Download Chapter 29: Maxwell`s Equation and EM Waves

Chapter 29: Maxwell’s Equation
and EM Waves
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 29-1
Equations of electromagnetism: a review
•  We’ve now seen the four fundamental equations of electromagnetism,
here listed together for the first time.
•  But one is incomplete: Ampère’s law needs refining
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Slide 29-2
CT 33.42
Current i
+
+
+
+
+
- Current i
B=?
According to Biot-Savart, is there a Magnetic
Field at the point labeled between the plates?
A: Yes
B: No (B=0 there)
©University of Colorado, Boulder (2008)
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 29-3
Maxwell’s Adjustment to Ampere’s Law
•  Applying Ampère’s law to a circuit
with a changing current results in an
ambiguity.
•  The result depends on which surface is
used to determine the encircled current.
•  Can’t have contradictory results – either there is
a B field or there isn’t!
•  Notice that electric field is changing inside
conductor. Ampere postulated a
�
dE
�
displacement current density: Jd = �0
dt
Q(t) = CVC =
�
�
�0 A
(Ed) = �0 EA = �0 ΦE
d
dΦE
dQ
�
�
Id = Jd · dA = �0
=
=I
dt
dt
•  Thus discrepancy goes away if add
displacement current to Ampere’s law
�
dΦE
�
�
B · d� = µ0 (I + Id )enc = µ0 I + �0
dt
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�
enc
Slide 29-4
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Slide 29-5
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Slide 29-6
•  Field between the plates:
�
dΦE
�
�
B · d� = µ0 �0
dt
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Slide 29-7
Maxwell’s equations
•  The four complete laws of electromagnetism are
collectively called Maxwell’s equations.
•  They describe all electromagnetic fields in the universe, outside
the realm of quantum physics.
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Slide 29-8
Maxwell’s equations in vacuum
•  In vacuum there’s no electric charge and therefore also no
electric current.
Maxwell’s equations in vacuum
A changing electric field is a source for a magnetic field, and
a changing magnetic field is a source for an electric field.
These equations infer the possibility of electromagnetic
waves!
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Slide 29-9
Wave Equation
•  Maxwell’s equations can be manipulated to derive the
following wave equations:
2�
E
∂
2�
∇ E = µ0 �0 2
∂t
2�
B
∂
2�
∇ B = µ0 �0 2
∂t
2
2
2
∂
∂
∂
∇2 =
x̂ + 2 ŷ + 2 ẑ
2
∂x
∂y
∂z
•  Very similar to wave equation for sound waves! Except
here it is the electric and magnetic field that is
“wiggling.”
•  For plane waves traveling in one direction (say the x
�
�
�
direction): ∂ 2 E�
∂2B
∂2B
∂2E
∂x2
= µ0 �0
∂t2
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
∂x2
= µ0 �0
∂t2
Slide 29-10
Plane electromagnetic waves
•  A plane electromagnetic wave are waves propagating in
one direction with one wavelength. (E and B do not vary
with respect to the other two dimensions)
•  The fields are perpendicular
to each other and to the
direction of propagation.
� ×B
� gives direction of propagation)
(E
•  Mathematically
2π
k=
λ
ω=
2π
T
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Slide 29-11
Clicker question
• 
At a particular point, the electric field of an
electromagnetic wave points in the
direction, while
the magnetic field points in the
direction. Which of
the following describes the propagation direction?
A. 
B. 
C.  either
or
but you can’t tell which
D. 
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Slide 29-12
Plane
Waves
as
Solutions
�
�
�
�
� · d�� = − d
E
dt
S
� · dA
�
B
∂E
∂B
=⇒
=−
∂x
∂t
� · d�� = µ0 �0 d
B
dt
=⇒
S
� · dA
�
E
∂B
∂E
= −�0 µ0
∂x
∂t
•  plane wave expression is a solution if
kEp = ωBp
kBp = �0 µ0 ωEp
fλ = c = c = 3.0 × 108 m/s
•  This also implies that
Ep
Bp =
c
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Slide 29-13
General Results for EM Radiation
•  Transverse waves (E and B perpendicular to direction of
propagation)
•  E and B perpendicular to each other.
•  E = cB; E and B oscillate in phase
•  Propagation speed is the speed of light in a vacuum
•  independent of wavelength:
c = λ f = ω/k
•  radio waves, light, infrared radiation, X-rays are all the same
phenomena!
1
c
•  In matter, the speed of light is v = √�µ = n
•  No medium required for propagation (no ether)
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Slide 29-14
Clicker question
•  A planar electromagnetic wave is propagating through
space. Its electric field vector is given by
� = Ep cos(kz − ωt)î
E
Its magnetic field vector is
� = Bp cos(kz − ωt)ĵ
1) B
� = Bp cos(ky − ωt)k̂
2) B
� = Bp cos(ky − ωt)ĵ
3) B
� = Bp cos(kz − ωt)k̂
4) B
� = Bp sin(kz − ωt)î
5) B
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Slide 29-15
Clicker Question
Two traveling waves 1 and 2 are described by the equations.
y1 ( x, t ) = 2 sin(2 x − t )
y 2 ( x, t ) = 4 sin( x − 2 t )
All the numbers are in the appropriate SI (mks) units.
Which wave has the higher speed?
A) Wave 1
B) Wave 2
C) Both have the same speed.
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Slide 29-16
The Electromagnetic Spectrum
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Slide 29-17
Clicker question
•  Which type of radiation travels with the highest speed?
1.  visible light
2.  X-rays
3.  Gamma-rays
4.  radio waves
5.  they all have the same speed
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Slide 29-18
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Slide 29-19
Producing electromagnetic waves
•  Electromagnetic waves are generated ultimately by accelerated
electric charge.
•  Details of emitting systems depend on wavelength, with most efficient
emitters being roughly a wavelength in size.
•  Radio waves are generated by alternating currents in metal antennas.
•  Molecular vibration and rotation produce infrared waves.
•  Visible light arises largely from atomic-scale processes.
•  X rays are produced in the rapid deceleration of electric charge.
•  Gamma rays result from nuclear processes.
A radio transmitter and antenna
Electric fields of an oscillating electric dipole
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Slide 29-20
Antennae
An electric field parallel to an antenna
(electric dipole) will “shake” electrons
and produce an AC current.
A magnetic dipole antenna (for AM
radios) should be oriented so that the
B-field passes into and out of the
plane of a loop, inducing a current in
the loop.
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Slide 29-21
Energy in EM waves
•  Electromagnetic waves transport energy
•  The Poynting vector describes the rate of energy flow per unit
area (W/m2 in SI):
2
S = uc = (uE + uB )c = �E c
•  For plane waves (traveling in x direction with E oriented in z direction):
� = 1 Ep Bp cos2 (kx − ωt)î
S
µ0
•  Averaging over the time variations of the oscillating fields gives the
average value, also called the average intensity:
1 1
1 Ep2
1
I =< S >=
Ep Bp =
= �0 Ep2 c
2 µ0
2 2µ0 c
2
•  Far from a localized source of radiation, electric field decreases as 1/r.
1 (as required by conservation of energy)
Thus,
I∝
r2
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Slide 29-22
Clicker Question
Two radio dishes are receiving signals from a radio station which is
sending out radio waves in all directions with power P. Dish 2 is twice
as far away as Dish 1, but has twice the diameter. Which dish receives
more power?
A: Dish 1
B: Dish 2
C: Both receive the same power
Dish 1
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Dish 2
Slide 29-23
Example: 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?
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Slide 29-24
Example continued:
(b) What is the intensity of sunlight incident on Mercury, which is
5.8x1010 m from the Sun?
(c) What is the maximum electric field (if monochromatic light)
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Slide 29-25
Flux and Solar Heating
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Slide 29-26
Clicker Question
How many solar collectors would you need to replace a
4.8 kWatt electric water heater?
Assume each collector is 2 meters2 and has an efficiency of 40%
for converting light energy to usable energy.
A) 1 panel
B) 2 panels
C) 4 panels
D) 8 panels
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Slide 29-27
Polarization
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Slide
Slide25-24
29-28
Light passed through a polarizing filter has an intensity of 2.0 W/m2.
How should a second polarizing filter be arranged to decrease the
intensity to 1.0 W/m2?
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Slide
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29-29
Clicker Question
An unpolarized beam of light passes through 2 Polaroid filters
oriented at 45o with respect to each other. The intensity of the
original beam is Io. What is the intensity of the light coming
I
through both filters?
o
A: (1/1.4)Io
B: (1/2)Io
C: (1/4)Io
D: (1/8)Io
E: None
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c
Slide 29-30
Clicker question
• 
Two polarizers are oriented at right angles, so no light
gets through the combination. A third polarizer is
inserted between the two, with its preferred direction at
45° to the others. How will this “sandwich” of
polarizers affect a beam of initially unpolarized light?
A.  All of the initial light will be blocked.
B.  Half of the initial light is blocked.
C.  One-quarter of the initial light is blocked.
D.  None of the initial light will be blocked.
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Slide 29-31
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Slide 29-32