電磁場與電磁波第1718講位函數(shù)波動方程時諧場

11.Faraday’sLawofElectromagneticInductionReview22.Maxwell’sEquations3.ElectromagneticBoundaryConditionsTheintegralformThedifferentialform

SignificanceFaraday’slaw(電磁感應定律)Ampere’scircuitallaw(全電流定律)Gauss’slaw(高斯定理)Noisolatedmagneticcharge(磁通連續(xù)性原理)3MaintopicTime-VaryingFieldsandMaxwell’sEquations1.PotentialFunctions2.WaveEquationsandTheirSolutions3.Time-HarmonicFields4Thedifferentialform

解方程：1）直接求解2）尋找電場和磁場分別滿足的方程（去耦）3）位函數(shù)的方法1.PotentialFunctions5

Supposethemediumislinear,homogeneous,andisotropic,fromMaxwell’sequationwefindWehave6ThesamewayWehave7

Therelationshipbetweenthefieldintensitiesandthesourcesisquite

complicated(復雜).Tosimplifytheprocess,itwillbehelpfultosolvethetime-varyingelectromagneticfieldsbyintroducingtwo

auxiliary

functions:the

scalar

andthe

vector

potentials.whereA

iscalledthe

vectorpotential.SubstitutingtheaboveequationintogivesDueto,hence

B

canbeexpressedintermsofthecurlofavectorfield

A,asgivenbyWehave8ThusitcanbeexpressedintermsofthegradientofascalarV

,sothatwhereViscalledthe

scalarpotential,andwehave

Thevectorpotential

A

andthescalarpotential

V

arefunctionsof

time

and

space.

Iftheyare

independentoftime,thentheresultsarethesameasthatofthe

static

fields.Therefore,thevectorpotential

A

isalsocalledthe

vectormagneticpotential

(矢量磁位)

andthescalarpotential

V

isalsocalledthe

scalarelectricpotential(標量電位).

9

Inordertoderivetherelationshipbetweenthepotentialsandthesources,fromthedefinitionofthepotentialsandMaxwell’sequationsweobtainUsing,theaboveequationsbecome10

Thecurlofthevectorfield

A

isgivenas,butthedivergencemustbespecified.Thentheabovetwoequationsbecome

Lorentzcondition(洛倫茲條件)

Afterthedivergenceofthevectorpotential

A

isgivenbytheLorentzcondition,theequationsaresimplified.Theoriginalequationsaretwo

coupled

equations,whilenewequationsare

decoupled.Inprinciple,thedivergencecanbetakenarbitrarily,buttosimplifytheapplicationoftheequations,wecanseethatiflet

Thevectorpotential

A

onlydependsonthe

current

J,whilethescalarpotential

V

isrelatedtothe

chargedensity

only.11

Ifthecurrentandthechargeareknown,thenthevectorpotential

A

andthescalarpotential

V

canbedetermined.After

A

and

V

arefound,theelectricandthemagneticfieldscanbeobtained.

TheoriginalEquationsaretwovectorequationswithcomplicatedstructure,andinthree-dimensionalspace,sixcoordinatecomponentsneedtobesolved.

Newpotentialequationsareavectorequationandascalarequation,respectively.

Consequently,thesolutionofMaxwell’sequationsisrelatedtothatoftheequationsforthe

potentialfunctions,andthesolutionis

simplified.

Inthree-dimensionalspace,onlyfourcoordinatecomponentsneedtobefound.12Inparticular,in

rectangular

coordinatesystemthevectorequationcanberesolvedintothreescalar

equations.132.WaveEquationsandTheirSolutionsItmeansthatonecansolvethenon-homogeneouswaveequationsforgivenchargeandcurrentdistributionsandJ.WithAandV

determined,EandBcanbefoundfrom14xPzyrO直接求解方程仍需要較多的數(shù)學知識，這里根據(jù)靜態(tài)場的結果，采用類比的方法，推出其解。1)點電荷的場2)疊加原理PzyrdV'OV'r'r-r'S'15

Herewefindthesolutionbyusingan

analogousmethod

basedontheresultsof

static

fields.

Ifthesourceisatime-varyingpointchargeplacedat

theorigin,thedistributionofthefieldshouldbeafunctionofthevariableR

only,andindependentoftheangles

and

.

Thescalarpotentialcausedbya

pointcharge

isobtainedfirst,thenuse

superpositionprinciple

toobtainthesolutionofthescalarpotentialduetoa

distribution

oftime-varyingcharge.whereIntheopenspace

excludingtheorigin,thescalarpotentialfunctionsatisfiesthefollowingequation16《數(shù)學物理方法》梁昆淼第三版P170－178一維波動方程的解達朗貝爾公式

定解問題弦振動方程、傳輸線方程通解：（a）作變量代換:（b）根據(jù)復合函數(shù)求導：（c）通解：17（d）通解的物理意義：波形波形f（x）以速度a向右傳播的行波波形f（x）以速度a向左傳播的行波行波Travelingwave波的入射、反射與透射在無限大均勻媒質中沒有反射波，即f1=0。1819

Theaboveequationisthe

homogeneous

waveequationforthefunction(VR),andthe

generalsolution

is

Wewillknowthatthe

secondterm

iscontrarytothephysicalsituation(違背客觀事實),anditshouldbe

excluded.Therefore,wefindthescalarelectricpotentialas

TheelectricpotentialproducedbythestaticelementalchargeattheoriginisComparingtheabovetwoequations,weknow20Hence,wefindtheelectricpotentialproducedbythetime-varyingelementalchargeattheoriginaswhere

R

isthedistancetothefieldpointfromthecharge

dV.

Fromtheaboveresult,theelectricpotentialproducedbythe

volumecharge

in

V

canbeobtainedasR'RzyxV(R,t)V'dV'R'-RO21

Tofindthevectorpotentialfunction

A,theaboveequationcanbeexpandedin

rectangular

coordinatesystem,withallcoordinatecomponentssatisfyingthe

same

inhomogeneouswaveequation,i.e.

Apparently,foreachcomponentwecanfindasolution

similar

tothatof

scalar

potentialequation.

Incorporating

thethreecomponentsgivesthesolutionofthevectorpotential

A

as22

Bothequationsshowthatthesolutionofthescalarorthevectorpotentialatthemoment

t

isrelatedtothesourcedistributionatthe

moment

.

Itmeansthatthefieldproducedbythesourceat

R

needsa

certaintime

toreach

R,andthistimedifferenceis.

Inotherwords,thefieldat

t

doesnotdependonthesourceatthesamemoment,butonthesourceat

anearliertime.

Thequantityisthe

distance

betweenthesourcepointandthefieldpoint,and

u

standsforthepropagationvelocity

oftheelectromagneticwave.23

Thechangewithrespect

totime

inhescalarelectricpotential

V

andthevectormagneticpotential

A

isalways

lagging

behindthesources.Hencethefunctions

V

and

A

arecalledthe

retardedpotentials(滯后位).

Sincethetimefactorimpliesthattheevolutionofthefieldprecedesthatofthesource,itviolates

causality,andmustbeabandoned(舍棄).

Thetimefactorcanberewrittenas

Forapointchargeplaced

inopenfreespace(自由空間)thisreflectivewavecannotexist.

Hence,thefunctioncanbeconsideredasawavetravelingtowardtheoriginas

areflectedwave(反射波)fromadistantlocation.24

Fromwecanseethatthepropagationvelocityofelectromagneticwaveisrelatedtothe

properties

ofthemedium.Invacuum,whichisthepropagation

velocityoflight(光速)

invacuum,alsocalledthespeedoflight,usuallydenotedas

c.

Itisworthnotingthatthefieldatapointawayfromthesourcemaystillbepresentatamoment

after

thesourceceasestoexist.

Energy

released

byasourcetravelsawayfromthesourceandcontinuoustopropagateevenafterthesourceis

takenaway.Thisphenomenonisaconsequenceof

electromagneticradiation(電磁輻射).

25

Radiation

isassociatedwitha

time-varyings(時變)

electromagneticfieldwhile

static(靜)fieldmustbetiedtoa

source,andthestaticfieldiscalledthe

bound

field(束縛場).

Thetransitionfromrear-fieldtofar-fielddependsnotonlyonthe

distance(距離)butalsothe

timerate

ofchange(時間變化率)

ofthesource.

Atapoint

close

toatime-varyingchargeorcurrent,thefieldvariesalmostinsynchronism(同步)withthesource.Thefieldinthisregioniscalledthe

nearfield,whichis

quasi-static(準靜態(tài))

innature.

Atapoint

veryfaraway

fromthesource,the

delayintheactionofthefieldwithrespecttothesourcewillbecomehighlynoticeable.Thefieldinthisregionisreferredtoasthe

farfield,anditiscalledradiationfield(輻射場).

Atransmissionantennaneedstobeexcitedbya

highfrequency(高頻)

currentinordertoradiateefficiently,whilethe

50Hz

powerlinecurrenthas

little

radiationeffect.

264.PotentialFunctions5.WaveEquationsandTheirSolutionsReview27homeworkThankyou!Bye-bye!P.7-13;7-14;28Maxwell’sequationsandalltheequationsderivedfromthemsofarinthischapterholdforelectromagneticquantitieswithanarbitrarytime-dependence(時間任意相關).Theactualtypeoftimefunctionsthatthefieldquantitiesassumedependson(取決于)thesource(源)functions

andJ.Inengineering,oneofthe

mostimportant

casesoftime-varyingelectromagneticfieldsisthe

time-harmonic(sinusoidal)field(時諧場、正弦場).Inthistypeoffield,the

excitation

sourcevaries

sinusoidally

intimewith

a

singlefrequency(單一頻率).In

alinearsystem(線性系統(tǒng)),asinusoidallyvarying

source

generates

fields

thatalsovarysinusoidallyintimeatallpointsinthesystem(正弦變化的源產生正弦變化的場).1)whatisTime-HarmonicFields3.Time-HarmonicFields292)討論時諧場（正弦信號）的原因Whenfieldsareexaminedinthismanner,thereisnolossingeneralityas(a)Theyareeasytogenerate(b)anytime-varyingperiodicfunctioncanberepresentedbyaFourierseriesintermsofsinusoidalfunctions(c)theprincipleofsuperpositioncanbeappliedunderlinearconditions.Inotherwords,wecanobtainthecompleteresponseoftimevaryingperiodicfieldsbyusinglinearcombinationsofmonochromaticresponses（a）正弦信號容易產生，50Hz交流電，通信的載波都是正弦信號（b）從信號分析的角度來看，正弦信號是一種簡單基本的信號。正弦信號進行各種運算（加減微分積分后仍為同頻率正弦信號）（c）傅立葉分析：任意周期信號分解為不同頻率的正弦之和（d）線性系統(tǒng)的疊加原理303.1

電路中的相量表達式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidallyintime(1)Instantaneous(time-dependent)expressionofasinusoidalscalarquantity(瞬時值）三角函數(shù)表達式3Parameters:

angularfrequency:

amplitude:Im

phase：(2)

復數(shù)的表示xjyP（x,y）復平面上一點P31(3)正弦表達式和相量表達式的對應關系相量的模正弦量的幅值初位相復角頻率是已知？頻率相量乘以ejt，再取實部32EXAMPLE7-6P337-338333.2

Time-harmonicElectromagneticsFieldvectorsthatvarywithspacecoordinatesandaresinusoidalfunctionsoftimecansimilarlyberepresentedbyvectorphasors(矢量相量)thatdependonspacecoordinatesbutnotontime.Asanexample,wecanwriteatime-harmonicE

fieldreferringtocostaswhereE(x,y,z)isavectorphasor(矢量相量)thatcontainsinformationondirection(方向),magnitude(振幅),andphase(相位).Phasorsare,ingeneral,complexquantities.weseethat,ifE(x,y,z,t)istoberepresentedbythevectorphasorE(x,y,z),thenE(x,y,z,t)/tandE(x,y,z,t)dtwouldberepresentedby,respectively,vectorphasorsjE(x,y,z)

andE(x,y,z)/j.Higher-orderdifferentiationsandintegrationswithrespecttowouldberepresented,respectively,bymultiplicationsanddivisionsofthephasorE(x,y,z)byhigherpowersofj.3435

已知正弦電磁場的場與源的頻率相同，因此可用復矢量形式表示麥克斯韋方程?？紤]到正弦時間函數(shù)的時間導數(shù)為或因此，麥克斯韋第一方程可表示為

上式對于任何時刻均成立，實部符號可以消去，即36瞬時值由相量值代替時間求導由jω代替Wenowwritetime-harmonicMaxwell’sequations(時諧麥克斯韋方程組)intermsofvectorfieldphasors(E,H)andsourcephasors(,J)inasimple(linear,isotropic,andhomogenous)mediumasfollows.37Thetime-harmonicwaveequations(時諧波動方程)forEandHbecome,respectively,Thetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letiscalledthewavenumber.38Then

Considerthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.

Weobtain39ThecomplexLorentzconditionis

Thecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsas

403.3

source-free(無源)fieldsinsimplemediaInasimple,nonconducting(非導電)source-freemediumcharacterizedby=0,J=0,=0,thetime-harmonicMaxwell’sequationsbecome

41whicharehomogeneousvectorHelmholtz’sequations(齊次矢量亥姆霍茲方程).andwaveequationsforAandV

becomeThetime-harmonicwaveequationsforEandHbecome,respectively,Letiscalledthewavenumber.42Ifthesimplemediumisconducting(0)(導電介質),acurrentJ=Ewillflow,andtheequationshouldbechangedtowithTheotherthreeequationsinMaxwell’sequationareunchanged.Hence,allthepreviousequationsfornonconducting(非導電)mediawillapplytoconductingmediaifisreplacedbythecomplexpermittivity

c.Meanwhile,thereal(實數(shù))wavenumberkinthehelmholtz’sequationsshouldbechangedtoacomplex(復數(shù))wavenumber:43Theratio’’/’

iscalledalosstangent(損耗正切)becauseitisameasureofthepowerlossinthemedium:Thequantityc

maybecalledthelossangle(損耗角).Amediumissaidtobeagoodconductor(良導體)if>>,andagoodinsulator(良絕緣體)if<<.Thus,amaterialmaybeagoodconductoratlowfrequencies(