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University Physics with Modern Physics, 13e
Young/Freedman
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H13 EM Waves (32.1-5)
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H13 EM Waves (32.1-5)
Due: 11:59pm on Monday, December 8, 2014
You will receive no credit for items you complete after the assignment is due. Grading Policy
Electric and Magnetic Field Vectors Conceptual Question
Description: Short conceptual questions about the direction the electromagnetic wave, electric field, and magnetic field
vectors point.
Part A
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 direction perpendicular to both of the field vectors. Since the two field vectors define a twodimensional plane, there are two distinct directions that are perpendicular 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 in your thumb pointing downward, in the –y direction. Therefore, the velocity of the
electromagnetic wave is in the –y direction.
ANSWER:
+x
–x
+y
–y
+z
–z
at a +45∘ angle in the xy plane
Part B
⃗
⃗
The electric and magnetic field vectors at a specific point in space and time are illustrated. (E and B 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?
ANSWER:
+x
–x
+y
–y
+z
–z
at a –45∘ angle in the xy plane
Part C
The magnetic field vector and the direction of propagation of an electromagnetic wave are illustrated. Based on this
information, in what direction does the 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 you that 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 xz plane
Part D
⃗
The electric field vector and the direction of propagation of an electromagnetic wave are illustrated. (E 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 you that 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 the magnetic field that your fingers can "curl"
into.
ANSWER:
+x
–x
+y
–y
+z
–z
at a –45∘ angle in the xz plane
Exercise 32.3
Description: A sinusoidal electromagnetic wave is propagating in a vacuum in the +z-direction. (a) If at a particular
instant and at a certain point in space the electric field is in the +x-direction and has a magnitude of E, what is the
magnitude of the magnetic ...
A sinusoidal electromagnetic wave is propagating in a vacuum in the +z-direction.
Part A
If at a particular instant and at a certain point in space the electric field is in the +x-direction and has a magnitude of
3.40V/m , what is the magnitude of the magnetic field of the wave at this same point in space and instant in time?
ANSWER:
B=
= 1.13×10−8
T
Part B
What is the direction of the magnetic field?
ANSWER:
- y - direction
- z - direction
+ y - direction
Traveling Electromagnetic Wave
Description: 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 comprise combinations 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, the electric and magnetic fields are given by
E ⃗ = E 0 sin(kx − ωt)^j ,
B⃗ = B 0 sin(kx − ωt)k^ .
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 E 0 and B 0 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:
maxima
amplitudes
wavelengths
velocities
Part B
The variable ω is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
velocity
angular frequency
wavelength
Part C
The variable k is called the __________ of the wave.
Choose the best answer to fill in the blank.
ANSWER:
wavenumber
wavelength
velocity
frequency
Part D
What is the mathematical expression for the electric field at the point x
= 0, y = 0, z at time t?
ANSWER:
E ⃗ = E 0 sin(−ωt)^j
E ⃗ = E sin(−ωt)k^
0
E⃗ = 0
E ⃗ = E 0 sin(kz − ωt)^i
E ⃗ = E 0 sin(kz − ωt)^j
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 would ordinarily be plotted as a function of these variables, one of which would be across
the x axis in a typical plot.
ANSWER:
x only
t only
k only
ω only
x and t
x and k
ω and t
k and ω
This is a plane wave; that is, it extends throughout all space. Therefore it exists for any values of the variables y
and z and can be considered a function of x , y , z , and t. Being an infinite plane wave, however, it is independent
of these variables. So whether they are considered independent variables 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 of light c), he would see constant electric and magnetic fields extending over a large
region of space with no time variation. He would not see any currents 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 λ of the wave described in the problem introduction?
Express the wavelength in terms of the other given variables and constants like π .
Hint 1. Finding the wavelength
The wave described in the introduction is sinusoidal. If we let t = 0, then the spatial dependence of the wave is
given by sin(kx) . The wavelength λ is defined to be the length in the x direction within which the wave repeats
itself. Mathematically, we require sin(kx)
= sin(kx + kλ) . To find λ , recall that the sine function repeats itself
when its argument changes by 2π : sin(θ) = sin(θ + 2π).
ANSWER:
λ=
Part G
What is the period T of the wave described in the problem introduction?
Express the period of this wave in terms of ω and any constants.
ANSWER:
T=
Part H
What is the velocity v of the wave described in the problem introduction?
Express the velocity in terms of quantities given in the introduction (such as ω and k ) and any useful
constants.
Hint 1. How to find v
You have found the wavelength and the period of this wave. Express the velocity in terms of these two
quantities: v = λ/T .
ANSWER:
v=
If this electromagnetic wave were traveling in a vacuum its velocity would be equivalent to c, the vacuum speed of
light.
Exercise 32.10
Description: The electric field of a sinusoidal electromagnetic wave obeys the equation E = (375 (V/m))cos ( (1.99 *
10^7 (rad/m))x+(5.97 * 10^15 (rad/s)) t). (a) What is the amplitude of the electric field of this wave? (b) What is the
amplitude of the...
The electric field of a sinusoidal electromagnetic wave obeys the equation
E = (375V/m) cos[(1.99 × 107 rad/m)x + (5.97 × 1015 rad/s)t].
Part A
What is the amplitude of the electric field of this wave?
ANSWER:
E = 375 V/m
Part B
What is the amplitude of the magnetic field of this wave?
ANSWER:
B = 1.25 µT
Part C
What is the frequency of the wave?
ANSWER:
f = 9.50×1014 Hz
Part D
What is the wavelength of the wave?
ANSWER:
λ = 316 nm
Part E
What is the period of the wave?
ANSWER:
T = 1.05×10−15 s
Part F
Is this light visible to humans?
ANSWER:
No
Yes
Part G
What is the speed of the wave?
ANSWER:
v = 3.00×108 m/s
Exercise 32.13
Description: Radio station WCCO in Minneapolis broadcasts at a frequency of 830 (kHz). At a point some distance
from the transmitter, the magnetic-field amplitude of the electromagnetic wave from WCCO is B_max. (a) Find the
wavelength. (b) Find the wave number. ...
Radio station WCCO in Minneapolis broadcasts at a frequency of 830 kHz. At a point some distance from the transmitter,
the magnetic-field amplitude of the electromagnetic wave from WCCO is 4.63×10−11T .
Part A
Find the wavelength.
ANSWER:
λ = 361 m
Part B
Find the wave number.
ANSWER:
k = 1.74×10−2 m −1
Part C
Calculate the angular frequency.
ANSWER:
ω = 5.22×106 rad/s
Part D
Calculate the electric-field amplitude.
ANSWER:
E=
= 1.39×10−2
V/m
Poynting Flux
Description: Given an expression for the electric field of an EM wave (travelling, in vacuum), choose the correct form of
the magnetic field and compute the Poynting vector. Very simple.
An electromagnetic wave is traveling through vacuum. Its electric field vector is given by
j is the unit vector in the y direction.
where ^
E ⃗ = E 0 sin (kx − ωt)^j ,
Part A
⃗
If B is the amplitude of the magnetic field vector, find the complete expression for the magnetic field vector B of the
wave.
Hint 1. Relative orientation of B⃗ and E⃗ 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's equations.
Hint 2. Orientation of E⃗ and B⃗ 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 the wave. This follows from Maxwell's equations.
Hint 3. Determine the direction of propagation of the wave
In what direction is this wave propagating?
i , ^j , and k^ .
Express your answer in terms of ^
Hint 1. How to approach the problem
The point that has the same phase as (x, t) , that is, phase
where ∆x
= kx − ωt at time t + ∆t , is x + ∆x,
= ω∆t/k.
ANSWER:
Hint 4. Phase relationship between E⃗ and B⃗
In free space, the electric and magnetic field vectors of an electromagnetic wave are exactly in phase. This
follows from Maxwell's equations.
ANSWER:
B 0 sin(kx − ωt)^i
B 0 sin(kx − ωt)^j
B 0 sin(kx − ωt)k^
B 0 cos(kx − ωt)^i
B 0 cos(kx − ωt)^j
B 0 cos(kx − ωt)k^
Part B
⃗ t), that is, the power per unit area associated with the electromagnetic wave
What is the Poynting vector S (x,
described in the problem introduction?
Give your answer in terms of some or all of the variables E 0 , B 0 , k , x , ω, t, and µ0 . Specify the direction of
i , ^j , and k^ as appropriate.
the Poynting vector using the unit vectors ^
Hint 1. Definition of the Poynting vector
⃗
⃗
The Poynting vector S of an electromagnetic wave in vacuum is given in terms of the electric field vector E and
⃗
the magnetic field vector B by the relation
S⃗ =
1
µ0
E⃗ × B⃗ .
ANSWER:
⃗ t) =
S (x,
Also accepted:
,
Solar Sail
Description: 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 made out of enormous reflecting panels. The area of the panels is maximized to catch the largest
number of incident photons, thus maximizing the momentum transfer 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 greater than 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 I is incident upon, and perpendicular to,
the reflective surface of the mirror. If the mirror has surface area A, what is Frad , 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 I , the mirror's surface area A, and the speed of light c.
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 total momentum transferred to the mirror by the photons that strike it, you can determine
the average force exerted on the mirror. Notice that when writing an expression for the momentum transfer you'll
need to take into account the fact that the mirror reflects the photons, rather than absorbs them.
Hint 2. Find the total momentum transfer
What is the total momentum ∆p transferred to the mirror by the photons in a time interval ∆t ?
Express your answer in terms of the time interval ∆t , the intensity I , the mirror's surface area A, and
the speed of light c.
Hint 1. Energy of the photons and their momentum
The momentum p of a photon can be expressed in terms of the photon energy U as
p=
U
c ,
where c 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 ∆t is given by
U = IA∆t ,
where I is the intensity of the radiation and A is the surface area of the mirror.
Hint 3. Reflection vs. absorption
When an object absorbs a photon of energy U , it receives momentum equal to U/c . When an object
reflects a photon of energy U , the object 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:
∆p =
Hint 3. Force and change in momentum
Let ∆p be the total momentum transferred to the mirror by the photons that strike the mirror during a time
interval ∆t . Then the magnitude of the average force exerted on the mirror is
F rad =
∆p
∆t
.
ANSWER:
Frad =
To solve the second part of this problem you will need to know the following:
the mass of the sun, M sun
= 2.0 × 1030 kg,
the intensity of sunlight as a function of the distance, R , from the sun,
I sun (R) =
and
the gravitational constant G
25
3.2×10
R2
W
,
= 6.67 × 10−11 m 3 /(kg ⋅ s 2 ) .
Part B
Suppose that the mirror described in Part A is initially at rest a distance R away from the sun. What is the critical value
of area density for the mirror at which 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 M . Find a general expression for Fgrav , 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 G, the mass of the sun, Msun ,
the mass of the mirror, M , and the mirror's distance from the sun, R .
ANSWER:
Fgrav =
Also accepted:
Hint 2. Solving for area density
By equating the force due to the sun's radiation (Frad found in Part A) and the force due to the sun's
gravitational pull, you should be able to solve for the area density of the mirror. Note that the expression for the
intensity, given in the problem, has a factor of 1/R 2 , just like the expression for the gravitational force, so the
critical value of the area density turns out to be independent of R .
ANSWER:
mass/area = 1.60×10−3
kg/m 2
In selecting the material for a solar sail, area density, strength, and reflectivity are the principal concerns. Given a
representative thickness of the sail of 1 µm, 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 g/cm 3 , roughly one-fifth
that of iron.
Exercise 32.27
Description: If the eye receives an average intensity greater than ## W/m^2, damage to the retina can occur. This
quantity is called the damage threshold of the retina. (a) What is the largest average power (in mW) that a laser beam d
in diameter can have so that ...
If the eye receives an average intensity greater than 100W/m 2 , damage to the retina can occur. This quantity is called the
damage threshold of the retina.
Part A
What is the largest average power (in mW) that a laser beam 1.49mm in diameter can have so that it can be
considered safe to view head-on?
ANSWER:
= 0.174
Pmax =
mW
Part B
What is the maximum value of the electric field for the beam in part A?
ANSWER:
= 274
E max =
V/m
Part C
What is the maximum value of the magnetic field for the beam in part A?
ANSWER:
B max = 9.13×10−7 T
Part D
How much energy would the beam in part A deliver per second to the retina?
ANSWER:
= 0.174
U=
mJ
Part E
Express the damage threshold in W/cm 2 .
ANSWER:
I=
= 1.00×10−2
W/cm 2
Exercise 32.35
Description: An electromagnetic standing wave in a certain material has a frequency of f and a speed of propagation v.
(a) What is the distance between a nodal plane of B_vec and the closest antinodal plane of B_vec? (b) What is the
distance between an antinodal ...
An electromagnetic standing wave in a certain material has a frequency of 1.15×1010Hz and a speed of propagation 2.60×108
m/s .
Part A
What is the distance between a nodal plane of B⃗ and the closest antinodal plane of B⃗ ?
ANSWER:
= 5.65×10−3
∆x 1 =
m
Part B
What is the distance between an antinodal plane of E ⃗ and the closest antinodal plane of B⃗ ?
ANSWER:
= 5.65×10−3
∆x 2 =
m
Part C
⃗
What is the distance between a nodal plane of E ⃗ and the closest nodal plane of B ?
ANSWER:
= 5.65×10−3
∆x 3 =
m
Problem 32.40
Description: 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 the frequency? (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 3.84
Part A
What is the frequency?
ANSWER:
f = 7.81×109 Hz
Part B
⃗
cm and an E ⃗-field amplitude of 1.35 V/m .
What is the B⃗ -field amplitude?
ANSWER:
B max = 4.50×10−9 T
Part C
What is the intensity?
ANSWER:
I = 2.42×10−3 W/m 2
Part D
What average force does this radiation exert on a totally absorbing surface with area 0.240
direction of propagation?
m2 perpendicular to the
ANSWER:
Fav = 1.93×10−12 N
Problem 32.52
Description: The 19th-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 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 a beam of cross-sectional area 100 m 2 .
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 and currents of 500 kV and 1000 A )?
ANSWER:
E max = 6.14×104 V/m
Part B
What is the amplitude of the magnetic field in the wave?
ANSWER:
B max = 2.05×10−4 T
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