電磁場(chǎng)與電磁波第1718講位函數(shù)波動(dòng)方程時(shí)諧場(chǎng)

11.Faraday’sLawofElectromagneticInductionReview22.Maxwell’sEquations3.ElectromagneticBoundaryConditionsTheintegralformThedifferentialform

SignificanceFaraday’slaw(電磁感應(yīng)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(chǎng)和磁場(chǎng)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(fù)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(biāo)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(xué)知識(shí)，這里根據(jù)靜態(tài)場(chǎng)的結(jié)果，采用類比的方法，推出其解。1)點(diǎn)電荷的場(chǎng)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(xué)物理方法》梁昆淼第三版P170－178一維波動(dòng)方程的解達(dá)朗貝爾公式

定解問題弦振動(dòng)方程、傳輸線方程通解：（a）作變量代換:（b）根據(jù)復(fù)合函數(shù)求導(dǎo)：（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(zhì)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(shí)),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(shí)變)

electromagneticfieldwhile

static(靜)fieldmustbetiedtoa

source,andthestaticfieldiscalledthe

bound

field(束縛場(chǎng)).

Thetransitionfromrear-fieldtofar-fielddependsnotonlyonthe

distance(距離)butalsothe

timerate

ofchange(時(shí)間變化率)

ofthesource.

Atapoint

close

toatime-varyingchargeorcurrent,thefieldvariesalmostinsynchronism(同步)withthesource.Thefieldinthisregioniscalledthe

nearfield,whichis

quasi-static(準(zhǔn)靜態(tài))

innature.

Atapoint

veryfaraway

fromthesource,the

delayintheactionofthefieldwithrespecttothesourcewillbecomehighlynoticeable.Thefieldinthisregionisreferredtoasthe

farfield,anditiscalledradiationfield(輻射場(chǎng)).

Atransmissionantennaneedstobeexcitedbya

highfrequency(高頻)

currentinordertoradiateefficiently,whilethe

50Hz

powerlinecurrenthas

little

radiationeffect.

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

andJ.Inengineering,oneofthe

mostimportant

casesoftime-varyingelectromagneticfieldsisthe

time-harmonic(sinusoidal)field(時(shí)諧場(chǎng)、正弦場(chǎng)).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(chǎn)生正弦變化的場(chǎng)).1)whatisTime-HarmonicFields3.Time-HarmonicFields292)討論時(shí)諧場(chǎng)（正弦信號(hào)）的原因Whenfieldsareexaminedinthismanner,thereisnolossingeneralityas(a)Theyareeasytogenerate(b)anytime-varyingperiodicfunctioncanberepresentedbyaFourierseriesintermsofsinusoidalfunctions(c)theprincipleofsuperpositioncanbeappliedunderlinearconditions.Inotherwords,wecanobtainthecompleteresponseoftimevaryingperiodicfieldsbyusinglinearcombinationsofmonochromaticresponses（a）正弦信號(hào)容易產(chǎn)生，50Hz交流電，通信的載波都是正弦信號(hào)（b）從信號(hào)分析的角度來看，正弦信號(hào)是一種簡(jiǎn)單基本的信號(hào)。正弦信號(hào)進(jìn)行各種運(yùn)算（加減微分積分后仍為同頻率正弦信號(hào)）（c）傅立葉分析：任意周期信號(hào)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(dá)式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidallyintime(1)Instantaneous(time-dependent)expressionofasinusoidalscalarquantity(瞬時(shí)值）三角函數(shù)表達(dá)式3Parameters:

angularfrequency:

amplitude:Im

phase：(2)

復(fù)數(shù)的表示xjyP（x,y）復(fù)平面上一點(diǎn)P31(3)正弦表達(dá)式和相量表達(dá)式的對(duì)應(yīng)關(guān)系相量的模正弦量的幅值初位相復(fù)角頻率是已知？頻率相量乘以ejt，再取實(shí)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(chǎng)的場(chǎng)與源的頻率相同，因此可用復(fù)矢量形式表示麥克斯韋方程?？紤]到正弦時(shí)間函數(shù)的時(shí)間導(dǎo)數(shù)為或因此，麥克斯韋第一方程可表示為

上式對(duì)于任何時(shí)刻均成立，實(shí)部符號(hào)可以消去，即36瞬時(shí)值由相量值代替時(shí)間求導(dǎo)由jω代替Wenowwritetime-harmonicMaxwell’sequations(時(shí)諧麥克斯韋方程組)intermsofvectorfieldphasors(E,H)andsourcephasors(,J)inasimple(linear,isotropic,andhomogenous)mediumasfollows.37Thetime-harmonicwaveequations(時(shí)諧波動(dòng)方程)forEandHbecome,respectively,Thetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letiscalledthewavenumber.38Then

Considerthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.

Weobtain39ThecomplexLorentzconditionis

Thecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsas

403.3

source-free(無源)fieldsinsimplemediaInasimple,nonconducting(非導(dǎo)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(dǎo)電介質(zhì)),acurrentJ=Ewillflow,andtheequationshouldbechangedtowithTheotherthreeequationsinMaxwell’sequationareunchanged.Hence,allthepreviousequationsfornonconducting(非導(dǎo)電)mediawillapplytoconductingmediaifisreplacedbythecomplexpermittivity

c.Meanwhile,thereal(實(shí)數(shù))wavenumberkinthehelmholtz’sequationsshouldbechangedtoacomplex(復(fù)數(shù))wavenumber:43Theratio’’/’

iscalledalosstangent(損耗正切)becauseitisameasureofthepowerlossinthemedium:Thequantityc

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