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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