1

EE750Advanced Engineering Electromagnetics

Lecture 1

2EE750, 2003, Dr. Mohamed Bakr

Notations

E: electric field intensity (V/M)

H: magnetic field intensity (A/M)

D: electric flux density (C/ M2)

B: magnetic flux density (Weber/M2)

Ji: impressed electric current density (A/M2)

Jc : conduction electric current density (A/M2)

Jd: displacement electric current density (A/M2)

µµµµi : impressed electric current density (V/M2)

µµµµd : impressed electric current density (A/M2)

3EE750, 2003, Dr. Mohamed Bakr

Notations (Cont’d)

qev: volumetric electric charge density (C/M3)

qmv : volumetric magnetic charge density (C/M3)

4EE750, 2003, Dr. Mohamed Bakr

Historical Background

• Ancient civilizations knew the effect of magneticmaterials

• Gauss’s law for electric fields

• Gauss’s law for magnetic fields

• Faraday’s law

QdVq evV

evS

=∫∫∫=∫∫ dSD.

0. =∫∫S

dSB

∫∫∂∂−=∫

SC

dSBdlE ..t

V

B

CdS

5EE750, 2003, Dr. Mohamed Bakr

Historical Background (Cont’d)

• Ampere’s law

• Ampere’s law in its original form could not beconsidered general

∫∫ ==∫SC

chargesofflowduecurrenttotal.. dSJdlH

I

IA

l

I

IA

l

I=∫C

dlH .

?0. =∫C

dlH

6EE750, 2003, Dr. Mohamed Bakr

Historical Background (Cont’d)

• Ampere’s law is modified by introducing thedisplacement current

• It follows that the 4 main laws are

∫∫ ∫∫∂∂+=∫

S SC tdSDdSJdlH ...

QdVq evV

evS

=∫∫∫=∫∫ dSD.

0. =∫∫S

dSB

∫∫∂∂−=∫

SC tdSBdlE ..

∫∫ ∫∫∂∂+=∫

S SC tdSDdSJdlH ...

7EE750, 2003, Dr. Mohamed Bakr

Maxwell’s Equations (the integral form)

• Maxwell’s equations are made symmetric by theintroduction of fictitious magnetic charges and currents

QdVq evV

evS

=∫∫∫=∫∫ dSD. QdVq mvV

mvS

=∫∫∫=∫∫ dSB.

∫∫ ∫∫∂∂+=∫

S SC tdSDdSJdlH ... ∫∫ ∫∫∂

∂−−=∫S SC t

dSBdSµdlE ...

JJJ ci +=

µµµ ci +=

8EE750, 2003, Dr. Mohamed Bakr

The Divergence Theorem

• The divergence of a vector at a point is defined as

• The divergence of a vector is a scalar value that isposition dependent

• Divergence Theorem converts a closed surfaceintegral to a volume integral over the enclosed volume

FF.dS

FV

.∆

limdiv ∆

0∆∇=

∫∫=

→ VS

∫∫∫∇=∫∫VS

dVFF.dS .

9EE750, 2003, Dr. Mohamed Bakr

The Divergence Theorem (Cont’d)

• For the ith element we have

• Summing over the N volumetric elements we get

• Notice that the flux cancels out between adjacentelements leaving only external surface flux

ViS i

∆div∆

FF.dS =∫∫

∑=∑ ∫∫==

N

ii

N

i SV

i 11∆∆div FF.dS

Iskandar 1992

10EE750, 2003, Dr. Mohamed Bakr

The Divergence Theorem (Cont’d)

• As N→∞, we get

• As an application of the divergence theorem, we have

∫∫∫∇=∫∫VS

dVFF.dS .

dVqV

evS

∫∫∫=∫∫ dSD. dVqdV.V

evV

∫∫∫=∫∫∫∇ D

q. ev=∇ D Gauss’s law in differential form

• Similarly, for the magnetic field we have

dVqmv∫∫∫=∫∫VS

dSB. dVqdV.V

mvV

∫∫∫=∫∫∫∇ B

q. vm=∇ B

11EE750, 2003, Dr. Mohamed Bakr

Stokes’ Theorem

• The curl is a measure of the rotation of a vector

Iskandar 1992

12EE750, 2003, Dr. Mohamed Bakr

Stokes’ Theorem (Cont’d)

∫ += ve.dlF

• The curl can measured through a line integral

∫ = 0.dlF ∫ −= ve.dlF

Iskandar 1992

13EE750, 2003, Dr. Mohamed Bakr

Stokes’ Theorem (Cont’d)

• The curl is a vector that have magnitude and phase

curl F= [curl F]x ax+ [curl F]y ay+[curl F]z az

zyC

zyx ∆∆

∫=×∇

→∆∆

1

0,lim][

F.dl

F

x

y

z

∆z

∆y

14EE750, 2003, Dr. Mohamed Bakr

Stokes’ Theorem (Cont’d)

• This theorem relates the value of a line integral over aclosed contour to a surface integral over the enclosedsurface

• For the ith element we have

or

nFdlF

.curl

.

lim0 Si

i

S i=

∆

∫

→∆

l

SFdlF i

i

∆=∫ .curl.l

∫∫ ×∇=∫S

dSFdlF ).(.l

15EE750, 2003, Dr. Mohamed Bakr

Stokes’ Theorem (Cont’d)

• Summing for all elements we get

• Notice that the internal line integrals cancel out and onlyintegration over the external contour remains

• It follows that as ∆Si → 0, we get

∑ ∆×∇=∑ ∫==

N

ii

N

i i 11).(. SFdlF

l

∫∫ ×∇=∫S

dSFdlF ).(.l

16EE750, 2003, Dr. Mohamed Bakr

Applications of Stokes’ Theorem

∫∫ ∫∫∂∂−−=∫

S SC tdSBdSµdlE ...

• Starting with the Maxwell’s integral equation

∫∫ ∫∫∂∂−−=∫∫ ×∇

S SS tdSBdSµdSE ..).(

apply Stokes’ Theorem

t∂∂−−=×∇ BµE)(

• Similarly, starting with

We get

∫∫ ∫∫∂∂+=∫

S SC tdSDdSJdlH ...

t∂∂+=×∇ D

JH )(

17EE750, 2003, Dr. Mohamed Bakr

Maxwell’s Equations

q. ev=∇ D

q. vm=∇ B

t∂∂−−=×∇ BµE)(

t∂∂+=×∇ D

JH )(

Differential form

QdVq evV

evS

=∫∫∫=∫∫ dSD.

QdVq mvV

mvS

=∫∫∫=∫∫ dSB.

∫∫ ∫∫∂∂−−=∫

S SC tdSBdSµdlE ...

∫∫ ∫∫∂∂+=∫

S SC tdSDdSJdlH ...

Integral form

18EE750, 2003, Dr. Mohamed Bakr

The Constitutive Relations

• A material is characterized by its constitutiveparameters ε, µ and σ

• For example,

• For frequency independent permittivity and forfrequency-domain analysis we have D=ε E

• For free space we have εo=10-9/(36π) F/M

• Similarly, B=µ*H B=µH (frequencyindependent or single frequency analysis)

• For free space we have µo=4π×10-7 H/M

τd)()()( ττtεt ED −∫=∞

∞−

19EE750, 2003, Dr. Mohamed Bakr

The Constitutive Parameters (Cont’d)

• Also, J=σ*E J=σE (frequency independent orsingle frequency analysis)

• For free space, σ o=0 S