马克斯威尔方程式和电磁波

29.Maxwell’sEquations&ElectromagneticWaves

馬克斯威爾方程式和電磁波TheFourLawsofElectromagnetism 電磁學的四個定律AmbiguityinAmpere’sLaw 安培定律的含糊Maxwell’sEquations 馬克斯威爾方程式ElectromagneticWaves 電磁波PropertiesofElectromagneticWaves 電磁波的性質TheElectromagneticSpectrum 電磁波譜ProducingElectromageticWaves 產生電磁波Energy&MomentuminElectromageticWaves 電磁波的能量和動量Howdoesaconversationtravelbetweencellphones?對話如何在手機間傳送？ByEMwaves. 由電磁波。29.1.TheFourLawsofElectromagnetism

電磁學的四個定律Law定律Mathematical數學Statement陳述Physical物理Meaning意義GaussforEE的高斯HowqproducesE; q怎樣產生E；Elinesbegin&endonq’s.E線始和終於q。GaussforBB的高斯Faraday法拉第Ampere安培(SteadyIonly)(限用於穩I)4LawsofEM(incomplete) 電磁學的四個定律(不完整)Nomagneticmonopole;無磁單極；Blinesformloops. B線必成圈。ChangingBgivesemf.B變動即產生emfMovingchargesgiveB.電荷移動即生B。NoteE-BasymmetrybetweentheFaraday&Amperelaws.留意法拉第和安培定律間E-B並不對稱。29.2.AmbiguityinAmpere’sLaw 安培定律的含糊BinaRCcircuit. RC線路中的B

Ampere’slaw安培定律:IiscurrentthroughanyopensurfaceSboundedbyC.I是通過任何被C圍住的開放式表面的電流。Currentflowsthroughsurfaces1,2,&4.1,2和4面有電流通過。Butnot3. 3

則沒有。

Ampere’slawfails(fornon-steadycurrent).安培定律用於非穩定電流時失敗。Maxwell’smodification馬克斯威爾的修正:ChangingEgivesI,whichinturngivesB.變動E產生I

，以產生B。位移電流Example29.1.ACapacitor 電容器AparallelplatecapacitorwithplateareaAandspacingdischargingattheratedV/dt.一個平行板式電容器(板面積為A

，板間隔為d

)以速率

dV/dt充電。Showthatthedisplacementcurrentisequaltothecurrentinthewiresfeedingthecapacitor.証明位移電流等於連往電容器的電線裏的電流。Capacitor:電容器：(Currentfeedingcapacitor)輸入電容器的電流Displacementcurrent:位移電流：QED29.3.Maxwell’sEquations

馬克斯威爾方程式Law定律MathematicalStatement數學陳述PhysicalMeaning物理意義GaussforEE的高斯HowqproducesE; q怎樣產生E；Elinesbegin&endonq’s.E線始和終於q。GaussforBB的高斯Faraday法拉第Ampere-Maxwell

安培-馬克斯威爾Nomagneticmonopole;無磁單極；Blinesformloops. B線必成圈。ChangingBgivesemf.B變動即產生emfMovingcharges&changingEgiveB.電荷移動和E變動都產生B。Maxwell’sEqs(1864). 馬克斯威爾方程式(1864)Classicalelectromagnetism. 古典電磁學Maxwell’sEquationsinVacuum

真空中的馬克斯威爾方程式GaussforEE的高斯GaussforBB的高斯Faraday法拉第Ampere-Maxwell

安培-馬克斯29.4.ElectromagneticWaves 電磁波 Electromagnetic(EM)waves

電磁波Faraday’slaw法拉第定律：Ampere-Maxwell’slaw

安培-馬克斯威爾定律：changingBgivesE.變B生E。changingEgivesB.變E生B。PlaneElectromagneticWave

平面電磁波EMwaveinvacuumistransverse:E

Bk(directionofpropagation).電磁波在真空中是橫的：E

Bk(進行方司)Foruniqueness,seeProb46唯一性請參考習題46。Sinusoidalplanewavesgoinginx-direction:正弦波朝x-方向走：Right-handrule 右手法則Gauss’sLaws 高斯定律BothE&Bfieldlinesarestraightlines, E和B的場線都是直線，sotheirfluxoveranyclosedsurfacesvanishidentically.因此它們通過任何密閉曲面的通量為零。HencetheGauss’slawsaresatisfied. 所以兩個高斯定律都滿足。Planewave:平面波：Faraday’sLaw 法拉第定律Forloopatxofheighth&widthdx:在x處長h寬dx的廻路：Faraday’sLaw:法拉第定律：Faraday’slawexpressedasadifferentialeq:法拉第定律以微分方程表示：Ampere-MaxwellLaw 安培-馬克斯威爾定律I=0Ampere-MaxwellLaw:安培-馬克斯威爾定律：Ampere-Maxwelllawexpressedasadifferentialeq:安培-馬克斯威爾定律以微分方程表示：invacuum真空中Forloopatxofheighth&widthdx:在x處長h寬dx的廻路：ConditionsonWaveFields 波場的條件Faraday’sLaw:法拉第定律：Ampere-MaxwellLaw:安培-馬克斯威爾定律：ForE=E(x,t)j&B=B(x,t)k,Foraplanewave平面波Faraday’sLaw:法拉第定律：Ampere-MaxwellLaw:安培-馬克斯威爾定律：29.5.PropertiesofElectromagneticWaves

電磁波的性質speedofwave=波的速率.

=speedoflightinvacuum光在真空中的速率=cMaxwell:lightisEMwave.馬克斯威爾：光是一種電磁波1983:meterisdefinedsothatcisexactly299,792,458m/s.1983:米的定義改成c剛好是299,792,458m/s。Hence,0=1/(4c2

107)C2/Nm2,wherec=299,792,458.因此0=1/(4c2

107)C2/Nm2,此處c=299,792,458。GOTIT?29.1AtaparticularpointEofanEMwavepointsinthe+ydirection,whileBpointsinthezdirection.一EM波在某點的E指向+y，其B則指向

z。Isthepropagationdirection 其進行方向+x; (a)+x; x; (b)x;either+xorxbutyoucan’ttellwhich; (c)不是+x就是x，很難說;y; (d)y; +z;or (e)+z;或notalonganyofthecoordinateaxes? (f)不在任何座標軸上Example29.2.LaserLight 雷射光Alaserbeamwithwavelength633nmispropagatingthroughairinthe+zdirection.一條波長633nm的雷射光束在空氣中朝+z方向進行。Itselectricfieldisparalleltothexaxisandhasmagnitude6.0kV/m.Find它的電場平行於x軸，大小則為6.0kV/m。求thewavefrequency, (a)波的頻率，theamplitudeofthemagneticfield,and (b)磁場的大小，和thedirectionofthemagneticfield. (c)磁場的方向。(a)(b)(c) yaxis

軸.Polarization 偏振(極)Polarization偏振

//E.Radiationfromantennasarepolarized.從天線發出的輻射是偏振的E.g.,radio,TV,…. 例：電台，電視，…Lightfromhotsourcesareunpolarized.從熱源發出的輻射無偏振E.g.,sun,lightbulb,… 例：太陽，燈泡Reflectionfromsurfacespolarizes.從表面反射會偏振化。E.g.,lightreflectingoffcarhoodsispartiallypolarizedinhorizontaldirection.例：光從車子的引擎蓋反射後會在水平方向偏振化。Transmissionthroughcrystal/someplasticspolarizes.穿透晶體或某些塑料會偏振化。E.g.,Polaroidsunglasses,… 例：寶麗來太陽鏡。LawofMalus:馬勒定律：OnlycomponentofE//preferreddirectione

istransmitted.祇有平行於傳輸軸e的E分量才能通過。=anglebetweenEinc&e.Einc與e的夾角or2polarizerswithmutuallyperpendiculartransmissionaxes.兩塊傳輸軸互相垂直的偏振片。Nolightgetsthroughwheretheyoverlap.它們重叠的地方光不能通過。PolarizationofEMwavegivesinfoaboutitssource&themediumitpassesthrough.EM波的偏振含有它的來源和它經過的介質的資訊。Applications:astronomy,geologicalsurvey,materialstressanalysis,…應用：天文學，地質測量，材料應力分析，…Liquidcrystaldisplay液晶顯示器(LCD)VerticalpolarizerpassesonlyEv.垂直偏振片祗通過Ev

。Unpolarizedlight無偏振光LCmoleculesrotatepolarizationtohorizontaldirection.液晶分子把偏振轉到水平方向Horizontalpolarizerpasseslight.水平偏振片讓光通過AppliedValignsmolecules;polarizationnotrotated.外加電壓使分子排序；偏振不被轉向。Horizontalpolarizerblockslight.水平偏振片把光擋下ConceptualExample29.1.CrossedPolarizers 交义偏振片Unpolarizedlightshinesonapairofpolarizerswiththeirtransmissionaxesperpendicular,sonolightgetsthroughthecombination.無偏振的光照在一對傳送軸互垂的偏振片上，故無光可穿透這組合。Whathappenswhenathirdpolarizerissandwichedinbetween,withitstransmissionaxesat45totheothers?把傳送軸與組合片都成45的第三塊偏振片插在組合片中間時會怎樣?1st&middlepolarizersnotsosomelightpassesthrough.第一和中間那塊偏振片不，所以有些光可以穿透。Passedlight’spolarizationnottoaxisoflastpolarizersosomelightpassesthrough.穿透的光的偏振不最後那塊偏振片，所以有些光可以穿透。MakingtheConnection 連起來Howdoestheintensityoflightemergingfromthispolarizer“sandwich”comparewiththeintensityoftheincidentunpolarizedlight?從這個偏振片“三明治”跑出來的光的強度，與入射無偏振光的強度比較如何?Intensityoflightemergingfrom1stpolarizer 從第一塊偏振片出來的光的強度:(polarizedalongaxisof1stpolarizer)偏振沿第一塊偏振片的軸Intensityoflightemergingfrommiddlepolarizer 從中間那塊偏振片出來的光的強度:Intensityoflightemergingfromensemble 從整組偏振片出來的光的強度:(polarizedalongaxisofmiddlepolarizer.)偏振沿中間那塊偏振片的軸(polarizedalongaxisof3rdpolarizer.)偏振沿第三塊偏振片的軸29.6.TheElectromagneticSpectrum 電磁波譜Earth’satmosphere地球的大氣:Transparentto: mostradio,visiblelight.

透明的：

大部份的無線電波，可見光。

Opaqueto: mostIR,upperUV,X-rays,rays.

不透明的：

大部份的紅外線，上紫外線，X光，伽瑪射線。 UVisabsorbedbyozonelayer 紫外線由臭氧層吸收 IRbygreenhousegases. 紅外線則由温室氣體29.7.ProducingElectromageticWaves 產生電磁波AnychangingEorBwillcreateEMwaves. E或B的任何變化都會產生電磁波Anyacceleratedchargeproducesradiation. 任何電荷加速就會產生輻射。Radiotransmitter:e’soscillateinantennadrivenbyLCcircuit.無線電發射器：e由LC電路驅動而在天線振動。X-raytube:acceleratede’sslammedintotarget. X-射線管：e加速後撞上標靶。MWmagnetrontube:e’scircleinB. 微波磁控管：e在B中繞圈。EMwave電磁波:f=fofqmotion運動Mostefficient:~dimensionofemitter/reciever効率最高：~發射/接收器大小Wavesemit/receiveaxisofdipole.發射/接收的波雙極的軸。Sourcereplenishesradiatedenergy能源補充輻射掉的能量LCoscillatordrivesIinantennaLC振動器驅動天線的I.OutgoingEMwaves外出電磁波29.8.Energy&MomentuminElectromageticWaves

電磁波的能量和動量Considerboxofthicknessdx,&faceA

kofEMwave.假想一個盒子，厚dx，其面A電磁波的k。Energydensities:能量密度：Energyinbox盒內能量:Rateofenergymovingthroughbox:能量通過盒子的速率：Intensity

S=rateofenergyflowperunitarea強度

S=每單位面積通過能量的速率Planewaves平面波:Planewaves:平面波：Ingeneral:通用：Poyntingvector坡印廷向量seeProb64Averageintensityforplanewaves:平面波的平均強度：E,Binphase同相GOTIT?29.3Lasers1&2emitlightofthesamecolor 雷射1&2發射同樣顏色的光，&Einthebeamfromlaser1istwiceasstrongasthatinlaser2’sbeam.且雷射1光束的E是雷射2光束的2倍。Howdotheir 比較一下它們的magneticfields, (a)磁場intensities,and (b)強度wavelengthscompare? (c)波長B1=2B2S1=4S21=2Example29.3.SolarEnergy 太陽能Theaverageintensityofnoontimesunlightonacleardayisabout1kW/m2.晴天午時陽光的平均強度約為1kW/m2。Whatarethepeakelectric&magneticfieldsinsunlight?陽光的電和磁場的峰值為何？Atthisintensity,whatareaof40%efficientsolarcollectorswouldyouneedtoreplacea4.8-kWwaterheater?據此強度，一個