電磁場(chǎng)與電磁波第9講靜電問題的解法wb

FieldandWaveElectromagnetics電磁場(chǎng)與電磁波2012.3.201Maintopic4.SolutionofElectrostaticProblems3.MethodofImages1.Poisson’sandLaplace’sEquations2.UniquenessofElectrostaticSolutions4.Boundary-ValueProblemsinCartesianCoordinates2

The

relationship

betweentheelectricpotentialVandtheelectricfieldintensity

E

is

Takingthedivergenceoperationforbothsidesoftheaboveequationgives

Ina

linear,andisotropic

medium,thedivergenceoftheelectricfieldintensity

E

is1.Poisson’sandLaplace’sEquations3Thedifferentialequationforthe

electricpotential

iswhichiscalled

Poisson’sequation.

Ina

nofreecharge(source-free)

region,andtheaboveequationbecomeswhichiscalled

Laplace’s

equation.

4

Poisson’sequation

statesthattheLaplacian(thedivergenceofthegradient)ofVequals–/forasimplemedium,where

isthepermittivityofthemedium(whichisaconstant)and

isthevolumechargedensityoffreecharges.Operator,

2,theLaplacianoperator,whichstandsfor“thedivergenceofthegradientof,”or“”.Sincebothdivergenceandgradientoperationsinvolve

first-orderspatialderivatives.

Poisson’sequation

isa

second-orderpartialdifferentialequation

thatholdsateverypointinspacewherethesecond-order

derivatives

exist.

Remarks5InCartesiancoordinates:Insphericalcoordinates:Incylindricalcoordinates:6邊值問題研究方法計(jì)算法解析法積分變換法分離變量法鏡像法（電軸法）微分方程法保角變換法實(shí)驗(yàn)法作圖法實(shí)測(cè)法模擬法定性定量數(shù)學(xué)模擬法物理模擬法數(shù)值法有限差分法有限元法邊界元法矩量法半解析法/半數(shù)值法格林函數(shù)法7Example1.一維泊松方程的解ThetwometalplateshavinganareaA

andaseparationdformaparallel-platecapacitor.TheupperplateisheldatpotentialofV0

,andthelowerplateisgrounded.Determine(a)thepotentialdistribution(b)theelectricfieldintensity(c)thechargedistributiononeachplate(d)thecapacitanceoftheparallel-platecapacitor8Solution:Choose

anappropriate

coordinatesystem

forthegivengeometry2.Governingequation

forproblemsand

boundarycondition.勻強(qiáng)電場(chǎng)，電位V只是隨高度z的變化而變化94.特解（帶入邊界條件求解未知系數(shù)）3.方程的通解1011Example2.

The

inner

conductorofradius

a

ofa

coaxialcable

isheldatapotentialof

V0

whiletheouterconductorofradius

b

is

groundedDetermine(a)the

potentialdistributionbetweentheconductors

(b)the

electricfieldintensity(c)the

chargedensity

ontheinnerconductor

(d)the

capacitanceofthe

perunitlength12Choose

anappropriate

coordinatesystem

forthegivengeometry2.Governingequation

forproblemsand

boundarycondition.Solution:134.特解（帶入邊界條件求解未知系數(shù)）

3.方程的通解141516Example3

Theupperandlowerconductingplatesofalargeparallel-platecapacitorareseparatedbyadistance

d

andmaintainedatpotentials

V0

and

0respectively.

Adielectricslabofdielectricrconstantanduniformthickness0.8d

isplacedoverthelowerplate.EandD

?yxD2D1E2E117(1)

求解區(qū)域：平行板電容器之間的區(qū)域(2)

分區(qū)：由于填充兩種介質(zhì)，因此場(chǎng)量在分界面上會(huì)發(fā)生突變，因此，分成兩個(gè)子區(qū)域(3)

建立坐標(biāo)系：豎直向上為y軸方向，建立坐標(biāo)系(4)場(chǎng)分布分析：在兩種介質(zhì)中都是勻強(qiáng)電場(chǎng)，電位V只是隨高度y的變化而變化V(y)，而與x，z無關(guān)，(5)

寫出場(chǎng)方程與邊界條件：待求量是兩個(gè)區(qū)域的電位V1、V2，場(chǎng)方程：泊松方程（有源）or拉普拉斯方程（無源）yxD2D1E2E118區(qū)域1：區(qū)域2：yxD2D1E2E119

寫出通解：一維邊值問題BVP電位的邊界條件，兩個(gè)介質(zhì)的銜接條件：2021yxD2D1E2E122uniquenesstheorem:meansthatasolutionofPoisson’sequation(ofwhichLaplace’sequationisaspecialcase)thatsatisfiesthegivenboundaryconditionsisauniquesolution.

Itdoesnotmeanthatonlyonemethodcanbeusedtoobtainthesolutionoftheelectrostaticproblem.Theimplicationoftheuniquenesstheoremisthatasolutionofanelectrostaticproblemwithitsboundaryconditionsistheonlypossiblesolution

irrespectiveofthemethodbywhichthesolutionisobtained.Asolutionobtainedevenbyintelligentguessingistheonlycorrectsolution2.UniquenessofElectrostaticSolutions23點(diǎn)電荷和帶電的球殼、球體在R>a的區(qū)域中產(chǎn)生的場(chǎng)是是相等的，稱為這三種源是相互等效的.注：在R<a的區(qū)域是不等效的，所以等效只是對(duì)某一區(qū)域等效，對(duì)另一區(qū)域是不等效的xyzxyzaxyza3.MethodofImages24yQdHalf－spaceproblemExample.

Considerthecaseofa

positivepointcharge

Q,locatedatadistancedabovealarge

grounded(zero-potential)conductingplane.

Theproblemistofindthepotentialateverypointabovetheconductingplane(y>0).(1)chap2：感應(yīng)電荷很難求(2)直接解方程:25yQdHalf－spaceproblem點(diǎn)電荷＆感應(yīng)電荷產(chǎn)生的場(chǎng)，靜態(tài)平衡后，導(dǎo)體表面是等勢(shì)面，電力線與其正交。而這種電力線的分布與以xoz平面為對(duì)稱面，在（0，d，0）處點(diǎn)電荷Q，（0，－d，0）處有－Q的一對(duì)點(diǎn)電荷在x>0空間的電力線分布相似。(3)另辟蹊徑：(等效原理)感應(yīng)（極化）電荷產(chǎn)生的場(chǎng)，由假想的簡(jiǎn)單電荷（像點(diǎn)電荷線電荷等）分布產(chǎn)生的場(chǎng)來等效(4)問題：引入像電荷后求得的場(chǎng)，是不是原問題的場(chǎng)？判斷的依據(jù)

（uniqueness

theorem）是不是滿足原問題的場(chǎng)方程＆邊界條件？26ImageChargeImagemethod

V(x,0,z)=0yQ–Q根據(jù)場(chǎng)疊加原理，寫出點(diǎn)電荷和像電荷在上半空間任意一點(diǎn)P處產(chǎn)生的場(chǎng)的表達(dá)式BVPB－C（判斷的條件）等效問題的場(chǎng)就是原問題的場(chǎng)27MethodofImage

Essence:

Theeffectoftheboundary

isreplacedbyoneorseveral

equivalentcharges,andtheoriginalinhomogeneousregionwithaboundarybecomes

aninfinitehomogeneous

space.

Basis：Theprincipleofuniqueness.Therefore,thesechargesshouldnotchangetheoriginalboundaryconditions.Theseequivalentchargesareattheimagepositionsoftheoriginalcharges,andarecalled

imagecharges,andthismethodiscalledthemethodofimages.

Key：Todetermine

thevalues

and

thepositions

oftheimagecharges.

Restriction：Theseimagechargesmaybedeterminedonlyforsome

specialboundaries

(infiniteplane,infinitelylongwedge,infinitelylongcylindrical,andsphericalboundaries)andchargeswith

certaindistributions.28q

Forthesemi-infinite

wedge

conductingboundary,themethodofimagesisalsoapplicable.However,theimagescanbefoundonlyforconductingwedgeswithanglegivenbywhere

n

isaninteger.Inordertokeepthewedgeboundaryatzero-potential,

several

imagechargesarerequired./3

Whenan

infiniteline

chargeisnearbyaninfiniteconductingplane,themethodofimagescanbeappliedaswell,basedonthe

principleof

superposition./3q29In

rectangularcoordinatesystem,Laplace’sEquationforelectricpotentialisLet

Substitutingitintotheaboveequation,anddividingbothsidesby

X(x)Y(y)Z(z),wehave

Whereeachterminvolves

onlyonevariable.Theonlywaytheequationcanbesatisfiedistohave

eachterm

equalto

aconstant.Lettheseconstantsbe,andwehave4.Boundary-ValueProblemsinCartesianCoordinates30

Thethreeseparationconstantsarenotindependentofeachother,andtheysatisfythefollowingequation

Thethree-dimensional

partial

differentialequationisseparatedtothree

ordinary

differentialequations,andthesolutionsoftheordinarydifferentialequationsareeasiertoobtain.

orwhereA,B,C,D

aretheconstantstobedetermined.where

kx

，ky

，kz

arecalledtheseparationconstants,andtheycouldbe

real

or

imaginary

numbers.

Ifkxis

anrealnumber,

Thesolutionoftheequationforthevariable

x

canbewrittenas31or

Thesolutionsoftheequationsforthevariables

y

and

z

havethe

sameforms.Theproductofthesesolutionsgivesthesolutionoftheoriginalpartialdifferentialequation.

Theseparationconstantscouldbeimaginarynumbers.Ifis

animaginarynumber,writtenas,thentheequationbecomes

Theconstantsinthesolutionsare

also

relatedtotheboundaryconditions.

Itisveryimportanttoselect

theforms

ofthesolutions,whichdependonthegiven

boundaryconditions.32Example.

Twosemi-infinite,groundedconductingplanesareparalleltoeachotherwithaseparationof

d.Thefiniteendisclosedbyaconductingplaneheldatelectricpotential

V0

,andisisolatedfromthesemi-infinitegroundedconductingplanewithasmallgap.Findthe

electricpotential

intheslot

constructedbythethreeconductingplanes.Solution:

Select

rectangular

coordinatesystem.Sincetheconductingplaneisinfinite

inthe

z-direction,thepotentialintheslotmustbe

independent

of

z,andthisisa

two-dimensional

problem.TheLaplace’sEquationfortheelectricpotentialbecomesdxyV=0V=0V=V0O33Usingthemethodof

separationofvariables,andlet

Theboundaryconditionsfortheelectricpotential

intheslot

canbeexpressedas

Inordertosatisfytheboundaryconditionsand,thesolutionof

Y(y)

shouldbeselectedas

Fromtheboundarycondition,wehave

V=0

aty=0,andtheconstant

B=0.Inordertosatisfy,theconstant

ky

shouldbe34WefindSince，weobtainTheconstant

kx

isanimaginarynumber,andthesolutionof

X(x)

shouldbeSince

V

=

0

at

x,theconstant

C=0,andThenWheretheconstant

C=AD

.35Since

V=V0

at

x=0

,andwehave

Therightsideoftheaboveequationisvariable,since

C

and

n

arenotfixed.Tosatisfytherequirementat

x=0,oneneedstotakethe

linearcombination

oftheequationasthesolution,leadingto

I