Chapter 33
When the amplitude of the oscillator in a series RLC circuit is doubled:
Electromagnetic Waves
A. the impedance is doubled.
Today’s information age is based almost entirely on the physics of
electromagnetic waves. The connection between electric and magnetic fields
to produce light is one of the greatest achievements produced by physics,
and electromagnetic waves are at the core of many fields in science and
engineering.
B. the voltage across the capacitor is halved
C. the capacitive reactance is halved
D. the power factor is halved
E. the current amplitude is doubled
In this chapter we introduce fundamental concepts and explore the
properties of electromagnetic waves.
33- 1
Maxwell’s Rainbow
2
The Travelling Electromagnetic (EM) Wave, Qualitatively
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
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
ω=
33- 3
Fig. 33-1
The Travelling Electromagnetic (EM) Wave, Qualitatively
33- 4
Mathematical Description of Travelling EM Waves
EM fields at P looking back
toward LC oscillator
Electric Field:
r
r
1. Electric E and magnetic B fields always
perpendicular to direction in which wave
is travelling → transverse wave (Ch. 16)
r
r
2. E always perpendicular to B
r r
3. E × B always gives direction of wave travel
r
r
4. E and B vary sinusoidally (in time and space)
and are in phase (in step) with each other
Magnetic Field:
E = Em sin ( kx − ω t )
B = Bm sin ( kx − ω t )
33- 5
1
c=
µ0ε 0
All EM waves travel a c in vacuum
Wavenumber:
k=
EM Wave Simulation
Angular frequency:
2π
λ
ω=
Vacuum Permittivity:
Amplitude Ratio:
Em
=c
Bm
Magnitude Ratio:
E (t )
=c
B (t )
2π
τ
ε0
Vacuum Permeability:
Fig. 33-5
Fig. 33-4
Wave Speed:
µ0
33- 6
1
A Most Curious Wave
The Travelling EM Wave, Quantitatively
• 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)!
Induced Electric Field
Changing magnetic fields produce electric fields, Faraday’s law of induction
r
c = 299 792 458 m/s
=0.9833192622 ft/ns
Fig. 33-6
33- 7
The Travelling EM Wave, Quantitatively
Changing electric fields produce magnetic fields, Maxwell’s law of induction
dΦE
r
∫ Bur d rs = µ ε dt
∫ B d s = − ( B + dB ) − Bh = h dB
0 0
Φ E = ( E )( h dx ) →

→ − h dB = µ0ε 0  h

∂B
∂E
−
= µ0ε 0
∂x
∂t
Fig. 33-7
dΦE
dE
= h dx
dt
dt
dB 
dx

dt 
− kBm cos ( kx − ω t ) = − µ0ε 0ω Em cos ( kx − ω t )
Em
1
1
=
=
=c→c=
Bm µ0ε 0 (ω k ) µ0ε 0c
dΦB
Φ B = ( B )( h dx )
dB
dE
dB
→ h dE = h dx
→
=−
dt
dx
dt
∂E
∂B
=−
∂x
∂t
∂E
∂B
= kEm cos ( kx − ω t ) and
= −ω Bm cos ( kx − ω t )
∂x
∂t
E
33- 8
kEm cos ( kx − ω t ) = −ω Bm cos ( kx − ω t ) → m = c
Bm
Energy Transport and the Poynting Vector
Induced Magnetic Field
r
r
∫ E d s = − dt
r r
∫ E d s = ( E + dE ) − Eh = h dE
• 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!
EM waves carry energy. The rate of energy transport in an EM wave
r
is characterized by the Poynting vector S
r 1 r r
Poynting Vector: S =
E×B
µ0
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

inst  area inst
r
The direction of S at any point gives the wave's travel direction
and the direction of energy transport at that point
1
µ0ε 0
33- 9
Energy Transport and the Poynting Vector
r r
r r
Since E ⊥ B → E × B = EB
1
Instantaneous
S=
EB
energy flow rate:
c µ0
1
E
S=
EB and since B =
µ0
c
33-10
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 energy of waves is conserved as they
spread from source.
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

avg  area avg
1
1
E 2  =
 Em2 sin 2 ( kx − ω t )
I = Savg =
avg
cµ0   avg cµ0 
Erms =
Em
2
I=
1 2
E rms
cµ0
1
1
1  1
2
uE = ε 0 E 2 = ε 0 ( cB ) = ε 0  
2
2
2   µ0ε 0
How does the intesnity
(power/area) change
with distance r?
I=
2
 
B2
B =
= uB
 
2 µ0
 
power
P
= S2
area
4π r
Fig. 33-8
33- 11
33-12
2
Radiation Pressure
Polarization
EM waves have linear momentum as well as energy→light can exert pressure
r
Sincident
Total absorption:
∆p
∆p =
r
Sreflected
r
Sincident
Total reflection
Back along path:
∆p
∆p
∆t
power energy/time
I=
=
area
area
∆U ∆t
=
A
F=
∆U
c
The polarization of light is describes
how the electric field in the EM wave
oscillates.
I
pr =
c
2 ∆U
∆p =
c
Vertically plane-polarized (or linearly
polarized)
2I
pr =
c
∆U = IA ∆t
IA
(total absorption)
c
2 IA
(total reflection back along path)
F=
c
F
33-13
Radiation Pressure
pr =
A
F=
Fig. 33-10
33-14
Polarized Light
Polarizing Sheet
Unpolarized or randomly polarized light has
its instantaneous polarization direction vary
randomly with time
I0
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.
Fig. 33-11
Polarized Light Simulation
I
Fig. 33-12
Only electric field component along polarizing direction of
polarizing sheet is passed (transmitted), the perpendicular
component is blocked (absorbed)
33-15
33-16
Intensity of Transmitted Polarized Light
Intensity of
transmitted light,
unpolarized
incident light:
I=
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
1
I0
2
What happens when a narrow beam of
light encounters a glass surface?
Since only the component of the
incident electric field E parallel to the
polarizing axis is transmitted
Law of Reflection
Etransmitted = E y = E cos θ
Fig. 33-13
Intensity of
transmitted light,
I
polarized
incident light:
Snell’s Law
= I 0 cos2 θ
Refraction:
For unpolarized light, θ varies randomly in time
I = ( I 0 cos θ )
2
= I 0 ( cos θ )
2
avg
avg
1
= I0
2
θ1 ' = θ1
Reflection:
Fig. 33-17
33-17
sin θ 2 =
n2 sin θ 2 = n1 sin θ1
n1
sin θ1
n2
n is the index of refraction of the material
33-18
3
Sound Waves
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
For light going from n1 to n2
•
n2 = n1 → θ2 = θ1
•
n2 > n1 → θ2<θ1, light bent towards
normal
•
n2 < n1 → θ2 > θ1, light bent away from
normal
Fig. 33-19
Fig. 33-18
33-19
Chromatic Dispersion
Fig. 33-20
n2blue>n2red
Chromatic dispersion can be good (e.g., used to analyze wavelength
33-20
composition of light) or bad (e.g., chromatic aberration in lenses)
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 the 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.
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)
What happens for rays that reflect twice off the back
surfaces of the droplets?
http://www.youtube.com/watch?v=OQ
SNhk5ICTI
Fig. 33-22
lens
33-21
Polarization by Reflection
Total Internal Reflection
For light that travels from a medium with a larger index of refraction to a
medium with a smaller medium of refraction n1>n1 → θ2>θ1, as θ1 increases, θ2
will reach 90o (the largest possible angle for refraction) before θ1 does.
n2
n1 sin θ c = n2 sin 90° = n2
Critical Angle: θ c
n1
Fig. 33-24
Total internal reflection can be used, for
example, to guide/contain light along an
optical fiber
33-22
= sin −1
n2
n1
When θ2> θc no light is
refracted (Snell’s Law does
not have a solution!) so no
light is transmitted → Total
Internal Reflection
33-23
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
θ B + θ r = 90°
n1 sin θ B = n2 sin θ r
n1 sin θ B = n2 sin ( 90° − θ B ) = n2 cos θ B
Fig. 33-27
−1 n2
In which direction does light reflecting Brewster Angle: θ B = tan
n1 33-24
off a lake tend to be polarized?
4
The Sun is about 1.5 × 1011 m away. The time for light to travel
this distance is about:
A. 4.5 × 1018 s
B. 8 s
C. 8 min
D. 8 hr
E. 8 yr
25
5