电磁场与电磁波第1718讲位函数波动方程时谐场

11.Faraday’sLawofElectromagneticInductionReview22.Maxwell’sEquations3.ElectromagneticBoundaryConditionsTheintegralformThedifferentialform

SignificanceFaraday’slaw(电磁感应定律)Ampere’scircuitallaw(全电流定律)Gauss’slaw(高斯定理)Noisolatedmagneticcharge(磁通连续性原理)3MaintopicTime-VaryingFieldsandMaxwell’sEquations1.PotentialFunctions2.WaveEquationsandTheirSolutions3.Time-HarmonicFields4Thedifferentialform

解方程：1）直接求解2）寻找电场和磁场分别满足的方程（去耦）3）位函数的方法1.PotentialFunctions5

Supposethemediumislinear,homogeneous,andisotropic,fromMaxwell’sequationwefindWehave6ThesamewayWehave7

Therelationshipbetweenthefieldintensitiesandthesourcesisquite

complicated(复杂).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(标量电位).

9

Inordertoderivetherelationshipbetweenthepotentialsandthesources,fromthedefinitionofthepotentialsandMaxwell’sequationsweobtainUsing,theaboveequationsbecome10

Thecurlofthevectorfield

A

isgivenas,butthedivergencemustbespecified.Thentheabovetwoequationsbecome

Lorentzcondition(洛伦兹条件)

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较多的数学知识，这里根据静态场的结果，采用类比的方法，推出其解。1)点电荷的场2)叠加原理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《数学物理方法》梁昆淼第三版P170－178一维波动方程的解达朗贝尔公式

定解问题弦振动方程、传输线方程通解：（a）作变量代换:（b）根据复合函数求导：（c）通解：17（d）通解的物理意义：波形波形f（x）以速度a向右传播的行波波形f（x）以速度a向左传播的行波行波Travelingwave波的入射、反射与透射在无限大均匀媒质中没有反射波，即f1=0。1819

Theaboveequationisthe

homogeneous

waveequationforthefunction(VR),andthe

generalsolution

is

Wewillknowthatthe

secondterm

iscontrarytothephysicalsituation(违背客观事实),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(滞后位).

Sincethetimefactorimpliesthattheevolutionofthefieldprecedesthatofthesource,itviolates

causality,andmustbeabandoned(舍弃).

Thetimefactorcanberewrittenas

Forapointchargeplaced

inopenfreespace(自由空间)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(电磁辐射).

25

Radiation

isassociatedwitha

time-varyings(时变)

electromagneticfieldwhile

static(静)fieldmustbetiedtoa

source,andthestaticfieldiscalledthe

bound

field(束缚场).

Thetransitionfromrear-fieldtofar-fielddependsnotonlyonthe

distance(距离)butalsothe

timerate

ofchange(时间变化率)

ofthesource.

Atapoint

close

toatime-varyingchargeorcurrent,thefieldvariesalmostinsynchronism(同步)withthesource.Thefieldinthisregioniscalledthe

nearfield,whichis

quasi-static(准静态)

innature.

Atapoint

veryfaraway

fromthesource,the

delayintheactionofthefieldwithrespecttothesourcewillbecomehighlynoticeable.Thefieldinthisregionisreferredtoasthe

farfield,anditiscalledradiationfield(辐射场).

Atransmissionantennaneedstobeexcitedbya

highfrequency(高频)

currentinordertoradiateefficiently,whilethe

50Hz

powerlinecurrenthas

little

radiationeffect.

264.PotentialFunctions5.WaveEquationsandTheirSolutionsReview27homeworkThankyou!Bye-bye!P.7-13;7-14;28Maxwell’sequationsandalltheequationsderivedfromthemsofarinthischapterholdforelectromagneticquantitieswithanarbitrarytime-dependence(时间任意相关).Theactualtypeoftimefunctionsthatthefieldquantitiesassumedependson(取决于)thesource(源)functions

andJ.Inengineering,oneofthe

mostimportant

casesoftime-varyingelectromagneticfieldsisthe

time-harmonic(sinusoidal)field(时谐场、正弦场).Inthistypeoffield,the

excitation

sourcevaries

sinusoidally

intimewith

a

singlefrequency(单一频率).In

alinearsystem(线性系统),asinusoidallyvarying

source

generates

fields

thatalsovarysinusoidallyintimeatallpointsinthesystem(正弦变化的源产生正弦变化的场).1)whatisTime-HarmonicFields3.Time-HarmonicFields292)讨论时谐场（正弦信号）的原因Whenfieldsareexaminedinthismanner,thereisnolossingeneralityas(a)Theyareeasytogenerate(b)anytime-varyingperiodicfunctioncanberepresentedbyaFourierseriesintermsofsinusoidalfunctions(c)theprincipleofsuperpositioncanbeappliedunderlinearconditions.Inotherwords,wecanobtainthecompleteresponseoftimevaryingperiodicfieldsbyusinglinearcombinationsofmonochromaticresponses（a）正弦信号容易产生，50Hz交流电，通信的载波都是正弦信号（b）从信号分析的角度来看，正弦信号是一种简单基本的信号。正弦信号进行各种运算（加减微分积分后仍为同频率正弦信号）（c）傅立叶分析：任意周期信号分解为不同频率的正弦之和（d）线性系统的叠加原理303.1

电路中的相量表达式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidallyintime(1)Instantaneous(time-dependent)expressionofasinusoidalscalarquantity(瞬时值）三角函数表达式3Parameters:

angularfrequency:

amplitude:Im

phase：(2)

复数的表示xjyP（x,y）复平面上一点P31(3)正弦表达式和相量表达式的对应关系相量的模正弦量的幅值初位相复角频率是已知？频率相量乘以ejt，再取实部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

已知正弦电磁场的场与源的频率相同，因此可用复矢量形式表示麦克斯韦方程。考虑到正弦时间函数的时间导数为或因此，麦克斯韦第一方程可表示为

上式对于任何时刻均成立，实部符号可以消去，即36瞬时值由相量值代替时间求导由jω代替Wenowwritetime-harmonicMaxwell’sequations(时谐麦克斯韦方程组)intermsofvectorfieldphasors(E,H)andsourcephasors(,J)inasimple(linear,isotropic,andhomogenous)mediumasfollows.37Thetime-harmonicwaveequations(时谐波动方程)forEandHbecome,respectively,Thetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letiscalledthewavenumber.38Then

Considerthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.

Weobtain39ThecomplexLorentzconditionis

Thecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsas

403.3

source-free(无源)fieldsinsimplemediaInasimple,nonconducting(非导电)source-freemediumcharacterizedby=0,J=0,=0,thetime-harmonicMaxwell’sequationsbecome

41whicharehomogeneousvectorHelmholtz’sequations(齐次矢量亥姆霍兹方程).andwaveequationsforAandV

becomeThetime-harmonicwaveequationsforEandHbecome,respectively,Letiscalledthewavenumber.42Ifthesimplemediumisconducting(0)(导电介质),acurrentJ=Ewillflow,andtheequationshouldbechangedtowithTheotherthreeequationsinMaxwell’sequationareunchanged.Hence,allthepreviousequationsfornonconducting(非导电)mediawillapplytoconductingmediaifisreplacedbythecomplexpermittivity

c.Meanwhile,thereal(实数)wavenumberkinthehelmholtz’sequationsshouldbechangedtoacomplex(复数)wavenumber:43Theratio’’/’

iscalledalosstangent(损耗正切)becauseitisameasureofthepowerlossinthemedium:Thequantityc

maybecalledthelossangle(损耗角).Amediumissaidtobeagoodconductor(良导体)if>>,andagoodinsulator(良绝缘体)if<<.Thus,amaterialmaybeagoodconductoratlowfrequencies(