1 EEE 498/598 Overview of Electrical Engineering Lecture 3: Electrostatics: Electrostatic Potential;...

1

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

EngineeringEngineering

Lecture 3:Lecture 3:Electrostatics: Electrostatic Electrostatics: Electrostatic

Potential; Charge Dipole; Potential; Charge Dipole; Visualization of Electric Fields; Visualization of Electric Fields;

Potentials; Gauss’s Law and Potentials; Gauss’s Law and Applications; Conductors and Applications; Conductors and

Conduction CurrentConduction Current

2Lecture 3

Lecture 3 ObjectivesLecture 3 Objectives

To continue our study of To continue our study of electrostatics with electrostatic electrostatics with electrostatic potential; charge dipole; potential; charge dipole; visualization of electric fields visualization of electric fields and potentials; Gauss’s law and and potentials; Gauss’s law and applications; conductors and applications; conductors and conduction current.conduction current.

3Lecture 3

Electrostatic Potential of Electrostatic Potential of a Point Charge at the a Point Charge at the

OriginOrigin

Q

P

r

r

Q

r

rdQ

rdar

QaldErV

r

r

rr

r

02

0

20

44

ˆ4

ˆ

spherically symmetric

4Lecture 3

Electrostatic Potential Electrostatic Potential Resulting from Multiple Resulting from Multiple

Point ChargesPoint Charges

Q1

P(R,)

r 1R

1rO

Q2

2r

n

k k

k

R

QrV

1 04

2R

No longer spherically symmetric!

5Lecture 3

Electrostatic Potential Electrostatic Potential Resulting from Continuous Resulting from Continuous

Charge DistributionsCharge Distributions

V

ev

S

es

L

el

R

vdrqrV

R

sdrqrV

R

ldrqrV

0

0

0

4

1

4

1

4

1

line charge

surface charge

volume charge

6Lecture 3

Charge DipoleCharge Dipole An An electric charge dipoleelectric charge dipole consists of a pair of equal consists of a pair of equal

and opposite point charges separated by a small and opposite point charges separated by a small distance (i.e., much smaller than the distance at distance (i.e., much smaller than the distance at which we observe the resulting field).which we observe the resulting field).

d

+Q -Q

7Lecture 3

Dipole MomentDipole Moment• Dipole moment p is a measure of the strength of the dipole and indicates its direction

dQp +Q

-Q

dp is in the direction from the negative point charge to the positive point charge

8Lecture 3

Electrostatic Potential Electrostatic Potential Due to Charge DipoleDue to Charge Dipole

observationpoint

d/2

+Q

-Q

z

d/2

P

Qdap zˆ

R

Rr

9Lecture 3

Electrostatic Potential Electrostatic Potential Due to Charge Dipole Due to Charge Dipole

(Cont’d)(Cont’d)

R

Q

R

QrVrV

00 44,

cylindrical symmetry

10Lecture 3

Electrostatic Potential Electrostatic Potential Due to Charge Dipole Due to Charge Dipole


d/2

d/2

cos)2/(

cos)2/(

22

22

rddrR

rddrR

R

R

r

P

11Lecture 3

Electrostatic Potential Electrostatic Potential Due to Charge Dipole in Due to Charge Dipole in

the Far-Fieldthe Far-Field• assume R>>d

• zeroth order approximation:

RR

RR

0V

not goodenough!

12Lecture 3

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the Far-to Charge Dipole in the Far-

Field (Cont’d)Field (Cont’d)• first order approximation from geometry:

cos2

cos2d

rR

drR

d/2

d/2

lines approximatelyparallel

R

R

r

13Lecture 3

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the Far-to Charge Dipole in the Far-

Field (Cont’d)Field (Cont’d)• Taylor series approximation:

cos2

111

cos2

11

cos2

11

cos2

111

r

d

rR

r

d

r

r

d

r

dr

R

1,11

:Recall

xnxx n

14Lecture 3

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field


20

0

4

cos

2

cos1

2

cos1

4,

r

Qd

r

d

r

d

r

QrV

15Lecture 3

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field


• In terms of the dipole moment:

20

ˆ

4

1

r

apV r

16Lecture 3

Electric Field of Charge Electric Field of Charge Dipole in the Far-FieldDipole in the Far-Field

sinˆcos2ˆ4

1ˆˆ

30

aar

Qd

V

ra

r

VaVE

r

r

17Lecture 3

Visualization of Electric Visualization of Electric FieldsFields

An electric field (like any vector field) can An electric field (like any vector field) can be visualized using be visualized using flux linesflux lines (also called (also called streamlinesstreamlines or or lines of forcelines of force).).

A A flux lineflux line is drawn such that it is is drawn such that it is everywhere tangent to the electric field.everywhere tangent to the electric field.

A A quiver plotquiver plot is a plot of the field lines is a plot of the field lines constructed by making a grid of points. An constructed by making a grid of points. An arrow whose tail is connected to the point arrow whose tail is connected to the point indicates the direction and magnitude of indicates the direction and magnitude of the field at that point.the field at that point.

18Lecture 3

Visualization of Electric Visualization of Electric PotentialsPotentials

The scalar electric potential can be The scalar electric potential can be visualized using visualized using equipotential surfacesequipotential surfaces..

An An equipotential surfaceequipotential surface is a surface over is a surface over which which VV is a constant. is a constant.

Because the electric field is the negative of Because the electric field is the negative of the gradient of the electric scalar the gradient of the electric scalar potential, the electric field lines are potential, the electric field lines are everywhere normal to the equipotential everywhere normal to the equipotential surfaces and point in the direction of surfaces and point in the direction of decreasing potential.decreasing potential.

19Lecture 3

Visualization of Electric Visualization of Electric FieldsFields

Flux linesFlux lines are suggestive of the flow of are suggestive of the flow of some fluid emanating from positive charges some fluid emanating from positive charges ((sourcesource) and terminating at negative charges ) and terminating at negative charges ((sinksink).).

Although electric field lines do NOT Although electric field lines do NOT represent fluid flow, it is useful to think of represent fluid flow, it is useful to think of them as describing the them as describing the fluxflux of something of something that, like fluid flow, is conserved.that, like fluid flow, is conserved.

20Lecture 3

Faraday’s ExperimentFaraday’s Experiment

charged sphere(+Q)

+

+

+ +

insulator

metal

21Lecture 3

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

Two concentric conducting spheres are Two concentric conducting spheres are separated by an insulating material.separated by an insulating material.

The inner sphere is charged to The inner sphere is charged to ++QQ. . The The outer sphere is initially uncharged.outer sphere is initially uncharged.

The outer sphere is The outer sphere is groundedgrounded momentarily.momentarily.

The charge on the outer sphere is The charge on the outer sphere is found to be found to be --QQ..

22Lecture 3

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

Faraday concluded there was a Faraday concluded there was a ““displacementdisplacement” from the charge on the inner ” from the charge on the inner sphere through the inner sphere through sphere through the inner sphere through the insulator to the outer sphere.the insulator to the outer sphere.

The The electric displacementelectric displacement (or (or electric fluxelectric flux) is ) is equal in magnitude to the charge that equal in magnitude to the charge that produces it, independent of the insulating produces it, independent of the insulating material and the size of the spheres.material and the size of the spheres.

23Lecture 3

Electric Displacement Electric Displacement (Electric Flux)(Electric Flux)

+Q

-Q

24Lecture 3

Electric (Displacement) Electric (Displacement) Flux DensityFlux Density

The density of electric displacement is the The density of electric displacement is the electric electric (displacement) flux density(displacement) flux density, , DD..

In free space the relationship between In free space the relationship between flux densityflux density and electric field is and electric field is

ED 0

25Lecture 3

Electric (Displacement) Electric (Displacement) Flux Density (Cont’d)Flux Density (Cont’d)

The electric (displacement) flux The electric (displacement) flux density for a point charge centered density for a point charge centered at the origin is at the origin is

26Lecture 3

Gauss’s LawGauss’s Law Gauss’s law states that “the net electric Gauss’s law states that “the net electric

flux emanating from a close surface flux emanating from a close surface SS is is equal to the total charge contained within equal to the total charge contained within the volume the volume VV bounded by that surface.” bounded by that surface.”

encl

S

QsdD

27Lecture 3

Gauss’s Law (Cont’d)Gauss’s Law (Cont’d)

V

Sds

By convention, dsis taken to be outward

from the volume V.

V

evencl dvqQ

Since volume chargedensity is the most

general, we can always write Qencl in this way.

28Lecture 3

Applications of Gauss’s Applications of Gauss’s LawLaw

Gauss’s law is an Gauss’s law is an integral equationintegral equation for the for the unknown electric flux density resulting unknown electric flux density resulting from a given charge distribution.from a given charge distribution.

encl

S

QsdD known

unknown

29Lecture 3

Applications of Gauss’s Applications of Gauss’s Law (Cont’d)Law (Cont’d)

In general, solutions to In general, solutions to integral integral equationsequations must be obtained using must be obtained using numerical techniques.numerical techniques.

However, for certain symmetric However, for certain symmetric charge distributions closed form charge distributions closed form solutions to Gauss’s law can be solutions to Gauss’s law can be obtained.obtained.

30Lecture 3

Applications of Gauss’s Applications of Gauss’s Law (Cont’d)Law (Cont’d)

Closed form solution to Gauss’s Closed form solution to Gauss’s law relies on our ability to law relies on our ability to construct a suitable family of construct a suitable family of Gaussian surfacesGaussian surfaces..

A A Gaussian surfaceGaussian surface is a surface to is a surface to which the electric flux density is which the electric flux density is normal and over which equal to normal and over which equal to a constant value.a constant value.

31Lecture 3

Electric Flux Density of a Electric Flux Density of a Point Charge Using Point Charge Using

Gauss’s LawGauss’s LawConsider a point charge at the origin:Consider a point charge at the origin:

Q

32Lecture 3

Electric Flux Density of a Electric Flux Density of a Point Charge Using Gauss’s Point Charge Using Gauss’s

Law (Cont’d)Law (Cont’d)(1) Assume from symmetry the form of (1) Assume from symmetry the form of

the fieldthe field

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

rDaD rrˆ

spheres of radius r where

r0

spherical symmetry

33Lecture 3


Law (Cont’d)Law (Cont’d)(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

34Lecture 3


Law (Cont’d)Law (Cont’d)

Q

R

Gaussian surface

QQencl

35Lecture 3


Law (Cont’d)Law (Cont’d)(4) For each Gaussian surface, (4) For each Gaussian surface,

evaluate the integralevaluate the integralDSsdD

S

24 rrDsdD r

S

magnitude of Don Gaussian

surface.

surface areaof Gaussian

surface.

36Lecture 3


Law (Cont’d)Law (Cont’d)(5) Solve for (5) Solve for DD on each Gaussian on each Gaussian

surfacesurface

S

QD encl

24ˆ

r

QaD r

2

00 4ˆ

r

Qa

DE r

37Lecture 3

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Spherical Shell of Charge

Using Gauss’s LawUsing Gauss’s LawConsider a spherical shell of uniform charge density:Consider a spherical shell of uniform charge density:

otherwise,0

,0 braqqev

a

b

38Lecture 3


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d)

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


RDaD rrˆ

spheres of radius r where

r0

39Lecture 3


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) Here, we shall need to treat Here, we shall need to treat

separately 3 sub-families of Gaussian separately 3 sub-families of Gaussian surfaces:surfaces:

ar 01)

bra 2)

br 3)

a

b

40Lecture 3

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Using Spherical Shell of Charge Using

Gauss’s Law (Cont’d)Gauss’s Law (Cont’d)Gaussian surfacesfor which

ar 0

Gaussian surfacesfor which

bra

Gaussian surfacesfor which

br

41Lecture 3


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d)(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume


V

evencl dvqQ

42Lecture 3



0enclQ For For

For For

ar 0

bra

330

30

300

3

4

3

4

3

4

arq

aqrqdvqQr

a

encl

43Lecture 3


Using Gauss’s Law (Cont’d)Using Gauss’s Law (Cont’d) For For

330

30

30

3

4

3

4

3

4

abq

aqbqdvqQb

a

evencl

br

44Lecture 3



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

DSsdDS

24 rrDsdD r

S


surface.


surface.

45Lecture 3



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

S

QD encl

46Lecture 3



brr

abqa

r

abqa

brar

ar

qa

r

arqa

ar

D

rr

rr

,3

ˆ4

34

ˆ

,3

ˆ4

34

ˆ

0,0

2

330

2

330

2

30

2

330

47Lecture 3



Notice that for Notice that for r > br > b

24ˆ

r

QaD tot

r

Total charge containedin spherical shell

48Lecture 3



0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R

Dr (

C/m

)

m 2

m 1

C/m 1 30

b

a

q

49Lecture 3

Electric Flux Density of an Electric Flux Density of an Infinite Line Charge Using Infinite Line Charge Using

Gauss’s LawGauss’s LawConsider a infinite line charge carrying charge perConsider a infinite line charge carrying charge per

unit length of unit length of qqelel::

z

elq

50Lecture 3



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


DaD ˆ

cylinders of radius where

0

51Lecture 3


Gauss’s Law (Cont’d)Gauss’s Law (Cont’d)(3) Evaluate the total charge within the volume (3) Evaluate the total charge within the volume


L

elencl dlqQ

lqQ elencl cylinder is infinitely long!

52Lecture 3



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

DSsdDS

lDsdDS

2


surface.


surface.

53Lecture 3



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

S

QD encl

2ˆ elqaD

54Lecture 3

Gauss’s Law in Integral Gauss’s Law in Integral FormForm

V

evencl

S

dvqQsdD

VS

sd

55Lecture 3

Recall the Divergence Recall the Divergence TheoremTheorem

Also called Also called Gauss’s theoremGauss’s theorem or or Green’s theoremGreen’s theorem..

Holds for Holds for anyany volume and volume and corresponding corresponding closed surface.closed surface.

dvDsdDVS

VS

sd

56Lecture 3

Applying Divergence Applying Divergence Theorem to Gauss’s LawTheorem to Gauss’s Law

V

ev

VS

dvqdvDsdD

Because the above must hold for any volume V, we must have

evqD Differential formof Gauss’s Law

57Lecture 3

Fields in MaterialsFields in Materials

Materials contain charged Materials contain charged particles that respond to applied particles that respond to applied electric and magnetic fields.electric and magnetic fields.

Materials are classified Materials are classified according to the nature of their according to the nature of their response to the applied fields.response to the applied fields.

58Lecture 3

Classification of Classification of MaterialsMaterials

ConductorsConductors SemiconductorsSemiconductors DielectricsDielectrics Magnetic materialsMagnetic materials

59Lecture 3

ConductorsConductors

A A conductorconductor is a material in which is a material in which electrons in the outermost shell electrons in the outermost shell of the electron migrate easily of the electron migrate easily from atom to atom.from atom to atom.

Metallic materials are in general Metallic materials are in general good conductors.good conductors.

60Lecture 3

Conduction CurrentConduction Current

In an otherwise empty universe, In an otherwise empty universe, a constant electric field would a constant electric field would cause an electron to move with cause an electron to move with constant acceleration.constant acceleration.

-e

em

Eea

aE

e = 1.602 e = 1.602 10 10-19-19 CCmagnitude of electron charge

61Lecture 3

Conduction Current Conduction Current (Cont’d)(Cont’d)

In a conductor, electrons are constantly In a conductor, electrons are constantly colliding with each other and with the colliding with each other and with the fixed nuclei, and losing momentum.fixed nuclei, and losing momentum.

The net macroscopic effect is that the The net macroscopic effect is that the electrons move with a (constant) drift electrons move with a (constant) drift velocity velocity vvdd which is proportional to the which is proportional to the electric field.electric field.

Ev ed Electron mobility

62Lecture 3

Conductor in an Conductor in an Electrostatic FieldElectrostatic Field

To have an electrostatic field, all To have an electrostatic field, all charges must have reached their charges must have reached their equilibrium positions (i.e., they equilibrium positions (i.e., they are stationary).are stationary).

Under such static conditions, Under such static conditions, there must be there must be zero electric fieldzero electric field within the conductor. (Otherwise within the conductor. (Otherwise charges would continue to flow.)charges would continue to flow.)

63Lecture 3

Conductor in an Conductor in an Electrostatic Field Electrostatic Field

(Cont’d)(Cont’d) If the electric field in which the conductor If the electric field in which the conductor

is immersed suddenly changes, charge is immersed suddenly changes, charge flows temporarily until equilibrium is flows temporarily until equilibrium is once again reached with the electric field once again reached with the electric field inside the conductor becoming zero.inside the conductor becoming zero.

In a metallic conductor, the In a metallic conductor, the establishment of equilibrium takes place establishment of equilibrium takes place in about 10in about 10-19-19 s - an extraordinarily short s - an extraordinarily short amount of time indeed.amount of time indeed.

64Lecture 3


(Cont’d)(Cont’d) There are two important consequences to There are two important consequences to

the fact that the electrostatic field inside the fact that the electrostatic field inside a metallic conductor is zero:a metallic conductor is zero: The conductor is an The conductor is an equipotentialequipotential body. body. The charge on a conductor must reside The charge on a conductor must reside

entirely on its surface.entirely on its surface.• A corollary of the above is that the A corollary of the above is that the

electric field just outside the conductor electric field just outside the conductor must be normal to its surface.must be normal to its surface.

65Lecture 3



++ + +

+

-- - -

-

66Lecture 3

Macroscopic versus Macroscopic versus Microscopic FieldsMicroscopic Fields

In our study of electromagnetics, In our study of electromagnetics, we use Maxwell’s equations which we use Maxwell’s equations which are written in terms of are written in terms of macroscopicmacroscopic quantities.quantities.

The lower limit of the classical The lower limit of the classical domain is about 10domain is about 10-8-8 m = 100 m = 100 angstroms. For smaller dimensions, angstroms. For smaller dimensions, quantum mechanics is needed.quantum mechanics is needed.

67Lecture 3

Boundary Conditions on the Boundary Conditions on the Electric Field at the Surface of a Electric Field at the Surface of a

Metallic ConductorMetallic Conductor

esnn

t

qDaD

E

ˆ

0

++ + +

+

-- - -

-

na

E = 0

68Lecture 3

Induced Charges on Induced Charges on ConductorsConductors

The BCs given above imply that if a The BCs given above imply that if a conductor is placed in an externally conductor is placed in an externally applied electric field, thenapplied electric field, then the field distribution is distorted so the field distribution is distorted so

that the electric field lines are normal that the electric field lines are normal to the conductor surfaceto the conductor surface

a surface charge is a surface charge is inducedinduced on the on the conductor to support the electric field conductor to support the electric field

69Lecture 3

Applied and Induced Applied and Induced Electric FieldsElectric Fields

The The applied electric fieldapplied electric field ( (EEappapp) is the field that ) is the field that exists in the absence of the metallic exists in the absence of the metallic conductor (conductor (obstacleobstacle).).

The The induced electric fieldinduced electric field ( (EEindind) is the field that ) is the field that arises from the induced surface charges.arises from the induced surface charges.

The The total fieldtotal field is the sum of the applied and is the sum of the applied and induced electric fields.induced electric fields.

indapp EEE

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1 EEE 498/598 Overview of Electrical Engineering Lecture 3: Electrostatics: Electrostatic Potential;...

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