UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 13 Prof. Steven Errede LECTURE NOTES 13

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1 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede LECTURE NOTES 13 ELECTROMAGNETIC RADIATION In P436 Le. Nes (Giffihs h. 9-10}, we disussed he ppagain f maspi EM waves, bu we have n ye disussed hw maspi EM waves ae eaed. Using wha we leaned in P436 Le. Nes 1, we an nw disuss hw maspi EM waves ae eaed. Enyped in Maxwell s equains:, 1) E, 3) E, B, E, ) B, 0 4) B, J, is he physis assiaed wih adiain f elemagnei waves/elemagnei enegy, aising fm he aeleain {and/ deeleain} f elei hages (and/ elei uens). In he P436 Leue Nes #1, we deived he eaded elemagnei fields assiaed wih a mving pin hage q fm he eaded Liénad-Wiehe penials: q 1 V, 4 q v A, 4 A, V, Wih: whee:, and: 1ˆv 1 ˆ = eadain fa v and: 1 We als deived he espnding eaded elei and magnei fields assiaed wih a mving pin elei hage q: A, E, V, em f genealized em f adiain/ Culmb field/veliy field aeleain field q E, 3 v u ua 4 u B A,, whee: u ˆ v and: B, ˆE, em f genealized em f adiain/ Culmb field/veliy field aeleain field 1 q B, ˆ 3 v u ua 4 u 1 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 1

2 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Mispially: The aeleain {and/ deeleain} f elei hages q and/ ime-vaying elei uen densiies e. g. J nqv; J nq v ~ nqa nves (a pin f he) viual phns (assiaed wih he sai Culmb field, whih individually have ze al enegy/ze-fequeny) eal phns (whih individually have finie al enegy/finie fequeny f ), whih hen feely ppagae uwad/away fm he sue f ime-vaying elei hage and/ elei uen a he speed f ligh, {in vauum / fee spae}. Sine eal phns individually ay enegy/linea mmenum/angula mmenum, maspi EM waves ay enegy/linea mmenum angula mmenum away fm he sue, in an ievesible manne hese EM waves ppagae away fm he sue ime. Enegy/mmenum mus be inpu he haged paile f his happen enegy/mmenum ae {bh} nseved in he adiain pess. {Ne als ha we an evese he aw f ime in his pess and hus lean abu he abspin f enegy/linea mmenum/angula mmenum by elei hages/uens fm inming/iniden EM waves....} The al insananeus pwe P, assiaed wih adiain f EM waves fm a sue S ve (assumed be lalized) is bained by inegaing he eaded Pyning s ve a lage spheial shell f adius a = haaeisi dimensin f a lalized sue his is knwn as he fa-field limi, when : 1 P S da E B da,,,, S S ad The insananeus pwe adiaed is he limi f P as : P lim P, The physial easn f his definiin is simple. In he s-alled nea-zne, when a, he (genealized) Culmb field(s) (mispially nsising f viual phns) ae dminan in his egin hus, ime-vaying bu nn-adiaing E and B fields ae pesen in pximiy he sue. The nea-zne EM fields fall ff/deease/diminish as ~1 fm he sue. In ealiy, f finie, hee is always a mixue f adiaing and nn-adiaing EM fields pesen ha is assiaed wih any sue. Expessed in a gaphial manne in ems f a:,, =1 a nea-zne egime: 1 a Genealized Culmb field(s) dminan (viual phns) fa-zne egime 1 a Radiain/aeleain field(s) dminan (feely ppagaing eal phns) Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

3 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede The insananeus EM pwe assiaed wih he Genealized Culmb field is: 1 P S da E B da,,,, GCF GCF GCF GCF S S GCF Bu: E, ~1 (even fase han his, if he ne hage = 0, e.g. f highe de EM mmens assiaed wih elei diples, quaduples, uples, e. ) GCF B, ~1 (even fase han his, if he ne hage = 0, e.g. f highe de EM mmens assiaed wih magnei diples, quaduples, uples, e. ) GCF S, ~1 (eve fase, f high-de EM mmens han a pin hage disibuin) And: 4 Bu: sphee A 4 = aea f sphee f adius. 1 1, ~ ~ GCF P 4 lim, 0 GCF Ne ha P sai sues d n adiae EM enegy. i.e. n EM pwe is assiaed wih G.C.F. a On he he hand, he insananeus EM pwe assiaed wih he adiain/aeleain fields is: 1 P S da E B da,,,, ad ad ad ad S S ad Bu: E ad ~1 and ~1, ~1 ad P ad B S ad (i.e. P,, ~1, GCF Thus, we an simply pik eliminae he P EM pwe assiaed wih Genealized Culmb fields is nly appeiable nea he sue. sphee A is independen f he adius f he enlsing sufae S ), ~ nibuin!!! {n.b. f {unphysial} nn-lalized sues f ime-vaying EM adiain e.g. infinie planes, infiniely lng wies, infinie slenids, e. his equies a diffeen appah algehe } In geneal, abiay nfiguains f lalized, ime-dependen elei hage and/ elei uen densiy disibuins, and J J an/d pdue EM adiain/feely-ppagaing EM waves. As we leaned in P435 (las semese), fm he pinipal f linea supepsiin, we an always dempse an abiay elei hage and/ uen disibuin in a linea mbinain f EM mmens f he elei hage/uen disibuin, i.e. elei mnple (elei hage), elei and magnei diple, elei and magnei quaduple, e. mmens. This is ue {sepaaely} f bh sai and ime-vaying EM mmens f he elei hage and/ uen disibuin(s). Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 3

4 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede 3 F a pin elei mnple field {E(0)}, i.e., q Whee q 1 V, d (E0), q 4 v' 4 al elei hage f he sue a he eaded ime. Bu elei hage is (always) nseved, and fuheme, (by definiin) a lalized sue is ne ha des n have elei hage q flwing in away fm i. Theefe, he elei mnple mmen nibuin/pin assiaed wih he (eaded) penial(s) and EM fields is f neessiy sai i.e. he elei mnple mmen q has n EM adiain assiaed wih i. In he wds, hee an be n ne ansvesely plaized EM adiain emied fm a spheiallysymmei hage disibuin! {See e.g. J. D. Jaksn Classial Eledynamis 3 d ed. p. 410 f addiinal/fuhe deails.} The lwes-de elei muliple mmen apable f pduing EM adiain is ha p, qd, q,, assiaed wih a ime-vaying elei diple mmen Elei diple (E1) adiain iginaes fm : The lwes-de magnei muliple mmen apable f pduing EM adiain is ha m, Ia, : I, a. J, assiaed wih a ime-vaying magnei diple mmen Magnei diple (M1) adiain iginaes fm Eah ime-vaying, lalized, highe-de EM mmen nibues in alenaing suessin, J, (i.e. elei vs. magnei) muliple mmen ems: beween and Time-vaying lalized elei mmens: Time-vaying lalized magnei mmens: J, E 0 elei mnple q NO! E elei diple p qd 1 E elei quaduple Q qdd E 3 elei uple E 4 elei sexuple e E, M 0 magnei mnple g NO! M magnei diple m Ia 1 M magnei quaduple Q Iaa M (3) magnei uple M 4 magnei sexuple M d. n magnei hages anyways 4 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

5 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede We will nside/disuss he ase f EM adiain fm an sillaing elei diple and hen disuss ase f adiain fm an abiay lalized sue nsising f an abiay linea mbinain f ime-vaying EM mmens, ae n n bm n n, whee En and M n n h -de ime-vaying elei and magnei muliple mmens, espeively. n1 Elei Diple Radiain: ae Cnside an sillaing (i.e. hamni/sinusidally ime-vaying) elei diple: p qd whee he hage sepaain disane vaies in ime: d d zˆ dszˆ, f p qd z p zˆ Then: s ˆ s, wih: p qd. Equivalenly, we an alenaively hink f his as: p q d and wih ime-vaying/sillaing elei hage: q qs p qd z p zˆ Then: s ˆ s, wih: p qd., wih: d dzˆ = nsan,. {n.b. same esul!} Eihe way ne views/hinks abu his, he physis assiaed wih a hamnially ime- p ps zˆ qds zˆ is unhanged. vaying/sillaing elei diple mmen n.b. an elei uen assiaed wih he sillaing elei diple: I dq d A piue f his, f a given mmen/insan/snapsh in ime 0 is shwn belw: z d p qd ẑ q I Obsevain / Field Pin P ŷ ˆ z, I 0 0 ˆx z d q n.b. The hie f igin is delibeaely hsen a he ene f he lalized hage disibuin a he ene f he sillaing elei diple. n.b. exis (as always) sme subleies assiaed wih he alulain f he eaded penials assiaed wih mving pin hages we will addess hese subsequenly, bu n igh hee / q qs vesin f nw igh nw s, we ll sik wih he sillaing hage Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 5

6 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede efes he ime-dependene assiaed wih iself. An bseve a Nw p psˆ z P a sees he effes f he ime-vaying p : due he eaded naue f his pblem. field pin ime lae, Thus, p manifes hemselves a a finie used in he fmulae f he eaded sala and ve penials mus be evaluaed a he eaded ime, i.e. p pqd qdszˆ. V, q q q q q s s s s hage a hage a hage a hage a + d zˆ d zˆ + d zˆ d zˆ I A, d 4 whee: dq sin and: d dzzˆ d I q Expliily inseing he eaded ime(s): : V, s q s 4 z d sin q A, dzzˆ 4 zd Le us fis fus u aenin n alulaing V,. Fm he law f sines {see P435 Leue Nes 8.e. he deivain f he sai muliple mmen expansin}: d d s Hweve, we wan invesigae EM adiain in he fa zne whee d. F his siuain we an make he fllwing appximain: d 1 d 1 s 4 d 1 1 s Bu: 11 f 1. Thus: 1 d d 1 s 1 s f d. Similaly/espndingly: d 1 s 1 d 1 s f d, sine: f 1. 6 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

7 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Likewise, f he s em, f he fa zne, when d we have: d d s s 1 s s s d d s s s sin sin s In de peed fuhe, we need make an addiinal simplifying assumpin, namely ha he haaeisi spaial dimensin a f he sue (hee, a = d) is wavelengh f he emied adiain, i.e. d { f }. Thus we have: d f { f }, : d : d. n.b. This assumpin is anamun/physially equivalen saying ha we will negle any/all ime-eadain effes assiaed wih finie EM ppagain delay imes ve he dimensins haaeisi f/assiaed wih he sue i.e. hanges in hage/uen ae essenially heen/insananeus ve he {small} spaial dimensins f he sue, elaive he wavelengh f he emied adiain. Suppse we have a sue (e.g. an am) wih a d 1 nm10 emiing a f 1 Hz sinewave. Sine EM adiain avels ppagaes a 1 f 30 m pe nansend, a 1 nm dimensin sue desn un in finie ppagain deay ime pblems unil: a d (hee) i.e. 1 nm se f 310 Hz d Thus, pvided ha we addiinally ae in he egime f d, d, i.e. 1. d d Ning ha if: 1, hen: s 1 { 0 }. Then fm he Tayl seies expansins f sx 1 and sin x d d d s s s0 1 and: sin s s d s s ssin Thus: Thus: x f vey small x 1, we see ha: q 1 d d V,, 1 s s ssin 4 1 d d 1 s s s si n Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 7

8 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Expanding his u: Thus: q 1 d V,, s 4 s s d d ssin s sin 4 d s s s d d ssin s sin 4 q 1 d d V,, ss ssin 4 qd s 1 s sin 4 Bu: p p qd. Hene in he fa-zne d and d : p s 1 V,, sin s 4 In he fa-zne d, wih he addiinal esiin ha we ve als impsed n he sue EM adiain: d. We nw addiinally equie/impse a hid esiin ha he fa-zne als be suh ha, hus we have he hieahial elain: d f fa-zne EM adiain, namely ha f, hen 1 i.e. 1 1 f. Thus f he fa-zne, when d we an negle he send em in he abve expessin f V,,. p s V,, sin 4 Then: in he fa-zne, f d. Ne ha in he sai limi, when 0 i is neessay eain he send em in he abve p s V, {f w/ P435 Le. Nes same!} expessin; we bain in his limi: 4 8 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

9 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Nw le us fus u aenin n alulaing A, : z d sin q zd A, dzzˆ 4 Beause he inegain iself indues a fa f d, hen fis de d in d 1: zs z wih: z zd sin sin Thus: dz d zd Then: qd 1 A, sin zˆ 4 1 ˆ, p A sin z 4 Thus: Ne ha in he sai limi, when 0 hen A, 0 as we expe. Nw ha we have bained he (eaded) sala and ve penials V,, and, : bu: p qd and A, i is a saigh fwad exeise mpue he assiaed (eaded) EM fields, E B E V,, In spheial dinaes: A, and: B, A, n.b. V, has n explii ˆ dependene 1 1 s, ˆ ˆ p V ˆ sin sin 4 p 1 1 s sin s ˆ 4 ˆ sin sin p 1 s sˆ ssin ˆ sinsin ˆ 4 Bu f fa-zne EM adiain, d we have: 1 1 s s ssin sinsin ~ 1 ~ 1 ~ 1 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 9

10 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede S we an negle/dp he 1 s sin ˆ sinsin ˆ ems. p s V ˆ, s 4 And: A, p 1 sin ˆ z s zˆ 4 4 Bu: zˆ sˆ sinˆ in spheial dinaes. A, p s sˆ sinˆ 4 A Then f fa-zne EM adiain, wih d : E, V, p p E, s sˆ s sˆ sinˆ 4 4, Bu: : 1 1, p E s sˆ 4 p s sinˆ 4 p s sˆ 4 sin E ˆ, p s 4 O: Then: B, A, p p wih: ˆ 1 1 A, sin z sin s ˆ sin ˆ 4 4 Thus: A B, sin A 1 1 A ˆ A ˆ 1 A A ˆ sin sin 10 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

11 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Thus: 1 B A, A ˆ 1 p s sin sin sin ˆ p 1 s sin sin sin ˆ 4 p 1 s sin sinˆ 4 negle Again, 1 hee, beause, hus sin, p B s ˆ 4 sin E ˆ, p s 4 and: Nw sine ˆ ˆ 1 ˆ, ne again we see ha: B, ˆ E,, i.e. B E and Bˆ Ne als ha: a.) E and B bh vay as ~1. b.) E, and B, ae in-phase wih eah he. E, and B, have he same angula dependene ( ~sin )..) ad The EM adiain enegy densiy, u, assiaed wih he sillaing elei diple f fa-zne EM adiain { d } is: ad Ead Mad 1 1 u, u, u, E, E, B, B, p sin p sin Jules s 16 s 16 3 m p sin p sin s s Ead Mad u, u, using: : 1. n.b. 1 4 sin, p u ad s 16 Jules 3 m f: d fa zne limi Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 11

12 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede The EM enegy adiaed by an sillaing elei diple, in he fa zne { d } limi is given by Pyning s ve: ˆ ˆ ˆ ad 1 S, E, B, ˆ ˆ ˆ ˆ ˆ ˆ ad 1 p p S, sin s ˆ ˆ sin O:, p S ad s ˆ Was 16 m Radial uwad flw f EM field enegy f: d fa zne limi The EM adiain linea mmenum densiy assiaed wih an sillaing elei diple, in he fa zne { d } is given by: ad ad 1 ad, S, S, 4 ad O:, p s sin ˆ kg 3 16 m -se Radial uwad EM field linea mmenum f: d fa zne limi The EM adiain angula mmenum densiy assiaed wih an sillaing elei diple, in he fa zne { d } is given by: ad ad,, 4 ad, p s sin 3 ˆˆ0 kg 16 m-se N EM field angula mmenum f: d fa zne limi ad n.b. The exa, 0 i.e. igne esiins n fa-zne limi, keep all highe-de ems... we have negleed E ˆ ~ em whih is nn-negligible in he nea-zne d ~ and als in he s-alled inemediae, induive zne ~. ˆ 1 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

13 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Time-Aveaged Quaniies f Radiain fm an Osillaing Elei Diple: A 1 A d 1 A 1 s d 0 0 A Reall he definiin f ime aveage: The ime-aveaged EM adiain enegy densiy assiaed wih an sillaing elei diple is: u 3 4 ad sin, p Jules 3 m f: d fa-zne limi ad The ime-aveaged Pyning s ve, whih is als he inensiy I f EM adiain assiaed wih an sillaing elei diple is: 4 ad ad 1 p sin I S, E, 3 We als see ha: I ad S ad, u ad, Was m. Was m f: d fa-zne limi The ime-aveaged EM adiaed pwe assiaed wih an sillaing elei diple is: P S da p 4 ad ad,, S 4 p sin sin 0 0 d s 4 p d 0 0 dd sin s sin ds 16 Le: u s, du d s, 0u 1, u 1, sin 1s 1 u u du u u The ime-aveaged adiaed pwe assiaed wih an sillaing elei diple is: P ad, 1 p 4 (Was) f: d fa-zne limi Ne ha ime-aveaged adiaed pwe vaies as he 4 h pwe f fequeny! ad n.b. P, The ime-aveaged EM adiain linea mmenum densiy assiaed wih an sillaing elei diple is: has n -dependene! sin ad, S ad, u ad, ˆ p ˆ 3 3 kg m -se f: d fa-zne limi Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 13

14 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede The ime-aveaged EM adiain angula mmenum densiy assiaed wih an sillaing elei diple is:,, ad ad p sin 3 ˆ ˆ kg m-se f: ad n.b. The exa 0 i.e. igne esiins n fa-zne limi, keep all highe-de ems... we have negleed he E ˆ ~ em whih is nn-negligible in he nea-zne d ~ and als in he s-alled inemediae, induive zne ~. 4 p sin ad ad Ne ha beause: I S, ad ad S S 3, 0,,, 0 sine: Was m sin 0 sin 0 i.e. n EM adiain us alng he axis f he elei diple ( ẑ axis) EM adiain f elei diple is peaked/maximum a (hen i.e. maximum EM adiain us he axis f he elei diple: S ad, d fa-zne limi sin 1) 4 p 3 ~sin 0 Thus, he inensiy pfile ad I in 3-D {f fixed } f elei diple adiain is dnu-shaped - ainally invaian in, as shwn in he figue belw: 14 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

15 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Giffihs Example 11.1: 4 ad The ime-aveaged pwe f elei diple adiain is p P. 1 ad Ne ha P ~ ( ~ f, ~ ) 8 F ed ligh: ed 780 nm fed Hz F vile ligh: vile 350 nm P f ed 14 P f ed vile f ed vile 9 vile vile Hene: ad P ~ explains why he sky is blue! Sunligh {unplaized ligh} iniden n O & N mleules in he eah s amsphee simulaes he O & N ams vibaes he {bund} ami elens a {angula} fequeny, ausing hem sillae as elei diples! Sla EM adiain a a given angula fequeny is hus absbed and e-emied in his EM adiain + am saeing pess. The abve fmula f EM pwe adiaed as elei diple adiain by suh ams, by ime-evesal invaiane f he EM ineain, is als he EM pwe absbed by ams, hus we 4 see ha beause f he -dependene f P, he highe fequeny/she wavelengh adiain (i.e. blue/vile ligh) is pefeenially saeed muh me s han he lwe fequeny/lnge wavelengh adiain (i.e. ed ligh). ad The Eah s sky appeas blue {e.g. an bseve n he gund, even e.g. a spae shule asnau in bi aund he eah} beause he ligh fm he sky is saeed (i.e. e-adiaed) ligh, whih is pefeenially in he blue/vile pin f he visible ligh EM speum. The saeing f EM adiain ff f ams is knwn as Rayleigh saeing. Ne ha peisely same physis als simulaneusly explains why he Sun appeas ed e.g. an bseve n he gund a sunise and sunse beause a hese imes f he day, pah ha he sunligh akes hugh he amsphee is he lnges, elaive ha assiaed e.g. wih is psiin a {lal} nn. If he highe-fequeny blue/vile ligh is pefeenially saeed u f he beam f sunligh, wha is lef in he beam f sunligh afe avesing he enie hikness f he Eah s amsphee is he lwe-fequeny, ange-ed ligh. 14 Hz Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 15

16 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Ne ha he Sun is a blak-bdy adia is EM speum peaks in he infa-ed egin hus i is NOT fla by any means {als is affeed by fequeny-dependen abspin in he amsphee}: Ne he lg sale n he veial axis! Thus, hee is n muh vile ligh in he Sun s EM speum, and hene hee is a deliae balaning a f flux f EM adiain fm he Sun {nvlued} wih is blak-bdy speum and he saeing f his adiain by ams in he Eah s amsphee hus we see he sky as blue. Thus, if he blak-bdy empeaue f he sun was diffeen, hen he l f he Eah s sky in he visible pin f he EM speum wuld als be diffeen mpae he blak-bdy spea f u Sun e.g. wih ha f Spia (60 ly away in he Vig nsellain) and Anaes (a ed gian 600 ly away in he Spi nsellain): 16 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

17 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Ligh fm he Sun is unplaized (i.e. i nsiss f all plaizains, andmly iened ve ime). Hweve, beause EM waves ae ansvesely plaized (defined by he ienain f he E -field ve) an iniden EM plane wave fm he Sun wih plaizain in a given diein ( k -ppagain diein) will (ansiily) indue elei diple mmens in p E is he mleula gas ams in eah s amsphee, via ml ml in, whee plaizabiliy a {angula} fequeny {see P435 Le. Nes 1 and P436 Le. Nes 7.5}. The axis f indued elei diple mmens will be he plane f plaizain f iniden wave a ha insan, hene he saeed adiain emied by he am will be pefeenially a 90 (i.e. ) he axis f he (indued) elei diple f gas ams in eah s amsphee. Thee ae w speifi/limiing ases nside (a) when he iniden E -field ve is veial and (b) when he iniden E -field ve is hiznal. Randm plaizain is hen an abiay linea mbinain f hese w limiing ases: (a.) E veial: in ml p indued E in B in B sa E sa Am 90 sa k in Ne: Esa pindued Ein f sa = 90 (max pbable diein f emissin). elei diples sillaing line -f-sigh pefeenially end adiae in he line-f-sigh diein. k sa = Line f sigh (b.) E in hiznal: Eah B in k in Same am and same bseve, bu bseve desn see his saeed adiain elei diples sillaing alng he line-f-sigh d n adiae in ha diein. = Line f sigh p indued B sa Am E sa k sa 90 sa E in Eah Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 17

18 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede Beause he blue ligh an bseve sees fm a given pin f he sky is due he pefeenial saeing f elei diple-ype Rayleigh saeing f sunligh/sla EM adiain ff f gas ams in he Eah s amsphee, wih Esa he line-f-sigh, his adiain has a ne plaizain i.e. he ligh fm he sky is plaized, espeially s away fm he sun, i.e. in he nhen pins f he sky {in he nhen hemisphee}!!! Yu an vey easily bseve/expliily veify his using a pai f plaid sunglasses y i sme ime!!! I is benefiial wea plaid sunglasses e.g. when u baing n a lake in de edue glae fm {plaized} sunligh efleed ff f he sufae f he wae!!! As menined abve, a sunise sunse, he sun appeas ed when an bseve is lking diely a he sun, beause he blue/vile ligh is ~ 5 me pefeenially saeed u f he beam f ligh iniden fm he sun {pe uni hikness f amsphee} han ed ligh. Thus sunligh when he sun is nea he hizn nsiss pedminanly f wha emains ed ligh. Ne ha his is als ue f mnise and mnse he mn will {likewise} have a eddish hue a hese imes, and ne ha his is als ue e.g. f he ase f an elipse f he mn by he Eah. One an als bseve his same phenmenn e.g. using a glass pihe f milk dilued wih wae beause milk mleules ae effiien Rayleigh saees f visible ligh! Hee s a simple expeimen ha yu an ay u a hme, e.g. using a flashligh: 18 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

19 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede The {sala} EM wave haaeisi adiain impedane f an anenna is exaly as we defined he haaeisi impedane f a waveguide; ning hee ha we ae dealing wih manifesly ansvese waves f EM wave adiain fm an elei diple anenna: Z anenna Le s hek he SI unis f his definiin: ad ad ad E E E ad H 1 ad 1 ad B B Vls Vls E m m Vls Ohms B Henys N N Amps Teslas Am - A m F elei diple adiain, he EM wave haaeisi adiain impedane in he fazne limi ( d ) wih 1 is: sin, p E s ˆ 4 and: sin B ˆ, p s 4 Z p 4 (1) anenna p 4 sin s sin s Z Whee: Henys/m = magnei pemeabiliy f fee spae / vauum Faads/m = elei pemiiviy f fee spae / vauum And: Z Henys/m Faads / m = {sala} haaeisi impedane f fee spae/he vauum. Thus we see ha elei diple anennae (in he fa-zne limi ( d b )) ae pefely impedane-mahed f ppagain f EM waves in fee spae / vauum! Ne als ha he fa-zne ( d ) EM wave haaeisi adiain impedane (1) Z has n spaial and/ fequeny dependene. anenna Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 19

20 UIUC Physis 436 EM Fields & Sues II Fall Semese, 015 Le. Nes 13 Pf. Seven Eede The EM Wave Radiain Resisane f an Anenna: The {sala} EM wave adiain esisane f an anenna Rad is defined in ems f he anenna pwe Pad and he ampliude f he uen I flwing in he anenna: P anenna ad : anenna I Rad R P I anenna anenna ad ad (Ohms) F an elei diple anenna: I = qω = ampliude f uen flwing in he diple. In he fa-zne limi, i.e. d : R (1) ad 4 p 3 1 I q 1 q 4 d d 1 (1) n.b. R ad is fequeny-dependen! In he fa-zne limi, i.e. d : d d d d R Z Bu: (1) (1) ad ad In he fa-zne limi, d : Z Z (1) ad (1) d 1 d ad R Z Z 1 1 d Hweve, in he fa-zne limi, d we have: 1 (1) Thus, we see ha he EM wave adiain esisane R ad assiaed wih elei diple anenna in he fa-zne limi ( d ) is muh less han he EM wave haaeisi (1),M(1) adiain impedane Z Z f an elei diple anenna: ad (1) 1 d ad R Z Z Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved