11.ElectricPotentialReview22.ConductorsinStaticElectricField3.DielectricsinStaticElectricFieldInsideaconductor(understaticconditions)BoundaryConditions(ataConductor/FreeSpaceInterface)34.ElectricFluxDensityandDielectricConstant4Maintopic1.

靜電場的邊界條件2.

電容和電容器3.

靜電場的能量51.BoundaryConditionsforElectrostaticFields

電磁問題通常涉及具有不同物理特性的媒質(zhì),并且需要關(guān)于兩種媒質(zhì)之間分界面的場量關(guān)系的知識.例如,我們希望知道場矢量E

和D

在穿過兩個界面時是如何變化的.現(xiàn)在我們來考慮兩種一般媒質(zhì)之間的分界面.6(1)電場強度E的切向分量

(tangentialcomponent)Letusconstructasmallpath

abcdawithsidesab

andcdinmedia1and2respectively,bothbeingparalleltotheinterfaceandequalto

w.Theintegralformisassumedtobevalidforregionscontainingdiscontinuousmedia,isappliedtothispath.Ifweletsidesbc=da=

h

approachzero,theircontributionstothelineintegralofEaroundthepathcanbeneglected.ThuswehaveE1E2

2

1at

w

hacdban27

電場強度在分界面處的切向分量是相等的.換句話說,電場強度的切向分量連續(xù).對于線性各向同性的一般介質(zhì),有:

an2

的方向是有媒質(zhì)②

指向媒質(zhì)

①.

8(2)電通密度D的法向分量(normalcomponent)Inordertofindarelationbetweenthenormalcomponentsofthefieldsataboundary,weconstructasmallpillboxwithitstopfaceinmedium1andbottomfaceinmedium2.Thefaceshaveanarea

S,andtheheightofthepillbox

hisvanishinglysmall.ApplyingGauss’slawtothepillbox,wehave

h

S

2

1an2D1D2

s9邊界的法線方向由媒質(zhì)②指向媒質(zhì)①.

S

是邊界上的面電荷密度.D場的法向分量在通過存在面電荷的分界面時是不連續(xù)的,不連續(xù)的量等于面電荷密度.10討論(1)、BoundaryConditionsfor

Dielectric-conductorInterface

討論(2)、

Intheabsenceofnetsurfacefreecharge,onehasTheboundaryconditionsthatmustbesatisfiedforstaticelectricfieldsareasfollows:11例1:兩理想介質(zhì)的分界面為Z=0的平面，如圖所示，在介質(zhì)2中的場強為求介質(zhì)1中分解面上的場分量。

2=0

r2

1=0

r1xzy12132.

電容和電容器導(dǎo)體比例常數(shù)C

稱為孤立導(dǎo)體的電容.電容是指電位每增加一單位所必須施加于物體的電荷量.14

電容器的電容是雙導(dǎo)體系統(tǒng)的一種物理性質(zhì),其依賴于導(dǎo)體的形狀和它們之間媒質(zhì)的介電常數(shù),與電荷Q

和導(dǎo)體之間的電位差V12都無關(guān).電容器即使沒有電壓提供或?qū)w上沒有自由電荷它也是存在的.15CapacitanceCcanbedeterminedfromaboveequationbyeither(1)assumingaV12

anddeterminingQintermsofV12,or(2)assumingaQanddeterminingV12intermsofQ.Atthisstage,sincewehavenotyetstudiedthemethodsforsolvingboundary-valueproblems(whichwillbetakenupinChapter4),wefindCbythesecondmethod.Theprocedureisasfollows:1.對已知的幾何形狀,選擇合適的坐標(biāo)系.2.假設(shè)導(dǎo)體上攜帶了電荷+Qand-Q.3.通過高斯定理或其它關(guān)系根據(jù)假設(shè)的電荷量Q來確定E.4.求出導(dǎo)體兩端的電位差V12.5.FindCbytakingtheratioQ/V12.16Example3-18P12417Example3-19P12518(1)Series＆ParallelConnectionsofCapacitorsParallelConnectionsofCapacitors電容器的電壓相等

Seriesconnectionsofcapacitors(電容器的電量相等)(2)CapacitancesInMulticonductorSystems19靜電能來源：外力克服電場力做功轉(zhuǎn)化而來，靜電場能僅與帶電體的最終帶電狀態(tài)有關(guān)而與到達(dá)這一狀態(tài)的中間過程無關(guān)。靜電能：當(dāng)電荷放入電場中，就會受到電場力的作用，電場力做功使電荷位移，這說明電場具有能量。靜電場內(nèi)儲存著能量，這種能量通常被稱為靜電能。電場越強，對電荷的力就越大，做功的能力就越強，說明電場具有的能量就越大。3.ElectrostaticEnergy能量的零點：

最初電荷都分散在彼此相距很遠(yuǎn)(無限遠(yuǎn))的位置上。通常規(guī)定，處于這種狀態(tài)下的靜電能為零。靜電場能量We等于于把各部分電荷從無限分散的狀態(tài)聚集成現(xiàn)有帶電體系時抵抗靜電力所作的全部功。20BringachargeQ2frominfinityagainstthefieldofachargeQ1

infreespacetoadistanceR12(1)TwochargeswhereV2isthepotentialatP2establishedbychargeQ1,chosethereferencepointforthepotentialatinfinity;Thisworkisstoredintheassemblyofthetwochargesaspotentialenergy.AnotherformwhereV1isthepotentialatP1establishedbychargeQ2.21Bringanothercharge

Q3frominfinitytoapointthatis

R13

tocharge

Q1

andR23

from

charge

Q2

infreespace,anadditionalworkisrequiredthatequals

whereV3

isthepotentialatP3

establishedbychargesQ1andQ2,W3,whichisstoredintheassemblyofthethreechargesQ1,Q2

,andQ3

,is(2)

Threecharges22WecanrewriteW3

inthefollowingformwhereV1

isthepotentialatQ1

establishedbychargesQ2andQ3,similarly,V2andV3arethepotentialsatQ2andQ3,respectively,inthethree-chargeassembly.23Extendingthisprocedureofbringinginadditionalcharges,wearriveatthefollowinggeneralexpressionfor

thepotentialenergyofagroupofN

discretepointchargesatrest.(ThepurposeofthesubscripteonWeistodenotethattheenergyisofanelectricnature.)WehavewhereVk

,

theelectricpotentialatQk,

iscausedbyalltheotherchargesandhasthefollowingexpression:(3)

AgroupofNdiscretepointchargesatrest24Tworemarksareinorderhere.First,Wecanbenegative.Inthatcase,workisdonebythefield(notagainstthefield).Second,Weinthisequationrepresentsonlytheinteractionenergy(mutalenergy)anddoesnotincludetheworkrequiredtoassembletheindividualpointchargesthemselves(self-energy).(4)acontinuous

chargedistributionWereplaceQkby

dvandthesummationbyanintegrationandobtain:NotethatWeinthisequationincludesthework(self-energy)requiredtoassemblethedistributionofmacroscopiccharges,becauseitistheenergyofinteractionofeveryinfinitesimalchargeelementwithallotherinfinitesimalchargeelements.25Example3-23P136Findtheenergyrequiredtoassembleauniformsphereofchargeofradiusbandvolumechargedensity

.

RbdR26(4)ElectrostaticEnergyinTermsatFieldQuantities

Recallingthevectoridentity,Wecanwriteas27SinceV’canbeanyvolumethatincludesallthecharges,wemaychooseittobeaverylargespherewithradiusR(R→

).Foralinearmedium,wehaveD=

E,28Wecanalwaysdefineanelectrostaticenergydensity

we

mathematically,However,thisdefinitionofenergydensityisar