in Fig. $32.5,$ but in which $\overrightarrow{\boldsymbol{E}}$ and $\overrightarrow{\boldsymbol{B}}$ also have components in the $x$-direction (along the direction of wave propagation). Use Gauss's law for electric and magnetic fields to show that the components $E_{x}$ and $B_{x}$ must both be equal to zero so that the fields $\vec{E}$ and $\vec{B}$ are both transverse. (Hint: Use a Gaussian surface like that shown in Fig. 32.6. Of the two faces parallel to the $y z$ -plane, choose one to be to the left of the wave front and the other to be to the right of the wave front)

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
07:50

Problem 43

The sun emits energy in the form of electromagnetic waves at a rate of $3.9 \times 10^{26} \mathrm{W}$ . This energy is produced by nuclear reactions deep in the sun's interior. (a) Find the intensity of electromagnetic radiation and the radiation pressure on an absorbing object at the surface of the sun (radius $r=R=6.96 \times 10^{5} \mathrm{km}$) and at $r=R / 2$ , in the sun's interior. Ignore any scattering of the waves as they move radially outward from the center of the sun. Compare to the values given in Section 32.4 for sunlight just before it enters the earth's atmosphere. (b) The gas pressure at the sun's surface is about $1.0 \times 10^{4} \mathrm{Pa}$ ; at $r=R / 2,$ the gas pressure is calculated from solar models to be about $4.7 \times 10^{33} \mathrm{Pa}$ Comparing with your results in part (a), would you expect that radiation pressure is an important factor in determining the structure of the sun? Why or why not?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:32

Problem 44

It has been proposed to place solar-power-collecting satellites in earth orbit. The power they collect would be beamed down to the earth as microwave radiation. For a microwave beam with a cross-sectional area of 36.0 $\mathrm{m}^{2}$ and a total power of 2.80 $\mathrm{kW}$ at the earth's surface, what is the amplitude of the electric field of the beam at the earth's surface?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
07:43

Problem 45

Two square reflectors, each 1.50 $\mathrm{cm}$ on a side and of mass 4.00 $\mathrm{g}$ , are located at opposite ends of a thin, extremely light, $1.00-\mathrm{m}$ rod that can rotate without friction and in a vacuum about an axle perpendicular to it through its center (Fig. 32.24$)$ . These reficctors are small enough to be treated as point masses in moment-of-inertia calculations. Both reflectors are illuminated on one face by a sinusoidal light wave having an electric field of amplitude 1.25 $\mathrm{N} / \mathrm{C}$ that falls uniformly on both surfaces and always strikes them perpendicular to the plane of their surfaces. One reflector is covered with a perfectly absorbing coating, and the other is covered with a perfectly reflecting coating. What is the angular acceleration of this device?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
06:30

Problem 46

The plane of a flat surface is perpendicular to the propagation direction of an electromagnetic wave of intensity $1 .$ The surface absorbs a fraction $w$ of the incident intensity, where $0 \leq w \leq 1,$ and reflects the rest. (a) Show that the radiation pressure on the surface equals $(2-w) I / c .$ (b) Show that this expression gives the correct results for a surface that is (i) totally absorbing and (ii) totally reflective. (c) For an incident intensity of $1.40 \mathrm{kW} / \mathrm{m}^{2},$ what is the radiation pressure for 90$\%$ absorption? For 90$\%$ reflection?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
16:16

Problem 47

A cylindrical conductor with a circular cross section has a radius $a$ and a resistivity $\rho$ and carries a constant current $L$ ( a) What are the magnitude and direction of the electric-field vector $\overrightarrow{\boldsymbol{E}}$ at a point just inside the wire at a distance $a$ from the axis? (b) What are the magnitude and direction of the magnetic-field vector $\overrightarrow{\boldsymbol{B}}$ at the same point? (c) What are the magnitude and direction of the Poynting vector $\overrightarrow{\mathbf{S}}$ at the same point? (The direction of $\overrightarrow{\boldsymbol{S}}$ is the direction in which electromagnetic energy flows into or out of the conductor.) (d) Use the result in part (c) to find the rate of flow of energy into the volume occupied by a length $l$ of the conductor. (Hint: Integrate $\vec{S}$ over the surface of this volume.) Compare your result to the rate of generation of thermal energy in the same volume. Discuss why the energy dissipated in a current-carrying conductor, due to its resistance, can be thought of as entering through the cylindrical sides of the conductor.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:38

Problem 48

A source of sinusoidal electromagnetic waves radiates uniformly in all directions. At 10.0 $\mathrm{m}$ from this source, the amplitude of the electric field is measured to be 1.50 $\mathrm{N} / \mathrm{C}$ . What is the electricfield amplitude at a distance of 20.0 $\mathrm{cm}$ from the source?

Supratim Pal
Supratim Pal
Numerade Educator
05:15

Problem 49

A circular loop of wire can be used us a radio antenna. If a 18.0-cm-diameter antenna is located 2.50 $\mathrm{km}$ from a 95.0 -MHz source with a total power of 55.0 $\mathrm{kW}$ , what is the maximum emf induced in the loop? (Assume that the plane of the antenna loop is perpendicular to the direction of the radiation's magnetic field and that the source radiates uniformly in all directions.)

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:00

Problem 50

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 $\mathrm{MHz}$ in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field? (b) If the standing wave pattern is determined to be in its eighth harmonic, how long is the cavity?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:09

Problem 51

You are the sole crew member of the interplanetary spaceship $T : 1339$ Vorga, which makes regular cargo runs between the earth and the mining colonies in the asteroid belt. You are working outside the ship one day while at a distance of 2.0 AU from the sun. II AU (astronomical unit) is the average distance from the earth to the sun, $149,600,000 \mathrm{km} .$ I Unfortunately, you lose contact with the ship's hull and begin to drift away into space. You use your spacesuit's rockets to try to push yourself back toward the ship, but they run out of fuel and stop working before you can return to the ship. You find yourself in an awkward position, floating 16.0 $\mathrm{m}$ from the spaceship with zero velocity relative to it. Fortunately, you are carrying a $200-\mathrm{W}$ flashlight. You turn on the flashlight and use its beam as a "light rocket" to push yourself back toward the ship. (a) If you, your spacesuit, and the flashlight have a combined mass of 150 $\mathrm{kg}$ , how long will it take you to get back to the ship? (b) Is there another way you could use the flashlight to accomplish the same job of returning you to the ship?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:20

Problem 52

The 19 th-century inventor Nikola Tesla proposed to transmit electric power via sinusoidal electromagnetic waves. Suppose power is to be transmitted in a beam of cross-sectional area $100 \mathrm{m}^{2} .$ What electric- and magnetic-field amplitudes are required to transmit an amount of power comparable to that handled by modern transmission lines (that carry voltages and currents of the order of 500 $\mathbf{k V}$ and 1000 $\mathrm{A} ) ?$

Ryan Hood
Ryan Hood
Numerade Educator
07:55

Problem 53

The GPS network consists of 24 satellites, each of which makes two orbits around the earth per day. Each satellite transmits a $50.0 .$ W (or even less) simu- soidal electromagnetic signal at two frequencies, one of which is 1575.42 $\mathrm{MHz}$ . Assume that a satellite transmits half of its power at each frequency and that the waves travel uniformly in a downward hemisphere. (a) What average intensity does a GPS receiver on the ground, directly below the satellite, receive? (Hint: First use Newton's laws to find the altitude of the satellite.) (b) What are the amplitudes of the electric and magnetic fields at the GPS receiver in part (a), and how long does it take the signal to reach the receiver? (c) If the receiver is a square panel 1.50 $\mathrm{cm}$ on a side that absorbs all of the beam, what average pressure does the signal exert on it? (d) What wavelength must the receiver be tuned to?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
06:08

Problem 54

NASA is giving serious consideration to the concept of solar sailing. A solar sailcraft uses a large, low-mass sail and the energy and momentum of sunlight for propulsion. (a) Should the sail be absorbing or reflective? Why? (b) The total power output of the sun is $3.9 \times 10^{26} \mathrm{W}$ . How large a sail is necessary to propel a $10,000-\mathrm{kg}$ spacecraft against the gravitational force of the sun? Express your result in square kilometers. (c) Explain why your answer to part ( b) is independent of the distance from the sun.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
09:56

Problem 55

Interplanetary space contains many small particles referred to as interplanetary dust. Radiation pressure from the sun sets a lower limit on the size of such dust particles. To see the origin of this limit, consider a spherical dust particle of radius $R$ and mass density $\rho$ (a) Write an expression for the gravitational force exerted on this particle by the sun (mass $M )$ when the particle is a distance $r$ from the sun. (b) Let $L$ represent the luminosity of the sun, equal to the rate at which it emits energy in electromagnetic radiation. Find the force exerted on the (totally absorbing) particle due to solar radiation pressure, remembering that the intensity of the sun's radiation also depends on the distance $r .$ The relevant area is the cross-sectional area of the particle, $n o t$ the total surface area of the particle. As part of your answer, explain why this is so. (c) The mass density of a typical interplanetary dust particle is about 3000 $\mathrm{kg} / \mathrm{m}^{3}$ . Find the particle radius $R$ such that the gravitational and radiation forces acting on the particle are equal in magnitude. The luminosity of the sun is $3.9 \times 10^{26} \mathrm{W}$ . Does your answer depend on the distance of the particle from the sun? Why or why not? (d) Explain why dust particles with a radius less than that found in part (c) are unlikely to be found in the solar system. [Hint: Construct the ratio of the two force expressions found in parts (a) and (b).]

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:13

Problem 56

The electron in a hydrogen atom can be considered to be in a circular orbit with a radius of 0.0529 $\mathrm{nm}$ and a kinetic energy of 13.6 $\mathrm{eV}$ . If the electron behaved classically, how much energy would it radiate per second (see Challenge Problem 32.57$) ?$ What does this tell you about the use of classical physics in describing the atom?

Mohit Khurana
Mohit Khurana
Texas A&M University
09:05

Problem 57

Electromagnetic radiation is emitted by accelerating charges. The rate at which energy is emitted from an accelerating charge that has charge $q$ and acceleration $a$ is given by
$$\frac{d E}{d t}=\frac{q^{2} a^{2}}{6 \pi \epsilon_{0} c^{3}}$$
where $c$ is the speed of light. (a) Verify that this equation is dimensionally correct. (b) If a proton with a kinetic energy of 6.0 MeV is traveling in a particle accelerator in a circular orbit of radius 0.750 $\mathrm{m}$ , what fraction of its energy does it radiate per second? (c) Consider an electron orbiting with the same speed and radius. What fraction of its energy does it radiate per second?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
10:34

Problem 58

Electromagnetic waves propagate much differently in conductors than they do in dielectrics or in vacuum. If the resistivity of the conductor is sufficiently low
(that is, if it is a sufficiently good conductor), the oscillating electric field of the wave gives rise to an oscillating conduction current that is much larger than the
displacement current. In this case, In this case, the wave equation for an electric field $\vec{E}(x, t)=E_{y}(x, t) \hat{\jmath}$ propagating in the $+x$ -direction
within a conductor is $\frac{\partial^{2} E_{y}(x, t)}{\partial x^{2}}=\frac{\mu \partial E_{y}(x, t)}{\rho t}$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator