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H13 EM Waves (32.1-5)Due: 11:59pm on Monday, December 9, 2013

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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

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H13 EM Waves (32.1-5) [ Edit ]

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s

m/s

= 210 m

kHzT

c m/s

= 361 m

= 1.74102 m1

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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

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

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

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

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

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

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

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

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

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 ^

.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

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