Lecue 4 Chape Wae Moion Plane waes 3D Diffeenial wae equaion Spheical waes Clindical waes
3-D waes: plane waes (simples 3-D waes) ll he sufaces of consan phase of disubance fom paallel planes ha ae pependicula o he popagaion diecion
3-D waes: plane waes (simples 3-D waes) ll he sufaces of consan phase of disubance fom paallel planes ha ae pependicula o he popagaion diecion Uni ecos n equaion of plane ha is ˆ i cons a pependicula o ˆj ˆ ll possible coodinaes of eco ae on a plane Can consuc a se of planes oe which aies in space hamonicall: sin o cos i o e
sin Plane waes The spaiall epeiie naue can be epessed as: In eponenial fom: i i / i e e e i e i Fo ha o be ue: e Veco is called popagaion eco
e i Plane waes: equaion This is snap-sho in ime no ime dependence To mae i moe need o add ime dependence he same wa as fo one-dimensional wae: Plane wae equaion i e
Plane wae: popagaion eloci Can simplif o -D case assuming ha wae popagaes along : i e î i e We hae shown ha fo -D wae phase eloci is: Tha is ue fo an diecion of + popagae wih - popagae opposie o Moe geneal case: see page 6
ample: wo plane waes Same waelengh: = ==/ Wie equaions fo boh waes. Soluion: i e Same speed : = == Do poduc: Wae : e i diecion Wae : e sin i sin cos cos cos sin cos Noe: in oelapping egion = + cos
Plane waes: Caesian coodinaes i e i e - diecion cosines of i e Wae eq-ns in Caesian coodinaes: Impoance of plane waes: eas o geneae using an hamonic geneao an 3D wae can be epessed as supeposiion of plane waes
Thee dimensional diffeenial wae equaion Taing second deiaies fo i e can deie he following: + + combine and use: 3-D diffeenial wae equaion
Thee dimensional diffeenial wae equaion lenaie epession Use Laplacian opeao: Using = we can ewie i e i e funcion of f I can be shown ha: g f g ae plane-wae soluions of he diff. eqn poided ha ae wice diffeeniable. No necessail hamonic! In moe geneal fom he combinaion is also a soluion: g C f C / /
ample Gien epession: a b c whee a> b> Does i coespond o a aeling wae? Wha is is speed? Soluion:. Funcion mus be wice diffeeniable a a b ca. Speed: a b a b b cb b a Diecion: negaie diecion
ample Gien epession whee a> b>: b a Does i coespond o a aeling wae? Wha is is speed? Soluion:. Funcion mus be wice diffeeniable 3 4 6 a b. Wae equaion: 6 4 a Is no soluion of wae equaion! This is no a wae aeling a consan speed!
Spheical waes Spheical waes oiginae fom a poin souce and popagae a consan speed in all diecions: waefoms ae concenic sphees. Isoopic souce - geneaes waes in all diecions. Smme: inoduce spheical coodinaes sin cos sin sin cos Smme: he phase of wae should onl depend on no on angles: -D concenic wae waes spheical wae
Spheical waes cos sin sin cos sin sin sin sin Since depends onl on : ealuaes o he same Wae equaion:
Spheical waes This is jus -D wae equaion In analog he soluion is: f f - popagaes ouwads (dieging) + popagaes inwad (coneging) Noe: soluion blows up a = In geneal supeposiion wos oo: g C f C
Hamonic Spheical waes f In analog wih D wae: Hamonic spheical wae cos i e Consan phase a an gien ime: =cons mpliude deceases wih - souce sengh Single popagaing pulse
Spheical hamonic waes cos Deceasing ampliude maes sense: Waes can anspo eneg (een hough mae does no moe) The aea oe which he eneg is disibued as wae moes ouwads inceases mpliude of he wae mus dop! Noe: spheical waes fa fom souce appoach plane waes:
Clindical waes Waefons fom concenic clindes of infinie lengh sin cos Smme: wo in clindical coodinaes I simila o essel s eq-n. lage he soluion can be appoimaed: Hamonic clindical wae cos i e
Clindical waes Hamonic clindical wae cos i e Can ceae a long wae souce b cuing a sli and diecing plane waes a i: emeging waes would be clindical.
Lecue 5 Chape 3 lecomagneic heo Phoons. and Ligh asic laws of elecomagneic heo Mawell s equaions lecomagneic waes Polaiaion of M waes neg and momenum
asic laws of elecomagneic heo lecic field Coulomb foce law: QQ F F 4 elecic pemiii of fee space Q Q lac bo F F F Q 4 Q ˆ Ineacion occus ia elecic field lecic field can eis een when chage disappeas (annihilaion in blac bo)
asic laws of elecomagneic heo Magneic field Moing chages ceae magneic field pemeabili of fee space The io-saa law fo moing chage 4 q ˆ Magneic field ineacs wih moing chages: F magneic q Chages ineac wih boh fields: F q q (Loen foce)
asic laws of elecomagneic heo Gauss s Law: elecic Kal Fiedich Gauss (777-855) lecic field flu fom an enclosed olume is popoional o he amoun of chage inside q S ds q Moe geneal fom: ds dv S V If hee ae no chages (no souces of field) he flu is eo: ds S Chage densi
asic laws of elecomagneic heo Gauss s Law: magneic Magneic field flu fom an enclosed olume is eo (no magneic monopoles) M S ds
asic laws of elecomagneic heo Faada s Inducion Law 8: Michael Faada Changing magneic field can esul in aiable elecic field d Fomal esion emf C dl d M d d ds nomal o aea ds aea nd ˆ Changing cuen in he solenoid poduces changing magneic field. Changing magneic field flu ceaes elecic field in he oue wie. d M ds nd ˆ d M dcos angle beween and nomal o he aea d
asic laws of elecomagneic heo mpèe s Cicuial Law 86: (Memoi on he Mahemaical Theo of lecodnamic Phenomena Uniquel Deduced fom peience) wie wih cuen ceaes magneic field aound i ll he cuens in he uniese conibue o bu onl ones inside he pah esul in noneo pah inegal mpee s law dl Iinside _ pah C Incomplee! dl J ds C Cuen densi
asic laws of elecomagneic heo mpèe s-mawell s Law Mawell consideed all nown laws and noiced asmme: Gauss s ds Gauss s Faada s mpèe s S S C C ds dl dl d d q ds J ds Changing magneic field leads o changing elecic field No simila em hee Hpohesis: changing elecic field leads o aiable magneic field
asic laws of elecomagneic heo mpèe s-mawell s Law mpèe s-mawell s Law: mpèe s law dl J ds C The will depend on aea: dl J ds i C C dl J ds Woaound: Include em ha aes ino accoun changing elecic field flu in aea : dl J ds C displacemen cuen densi
Gauss s Gauss s Faada s mpèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d q ds J q In acuum (fee space) ds fields ae defined hough ineacion wih chages Inside he media elecic and magneic fields ae scaled. To accoun fo ha he fee space pemiii and ae eplaced b and : K KM dielecic consan K > elaie pemeabili
Gauss s Gauss s Faada s mpèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d q ds J q ds In mae fields ae defined hough ineacion wih chages
Mawell equaions: fee space no chages Cuen J and chage ae eo Inegal fom of Mawell equaions in fee space: ds no magneic chages S no elecic chages ds changing magneic field ceaes changing elecic field changing elecic field ceaes changing magneic field S C C dl dl d ds d ds Thee is emaable smme beween elecic and magneic fields!
Mawell equaions: diffeenial fom (fee space) Noaion: j i ˆ ˆ ˆ Laplacian: ) ( di j i ˆ ˆ ˆ cul()
lecomagneic waes (fee space) Changing field ceaes field Changing field ceaes field Is i possible o ceae self-susaining M field? Can manipulae mahemaicall ino:
lecomagneic waes ˆ ˆ ˆ j i Resembles wae equaion: ach componen of he M field obes he scala wae equaion poided ha
Ligh - elecomagneic wae? Mawell in ~865 found ha M wae mus moe a speed ha ime pemiii and pemeabili wee nown fom elecic/magneic foce measuemens and Mawell calculaed 374 m/s Speed of ligh was also measued b Fieau in 949: 353 m/s Mawell woe: This eloci is so neal ha of ligh ha i seems we hae song eason o conclude ha ligh iself (including adian hea and ohe adiaions if an) is an elecomagneic disubance in he fom of waes popagaed hough he elecomagneic field accoding o elecomagneic laws. cele (la. - fas) ac alue of speed of ligh: c =.997 94 58 8 m/s
lecomagneic wae ssume: efeence fame is chosen so ha =( ) longiudinal wae popagaes along does no a wih This canno be a wae! Conclusion: i mus be ansese wae i.e. =. Similal =. Since is pependicula o we mus specif is diecion as a funcion of ime Diecion of eco in M wae is called polaiaion Simple case: polaiaion is fied i.e. diecion of does no change
Polaied elecomagneic wae We ae fee o chose -ais so ha field popagaing along is polaied along : ( ). lso: = =cons (=) -field of wae has onl componen -field of wae has onl componen (fo polaied wae popagaing along ) In fee space he plane M wae is ansese
Hamonic polaied elecomagneic wae Hamonic funcions ae soluion fo wae equaion: cos c / Find : polaied along ais c d popagaes along ais cos c / c This is ue fo an wae: - ampliude aio is c - and ae in-phase
Hamonic polaied elecomagneic wae lecomagneic waes * diecion of popagaion is in he diecion of coss-poduc: * M field does no moe in space onl disubance does. Changing field ceaes changing field and ice esa
neg of M wae I was shown (in Phs 7) ha field eneg densiies ae: u u Since =c and c=( ) -/ : u u - he eneg in M wae is shaed equall beween elecic and magneic fields Toal eneg: u u u (W/m )
The Poning eco M field conains eneg ha popagaes hough space a speed c neg anspoed hough aea in ime : uc S uc neg S anspoed b a wae hough uni aea in uni ime: uc c c c c John Hen Poning (85-94) The Poning eco: S powe flow pe uni aea fo a wae diecion of popagaion is diecion of S. (unis: W/m )
The Poning eco: polaied hamonic wae S Polaied M wae: cos cos Poning eco: S cos This is insananeous alue: S is oscillaing Ligh field oscillaes a ~ 5 H - mos deecos will see aeage alue of S.
Iadiance eage alue fo peiodic funcion: need o aeage one peiod onl. S cos I can be shown ha aeage of cos is: cos nd aeage powe flow pe uni ime: I Iadiance: c S T lenaie eq-ns: c I c T S T T T c Iadiance is popoional o he squae of he ampliude of he field Usuall mosl -field componen ineacs wih mae and we will efe o as opical field and use eneg eq-ns wih Fo linea isoopic dielecic: I Opical powe adian flu oal powe falling on some aea (Was) T
Spheical wae: inese squae law Spheical waes ae poduced b poin souces. s ou moe awa fom he souce ligh inensi dops cos Spheical wae eq-n: cos cos S cos c S I T Inese squae law: he iadiance fom a poin souce dops as /
Classical M waes esus phoons The eneg of a single ligh phoon is =h The Planc s consan h = 6.66-34 Js Visible ligh waelengh is ~.5 m h c h 4 9 J ample: lase poine oupu powe is ~ mw numbe of phoons emied ee second: P 4 5.5 9 3 J/s J/phoon phoons/s Conclusion: in man ee da siuaions he quanum naue of ligh is no ponounced and ligh could be eaed as a classical M wae