Lect09

1

EEE 498/598EEE 498/598Overview of Electrical Overview of Electrical

EngineeringEngineering

Lecture 9: Faraday’s Law Of Lecture 9: Faraday’s Law Of Electromagnetic Induction; Electromagnetic Induction;

Displacement Current; Complex Displacement Current; Complex Permittivity and PermeabilityPermittivity and Permeability

2 Lecture 9

Lecture 9 ObjectivesLecture 9 Objectives

To study Faraday’s law of electromagnetic To study Faraday’s law of electromagnetic induction; displacement current; and induction; displacement current; and complex permittivity and permeability.complex permittivity and permeability.

3 Lecture 9

Fundamental Laws of Fundamental Laws of ElectrostaticsElectrostatics

Integral formIntegral form Differential formDifferential form

∫∫

∫=⋅

=⋅

V

ev

S

C

dvqsdD

ldE 0

evqD

E

=⋅∇=×∇ 0

ED ε=

4 Lecture 9

Fundamental Laws of Fundamental Laws of MagnetostaticsMagnetostatics

Integral formIntegral form Differential formDifferential form

0=⋅

⋅=⋅

∫

∫∫

S

SC

sdB

sdJldH

0=⋅∇=×∇

B

JH

HB µ=

5 Lecture 9

Electrostatic, Magnetostatic, and Electrostatic, Magnetostatic, and Electromagnetostatic FieldsElectromagnetostatic Fields

In the static case (no time variation), the electric In the static case (no time variation), the electric field (specified by field (specified by EE and and DD) and the magnetic ) and the magnetic field (specified by field (specified by BB and and HH) are described by ) are described by separate and independent sets of equations.separate and independent sets of equations.

In a conducting medium, both electrostatic and In a conducting medium, both electrostatic and magnetostatic fields can exist, and are coupled magnetostatic fields can exist, and are coupled through the Ohm’s law (through the Ohm’s law (JJ = = σσEE). Such a ). Such a situation is called situation is called electromagnetostaticelectromagnetostatic ..

6 Lecture 9

Electromagnetostatic FieldsElectromagnetostatic Fields

In an electromagnetostatic field, the electric field In an electromagnetostatic field, the electric field is completely determined by the stationary is completely determined by the stationary charges present in the system, and the magnetic charges present in the system, and the magnetic field is completely determined by the current.field is completely determined by the current.

The magnetic field does not enter into the The magnetic field does not enter into the calculation of the electric field, nor does the calculation of the electric field, nor does the electric field enter into the calculation of the electric field enter into the calculation of the magnetic field.magnetic field.

7 Lecture 9

The Three Experimental Pillars The Three Experimental Pillars of Electromagneticsof Electromagnetics

Electric charges attract/repel each other as Electric charges attract/repel each other as described by described by Coulomb’s lawCoulomb’s law..

Current-carrying wires attract/repel each other Current-carrying wires attract/repel each other as described by as described by Ampere’s law of forceAmpere’s law of force..

Magnetic fields that change with time induce Magnetic fields that change with time induce electromotive force as described by electromotive force as described by Faraday’s Faraday’s lawlaw..

8 Lecture 9

Faraday’s ExperimentFaraday’s Experiment

battery

switch

toroidal ironcore

compass

primarycoil

secondarycoil

9 Lecture 9

Faraday’s Experiment (Cont’d)Faraday’s Experiment (Cont’d)

Upon closing the switch, current begins to flow Upon closing the switch, current begins to flow in the in the primary coilprimary coil..

A momentary deflection of the A momentary deflection of the compasscompass needleneedle indicates a brief surge of current flowing in the indicates a brief surge of current flowing in the secondary coilsecondary coil..

The The compass needlecompass needle quickly settles back to quickly settles back to zero.zero.

Upon opening the switch, another brief Upon opening the switch, another brief deflection of the deflection of the compass needlecompass needle is observed. is observed.

10 Lecture 9

Faraday’s Law of Faraday’s Law of Electromagnetic InductionElectromagnetic Induction

““The electromotive force induced around a The electromotive force induced around a closed loop closed loop CC is equal to the time rate of is equal to the time rate of decrease of the magnetic flux linking the loop.”decrease of the magnetic flux linking the loop.”

C

S

dt

dVind

Φ−=

11 Lecture 9

Faraday’s Law of Electromagnetic Faraday’s Law of Electromagnetic Induction (Cont’d)Induction (Cont’d)

∫ ⋅=ΦS

sdB• S is any surface bounded by C∫ ⋅=

C

ind ldEV

∫∫ ⋅−=⋅SC

sdBdt

dldE

integral form of Faraday’s

law

12 Lecture 9

Faraday’s Law (Cont’d)Faraday’s Law (Cont’d)

∫∫

∫∫

⋅∂∂−=⋅−

⋅×∇=⋅

SS

SC

sdt

BsdB

dt

d

sdEldE

Stokes’s theorem

assuming a stationary surface S

13 Lecture 9


Since the above must hold for any Since the above must hold for any SS, we have, we have

t

BE

∂∂−=×∇

differential form of Faraday’s law

(assuming a stationary frame

of reference)

14 Lecture 9


Faraday’s law states that a changing Faraday’s law states that a changing magnetic field induces an electric field.magnetic field induces an electric field.

The induced electric field is The induced electric field is non-non-conservativeconservative..

15 Lecture 9

Lenz’s LawLenz’s Law

““The sense of the emf induced by the time-The sense of the emf induced by the time-varying magnetic flux is such that any current it varying magnetic flux is such that any current it produces tends to set up a magnetic field that produces tends to set up a magnetic field that opposes the opposes the changechange in the original magnetic in the original magnetic field.”field.”

Lenz’s law is a consequence of conservation of Lenz’s law is a consequence of conservation of energy.energy.

Lenz’s law explains the minus sign in Faraday’s Lenz’s law explains the minus sign in Faraday’s law.law.

16 Lecture 9

Faraday’s LawFaraday’s Law ““The electromotive force induced around a The electromotive force induced around a

closed loop closed loop CC is equal to the time rate of is equal to the time rate of decrease of the magnetic flux linking the decrease of the magnetic flux linking the loop.”loop.”

For a coil of N tightly wound turnsFor a coil of N tightly wound turns

dt

dVind

Φ−=

dt

dNVind

Φ−=

17 Lecture 9

∫ ⋅=ΦS

sdB

• S is any surface bounded by C∫ ⋅=

C

ind ldEV


C

S

18 Lecture 9


Faraday’s law applies to situations whereFaraday’s law applies to situations where (1) the (1) the BB-field is a function of time-field is a function of time (2) (2) ddss is a function of time is a function of time (3) (3) BB and and ddss are functions of time are functions of time

19 Lecture 9


The induced The induced emfemf around a circuit can be around a circuit can be separated into two terms:separated into two terms: (1) due to the time-rate of change of the B-(1) due to the time-rate of change of the B-

field (field (transformer emftransformer emf)) (2) due to the motion of the circuit ((2) due to the motion of the circuit (motional motional

emfemf))

20 Lecture 9


( )∫∫

∫

⋅×+⋅∂∂−=

⋅−=

CS

S

ind

ldBvsdt

B

sdBdt

dV

transformer emfmotional emf

21 Lecture 9

Moving Conductor in a Static Moving Conductor in a Static Magnetic FieldMagnetic Field

Consider a conducting bar moving with Consider a conducting bar moving with velocity velocity vv in a magnetostatic field: in a magnetostatic field:

Bv

2

1+

-• The magnetic force on an electron in the conducting bar is given by

BveF m ×−=

22 Lecture 9

Moving Conductor in a Static Moving Conductor in a Static Magnetic Field (Cont’d)Magnetic Field (Cont’d)

Electrons are pulled Electrons are pulled toward end toward end 22. End . End 22 becomes negatively becomes negatively charged and end charged and end 11 becomes becomes ++ charged. charged.

An electrostatic force An electrostatic force of attraction is of attraction is established between the established between the two ends of the bar.two ends of the bar.

Bv

2

1+

-

23 Lecture 9


The electrostatic force on an electron The electrostatic force on an electron due to the induced electrostatic field is due to the induced electrostatic field is given bygiven by

The migration of electrons stops The migration of electrons stops (equilibrium is established) when(equilibrium is established) when

EeF e −=

BvEFF me ×−=⇒−=

24 Lecture 9


A A motionalmotional (or “flux cutting”) (or “flux cutting”) emfemf is is produced given byproduced given by

( ) ldBvVind ⋅×= ∫1

2

25 Lecture 9

Electric Field in Terms of Electric Field in Terms of Potential FunctionsPotential Functions

Electrostatics:Electrostatics:

Φ−∇=⇒=×∇ EE 0

scalar electric potential

26 Lecture 9

Electric Field in Terms of Electric Field in Terms of Potential Functions (Cont’d)Potential Functions (Cont’d)

Electrodynamics:Electrodynamics:

( )

Φ−∇=∂∂+⇒=

∂∂+×∇

×∇∂∂−=

∂∂−=×∇

t

AE

t

AE

Att

BE

0

AB ×∇=

27 Lecture 9

Electric Field in Terms of Electric Field in Terms of Potential Functions (Cont’d)Potential Functions (Cont’d)

Electrodynamics:Electrodynamics:

t

AE

∂∂−Φ−∇=

scalar electric

potential

vector magnetic potential

• both of these potentials are now functions of time.

28 Lecture 9

The differential form of Ampere’s law in The differential form of Ampere’s law in the static case isthe static case is

The continuity equation isThe continuity equation is

JH =×∇

0=∂

∂+⋅∇t

qJ ev

Ampere’s Law and the Continuity Ampere’s Law and the Continuity EquationEquation

29 Lecture 9

Ampere’s Law and the Continuity Ampere’s Law and the Continuity Equation (Cont’d)Equation (Cont’d)

In the time-varying case, Ampere’s law in In the time-varying case, Ampere’s law in the above form is inconsistent with the the above form is inconsistent with the continuity equationcontinuity equation

( ) 0=×∇⋅∇=⋅∇ HJ

30 Lecture 9


To resolve this inconsistency, Maxwell To resolve this inconsistency, Maxwell modified Ampere’s law to readmodified Ampere’s law to read

t

DJH c ∂

∂+=×∇

conduction current density

displacement current density

31 Lecture 9


The new form of Ampere’s law is The new form of Ampere’s law is consistent with the continuity equation as consistent with the continuity equation as well as with the differential form of well as with the differential form of Gauss’s lawGauss’s law

( ) ( ) 0=×∇⋅∇=⋅∇∂∂+⋅∇ HDt

J c

qev

32 Lecture 9

Displacement CurrentDisplacement Current

Ampere’s law can be written asAmpere’s law can be written as

dc JJH +=×∇

where

)(A/mdensity current nt displaceme 2=∂∂=

t

DJ d

33 Lecture 9

Displacement Current (Cont’d)Displacement Current (Cont’d)

Displacement currentDisplacement current is the type of current is the type of current that flows between the plates of a capacitor.that flows between the plates of a capacitor.

Displacement currentDisplacement current is the mechanism is the mechanism which allows electromagnetic waves to which allows electromagnetic waves to propagate in a non-conducting medium.propagate in a non-conducting medium.

Displacement currentDisplacement current is a consequence of is a consequence of the three experimental pillars of the three experimental pillars of electromagnetics.electromagnetics.

34 Lecture 9

Displacement Current in a Displacement Current in a CapacitorCapacitor

Consider a parallel-plate capacitor with plates of Consider a parallel-plate capacitor with plates of area area AA separated by a dielectric of permittivity separated by a dielectric of permittivity εε and thickness and thickness dd and connected to an and connected to an acac generator:generator:

tVtv ωcos)( 0=+

-z = 0

z = d εicA

id

z

35 Lecture 9

Displacement Current in a Displacement Current in a Capacitor (Cont’d)Capacitor (Cont’d)

The electric field and displacement flux The electric field and displacement flux density in the capacitor is given bydensity in the capacitor is given by

The displacement current density is The displacement current density is given bygiven by

td

VaED

td

Va

d

tvaE

z

zz

ωεε

ω

cosˆ

cosˆ)(

ˆ

0

0

−==

−=−= • assume fringing is negligible

td

Va

t

DJ zd ωωε

sinˆ 0=∂∂=

36 Lecture 9

c

d

S

dd

idt

dvCtCV

tVd

AAJsdJi

==−=

−=−=⋅= ∫

ωω

ωωε

sin

sin

0

0

Displacement Current in a Displacement Current in a Capacitor (Cont’d)Capacitor (Cont’d)

The displacement current is given byThe displacement current is given by

conduction current in

wire

37 Lecture 9

Conduction to Displacement Conduction to Displacement Current RatioCurrent Ratio

Consider a conducting medium characterized by Consider a conducting medium characterized by conductivity conductivity σσ and permittivity and permittivity εε..

The conduction current density is given byThe conduction current density is given by

The displacement current density is given byThe displacement current density is given by

EJ c σ=

t

EJ d ∂

∂= ε

38 Lecture 9

Conduction to Displacement Conduction to Displacement Current Ratio (Cont’d)Current Ratio (Cont’d)

Assume that the electric field is a sinusoidal Assume that the electric field is a sinusoidal function of time:function of time:

Then, Then, tEE ωcos0=

tEJ

tEJ

d

c

ωωεωσsin

cos

0

0

−==

39 Lecture 9


We haveWe have

Therefore Therefore

0max

0max

EJ

EJ

d

c

ωε

σ

=

=

ωεσ=

max

max

d

c

J

J

40 Lecture 9


The value of the quantity The value of the quantity σ/ωεσ/ωε at a specified at a specified frequency determines the properties of the frequency determines the properties of the medium at that given frequency.medium at that given frequency.

In a metallic conductor, the displacement In a metallic conductor, the displacement current is negligible below optical frequencies.current is negligible below optical frequencies.

In free space (or other perfect dielectric), the In free space (or other perfect dielectric), the conduction current is zero and only conduction current is zero and only displacement current can exist. displacement current can exist.

41 Lecture 9

100

102

104

106

108

1010

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

Frequency (Hz)

Humid Soil (εr = 30, σ = 10-2 S/m)

ωεσ

goodconductor

good insulator


42 Lecture 9

Complex PermittivityComplex Permittivity

In a good insulator, the conduction current (due to In a good insulator, the conduction current (due to non-zero non-zero σσ) is usually negligible.) is usually negligible.

However, at high frequencies, the rapidly varying However, at high frequencies, the rapidly varying electric field has to do work against molecular forces electric field has to do work against molecular forces in alternately polarizing the bound electrons.in alternately polarizing the bound electrons.

The result is that The result is that PP is not necessarily in phase with is not necessarily in phase with EE, and the electric susceptibility, and hence the , and the electric susceptibility, and hence the dielectric constant, are complex.dielectric constant, are complex.

43 Lecture 9

Complex Permittivity (Cont’d)Complex Permittivity (Cont’d)

The The complex dielectric constantcomplex dielectric constant can be written can be written asas

Substituting the complex dielectric constant into Substituting the complex dielectric constant into the differential frequency-domain form of the differential frequency-domain form of Ampere’s law, we haveAmpere’s law, we have

εεε ′′−′= jc

EEjEH εωεωσ ′′+′+=×∇

44 Lecture 9

Complex Permittivity (Cont’d)Complex Permittivity (Cont’d) Thus, the imaginary part of the complex Thus, the imaginary part of the complex

permittivity leads to a volume current density permittivity leads to a volume current density term that is in phase with the electric field, as term that is in phase with the electric field, as if the material had an effective conductivity if the material had an effective conductivity given bygiven by

The power dissipated per unit volume in the The power dissipated per unit volume in the medium is given bymedium is given by

εωσσ ′′+=eff

222 EEEeff εωσσ ′′+=

45 Lecture 9


The term The term ωε′′ ωε′′ EE22 is the basis for microwave is the basis for microwave heating of dielectric materials.heating of dielectric materials.

Often in dielectric materials, we do not Often in dielectric materials, we do not distinguish between distinguish between σσ and and ωε′′ωε′′, and lump them , and lump them together in together in ωε′′ωε′′ as as

effσεω =′′• The value of σeff is often determined by

measurements.

46 Lecture 9

Complex Permittivity (Cont’d)Complex Permittivity (Cont’d) In general, both In general, both ε′ε′ and and ε′′ε′′ depend on depend on

frequency, exhibiting resonance frequency, exhibiting resonance characteristics at several frequencies. characteristics at several frequencies.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Normalized Frequency

Rea

l Par

t of

Die

lect

ric C

ons

tant

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Frequency

Imag

Par

t of

Die

lect

ric C

ons

tant

47 Lecture 9


In tabulating the dielectric properties of In tabulating the dielectric properties of materials, it is customary to specify the real part materials, it is customary to specify the real part of the dielectric constant (of the dielectric constant (ε′ε′ / / εε00) and the loss ) and the loss

tangent (tangent (tantanδδ) defined as) defined as

εεδ

′′′

=tan

48 Lecture 9

Complex PermeabilityComplex Permeability

Like the electric field, the magnetic field Like the electric field, the magnetic field encounters molecular forces which require work encounters molecular forces which require work to overcome in magnetizing the material.to overcome in magnetizing the material.

In analogy with permittivity, the permeability In analogy with permittivity, the permeability can also be complex can also be complex

µµµ ′′−′= jc

49 Lecture 9

Maxwell’s Equations in Differential Form for Maxwell’s Equations in Differential Form for Time-Harmonic Fields in Simple MediumTime-Harmonic Fields in Simple Medium

( )( )

µ

ε

σωεσωµ

mv

ev

ie

im

qH

qE

JEjH

KHjE

=⋅∇

=⋅∇

++=×∇−+−=×∇

50 Lecture 9

Maxwell’s Curl Equations for Time-Harmonic Maxwell’s Curl Equations for Time-Harmonic Fields in Simple Medium Using Complex Fields in Simple Medium Using Complex

Permittivity and PermeabilityPermittivity and Permeability

i

i

JEjH

KHjE

+=×∇−−=×∇

ωεωµ

complexpermittivity

complexpermeability

Lect09

Engineering

Transcript of Lect09