Download Chapter 35. Electromagnetic Fields and Waves

Chapter 35. Electromagnetic Fields and
Waves
To understand a laser beam,
we need to know how electric
and magnetic fields change
with time. Examples of timedependent electromagnetic
phenomena include highspeed circuits, transmission
lines, radar, and optical
communications.
Chapter Goal: To study the
properties of electromagnetic
fields and waves.
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E or B? It Depends on Your Perspective
Whether a field is seen as “electric” or “magnetic” depends
on the motion of the reference frame relative to the sources of
the field.
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Transformations
The Galilean field transformation equations are
where V is the velocity of frame S' relative to frame S and
where the fields are measured at the same point in space by
experimenters at rest in each reference frame.
NOTE: These equations are only valid if V << c.
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Example transformation
Consider a charge at rest at the origin in S where B=0 and E
is given by Coulomb’s Law.
In S’, the charge is moving and there is both an electric field
E’=E and a magnetic field B’.
Biot Savart Law
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Maxwell’s Equations and Electromagnetic
Waves
Maxwell’s equations provide a unified description of the
electromagnetic field and predict that
•  Electromagnetic waves can exist at any frequency, not
just at the frequencies of visible light. This prediction
was the harbinger of radio waves.
•  All electromagnetic waves travel in a vacuum with the
same speed, a speed that we now call the speed of
light.
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Light speed
Maxwell’s equations predict EM waves move at a unique
(light) speed relative to ANY frame of reference.
That is impossible if the Galilean velocity addition rule
v’ = v+vrel holds.
This paradox is resolved by Einstein’s Special Theory of
Relativity.
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Discovery of artificial EM waves by Hertz
Receiver spark
gap
Transmitter
spark gap
EM wave
Magnified view of the spark gap
and dipole transmitting ("feed")
antenna at the focal point of
the reflector. The high voltage
spark jumped the gap between
the spherical electrodes. The
electrical impulse produced by
the spark generated damped
oscillations in the dipole
antenna.
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Magnified view of the spark gap and
dipole receiving antenna at the focal
point of a receiving reflector similar
to the transmitting one. The width of
the small spark gap on the right is
controlled by the screw below it. The
vertical dipole antenna at the left was
about 40 centimeters long.
Schematic of operation of antenna
An oscillating electric dipole is surrounded by an oscillating
electric (and magnetic) field. Field disturbances propagate
away at light speed perpendicular to the dipole. Operating in
reverse, an EM wave excites charges oscillation in a dipole.
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Marconi’s transatlantic signal experiment
Left to right: Kemp, Marconi, and Paget pose in front of a kite that
was used to keep aloft the receiving aerial wire used in the
transatlantic radio experiment.
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Marconi’s transatlantic signal generator
Induction
coils
Capacitor
banks
Spark gap
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Radio
Like two identical tuning forks coupled resonantly through a
sound wave, two tuned LC or other resonant circuits can be
coupled through an EM wave. A high frequency EM can carry
audio frequency analog information by warbling the frequency
(FM) or amplitude (AM).
An optimal antenna has a size of order a wavelength. More
compact loop antennas work through induction.
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Harmonic Waves
A harmonic plane wave is
generated by a single frequency
source current distribution and is
characterized by frequency f and
wavelength c/f.
E, B, v form a right handed
coordinate system.
A “linearly polarized” wave has
the orientation of E fixed.
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Generation of electromagnetic waves
EM waves are emitted in general by accelerated charges which
shed free force fields.
Synchrotron light source
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Dipole antenna
EM wave spectrum
EM waves are observed over a
wide range of frequencies.
Megahertz natural and artificial
sources produce radio waves.
Ultra high frequency motions
inside nuclei source gamma ray
radiation.
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Properties of Electromagnetic Waves
Any electromagnetic wave must satisfy four basic
conditions:
1.  The fields E and B and are perpendicular to the direction
of propagation vem.Thus an electromagnetic wave is a
transverse wave.
2.  E and B are perpendicular to each other in a manner such
that E × B is in the direction of vem.
3.  The wave travels in vacuum at speed vem = c
4.  E = cB at any point on the wave.
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Energy of Electromagnetic Waves
EM waves carry energy density u and momentum density u/c.
Energy density in E-field
Energy density in B-field
uE = εo E 2 ( r,t ) /2
uB = B 2 ( r,t ) /2µo
2
2
u
=
ε
E
/2
+
B
/2µo
Total Tot
o
= εo E 2 /2 + E 2 /2c 2µo = εo E 2 ( r,t ) = B 2 ( r,t ) / µo
uTot = εo E 2 = εo E o2 cos2 ( kz − ω t) moves w/ EM wave
at speed c
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Poynting vector
The energy flow of an electromagnetic wave is described
by the Poynting vector defined as
The magnitude of the Poynting vector is
The intensity of an electromagnetic wave whose electric
field amplitude is E0 is
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EXAMPLE 35.4 The electric field of a laser
beam
Laser light is monochromatic and
sourced by coherent high
frequency motion of atomic
electrons.
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EXAMPLE 35.4 The electric field of a laser
beam
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Radiation Pressure
A electromagnetic wave carries momentum density U/c and
if the momentum is absorbed or reflected a pressure is
exerted called the radiation pressure prad. The radiation
pressure on an object that absorbs all the light is
where I is the intensity of the light wave. Note reflection
implies twice the momentum transfer and twice the
pressure.
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EXAMPLE 35.5 Solar sailing
QUESTION:
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EXAMPLE 35.5 Solar sailing
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Polarization
Plane Polarized

x
z
y
E = E o cos( kz − ω t) xˆ

B = Bo cos( kz − ω t) yˆ
Unpolarized
Superposition of
plane polarized
waves
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Polarization filters
An array of linear
conductors absorbs energy
only from the component
of electric field along the
conducting direction
transmitting the orthogonal
component.
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Polarization filters
Plane-polarized
incident wave
y


E inc = E o cos( kx − ωt )
x
polarizer
( E inc cosθ ) xˆ + ( E inc sin θ ) yˆ
transmitted
 
absorbed
Transmitted wave =

E trans = E o cosθ cos( kx − ωt ) xˆ
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transmission
Malus’s Law
Suppose a polarized light wave of intensity I0 approaches a
polarizing filter. θ is the angle between the incident plane of
polarization and the polarizer axis. The transmitted intensity
is given by Malus’s Law:
If the light incident on a polarizing filter is unpolarized, the
transmitted intensity is
In other words, a polarizing filter passes 50% of unpolarized
light and blocks 50%.
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Malus’s Law
Here we see the effect of two filters.
When parallel, light transmitted by the first is transmitted by
the second.
When orthogonal, light transmitted by the first is absorbed by
the second.
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Polarization by reflection
 
 
Unpolarized light reflected
from a surface becomes
partially polarized
Unpolarized
Incident light
Degree of polarization
depends on angle of incidence
Reflection
polarized with
E-field
parallel to
surface
n
Refracted
light
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Glare reduction
 
 
Reflected sunlight partially polarized.
Horizontal reflective surface ->the Efield vector of reflected light has
strong horizontal component.
Transmission axis
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