ELECTROMAGNETIC WAVES

Presented by : Anup Kr Bordoloi

ECE Department ,Tezpur University11/11/2008

Electromagnetic Waves in homogeneous medium:The following field equation must be satisfied for solution of electromagnetic problem there are three constitutional relation which

determines characteristic of the medium in which the fields

exist.

Solution for free space condition:in particular case of e.m. phenomena in free space or in a perfect dielectric containing

no charge an no conduction current

Differentiating 1st

Jt

DH

t

BE

D.

0. B EJ

HB

ED

t

DH

t

BE

0. D

0. B

t

H

t

H

Also since and are independent of time

Now the 1st equation becomes on differentiating it

Taking curl of 2nd equation

(But )

this is the law that E must obey

lly for H

these are wave equation so E and H satisfy wave equation.

t

H

t

B

t

E

t

D

2

2

t

E

t

H

t

H

t

BE

t

EEE

2

22.

EEE 2.

0.1

. DE

2

22

t

EE

2

22

t

HH

t

EE

2

2

For charge free region

for uniform plane wave

There is no component in X direction be either zero, constant in

time or increasing uniformly with time .similar analysis holds for H Uniform plane electromagnetic waves are transverse and have components

in E and H only in the direction perpendicular to direction of propagationRelation between E and H in a uniform plane wave:For a plane uniform wave travelling in x direction a)E and H are both independent of y and z b)E and H have no x component

From Maxwell’s 1st equation

From Maxwell’s 2nd equation

0

0.1

.

z

E

y

E

x

E

DE

zyx

0x

Ex

xE

zx

Hy

x

HH

zx

Ey

x

EE

yz

yz

ˆˆ

ˆˆ

t

DH

yt

Ez

t

Ez

x

Hy

x

H zyyz ˆˆˆˆ

t

BE

zt

Hy

t

Hz

x

Ey

x

E yzyz ˆˆˆˆ

Comparing y and z terms from the above equations

on solving finally we get

lly

Since

The ratio has the dimension of impedance or ohms , called characteristic impedance or intrinsic impedance

of the (non conducting) medium. For space

t

H

x

Et

H

x

E

t

E

x

Ht

E

x

H

zy

yz

zy

yz

y

z

z

y

yz

H

E

H

E

EH

22

22

zy

zy

HHH

EEE

H

E

ohms

mhenry

v

v

v

v

377120

1036

1

/104

9

7

ohmsv

vv 377

The relative orientation of E and H may be determined by taking their dot product and using above relation

In a uniform plane wave ,E and H are at right angles to each other.

electric field vector crossed into the magnetic field vector gives the direction in which the wave travels.

0. zyzyzzyy HHHHHEHEHE

222 ˆˆˆ HxHHxHEHExHE yzyzzy

The wave equation for conducting medium:From Maxwell’s equation if the medium has conductivity

Taking curl of 2nd eq. ( )

For any homogeneous medium in which is constant But there is no net charge within a conductorHence wave equation for E. lly , wave equation for H.

Sinusoidal time variations: where is the frequency of

variation. time factor may be suppressed through the

use phasor notation.Time varying field may be expressed in terms of corresponding phasor

quan-tity

HE

JEH

EJ

EEH

HE

EE

EEE 2.

EEEE .2

0.

.1

.

D

DE

02 EEE 02 HHH

2

f

tEE

tEE

sin

cos

0

0

trE ,~

rE

as

Phasor is defined by

realPhase is determined by of the complex number ,time varying field quantity may be expressed as

Maxwell’s equation in phasor form:

for sinusoidal steady state we may substitute the phasor

relation as

tjerEtrE Re,~

xE tjxx erEtrE Re,~

Real axis

Imaginary axis

tj

xx eEE

xE

tjxeE

xE

tjjxx eeEE Re

~

tEx cos

Jt

DH

~

~

0Re

ReReRe

tj

tjtjtj

eJDjH

JeDet

He

JDjH

which is the differential equation in phasor form.Observation point:Time varying quantity is replaced by phasor quantityTime derivative is replaced with a factor

Maxwell’s equation becomes

The above equations contain the equation of continuity

The constitutive relation retain their forms

For sinusoidal time variations the wave equation for electric field in lossless

medium

HH ~

jt

j

0.

.

B

D

BjE

JDjH

0.

.

..

..

daB

dVdaD

daBjdsE

daJDjdsH

jJ . dVjdaJ .

EJ

HB

ED

becomes

In a conducting medium the wave equation becomes

Wave propagation in lossless medium:For uniform plane wave there is no variation w.r.t. Y or Z.

For Ey component solution may be written as

The time varying field is

real

2

22

t

EE

t

E

tE 2

E

Ejj

2

022 EjE

Ex

E

Ex

E

22

2

22

2

xjxjy eCeCE

21

xtjxtj

tjyy

eCeC

exEtxE

21Re

Re,~

When c1 and c2 are real,

if c1 = c2 the two travelling waves combine to form standing wave which

does not progress. Wave velocity: if velocity is given by

or phase –shift

constant.

From fig.

Again

xtCxtCtxEy coscos,~

21

vtxfEy

v

dt

dxv

constxt .

0t4

t

2

t

v

fv

vf

v

22

2

2

0

1vv

Thank you