Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 finitesized wave).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§22.2 Maxwell’s Equations
Gauss’s Law
Gauss’s Law for magnetism
Faraday’s Law
Ampère-Maxwell Law
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§22.3 Antennae
An electric field parallel to an antenna (electric dipole)
will “shake” electrons and produce an AC current.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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?
N
Looking down from
above the Electric
Dipole antenna
W
E
S
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§22.4 The EM Spectrum
EM waves of any frequency can exist.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The EM Spectrum:
Energy increases with increasing frequency.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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.
c

1
 0 0
8.85 10
1
12

C 2 /Nm 2 4 10 7 Tm/A

 3.00 108 m/s
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 2.25108 m/s.
Fizeau’s experiment of 1849 measured the value to be
about 3108 m/s. (done before Maxwell’s work)
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
When light travels though a material medium, its speed is
reduced.
c
v
n
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
A dispersive medium is one in which the index of refraction
depends on the wavelength of light.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The fields are related by the relationship
E ( x, y, z , t )  cB( x, y, z , t )
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The direction of propagation is given by
E B.
The wave carries one-half of its energy in its electric field
and one-half in its magnetic field.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Ez  Em sin ky  t   
phase
constant
The
amplitude
wave
number
k
angular
frequency
2
  2f

The wave speed is c  f 

k
.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 22.27): The electric field of an EM
wave is given by:


E z  Em sin  ky  t  
6

Ex  0
Ey  0
(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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) Write expressions for the magnetic field of this wave.
EB
must be in the +y-direction
(E is in the z-direction).
Therefore, B must be along the x-axis.
Bz  0, B y  0


Bx  Bm sin  ky  t  
6

Em
with Bm 
c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§22.7 Energy Transport by EM
Waves
The intensity of a wave is
Pav
I
.
A
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Also,
E uavV uav Ax
I


 uav c
At At
At
where uav is the average energy density (energy per unit
volume) contained in the wave.
For EM waves:
uav   0 E
2
rms

1
0
2
Brms
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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?

Pav  IA  I 4R 2



 4 1400 W/m 1.50 10 m
2
11

2
 4.0 10 26 W
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) What is the intensity of sunlight incident on Mercury, which
is 5.81010 m from the Sun?
Pav
Pav
I

A 4r 2
4.0 10 26 W

4 5.8 1010 m


2
 9460 W/m 2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What if the EM waves strike at non-normal incidence?
Replace A with Acos.

Pav  IA cos 
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The EM waves emitted by an antenna are polarized; the Efield 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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
Direction of
linear
polarization
Transmission axis
of 2nd polarizer.
45
I 2  I1 cos 2 
1
1  2
  I 0  cos 45  I 0
4
2 
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§22.9 The Doppler Effect
For EM waves, the Doppler shift formula is
fo  f s
v
c
v
1
c
1
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
If the source and observer are approaching each other, then
v is positive, and v is negative if they are receding.
When v/c<<1, the previous expression can be approximated
as:
 v
f o  f s 1  
 c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
 v
f o  f s 1  
 c
c / o
s
v fo
 1 
 1   1  0.0023
c fs
c / s
o
v  6.8 105 m/s  680 km/s
Star is receding.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Summary
•Maxwell’s Equations
•EM Spectrum
•Properties of EM Waves
•Energy Transport by EM Waves
•Polarization
•Doppler Effect
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