Assignment 24: Electromagnetic Waves

Due: 8:00am on Wednesday, April 11, 2012

Note: To understand how points are awarded, read your instructor's Grading Policy.

Traveling Electromagnetic 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

,

.

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.

Hint A.1 What are parameters?

Hint not displayed

Choose the best answer to fill in the blank.

ANSWER:

maxima

amplitudes

wavelengths

velocities

Correct

Part B

The variable is called the __________ of the wave.

Choose the best answer to fill in the blank.

ANSWER:

velocity

angular frequency

wavelength

Correct

Part C

The variable is called the __________ of the wave.

Choose the best answer to fill in the blank.

ANSWER:

wavenumber

wavelength

velocity

frequency

Correct

Part D

What is the mathematical expression for the electric field at the point at time ?

ANSWER:

Correct

Part E

For a given wave, what are the physical variables to which the wave responds?

Hint E.1 What are independent variables?

Hint not displayed

ANSWER:

only

only

only

only

and

and

and

and

Correct

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 a function of , , , and . 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 ), 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 of the wave described in the problem introduction?

Hint F.1 Finding the wavelength

Hint not displayed

Express the wavelength in terms of the other given variables and constants like .

ANSWER:

= Correct

Part G

What is the period of the wave described in the problem introduction?

Express the period of this wave in terms of and any constants.

ANSWER:

= Correct

Part H

What is the velocity of the wave described in the problem introduction?

Hint H.1 How to find

Hint not displayed

Express the velocity in terms of quantities given in the introduction (such as and ) and any useful constants.

ANSWER:

=

Correct

If this electromagnetic wave were traveling in a vacuum its velocity would be equivalent to , the vacuum speed of

light.

Electric and Magnetic Field Vectors Conceptual Question

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 A.1 Right-hand rule for electromagnetic wave velocity

Hint not displayed

ANSWER:

+x

–x

+y

–y

+z

–z

at a +45 angle in the xy plane

Correct

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?

ANSWER:

+x

–x

+y

–y

+z

–z

at a –45 angle in the xy plane

Correct

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 C.1 Working backward with the right-hand rule

Hint not displayed

ANSWER:

+x

–x

+y

–y

+z

–z

at a +45 angle in the xz plane

Correct

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 D.1 Working backward with the right-hand rule

Hint not displayed

ANSWER:

+x

–x

+y

–y

+z

–z

at a –45 angle in the xz plane

Correct

Exercise 32.12

An electromagnetic wave has a magnetic field given by .

Part A

In which direction is the wave traveling?

ANSWER:

+x-direction

-x-direction

+y-direction

Correct

Part B

What is the frequency of the wave?

ANSWER:

= 6.59×10

11

Correct

Part C

Write the vector equation for .

ANSWER:

Correct

Problem 32.38

Consider a sinusoidal electromagnetic wave with fields and , with

. If and are to satisfy equations and , find and

.

Part A

Express your answer in terms of the appropriate constants ( , , , and ).

ANSWER:

=

Correct

Part B

Express your answer in terms of the appropriate constants ( , , , and ).

ANSWER:

= 0

Correct

Triangle Electromagnetic Wave

Learning Goal: To show how a propagating triangle electromagnetic wave can satisfy Maxwell's equations if the

wave travels at speed c.

Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves.

Electromagnetic waves consist of mutually compatible combinations of electric and magnetic fields ("mutually

compatible" in the sense that changes in the electric field generate the magnetic field, and vice versa).

The simplest form for a traveling electromagnetic wave is a plane wave. One particularly simple form for a plane wave

is known as a "triangle wave," in which the electric and magnetic fields are linear in position and time (rather than

sinusoidal). In this problem we will investigate a triangle wave traveling in the x direction whose electric field is in the

y direction. This wave is linearly polarized along the y axis; in other words, the electric field is always directed along

the y axis. Its electric and magnetic fields are given by the following expressions:

and ,

where , , and are constants. The constant , which has dimensions of length, is introduced so that the constants

and have dimensions of electric and magnetic field respectively. This wave is pictured in the figure at time .

Note that we have only drawn the field vectors along the x axis. In

fact, this idealized wave fills all space, but the field vectors only vary in the x direction.

We expect this wave to satisfy Maxwell's equations. For it to do so, we will find that the following must be true:

1. The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field.

2. The wave must travel at a particular velocity (namely, the speed of light).

Part A

What is the propagation velocity of the electromagnetic wave whose electric and magnetic fields are given by the

expressions in the introduction?

Hint A.1 Phase velocity

Hint not displayed

Express in terms of and the unit vectors , , and . The answer will not involve ; we have not yet shown that

this wave travels at the speed of light.

ANSWER:

= Correct

In the next few parts, we will use Faraday's law of induction to find a relationship between and .

Faraday's law relates the line integral of the electric field around a closed loop to the rate of change in magnetic flux

through this loop:

.

Part B

To use Faraday's law for this problem, you will need to constuct a suitable loop, around which you will integrate the

electric field. In which plane should the loop lie to get a nonzero electric field line integral and a nonzero magnetic

flux?

ANSWER:

the xy plane

the yz plane

the zx plane

Correct

Part C

Consider the loop shown in the figure. It is a square loop with sides of length , with one corner at the origin and

the opposite corner at the coordinates , . Recall that

. What is the value of the line integral of the electric field around loop at arbitrary time ?

Hint C.1 Integrating along segments 1 and 2

Hint not displayed

Hint C.2 Integrating along segments 3 and 4

Hint not displayed

Hint C.3 Integrating around the entire loop

Hint not displayed

Express the line integral in terms of , , , , and/or .

ANSWER:

= Correct

Part D

Recall that . Find the value of the magnetic flux through the surface in the xy plane that is

bounded by the loop , at arbitrary time .

Hint D.1 Simplifying the integrand

Hint not displayed

Hint D.2 Evaluating the integral

Hint not displayed

Express the magnetic flux in terms of , , , , and/or .

ANSWER:

= Correct

Part E

Now use Faraday's law to establish a relationship between and .

Hint E.1 Using Faraday's law

Hint not displayed

Express in terms of and other quantities given in the introduction.

ANSWER:

= Correct

If the electric and magnetic fields given in the introduction are to be self-consistent, they must obey all of Maxwell's

equations, including the Ampère-Maxwell law. In these last few parts (again, most of which are hidden) we will use the

Ampère-Maxwell law to show that self-consistency requires the electromagnetic wave described in the introduction to

propagate at the speed of light.

The Ampère-Maxwell law relates the line integral of the magnetic field around a closed loop to the rate of change in

electric flux through this loop:

.

In this problem, the current is zero. (For to be nonzero, we would need charged particles moving around. In this

problem, there are no charged particles present. We assume that the electromagnetic wave is propagating through a

vacuum.)

Part F

To use the Ampère-Maxwell law you will once again need to construct a suitable loop, but this time you will integrate

the magnetic field around the loop. In which plane should the loop lie to get a nonzero magnetic field line integral and

hence nonzero electric flux?

ANSWER:

the xy plane

the yz plane

the zx plane

Correct

Part G

Use the Ampère-Maxwell law to find a new relationship between and .

Hint G.1 How to approach the problem

Hint not displayed

Hint G.2 Find an expression for the left-hand side of the equation

Hint not displayed

Hint G.3 Find an expression for the right-hand side of the equation

Hint not displayed

Hint G.4 Use the Ampère-Maxwell law

Hint not displayed

Express in terms of , , , and other quantities given in the introduction.

ANSWER:

= Correct

Part H

Finally we are ready to show that the electric and magnetic fields given in the introduction describe an

electromagnetic wave propagating at the speed of light. If the electric and magnetic fields are to be self-consistent,

they must obey all of Maxwell's equations. Using one of Maxwell's equations, Faraday's law, we found a certain

relationship between and . You derived this in Part E. Using another of Maxwell's equations, the Ampère-

Maxwell law, we found what appears to be a different relationship between and . You derived this in Part I. If

the results of Parts E and I are to agree, what does this imply that the speed of propagation must be?

Express in terms of only and .

ANSWER:

=

Correct

You have just worked through the details of one of the great triumphs of physics: Maxwell's equations predict a form

of traveling wave consisting of a matched pair of electric and magnetic fields moving at a very high velocity

. We can measure and independently in the laboratory, and these experimentally determined values

lead to a speed of , the speed of light . After thousands of years of speculation about the nature of

light, Maxwell had developed a plausible and quantitatively testable theory about it.

Faraday had a hunch that light and magnetism were related, as demonstrated by the Faraday effect. (Glass, put in a

large magnetic field, will rotate the plane of polarization of light that passes through it.) Now Maxwell had predicted

an electromagnetic wave with the following properties:

1. It was transverse, with two possible polarizations (which agreed with an already known characteristic of light).

2. It had an extraordinarily high velocity (relative to waves in air or on strings) that agreed with the

experimentally determined value for the speed of light.

Any doubt that light waves were in fact electromagnetic waves vanished as various optical phenomena (such as the

behavior of electromagnetic waves at glass surfaces) were predicted and found to agree with the behavior of light.

This theory showed that lower frequency waves could be created and detected by their interactions with currents in

wires (later called antennas) and paved the way to the creation and detection of radio waves.

The Electromagnetic Spectrum

Electromagnetic radiation is more common than you think. Radio and TV stations emit radio waves when they

broadcast their programs; microwaves cook your food in a microwave oven; dentists use X rays to check your teeth.

Even though they have different names and different applications, these types of radiation are really all the same thing:

electromagnetic (EM) waves, that is, energy that travels in the form of oscillating electric and magnetic fields.

Consider the following:

radio waves emitted by a weather radar system to detect raindrops and ice crystals in the atmosphere to study

weather patterns;

microwaves used in communication satellite transmissions;

infrared waves that are perceived as heat when you turn on a burner on an electric stove;

the multicolor light in a rainbow;

the ultraviolet solar radiation that reaches the surface of the earth and causes unprotected skin to burn; and

X rays used in medicine for diagnostic imaging.

Part A

Which of the following statements correctly describe the various forms of EM radiation listed above?

Hint A.1 The electromagnetic spectrum

Hint not displayed

Hint A.2 Frequency and wavelength of an EM wave

Hint not displayed

Check all that apply.

ANSWER:

They have different wavelengths.

They have different frequencies.

They propagate at different speeds through a vacuum depending on their frequency.

They propagate at different speeds through nonvacuum media depending on both their frequency

and the material in which they travel.

They require different media to propagate.

Correct

The frequency and wavelength of EM waves can vary over a wide range of values. Scientists refer to the full range of

frequencies that EM radiation can have as the electromagnetic spectrum.

Electromagnetic waves are used extensively in modern technology. Many devices are built to emit and/or receive EM

waves at a very specific frequency, or within a narrow band of frequencies. Here are some examples followed by their

frequencies of operation:

garage door openers: 40.0 ,

standard cordless phones: 40.0 to 50.0 ,

baby monitors: 49.0 ,

FM radio stations: 88.0 to 108 ,

cell phones: 800 to 900 ,

Global Positioning System: 1227 to 1575 ,

microwave ovens: 2450 ,

wireless Internet technology: 2.4 to 2.6 .

Part B

Which of the following statements correctly describe the various applications listed above?

Hint B.1 Frequency and wavelength of an EM wave

Hint not displayed

Hint B.2 Hertz, megahertz, and gigahertz

Hint not displayed

Hint B.3 Meters and kilometers

Hint not displayed

Check all that apply.

ANSWER:

All these technologies use radio waves, including low-frequency microwaves.

All these technologies use radio waves, including high-frequency microwaves.

All these technologies use a combination of infrared waves and high-frequency microwaves.

Microwave ovens emit in the same frequency band as some wireless Internet devices.

The radiation emitted by wireless Internet devices has the shortest wavelength of all the

technologies listed above.

All these technologies emit waves with a wavelength in the range 0.10 to 10.0 .

All the technologies emit waves with a wavelength in the range 0.01 to 10.0 .

Correct

The frequency band used in wireless technology is strictly regulated by government agencies to avoid undesired

interference effects. In the United States, the Federal Communications Commission (FCC) is responsible for assigning

specific radio frequency bands to different wireless communication systems.

Despite their extensive applications in communication systems, radio waves are not the only form of EM waves present

in our atmosphere. Another form of EM radiation plays an even more important role in our life (and the life of our

planet): sunlight.

The sun emits over a wide range of frequencies; however, the fraction of its radiation that reaches the earth's surface is

mostly in the visible spectrum. (Note that about 35% of the radiation coming from the sun is absorbed directly by the

atmosphere before even reaching the earth's surface.) The earth, then, absorbs this radiation and reemits it as infrared

waves.

Part C

Based on this information, which of the following statements is correct?

Hint C.1 Relation between frequency and wavelength

Hint not displayed

Check all that apply.

ANSWER: The earth absorbs visible light and emits radiation with a shorter wavelength.

The earth absorbs visible light and emits radiation with a longer wavelength.

The earth absorbs visible light and emits radiation with a lower frequency.

The earth absorbs visible light and emits radiation with a higher frequency.

Correct

Even though our atmosphere absorbs a very small amount of visible light, it strongly reflects and absorbs infrared

waves. Therefore the radiation emitted by the earth does not leave the atmosphere. Instead, it is reflected back into it,

contributing to a warming effect known as the greenhouse effect.

Part D

A large fraction of the ultraviolet (UV) radiation coming from the sun is absorbed by the atmosphere. The main UV

absorber in our atmosphere is ozone, . In particular, ozone absorbs radiation with frequencies around 9.38×1014

.

What is the wavelength of the radiation absorbed by ozone?

Hint D.1 Frequency and wavelength of an EM wave

Hint not displayed

Hint D.2 Meters and nanometers

Hint not displayed

Express your answer in nanometers.

ANSWER:

= 320

Correct