Maxwell’s Equations & The Electromagnetic Wave Equation.indico.ictp.it/event/a06180/session/1/contribution/1/material/0/0.pdf · Maxwell’s Equations & The Electromagnetic Wave

SMR 1826 - 2

Preparatory School��

Winter College on Fibre Optics, Fibre Lasers and Sensors

� � � ��

Maxwell’s Equations&

The Electromagnetic Wave Equation

Imrana Ashraf Zahid��

��

5-02-2007 Preparatory School on Fiber

Optics, Fiber Lasers and Sensors

1

Maxwell’s Equations

&

The Electromagnetic Wave

Equation

Dr. Imrana Ashraf ZahidQuaid-i-Azam University, Islamabad

Pakistan



2


• Introduction

• Historical background

• Electrodynamics before Maxwell

• Maxwell’s correction to Ampere’s law

• General form of Maxwell’s equations

• Maxwell’s equations in vacuum

• Maxwell’s equations inside matter



3

Introduction

• In electrodynamics Maxwell’s equations are a set of four equations, that describes the behavior of both the electric and magnetic fields as well as their interaction with matter

• Maxwell’s four equations express

– How electric charges produce electric field (Gauss’s law)

– The absence of magnetic monopoles

– How currents and changing electric fields produces magnetic fields (Ampere’s law)

– How changing magnetic fields produces electric fields (Faraday’s law of induction)



4

Historical Background

• 1864 Maxwell in his paper “A Dynamical Theory of the Electromagnetic Field” collected all four equations

• 1884 Oliver Heaviside and Willard Gibbs gave the modern mathematical formulation using vector calculus.

• The change to vector notation produced a symmetric mathematical representation, that reinforced the perception of physical symmetries between the various fields.



5

Nomenclature

• E = Electric field

• D = Electric displacement

• B = Magnetic flux density

• H = Auxiliary field

• = Charge density

• j = Current density

• 0 (permeability of free space) = 4 10-7

• 0 (permittivity of free space) = 8.854 10-12

• c (speed of light) = 2.99792458 108 m/s



6

Electrodynamics Before Maxwell

JBiv

t

BEiii

Bii

Ei

o

o

)(

)(

0)(

)(

AB

t

AVE

Gauss’s Law

No name

Faraday’s Law

Ampere’s Law



7


(Cont’d)

Apply divergence to (iii)

.0becausezeroissidehandrightThe

zero.iscurlaofdivergencebecausezero,issidehandleftThe

B

Btt

BE

Apply divergence to (iv)

JB o



8

• The left hand side is zero, because divergence of a curl is zero.

• The right hand side is zero for steady currents i.e.,

• In electrodynamics from conservation of charge


(Cont’d)

0J

0t

tJ

is constant at any point in space which is

wrong.



9

Maxwell’s Correction to Ampere’s Law

Consider Gauss’s Law

t

E

t

tE

t

E

o

o

o

t

E

t

Do Displacement current

This result along with Ampere’s law and the conservation of charge

equation suggest that there are actually two sources of magnetic field.

The current density and displacement current.



10

Maxwell’s Correction to Ampere’s

Law (Cont’d)

Amperes law with Maxwell’s correction

t

EJB ooo



11

General Form of Maxwell’s Equations

Differential Form Integral Form

S

ooenco

C

SC

S

VoS

SdEdt

dIldB

SdBdt

dldE

SdB

dVSdE

0

1

0

t

EJB

t

BE

B

E

ooo

o



12

Maxwell’s Equations in vacuum

t

EB

t

BE

B

E

oo

0

0

• The vacuum is a linear, homogeneous, isotropic and dispersion less medium

• Since there is no current or electric charge is present in the vacuum, hence Maxwell’s equations reads as

• These equations have a simple solution interms of traveling sinusoidal waves, with the electric andmagnetic fields direction orthogonal to each other and the direction of travel



13

Maxwell’s Equations Inside Matter

Maxwell’s equations are modified for polarized and

magnetized materials.

For linear materials the polarization P and magnetization M

is given by

HM

E

m

eo

And the D and B fields are related to E and H by

.andmaterialoflitysusceptibimagnetictheis

material,oflitysusceptibielectrictheisWhere

1

1

m

e

omo

oeo

HHMHB

EEPED



14


(Cont’d)

• For polarized materials we have bound charges in addition to free charges

P

nP

b

b

MJ

nMK

b

b

• For magnetized materials we have bound currents



15


(Cont’d)

• In electrodynamics any change in the electric

polarization involves a flow of bound charges

resulting in polarization current JP

Polarization current density is due

to linear motion of charge when the

Electric polarization changes t

PJ p

pbf

bft

JJJJ t

densitycurrentTotal

densitychargeTotal



16


(Cont’d)

• Maxwell’s equations inside matter are written as

t

EJJJB

t

BE

B

E

oobopofo

o

t

0

Dt

JH

PEt

JMB

t

EM

t

PJ

B

f

of

o

of

o



17


(Cont’d)

• In non-dispersive, isotropic media and µ are time-independent scalars, and Maxwell’s equations reduces to

t

EJH

t

HE

H

E

0



18


(Cont’d)

• In uniform (homogeneous) medium and µ areindependent of position, hence Maxwell’s equations reads as

Generally, and

µ can be rank-2

tensor (3X3

matrices)

describing

birefringent

anisotropic

materials.

00

C

C

S

S

S

encff

S

encff

SdDdt

dIldH

t

EJH

SdHdt

dldE

t

HE

SdHH

QSdDD



19

The Electromagnetic Wave Equation

(EM Wave)

• The EM wave from Maxwell’s Equation

• Solution of EM wave in vacuum

• EM plane wave

• Polarization

• Energy and momentum of EM wave

• Inhomogeneous wave equation



20

The Electromagnetic Wave from


Take curl of

][t

BE

t

BE

Change the order of differentiation on the R.H.S

][ Bt

E



21


Maxwell’s Equations (cont’d)

As

t

EB oo

Substituting for we haveB

2

2

][

][][][

t

EE

t

E

tE

t

BE

oo

oo

•Assuming that µo and o are constant in time



22


Maxwell’s Equations (cont’d)

2

2

t

EE

2

22)(

t

EEE oo

0E

Using the vector identity

becomes,

In free space

And we are left with the wave equation

02

22

t

EE oo



23

The Electromagnetic Wave

from Maxwell’s Equations (cont’d)

Similarly the wave equation for magnetic field

02

22

t

BB oo

oo

c1where,



24

Solution of Electromagnetic Waves

in Vacuum

02

22

t

BB oo

The solutions to the wave equations, where there is no

source charge is present

02

22

t

EE oo

can be plane waves, obtained by method

of separation of variables



25

Solution of Electromagnetic Waves

in Vacuum (Cont’d)

ti

o

ti

o

eBB

eEE

rk

rk

Where Eo and Bo are the complex amplitudes of electric

and magnetic fields and related to each other by

relation

)ˆ(1

oo Ekc

B

Where is a propagation vector.k̂



26

Electromagnetic Plane waves

• Plane electromagnetic waves can be expressed as

Ec

neEc

B

neEE

ti

o

ti

o

kkrk

rk

ˆ1ˆˆ1

ˆ

Where is the polarization vector.n̂



27

Electromagnetic Plane waves

The real electric and magnetic fields in a

monochromatic plane wave with propagation vectorkˆ and polarization nˆ are therefore

nktrkEc

trB

ntrkEtrE

o

o

ˆ)cos(1

,

ˆ)cos(,



28

Polarization

•The polarization is specified by the orientation of the

electromagnetic field.

•The plane containing the electric field is called the

plane of polarization.



29

Polarization (Cont’d)

• Can be horizontal, vertical, circular, or elliptical

x

y

zE

Horizontal Polarization

Electric Field

Magnetic Field

Electromagnetic

Wave

x

y

z

E

Vertical Polarization



30

Polarization (Cont’d)



31

Energy and Momentum of

Electromagnetic Waves

The energy per unit volume stored in electromagnetic field is

22 1

2

1BEU

o

o

In the case of monochromatic plane wave

)(cos

1

222

22

2

2

tkxEEU

EEc

B

ooo

oo



32

Energy and Momentum of

Electromagnetic Waves (Cont’d)

• As the wave propagates, it carries this energy along with it.

The energy flux density (energy per unit area per unit time)

transported by the field is given by the poynting vector

BESo

1

For monochromatic plane waves

icUitkxEcS ooˆˆcos2

2



33

Homogenous Wave Equations Inside

Matter

MatterVacuum

2

22

2

22

2

22

2

22

11

11

t

BB

t

BB

t

EE

t

EE

oo

oo



34

Homogenous Wave Equations Inside

Matter (cont..)

rrrr

v1111

0000

n

c

v

=c = n

Permittivity: = r o ( r is dielectric constant)

Permeability: µ=µrµo (µr is relative permeability 1

n=Refractive Index



35

Inhomogeneous Electromagnetic

Wave Equation

Inside linear dielectric medium with no free charge present,

Maxwell’s equations reads as

(iv)

(iii)

(ii)0

(i)0

t

DH

t

BE

B

D

Where, HBPED oo and



36

Inhomogeneous Electromagnetic

Wave Equation (Cont’d)

Taking curl of (iii)

][][][ Ht

Bt

E o

Using (iv)

2

2

22

2

2

2

2

2

22

2

2

2

2

2

2

22

2

22

11

11

][

t

P

ct

E

cE

t

P

ct

E

cE

t

P

t

EE

t

DEE

o

o

ooo

o

Source term



37

Solution of Inhomogeneous

Electromagnetic Wave Equation

2

2

22

2

2

2 11

t

P

ct

E

cE

o

Inhomogeneous wave equation can be solved

with the help of Green’s Theorem



38

THANK YOU

Maxwell’s Equations & The Electromagnetic Wave Equation.indico.ictp.it/event/a06180/session/1/contribution/1/material/0/0.pdf · Maxwell’s Equations & The Electromagnetic Wave

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