ShortVersion: 29.Maxwell’sEquations&EMWaves

短版: 29.麥克斯韋方程和電磁波ShortVersion: 29.Maxwell29.1.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。29.1.TheFourLawsofElectro29.2.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29.2.AmbiguityinAmpere’sLa29.3.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29.3.Maxwell’sEquations

馬克斯威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Maxwell’sEquationsinVacuum

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。29.4.ElectromagneticWaves 電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PlaneElectromagneticWave

平面電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：Gauss’sLaws 高斯定律BothE&BfiFaraday’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：Faraday’sLaw 法拉第定律ForloopatAmpere-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：Ampere-MaxwellLaw 安培-馬克斯威爾定律IConditionsonWaveFields 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：ConditionsonWaveFields 波場的條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。29.5.PropertiesofElectromagExample29.2.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.Example29.2.LaserLight 雷射光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：電臺，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orPolarization 偏振(極)Polarizatio2polarizerswithmutuallyperpendiculartransmissionaxes.兩塊傳輸軸互相垂直的偏振片。Nolightgetsthroughwheretheyoverlap.它們重疊的地方光不能通過。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2polarizerswithmutuallyperConceptualExample29.1.CrossedPolarizers 交義偏振片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。ConceptualExample29.1.CrosMakingtheConnection 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MakingtheConnection 連起來Howd29.6.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. 紅外線則由溫室氣體29.6.TheElectromagneticSpec29.7.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29.7.ProducingElectromageti29.8.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:29.8.Energy&MomentuminElPlanewaves: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Planewaves:平面波：Ingeneral:PoyExample29.3.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40%効率的陽光收集器需要多大面積才能取替一個4.8-kW的熱水器？(a)(b)Areaneededis需要面積Example29.3.SolarEnergy 太陽WavesfromLocalizedSources

來自小範圍源頭的波Afarfromlocalizedsource,waveisspherical:離小範圍源頭甚遠處，波呈球狀：Intensity=power/area強度=功率/面積wavefieldsdominatesstaticfieldsawayfromthesources.

離源頭甚遠處，波的場蓋過靜態場。WavesfromLocalizedSources

來Example29.4.CellPhoneReception 手機收訊Acellphone’stypicalaveragepoweroutputisabout0.6W.一個手機的平均輸出功率通常是0.6W左右。Ifthereceiveratacelltowercanhandlesignalswithpeakelectricfieldsasweakas1.2mV/m,如果基地臺的接收器可以處理峰值低至1.2mV/m的電場，whatisthemaximumallowabledistancefromcellphonetotower?基地臺的最大收發距離是多少？Example29.4.CellPhoneReceApplication:CellPhones 手機Hexagonalcellarea25km2.六角細胞的面積25km2。~circleofradius 同面積的圓的半徑為Transmission&receptionareatdifferentfrequencies.發射和接收的頻率不相同。Application:CellPhones 手機HeMomentum&RadiationPressure 動量和輻射壓力Maxwell馬克斯威爾:radiationmomentum輻射動量radiationpressureonabsorbingsurface吸收面的輻射壓力radiationpressureonreflectingsurface反射面的輻射壓力Cosmos1,asolarlight-sailingspacecraft,failedatlaunchin2005.宇宙一號，史上第一艘陽光帆太空船，在2005年升空失敗。Momentum&RadiationPressure ShortVersion: 29.Maxwell’sEquations&EMWaves

短版: 29.麥克斯韋方程和電磁波ShortVersion: 29.Maxwell29.1.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。29.1.TheFourLawsofElectro29.2.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29.2.AmbiguityinAmpere’sLa29.3.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29.3.Maxwell’sEquations

馬克斯威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Maxwell’sEquationsinVacuum

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。29.4.ElectromagneticWaves 電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PlaneElectromagneticWave

平面電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：Gauss’sLaws 高斯定律BothE&BfiFaraday’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：Faraday’sLaw 法拉第定律ForloopatAmpere-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：Ampere-MaxwellLaw 安培-馬克斯威爾定律IConditionsonWaveFields 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：ConditionsonWaveFields 波場的條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。29.5.PropertiesofElectromagExample29.2.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.Example29.2.LaserLight 雷射光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：電臺，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orPolarization 偏振(極)Polarizatio2polarizerswithmutuallyperpendiculartransmissionaxes.兩塊傳輸軸互相垂直的偏振片。Nolightgetsthroughwheretheyoverlap.它們重疊的地方光不能通過。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2polarizerswithmutuallyperConceptualExample29.1.CrossedPolarizers 交義偏振片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。ConceptualExample29.1.CrosMakingtheConnection 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MakingtheConnection 連起來Howd29.6.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. 紅外線則由溫室氣體29.6.TheElectromagneticSpec29.7.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29.7.ProducingElectromageti29.8.Energy&MomentuminElectromageticWaves

電磁波的能量和動量Considerboxofthicknessdx,