1 EEE 498/598 Overview of Electrical Engineering Lecture 4: Electrostatics: Electrostatic Shielding; Poisson’s and Laplace’s Equations; Capacitance; Dielectric

1

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

EngineeringEngineering

Lecture 4:Lecture 4:Electrostatics: Electrostatic Electrostatics: Electrostatic

Shielding; Poisson’s and Shielding; Poisson’s and Laplace’s Equations; Laplace’s Equations;

Capacitance; Dielectric Capacitance; Dielectric Materials and PermittivityMaterials and Permittivity

2Lecture 4

Lecture 4 ObjectivesLecture 4 Objectives

To continue our study of To continue our study of electrostatics with electrostatic electrostatics with electrostatic shielding; Poisson’s and shielding; Poisson’s and Laplace’s equations; Laplace’s equations; capacitance; and dielectric capacitance; and dielectric materials and permittivity.materials and permittivity.

3Lecture 4

Ungrounded Spherical Ungrounded Spherical Metallic ShellMetallic Shell

Consider a point charge at the Consider a point charge at the center of a spherical metallic shell:center of a spherical metallic shell:

Qa

b

Electricallyneutral

4Lecture 4

Ungrounded Spherical Ungrounded Spherical Metallic Shell (Cont’d)Metallic Shell (Cont’d)

The applied electric field is given byThe applied electric field is given by

204

ˆr

QaE rapp

5Lecture 4


The total electric field can be obtained using Gauss’s law together The total electric field can be obtained using Gauss’s law together with our knowledge of how fields behave in a conductor.with our knowledge of how fields behave in a conductor.

6Lecture 4


(1) Assume from symmetry the form (1) Assume from symmetry the form of the fieldof the field

(2) Construct a family of Gaussian (2) Construct a family of Gaussian surfacessurfaces

rDaD rrˆ

spheres of radius r where

r0

7Lecture 4


Here, we shall need to treat Here, we shall need to treat separately 3 sub-families of separately 3 sub-families of Gaussian surfaces:Gaussian surfaces:

1)

bra 2)

br 3)

0)( rDr

a

b

8Lecture 4


(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume enclosed by each Gaussian surface enclosed by each Gaussian surface

V

evencl dvqQ

9Lecture 4


Gaussian surfacesfor which

ar 0


bra


br

10Lecture 4


ForFor

For For

QQencl ar 0

QQencl br

Shell is electrically neutral:The net charge carried by shell is zero.

11Lecture 4


ForFor

since the electric field is zero since the electric field is zero inside conductor. inside conductor.

A surface charge must exist on A surface charge must exist on the inner surface and be given the inner surface and be given byby

bra 0enclQ

24 a

Qqesa

12Lecture 4


Since the conducting shell is Since the conducting shell is initially neutral, a surface initially neutral, a surface charge must also exist on the charge must also exist on the outer surface and be given byouter surface and be given by

24 b

Qqesb

13Lecture 4


(4) For each Gaussian surface, (4) For each Gaussian surface, evaluate the integralevaluate the integral

DSsdDS

24 rrDsdD r

S

magnitude of Don Gaussian

surface.

surface areaof Gaussian

surface.

14Lecture 4


(5) Solve for (5) Solve for DD on each Gaussian surface on each Gaussian surface

S

QD encl

(6) Evaluate E as

0D

E

15Lecture 4

brr

Qa

bra

arr

Qa

E

r

r

,4

ˆ

,0

0,4

ˆ

20

20


16Lecture 4


The induced field is given byThe induced field is given by

br

brar

Qa

ar

EEE rappind

,0

,4

ˆ

0,0

20

17Lecture 4


r

E

Eind

totalelectric

field Eapp

a b

18Lecture 4


The electrostatic potential is The electrostatic potential is obtained by taking the line integral obtained by taking the line integral of of EE. To do this correctly, we must . To do this correctly, we must start at infinity (the reference point start at infinity (the reference point or or groundground) and “move in” back toward ) and “move in” back toward the point charge.the point charge.

For For r > br > b r

QdrErV

r

r04

19Lecture 4


Since the conductor is an Since the conductor is an equipotential bodyequipotential body (and potential is (and potential is a continuous function), we have a continuous function), we have for for bra

b

QbVrV

04

20Lecture 4


For For ar 0

arb

Q

drEbVrVr

a

r

111

4 0

21Lecture 4


r

V

No metallic shell

a b

22Lecture 4

Grounded Spherical Grounded Spherical Metallic ShellMetallic Shell

When the conducting sphere is When the conducting sphere is grounded, we can consider it and grounded, we can consider it and ground to be one huge conducting body ground to be one huge conducting body at ground (zero) potential.at ground (zero) potential.

Electrons migrate from the ground, so Electrons migrate from the ground, so that the conducting sphere now has an that the conducting sphere now has an excess charge exactly equal to excess charge exactly equal to --QQ. This . This charge appears in the form of a surface charge appears in the form of a surface charge density on the inner surface of charge density on the inner surface of the sphere.the sphere.

23Lecture 4

Grounded Spherical Grounded Spherical Metallic ShellMetallic Shell

There is no longer a surface charge on the There is no longer a surface charge on the outer surface of the sphere.outer surface of the sphere.

The total field outside the sphere is zero.The total field outside the sphere is zero. The electrostatic potential of the sphere is The electrostatic potential of the sphere is

zero.zero.

Qa

b

--

--

-----

-

24Lecture 4

Grounded Spherical Grounded Spherical Metallic Shell (Cont’d)Metallic Shell (Cont’d)

R

E

Eind

totalelectric

field Eapp

a b

25Lecture 4

Grounded Spherical Grounded Spherical Metallic Shell (Cont’d)Metallic Shell (Cont’d)

R

V

a b

Grounded metallic shell acts as a shield.

26Lecture 4

The Need for Poisson’s The Need for Poisson’s and Laplace’s Equationsand Laplace’s Equations

So far, we have studied two So far, we have studied two approaches for finding the approaches for finding the electric field and electrostatic electric field and electrostatic potential due to a given charge potential due to a given charge distribution.distribution.

27Lecture 4

The Need for Poisson’s The Need for Poisson’s and Laplace’s Equations and Laplace’s Equations

(Cont’d)(Cont’d) Method 1Method 1: given the position of all : given the position of all

the charges, find the electric field the charges, find the electric field and electrostatic potential usingand electrostatic potential using (A)(A)

P

V

ev

ldErV

R

vdRrqrE

304

28Lecture 4

The Need for Poisson’s The Need for Poisson’s and Laplace’s Equations and Laplace’s Equations

(Cont’d)(Cont’d) (B)(B)

rVrE

R

vdrqrV

V

ev

04

Method 1 is valid only for charges in free space.

29Lecture 4

The Need for Poisson’s The Need for Poisson’s and Laplace’s and Laplace’s

Equations (Cont’d)Equations (Cont’d) Method 2Method 2: Find the electric field : Find the electric field and electrostatic potential usingand electrostatic potential using

P

V

ev

S

ldErV

dvqsdD

Method 2 works only for symmetric charge distributions, but we can have materials other than free space present.

Gauss’s Law

30Lecture 4


Equations (Cont’d)Equations (Cont’d) Consider the following problem:Consider the following problem:

Conductingbodies

r

What are E and V in the region?

2VV 1VV

Neither Method 1 nor Method 2 can be used!

31Lecture 4


Equations (Cont’d)Equations (Cont’d) Poisson’s equationPoisson’s equation is a differential equation for is a differential equation for

the electrostatic potential the electrostatic potential VV. Poisson’s . Poisson’s equation and the boundary conditions equation and the boundary conditions applicable to the particular geometry form a applicable to the particular geometry form a boundary-value problem that can be solved boundary-value problem that can be solved either analytically for some geometries or either analytically for some geometries or numerically for any geometry.numerically for any geometry.

After the electrostatic potential is evaluated, After the electrostatic potential is evaluated, the electric field is obtained usingthe electric field is obtained using

rVrE

32Lecture 4

Derivation of Poisson’s Derivation of Poisson’s EquationEquation

For now, we shall assume the For now, we shall assume the only materials present are free only materials present are free space and conductors on which space and conductors on which the electrostatic potential is the electrostatic potential is specified. However, Poisson’s specified. However, Poisson’s equation can be generalized for equation can be generalized for other materials (dielectric and other materials (dielectric and magnetic as well).magnetic as well).

33Lecture 4

Derivation of Poisson’s Derivation of Poisson’s Equation (Cont’d)Equation (Cont’d)

0

0

ev

evev

qVVE

qEqD

V2

34Lecture 4

Derivation of Poisson’s Derivation of Poisson’s Equation (Cont’d)Equation (Cont’d)

0

2

evq

V Poisson’sequation

2 is the Laplacian operator. The Laplacian of a scalarfunction is a scalar function equal to the divergence of thegradient of the original scalar function.

35Lecture 4

Laplacian Operator in Laplacian Operator in Cartesian, Cylindrical, and Cartesian, Cylindrical, and

Spherical CoordinatesSpherical Coordinates

36Lecture 4

Laplace’s EquationLaplace’s Equation Laplace’s equationLaplace’s equation is the homogeneous form is the homogeneous form

of of Poisson’s equationPoisson’s equation.. We use Laplace’s equation to solve We use Laplace’s equation to solve

problems where potentials are specified problems where potentials are specified on conducting bodies, but no charge on conducting bodies, but no charge exists in the free space region.exists in the free space region.

02 V Laplace’sequation

37Lecture 4

Uniqueness TheoremUniqueness Theorem

A solution to Poisson’s or A solution to Poisson’s or Laplace’s equation that satisfies Laplace’s equation that satisfies the given boundary conditions is the given boundary conditions is the the uniqueunique (i.e., the one and only (i.e., the one and only correct) solution to the problem.correct) solution to the problem.

38Lecture 4

Potential Between Coaxial Potential Between Coaxial Cylinders Using Laplace’s Cylinders Using Laplace’s

EquationEquation Two conducting coaxial cylinders exist such thatTwo conducting coaxial cylinders exist such that

0

0

bV

VaV

a

b

x

y

+V0

39Lecture 4


Equation (Cont’d)Equation (Cont’d) Assume from symmetry thatAssume from symmetry that

VV

012

d

d

d

dV

40Lecture 4


Equation (Cont’d)Equation (Cont’d) Two successive integrations yieldTwo successive integrations yield

The two constants are obtained from The two constants are obtained from the two BCs:the two BCs:

21 ln CCV

21

210

ln0

ln

CbCbV

CaCVaV

41Lecture 4


Equation (Cont’d)Equation (Cont’d) Solving for Solving for CC11 and and CC22, we obtain:, we obtain:

The potential isThe potential is

ba

VC

/ln0

1 ba

bVC

/ln

ln02

bba

VV

ln/ln

0

42Lecture 4


Equation (Cont’d)Equation (Cont’d) The electric field between the The electric field between the

plates is given by:plates is given by:

The surface charge densities on The surface charge densities on the inner and outer conductors the inner and outer conductors are given byare given by

abb

VbEaq

aba

VaEaq

esb

esa

/lnˆ

/lnˆ

000

000

ab

Va

d

dVaVE

/lnˆˆ 0

43Lecture 4

Capacitance of a Two Capacitance of a Two Conductor SystemConductor System

The The capacitancecapacitance of a two conductor system is of a two conductor system is the ratio of the total charge on one of the the ratio of the total charge on one of the conductors to the potential difference conductors to the potential difference between that conductor and the other between that conductor and the other conductor.conductor.

+

V12 = V2-V1

V2 V1

+ -

12V

QC

44Lecture 4


CapacitanceCapacitance is a positive quantity is a positive quantity measured in units of measured in units of FaradsFarads..

CapacitanceCapacitance is a measure of the ability is a measure of the ability of a conductor configuration to store of a conductor configuration to store charge.charge.

45Lecture 4

V

QC


The capacitance of an isolated The capacitance of an isolated conductor can be considered to conductor can be considered to be equal to the capacitance of a be equal to the capacitance of a two conductor system where the two conductor system where the second conductor is an infinite second conductor is an infinite distance away from the first and distance away from the first and at ground potential.at ground potential.

46Lecture 4

CapacitorsCapacitors

A A capacitorcapacitor is an electrical device consisting is an electrical device consisting of two conductors separated by free space of two conductors separated by free space or another conducting medium.or another conducting medium.

To evaluate the capacitance of a two To evaluate the capacitance of a two conductor system, we must find either the conductor system, we must find either the charge on each conductor in terms of an charge on each conductor in terms of an assumed potential difference between the assumed potential difference between the conductors, or the potential difference conductors, or the potential difference between the conductors for an assumed between the conductors for an assumed charge on the conductors.charge on the conductors.

47Lecture 4

Capacitors (Cont’d)Capacitors (Cont’d)

The former method is the more The former method is the more general but requires solution of general but requires solution of Laplace’s equation.Laplace’s equation.

The latter method is useful in The latter method is useful in cases where the symmetry of the cases where the symmetry of the problem allows us to use Gauss’s problem allows us to use Gauss’s law to find the electric field from law to find the electric field from a given charge distribution.a given charge distribution.

48Lecture 4

Parallel-Plate CapacitorParallel-Plate Capacitor

Determine an approximate expression Determine an approximate expression for the capacitance of a parallel-plate for the capacitance of a parallel-plate capacitor by capacitor by neglecting fringingneglecting fringing..

d AConductor 1

Conductor 2

49Lecture 4

Parallel-Plate Capacitor Parallel-Plate Capacitor (Cont’d)(Cont’d)

““Neglecting fringingNeglecting fringing” means to assume that ” means to assume that the field that exists in the real problem is the field that exists in the real problem is the same as for the infinite problem.the same as for the infinite problem.

V = V12

V = 0

z

z = 0

z = d

50Lecture 4


Determine the potential between Determine the potential between the plates by solving Laplace’s the plates by solving Laplace’s equation.equation.

12

2

22

00

0

VdzV

zVdz

VdV

51Lecture 4


zd

VzV

d

VcdcVdzV

czV

czczVdz

Vd

12

121112

2

212

2

00

0

52Lecture 4


d

Va

dz

dVaVE zz

12ˆˆ

• Evaluate the electric field between the plates

53Lecture 4


d

V

d

VaaEaq zznes

12012002 ˆˆˆ

• Evaluate the surface charge on conductor 2

• Evaluate the total charge on conductor 2

d

AVAqQ es

1202

54Lecture 4


• Evaluate the capacitance

d

A

V

QC 0

12

55Lecture 4

Dielectric MaterialsDielectric Materials

A A dielectricdielectric (insulator) is a medium which (insulator) is a medium which possess no (or very few) free electrons to possess no (or very few) free electrons to provide currents due to an impressed provide currents due to an impressed electric field.electric field.

Although there is no macroscopic migration Although there is no macroscopic migration of charge when a dielectric is placed in an of charge when a dielectric is placed in an electric field, microscopic displacements electric field, microscopic displacements (on the order of the size of atoms or (on the order of the size of atoms or molecules) of charge occur resulting in the molecules) of charge occur resulting in the appearance of induced electric dipoles.appearance of induced electric dipoles.

56Lecture 4

Dielectric Materials Dielectric Materials (Cont’d)(Cont’d)

A A dielectricdielectric is said to be is said to be polarizedpolarized when when induced electric dipoles are present.induced electric dipoles are present.

Although all substances are Although all substances are polarizablepolarizable to to some extent, the effects of polarization some extent, the effects of polarization become important only for insulating become important only for insulating materials.materials.

The presence of induced electric dipoles The presence of induced electric dipoles within the dielectric causes the electric within the dielectric causes the electric field both inside and outside the material field both inside and outside the material to be modified.to be modified.

57Lecture 4

PolarizabilityPolarizability

PolarizabilityPolarizability is a measure of the is a measure of the ability of a material to become ability of a material to become polarized in the presence of an polarized in the presence of an applied electric field.applied electric field.

Polarization occurs in both Polarization occurs in both polarpolar and and nonpolarnonpolar materials. materials.

58Lecture 4

Electronic PolarizabilityElectronic Polarizability In the absence of an In the absence of an

applied electric applied electric field, the positively field, the positively charged nucleus is charged nucleus is surrounded by a surrounded by a spherical electron spherical electron cloud with equal and cloud with equal and opposite charge.opposite charge.

Outside the atom, Outside the atom, the electric field is the electric field is zero.zero.

electroncloud nucleus

59Lecture 4

Electronic Polarizability Electronic Polarizability (Cont’d)(Cont’d)

In the presence of In the presence of an applied electric an applied electric field, the electron field, the electron cloud is distorted cloud is distorted such that it is such that it is displaced in a displaced in a direction (w.r.t. the direction (w.r.t. the nucleus) opposite to nucleus) opposite to that of the applied that of the applied electric field.electric field.

Eapp

60Lecture 4

Electronic Polarizability Electronic Polarizability (Cont’d)(Cont’d)

The net effect is The net effect is that each atom that each atom becomes a small becomes a small charge dipole charge dipole which affects which affects the total electric the total electric field both inside field both inside and outside the and outside the material.material.

e e

loce Ep

dipolemoment(C-m)

polarizability (F-m2)

61Lecture 4

Ionic PolarizabilityIonic Polarizability In the absence of In the absence of

an applied an applied electric field, the electric field, the ionic molecules ionic molecules are randomly are randomly oriented such that oriented such that the net dipole the net dipole moment within moment within any small volume any small volume is zero.is zero.

negativeion

positiveion

62Lecture 4

Ionic Polarizability Ionic Polarizability (Cont’d)(Cont’d)

In the presence In the presence of an applied of an applied electric field, electric field, the dipoles tend the dipoles tend to align to align themselves with themselves with the applied the applied electric field.electric field.

Eapp

63Lecture 4

Ionic Polarizability Ionic Polarizability (Cont’d)(Cont’d)

The net effect is The net effect is that each ionic that each ionic molecule is a small molecule is a small charge dipole which charge dipole which aligns with the aligns with the applied electric applied electric field and influences field and influences the total electric the total electric field both inside field both inside and outside the and outside the material.material.

e e

loci Ep

dipolemoment(C-m)


64Lecture 4

Orientational Orientational PolarizabilityPolarizability

In the absence of In the absence of an applied an applied electric field, the electric field, the polar molecules polar molecules are randomly are randomly oriented such that oriented such that the net dipole the net dipole moment within moment within any small volume any small volume is zero.is zero.

65Lecture 4

Orientational Polarizability Orientational Polarizability (Cont’d)(Cont’d)

In the presence In the presence of an applied of an applied electric field, electric field, the dipoles tend the dipoles tend to align to align themselves with themselves with the applied the applied electric field.electric field.

Eapp

66Lecture 4

Orientational Polarizability Orientational Polarizability (Cont’d)(Cont’d)

The net effect is The net effect is that each polar that each polar molecule is a small molecule is a small charge dipole which charge dipole which aligns with the aligns with the applied electric applied electric field and influences field and influences the total electric the total electric field both inside field both inside and outside the and outside the material.material.

e e

loco Ep

dipolemoment(C-m)


67Lecture 4

Polarization Per Unit Polarization Per Unit VolumeVolume

The total polarization of a given The total polarization of a given material may arise from a material may arise from a combination of electronic, ionic, combination of electronic, ionic, and orientational polarizability.and orientational polarizability.

The The polarization per unit volumepolarization per unit volume is is given bygiven by

locT ENpNP

68Lecture 4

Polarization Per Unit Volume Polarization Per Unit Volume (Cont’d)(Cont’d)

PP is the polarization per unit volume. is the polarization per unit volume. (C/m(C/m22))

NN is the number of dipoles per unit is the number of dipoles per unit volume. (mvolume. (m-3-3))

pp is the average dipole moment of the is the average dipole moment of the dipoles in the medium. (C-m)dipoles in the medium. (C-m)

TT is the average polarizability of the is the average polarizability of the dipoles in the medium. (F-mdipoles in the medium. (F-m22))oieT

69Lecture 4


EElocloc is the total electric field that is the total electric field that actually exists at each dipole actually exists at each dipole location.location.

For gases For gases EElocloc = = EE where where EE is the is the total macroscopic field.total macroscopic field.

For solids For solids

1

031

T

loc

NEE

70Lecture 4


From the macroscopic point of From the macroscopic point of view, it suffices to useview, it suffices to use

EP e 0

electronsusceptibility

(dimensionless)

71Lecture 4

Dielectric MaterialsDielectric Materials

The effect of an applied electric field The effect of an applied electric field on a dielectric material is to create a on a dielectric material is to create a net dipole moment per unit volumenet dipole moment per unit volume PP..

The dipole moment distribution sets The dipole moment distribution sets up induced secondary fields:up induced secondary fields:

indapp EEE

72Lecture 4

Volume and Surface Volume and Surface Bound Charge DensitiesBound Charge Densities

A volume distribution of dipoles may be A volume distribution of dipoles may be represented as an equivalent volume (represented as an equivalent volume (qqevbevb) ) and surface and surface ((qqesbesb)) distribution of distribution of boundbound charge.charge.

These charge distributions are related to These charge distributions are related to the dipole moment distribution:the dipole moment distribution:

nPq

Pq

esb

evb

ˆ

73Lecture 4

Gauss’s Law in Gauss’s Law in DielectricsDielectrics

Gauss’s law in differential form in free space:Gauss’s law in differential form in free space:

Gauss’s law in differential form in dielectric:Gauss’s law in differential form in dielectric:

evqE 0

evbev qqE 0

74Lecture 4

Displacement Flux Displacement Flux DensityDensity

ev

evevbev

qPE

PqqqE

0

0

• Hence, the displacement flux density vector is given by

PED 0

75Lecture 4

General Forms of General Forms of Gauss’s LawGauss’s Law

Gauss’s law in differential form:Gauss’s law in differential form:

Gauss’s law in integral form:Gauss’s law in integral form:

evqD

encl

S

QsdD

76Lecture 4

Permittivity ConceptPermittivity Concept

Assuming thatAssuming that

we havewe have

The parameter The parameter is the is the electric electric permittivitypermittivity or the or the dielectric constantdielectric constant of the material.of the material.

EP e 0

EED e 10

77Lecture 4

Permittivity Concept Permittivity Concept (Cont’d)(Cont’d)

The concepts of polarizability and dipole The concepts of polarizability and dipole moment distribution are introduced to moment distribution are introduced to relate microscopic phenomena to the relate microscopic phenomena to the macroscopic fields.macroscopic fields.

The introduction of The introduction of permittivitypermittivity eliminates the eliminates the need for us to explicitly consider need for us to explicitly consider microscopic effects.microscopic effects.

Knowing theKnowing the permittivitypermittivity of a dielectric tells us of a dielectric tells us all we need to know from the point of view all we need to know from the point of view of macroscopic electromagnetics.of macroscopic electromagnetics.

78Lecture 4

Permittivity Concept Permittivity Concept (Cont’d)(Cont’d)

For the most part in macroscopic For the most part in macroscopic electromagnetics, we specify the electromagnetics, we specify the permittivity of the material and if permittivity of the material and if necessary calculate the dipole necessary calculate the dipole moment distribution within the moment distribution within the medium by usingmedium by using EEDP 00

79Lecture 4

Relative PermittivityRelative Permittivity

The The relative permittivityrelative permittivity of a of a dielectric is the ratio of the dielectric is the ratio of the permittivity of the dielectric to permittivity of the dielectric to the permittivity of free spacethe permittivity of free space

0 r

Documents

1 EEE 498/598 Overview of Electrical Engineering Lecture 4: Electrostatics: Electrostatic Shielding; Poisson’s and Laplace’s Equations; Capacitance; Dielectric