p212c33: 1

Electromagnetic Waves

Maxwell’s Equations

om

o

M

Ec

enc

KK

dtdldE

dtdIldB

AdB

QAdE

)(

0

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“Sourceless” Maxwell’s Equations

AdBdtd

dtdldE

AdEdtd

dtdldB

AdB

AdE

M

E

0

0

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A simple Electromagnetic Wave

Pulse: E and B constant within a “sheet” moving at velocity v

z

x

y

E

Bv

... need to verify consistency with Maxwell’s Equations

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E dA

B dA

0

0

Field lines continue forever: each field line which enters (exits) a closed surface must also enter (exit), so net number of field lines entering (exiting) a closed surface must be zero.

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EvBELvBL

AdEdtdldB

ELvAdEdtd

ELvdtAdEd

BLldB

0)(

dldA

v dt

L

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dl

v dt

LdA

BvEBLvEL

AdBdtdldE

BLvAdBdtd

BLvdtAdBd

ELldE

0)(

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BvEBEv

smc

v

BvEvEB

oo

||

featuresimportant other

1099792458.21in vacuumlight of speed

1

Equations sMaxwell' with consistent is Pulse

8

iff

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x

y

a

x

tB

xE

x

xaBAdB

axExxEldE

AdBdtdldE

zy

z

yy

0lim take

))()((

General relations between (crossed) E and B fields creating EM waves.

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x

x

y

a

tE

xB

xaEAdE

axBxxBldB

AdEdtdldB

yz

y

zz

0xlim take

))()((

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1;1

1

Start with

2

2

22

2

2

2

2

2

2

2

2

2

vtE

vxE

tE

xE

xB

ttE

tE

t

tB

xxE

xE

x

tB

xE

tE

xB

yy

yy

zyy

zyy

zy

yz

Classical Wave Equation!

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Ey

Bz

x

Sinusoidal Electromagnetic Waves

maxmax

max

max

22)sin(ˆ

)sin(ˆ

vBE

vk

f

fk

tkxBzB

tkxEyE

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Energy in an electromagnetic wave

2

)(sin

WaveHarmonic aFor so

but

21

21

2max

22max

2

2

22

Eu

tkxEEu

Eu

EvEB

BEu

av

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eous)(instantanIntensity =Vector Poynting

1

:FlowEnergy )(

:Energy

22

2

SS

BES

EBEvEAdt

dUS

AvdtEudVdU

Energy in an electromagnetic wave

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Intensity222

)(sinˆ

1

2max

2maxmaxmax

2maxmax

IvEv

EBES

tkxBEx

BES

av

Energy Flow and Harmonic Waves

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Example: A radio station the surface of the earth emits 50 kW sinusoidal waves. Determine the intensity, and the Electric and Magnetic field amplitudes for an orbiting satellite at a distance of 100 km from the station.

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cS

AFP

cS

AFP

cS

Vp

Sp

avref

avabs

avgavg

2:PressureRadiation

c=

V

Density Momentum

2

2

Momentum in an electromagnetic wave

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Example: Satellite in previous example has a 2m diameter antenna. What is the force of the radiation on the antenna assuming perfect reflection?

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Standing Waves:Superposition of equal amplitude traveling waves

of opposite directions.

Lcncf

nL

nL

nx

tkxBtkxBtkxBB

tkxEtkxEtkxEE

nnn 2

;22

when mode waveStanding

2,

23,,

2

E)(for planes nodalsincos2

)sin()sin(cossin2

)sin()sin(

max

maxmax

max

maxmax

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Example: EM standing waves are set up in a cavity used for electron spin resonance studies. The cavity has two parallel conducting plates separated by 1.50 cm.

a) Calculate the longest wavelength and lowest frequency of EM standing waves between the walls.

b) Where in the cavity is the maximum magnitude electric field and magnetic field?

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Electromagnetic Spectrum

(see graphic)

in vacuum, v = c = 2.99792458x108 m/s

f = v increasing frequency <=> decreasing wavelength

visible spectrum: 400 nm (violet) to 700 nm (red)

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Radiation from a Dipole

Q

Q

Ep k

rkr t

Bp k

v rkr t

Ir

02

02

2

4

4

sin sin( )

sin sin( )

sin

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