20140428170425 Ch 32 A

2005 Pearson Education South Asia Pte Ltd

ELECTROMAGNETISM

FE1001 Physics I NTU - College of Engineering

31. Alternating Current

32. Electromagnetic Waves

37. Relativity

38. Photons, Electrons, and

Atoms

39. The Wave Nature of

Particles


ELECTROMAGNETISM

FE1001 Physics I NTU - College of Engineering

40. Quantum Mechanics

41. Atomic Structure

42. Molecules and

Condensed Matter

43. Nuclear Physics

44. Particle Physics and

Cosmology



Chapter Objectives

• Maxwell’s equations for understanding

electromagnetic waves

• Properties of sinusoidal electromagnetic waves

• Types of electromagnetic waves



Chapter Outline

1. Maxwell’s Equations and Electromagnetic Waves

2. Plane Electromagnetic Waves and the Speed of

Light

3. Sinusoidal Electromagnetic Waves

4. Energy and Momentum in Electromagnetic

Waves

5. Standing Electromagnetic Waves

6. The Electromagnetic Spectrum



32.1 Maxwell’s Equations and Electromagnetic Waves

• An electromagnetic wave is an electromagnetic

disturbance, consisting of time-varying electric and

magnetic fields, that can propagate through space

from one region to another, even when there is no

matter in the intervening region.

• Such a disturbance will have the properties of a

wave.

• The basic principles of electromagnetism can be

expressed in terms of the four equations that we

now call Maxwell’s equations.




• Because the electric and magnetic disturbances

spread or radiate away from the source, the name

electromagnetic radiation is used

interchangeably with the phrase “electromagnetic

waves”.

• Electromagnetic waves can be used for long-

distance communication via devices such as a

radio transmitter.

• Fig. 32.2 shows electric field lines of a point charge

oscillating in simple harmonic motion.




Fig. 32.2



32.2 Plane Electromagnetic Waves and the Speed of Light

• A plane wave is a wave in which at any instant the

electric and magnetic fields are uniform over any

plane perpendicular to the direction of propagation.

• Fig. 32.3 shows an electromagnetic wave front.




Fig. 32.3




• To satisfy Maxwell’s first and second equations, the

electric and magnetic fields must be perpendicular

to the direction of propagation; that is, the wave

must be transverse.

• Fig. 32.4 shows the Gaussian surface for a plane

electromagnetic wave.




Fig. 32.4




• The wave must be consistent with Faraday’s law,

where the wave speed c and the magnitudes of the

perpendicular vectors and are related as in

Eq. (32.4):

• Fig. 32.5 shows the application of Faraday’s law to

a plane wave.

E B




Fig. 32.5




• The wave must also be consistent with Ampere’s law where B, c, and E are related as in Eq. (32.8):

• The basis of the plane wave obeying Maxwell’s equations is Eq. (32.9):

• Fig. 32.6 shows the application of Ampere’s law to a plane wave.




Fig. 32.6




• Electromagnetic waves have the property of

polarization.

• A wave in which is always parallel to a certain

axis is said to be linearly polarized along that axis.

• An alternative derivation of Eq. (32.9) for the speed

of electromagnetic waves includes the derivation of

the wave equation.

• Fig. 32.7 shows how Faraday’s law is also applied

in this alternative method.

E




Fig. 32.7




• Fig. 32.8 shows how

Ampere’s law is also

applied in this

alternative method.

Fig. 32.8



32.3 Sinusoidal Electromagnetic Waves

• In a sinusoidal electromagnetic wave, and at

any point in space are sinusoidal functions of time,

and at any instant of time the spatial variation of the

fields is also sinusoidal.

• Waves passing through a small area at a

sufficiently great distance from a source can be

treated as plane waves (Fig. 32.9).

E B




Fig. 32.9




• Fig. 32.10 shows a linearly polarized sinusoidal

electromagnetic wave traveling in the +x-direction,

where the electric and magnetic fields oscillate in

phase.




Fig. 32.10




• We can describe electromagnetic waves by means

of wave functions.

• Eq. (32.17) shows in vector form the wave function

for a sinusoidal electromagnetic wave propagating

in +x-direction:

• Together with Eq. (32.4), we now get




• Fig. 32.11 shows the electric and magnetic fields of

a wave traveling in the negative x-direction.

• Note that as with the wave traveling in the +x-

direction, at any point the sinusoidal oscillations of

the and fields are in phase. E B




Fig. 32.11



Example 32.1 Fields of a laser beam

A carbon dioxide laser emits a sinusoidal

electromagnetic wave that travels in vacuum in the

negative x-direction. The wavelength is 10.6 m and

the field is parallel to the z-axis, with maximum

magnitude of 1.5 MV/m. Write vector equations for

and as functions of time and position.

E

E

B



Example 32.1 (SOLN)

Identify and Set Up

Eqs. (32.19) describe a wave traveling in the negative

x-direction with along the y-axis – that is, a wave

that is linearly polarized along the y-axis. By contrast,

the wave in this example is linearly polarized along

the z-axis. At points where is in the positive z-

direction, must be in the positive y-direction for the

vector product to be in the negative x-direction

(the direction of propagation). Fig. 32.12 shows a

wave that satisfies these requirements.

E

E

B

E B



Example 32.1 (SOLN)

Fig. 32.12



Example 32.1 (SOLN)

Execute

A possible pair of wave functions that describe the

wave shown in Fig. 32.12 are

The plus sign in the arguments of the cosine

functions indicates that the wave is propagating in the

negative x-direction, as it should. Faraday’s law

requires that Emax = cBmax [Eq. (32.18)], so

To check unit consistency, note that 1 V = 1 Wb/s and

1 Wb/m2 = 1 T.

max maxˆ ˆ( , ) cos( ) ( , ) cos( )E x t kE kx t B x t jB kx t

63max

max 8

1.5 10 /5.0 10

3.0 10 /

E V mB T

c m s



Example 32.1 (SOLN)

We have = 10.6 x 10-6 m, so the wave number and

angular frequency are

Substituting these values into the above wave

functions, we get

6 5

8 5

14

2 / (2 ) /(10.6 10 ) 5.93 10 /

(3.00 10 / )(5.93 10 / )

1.78 10 /

k rad m rad m

ck m s rad m

rad s

6 14

5

3 14

5

ˆ( , ) (1.5 10 / )cos[(1.78 10 / )

(5.93 10 / ) ]

ˆ( , ) (5.0 10 )cos[(1.78 10 / )

(5.93 10 / ) ]

E x t k V m rad s t

rad m x

B x t j T rad s t

rad m x



Example 32.1 (SOLN)

With these equations we can find the fields in the

laser beam at any particular position and time by

substituting specific values of x and t.



Example 32.1 (SOLN)

Evaluate

As we expect, the magnitude Bmax in teslas is much

smaller than the magnitude Emax in volts per meter.

To check the directions of and , note that

is in the direction of . This is as it should be for

a wave that propagates in the negative x-direction.

E B E Bˆ ˆ ˆk j i



Example 32.1 (SOLN)

Evaluate

Our expressions for and are not the

only possible solutions. We could always add a phase

to the arguments of the cosine function, so that kx +

t would become kx + t + . To determine the value

of we would need to know and either as

functions of x at a given time t or as functions of t at a

given coordinate x. However, the statement of the

problem doesn’t include this information.

( , )E x t ( , )B x t

E B




• Other than traveling in a vacuum, electromagnetic

waves can also travel in matter, including

nonconducting materials such as dielectrics.

• The wave speed in a dielectric is given by:



Example 32.2 Electromagnetic waves in different materials

a) While visiting a jewelry store one evening, you hold

a diamond up to the light of a street lamp. The heated

sodium vapor in the street lamp emits yellow light with

a frequency of 5.09 x 1014 Hz. Find the wavelength in

vacuum, the speed of wave propagation in diamond,

and the wavelength in diamond. At this frequency,

diamond has properties K = 5.84 and Km = 1.00.



Example 32.2 Electromagnetic waves in different materials

b) A radio wave with a frequency of 90.0 MHz (in the

FM radio broadcast band) passes from vacuum into

an insulating ferrite (a ferromagnetic material used in

computer cables to suppress radio interference). Find

the wavelength in vacuum, the speed of wave

propagation in the ferrite, and the wavelength in the

ferrite. At this frequency, the ferrite has properties K =

10.0 and Km = 1000.



Example 32.2 (SOLN)

Identify and Set Up

In each case we find the wavelength in vacuum using

c = f. The wave speed v is given in terms of c, the

dielectric constant K, and the relative permeability Km

by Eq. (32.21). Once we know the value of v, we use

v = f to find the wavelength in the material in the

question.



Example 32.2 (SOLN)

Execute

a) The wavelength in vacuum of the sodium light is

The wave speed in diamond is

This is about two-fifths of the speed in vacuum. The

wavelength is proportional to the wave speed and so

is reduced by the same factor:

87

14

3.00 10 /5.89 10 589

5.09 10vacuum

c m sm nm

f Hz

883.00 10 /

1.24 10 /(5.84)(1.00)

diamondm

c m sv m s

KK

8

14

7

1.24 10 /

5.09 10

2.44 10 244

diamonddiamond

v m s

f Hz

m nm



Example 32.2 (SOLN)

Execute

b) Following the same steps as in part (a), we find

that the wavelength in vacuum of the radio wave is

The wave speed in the ferrite is

This is only 1% of the speed of light in a vacuum, so

the wavelength is likewise 1% as large as the

wavelength in vacuum:

8

6

3.00 10 /3.33

90.0 10vacuum

c m sm

f Hz

863.00 10 /

3.00 10 /(10.0)(1000)

ferritem

c m sv m s

KK

92

6

3.00 10 /3.33 10 3.33

90.0 10

ferriteferrite

v m sm cm

f Hz



Example 32.2 (SOLN)

Evaluate

The speed of light in transparent materials like

diamond is typically between c and several percent of

c. As our results in part (b) show, the speed of

electromagnetic waves in dense materials like ferrite

can be far slower than in vacuum.

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20140428170425 Ch 32 A