Plane Waves, Polarization and the Poynting Vector

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Laureen Kathryn Tate
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1 Plan Wavs, Plaiain and h Pning Vc H = - h h + -

2 Unifm Plan Wav in F Spac W hav pviusl sablishd h fllwing ppis f plan wavs in f spac: lcic and magnic fild cmpnns in ppagain dicin a. lcic and magnic filds ( ) a lad. ach is h, B, B suc f h h. A s f h scnd d diffnial quains appl, n f ach fild cmpnn in cangula cdinas. F ampl: m =, f hamnic filds, in phas nain: =-w m A gnal sluin h Hlmhl quain is win: jb = C + C ( can b cmpl), 1 b = w m = w/ c = wh, spd f ligh. c C i Ppagain f Magnic Fild Supps w hav an lcic fild wav avlling in h psiiv dicin. Rcall ha cul =- B, f u avlling wav, = + =- B d + i =- B b + = - B fi B = + w w = + cw

F ampl: m =, f hamnic filds, in phas nain: =-w m A gnal sluin h Hlmhl quain is win: jb = C + C ( can b cmpl), 1 b = w m = w/ c = wh, spd f ligh.

3 I is v usful pss his ai f lcic and magnic filds in ms f H, ah han B. m H = = m, c m m, af aanging: = m c = = H m Dimnsinal analsis: Hn / m Faad / m : lcic filds a vls/m and H filds a amps/m. m = h Ohms, h ª 1p = 377 Ohms (impdanc f fspac) H F a wav avlling in h - $ dicin, =-h. This jus mans ha H is ind alng h - $ dicin ( quivalnl wih H ). lcic and magnic ( B ) fild a hgnal (ppndicula, Righ Hand Rul), in-phas, and h ai f h fild magniuds is h impdanc. Summa: and H a ppndicula ach h and ppagain dicin. Righ Hand Rul givs dicin. Rai f H is h ininsic wav impdanc, h. Th wavlngh is h disanc ha h wav avls s ha h phas changs b p adians. bl = p, l Jul / amp vl = = = Ohm!!! Jul / vl amp p p p c = = = = b w/ c w c f

m = h Ohms, h ª 1p = 377 Ohms (impdanc f fspac) H F a wav avlling in h - $ dicin, =-h. This jus mans ha H is ind alng h - $ dicin ( quivalnl wih H ).

4 Picu: Phas vlci, v p T undsand h phas vlci, w mus un h al psnain f h fild wih bh spac and im dpndnc; ha is, = + cs w-b Cnsid an bsv mving alng in a h sam pin, sa h pak, f h scillaing fild. This mans mahmaicall ha: w - b = an cnsan. Thus, wd = b d w = vp, h phas vlci. d b Plaiain f Plac Wavs Cnsid h ppagain chaacisics f a plan wav in which h lcic fild has cmpnns in bh h and dicins: - j = $ + $, wh h fild cmpnns ma b cmpl. Tha is, ja jb = and = In phas a = b, = + b $ $ b d i - ag j b g d,

5 in al fm: j b g = + cs w-b -a This is lina plaiain. Nw w cnsid a m gnal cas. llipical Plaiain In his cas w allw abia phas lainships a and b: ja - b = + I is asi s wha his mans if w wi ach cmpnn u in al fm: = csaw+ a-bf = cs w+ b-b Supps w l a = and b = p /. Thn: Wha can w sa abu,? = cs w-b =- sin w-b a a f = + a a j jb j f f f

asi s wha his mans if w wi ach cmpnn u in al fm: = csaw+ a-bf = cs w+ b-b Supps w

6 1 Mak a pl in h - plan wih as a paam and = 1, = w.77 p /4 p/ p 5p/ 4 3p / 7.77 p / / 1 /. 77/ 1 /. 77/ This clckwis ain dscibs an llips, wih maj as paalll h ais. As h wav ppagas alng, h llips spads u an llipical hli. This is calld llipical plaiain. F h spcial cas f =, w hav cicula plaiain. If w had chsn b =-p /, w wuld hav fund ha an idnical llips wuld b fmd cp ha h ain wuld b cun-clckwis. Pning Thm W knw ha ng is ppagad b wavs, in gnal, and lcmagnic wavs, in paicula. W nd quanif h assciad pw flw. W asil bain a hin f hw calcula pw flw b calling u cicui * h, wh P = VI, b a dimnsinal analsis f h filds. Rmmb ha h unis f H a Amp/m and h unis f a V/m. Thi pduc Amp Vls/m has unis f Was/aa, which is a pw dnsi, jus wha w wan. Th pduc mus clal hav a dicin assciad wih i, and i ugh smhw pin in a asnabl dicin. A asnabl dicin f pw flw in lsslss, hmgnus, lina mdia, such as f spac, wuld b in h dicin f ppagain. W migh hf guss ha H wuld b a asnabl dfiniin f pw dnsi. W will cnsid h cmpl Pning vc f im hamnic plan lcmagnic wavs in phas nain. This quis sm hugh bcaus

If w had chsn b =-p /, w wuld hav fund ha an idnical llips wuld b fmd cp ha h ain wuld b cun-clckwis. Pning Thm W knw ha ng is ppagad b wavs, in gnal, and lcmagnic wavs, in paicula.

7 f h nnlina nau f H. L a f a f a f a f, = R $ and H, = R H $ W will wi u h al and imagina pas f and H. a, f = R af + jiaf $ and H, = R H + jhi $ a f a f a f Nw pand h cmpl pnnial and ak h al pa: a, f = af csw- iaf sin w $ H, = H csw- H sin w $ a f af af i a f = = Th Pning vc is: P, H H cs w+ H sin w- H cswsinw- H sinwcs w $ i i i i Th im avag Pning vc is: T Pavaf = 1 Pa fd = 1 1, H + H T i i Th v sam pssin can b baind dicl fm h phass b h fllwing ul: P a, f = 1 1 Rd H* i = R d + jii$ dh- jhii$ 1 = d H + i Hii $

csw- H sin w $ a f af af i a f = = Th Pning vc is: P, H H cs w+ H sin w- H cswsinw- H sinwcs w $ i i i i Th im avag Pning vc is:

8 ampl Wha is h im avag Pning vc f a plan wav ppagaing in f spac wih h fllwing (phas) filds: + and H af = af = $ + $? h Thn : P a f d i 1 F jb R + HG 1 Pav, = R H* = av a f = +, h $ 1 jb $ $ R $ h h I = KJ,

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