Transcript 1700_Maxwell_2013aug

Maxwell’s Equations &
Hertz Waves
Dr. Bill Pezzaglia
Updated: 2013Aug15
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XI. Maxwell & Electromagnetic Waves
A. Maxwell’s Equations
B. Hertz Waves & Poynting
C. Polarization
A. Maxwell’s Equations
1) Light is Electromagnetic
2) Displacement Current
3) Maxwell’s Equations
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1. Hints that Light is Electromagnetic
A number of experiments suggested that there was a
connection between electricity, magnetism and the
phenomena of light.
•
1834, 1857 Speed of electricity in wire measured to be very
fast (close to speed of light)
•
1850 Speed of light is measured accurately 1850 by Foucault.
•
1844 Faraday rotates the polarization of light with a magnetic
field (implies light has magnetic properties).
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Charles Wheatstone (1802-1875)
•1834 discovery by English
physicist Charles
Wheatstone that current
traveled through long
lengths of wire with great
velocity – almost 288,000
miles/second
•A bit off, it can’t travel
faster than speed of light
(186, 282 miles/second)
•1837 Developed an early
telegraph (5 needles)
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Michael Faraday
•1844 Faraday rotates the
polarization of light using a
magnetic field. Suggests light is
a transverse magnetic
disturbance
•1857 Wilhem Weber shows
Amp of current is a Coulomb per
second, gets characteristic speed
of electrical signals to be speed
of light.
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Gustav Kirchhoff (1824-1887)
•1857 Telegraphy
Equations
•Derives (based on
earlier work by Faraday
& Thomson 1854) that
speed of electrical
signal in cable should
be close to speed of
light.
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2. Field Induction
•Recall Faraday’s law is that
voltage (emf=electric firld
times circumference) in a
wire loop was generated by
changing magnetic flux 
through the loop.
•Maxwell shows that the law
is more general. A changing
magnetic field generates a
circular electric field even if
there is no wire!

V 
t
James Maxwell (1831-1879)
b. Ampere’s circulation Law
• Recall : Ampere’s law says that a
circular magnetic field is generated
by a current.
• Or: B field multiplied by
circumference of a circle is
proportional to current flowing
through circle
B ( 2 r )   0 I
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c. Maxwell’s Displacement Current
1861: Maxwell makes Ampere’s law look like
the complement of Faradays Law. A changing
electric flux will generate circular B field.
B ( 2r )   0 I 
1
c2

t
[details, you can
ignore equation]
Note “c” is the speed of light, and Electric Flux is
defined:
 
 EA
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3. Maxwell’s Equations
(a) The General Laws of Maxwell
•
Gauss’s Law shows that charge is the source of electric fields
(electric flux through a closed surface is proportional to net
enclosed charge)
•
Gauss’s Law for magnetism states that there are no magnetic
charges (magnetic flux through a closed surface is zero).
•
Faraday’s Law: changing magnetic fields create electric fields
•
Ampere’s Circulation Law: current is the source of magnetic
fields. Maxwell adds the “displacement current” to this equation
such that changing electric fields create magnetic fields.
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b. Differential Form of Maxwell’s Equations
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•1884 (with Gibbs) Heaviside
reorganizes Maxwell’s equations
compactly into 4 “vector”
equations
For completeness, here they are, but
don’t worry about them.
B  0

E 
0
B
 E  
t
 E

  B   0   0
J
 t

Oliver Heaviside (1850-1925)
c. Relativity and Maxwell’s Equations
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• 1905 Einstein’s Relativity shows that time is
the 4th dimension.
• In our ordinary “3D” view of the world,
electric fields are different than magnetic
fields, however we see they are
complementary
• In “4D” we see that they are both the same
thing, i.e. we “unify” electricity with
magnetism.
• We can write Maxwell’s 4 equations in just 2:
 F

 j

 [  F ]  0
For completeness, here they are, but
don’t worry about them.
B. Hertz Waves
1) Equations predict waves
2) Hertz Experiment
3) Energy in Waves
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1. EM Wave Equation
(a) 1865 Maxwell shows his equations predict that
electromagnetic waves can exist in vacuum (note E & B
are perpendicular to each other and direction of wave)
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1b. Prediction of Electromagnetic Waves
v
1
•
The Theoretical speed:
comes out very close to
known speed of light “c”
•
Magnitude of electric and magnetic fields are simply
related by wavespeed:
E  cB
 0 0
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2. Making EM Waves
(a) 1891 (1888?) Hertz
demonstration that electromagnetic
waves can be transmitted and then
received. Proves existence of
waves with frequencies of 100
million cycles per second.
Heinrich Hertz (1857-1894)
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2b. Nikola Tesla (1856-1943)
•1891 (1893?) Chicago World’s fair,
demonstrates wireless telegraphy (30 feet)
•1894 Lodge transmits 150 yards
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2c Guglielmo Marconi (1874-1937
•1899 Marconi “steals” Tesla’s
design and broadcasts across the
English Channel
•1901 Across the Atlantic
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3. Wavespeed Phenomena
(a) Index of refraction:
• Speed of light “v” in media is slower
where “n” is index of refraction (about
1.5 for glass).
• Index can be calculated from the
electrical permittivity () and magnetic
permeability () properties of the media.
• Index usually depends upon wavelength
of the light (e.g. in glass red might have
n=1.50 while for blue n=1.53)
c
v
n
n
1

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(b) Reflection & Transmission
• As a wave (such as light) in media 1,
with index n1, enters a denser media
(index n2) where the speed changes,
part of the wave will be reflected.
• Proportion given by formula:
• The rest is transmitted.
• For glass (n=1.5) we calculate that 4%
is reflected, 96% transmitted
 n2  n1 

R  
 n2  n1 
2
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(c) Absorption
• Good conductors: reflect nearly 100%
• Poor conductors: wave penetrates into media to
“skin depth” and is absorbed (energy turned into
heat). Wave exponentially decays with distance.
• For AC Signals traveling through a wire, at higher
frequencies the skin depth is very small, and so
electricity will travel only on the outside of a
conductor (hence increasing its resistance).
• At 60 cycles the skin depth is 8.5 mm for copper, so
making a bigger diameter wire is a waste of metal.
• Instead, use Litz wire, made of many small wires.
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C. Polarization
1) Linear Polarization
2) Birefringence
3) Circular Polarization
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Linear Polarization
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[1812 Fresnel develops wave
theory of transverse polarized
light, well before the
electromagnetic nature was
known]
light has two perpendicular
linear polarizations (electric
field) can be horizontal or
vertical
1888 Hertz shows
electromagnetic waves have
transverse polarization
(equivalent to “light”)
Polarization by Reflection
• 1808 Malus’s Law: Reflected light is often
polarization
• 1812 Fresnel develops wave theory of
transverse polarized light
• 1815 Brewster’s angle: at this angle of
incidence the reflected light is entirely “s”
polarized such that electric field is parallel to
the interface surface
tan  b  n
Note: “Plane of incidence” is the plane defined by the three beams above.
The normal also lies in this plane. The plane is perpendicular to the surface.
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Detecting Polarized Light
• Linear polarizer can be used to detect polarized light,
only lets one polarization through!
• 1808 Malus’s Law: Linear polarized light passing through
a second polarizer tilted at angle  to first will be
attenuated:
I  I 0 [cos  ]
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• Hence no light gets through “crossed polarizers” (=±90°)
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Optical Activity
•Optically active materials can rotate the polarization
•If such a substance is put between “crossed polarizers” (90º angle) you will
often see interesting colors.
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Birefringence
•1669 Erasmus Bartolinus (Denmark)
discovers the birefringence (double refraction)
of calcite crystals.
•When polarization was understood better, it
was realized the two different polarizations
took different paths (they are “refracted”
differently, or the index of refraction is
dependent upon polarization)
•Index of refraction: n=c/v, so the different
polarizations travel at slightly different speeds.
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Quarter Wave Plates
A quarter wave plate retards horizontal polarization by 90º to vertical. It
can be used to make circular polarized light from linear polarized light.
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Circular Polarized Light
Another type
of polarized
light can be
left or right
handed
circular
polarized
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Detecting Circular Polarized Light
A quarter wave plate will turn circular back into
linear, which can be detected by a linear polarizer
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References
•http://maxwell.byu.edu/~spencerr/phys442/node4.html
•http://en.wikipedia.org/wiki/Timeline_of_Fundamental_Physics_Discoveries
•http://www.sparkmuseum.com/GLASS.HTM
•http://keelynet.com/spider/b-103e.htm
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Things to Do
•Find tesla museum stuff
•Who first predicted circular polarized light?
•Can we make a 3D image for students using polarized light? Need two
projectors?
•Ideally we’d use circular polarized light, but one test so far shows either the
transparency projector or the screen does not preserve the circular polarization.