EEL 3472 ElectromagneticWaves. 2 Electromagnetic Waves Spherical Wavefront Direction of Propagation Plane-wave approximation

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ElectromagneticElectromagnetic

WavesWaves

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Electromagnetic WavesElectromagnetic Waves

SphericalWavefront

Direction ofPropagation

Plane-waveapproximatio

n

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Electromagnetic Waves

In the case of (fields inside a good insulator such

as air or vacuum) we have

vector Helmholtz

equation

This equation has a rich variety of solutions.

Let us assume that has only an x component and varies

only in the z direction. In this case the vector Helmholtz

equation simplifies to

scalar Helmholtz

equation

The latter equation is similar to the voltage wave equation

for a lossless transmission line.

EjjE E )(2

E

E

022 EE

022

2

xEz

xE


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The solution of the scalar Helmholtz equation is

where , to be found from boundary conditions; the minus and the plus correspond to waves moving in the+z and –z directions, respectively.

In terms of propagation constant k

The position of a field maximum is given by

zzjox eEeEE 0

ojeAE 00

)cos(]Re[),( ootjzj

ox ztAeeEtzE

omaxomax

tz0zt

1max

dt

dzU p

s/m10x99793.2/1cU

, If8

oop

oo

)cos(),( oox kztAtzE k


k

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Characteristics of Plane WavesFor a plane wave which propagates in the +z direction and has an electric field directed in the x direction

where The time-varying electric field of the wave must, according to Faraday’s law, be accompanied by a magnetic field. Thus,

when and is called the characteristicimpedance of vacuum.

xjkz

o eeEE

/2k

yjkz

oyx eeEjke

zE

EHj

yjkzee

EH

0

377 , oo

wave impedance (intrinsic impedance of the medium

dt

dB


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If

1. , , and the direction of propagation are allperpendicular to each other.

2. and are in phase

3. The wave appears to move in the +z direction as though it were a rigid structure moving toward the right

(in exactly the same way as the transmission-line waves).

oj

oo eAE

xoo e)kztcos(AE

yoo e)kztcos(

AH

E H

HE


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)4

cos(5.1),(

0

kzttzEoA

x

jkz

E

j

x eeE

0

45.1


)0( t

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wave plane

jkz4j

ox eeAE

kz o

kzo const defines the equiphase surfaces

= const - planes

If the amplitude is the same throughout any transverse plane, a plane wave is called uniform.


2

t

kz0

kz 0

waveplane

00 jkzj

x eeAE

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In general,kHE

zyx

eee

eee

unit vector along the direction of propagation


In the x-z plane

In the y-z plane

0tt

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When the electric field of a wave is always directed alongthe same line, the wave is said to be linearly polarized.


20 tt

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)( yxjkz

o ejeeEE

sine-tcoseE)(Re(t)E 0 yx0 teejeEz tjyxo


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Poynting’s TheoremPoynting’s theorem is an identity based on Maxwell’sequations, which can often be used as an energy-balanceequation

JEtD

EtB

H)HE(

)H(E)E(H)HE(tD

JH

tB

E

Faraday’s law

Ampere’s law

vector identity


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If do not change with time,E and ,

2E

21

tt)EE(

21

t)E(

EtD

E

2H

21

tt)HH(

21

t)H(

HtB

H

2EE E)E(EJE

2E

22 EH21

E21

t)HE(

Magnetic energy density

Electric energy density


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Integrating over the volume of concern and using the divergence theorem to convert the volume integral ofto the closed surface integral of we have equation referred to as Poynting’s theorem:

Total power leaving the volume

Rate of decrease in energy stored in electric and magnetic fields

Ohmic power dissipated as heat

)HE(

)HE(

Poynting vector (represents the power flow per unit area)


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Net power entering the volume

rate of increase in stored We

rate of increase in stored

ohmic losses


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xz

o a)ztcos(eE)t,z(E

)BAcos()BAcos(

21

BcosAcos

The time-average Poynting vector:

The total time-average power crossing a given surface is

obtained by integrating over that surface.aveS


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where V is the voltage across the volume and I is the current flowing through it.

V=LA

vv

2e dvJEdvE

cosLAJEdvJEv


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Example. Using Poynting’s theorem to find the power dissipated in the wire carrying direct current I

eeez


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Reflection and Transmission at Normal IncidenceWhen a plane wave from one medium meets a differentmedium, it is partly reflected and partly transmitted.If the boundary between the two media is planar and perpendicular to the wave’s direction of propagation, the wave is said to be normally incident on the boundary.Let us consider a wave linearly polarized in the x directionand propagating in the +z direction, that strikes a perfectlyconducting wall at normal incidence. The wall is located at z=0

zyyjkz

r

yjkzo

i

xjkz

r

xjkz

oi

eeeeE

H

eeE

H

eeEE

eeEE

)(e x1

1

incident wave

reflected wave


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At the interface z=0, the boundary conditions require thatthe tangential component of E must vanish (since forz>0)

(This result is similar to the expression for the voltage on a short-circuited transmission line)

The wave impedance is defined as

This impedance is analogous to the input impedance Z(z)=V(z)/i(z) on a transmission line. For a single x-polarized wave moving in the +z direction , intrinsic

impedance of the medium.

E

o1x1o EE0e)EE(

xoxjkzjkz

ori ekzjEe)e(eEEEE sin2

yo

yjkzjkzo

ri ekzE

eeeE

HHH cos2

)(

(z)H

(z)EZ(z)

y

x

)z(Z


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reflection coefficient

transmission coefficient

The incident wave is totally reflected with a phase reversal,and no power is transmitted across a perfectly conductingboundary. The conducting boundary acts as a short circuit.

standing wave ratio

1)0z(E)0z(E

i

ro

0)0z(E)0z(E

i

t

o

o

min

max

1

1

E

ESWR


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Let us next consider the case of a plane wave normally incident on a dielectric discontinuity

)(zH)(zH)(zH

)(zE)(zE)(zE

tri

tri

000

000

xjkz

t

xjkz

r

xjkz

oi

eeEE

eeEE

eeEE

2

1


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2

2

1

1

1

o

21o

EEE

EEE

202

101

/

/

12

120

0

1

E

E

12

2

0

2 2

E

E

01

reflection coefficient

transmission coefficient

z=0


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avSavS

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EEL 3472 ElectromagneticWaves. 2 Electromagnetic Waves Spherical Wavefront Direction of Propagation Plane-wave approximation