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Chapter 22: Electromagnetic Waves

•Production of EM waves

•Maxwell’s Equations

•Antennae

•The EM Spectrum

•Speed of EM Waves

•Energy Transport

•Polarization

•Doppler Effect

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§22.1 Production of EM Waves

A stationary charge produces an electric field.

A charge moving at constant speed produces electric and magnetic fields.

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A charge that is accelerated will produce variable electric and magnetic fields. These are electromagnetic waves.

If the charge oscillates with a frequency f, then the resulting EM wave will have a frequency f. If the charge ceases to oscillate, then the EM wave is a pulse (a finite-sized wave).

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§22.2 Maxwell’s Equations

Gauss’s Law

Gauss’s Law for magnetism

Faraday’s Law

Ampère-Maxwell Law

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Gauss’s Law:

Electric fields (not induced) must begin on + charges and end on – charges.

Gauss’s Law for magnetism:

There are no magnetic monopoles (a magnet must have at least one north and one south pole).

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Faraday’s Law:

A changing magnetic field creates an electric field.

Ampère-Maxwell Law

A current or a changing electric field creates a magnetic field.

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When Maxwell’s equations are combined, the solutions are electric and magnetic fields that vary with position and time. These are EM waves.

An electric field only wave cannot exist, nor can a magnetic field only wave.

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§22.3 Antennae

An electric field parallel to an antenna (electric dipole) will “shake” electrons and produce an AC current.

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An EM wave also has a magnetic component. A magnetic dipole antenna can be oriented so that the B-field passes into and out of the plane of a loop, inducing a current in the loop.

The B-field of an EM wave is perpendicular to its E-field and also the direction of travel.

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Example (text problem 22.5): A dipole radio antenna has its rod-shaped antenna oriented vertically. At a point due south of the transmitter, what is the orientation of the emitted wave’s B-field?

Looking down from above the Electric Dipole antenna

N

W

S

E

South of the transmitter, the E-field is directed into/out of the page. The B-field is perpendicular to this direction and also to the direction of travel (South). The B-field must be east-west.

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§22.4 The EM Spectrum

EM waves of any frequency can exist.

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The EM Spectrum:

Energy increases with increasing frequency.

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§22.5 Speed of Light

Maxwell was able to derive the speed of EM waves in vacuum. EM waves do not need a medium to travel through.

m/s 1000.3

Tm/A 104/NmC 1085.8

1

1

8

72212

00

c

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In 1675 Ole Römer presented a calculation of the speed of light. He used the time between eclipses of Jupiter’s Gallilean Satellites to show that the speed of light was finite and that its value was 2.25108 m/s.

Fizeau’s experiment of 1849 measured the value to be about 3108 m/s. (done before Maxwell’s work)

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When light travels though a material medium, its speed is reduced.

n

cv

where v is the speed of light in the medium and n is the refractive index of the medium.

When a wave passes from one medium to another the frequency stays the same, but the wavelength is changed.

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A dispersive medium is one in which the index of refraction depends on the wavelength of light.

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§22.6 Properties of EM Waves

All EM waves in vacuum travel at the “speed of light” c.

Both the electric and magnetic fields have the same oscillation frequency f.

The electric and magnetic fields oscillate in phase.

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The fields are related by the relationship

),,,(),,,( tzyxcBtzyxE

EM waves are transverse. The fields oscillate in a direction that is perpendicular to the wave’s direction of travel. The fields are also perpendicular to each other.

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The direction of propagation is given by .BE

The wave carries one-half of its energy in its electric field and one-half in its magnetic field.

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tkyEE mz sin

The amplitude wave

numberangular frequency

2

k f 2

The wave speed is .k

fc

phase constant

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Example (text problem 22.27): The electric field of an EM wave is given by:

0

0

6sin

y

x

mz

E

E

tkyEE

(a) In what direction is this wave traveling?

The wave does not depend on the coordinates x or z; it must travel parallel to the y-axis. The wave travels in the +y direction.

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(b) Write expressions for the magnetic field of this wave.

BE must be in the +y-direction (E is in the z-direction).

Therefore, B must be along the x-axis.

c

EB

tkyBB

BB

mm

mx

yz

with

6sin

0,0

Example continued:

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§22.7 Energy Transport by EM Waves

The intensity of a wave is .A

PI av

This is a measure of how much energy strikes a surface of area A every second for normal incidence.

Surface

The rays make a 90 angle with the surface.

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cutA

xAu

tA

Vu

tA

EI av

avav

where uav is the average energy density (energy per unit volume) contained in the wave.

Also,

For EM waves:

2rms

0

2rms0

1BEuav

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Example (text problem 22.35): The intensity of the sunlight that reaches Earth’s upper atmosphere is 1400 W/m2.

(a) What is the total average power output of the Sun, assuming it to be an isotropic source?

W100.4

m 1050.1 W/m14004

4

26

2112

2

RIIAPav

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(b) What is the intensity of sunlight incident on Mercury, which is 5.81010 m from the Sun?

2

210

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2

W/m9460

m 108.54

W100.4

4

rP

A

PI avav

Example continued:

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What if the EM waves strike at non-normal incidence?

Replace A with Acos.

cosIAPav

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§22.8 Polarization

A wave on a string is linearly polarized. The vibrations occur in the same plane. The orientation of this plane determines the polarization state of a wave.

For an EM wave, the direction of polarization is given by the direction of the E-field.

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The EM waves emitted by an antenna are polarized; the E-field is always in the same direction.

A source of EM waves is unpolarized if the E-fields are in random directions.

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A polarizer will transmit linear polarized waves in the same direction independent of the incoming wave.

It is only the component of the wave’s amplitude parallel to the transmission axis that is transmitted.

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If unpolarized light is incident on 1 polarizer, the intensity of the light passing through is I= ½ I0.

If the incident wave is already polarized, then the transmitted intensity is I=I0cos2 where is the angle between the incident wave’s direction of polarization and the transmission axis of the polarizer. (Law of Malus)

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Example (text problem 22.40): Unpolarized light passes through two polarizers in turn with axes at 45 to each other. What is the fraction of the incident light intensity that is transmitted?

After passing through the first polarizer, the intensity is ½ of its initial value. The wave is now linearly polarized.

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Transmission axis of 2nd polarizer.

Direction of linear polarization

02

0

212

4

145cos

2

1

cos

II

II

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§22.9 The Doppler Effect

cvcv

ff so

1

1For EM waves, the Doppler shift formula is

where fs is the frequency emitted by the source, fo is the frequency received by the observer, v is the relative velocity of the source and the observer, and c is the speed of light.

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When v/c<<1, the previous expression can be approximated as:

c

vff so 1

If the source and observer are approaching each other, then v is positive, and v is negative if they are receding.

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Example (text problem 22.48): Light of wavelength 659.6 nm is emitted by a star. The wavelength of this light as measured on Earth is 661.1 nm. How fast is the star moving with respect to the Earth? Is it moving toward Earth or away from it?

The wavelength shift is small (<<) so v<<c.

km/s 680m/s 108.6

0023.011/

/1

1

5

v

c

c

f

f

c

v

c

vff

o

s

s

o

s

o

so

Star is receding.

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Summary

•Maxwell’s Equations

•EM Spectrum

•Properties of EM Waves

•Energy Transport by EM Waves

•Polarization

•Doppler Effect