1、Short Version : 29. Maxwells Equations & EM Waves短版 :29. 麥克斯韋方程和電磁波29.1. The Four Laws of Electromagnetism電磁學的四個定律Law定律定律Mathematical 數學數學Statement 陳陳述述Physical 物理物理Meaning 意義意義Gauss for EE 的高斯0qdEAHow q produces E; q 怎樣產生 E ；E lines begin & end on qs. E 線始和終於 q 。Gauss for BB 的高斯Faraday法拉第Ampere 安培(
2、Steady I only)(限用於穩 I )4 Laws of EM (incomplete)電磁學的四個定律 (不完整)0dBABdddt Er0dIBrNo magnetic monopole; 無磁單極；B lines form loops. B 線必成圈。Changing B gives emf. B 變動即產生 emfMoving charges give B.電荷移動即生 B 。Note E-B asymmetry between the Faraday & Ampere laws.留意法拉第和安培定律間 E-B 並不對稱。29.2. Ambiguity in Amperes L
3、aw安培定律的含糊B in a RC circuit. RC 線路中的 B Amperes law 安培定律:0CdIBrI is current through any open surface S bounded by C.I 是通過任何被 C 圍住的開放式表面的電流。Current flows through surfaces 1,2,& 4. 1, 2 和 4 面有電流通過。But not 3.3 則沒有。 Amperes law fails ( for non-steady current ). 安培定律用於非穩定電流時失敗。Maxwells modification 馬克斯威爾的修
4、正 :000ECddIdt Br0Displacement curre ntEddtChanging E gives I , which in turn gives B.變動 E 產生 I ，以產生 B。位移電流29.3. Maxwells Equations馬克斯威爾方程式Law定律定律Mathematical Statement數學陳數學陳述述Physical Meaning物理意義物理意義Gauss for EE 的高斯0qdEAHow q produces E; q 怎樣產生 E ；E lines begin & end on qs. E 線始和終於 q 。Gauss for BB 的
5、高斯Faraday法拉第Ampere-Maxwell 安培-馬克斯威爾0dBABdddt ErNo magnetic monopole; 無磁單極；B lines form loops. B 線必成圈。Changing B gives emf. B 變動即產生 emfMoving charges & changing E give B.電荷移動和 E 變動都產生 B。Maxwells Eqs (1864).馬克斯威爾方程式 (1864)Classical electromagnetism.古典電磁學000EddIdt BrMaxwells Equations in Vacuum真空中的馬克斯威
6、爾方程式Gauss for EE 的高斯0dEAGauss for BB 的高斯Faraday法拉第Ampere-Maxwell 安培-馬克斯0dBABdddt Er00Edddt Br29.4. Electromagnetic Waves電磁波 Electromagnetic (EM) waves電磁波Faradays law 法拉第定律：Ampere-Maxwells law 安培-馬克斯威爾定律：changing B gives E. 變 B 生 E。changing E gives B. 變 E 生 B。Plane Electromagnetic Wave平面電磁波EM wave in
7、 vacuum is transverse: E B k (direction of propagation).電磁波在真空中是橫的：E B k (進行方司)For uniqueness, see Prob 46唯一性請參考習題 46 。Sinusoidal plane waves going in x-direction:正弦波朝 x-方向走：,yx tEx tEj,zx tBx tBkkEBRight-hand rule右手法則sinpEkxtjsinpBkxtkGausss Laws高斯定律sinpEkxtjBoth E & B field lines are straightlines
8、, E 和 B 的場線都是直線，so their flux over any closed surfaces vanish identically. 因此它們通過任何密閉曲面的通量為零。Hence the Gausss laws are satisfied. 所以兩個高斯定律都滿足。Plane wave :平面波：,yx tEx tEj,zx tBx tBksinpBkxtkFaradays Law法拉第定律For loop at x of height h & width dx :在 x 處長 h 寬 dx 的廻路：,dE x t hE xdx t h Er,EE x t hE x td x
9、 hx Ed xhxBB h dxBdBh dxdttFaradays Law :法拉第定律：EBxt Faradays law expressed as a differential eq :法拉第定律以微分方程表示：t BEBdddt ErEEyBBzyzEBxt xyzxyzVVV ijkVd EAdt BAAmpere-Maxwell Law安培-馬克斯威爾定律00ECdddt Br CdB x hB xdx hBrI = 0,BB x t hB x td x hxBd xhx EE h dxEdEh dxdttAmpere-Maxwell Law :安培-馬克斯威爾定律：00BExt
10、 Ampere-Maxwell law expressed as a differential eq :安培-馬克斯威爾定律以微分方程表示：00t EBin vacuum真空中For loop at x of height h & width dx :在 x 處長 h 寬 dx 的廻路：EEyBBz00yzEBxt Conditions on Wave Fields波場的條件Faradays Law :法拉第定律：EBxt Ampere-Maxwell Law :安培-馬克斯威爾定律：00BExt For E = E(x,t) j & B = B(x,t) k, For a plane wav
11、e平面波,sinpx tEkxtEj,sinpx tBkxtBkcoscosppk Ek xtBk xtFaradays Law :法拉第定律：ppk EBAmpere-Maxwell Law :安培-馬克斯威爾定律：00coscosppk Bk xtEk xt 00ppk BE xyzxyzVVV ijkV29.5. Properties of Electromagnetic Waves電磁波的性質ppk EB00ppk BE 2200k 001k speed of wave =波的速率 . 7222911410/4910NACN m83 10/ms= speed of light in v
12、acuum 光在真空中的速率= c Maxwell : light is EM wave.馬克斯威爾：光是一種電磁波1983: meter is defined so that c is exactly 299,792,458 m/s.1983: 米的定義改成 c 剛好是 299,792,458 m/s 。Hence, 0 = 1 / (4 c2 107 ) C2/Nm2, where c = 299,792,458.因此 0 = 1 / (4 c2 107 ) C2/Nm2, 此處 c = 299,792,458。Example 29.2. Laser Light雷射光A laser bea
13、m with wavelength 633 nm is propagating through air in the +z direction.一條波長 633 nm 的雷射光束在空氣中朝 +z 方向進行。Its electric field is parallel to the x axis and has magnitude 6.0 kV/m. Find它的電場平行於 x 軸，大小則為 6.0 kV/m 。求(a) the wave frequency,(a) 波的頻率，(b) the amplitude of the magnetic field, and(b) 磁場的大小，和(c) t
14、he direction of the magnetic field.(c) 磁場的方向。cf(a)893.010/633 10msm144.710HzEBc(b)(c)y axis 軸.386.010/3.010/Vmms52.010T20TPolarization偏振(極)Polarization 偏振 / E.Radiation from antennas are polarized.從天線發出的輻射是偏振的E.g., radio, TV, . 例：電台，電視，Light from hot sources are unpolarized.從熱源發出的輻射無偏振E.g., sun, lig
15、ht bulb, 例：太陽，燈泡Reflection from surfaces polarizes. 從表面反射會偏振化。E.g., light reflecting off car hoods is partially polarized in horizontal direction.例：光從車子的引擎蓋反射後會在水平方向偏振化。Transmission through crystal / some plastics polarizes.穿透晶體或某些塑料會偏振化。E.g., Polaroid sunglasses, 例：寶麗來太陽鏡。Law of Malus :馬勒定律：Only co
16、mponent of E / preferred direction e is transmitted.祇有平行於傳輸軸 e 的 E 分量才能通過。222costransincEE transincEEe ecosincEe = angle between Einc & e. Einc 與 e 的夾角2costransincSSor j k20cosSS2 polarizers with mutually perpendicular transmission axes.兩塊傳輸軸互相垂直的偏振片。No light gets through where they overlap.它們重叠的地方光
17、不能通過。Polarization of EM wave gives info about its source & the medium it passes through.EM 波的偏振含有它的來源和它經過的介質的資訊。Applications: astronomy, geological survey, material stress analysis, 應用：天文學，地質測量，材料應力分析，Liquid crystal display 液晶顯示器 (LCD)Vertical polarizer passes only Ev .垂直偏振片祗通過 Ev 。Unpolarized light
18、無偏振光LC molecules rotate polarization to horizontal direction.液晶分子把偏振轉到水平方向Horizontal polarizer passes light.水平偏振片讓光通過Applied V aligns molecules; polarization not rotated.外加電壓使分子排序；偏振不被轉向。Horizontal polarizer blocks light.水平偏振片把光擋下Conceptual Example 29.1. Crossed Polarizers交义偏振片Unpolarized light shin
19、es on a pair of polarizers with their transmission axes perpendicular, so no light gets through the combination.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。What happens when a third polarizer is sandwiched in between, with its transmission axes at 45 to the others?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 & middle pol
20、arizers not so some light passes through.第一和中間那塊偏振片不 ，所以有些光可以穿透。Passed lights polarization not to axis of last polarizer so some light passes through. 穿透的光的偏振不 最後那塊偏振片，所以有些光可以穿透。Making the Connection連起來How does the intensity of light emerging from this polarizer “sandwich” compare with the intensity
21、 of the incident unpolarized light?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?2costransincSSIntensity of light emerging from 1st polarizer 從第一塊偏振片出來的光的強度 :22101cos2incSSd12incS( polarized along axis of 1st polarizer )偏振沿第一塊偏振片的軸Intensity of light emerging from middle polarizer 從中間那塊偏振片出來的光的強度 :221cos 45SS
22、112S14incSIntensity of light emerging from ensemble從整組偏振片出來的光的強度 :232cos 45SS212S18incS( polarized along axis of middle polarizer.)偏振沿中間那塊偏振片的軸( polarized along axis of 3rd polarizer.)偏振沿第三塊偏振片的軸29.6. The Electromagnetic Spectrum 電磁波譜Earths atmosphere 地球的大氣:Transparent to :most radio, visible light.
23、透明的： 大部份的無線電波，可見光。 Opaque to:most IR, upper UV, X-rays, rays. 不透明的： 大部份的紅外線，上紫外線， X 光，伽瑪射線。UV is absorbed by ozone layer紫外線由臭氧層吸收IR by green house gases.紅外線則由温室氣體29.7. Producing Electromagetic Waves產生電磁波Any changing E or B will create EM waves. E 或 B 的任何變化都會產生電磁波Any accelerated charge produces radia
24、tion.任何電荷加速就會產生輻射。Radio transmitter: es oscillate in antenna driven by LC circuit.無線電發射器：e由LC電路驅動而在天線振動。X-ray tube: accelerated es slammed into target.X-射線管：e 加速後撞上標靶。MW magnetron tube: es circle in B. 微波磁控管：e 在 B 中繞圈。EM wave 電磁波: f = f of q motion 運動Most efficient: dimension of emitter / reciever効率
25、最高： 發射 / 接收器 大小 Waves emit / receive axis of dipole.發射 / 接收的波 雙極的軸。Source replenishes radiated energy能源補充輻射掉的能量LC oscillator drives I in antennaLC 振動器驅動天線的 I.Outgoing EM waves外出電磁波29.8. Energy & Momentum in Electromagetic Waves電磁波的能量和動量Consider box of thickness dx, & face A k of EM wave.假想一個盒子，厚 dx
26、，其面 A 電磁波的 k。2012EuEEnergy densities:能量密度：2012BuBEnergy in box 盒內能量 :EBdUuuA dx2200112A dxEBRate of energy moving through box:能量通過盒子的速率：/dUd Udtdx c2200112A cEBIntensity S = rate of energy flow per unit area 強度 S = 每單位面積通過能量的速率2200112ScEBPlane waves 平面波:EBc00112ScE B cc2000112cE B 01E B01SE BPlane w
27、aves:平面波：In general:通用：01SEBPoynting vector坡印廷向量see Prob 64Average intensity for plane waves :平面波的平均強度：01SE B012pkpkEBE, B in phase 同相2012pkEc2012pkBcExample 29.3. Solar Energy太陽能The average intensity of noontime sunlight on a clear day is about 1 kW/m2.晴天午時陽光的平均強度約為 1 kW/m2。(a) What are the peak el
28、ectric & magnetic fields in sunlight ?陽光的電和磁場的峰值為何？(b) At this intensity, what area of 40% efficient solar collectors would you need to replace a 4.8-kW water heater ? 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？62.910T(a)2012pkBSc02pkSBc73282 410/10/3 10/HmWmms2.9TpkpkEc B28.710/Vm0.87/kVm(b)Area
29、needed is需要面積224.81240%1.0/kWmkWmWaves from Localized Sources來自小範圍源頭的波Afar from localized source, wave is spherical :離小範圍源頭甚遠處，波呈球狀：24PSrIntensity = power / area 強度 = 功率 / 面積22,SEB1,EBr wave fields dominates static fields away from the sources. 離源頭甚遠處，波的場蓋過靜態場。Example 29.4. Cell Phone Reception 手機收訊
30、A cell phones typical average power output is about 0.6 W.一個手機的平均輸出功率通常是 0.6 W 左右。If the receiver at a cell tower can handle signals with peak electric fields as weak as 1.2 mV/m,如果基地台的接收器可以處理峰值低至 1.2 mV/m 的電場，what is the maximum allowable distance from cell phone to tower ?基地台的最大收發距離是多少？24PSr0224pk
31、c PrE2012pkEc78232 410/3 10/0.641.210/HmmsWVm5 kmApplication: Cell Phones手機Hexagonal cell area 25 km2.六角細胞的面積 25 km2。 circle of radius同面積的圓的半徑為622510 mr2.8 kmTransmission & reception are at different frequencies.發射和接收的頻率不相同。Momentum & Radiation Pressure動量和輻射壓力Maxwell 馬克斯威爾:1radUcpS1radcPSradiation momentum輻射動量radiation pressureon absorbing surface吸收面的輻射壓力12radcPSradiation pressureon reflecting surface反射面的輻射壓力Cosmos 1, a solar light-sailing spacecraft, failed at launch in 2005.宇宙一號，史上第一艘陽光帆太空船，在 2005年升空失敗。
