Post on 04-Apr-2018
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f( )x
x
x
z
y
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Lecture Outline
Recap
Properties of E&M waves
Polarization
Poynting vector S Intensity
Energy density Momentum Light Pressure Frequency E&M spectrum
How to change colors Nonlinear optics
The Doppler shift
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Summary of Electromagnetic Radiation
Combined Faradays Law and Amperes Law
time varyingB-field inducesE-field time varyingE-field inducesB-field
x
z
y
E-field and B-field are perpendicular
2 2
2 2x xo o
E E
z t
E
B
S
sbe
By is in phase withEx
Bmax =Emax /c
max sin( )xE E kz t (plane wave solution)
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(Linear) Polarization We have been discussing the plane electromagnetic waves.
Our main example has been harmonic:
)sin(0 tkzEEx 0yE 0zE0 sin( )y
EB kz t
c 0zB0xB
in terms of components:
q
e
x
y0 cos sin( )xE E kz tq
0 sin sin( )yE E kz tq
0z
E
)sin( 0 tkzEeE
is constantphase
This wave is an example of a linearlypolarized wave.
We always define the direction of polarization as the direction ofthe oscillation of the Electricfield vector (x-direction in this case)
In general, a linearly polarized wave traveling in the +z directioncan be written as:
1
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a) Ex =Eosin (kz + t)
b) Ey =Eosin (kz - t)
c) By =Bo sin (kz - t)
6) Which equation correctly describes this electromagnetic wave?
7) In which direction is this wave polarized?
a) x
b) y
c) z
Preflight 22:
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Right:
polarization is the direction of oscillation of the E field.
Ey is obviously wrong because E is in the x direction. Ex is wrong because
(kz + wt) would mean it was moving in the -z direction, but it's moving in the
+z direction.
a) Ex =Eosin (kz + t): NO: this wave moves in z direction
b) Ey =Eosin (kz - t): NO: the wave in the picture hasEy = 0
c) By =Bo sin (kz - t): YES: +zdirection
7) Direction of polarization = direction of E
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Lecture 22, ACT 1 Att = 0,z = 0, the electric field of an electro-
magnetic wave is oriented at an angle with
respect to thex-axis, as shown. Which arrow indicates the direction of
the magnetic field at the same locationand instant of time?
(a) A (b) B
This question cannot be answered unless thedirection of propagation is specified:
If the wave propagates in the +z direction,
then B-field is along A If the wave propagates in the z direction,then B-field is along B
q
e
x
y
A
B
Th P i V
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The Poynting Vector The direction of the propagation of the electromagnetic wave
is given by: bes
0
E BS
This wave carries energy. This energy transport is definedby the Poynting vector Sas:
2
377
E
S
The direction of Sis the direction of propagation of the wave
The magnitude of Sis directly related to the energy beingtransported by the wave:
2 222max max
0
1sin ( )
377 2 377
E EEI S kz t
Z
The Intensity of a wave is the spatial- and time-average of S:
Define
37700 cZ
2
0 0
E B ES
c
[Watts/m2]
E D it i E M W t
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Energy Density in E-M Waves, cont.
Usually we are interested in the averageenergy density:
Note: We can also define the Intensityof a wave asI= average power transmitted per unit area
= average energy density times wave velocity:
Same as before! :-)
22 2 2 0 max
0 0 max sin ( ) 2
E
u E E t kx
If we introduce , we havemaxrms
2
EE
220 max
0 rms2
Eu E
2 2 2 22 max max max rms
0 0
0
1
2 2 2 377 377
E E E EI c u c E c
c
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Light PressureEnergy transport momentum
Momentum =energy carried by wave
speed
total U
c
Intensity = energy
time-area
energy/ momentum
time-area time-area
I c
c
momentumForce =time
Forcearea
Ic
Radiation Pressure
Light pressure, though
light, has noticeableeffects comets tail
pushed away from the sun*.
2*Note: The dust tail is pushed away
by radiation; the ion tail is pushedawa b the solar wind!
P fli ht 22
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5) An absorbing black disk of radiusr, massm, is hanging
by a thread. A laser beam with radiusa, intensityI, and
frequency , is incident on the disk (centered on it) from
the left. If we increase ____ (keeping all other parameters
the same), the light force on the disk will increase.
a. disk radiusr
b. disk massm
c. laser beam radiusa
d. laser beam intensityI
e. laser frequency
force = pressure x area
pressure =I/c
r a: increasinga more of disk is
hit by the beam more force.
increasingr does nothing
IncreasingIdefinitely increases force.
Intensity is independent of frequency.
Preflight 22:
E l
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ExampleThe intensity of sunlight in Urbana, IL is 100 mW/cm2.
What is the amplitudeEmaxof the electric field?
What is the pressureon the blackground (pretend it is perfectly absorbing)?On the white sidewalk (pretend it is perfectly reflecting)?
Finally, what is the maximum amount ofenergya solar cell could absorb in1 sec, assuming an area of1m2?
Solution:2
2 21 W 100 cm100 mW/cm 1000 W/m
1000 mW 1 m
I
2
max1
2 377
E
Energy = averageenergy density volume = 1000 J
26 30 max 3.3 10 J/m
2
E Iu
c
2 8 8 3Area 1m 3 10 m/s 1s 3 10 mct
Black ground: Pressure =
2
8
1000 W/m
3 10 m/s
I
c
6 2
3.3 10 N/m
Reflective surface: 2 enhancement Pressure = 6 26.7 10 N/m
2
W
m2 377 1000 868 N/C 868 V/m
maxE
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M th d t G t Diff t C l
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Methods to Get Different Colors:
Why would you want to do this?
More information storage!It is a consequence of the quantum
Uncertainty Principle (itself due to thewave nature of everything) that youcannot focus light to a spot smallerthan ~l. Therefore, with ultravioletphotons (300nm) you can get aminimum spot ~1/2 the diameter of a
spot with red photons (600nm)
afactor of4 in area 4x informationstorage on an optical CD or DVD(and faster too)!
Ex.: two infrared photons (l= 1064nm) one green photon (532nm)
How else to change the frequency? 5. Use the Doppler Shift
Classical Doppler shift (sound waves)
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Classical Doppler shift (sound waves)
Source moving towardyou at speed vs:
Let c be speed of sound (in stationary air).
0 0t 1d
tc
2t T 3sd v Tt T
c
3 1 (1 )s s
v Td d vt t T T
cc c c
1'
'f
T
1f
T 1'
1 sv
c
f f f
Receive 2nd crest
my interval between crests T
vs
S
d
c
I receive first crest
d - v
sT
vsS
c
If instead the listener moves with
speed votoward the source: ' 1ov
cf f f
Preflight 22:
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9) You are riding your bicycle south on Lincoln. A police car
is coming in your direction, traveling north on Lincoln, with
its sirens blaring (trying to catch your physics professor?!).
Compared to how it would sound if you and the car were
stationary, the pitch is
a) higher b) lower c) the same
10) The change in the pitch is due to
a) the motion of the police car
b) your motion on the bicycle
c) only the relative motion
d) a complicated combination of your motion and the
motion of the car
Preflight 22:
Doppler Shifts
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Doppler ShiftsClassical Waves
Note that the exact value of the shift depends on whether the source or thereceiver is moving (though the signdepends only on relative motion).
This is because the medium in which the sound wave propagates gives apreferred frame of referenceyou can tell whether you are movingthrough the air, or if the approaching fire truck is moving through the air.
However, according to special relativity there can be nopreferred frame ofreferencethe speed of light is identical forallobservers (c.f., theMichelson-Morley experiments). The consequence of this is that theDoppler shift for electromagnetic radiation is the same whether or not thesource is moving or the receiver is moving (see appendix for derivation):
1
1
vc vcf f f
v c vc
- Top signs for decreasing separation
- Bottom signs for increasing separation
- Ifv
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Applications: Police Radar
Here there is a double effect:First, my car is moving toward the radar
transmitter as an observer; second, my car acts as a
source which is moving toward the police car receiver.
2 2(1 ) (1 ) (1 ) v v vc c cf f f f
100 miles/hour = 44.7 m/sv
10.6 GHzf " 10.60000316 GHzf
Ex.
3160 Hz f
Example 2: Red Shift
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Example 2: Red Shift
It has been known since 1930s
that the universe is expanding.
How do we know?
Distant galaxies are moving
away from us (and the farther
away they are, the faster theyappear to be moving!)
The speed of distant objects is determined by
the Doppler shift of their atomic emission lines.Example: Hydrogen 434 nml 0.1v c
Hydrogen Hydrogen Hydrogen
1 1.1
1 0.9
vc
vc
l l l
479.8 nm
Summary
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Summary Properties of electromagnetic waves, e.g., light
Polarization (oscillation) direction ofelectric field
Poynting vector, intensity, energy density
Momentum, light pressure
absorption vs. reflecting
Frequency, wavelength, spectrum, color
Doppler shift
Force
area
Iu
c
0
E BS
1
1
1
v c
vc vf f f
cvc
220 max
0 rms
2
Eu E
22
max
0
1
2 377
EEI S
Z
Appendix: Relativistic Doppler Shift
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Appendix: RelativisticDoppler Shift
' sv TT Tc
2
2
1
1 vc
T T
1 (1 )svT Tf c
2
211 1
1 (1 )
s
s s
v
c
v vc c
fT T
2
21
1 s
v
c
vc
f
(1 )(1 )
1
s s
s
v v
c c
vc
f
(1 )
(1 )
s
s
v
c
v
c
f f
Forvs