Lect09

1
EEE 498/598EEE 498/598
Overview of ElectricalOverview of Electrical
EngineeringEngineering
Lecture 9: Faraday’s Law OfLecture 9: Faraday’s Law Of
Electromagnetic Induction;Electromagnetic Induction;
Displacement Current; ComplexDisplacement Current; Complex
Permittivity and PermeabilityPermittivity and Permeability

2 Lecture 9
Lecture 9 ObjectivesLecture 9 Objectives
 To study Faraday’s law of electromagneticTo study Faraday’s law of electromagnetic
induction; displacement current; andinduction; displacement current; and
complex permittivity and permeability.complex permittivity and permeability.

3 Lecture 9
Fundamental Laws ofFundamental 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 ofFundamental Laws of
MagnetostaticsMagnetostatics
 Integral formIntegral form  Differential formDifferential form
0=⋅
⋅=⋅
∫
∫∫
S
SC
sdB
sdJldH
0=⋅∇
=×∇
B
JH
HB µ=

5 Lecture 9
Electrostatic, Magnetostatic, andElectrostatic, Magnetostatic, and
Electromagnetostatic FieldsElectromagnetostatic Fields
 In the static case (no time variation), the electricIn the static case (no time variation), the electric
field (specified byfield (specified by EE andand DD) and the magnetic) and the magnetic
field (specified byfield (specified by BB andand 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 andIn a conducting medium, both electrostatic and
magnetostatic fields can exist, and are coupledmagnetostatic fields can exist, and are coupled
through the Ohm’s law (through the Ohm’s law (JJ == σσEE). Such a). Such a
situation is calledsituation is called electromagnetostaticelectromagnetostatic..

6 Lecture 9
Electromagnetostatic FieldsElectromagnetostatic Fields
 In an electromagnetostatic field, the electric fieldIn an electromagnetostatic field, the electric field
is completely determined by the stationaryis completely determined by the stationary
charges present in the system, and the magneticcharges 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 theThe magnetic field does not enter into the
calculation of the electric field, nor does thecalculation of the electric field, nor does the
electric field enter into the calculation of theelectric field enter into the calculation of the
magnetic field.magnetic field.

7 Lecture 9
The Three Experimental PillarsThe Three Experimental Pillars
of Electromagneticsof Electromagnetics
 Electric charges attract/repel each other asElectric charges attract/repel each other as
described bydescribed by Coulomb’s lawCoulomb’s law..
 Current-carrying wires attract/repel each otherCurrent-carrying wires attract/repel each other
as described byas described by Ampere’s law of forceAmpere’s law of force..
 Magnetic fields that change with time induceMagnetic fields that change with time induce
electromotive force as described byelectromotive force as described by Faraday’sFaraday’s
lawlaw..

8 Lecture 9
Faraday’s ExperimentFaraday’s Experiment
battery
switch
toroidal iron
core
compass
primary
coil
secondary
coil

9 Lecture 9
Faraday’s Experiment (Cont’d)Faraday’s Experiment (Cont’d)
 Upon closing the switch, current begins to flowUpon closing the switch, current begins to flow
in thein the primary coilprimary coil..
 A momentary deflection of theA momentary deflection of the compasscompass needleneedle
indicates a brief surge of current flowing in theindicates a brief surge of current flowing in the
secondary coilsecondary coil..
 TheThe compass needlecompass needle quickly settles back toquickly settles back to
zero.zero.
 Upon opening the switch, another briefUpon opening the switch, another brief
deflection of thedeflection of the compass needlecompass needle is observed.is observed.

10 Lecture 9
Faraday’s Law ofFaraday’s Law of
Electromagnetic InductionElectromagnetic Induction
 ““The electromotive force induced around aThe electromotive force induced around a
closed loopclosed loop CC is equal to the time rate ofis 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
d
Vind
Φ
−=

11 Lecture 9
Faraday’s Law of ElectromagneticFaraday’s Law of Electromagnetic
Induction (Cont’d)Induction (Cont’d)
∫ ⋅=Φ
S
sdB
• S is any surface
bounded by C
∫ ⋅=
C
ind ldEV
∫∫ ⋅−=⋅
SC
sdB
dt
d
ldE
integral form
of Faraday’s
law

12 Lecture 9
Faraday’s Law (Cont’d)Faraday’s Law (Cont’d)
∫∫
∫∫
⋅
∂
∂
−=⋅−
⋅×∇=⋅
SS
SC
sd
t
B
sdB
dt
d
sdEldE
Stokes’s theorem
assuming a stationary surface S

13 Lecture 9
 Since the above must hold for anySince the above must hold for any SS, we have, we have
t
B
E
∂
∂
−=×∇
differential form
of Faraday’s law
(assuming a
stationary frame
of reference)

14 Lecture 9
 Faraday’s law states that a changingFaraday’s law states that a changing
magnetic field induces an electric field.magnetic field induces an electric field.
 The induced electric field isThe 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 itvarying magnetic flux is such that any current it
produces tends to set up a magnetic field thatproduces tends to set up a magnetic field that
opposes theopposes the changechange in the original magneticin the original magnetic
field.”field.”
 Lenz’s law is a consequence of conservation ofLenz’s law is a consequence of conservation of
energy.energy.
 Lenz’s law explains the minus sign in Faraday’sLenz’s law explains the minus sign in Faraday’s
law.law.

16 Lecture 9
Faraday’s LawFaraday’s Law
 ““The electromotive force induced around aThe electromotive force induced around a
closed loopclosed loop CC is equal to the time rate ofis equal to the time rate of
decrease of the magnetic flux linking thedecrease of the magnetic flux linking the
loop.”loop.”
 For a coil of N tightly wound turnsFor a coil of N tightly wound turns
dt
d
Vind
Φ
−=
dt
d
NVind
Φ
−=

17 Lecture 9
∫ ⋅=Φ
S
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 timeis a function of time
 (3)(3) BB andand ddss are functions of timeare functions of time

19 Lecture 9
 The inducedThe induced emfemf around a circuit can bearound 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 (motionalmotional
emfemf))

20 Lecture 9
( )∫∫
∫
⋅×+⋅
∂
∂
−=
⋅−=
CS
S
ind
ldBvsd
t
B
sdB
dt
d
V
transformer emf
motional emf

21 Lecture 9
Moving Conductor in a StaticMoving Conductor in a Static
Magnetic FieldMagnetic Field
 Consider a conducting bar moving withConsider a conducting bar moving with
velocityvelocity vv in a magnetostatic field:in a magnetostatic field:
B
v
2
1
+
-
• The magnetic force on an
electron in the conducting
bar is given by
BveFm ×−=

22 Lecture 9
Magnetic Field (Cont’d)Magnetic Field (Cont’d)
 Electrons are pulledElectrons are pulled
toward endtoward end 22. End. End 22
becomes negativelybecomes negatively
charged and endcharged and end 11
becomesbecomes ++ charged.charged.
 An electrostatic forceAn electrostatic force
of attraction isof attraction is
established between theestablished between the
two ends of the bar.two ends of the bar.
B
v
2
1
+
-

23 Lecture 9
 The electrostatic force on an electronThe electrostatic force on an electron
due to the induced electrostatic field isdue to the induced electrostatic field is
given bygiven by
 The migration of electrons stopsThe migration of electrons stops
(equilibrium is established) when(equilibrium is established) when
EeFe −=
BvEFF me ×−=⇒−=

24 Lecture 9
 AA motionalmotional (or “flux cutting”)(or “flux cutting”) emfemf isis
produced given byproduced given by
( ) ldBvVind ⋅×= ∫
1
2

25 Lecture 9
Electric Field in Terms ofElectric Field in Terms of
Potential FunctionsPotential Functions
 Electrostatics:Electrostatics:
Φ−∇=⇒=×∇ EE 0
scalar electric potential

26 Lecture 9
Potential Functions (Cont’d)Potential Functions (Cont’d)
 Electrodynamics:Electrodynamics:
( )
Φ−∇=
∂
∂
+⇒=





∂
∂
+×∇
×∇
∂
∂
−=
∂
∂
−=×∇
t
A
E
t
A
E
A
tt
B
E
0
AB ×∇=

27 Lecture 9
Potential Functions (Cont’d)Potential Functions (Cont’d)
 Electrodynamics:Electrodynamics:
t
A
E
∂
∂
−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 inThe differential form of Ampere’s law in
the static case isthe static case is
 The continuity equation isThe continuity equation is
JH =×∇
0=
∂
∂
+⋅∇
t
q
J ev
Ampere’s Law and the ContinuityAmpere’s Law and the Continuity
EquationEquation

29 Lecture 9
Equation (Cont’d)Equation (Cont’d)
 In the time-varying case, Ampere’s law inIn the time-varying case, Ampere’s law in
the above form is inconsistent with thethe above form is inconsistent with the
continuity equationcontinuity equation
( ) 0=×∇⋅∇=⋅∇ HJ

30 Lecture 9
 To resolve this inconsistency, MaxwellTo resolve this inconsistency, Maxwell
modified Ampere’s law to readmodified Ampere’s law to read
t
D
JH c
∂
∂
+=×∇
conduction
current density
displacement
current density

31 Lecture 9
 The new form of Ampere’s law isThe new form of Ampere’s law is
consistent with the continuity equation asconsistent with the continuity equation as
well as with the differential form ofwell as with the differential form of
Gauss’s lawGauss’s law
( ) ( ) 0=×∇⋅∇=⋅∇
∂
∂
+⋅∇ HD
t
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/mdensitycurrentntdisplaceme 2
=
∂
∂
=
t
D
J d

33 Lecture 9
Displacement Current (Cont’d)Displacement Current (Cont’d)
 Displacement currentDisplacement current is the type of currentis 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 mechanismis the mechanism
which allows electromagnetic waves towhich allows electromagnetic waves to
propagate in a non-conducting medium.propagate in a non-conducting medium.
 Displacement currentDisplacement current is a consequence ofis a consequence of
the three experimental pillars ofthe three experimental pillars of
electromagnetics.electromagnetics.

34 Lecture 9
Displacement Current in aDisplacement Current in a
CapacitorCapacitor
 Consider a parallel-plate capacitor with plates ofConsider a parallel-plate capacitor with plates of
areaarea AA separated by a dielectric of permittivityseparated by a dielectric of permittivity εε
and thicknessand thickness dd and connected to anand connected to an acac
generator:generator:
tVtv ωcos)( 0=
+
-
z = 0
z = d
ε
icA
id
z

35 Lecture 9
Capacitor (Cont’d)Capacitor (Cont’d)
 The electric field and displacement fluxThe electric field and displacement flux
density in the capacitor is given bydensity in the capacitor is given by
 The displacement current density isThe displacement current density is
given bygiven by
t
d
V
aED
t
d
V
a
d
tv
aE
z
zz
ω
ε
ε
ω
cosˆ
cosˆ
)(
ˆ
0
0
−==
−=−= • assume
fringing is
negligible
t
d
V
a
t
D
J zd ω
ωε
sinˆ 0
=
∂
∂
=

36 Lecture 9
c
d
S
dd
i
dt
dv
CtCV
tV
d
A
AJsdJi
==−=
−=−=⋅= ∫
ωω
ω
ωε
sin
sin
0
0
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 DisplacementConduction to Displacement
Current RatioCurrent Ratio
 Consider a conducting medium characterized byConsider a conducting medium characterized by
conductivityconductivity σσ and permittivityand 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
E
J d
∂
∂
= ε

38 Lecture 9
Current Ratio (Cont’d)Current Ratio (Cont’d)
 Assume that the electric field is a sinusoidalAssume that the electric field is a sinusoidal
function of time:function of time:
 Then,Then,
tEE ωcos0=
tEJ
tEJ
d
c
ωωε
ωσ
sin
cos
0
0
−=
=

39 Lecture 9
 We haveWe have
 ThereforeTherefore
0max
0max
EJ
EJ
d
c
ωε
σ
=
=
ωε
σ
=
max
max
d
c
J
J

40 Lecture 9
 The value of the quantityThe value of the quantity σ/ωεσ/ωε at a specifiedat a specified
frequency determines the properties of thefrequency determines the properties of the
medium at that given frequency.medium at that given frequency.
 In a metallic conductor, the displacementIn a metallic conductor, the displacement
current is negligible below optical frequencies.current is negligible below optical frequencies.
 In free space (or other perfect dielectric), theIn free space (or other perfect dielectric), the
conduction current is zero and onlyconduction current is zero and only
displacement current can exist.displacement current can exist.

41 Lecture 9
10
0
10
2
10
4
10
6
10
8
10
10
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Frequency (Hz)
Humid Soil (εr
= 30, σ = 10-2
S/m)
ωε
σ
good
conductor
good insulator

42 Lecture 9
Complex PermittivityComplex Permittivity
 In a good insulator, the conduction current (due toIn a good insulator, the conduction current (due to
non-zeronon-zero σσ) is usually negligible.) is usually negligible.
 However, at high frequencies, the rapidly varyingHowever, at high frequencies, the rapidly varying
electric field has to do work against molecular forceselectric field has to do work against molecular forces
in alternately polarizing the bound electrons.in alternately polarizing the bound electrons.
 The result is thatThe result is that PP is not necessarily in phase withis 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)
 TheThe complex dielectric constantcomplex dielectric constant can be writtencan be written
asas
 Substituting the complex dielectric constant intoSubstituting the complex dielectric constant into
the differential frequency-domain form ofthe differential frequency-domain form of
Ampere’s law, we haveAmpere’s law, we have
εεε ′′−′= jc
EEjEH εωεωσ ′′+′+=×∇

44 Lecture 9
 Thus, the imaginary part of the complexThus, the imaginary part of the complex
permittivity leads to a volume current densitypermittivity leads to a volume current density
term that is in phase with the electric field, asterm that is in phase with the electric field, as
if the material had an effective conductivityif the material had an effective conductivity
given bygiven by
 The power dissipated per unit volume in theThe power dissipated per unit volume in the
medium is given bymedium is given by
εωσσ ′′+=eff
222
EEEeff εωσσ ′′+=

45 Lecture 9
 The termThe term ωε′′ωε′′ EE22
is the basis for microwaveis the basis for microwave
heating of dielectric materials.heating of dielectric materials.
 Often in dielectric materials, we do notOften in dielectric materials, we do not
distinguish betweendistinguish between σσ andand ωε′′ωε′′, and lump them, and lump them
together intogether in ωε′′ωε′′ asas
effσεω =′′
• The value of σeff is
often determined by
measurements.

46 Lecture 9
 In general, bothIn general, both ε′ε′ andand ε′′ε′′ depend ondepend on
frequency, exhibiting resonancefrequency, exhibiting resonance
characteristics at several frequencies.characteristics at several frequencies.
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
2.5
Normalized Frequency
RealPartofDielectricConstant
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Frequency
ImagPartofDielectricConstant

47 Lecture 9
 In tabulating the dielectric properties ofIn tabulating the dielectric properties of
materials, it is customary to specify the real partmaterials, 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
ε
ε
δ
′
′′
=tan

48 Lecture 9
Complex PermeabilityComplex Permeability
 Like the electric field, the magnetic fieldLike the electric field, the magnetic field
encounters molecular forces which require workencounters molecular forces which require work
to overcome in magnetizing the material.to overcome in magnetizing the material.
 In analogy with permittivity, the permeabilityIn analogy with permittivity, the permeability
can also be complexcan also be complex
µµµ ′′−′= jc

49 Lecture 9
Maxwell’s Equations in Differential Form forMaxwell’s Equations in Differential Form for
Time-Harmonic Fields in Simple MediumTime-Harmonic Fields in Simple Medium
( )
( )
µ
ε
σωε
σωµ
mv
ev
ie
im
q
H
q
E
JEjH
KHjE
=⋅∇
=⋅∇
++=×∇
−+−=×∇

50 Lecture 9
Maxwell’s Curl Equations for Time-HarmonicMaxwell’s Curl Equations for Time-Harmonic
Fields in Simple Medium Using ComplexFields in Simple Medium Using Complex
Permittivity and PermeabilityPermittivity and Permeability
i
i
JEjH
KHjE
+=×∇
−−=×∇
ωε
ωµ
complex
permittivity
complex
permeability

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Lect09