Int[1].to EM Theory

8/8/2019 Int[1].to EM Theory

1/78

1

Overview of ElectromagneticOverview of Electromagnetic--

TheoryTheory

Dr. Vijay Kumar KDr. Vijay Kumar K

Introduction to Electromagnetic Fields;Introduction to Electromagnetic Fields;

Maxwells Equations; Electromagnetic Fields in Materials;Maxwells Equations; Electromagnetic Fields in Materials;PhasorPhasor Concepts;Concepts;Electrostatics: Coulombs Law, Electric Field, DiscreteElectrostatics: Coulombs Law, Electric Field, Discrete

and Continuous Charge Distributions; Electrostaticand Continuous Charge Distributions; ElectrostaticPotentialPotential


2/78

Lecture 22

Lecture 2 ObjectivesLecture 2 Objectives

To provide an overview of classicalTo provide an overview of classicalelectromagnetics, Maxwells equations,electromagnetics, Maxwells equations,electromagnetic fields in materials, and phasorelectromagnetic fields in materials, and phasor

concepts.concepts. To begin our study of electrostatics withTo begin our study of electrostatics with

Coulombs law; definition of electric field;Coulombs law; definition of electric field;

computation of electric field from discrete andcomputation of electric field from discrete andcontinuous charge distributions; and scalarcontinuous charge distributions; and scalarelectric potential.electric potential.


3/78

Lecture 23

Introduction to ElectromagneticIntroduction to Electromagnetic

FieldsFields ElectromagneticsElectromagnetics is the study of the effect ofis the study of the effect of

charges at rest and charges in motion.charges at rest and charges in motion.

Some special cases of electromagnetics:Some special cases of electromagnetics: ElectrostaticsElectrostatics: charges at rest: charges at rest

MagnetostaticsMagnetostatics: charges in steady motion (DC): charges in steady motion (DC)

Electromagnetic wavesElectromagnetic waves: waves excited by charges: waves excited by chargesin timein time--varying motionvarying motion


4/78

Lecture 24


FieldsFields

Maxwells

equations

Fundamental laws of

classical electromagnetics

Special

cases

Electro-

statics

Magneto-

staticsElectro-

magnetic

waves

Kirchoffs

Laws

Statics: 0|x

x

t

Pd

GeometricOptics

Transmission

LineTheory

Circuit

Theory

Input from

other

disciplines


5/78

Lecture 25


FieldsFields

transmitter and receiver

are connected by a field.


6/78

Lecture 26


FieldsFields

1

2 3

4

consider an interconnect between points 1 and 2

High-speed, high-density digital circuits:


7/78

Lecture 27


FieldsFields

0 10 20 30 40 50 60 70 80 90 100

0

1

2

t (ns)

v1

(t),

V

0 10 20 30 40 50 60 70 80 90 1000

1

2

t (ns)

v2

(t),

V

0 10 20 30 40 50 60 70 80 90 1000

1

2

t (ns)

v3

(t),

V

PropagationPropagationdelaydelay

ElectromagneticElectromagnetic

couplingcouplingSubstrate modesSubstrate modes


8/78

Lecture 28


FieldsFields When an event in one place has an effectWhen an event in one place has an effect

on something at a different location, weon something at a different location, we

talk about the events as being connected bytalk about the events as being connected bya field.a field.

AAfieldfieldis a spatial distribution of ais a spatial distribution of a

quantity; in general, it can be eitherquantity; in general, it can be eitherscalarscalaroror vectorvectorin nature.in nature.


9/78

Lecture 29


FieldsFields Electric and magnetic fields:Electric and magnetic fields:

Are vector fields with three spatialAre vector fields with three spatial

components.components.Vary as a function of position in 3D space asVary as a function of position in 3D space as

well as time.well as time.

Are governed by partial differential equationsAre governed by partial differential equationsderived from Maxwells equations.derived from Maxwells equations.


10/78

Lecture 210


FieldsFieldsAA scalarscalaris a quantity having only an amplitudeis a quantity having only an amplitude

(and possibly phase).(and possibly phase).

AA vectorvectoris a quantity having direction inis a quantity having direction in

addition to amplitude (and possibly phase).addition to amplitude (and possibly phase).

Examples: voltage, current, charge, energy, temperature

Examples: velocity, acceleration, force


11/78

Lecture 211


FieldsFields Fundamental vector field quantities inFundamental vector field quantities in

electromagnetics:electromagnetics:

Electric field intensityElectric field intensity

Electric flux density (electric displacement)Electric flux density (electric displacement)

Magnetic field intensityMagnetic field intensity

Magnetic flux densityMagnetic flux density

units = volts per meter (V/m = kg m/A/s3)

units = coulombs per square meter (C/m2 = A s /m2)

units = amps per meter (A/m)

units = teslas = webers per square meter (T =

Wb/ m2 = kg/A/s3)

E

D

H

B


12/78

Lecture 212


FieldsFields

Universal constants in electromagnetics:Universal constants in electromagnetics:Velocity of an electromagnetic wave (e.g., light)Velocity of an electromagnetic wave (e.g., light)

in free space (perfect vacuum)in free space (perfect vacuum)

Permeability of free spacePermeability of free space

Permittivity of free space:Permittivity of free space:

Intrinsic impedance of free space:Intrinsic impedance of free space:

m/s103

8v}

c

H/m1047

0

v! TQ

F/85.82

v}I

;} TL 2


13/78

Lecture 213


FieldsFields

0

0

0

00

1

I

QL

IQ!!c

In free space:

HB Q!

ED0

I!

Relationships involving the universalRelationships involving the universalconstants:constants:


14/78

Lecture 214


FieldsFieldssources

Ji, Ki

Obtained

by assumption

from solution to IE

fields

E, H

Solution to

Maxwells equations

Observable

quantities


15/78

Lecture 215

Maxwells EquationsMaxwells Equations

Maxwells equations in integral formMaxwells equations in integral form are theare thefundamental postulates of classical electromagneticsfundamental postulates of classical electromagnetics --all classical electromagnetic phenomena are explainedall classical electromagnetic phenomena are explained

by these equations.by these equations. Electromagnetic phenomena include electrostatics,Electromagnetic phenomena include electrostatics,

magnetostatics, electromagnetostatics andmagnetostatics, electromagnetostatics andelectromagnetic wave propagation.electromagnetic wave propagation.

The differential equations and boundary conditions thatThe differential equations and boundary conditions thatwe use to formulate and solve EM problems are allwe use to formulate and solve EM problems are allderived fromderived from Maxwells equations inintegral formMaxwells equations inintegral form..


16/78

Lecture 216

Maxwells EquationsMaxwells Equations

VariousVarious equivalence principlesequivalence principles consistentconsistentwith Maxwells equations allow us towith Maxwells equations allow us to

replace more complicated electric currentreplace more complicated electric currentand charge distributions withand charge distributions with equivalentequivalentmagnetic sourcesmagnetic sources..

TheseThese equivalentmagnetic sourcesequivalentmagnetic sources can becan betreated by a generalization of Maxwellstreated by a generalization of Maxwellsequations.equations.


17/78

Lecture 217

Maxwells Equations in Integral Form (Generalized toMaxwells Equations in Integral Form (Generalized to

Include Equivalent Magnetic Sources)Include Equivalent Magnetic Sources)

!

!

!

!

Vmv

S

V

ev

S

Si

Sc

SC

Si

Sc

SC

dvqSdB

dvqSdD

SdJSdJSdDdt

dldH

SdKSdKSdBdt

dldE

Adding the fictitious magnetic sourceterms is equivalent to living in a universewhere magnetic monopoles (charges)exist.


18/78

Lecture 218

Continuity Equation in Integral Form (GeneralizedContinuity Equation in Integral Form (Generalized

to Include Equivalent Magnetic Sources)to Include Equivalent Magnetic Sources)

x

x

x

x

V

mv

S

V

ev

S

dvq

t

sdK

dvq

t

sdJ The continuity

equations areimplicit in

Maxwells

equations.


19/78

Lecture 219

Contour, Surface and VolumeContour, Surface and Volume

C

onventionsC

onventionsCS

dS

open surface Sbounded by

closed contourC

dSin direction given by

RH rule

V

S

dS

volume Vbounded by

closed surface S dSin direction outward

from V


20/78

Lecture 220

Electric Current and ChargeElectric Current and Charge

DensitiesDensities JJcc = (electric) conduction current density= (electric) conduction current density

(A/m(A/m22))

JJii = (electric) impressed current density= (electric) impressed current density(A/m(A/m22))

qqevev = (electric) charge density (C/m= (electric) charge density (C/m33))


21/78

Lecture 221

Magnetic Current and ChargeMagnetic Current and Charge

DensitiesDensities KKcc = magnetic conduction current density= magnetic conduction current density

(V/m(V/m22))

KKii = magnetic impressed current density= magnetic impressed current density(V/m(V/m22))

qqmvmv = magnetic charge density (Wb/m= magnetic charge density (Wb/m33))


22/78

Lecture 222

Maxwells EquationsMaxwells Equations -- SourcesSources

and Responsesand Responses Sources of EM field:Sources of EM field:

KKii,,JJii, q, qevev, q, qmvmv

Responses to EM field:Responses to EM field:

E,,HH,,DD,,BB,,JJcc,,KKcc


23/78

Lecture 223

Maxwells Equations in Differential Form (Generalized toMaxwells Equations in Differential Form (Generalized to


mv

ev

ic

ic

q

qD

JJt

DH

KK

t

BE

!

!

x

x!v

x

x!v


24/78

Lecture 224

Continuity Equation in Differential Form (Generalized toContinuity Equation in Differential Form (Generalized to


t

qK

t

qJ

mv

ev

x

x!

x

x!

The continuity

equations areimplicit in

Maxwells

equations.


25/78

Lecture 225

Electromagnetic BoundaryElectromagnetic Boundary

C

onditionsC

onditions

Region 2

Region 1 n


26/78

Lecture 226

Electromagnetic BoundaryElectromagnetic Boundary

C

onditionsC

onditions

ms

es

S

S

qBB

qDD

JHH

KEE

!

!

!v

!v

21

21

21

21


27/78

Lecture 227

Surface Current and ChargeSurface Current and Charge

DensitiesDensities Can be eitherCan be eithersources ofsources oforor responses toresponses to

EM field.EM field.

Units:Units:KKss -- V/mV/m

JJss -- A/mA/m

qqeses -- C/mC/m22

qqmsms -- W/mW/m22


28/78

Lecture 228

Electromagnetic Fields inElectromagnetic Fields in

MaterialsMaterials In timeIn time--varying electromagnetics, we considervarying electromagnetics, we consider EEandandHH to be the primary responses, and attempt toto be the primary responses, and attempt towrite the secondary responseswrite the secondary responses DD,, BB,, JJcc, and, and KKcc inin

terms ofterms ofEEandand HH..

The relationships between the primary andThe relationships between the primary andsecondary responses depends on thesecondary responses depends on the mediummedium ininwhich the field exists.which the field exists.

The relationships between the primary andThe relationships between the primary and

secondary responses are calledsecondary responses are called constitutiveconstitutiverelationshipsrelationships..


29/78

Lecture 229


M

aterialsM

aterials Most generalMost general constitutive relationshipsconstitutive relationships::

),(),(

),(),(

HEKK

HEJJ

HEBBHEDD

cc

cc

!

!

!

!


30/78

Lecture 230


M

aterialsM

aterials In free space, we have:In free space, we have:

0

0

0

0

!

!

!

!

c

c

K

J

B

ED

Q

I


31/78

Lecture 231


Materials

Materials

In aIn a simple mediumsimple medium, we have:, we have:

HK

EJ

HB

E

mc

c

W

W

Q

I linear(independent of field(independent of field

strength)strength) isotropic (independent of position(independent of position

within the medium)within the medium)

homogeneous (independent of(independent of

direction)direction)

time-invariant(independent of(independent oftime)time)

non-dispersive (independent of

frequency)


32/78

Lecture 232

Electromagnetic Fields in MaterialsElectromagnetic Fields in Materials

II= permittivity== permittivity=IIrrII00 (F/m)(F/m)

QQ = permeability== permeability=QQrrQQ00 (H/m)(H/m) WW= electric conductivity== electric conductivity=IIrrII00 (S/m)(S/m)

WWmm = magnetic conductivity== magnetic conductivity=IIrrII00 ((;;/m)/m)


33/78

Lecture 233

Phasor Representation of a TimePhasor Representation of a Time--

Harmonic FieldHarmonic FieldAAphasorphasoris a complex numberis a complex number

representing the amplitude and phase of arepresenting the amplitude and phase of a

sinusoid of known frequency.sinusoid of known frequency.

U

U[

j

AetA cos

time domain frequency domain

phasor


34/78

Lecture 234


Harmonic FieldHarmonic Field PhasorsPhasors are an extremely important concept in theare an extremely important concept in thestudy of classical electromagnetics, circuit theory, andstudy of classical electromagnetics, circuit theory, andcommunications systems.communications systems.

Maxwells equations in simple media, circuitsMaxwells equations in simple media, circuitscomprising linear devices, and many components ofcomprising linear devices, and many components ofcommunications systems can all be represented ascommunications systems can all be represented aslinear timelinear time--invariantinvariant((LTILTI ) systems. (Formal ) systems. (Formaldefinition of these later in the course )definition of these later in the course )

The eigenfunctions of any L

TI system are the complex

The eigenfunctions of any L

TI system are the complexexponentials of the form:exponentials of the form:

tje

[


35/78

Lecture 235


Harmonic FieldHarmonic Field

If the input to an LTIIf the input to an LTIsystem is a sinusoid ofsystem is a sinusoid offrequencyfrequency[[, then the, then the

output is also a sinusoidoutput is also a sinusoidof frequencyof frequency[[ (with(withdifferent amplitude anddifferent amplitude andphase).phase).

tje [ LTI tjejH [[

A complex constant (for fixed [);as a function of [ gives thefrequency response of the LTI

system.


36/78

Lecture 236


Harmonic FieldHarmonic Field The amplitude and phase of a sinusoidalThe amplitude and phase of a sinusoidal

function can also depend on position, and thefunction can also depend on position, and thesinusoid can also be a vector function:sinusoid can also be a vector function:

)()()(cos)( rjAA erAartrAaUU


37/78

Lecture 237


Harmonic FieldHarmonic Field Given the phasor (frequencyGiven the phasor (frequency--domain)domain)

representation of a timerepresentation of a time--harmonic vector field,harmonic vector field,the timethe time--domain representation of the vectordomain representation of the vector

field is obtained using the recipe:field is obtained using the recipe:

_ atj

erEtrE

[

Re, !


38/78

Lecture 238


Harmonic FieldHarmonic Field PhasorsPhasors can be used provided all of the mediacan be used provided all of the media

in the problem arein the problem are linearlinear no frequencyno frequencyconversionconversion..

When phasors are used, integroWhen phasors are used, integro--differentialdifferentialoperators in time become algebraic operations inoperators in time become algebraic operations in

frequency, e.g.:frequency, e.g.:

rEj

t

trE[

x

x ,


39/78

Lecture 239

TimeTime--HarmonicMaxwellsHarmonicMaxwells

EquationsEquations If the sources are timeIf the sources are time--harmonic (sinusoidal), and allharmonic (sinusoidal), and all

media are linear, then the electromagnetic fields aremedia are linear, then the electromagnetic fields aresinusoids of the same frequency as the sources.sinusoids of the same frequency as the sources.

In this case, we can simplify matters by usingIn this case, we can simplify matters by usingMaxwells equations in theMaxwells equations in thefrequencyfrequency--domaindomain..

Maxwells equations in the frequencyMaxwells equations in the frequency--domain aredomain are

relationships between the phasor representations ofrelationships between the phasor representations ofthe fields.the fields.


40/78

Lecture 240

Maxwells Equations in DifferentialMaxwells Equations in Differential

Form for TimeForm for Time--Harmonic FieldsHarmonic Fields

mv

ev

ic

ic

qB

qD

JJDjH

KKBjE

!

!

!v

!v

[

[


41/78

Lecture 241

Maxwells Equations in Differential Form forMaxwells Equations in Differential Form for

TimeTime--Harmonic Fields in Simple MediumHarmonic Fields in Simple Medium

Q

I

W[I

W[Q

mv

ev

i

im

qH

qE

JEjH

KHjE

!

!

!v

!v


42/78

Lecture 242

Electrostatics as a Special Case ofElectrostatics as a Special Case of

ElectromagneticsElectromagnetics

Maxwells

equations

Fundamental laws of

classical

electromagnetics

Specialcases

Electro-statics

Magneto-statics

Electro-magnetic

waves

Kirchoffs

Laws

Statics: 0|x

x

t

Pd

GeometricOptics

Transmission

Line

Theory

Circuit

Theory

Input from

other

disciplines


43/78

Lecture 243

ElectrostaticsElectrostatics

ElectrostaticsElectrostatics is the branch ofis the branch ofelectromagnetics dealing with the effectselectromagnetics dealing with the effects

of electric charges at rest.of electric charges at rest.

The fundamental law ofThe fundamental law ofelectrostaticselectrostatics isisCoulombs lawCoulombs law..


44/78

Lecture 244

Electric ChargeElectric Charge

Electrical phenomena caused by friction are partElectrical phenomena caused by friction are part

of our everyday lives, and can be understood inof our everyday lives, and can be understood in

terms ofterms ofelectrical chargeelectrical charge..

The effects ofThe effects ofelectrical chargeelectrical charge can becan beobserved in the attraction/repulsion of variousobserved in the attraction/repulsion of various

objects when charged.objects when charged.

Charge comes in two varieties called positiveCharge comes in two varieties called positiveand negative.and negative.


45/78

Lecture 245


Objects carrying a net positive charge attractObjects carrying a net positive charge attract

those carrying a net negative charge and repelthose carrying a net negative charge and repelthose carrying a net positive charge.those carrying a net positive charge.

Objects carrying a net negative charge attractObjects carrying a net negative charge attractthose carrying a net positive charge and repelthose carrying a net positive charge and repel

those carrying a net negative charge.those carrying a net negative charge.

On an atomic scale, electrons are negativelyOn an atomic scale, electrons are negativelycharged and nuclei are positively charged.charged and nuclei are positively charged.


46/78

Lecture 246


Electric charge is inherently quantized such thatElectric charge is inherently quantized such that

the charge on any object is an integer multiple ofthe charge on any object is an integer multiple ofthe smallest unit of charge which is thethe smallest unit of charge which is the

magnitude of the electron chargemagnitude of the electron charge

ee = 1.602= 1.602 vv 1010--1919 CC..

On the macroscopic level, we can assume thatOn the macroscopic level, we can assume that

charge is continuous.charge is continuous.


47/78

Lecture 247

Coulombs LawCoulombs Law

Coulombs lawCoulombs law is the law of action betweenis the law of action betweencharged bodies.charged bodies.

Coulombs lawCoulombs lawgives the electric force betweengives the electric force betweentwotwopoint chargespoint chargesin an otherwise empty universe.in an otherwise empty universe.

AApoint chargepoint charge is a charge that occupies ais a charge that occupies aregion of space which is negligibly smallregion of space which is negligibly small

compared to the distance between the pointcompared to the distance between the pointcharge and any other object.charge and any other object.


48/78

Lecture 248


2

120

2112

4

12 r

QQaF R

IT

!

Q1

Q212r

12F

Force due to Q1acting on Q2

Unit vector in

direction ofR12


49/78

Lecture 249


The force onThe force on QQ11 due todue to QQ22 is equal inis equal inmagnitude but opposite in direction to themagnitude but opposite in direction to the

force onforce on QQ22 due todue to QQ11..

1221 FF!


50/78

Lecture 250

Electric FieldElectric Field

Consider a point chargeConsider a point chargeQQ placed at theplaced at the originorigin of aof acoordinate system in ancoordinate system in an

otherwise empty universe.otherwise empty universe.A test chargeA test charge QQtt broughtbrought

nearnear QQ experiences aexperiences aforce:force:

2

04

r

QQa trQt

TI

!

Q

Qt

r


51/78

Lecture 251


The existence of the force onThe existence of the force on QQtt can becan be

attributed to anattributed to an electric fieldelectric fieldproduced byproduced byQQ..

T

heT

heelectric fieldelectric field

produced byproduced byQQ

at a point inat a point inspace can be defined as the force per unit chargespace can be defined as the force per unit charge

acting on a test chargeacting on a test charge QQtt placed at that point.placed at that point.

t

Q

Q QFE t

t 0

limp

!


52/78

Lecture 252


The electric field describes the effect of aThe electric field describes the effect of astationary charge on other charges and isstationary charge on other charges and isan abstract actionan abstract action--atat--aa--distance concept,distance concept,very similar to the concept of a gravityvery similar to the concept of a gravityfield.field.

T

he basic units of electric field areT

he basic units of electric field arenewtonsnewtons

per coulombper coulomb..

In practice, we usually useIn practice, we usually use volts per metervolts per meter..


53/78

Lecture 253


For a point charge at theFor a point charge at the originorigin, the electric, the electric

field at any point is given byfield at any point is given by

3

0

2

044

r

rQ

r

QarE r

TITI

!!


54/78

Lecture 254


For a point charge located at a pointFor a point charge located at a point PPdescribed by a position vectordescribed by a position vector

the electric field atthe electric field at PPis given byis given by

rrR

rrR

R

RQrE

d!

d!

!

where

43

0TI

rd

Q

P

r R

rdO


55/78

Lecture 255


In electromagnetics, it is very popular toIn electromagnetics, it is very popular todescribe the source in terms ofdescribe the source in terms ofprimedprimed

coordinatescoordinates, and the observation point in, and the observation point interms ofterms ofunprimed coordinatesunprimed coordinates..

As we shall see, for continuous sourceAs we shall see, for continuous source

distributions we shall need to integratedistributions we shall need to integrateover the source coordinates.over the source coordinates.


56/78

Lecture 256


Using the principal ofUsing the principal ofsuperpositionsuperposition, the, theelectric field at a point arising fromelectric field at a point arising from

multiple point charges may be evaluated asmultiple point charges may be evaluated as

!

!

n

k k

kk

R

RQrE

1

3

04TI


57/78

Lecture 257

Continuous Distributions ofContinuous Distributions of

ChargeCharge

Charge can occur asCharge can occur as

pointchargespointcharges (C)(C)

volume chargesvolume charges (C/m(C/m33

))surface chargessurface charges (C/m(C/m22))

line chargesline charges (C/m)(C/m)

most general


58/78

Lecture 258


ChargeCharge

Volume charge densityVolume charge density

VQrqencl

Vev

d(!d

p( 0lim

Qencl

rd (V

C i Di ib i fC i Di ib i f


59/78

Lecture 259


ChargeCharge

Electric field due to volume chargeElectric field due to volume chargedensitydensity

QenclrddV V Pr

3

04 R

RvdqEd ev

TI!


60/78

Lecture 260

Electric Field Due to VolumeElectric Field Due to Volume

Charge DensityCharge Density

d

dd!V

ev vdR

RqE3

041TI


61/78

Lecture 261


ChargeCharge

Surface charge densitySurface charge density

SQrqencl

Ses

d(!d

pd( 0lim

Qencl

rd (S


62/78

Lecture 262


ChargeCharge

Electric field due to surface chargeElectric field due to surface chargedensitydensity

QenclrddS S Pr

30

4 RRsdqEd es

TI!


63/78

Lecture 263

Electric Field Due to SurfaceElectric Field Due to Surface

Charge DensityCharge Density

d

dd!S

es sdR

RqE3

041TI


64/78

Lecture 264


Charge

Charge

Line charge densityLine charge density

L

Qrq encl

Lel

d(!d

pd( 0

lim

Qenclrd (L


65/78

Lecture 265


Charge

Charge

Electric field due to line charge densityElectric field due to line charge density

Qenclrd (L r

3

04 R

RldrqrEd el

TI!

P


66/78

Lecture 266

Electric Field Due to Line ChargeElectric Field Due to Line Charge

DensityDensity

d

dd!L

el ldR

RqE3

041TI


67/78

Lecture 267

Electrostatic PotentialElectrostatic Potential

An electric field is aAn electric field is aforce fieldforce field..

If a body being acted on by a force isIf a body being acted on by a force is

moved from one point to another, thenmoved from one point to another, thenworkworkis done.is done.

The concept ofThe concept ofscalar electric potentialscalar electric potential

provides a measure of the work done inprovides a measure of the work done inmoving charged bodies in anmoving charged bodies in anelectrostatic field.electrostatic field.


68/78

Lecture 268


The work done in moving a test charge fromThe work done in moving a test charge fromone point to another in a region of electricone point to another in a region of electric

field:field:

!!pb

a

b

a

ba ldEqldFW

ab

q

F

ld


69/78

Lecture 269


In evaluating line integrals, it is customary to takeIn evaluating line integrals, it is customary to takethethe ddllin the direction of increasing coordinate valuein the direction of increasing coordinate valueso that the manner in which the path of integrationso that the manner in which the path of integration

is traversed is unambiguously determined by theis traversed is unambiguously determined by thelimits of integration.limits of integration.

y!p3

5

dxaEqW xa

x

3 5

b a


70/78

Lecture 270


The electrostatic field isThe electrostatic field is conservativeconservative::

The value of the line integral depends onlyThe value of the line integral depends only

on the end points and is independent ofon the end points and is independent ofthe path taken.the path taken.

The value of the line integral around anyThe value of the line integral around any

closed path is zero.closed path is zero.0!

C

ldEC


71/78

Lecture 271


The work done per unit charge in moving aThe work done per unit charge in moving a

test charge from pointtest charge from point aato pointto point bbis theis the

electrostatic potential differenceelectrostatic potential difference betweenbetween

the two points:the two points:

!|p

b

a

baab ldE

q

WV

electrostatic potentialdifference

Units are volts.


72/78

Lecture 272


Since the electrostatic field isSince the electrostatic field isconservative we can writeconservative we can write

aVbV

ldE

ldE

ldEldEldEV

a

P

b

P

b

P

P

a

b

a

ab

!

yy!

yy!y!

00

0

0


73/78

Lecture 273


Thus theThus the electrostatic potentialelectrostatic potentialVV is ais ascalar field that is defined at every point inscalar field that is defined at every point in

space.space. In particular the value of theIn particular the value of the electrostaticelectrostaticpotentialpotentialat any pointat any point PP is given byis given by

y!P

P

ldErV

0 reference point


74/78

Lecture 274


TheThe reference pointreference point((PP00) is where the potential) is where the potential

is zero (analogous tois zero (analogous togroundgroundin a circuit).in a circuit).

Often the reference is taken to be at infinity soOften the reference is taken to be at infinity so

that the potential of a point in space is definedthat the potential of a point in space is definedasas

g

y!

P

ldrV

El i P i l dEl i P i l d


75/78

Lecture 275

Electrostatic Potential andElectrostatic Potential and


The work done in moving a point chargeThe work done in moving a point chargefrom pointfrom point aa to pointto point bb can be written ascan be written as

_ a

y!

!!b

a

abba

ldE

aVbVQVQW

El t t ti P t ti l dEl t t ti P t ti l d


76/78

Lecture 276



Along a short path of lengthAlong a short path of length ((llwe havewe have

lEV

lEQQW

(!(

(!(!(

or



77/78

Lecture 277



Along an incremental path of lengthAlong an incremental path of length dldlwewehavehave

Recall from the definition ofRecall from the definition ofdirectionaldirectionalderivativederivative::

lddV!

ldVdV !



78/78

Lecture 2



Thus:Thus:

VE !

the del or nabla operator

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Int[1].to EM Theory

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