Plane Wave

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Plane Wave EquationsAlan Murray

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Alan Murray University of Edinburgh

Maxwell's Equations Completed!

.

. 0

DB

DH J

BE

C t

t

Gauss( D )Gauss ( B)

Ampere

Faraday

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What does this mean?

. .t

dlE s

Bd

. .C t

sH Ddl J d

a changing magnetic field causes an electric field

a changing electric field/flux causes an magnetic field

Question : If we put these together, can we get electric andmagnetic fields that, once created, sustain one another?

Fa ra d a y

Am p e re

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Cross-breed Ampere and Faraday!C t

t t

t

... all in terms of E and H

... all in terms of

D

E

EH J

aB H

E

E nd H

dt d t

d

t

d

t

... differentiate both sides

... curl of both

EH E

HE

sides

2

2

d

d

d d

dt dt

d

t d

t

H E

HE

E

2

2

d

dt

d

dt

E

EE

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Cross-breed Ampere and Faraday!

C t

t t

t

... all in terms of E and H

... all in terms of

D

E

EH J

aB H

E

E nd H

t

t

t

... curl of b

HE

oth sid sEE E eE

H

2

2t t H HH Same equation as acquired for E

2

2

d dt

d dt

EE E

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Now some simplifications

E = (0,E Y,0) only

x

y

z

EY = E Y0sin( t - x)

Align y-axis with electric field and the x-axis with thedirection of (wave) propagation.

8/4/2019 Plane Wave

7/25Alan Murray University of Edinburgh

Travelling Waves

EY = E Y0sin( t)

EY = E Y0sin( t) EY = E Y0sin( t - f )

EY = E Y0sin( t -x)

Take a time-varying electric field,E , at a point

Add a second one with a smallphase difference, nearby

Now lets have a lot of them, with a sinusoidal variationof phase with direction x.
http://twave.ppt/http://twaveb.ppt/http://twavea.ppt/

8/4/2019 Plane Wave


Plane WaveWe will also look for a plane wave solution where the field E Y is the same (at an instant in time) across the entire zy plane.

Here is an animation to seewhat this means - looking at theyz plane, down the direction of travel

Lookdownhere

E = (0,E Y,0) only

x

y

z

EY = E Y0sin( t -x)
http://plane_anim.ppt/

8/4/2019 Plane Wave



,0 ,

0 0

y y

y

dE dE d d d dx dy dz dz dx

E

i j k

E

2 2 2 2

2 2, ,

0

y y y y

y y

d E d E d E d E d d d dx dy dz dxdy dzdy dz dx dE dE dz dx

i j kE

And, as we have simplified down to E=(0,E y,0), with | EY| constantin the zy plane, this reduces to

2

2y

y d E

dx E

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Plane wave equation for E describes the variation in time and space of an electric plane wave

With a y-component only (we have aligned the y-axis with E)propagating in the x-direction.

There is an exactly equivalent equation for H Eliminate E, not H , from the combination of Ampere and Faraday.

rather a waste of our time : in notes, but not lectured.We can, however, infer that whatever behaviour we get for E y willapply to H , although we do not yet know the direction of H .

Watch this space

2 2

2 2y y y d E dE d E

dt dx dt

Becomes the 1D equation

2

2

d d dt dt

E E

ESo (in 3D)

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What have we here?

2 2

2 2y y y d E dE d E

dt dx dt

Variation of E y in space(x=direction of propagation)

Variation of E y with time

Magnetic permeability(4 px10 7 in vacuum, larger in a magnet)

Conductivity(0 in an insulator, much larger in a conductor)

Dielectric constant(8.85x10 -12 in a vacuum, larger in a dielectric)

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Start with an insulatorto make life easy ( =0)

2 2

2 2y y d E d E

dx dt

( )0

j t x y y E E e

Look for a solution of the form

Where and depend upon and the characteristics of the insulator

2 2

2 2y y y d E dE d E

dt dx dt becomes

2 2 22 2

2 2 2

1,y y y y

d E d E E E

dx dt

2

2

1 , what does this mean??

,22

22

Remember, = =waveleng

ft

requency dh

an= f v f p

p

p p

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Still dont know what it means

Travelling waveof the form

( )0 0 cos

j t x y y y E E e E t x

212

It travels with a velocity f v p

p

In a vacuum, = 0=4 px10 -7 , = 0=8.85x10 -12

8

0 0

13 10 / ... a familiar speed?v m s

In (eg) glass, = 0=4 px10 -7 , = r 0= 5x 8.85x10 -12

8

0 0

11.43 10 / ... light slows down in glass

r

v m s

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This is why lenses work

V=3x10 8 m/s V=1.43x10 8m/s V=3x10 8m/s

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What is H up to?( )

(0, ,0) j t x

y E e

E ( )00,0 , 0,0, , j t x y y dE d j x E t e

HFaraday says E E

0 0(0,0, ) , 0,0,HH j t j x z z z t x z H H H e t

j H e So and if

( )0 0

j t x j t x z y H e E e

H E time-phaand are in in a non-conduse ctor

0 0 0 01 1

Also, z y y y H E E E

(0,0, ) (0, ,0)So and are at 90 to one another ... and z y H E H E

i Z

, the intrinsic impedance ( )of t realhe medium, is for an insulator

Lookie here

And here
http://eh_anim.ppt/http://vacuum_anim.ppt/

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Summary so far : Insulator

H and E both obey e j( t- x) H and E are in time-phase| E |=Z i| H |, Z i is the characteristicimpedance Z i is real in an insulator Z i = 377 in free space (air!)

Z i 150 in glassWave travels at a velocity v= 3x10 8 m/s in free space

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Now a conductor

Fields lead to currentsCurrents cause Joule heating (I 2R)Leads to loss of energyFields still oscillate, but they decayMultiply the solution we have alreadyby a term e - a x?

e - a x e - a x sin( t - x) HEAT! HEAT!

HEAT!

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Now a conductor >02 2

2 2 y y y d E dE d E

dt dx dt ( )

0 a j t x

y x

y E e e E Look for a solution of the form

0

j x j t y y E E e e

a

2 2 20 0 0 0a y y y y j E E j E E

. j a For tidiness, write is called the propagation constant

2

, j j j j

0x j t

y y E E e e

XX X X

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Example : Good Conductor

f a v

6x10 7 (S/m) 100MHz 6.28x10 8 8.85x10 -12 1.26x10 -6 1.54x10 5 1.54x10 5 4x10 3m/s

0 ,x j t

y y E E e e j j

3 3 5790 6 10 0.006 790 6 10 1.54 10 (1 ) j x j x j x j

Comments :v>1 rapid attenuation via e -a x

Lets have a look at e -ax

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Example : Good Conductor

e -x

0

0.10.20.30.40.5

0.60.70.8

0m 10 m 20 m 30 m

0.36=1/e

Amplitude falls by 0.36=1/e in 6 mi.e. the wave doesnt get far in copper!
http://../spreadsheets/teaching/em3/toybox/waves.xls

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Example : Good Conductor,E =Z iH . Intrinsic Impedance

00,0 , 0 ,, ,0HFaraday says E E y j t x y d e E E dx t

0 0(0,0, ) , 0,0, So and if j t j t x z x z z z j H e H H H e t HH

0 0 0y z i z

i

j E H Z H

j j j Z j j j

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Example : Good Conductor,E =Z i H . Intrinsic Impedance

40 0 0 0 0

j

y i z z z z j j

E Z H H H e H j

p

00

4

H Ey z j

E H

e p

So relates the magnitudes of and

0 0 4y z E H p and leads by It looks like this
http://conduct_anim.ppt/

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Poynting Vector (introduction only)

P = ExH is calledthe PoyntingVector

direction of travel power

Actually power/area

2

No proof but check dimensions[P] = [E][H]

[V] [I] [VI][P] = =

[L] [L] [L ][P] = Power/area

E I

H I

P

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Reflection at a Boundary

E I

H I

ET

H TER

H R

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Reflection at a BoundaryE E EH H H

I R T

I R T

1 1 2H H HH H H

I R T

I R T

Z Z Z

2 1 2 1

1 2 1 2

, H H reflection coefficientR I

Z Z Z Z

Z Z Z Z 1 1

1 2 1 2

2 2,H H transmission coefficientT I

Z Z Z Z Z Z

2 1 , 0 reflection coefficientZ Z

2 1 , 1 transmission coefficientZ Z

2 1 , 1 reflection coefficientZ Z

2 1 , 0 transmission coefficientZ Z

Plane Wave

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