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H13 EM Waves (32.1-5)Due: 11:59pm on Monday, December 9,
2013
You will receive no credit for items you complete after the
assignment is due. Grading Policy
Problem 32.2Description: TV Ghosting. In a TV picture, ghost
images are formed when the signal from the transmitter travels to
the receiver both directly andindirectly after reflection from a
building or other large metallic mass. In a 25-inch set, the ghost
is about 1.0 cm...
TV Ghosting. In a TV picture, ghost images are formed when the
signal from the transmitter travels to the receiver both directly
and indirectly afterreflection from a building or other large
metallic mass. In a 25-inch set, the ghost is about 1.0 cm to the
right of the principal image if the reflected signalarrives 0.700
after the principal signal.
Part A
In this case, what is the difference in path lengths for the two
signals?
Use 3.00108 for the speed of light in a vacuum.
ANSWER:
Problem 32.7Description: Radio station WCCO in Minneapolis
broadcasts at a frequency of f. At a point some distance from the
transmitter, the magnetic-fieldamplitude of the electromagnetic
wave from WCCO is B_max. (a) Find the wavelength. (b) Find the wave
number. (c)...
Radio station WCCO in Minneapolis broadcasts at a frequency of
830 . At a point some distance from the transmitter, the
magnetic-field amplitude ofthe electromagnetic wave from WCCO is
4.091011 .
Part A
Find the wavelength.
Use = 3.00108 for the speed of light in a vacuum.
ANSWER:
Part B
Find the wave number.
ANSWER:
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University Physics with Modern Physics, 13eYoung/Freedman
Instructor Resources eText Study Area
H13 EM Waves (32.1-5) [ Edit ]
Overview Summary View Diagnostics View Print View with
Answers
s
m/s
= 210 m
kHzT
c m/s
= 361 m
= 1.74102 m1
Course Home Assignments Roster Gradebook Item Library
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Part C
Calculate the angular frequency.
ANSWER:
Part D
Calculate the electric-field amplitude.
ANSWER:
Poynting FluxDescription: Given an expression for the electric
field of an EM wave (travelling, in vacuum), choose the correct
form of the magnetic field andcompute the Poynting vector. Very
simple.
An electromagnetic wave is traveling through vacuum. Its
electric field vector is given by
,where is the unit vector in the y direction.
Part A
If is the amplitude of the magnetic field vector, find the
complete expression for the magnetic field vector of the wave.
Hint 1. Relative orientation of and for a wave in vacuum
In free space, the electric and magnetic field vectors of an
electromagnetic wave are perpendicular to each other. This follows
from Maxwell'sequations.
Hint 2. Orientation of and relative to the direction of
propagation
In free space, the electric and magnetic field vectors of an
electromagnetic wave are both perpendicular to the direction of
propagation of thewave. This follows from Maxwell's equations.
Hint 3. Determine the direction of propagation of the wave
In what direction is this wave propagating?
Express your answer in terms of , , and .
Hint 1. How to approach the problem
The point that has the same phase as , that is, at time , is ,
where .
ANSWER:
Hint 4. Phase relationship between and
In free space, the electric and magnetic field vectors of an
electromagnetic wave are exactly in phase. This follows from
Maxwell's equations.
ANSWER:
= 5.22106 rad/s
= 1.23102 V/m
= sin(kx t)E E0 j^j^
B B
B E
E B
i^ j^ k^
(x, t) phase = kx t t+t x+x x = t/k
E B
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Part B
What is the Poynting vector , that is, the power per unit area
associated with the electromagnetic wave described in the problem
introduction?
Give your answer in terms of some or all of the variables , , ,
, , , and . Specify the direction of the Poynting vector using
the
unit vectors , , and as appropriate.
Hint 1. Definition of the Poynting vector
The Poynting vector of an electromagnetic wave in vacuum is
given in terms of the electric field vector and the magnetic field
vector bythe relation
.
ANSWER:
Problem 32.10Description: An electromagnetic wave with frequency
65.0 Hz travels in an insulating magnetic material that has
dielectric constant 3.64 and relativepermeability 5.18 at this
frequency. The electric field has amplitude 7.20 * 10^( - 3) V/m.
(a) What is the speed ...
An electromagnetic wave with frequency 65.0 Hz travels in an
insulating magnetic material that has dielectric constant 3.64 and
relative permeability 5.18 atthis frequency. The electric field has
amplitude .
Part A
What is the speed of propagation of the wave?
ANSWER:
Part B
What is the wavelength of the wave?
ANSWER:
Part C
sin(kx t)B0 i^
sin(kx t)B0 j^
sin(kx t)B0 k^
cos(kx t)B0 i^
cos(kx t)B0 j^
cos(kx t)B0 k^
(x, t)S
E0 B0 k x t 0
i^ j^ k^
S E B
= S 10 E B
=
Also accepted: ,
(x, t)S
7.20 V/m103
6.91107 m/s
1.06106 m
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What is the amplitude of the magnetic field?
ANSWER:
Part D
What is the intensity of the wave?
ANSWER:
Problem 32.18Description: A sinusoidal electromagnetic wave
emitted by a cellular phone has a wavelength of lambda and an
electric-field amplitude of E_max at adistance of x from the
antenna. (a) Calculate the frequency of the wave. (b) Calculate the
magnetic-field...
A sinusoidal electromagnetic wave emitted by a cellular phone
has a wavelength of 36.1 and an electric-field amplitude of 5.00102
at a distanceof 290 from the antenna.
Part A
Calculate the frequency of the wave.
Use 3.00108 for the speed of light in a vacuum.
ANSWER:
Part B
Calculate the magnetic-field amplitude.
ANSWER:
Part C
Find the intensity of the wave.
Use 3.00108 for the speed of light in a vacuum and 8.851012 for
the permittivity of free space.
ANSWER:
Problem 32.40Description: A plane sinusoidal electromagnetic
wave in air has a wavelength of 3.84 (cm) and an E_vec-field
amplitude of 1.35 (V/m). (a) What is thefrequency? (b) What is the
B_vec-field amplitude? (c) What is the intensity? (d) What average
force does this...
A plane sinusoidal electromagnetic wave in air has a wavelength
of and an -field amplitude of .
1.041010 T
5.75108
Also accepted: 5.77108
W/m2
cm V/mm
m/s
= 8.31108 Hz
= 1.671010 T
m/s
= 3.32106 W/m2
3.84 cm E 1.35 V/m
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Part A
What is the frequency?
ANSWER:
Part B
What is the -field amplitude?
ANSWER:
Part C
What is the intensity?
ANSWER:
Part D
What average force does this radiation exert on a totally
absorbing surface with area perpendicular to the direction of
propagation?
ANSWER:
Problem 32.29Description: A standing electromagnetic wave in a
certain material has a frequency of f. The nodal planes of B_vec
are separated by a distance ofDeltax. (a) Find the wavelength of
the wave in this material. (b) Find the distance between adjacent
nodal planes of ...
A standing electromagnetic wave in a certain material has a
frequency of 2.701010 . The nodal planes of are separated by a
distance of 3.45 .
Part A
Find the wavelength of the wave in this material.
ANSWER:
Part B
Find the distance between adjacent nodal planes of the
field.
ANSWER:
Part C
Find the speed of propagation of the wave.
ANSWER:
= 7.81109 f Hz
B
= 4.50109 Bmax T
= 2.42103 I W/m2
0.240 m2
= 1.931012 Fav N
Hz B mm
= 6.90 mm
E
= 3.45 mm
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Problem 32.47Description: (a) A circular loop of wire can be
used as a radio antenna. If an antenna with a diameter of d is
located a distance of r away from a from asource with a total power
of P at a frequency of f, what is the maximum emf induced in the
loop? (Assume...
Part A
A circular loop of wire can be used as a radio antenna. If an
antenna with a diameter of 17.0 is located a distance of 2.30 away
from a from asource with a total power of 61.0 at a frequency of
101 , what is the maximum emf induced in the loop? (Assume that the
plane of theantenna loop is perpendicular to the direction of the
radiation's magnetic field and that the source radiates uniformly
in all directions.)
Use 3.00108 for the speed of light in a vacuum and 1.26106 for
the permeability of free space.
ANSWER:
Problem 32.53: Global Positioning System (GPS)Description: 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) sinusoidal electromagnetic signal at two
frequencies, one of which is 1575.42 MHz. Assume that a...
The GPS network consists of 24 satellites, each of which makes
two orbits around the earth per day. Each satellite transmits a
50.0 (or even less)sinusoidal electromagnetic signal at two
frequencies, one of which is 1575.42 . Assume that a satellite
transmits half of its power at each frequencyand that the waves
travel uniformly in a downward hemisphere.
Part 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 thesatellite.)
ANSWER:
Part B
What is the amplitude of the electric field at the GPS receiver
in part (a)?
ANSWER:
Part C
What is the amplitude of the magnetic field at the GPS receiver
in part (a)?
ANSWER:
Part D
How long does it take the signal to reach the receiver?
ANSWER:
= 1.86108 m/s
cm kmkW MHz
m/s N/A2
= 4.00102 V
WMHz
= 9.751015 I W/m2
= 2.71106 Emax N/C
= 9.031015 Bmax T
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Part E
If the receiver is a square panel 1.00 on a side that absorbs
all of the beam, what average pressure does the signal exert on
it?
ANSWER:
Part F
What wavelength must the receiver be tuned to?
ANSWER:
Solar SailDescription: Find the force due to radiation pressure
on a solar sail. Then, find the area density needed for effective
propulsion.
A solar sail allows a spacecraft to use radiation pressure for
propulsion, similar to the way wind propels a sailboat. The sails
of such spacecraft are madeout of enormous reflecting panels. The
area of the panels is maximized to catch the largest number of
incident photons, thus maximizing the momentumtransfer from the
incident radiation.If such a spacecraft were to be simply pushed
away from a star by the incident photons, the force of the
radiation pressure would have to be be greaterthan the
gravitational attraction from the star emitting the photons. The
critical parameter is the area density (mass per unit area) of the
sail.
Part A
Consider a perfectly reflecting mirror oriented so that solar
radiation of intensity is incident upon, and perpendicular to, the
reflective surface of themirror. If the mirror has surface area ,
what is , the magnitude of the average force due to the radiation
pressure of the sunlight on the mirror?
Express your answer in terms of the intensity , the mirror's
surface area , and the speed of light .
Hint 1. How to approach the problem
Radiation pressure arises from the photon momentum transfer as
the photons strike the mirror. Thus, if you find an expression for
the totalmomentum transferred to the mirror by the photons that
strike it, you can determine the average force exerted on the
mirror. Notice that whenwriting an expression for the momentum
transfer you'll need to take into account the fact that the mirror
reflects the photons, rather than absorbsthem.
Hint 2. Find the total momentum transfer
What is the total momentum transferred to the mirror by the
photons in a time interval ?
Express your answer in terms of the time interval , the
intensity , the mirror's surface area , and the speed of light
.
Hint 1. Energy of the photons and their momentum
The momentum of a photon can be expressed in terms of the photon
energy as
,
where is the speed of light in vacuum. This ratio also holds for
the total momentum and energy of the photons striking the
mirror.
Hint 2. Radiation intensity and energy
The total energy of the photons striking the mirror during a
time interval is given by
,
where is the intensity of the radiation and is the surface area
of the mirror.
= 6.73102 t s
cm
= 3.251023 Pav Pa
= 0.190 m
IA Frad
I A c
p t
t I A c
p U
p= Ucc
t
U = IAt
I A
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Hint 3. Reflection vs. absorption
When an object absorbs a photon of energy , it receives momentum
equal to . When an object reflects a photon of energy , theobject
must not only stop the photon (as is the case when the photon is
absorbed) but also send it back in the opposite direction. Thus,the
total momentum transfer for photon reflection is twice as much as
in the case of photon absorption.
ANSWER:
Hint 3. Force and change in momentum
Let be the total momentum transferred to the mirror by the
photons that strike the mirror during a time interval . Then the
magnitude ofthe average force exerted on the mirror is
ANSWER:
To solve the second part of this problem you will need to know
the following:
the mass of the sun, ,the intensity of sunlight as a function of
the distance, , from the sun,
,
andthe gravitational constant .
Part B
Suppose that the mirror described in Part A is initially at rest
a distance away from the sun. What is the critical value of area
density for the mirror atwhich the radiation pressure exactly
cancels out the gravitational attraction from the sun?
Express your answer numerically, to two significant figures, in
units of kilograms per meter squared.
Hint 1. Find the force due to gravity
Suppose the mirror has mass . Find a general expression for ,
the magnitude of the gravitational force due to the sun that acts
on the
mirror.
Express your answer symbolically in terms of the gravitational
constant , the mass of the sun, , the mass of the mirror, , andthe
mirror's distance from the sun, .
ANSWER:
Hint 2. Solving for area density
By equating the force due to the sun's radiation ( found in Part
A) and the force due to the sun's gravitational pull, you should be
able to
U U/c U
= p
p t
= .Fradpt
= Frad
= 2.0 kgMsun 1030
R
(R) =Isun 3.2 W1025
R2
G = 6.67 /(kg )1011 m3 s2
R
M Fgrav
G Msun M
R
=
Also accepted:
Fgrav
Frad
1/ 2
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solve for the area density of the mirror. Note that the
expression for the intensity, given in the problem, has a factor of
, just like theexpression for the gravitational force, so the
critical value of the area density turns out to be independent of
.
ANSWER:
In selecting the material for a solar sail, area density,
strength, and reflectivity are the principal concerns. Given a
representative thickness of thesail of 1 , one of the few currently
existing materials with a sufficiently low density and high
strength can be made from carbon fibers. These
have a density of 1.60 , roughly one-fifth that of iron.
Problem 32.48Description: A source of sinusoidal electromagnetic
waves radiates uniformly in all directions. At d from this source,
the amplitude of the electric fieldis measured to be E. (a) What is
the electric-field amplitude at a distance of r from the
source?
A source of sinusoidal electromagnetic waves radiates uniformly
in all directions. At 15.0 from this source, the amplitude of the
electric field is measuredto be 1.30 .
Part A
What is the electric-field amplitude at a distance of 30.0 from
the source?
ANSWER:
Problem 32.52Description: The 19th-century inventor Nikola Tesla
proposed to transmit electric power via sinusoidal electromagnetic
waves. Suppose power is to betransmitted in a beam of
cross-sectional area 100 m^2. (a) What electric-field amplitude is
required to transmit...
The 19th-century inventor Nikola Tesla proposed to transmit
electric power via sinusoidal electromagnetic waves. Suppose power
is to be transmitted in abeam of cross-sectional area .
Part A
What electric-field amplitude is required to transmit an amount
of power equal to that handled by modern transmission lines (that
carry voltages andcurrents of and )?
ANSWER:
Part B
What is the amplitude of the magnetic field in the wave?
ANSWER:
Electric and Magnetic Field Vectors Conceptual
QuestionDescription: Short conceptual questions about the direction
the electromagnetic wave, electric field, and magnetic field
vectors point.
Part A
1/R2
R
mass/area = 1.60103 kg/m2
mg/cm3
mN/C
cm
= = 65.0 Emax N/C
100 m2
500 kV 1000 A
= 6.14104 Emax V/m
= 2.05104 Bmax T
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The electric and magnetic field vectors at a specific point in
space and time are illustrated.Based on this information, in what
direction does the electromagnetic wave propagate?
Hint 1. Right-hand rule for electromagnetic wave velocity
In an electromagnetic wave, the electric and magnetic field
vectors are perpendicular to each other. The wave propagates in a
directionperpendicular to both of the field vectors. Since the two
field vectors define a two-dimensional plane, there are two
distinct directions that areperpendicular to the plane. The
right-hand rule specifies in which of these two directions the wave
travels.To employ the right hand rule, do the following:
1. Point the fingers of your right hand in the direction of the
electric field vector.2. Rotate your hand until you can curl your
fingers in the direction of the magnetic field vector.
The direction of your thumb is then the direction of the
velocity of the electromagnetic wave.
If the electric and magnetic field vectors at a specific point
in space and time are as shown below, applying the right-hand rule
should result inyour thumb pointing downward, in the y direction.
Therefore, the velocity of the electromagnetic wave is in the y
direction.
ANSWER:
Part B
The electric and magnetic field vectors at a specific point in
space and time are illustrated. ( and are in the xy plane. Both
vectors make 45 angles with the y axis.) Based on
this information, in what direction does the electromagnetic
wave propagate?
+x
x
+y
y
+z
z
at a +45 angle in the xy plane
E B +
-
ANSWER:
Part C
The magnetic field vector and the direction of propagation of an
electromagnetic wave are illustrated. Based on this information, in
what direction doesthe electric field vector point?
Hint 1. Working backward with the right-hand rule
Since the velocity of the wave is given, the orientation of your
right thumb is known. Placing your right thumb along the +x axis
should inform youthat the electric field vector must be in the yz
plane. Since the electric field must also be perpendicular to the
magnetic field, and be "curlable"into the magnetic field, only one
option remains for the orientation of the electric field
vector.
ANSWER:
+x
x
+y
y
+z
z
at a 45 angle in the xy plane
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Part D
The electric field vector and the direction of propagation of an
electromagnetic wave are illustrated. ( is in xz plane and makes a
45 angle with the x axis.) Based on this information, in what
direction does the magnetic field vector point?
Hint 1. Working backward with the right-hand rule
Since the velocity of the wave is given, the orientation of your
right thumb is known. Placing your right thumb along the +z axis
should inform youthat the magnetic field vector must be in the xz
plane. With your fingers in the direction of the electric field,
there is only one orientation of themagnetic field that your
fingers can "curl" into.
ANSWER:
Traveling Electromagnetic WaveDescription: Understand the
standard formula for a traveling E&M wave
Learning Goal:
To understand the formula representing a traveling
electromagnetic wave.
Light, radiant heat (infrared radiation), X rays, and radio
waves are all examples of traveling electromagnetic waves.
Electromagnetic waves comprisecombinations of electric and magnetic
fields that are mutually compatible in the sense that the changes
in one generate the other.The simplest form of a traveling
electromagnetic wave is a plane wave. For a wave traveling in the x
direction whose electric field is in the y direction, theelectric
and magnetic fields are given by
,
+x
x
+y
y
+z
z
at a +45 angle in the xz plane
E +
+x
x
+y
y
+z
z
at a 45 angle in the xz plane
= sin(kx t)E E0 j^
= sin(kx t) 0 ^
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.This wave is linearly polarized in the y direction.
Part A
In these formulas, it is useful to understand which variables
are parameters that specify the nature of the wave. The variables
and are the__________ of the electric and magnetic fields.
Choose the best answer to fill in the blank.
Hint 1. What are parameters?
Parameters are variables in any function that determine the
characteristics of a particular wave. For a wave these can include
the amplitude,frequency, wavelength, period, velocity, etc. of the
wave, depending on the particular mathematical formulation.
ANSWER:
Part B
The variable is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
Part C
The variable is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
Part D
What is the mathematical expression for the electric field at
the point at time ?
ANSWER:
= sin(kx t)B B0 k^
E0 B0
maxima
amplitudes
wavelengths
velocities
velocity
angular frequency
wavelength
k
wavenumber
wavelength
velocity
frequency
x = 0,y = 0,z t
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Part E
For a given wave, what are the physical variables to which the
wave responds?
Hint 1. What are independent variables?
The independent variables are those that may be freely varied
over the defined range of the function to control its value. The
formula wouldordinarily be plotted as a function of these
variables, one of which would be across the x axis in a typical
plot.
ANSWER:
This is a plane wave; that is, it extends throughout all space.
Therefore it exists for any values of the variables and and can be
considered afunction of , , , and . Being an infinite plane wave,
however, it is independent of these variables. So whether they are
considered independentvariables is a question of semantics.
When you appreciate this you will understand the conundrum
facing the young Einstein. If he traveled along with this wave
(i.e., at the speed oflight ), he would see constant electric and
magnetic fields extending over a large region of space with no time
variation. He would not see anycurrents or charge, and so he could
not see how these fields could satisfy the standard electromagnetic
equations for the production of fields.
Part F
What is the wavelength of the wave described in the problem
introduction?
Express the wavelength in terms of the other given variables and
constants like .
Hint 1. Finding the wavelength
The wave described in the introduction is sinusoidal. If we let
, then the spatial dependence of the wave is given by .
Thewavelength is defined to be the length in the x direction within
which the wave repeats itself. Mathematically, we require
. To find , recall that the sine function repeats itself when
its argument changes by : .
ANSWER:
Part G
What is the period of the wave described in the problem
introduction?
= sin(t)E E0 j^
= sin(t)E E0 k^
= 0E
= sin(kz t)E E0 i^
= sin(kz t)E E0 j^
only
only
only
only
and
and
and
and
x
t
k
x t
x k
t
k
y z
x y z t
c
t = 0 sin(kx)
sin(kx) = sin(kx+ k) 2 sin() = sin(+ 2)
=
T
-
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Express the period of this wave in terms of and any
constants.
ANSWER:
Part H
What is the velocity of the wave described in the problem
introduction?
Express the velocity in terms of quantities given in the
introduction (such as and ) and any useful constants.
Hint 1. How to find
You have found the wavelength and the period of this wave.
Express the velocity in terms of these two quantities: .
ANSWER:
If this electromagnetic wave were traveling in a vacuum its
velocity would be equivalent to , the vacuum speed of light.
= T
v
k
v
v = /T
= v
c