FieldandWaveElectromagnetic電磁場與電磁波第17講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)34.PotentialFunctions5.WaveEquationsandTheirSolutions4Maxwell’sequationsandalltheequationsderivedfromthemsofarinthischapterholdforelectromagneticquantitieswithanarbitrarytime-dependence(時間任意相關(guān)).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(chǎn)生正弦變化的場).1)whatisTime-HarmonicFields3.Time-HarmonicFields52)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(chǎn)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63.1

電路中的相量表達式Incircuittheory,youhavealreadyusedthephasornotation(相量)torepresentvoltagesandcurrentsvaryingsinusoidally

intime(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P7(3)正弦表達式和相量表達式的對應關(guān)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8EXAMPLE7-6P337-33893.2

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)istoberepresentedbythevectorphasor

E(x,y,z),thenE(x,y,z,t)/tandE(x,y,z,t)dtwouldberepresentedby,respectively,vectorphasors

jE(x,y,z)

andE(x,y,z)/j.Higher-orderdifferentiationsandintegrationswithrespecttowouldberepresented,respectively,bymultiplicationsanddivisionsofthephasor

E(x,y,z)byhigherpowersofj.1011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12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.13Thetime-harmonicwaveequations(時諧波動方程)forEandHbecome,respectively,Thetime-harmonicwaveequationsforscalarpotentialVandvectorpotentialAbecome,respectively,Letiscalledthewavenumber.14Then

Considerthetimedelayfactor,forasinusoidalfunctionitleadstoaphasedelayof.

Weobtain15ThecomplexLorentzconditionis

Thecomplexelectricandmagneticfieldscanbeexpressedintermsofthecomplexpotentialsas

163.3

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

17whicharehomogeneousvectorHelmholtz’sequations(齊次矢量亥姆霍茲方程).andwaveequationsforAandV

becomeThetime-harmonicwaveequationsforEandHbecome,respectively,Letiscalledthewavenumber.18Ifthesimplemediumisconducting(0)(導電介質(zhì)),acurrentJ=Ewillflow,andtheequationshouldbechangedtowithTheotherthreeequationsinMaxwell’sequationareunchanged.Hence,allthepreviousequationsfornonconducting(非導電)mediawillapplytoconductingmediaifisreplacedbythecomplexpermittivity

c.Meanwhile,thereal(實數(shù))

wavenumber

kinthehelmholtz’sequationsshouldbechangedtoacomplex(復數(shù))

wavenumber:19Theratio’’/’

iscalledalosstangent(損耗正切)becauseitisameasureofthepowerlossinthemedium:Thequantityc

maybecalledthelossangle(損耗角).Amediumissaidtobeagoodconductor(良導體)if>>,andagoodinsulator(良絕緣體)if<<.Thus,amaterialmaybeagoodconductoratlowfrequencies(低頻)butmayhavethepropertiesofalossydielectricatveryhighfrequencies(高頻).201.Faraday’sLawofElectromagneticInductionReview212.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)224.PotentialFunctions5.WaveEquationsandTheirSolutions236.Time-HarmonicFields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24dx25P.7-7P34926P.7-13P35127梯度運算符合以下規(guī)則：C為常數(shù)散度運算規(guī)則旋度運算規(guī)則28P.7-25P3522930P.7-30P35331Theelectricfieldintensityinasource-freedielectric()regionisgivenas(V/m),whereangularfrequency，allareconstants.Find:Example.(1)Thephasorrepresentationofelectricfieldintens