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PH 222-2C Fall 2012
Electromagnetic Waves
Lectures 21-22
Chapter 33
(Halliday/Resnick/Walker, Fundamentals of Physics 8th edition)
1
Chapter 33
Electromagnetic Waves
Today’s information age is based almost entirely on the physics of
electromagnetic waves. Electromagnetic waves are at the core of many
fields in science and engineering.
In this chapter we introduce fundamental concepts and explore the
properties of electromagnetic waves.
2
Maxwell’s Rainbow
The wavelength/frequency range in which electromagnetic (EM) waves (light)
are visible is only a tiny fraction of the entire electromagnetic spectrum.
Fig. 33-2
Fig. 33-1
3
The Traveling Electromagnetic (EM) Wave, Qualitatively
An LC oscillator causes currents to flow sinusoidally, which in turn produces
oscillating electric and magnetic fields, which then propagate through space as
EM waves.
Next slide
Fig. 33-3
Oscillation Frequency:

1
LC
4
The Traveling Electromagnetic (EM) Wave, Qualitatively
EM fields at P looking back
toward LC oscillator


1. Electric E and magnetic B fields are always
perpendicular to direction in which wave
is traveling  transverse wave (Ch. 16).


2. E is always perpendicular to B.
 
3. E  B always gives direction of wave travel.


4. E and B vary sinusoidally (in time and space)
and are in phase (in step) with each other.
Fig. 33-4
5
Mathematical Description of Traveling EM Waves
Electric Field:
Magnetic Field:
E  Em sin  kx   t 
B  Bm sin  kx   t 
Wave Speed:
c
1
0 0
All EM waves travel a c in vacuum
Wavenumber:
k
EM Wave Simulation
Angular frequency:
2


2

Vacuum Permittivity:  0
Vacuum Permeability:
Fig. 33-5
Em
Amplitude Ratio:
c
Bm
E t 
Magnitude Ratio:
c
B t 
0
6
A Most Curious Wave
• Unlike all the waves discussed in Chs. 16 and 17, EM waves require no
medium through/along which to travel. EM waves can travel through empty
space (vacuum)!
• Speed of light is independent of speed of observer! You could be heading
toward a light beam at the speed of light, but you would still measure c as the
speed of the beam!
c  299 792 458 m/s
7
The Traveling EM Wave, Quantitatively
Induced Electric Field
Changing magnetic fields produce electric fields, Faraday’s law of induction:
 
dB
 E d s   dt
 
 E d s   E  dE  h  Eh  h dE
Fig. 33-6
 B   B  h dx 
dB
dE
dB
 h dE  h dx


dt
dx
dt
E
B

x
t
E
B
 kEm cos  kx   t  and
  Bm cos  kx   t 
x
t
Em
8
c
kEm cos  kx   t    Bm cos  kx   t  
Bm
The Traveling EM Wave, Quantitatively
Induced Magnetic Field
Changing electric fields produce magnetic fields, Maxwell’s law of induction:
Fig. 33-7
 
dE
Bd s  0 0 dt
 Bd s    B  dB  h  Bh  h dB
dE
dE
 E   E  h dx  
 h dx
dt
dt
dE 

  h dB  0 0  h dx

dt


B
E

 0 0
x
t
 kBm cos  kx   t    0 0 Em cos  kx   t 
Em
1
1


cc
Bm 0 0  k  0 0c
1
0 0
9
Energy Transport and the Poynting Vector
EM waves carry energy. The rate of energy transport in an EM wave

is characterized by the Poynting vector, S :
 1  
Poynting Vector: S 
EB
0
The magnitude of S is related to the rate at which energy is transported by
a wave across a unit area at any instant (inst). The unit for S is (W/m2).
 energy/time 
 power 
S 
 

area
area

inst 
inst

The direction of S at any point gives the wave's travel direction
and the direction of energy transport at that point.
10
Energy Transport and the Poynting Vector
 
 
Since E  B  E  B  EB
1
Instantaneous
S
EB
energy flow rate:
0
1
E
S
EB and since B 
c
0
Note that S is a function of time. The time-averaged value for S, Savg is also
called the intensity I of the wave.
 energy/time 
 power 
I  Savg  




area
area

avg 
avg
1
1
2
 E  
 Em2 sin 2  kx   t  
I  Savg 
avg
avg
c0
c0
1 2
Em
I
Erms
Erms 
c0
2
1
1
1  1
2
2
uE   0 E   0  cB    0  
2
2
2   0 0
2
 
B2
 uB
 B  

2 0
 
11
Variation of Intensity with Distance
Consider a point source S that is emitting EM waves isotropically (equally in
all directions) at a rate PS. Assume that the energy of waves is conserved
as they spread from source.
How does the intensity
(power/area) change
with distance r?
power
PS
I

area
4 r 2
Fig. 33-8
12
Radiation Pressure
EM waves have linear momentum as well as energylight can exert pressure.

Sincident
p
U
Total absorption: p 
c

Sreflected

Sincident
p
p
F
t
power energy/time
I

area
area
 U t

A
Total reflection
back along path:
2 U
p 
c
I
pr 
c
2I
pr 
c
U  IA t
IA
F
(total absorption)
c
2 IA
F
(total reflection back along path)
c
F
13
pr 
Radiation Pressure
A
Polarization
The polarization of light describes
how the electric field in the EM wave
oscillates.
Vertically planepolarized (or linearly
polarized)
Fig. 33-10
14
Polarized Light
Unpolarized or randomly polarized light has
its instantaneous polarization direction vary
randomly with time.
Fig. 33-11
One can produce unpolarized light by the
addition (superposition) of two
perpendicularly polarized waves with
randomly varying amplitudes. If the two
perpendicularly polarized waves have fixed
amplitudes and phases, one can produce
different polarizations such as circularly or
elliptically polarized light.
15
Polarizing Sheet
I0
I
Fig. 33-12
Only the electric field component along the polarizing
direction of polarizing sheet is passed (transmitted); the
perpendicular component is blocked (absorbed).
16
Intensity of Transmitted Polarized Light
Intensity of
transmitted light,
unpolarized
incident light:
1
I  I0
2
Since only the component of the
incident electric field E parallel to the
polarizing axis is transmitted.
Etransmitted  E y  E cos
Fig. 33-13
Intensity of
transmitted light,
I
polarized
incident light:
 I 0 cos2 
For unpolarized light,  varies randomly in time:
I   I 0 cos  
2
 I 0  cos  
2
avg
avg
1
 I0
2
17
Reflection and Refraction
Although light waves spread as they move from a source, often we can
approximate its travel as being a straight line  geometrical optics.
What happens when a narrow beam of
light encounters a glass surface?
Law of Reflection
Reflection:
1 '   1
Snell’s Law
Refraction: n2 sin  2
Fig. 33-17
sin  2 
n1
sin 1
n2
n is the index of refraction of the material.
 n1 sin 1
18
Refraction of Light traveling from medium with n1 to
medium with n2
n1
sin  2  sin 1
n2
For light going from n1 to n2:
•
n2 = n1  2 = 1
•
n2 > n1  2<1, light bent toward
normal
•
n2 < n1  2 > 1, light bent away from
normal
Fig. 33-18
19
Chromatic Dispersion
The index of refraction n encountered by light in any medium except vacuum
depends on the wavelength of the light. So if light consisting of different
wavelengths enters a material, the different wavelengths will be refracted
differently  chromatic dispersion.
Fig. 33-19
Fig. 33-20
n2,blue>n2,red
Chromatic dispersion can be good (e.g., used to analyze wavelength
composition of light) or bad (e.g., chromatic aberration in lenses).
20
Chromatic Dispersion
Chromatic dispersion can be good (e.g., used to analyze wavelength
composition of light)
prism
Fig. 33-21
or bad (e.g., chromatic aberration in lenses)
lens
21
Rainbows
Sunlight consists of all visible colors and water is
dispersive, so when sunlight is refracted as it enters
water droplets, is reflected off the back surface, and
again is refracted as it exits the water drops, the range of
angles for the exiting ray will depend on the color of the
ray. Since blue is refracted more strongly than red, only
droplets that are closer to the rainbow center (A) will
refract/reflect blue light to the observer (O). Droplets at
larger angles will still refract/reflect red light to the
observer.
What happens for rays that reflect twice off the back
surfaces of the droplets?
Fig. 33-22
22
Total Internal Reflection
For light that travels from a medium with a larger index of refraction to a
medium with a smaller index of refraction n1>n2  2>1, as 1 increases, 2
will reach 90o (the largest possible angle for refraction) before 1 does.
n2
n1 sin  c  n2 sin 90  n2
n2
Critical Angle:  c  sin
n1
1
n1
Fig. 33-24
Total internal reflection can be used, for
example, to guide/contain light along an
optical fiber.
When 2> c no light is
refracted (Snell’s law does
not have a solution!) so no
light is transmitted  Total
Internal Reflection
23
Polarization by Reflection
When the refracted ray is perpendicular to the reflected ray, the electric field
parallel to the page (plane of incidence) in the medium does not produce a
reflected ray since there is no component of that field perpendicular to the
reflected ray (EM waves are transverse).
Applications
1. Perfect window: since parallel polarization
is not reflected, all of it is transmitted
2. Polarizer: only the perpendicular component
is reflected, so one can select only this
component of the incident polarization
Brewster’s Law
 B   r  90
n1 sin  B  n2 sin  r
n1 sin  B  n2 sin  90   B   n2 cos B
Fig. 33-27
1 n2
In which direction does light reflecting Brewster Angle:  B  tan
n1 24
off a lake tend to be polarized?
s
p