Electromagnetic Fields Electromagnetic Fields

First Semester 2008/2009First Semester 2008/2009

Electronic and Comm. Eng. Dept.Electronic and Comm. Eng. Dept.

Course Outlines

Chapter (1):Quasi Stationary Magnetic Fields

1.1 Faraday’s law.

1.2 Induced e.m.f due to motion.

Chapter (2): Maxwell’s Equations and Plane waves

2.1 Displacement currents.

2.2 Differential and integral forms (time domain).

2.3 Sinusoidal time varying fields.

2.4 Derivation and solution of wave equation in

unbounded media.

2.5 Plane waves in different media.

2.6 Power, energy and poynting theorem.

Course Outlines (Continued)

Chapter (2): Maxwell’s Equations and Plane waves

2.7 Wave polarization and propagation modes.

2.8 Reflection and refraction of plane waves.

Chapter (3): Guided Waves

3.1 Solution of wave Eqn. in bounded media (W.G.).

3.2 Rectangular waveguides (RWG).

3.3 Transverse Electric (TE) modes.

3.4 Transverse Magnetic (TM) modes.

3.5 Power transmission and attenuation inside RWG.

3.6 RWG currents and excitation techniques.

References

1. William H. Hayt, “Engineering

Electromagnetic,” McGraw-Hill, 1989.

2. Plonsey & Collin, “ Principles and

Applications of Electromagnetic Fields,”

McGraw-Hill, 1962.

3. F.T. Ulaby, “Fundamentals of Applied

Electro-magnetic,” Prentice-Hall, 1997.

Classification of fields w.r.t. time:

Static fields: source at rest w.r.t time (static charges) .

Stationary fields: source with uniform motion w.r.t. time, (i.e. D.C.

current or t =0).

Quasi-Stationary fields: by quasi-stationary field we mean field that

is slowly varying with time (t ≠0) in such a way that all radiation

effects on the system can be neglected (for circuit of maximum

dimension D; the wave length of the operating frequency must

satisfy the condition ≥D ).

General time-varying fields: the variation of frequency is not limited

by the dimensions of the circuit.

Introduction:

(1.1) Faraday’s law of induction: In 1820 C.H. Oersted demonstrated that an electric current affected

a compass needle.

After this, Faraday professed his belief that if a current could produce a magnetic effect, then the magnetic effect should be able to produce a current (magnetism).

In 1931, the electric induction phenomenon was discovered as a results of Faraday’s experiments.

Faraday’s first experiment:If two separate coils are wound

on an iron ring. One of them is

connected through a switch to

D.C. battery. It was observed

that whenever the current was

changed, an induced current

would flow in the other coil.

Faraday’s second experiment:If a magnet moves near a coil, an induced current will be produced in the

galvanometer.

Generally, for any closed path C in space

which is linked by a changing magnetic field,

the induced voltage (e.m.f) around this path is

equal to the negative time rate of change of

the total magnetic flux through the closed

path.

t

)t( Vf.m.e ind

For N-turns loop:

Vind= - / t

.... Depend on the distribution of the

flux in each turn for N-turns loop.

S

C

(t)

Faraday’s law of induction (Basic form)

=1 +2 +…..

(different in each turn)

N-turns

= N

(same in each turn)

(t)

N-turns

The minus sign is according to Lenz’s law which states that:

“The induced voltage is in such direction that it resists the original change”

Faraday’s law in integral formintegral form:

tVind

Faraday’s law in differential formdifferential form:

SC

SdBt

dE

..

Notes:Notes:

The electric field has two sources (charges and time varying magnetic field).

If there is no time variation ( / t =0), gives (Static case).

SSC

Sd.Bt

Sd).E(d.E

t

B)E(

Faraday’s law in circuit formcircuit form:

LI I

L ; t

Vind

dt

dILV ind

(1.2) Induced e.m.f. due to motion:

(1.2.1) Moving conductor in static magnetic field:When a conductor is moving through a static magnetic field, an induced voltage is produced in the conductor. The magnitude of this voltage is found from:

Lorentz force law;

“A particle of charge ( q ) moving with velocity ( ) in magnetic

field ( ) experience a force given by, ( ) ”BvqF

B v

Example: Consider a conducting wire of length L moving with velocity through a magnetic field ( ).Bv

B v

B

)evB(F Bv eF mm v

LEach electron of charge (–e) in the

conductor experience a Lorentz force:

Which force the electron to move toward P1,

leaving positive charges at P2.

P1

P2

The displaced charges setup an induced electric field which opposes the

displacement of the charges due to Lorentz force. Then,

)eE(F EeEqF indeindinde

When sufficient charges have been built up equilibrium is established,

em FF Bv Eind The induced voltage between the ends of the conductor is given by:

2

1

2

1

P

P

P

Pind B.E

dvdVind LBvVind

mF

eF

indE

BVEin

LdBVd.E V c

P

Pindind

2

1

(1.2.2) Moving conductor in time varying magnetic field:

)t(B

v

S

C

dtv

d

dS

)t(B

After dt

d.)Bv(Sd.t

BV

CSind

Contribution due to time variation

Contribution due to motion

Example:Example: Within a certain, ε=10-11 F/m and µ=10-5 H/m. If Bx=2*10-4cos

105t sin 10-3y T: (a) Use Δ x H = ε ∂E/ ∂t to prove that:

E=-20000 cos10-3y sin105t v/m.solution

10

1010105

354 sincos2

ytHB B

Hx

x

HHH

aaa

zyx

zyx

zyxH

ˆˆˆ

0 0

aH

aH

zx

yx

yzˆˆ

= =

0

yt 1010 35 sincos20 =

azyt ˆ101010 353 coscos20

azytt

Eˆ101010 353 coscos20

dtt10cosy10cos2010

E 533

z10

1010

10

105

53

11

3 sincos

20 ty

=

at10siny10cos20000E z53

v/m

(2.1) Displacement current:The British physicist Maxwell’s was the first one who postulated the

electromagnetic wave propagation, his first study starts from the basic equations of the electric and magnetic fields including the time variation.

Introduction:

In this chapter we will concern with time varying electromagnetic field, and we shall then find that the electric and magnetic field are related to each other. I.e. a time varying magnetic field producing an electric field and a time varying electric field producing magnetic field, results in a phenomena of wave propagation.

)1(t

BE

D. )3(

JH )2(

0B. )4(

tJ. )5(

……………….

……………….

……………….

……………….

……………….

Faraday’s law of induction.

Gauss’ flux theorem.

Ampere’s circuital law.

Law of continuity of B-lines (magnetic Gauss’ law).

Continuity equation (law of conservation of charges).

0B.

).().( Bt

E

Maxwell’s pointed out that the above equations form a set which is inconsistent (He shows its inadequacy for time varying conditions). How ?

Taking the divergence of both sides of eqn.(1):

which gives the same result as eqn.(4) (i.e dependant equation).

0

)1(t

BE

………………. Faraday’s law of induction.

Taking the divergence of both sides of eqn.(2):

0J.

)D.(t

J.)H.(

0

J.)H.(

0

)D.(

tJ.

which is in contradiction with eqn.(5). For this reason, Maxwell add term to eqn.(2) which gives:

t

D

which gives the same result as eqn.(5) (i.e dependant equation).

….. Maxwell’s current density (Displacement current density)

….. Its unit is [A/m2].

dJ

t

D

….. Displacement Current [A].

s S

dd Sd.t

DSd.JI

JH )2(

………………. Ampere’s circuital law.

Differential formDifferential form Integral formIntegral form

t

DJH)2

SSCSd.D

tSd.Jd.H

Ampere’s circuital law......

RemarksRemarks

To discuss the displacement current density:

SS

dC

Sd.Dt

Sd.JId.H I

I

Example:Example: to illustrate the physical nature of the displacement current.

d.c

a.c

The –ve charges accumulate in one plate and the +ve charges accumulate at the other plate, so the dielectric material will polarized and there is no movements of dipoles (no current).

The polarity of the capacitor plates is changed which change the direction of dipoles that represent a displacement for the electrons and a current Id will flow.

Ia.c

CScC

Ia.c

Sc

ISd.Jd.HSc

)Ed(

dt

d

d

S

dt

dvCI o

oc

??d.H

c

From Ampere’s circuital law: From Ampere’s circuital law:

t

DJ

S

I d

c

ddoo

SS

(2.2) Maxwell’s equations differential and integral form (time domain):

Differential formDifferential form Integral formIntegral form

t

BE )1

D.)3

t

DJH)2

0B. )4

tJ.&

SC

S.dB t

.dE

VS

dvSd.D

SSC

Sd.Dt

Sd.Jd.H

0Sd.BS

VS

dvt

Sd.J

Faraday’s law of induction.

Gauss’ flux theorem.

Ampere’s circuital law.

Law of continuity of B-lines (magnetic Gauss’ law)

Continuity equation(law of conservation of charges)

…………

…………….

.....

…………………........

…………........

Constitutive Relations:

HB ; EJ ; ED

Where , and are the medium parameters.

RemarksRemarks

Conduction current density (Jcond):

Motion of charges usually electrons in a region of zero net charge density.

Convection current density (Jconv): Motion of volume charge density ()

J

Note:

Jcond = E

Jconv = v

Displacement current density (Jd): Third type of current density.

dJ

t

D

Electromagnetic quantities (review): Electromagnetic quantities (review):

In free-space:In free-space:**

[H/m]. 104

]. [F/m 10854.87

o

12

o

V/m]. ;er [Volts/met intensity field .Electric.......... E

A/m]. ; [Amperes/m intensity field .Mgnetic.......... H

T]. ; Teslaor wb/m; [Webers/m density flux .Magnetic.......... B 22

]C/m ; /m[Coulombsdensity current)ent (Displacemflux .Electric.......... D 22

].A/m ; [Amperes/mdensity current ..Electric.......... J 22

].C/m ; m[Coulombs/ density charge ..Electric.......... 33

H/m]. ; [Henery/mty permeabili .Magnetic..........

F/m]. ; [Farad/my permitivit ic..Dielectr..........

/m].; [Moh/mty conductivi .Electric..........

Source-free wave equation :Source-free wave equation :

Source-freeSource-free:: the solution region does not include any sources.

0 & 0Jimp

How Maxwell’s equations used to show wave equation?How Maxwell’s equations used to show wave equation?

Consider the electric and magnetic fields in a region does not include any sources, which called:

Source-freeSource-free

(2.4) Derivation and solution of wave equations in unbounded media:

2.4.1 Types of media ( according to the values of 2.4.1 Types of media ( according to the values of , µ and, µ and ). ).

2.4.2 Source-free wave equation.2.4.2 Source-free wave equation.

2.4.2.1 Time form.2.4.2.1 Time form.

a. In free-apace ( = o , µ = µo and = 0 ).

b. Lossless dielectric ( = o r , µ = µoµr and = 0 ).

c. Lossy dielectric ( =o r , µ = µoµr and ≠ 0 ).

2.4.2.2 Complex form.2.4.2.2 Complex form.

2.4.3 Properties of plane wave.2.4.3 Properties of plane wave.

There are many media to derive and solve the wave equation. Let’s start by free space.

a) In free-apace: ( = o , µ = µo and = 0 )

(1) ..... t

HE o

(3) ..... 0D.

(2) ..... t

EH o

(4) ..... 0B.

(5) ..... 0J. &

)H(t

E o

To derive the wave equation for the electric field:

Take the curl of both sides of eqn. (1) :

(2.4.2.1)Time form:(2.4.2.1)Time form:

)t

E(

tE)E.( oo

2

0

0t

HH 2

2

oo2

0

t

EE 2

2

oo2

Similarly;

0H

E

t2

2

oo2

Generally:

Source-free wave equation in free-spaceSource-free wave equation in free-space(2nd order P.D.E.)

Using vector relationship, we get

Vector relationship:

0. D

c) Lossy dielectric: (dielectric with finite conductivity)

( =o r , µ = µo µr and ≠ 0 )

Sinusoidal time varying fields:

Types of media ( according to the values of Types of media ( according to the values of , µ and, µ and ): ):

Complex form (general medium):Complex form (general medium):

Solution of source-free wave equation (complex form):Solution of source-free wave equation (complex form):

Properties of plane wave :Properties of plane wave :

Classification of mediaClassification of media

(3.1) Solution of wave equation in bounded media