Chapter 7: Plane Electromagnetic Waves and Wave Propagation

Transcription

1 Chapr 7: Plan lcromagnc Wavs and Wav Propagaon An Hsorcal Prspcv: Faraday:Tm-varyng magnc fld gnras lcrc fld. Mawll:Tm-varyng lcrc fld gnras magnc fld. Hr dscovrd rado wavs; nsn's spcal hory Mawll's hory accpd of rlavy Faraday Faraday's law; orn Mawll Mawll orn Mawll dd; quaons nsn orn A No aou Oscllaory Bhavor: Common faur of oscllaory havor: Oscllaons rqur nrgy nrgy yp yp nrgy sorng mchansms nrgy chang mchansm(s nrgy sorng nrgy chang ampl mdum mchansms mchansm(s mass-sprng sysm mv, rsorng forc mass & sprng LC oscllaor LI, CV Q, I L, C, & wr M wav B, db, d no rqurd d d 3 Organaon of Lcur Nos on Ch. 7: In Jacson, plan wavs n dlcrc mda ar rad n Scs. 7. and 7.. Varous spcal cass (plasma mdum and hgh-frquncy lm ar rad n Sc Plan wavs n conducors ar rad n Sc. 5.8 [.g. qs. (5.63-(5.69] and Sc. 8. [.g. qs. (8.9, (8., (8., (8.4, and (8.5] y mhods dffrn from hos n Scs. 7. and 7.. Hr, w wll covr hs scons n Jacson wh a unfd ramn of plan wavs n oh dlcrcs and conducors, and a all frquncs. quaons n Jacson wll amnd n grar dal, u n somwha dffrn ordr. So, n h lcur nos, h hr scons on hs marals wll numrd Scs. I, II, and III rahr han followng Jacson s scon numrs. Howvr, Scs. 7.3, 7.4, 7.8, and 7.9 of Jacson wll followd closly n ssqnlc susqun lcur nos (and numrd as n Jacson. W gn wh a drvaon of h gnrald dlcrc consan /, whch s applcal o oh dlcrc and conducng mda. 4

2 I. Drvaon of h Gnrald Dlcrc Consan / [Sc. 7.5 (par A] Dpol Momn of a Sngl lcron: Th quaon of moon for an aomc or molcular lcron wh mass m and charg n h prsnc of an rnal lcrc fld (, can wrn: rsorng forc du o lcron dsplacmn F rsorng forc m (, m m (7.49 : dsplacmn of : lcron collson frquncy h lcron from m : dampng forc F F( F' ( s qulrum (ra of chang of m poson =. lcron momnum As n Sc. 4.6, w nglc du o collsons hghr-ordr rms. Th "ndng frquncy" s h naural oscllaon frquncy of h lcron f s s o osclla aou undr h rsorng forc m. Snc / m, h rsorng forc s ndpndn of m. 5 I. Drvaon of h Gnrald Dlcrc Consan / (connud Rwr (7.49, m (, m m, as m( (, L* (, = and pand aou h qulrum poson, w oan ( ( (, of h ordr of ( f whr s h scal lngh of. For ampl, f s a wav fld, hn wavlngh. By nglcng (, w hav assumd ha h lcron dsplacmn s oo small for h lcron o s any spaal fld varaon. Thus, w assum ha h lcron s acd on y a spaally unform fld: (, (, and s undrsood ha (, s gvn y h ral par of h RHS. *Ths s quvaln o a Fourr rnasformaon o h spac and s a compl quany calld h phasor [s Appnd A] 6 I. Drvaon of h Gnrald Dlcrc Consan / (connud L ( and susu ( no m( (,, (, ( w oan m( ( wh h soluon: ( m ( m ( ( rprsns h forcd oscllaon of asmplharmoncoscllaor oscllaor wh naural oscllaon frquncy. Th m-dpndn rsuls n a m-dpndn dpol momn a gvn y whr p m p( p, ( Ths rducs o (4.7 n h sac lm:. 7 I. Drvaon of h Gnrald Dlcrc Consan / (connud ( (, ( m Rwr ( and ( ( p p p m In hs quaons, (,, and p ar phasors conanng phas and amplud nformaon of (,,, and p, rspcvly. Th suscrp "" n and p rfrs o h fac ha h oscllaon s cnrd a, whr (, s appromad y a spaally unform fld ( (s valu a. If h oscllaon s cnrd a an arrary pon, h only dffrnc s ha h lcron would s a spally consan fld gvn y. Thus, n gnral, p wh p p m (7.5 No ha, n (7.5, s a spaal varal (no h lcron dsplacmn, and p and ar phasors. 8

3 I. Drvaon of h Gnrald Dlcrc Consan / (connud Th Gnrald Dlcrc Consan : Assum hr ar N molculs pr un volum and Z lcrons pr molcul. Dvd h lcrons of a molcul no groups, ach wh lcron numr f ( f Z, ndng frquncy nc, and collson frquncy nc [Thr may on or mor fr lcrons ( pr molcul.] Thn, h lcrc polaraon (oal dpol momn pr un volum s P N f p N f m a macroscopc quany (7.5 (4.36 a spaal varal ndng h dfnons of h sac lcrc dsplacmn ( D D P (4.34 (4.37 and prmvy ( : ( (4.38 o flds wh p dpndnc, w oan D ( gnrald wh N f m (7.5 dlcrc consan 9 I. Drvaon of h Gnrald Dlcrc Consan / (connud Dvd h lcrons n h mdum no For coppr, f ound lcrons: 3 and 4 / s. fr lcrons:, f f, (7.5 N f N f m (ound m ( du o fr lcrons (7.56 f N Drud modl for h whr (7.58 m ( lcrcal conducvy In gnral, (s p. 3. Hnc, s prdomnanly ral. Whn, Im com s larg. rsonan asorpon Qusons:. as. Hnc, h drvaon ras down. Why?. Wha mas h mdum dsprsv (.. dpnds on? I. Drvaon of h Gnrald Dlcrc Consan / (connud Dscusson : ( D mpls a lnar rlaon wn D and. Th lnary rsuls from h assumpon ha h lcron dsplacmn s suffcn samll so ha, n (7.49, f( and can appromad y a consan (. ( / n (7.5 or (7.56 s a gnrald dlcrc consan, whch ncluds conruons from oh ound and fr lcrons. I s hus applcal o oh nsulang and conducng marals. In h wav flds, fr lcrons osclla aou an qulrum poson us l h ound lcron. Hnc, oh yps of lcrons can rad on qual foong. Th gnrald s an rmly usful quany. As wll shown, allows a unfd ramn of M wavs n oh nsulang and conducng marals. I. Drvaon of h Gnrald Dlcrc Consan / (connud ( Wr [ R(, Im( ]. From (7.56, can sn ha s du o [.. h dampng rm n (7.49]. Hnc, s rsposl for h anuaon of M wavs n h maral. For h nsulang maral,, h anuaon consan s gvn y Jacson (7.55 n rms of. For a good conducor,, h anuaon consan s gvn y Jacson (5.64 n rms of. Th anuaon consans n dlcrc and conducng marals wll drvd lar n hs chapr. No ha oh ound and fr lcrons conru o [s (7.56 ], u conruon from fr lcrons s usually far mor mporan han ound lcrons ( why?. vn h nsulang maral conans a small numr of fr lcrons o gv h maral a small conducvy.

4 I. Drvaon of h Gnrald Dlcrc Consan / (connud (v s drvd n h -spac for a harmonc fld of arrary frquncy. Hnc, D s a consuv rlaon n -spac vald for all. For mul-frquncy flds, w may oan h -spac D hrough a Fourr ransformaon D( D d n gnral d [ ] (3 N f N f m (ound m ( Snc d, w fnd from (3 ha, n gnral, D caus s a funcon of. Thr ar, howvr, spcal cass for whch (3 wll yld D n -spac, as dscussd n (v and ( v low. 3 I. Drvaon of h Gnrald Dlcrc Consan / (connud (vconsdrasac( sac ( lcrcfld nadlcrcmdum whou fr lcrons ( f, w hav ( d d ( f N N f m (ound m (, f N f m (ound s ral. Thus, n -spac, w hav a sac D gvn y D d d, Ths rcovrs h sac rlaon n (4.37 whou mang any appromaon. 4 I. Drvaon of h Gnrald Dlcrc Consan / (connud (v For m-dpndn flds n a mdum wh nglgl dsprson [.. ] and nglgl loss (.., w hav D ( D ( d ( ( d ( (, whr N f N f w m m Ths plans assumpon ( on p. 59 for h drvaon of (6.7; namly, h macroscopc mdum s lnar n s lcrcal propry and has nglgl dsprson and nglgl loss. Undr hs assumpon, w may wr D( (. Hnc, n (6.5, w hav D = = D. Qusons :. Assum an lcromagnc sgnal s propagang n h mdum. Wha s h condon on h sgnal ln ordr for?. Why s h assumpon of "nglgl loss" also rqurd? 5 I. Drvaon of h Gnrald Dlcrc Consan / (connud A no aou rmnology : In gnral, h lcrc prmvy s a nsor (dno y ε and w may wr 3 D ε, whr ε Th lcrcal lpropry f of h mdum s unform (or homognous ε s ndp. of lnar ε s ndp. of nondsprsv ε s ndp. of 33, soropc f 6

5 II. Plan Wav quaons n Dlcrcs and Conducors - A Unfd Formalsm Basc quaons : Macroscopc Mawll quaons: fr, J fr ar du o D (, fr(, fr lcrons. Thy ar nglcd n (7.. B (, ( D ( B( (, B (, and H (, hr ar, D, H (, J fr(, D (, B, and H n (7.. quaon of connuy (consrvaon of fr chargs: fr (,, (,, (,, (, J (, (5 fr As dscussd arlr, h consuv rlaons D (for ound lcrons and D (for oh ound and fr lcrons ar n gnral applcal only n h -spac. Smlarly, B H and J ar also -spac rlaons. To ul hs rlaon, w go o h -spac y assumng harmonc m dpndnc for h flds. 7 (4 II. Plan Wav quaons n Dlcrcs and Conducors (connud Assumpon : harmonc m dpndnc ( : ral and posv (, (, By convnon, h LHS s D D h ral par of h RHS. B (, B L R wh. H (, H (,, B hr ar B, J J n (7. and (7.3 (, ( ral compl (calld h phasor D (, fr(, D fr B (, B (6 (, B (, B (, (, (, fr ( fr H J D H J D (, J (, J (7 8 fr fr fr fr II. Plan Wav quaons n Dlcrcs and Conducors (connud Ohm's law: (5.59 Assumpon : lnar and soropc mdum,.. and P. 3 D, B H, J fr or ( v. No: W hav usd dfnons of D. Hr, D. In (, D(. ( D has no physcal sgnfcanc. Rwr (7: fr J fr ( fr fr Hnc, D fr (, (8 whr as h form of h gnrald drvd d n (7.5 and (7.56. Smlarly, H J D gvs H [ ], (9 whr agan and ar comnd n h sam mannr as n (8. Ths gvs an alrnav drvaon of h gnrald. Howvr, n (7.5 and (7.56 gvs h plc prssons for and. 9 fr II. Plan Wavs n Dlcrcs and Conducors (connud Usng(8and(9,wwrhmacroscopc (9, wr h Mawll quaons for harmonc flds n a lnar and soropc mdum n rms of phasor flds and h gnrald : B B H Dscusson : ( ( Bound lcrons and fr lcrons ar sparad dn h Mawll quaons n (4 and (6, whr conans h ffcs of ound lcrons and conans h ffcs of ffr lcrons. ( Bound lcrons and fr lcrons ar comnd n h Mawll quaons n (, whr ( conans h ff cs of foh ound and fr lcrons.

6 II. Plan Wav quaons n Dlcrcs and Conducors (connud Assumpon 3: unform mdum (.., ndpndn of ( B B ( ( B B (3 H B (4 (3 (5 (4 B B (5 has h sam form as (7.3, whch s drvd from h sourc urcfr Mawll quaons [(7.] for a non-conducng mdum (. Howvr, (5 s applcal o oh dlcrc and conducng mda. In (7.3,. In (5,. Soluon for (5 and (7. 3 as h sam algrac sps. Bu wh, h soluon for (5 wll applcal o oh dlcrc and conducng mda. II. Plan Wav quaons n Dlcrcs and Conducors (connud B B n (7.8-(7. Assumpon 4 :, B hr ar, B B B B (dsprson rlaon (6 * No:. ;.. and unlss s ral. 3. can compl, u s always ral and posv. (7 (-(3 B (8 B (9 N o : : (4 gvs B, whch s mplc n ( 7 and (9. S II. Plan Wav quaons n Dlcrcs and Conducors (connud m avragd powr flow pr un ara (calld nnsy (, H (, ral quans phasors R H R ( R H H B H (9 R ( No : (, H (, R[ H ] s drvd n Sc. 6.9 of lcur nos. 3 II. Plan Wav quaons n Dlcrcs and Conducors (connud Dscusson: : ( Assumng, ar gvn, (6-(9 ar condons mposd on,,, B y h Mawll quaons. ( Th drvaon of (6-(9 only rqurs,,,,, and B o consans, u no ncssarly ral (w hav assumd o ral. Thus, any s of compl consans,,,,, and B can a vald soluon of h Mawll quaons provdd hy sasfy (6-(9 and h oundary condons (f applcal. ( Th gnrald s n gnral a compl numr. can also a compl numr. hr compl or compl can lad o compl soluons for,, and B. vn whn and ar ral, oundary condons (f applcal can lad o compl soluons for,, and B [o shown n Sc. 7.4, q. (48]. 4

7 II. Plan Wav quaons n Dlcrcs and Conducors (connud (v Undr assumpons and 4, h flds ar hos of a plan wav; namly, h surfac of consan phas s a plan (s followng ampls. Thr ar yps of plan wavs dpndng on h form of h wav vcor (also calld h propagaon consan. a. Homognous plan wav Consdr h soluon: B wh B B B y y whr, y, and ar ral un vcors, u, B, and can all compl. Ths clarly sasfs (6-(9 and s h mos famlar yp of plan wavs. Any plan prpndcular p o h -as s a plan of consan phas. 5 Oponal. Inhomognous plan wav Consdr anohr soluon sasfyng (6-(9: II. Plan Wav quaons n Dlcrcs and Conducors (connud wh B B y y B y ( / whr,,,, and B ar all ral consans. y ( dfnd hr can convrd o h form n( nr ni as usd on p. 98 of Jacson. Hr, w rsrv h noaon n for lar us as a ral un vcor. Th physcal manng of such a soluon coms clar whn w consruc h physcal quany (, from h phasor. (, R R cos sn 6 Inrsng phnomnon II. Plan Wav quaons n Dlcrcs and Conducors (connud Rwr (, cos sn Ths rprsns a surfac wav n h half spac. I propagas along h -drcon wh an amplud pang a and dcrasng ponnally along h posv -drcon. Th surfac wav s also calld an nhomognous plan wav (p.98. Any plan prpndcular o h -as s a plan of consan phas. Whn a plan wav ncdnd from a dns mdum ono a surfac wav ar nuous mdum (.g. war o war ar s oally rflcd from h nrfac, flds n h nuous ncdn rflcd plan wav plan wav mdum form such a surfac wav du o oundary condons a. Ths wll dscussd n Sc II. Plan Wav quaons n Dlcrcs and Conducors (connud (v ( Orhogonaly of vcors,, and B n (7-(9, B, and ar algracally (7-(9 B orhogonal o on anohr B For h homognous og ous plan wav, (, B( By, and ar also gomrcally orhogonal. B y For h nhomognous plan wav, h algrac orhogonaly of (, (, and B ( B dos y y no mply gomrc orhogonaly caus and do no hav cl ar gomrc drcons. In -spac, w hav us shown (, cos sn, whch h shows ha h wav propagas along h -drcon, u s -fld also has an -componn. 8

8 II. Plan Wav quaons n Dlcrcs and Conducors (connud (v dos no ncssarly mply. (A smlar commn s mad n Jacson, s foono on p. 98. For h homognous plan wav (,, Bu for h nhomognous plan wav: rm mu Thus, n gnral, h rm mu s p n ( [s qs. (53 and (54 n Sc. 7.4.] 9 II. Plan Wav quaons n Dlcrcs and Conducors (connud (6 (7 (v Rwr (6-(9: (9: B (8 B (9 Ths s of quaons s quvaln o (7.9-(9. n Jacson, wh n (7.9-(7. nrprd as h gnrald. Th dffrnc s n noaons. In (7.9-(7., n s n gnral a compl un vcor suc o h condon nn, whch lads o condon (7.5. Hr, w ra as compl vcor [as n (] whou any addonal condon cp for hos mposd dy h Mawll quaons [(6-(9].(9] Thus, h compl s mor convnn o us, as has n dmonsrad n ( and wll sn agan n Sc II. Plan Wav quaons n Dlcrcs and Conducors (connud Assumpon 5: n ( r n : compl consan n: ral un vcor Thn, (7-(9 can wrn n (6, (-(4 hr ar quvaln o ( nb (7.9-(7. whn n n (7.9-(7. s (3 a ral un vcor and n (7.9-(7. B n s nrprd as h gnrald. (4 R n and. Thus, ( rducs o S n (5 Undr assumpon 5, h wav vcor has a gomrc drcon ( n. Hnc, (-(4 now rprsn homognous plan wavs wh gomrcally orhogonal,, and B. In ( r n, r( g gvs h wavlngh, gvs h ra of anuaon, and n gvs h drcon of wav propagaon. 3 S Chap. 6.9 II. Plan Wav quaons n Dlcrcs and Conducors (connud Dfnon of mpdanc and admanc of h mdum : Rwr B n (4 In ngnrng lraur, hs quaon s ofn wrn B n H, (7. Z whr Z s h mpdanc of h mdum (p. 97. Th admanc of h mdum s dfnd as Y = Z. Z and Y ar nrnsc proprs of h mdum. L ε and B Bε. Bcaus n, ε, and ε ar muually prpndcular, w hav Z / H Z s h (compl amplud rao of and H n h mdum (Th dfnon s vald vn f, ar compl. In vacuum, Z Z

9 III. Proprs of Plan Wavs n Dlcrcs and Conducors [A unfd ramn of Scs. 5.8, 7., 7., 7.5, and 8. usng h gnrald n (7.5] In Sc. II, undr assumpons -5, w hav oand h famlar homognous plan-wav quaons: : wav numr or propagaon o consan (6 (6 n ( nb (3 B n (4 n R (5 S n for a unform and soropc mdum, whr and B ar (compl amplud consans of h flds: (, R (, B B and n s a (ral ldrcon un vcor of fh h (compl wav vcor or propagaon vcor: n( n 33 r III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud On h ass of hs quaons, w consdr low 4 radcally dffrn cass whch ar dsngushal y h wav frquncy and h mdum propry characrd y h gnrald prmvy: N f N f m (ound m Cas. Wavs n a dlcrc mdum Cas. Wavs n a good conducor Cas 3. Wavs a opcal frquncs and yond Cas 4. Wavs n a plasma (7.5, ( III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas : Wavs n a dlcrc mdum m [ 7., 7., 7.5 (Par B] N f N f (7.5 m (ound ( m Proprs of : nglgl ( f or vry small. In gnral, << (s p.3, hnc Im<< R.. Whn s nar ach (ndng frquncy of h h group of lcrons, hs rsonan havor n h form of anomalous dsprson and rsonan asorpon. 3A As passs mor s, R dcrass. R nd of rfracon of war vs frquncy Im 35 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas.: Losslss dlcrc ( and ar ral. Scs. 7. and 7. Plan wavs n a dlcrc mdum govrnd y qs. (6, (-(5 ar s amplfd y h smpl cas of no mdum loss (.. and ar oh ral. Tm- avragd quans: nnsy: m avragd (5 S (7.3 n Poynng vcor, B B n u m avragd nrgy dnsy 4[ B B ] (7.4 Ths rms ar qual [ B n (4]. quparon of -fld and B-fld nrgs d (7.3 and (7.4, whr S n u vg vg ( d 36

10 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Tm- dpndn flds : ε L B n ε whr ε, ε, ar muually prpndcular and h flds ar lnarly polard. Furhr l, hn ε ε, n ε ε (, R[ ] cos ε B (, R[ n ] cos ε and ar ral. (, and B (, ar n phas. S (, (, H (, nsananous Poynng vcor [(6.9] cos n A a fd poson, S vars wn and h mamum (posv valu a h frquncy. 37 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Two lnarly polard wavs can comnd o gv (, (, (, εε (7.9 (7.9 consss of h followng 3 cass:. (7.9 s a lnarly polard plan wav f and ar n phas,.. f and. (7.9 s an llpcally polard plan wav f and ar no n phas,.. f and. 3. (7.9 s a crcularly polard plan wav (a spcal cas of llpcal polaraon f ( and. Hnc, (, ( ε ε (7. For an alrnav rprsnaon, w dfn ( ε ε, (7. * * whr and. Thn,,(7.9[no (7.] can wrn (, ( ε ε ( III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud A spcfc ampl of crcularly polard wav: y Rwr (7.: (, R ε ε L ε, ε y, and n. W hav (, cos y ngav hlcy y y (, sn rcs: Show ha h nsananous Poynng vcor of a crcularly polard plan wav s ndpndn of m. y posv hlcy Mdum propry : [(6] gvs h phasvlocy( v c v, whr n (nd of rfracon (7.5 n N, w consdr plan wavs n a lossy dlcrc, whr h flds dffr only slghly from hos n a losslss cas dlcrc (.g., B ar slghly ou of phas. Howvr, as a qualav dffrnc, h mdum asors h wav. So, our mphss wll on h mdum proprs. 39 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas.: Lossy dlcrc [μand/or ar compl, Sc. 7.5 (Par B] can wrn: R Im (7.53 whr r gvs (for arrary and h wavlngh h phas vlocy v R c h nd of rfracon n v R usd on p34 p. 34. To fnd h manng of, w s and n n R n S (5 n nnsy (avra g y( P S n R, powr/un ara Hnc, s h powr anuaon consan gvn y P P [ Im ] usd on p

11 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud For h common cas of wa anuaon, w l ral, wh R Im (for ral and small r (phas consan c rducs o h prsson on v c n (phas vlocy p. 3 whn μ=μ. c n v c (nd of rfracon (powr anuaon consan (7.55 P R (nnsy In (7.55, ( an l s commonly rfrrd o as h loss an gn. 4 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud (R and loss angn (an or l of som marals a dffrn frquncs from Ramo, Whnnry, and VanDur Dur, p.334. p334 4 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud A mraculous propry of war: Th nd of rfracon (op and asorpon coffcn (oom for lqud war as a funcon of frquncy n H [Sc. 7.5 (Par ] III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas : Wavs n a good conducor [Scs. 5.8 and 8., applcal o wavs n mals undr h condon ω<< (~4 3 /s. s p. 3,.. for vry low frquncy (.g. 6 H up o nar rahr frquncs] Df Dfnon of good conducor : N f N f (7.5 m (ound m ( In gnral,, s p. 3. In gnral, R( I m( N f (7.58 m( (

12 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Up o low rahr rgon, w hav 3 ( s of h ordr of 4 / s. Hnc, Whn, ral N f N f n and s ndpndn of. m( m m ( n : fr lcron dnsy In / [(7.56], / Im(. So w may assum o ral. A good conducor s dfnd y: ( farad/m 7 4 Quanav coppr /Ω-m m, graph 6 /Ω-m 3 ampls: sa war 6/Ω-m, ground -3.5 / Ω-m 6 H for houshold currn f.3 3 GH for mcrowavs Quson: : Why s dangrous f an lcrcal applanc falls no your ah u? 45 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Flds n a good conducor : For a good conducor (, w hav ( ( ( (for forward wav (5.64 : sn dph whr (5.65 and (8.8 s ral y assumpon. L, n. Thn, H n y (, (7 H (, H y (8 y (7 and (8 ar quvaln o (8. and ( III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud ampls: coppr.85 cm a f 6 H (houshold currn 5 7 cm a f H (mcrowav 47 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Dscusson : (, (7 ( Rwr (, y (8 H Insd h good conducor, h wav has a wavlngh of and damps y a facor of / ovr a dsanc of. and H n a good conducor ar 45 ou of phas. ( Th flds n a good conducor ar smlar o hos n a lossy dlcrc n ha hy oh rprsn an anuad plan wav wh,, H, muually orhogonal. Howvr, a h sam frquncy, h wavlngh s much shorr and h anuaon consan much grar n h conducor han n h dlcrc. o 48

13 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud p f ampls: L f H (ypcal (yp mcrowav frquncy 4 5 glass (, 4,. coppr 7 cm.5 cm (Cas. 4.4 cm cm (7.55 dp P d 4.5 cm (v A wav ncdn from h ousd no a good conducor (a any ncdn angl wll propaga and anua nsd h conducor appromaly along h normal o h surfac (s Jacson Sc. 8.. Th rason s shown n h fgur low. ar conducor Wav propagas appromaly along III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Hnc, w may appromaly wr h wav flds nsd h conducor as (7 and (8,.. and H ar paralll o h surfac, vn f h wav s ncdn a an olqu angl no h conducor. Quson: Dos ma sns o us powr lns of vry larg damr (.g. cm n ordr o conduc hghr currn and hnc ransm mor powr? (v Th homognous Mawll quaons rqur ha and B connuous across h conducor surfac., H( snc, wha happn o h surfac currn K? ar conducor H H No: Th currn dnsy n a good conducor s fn unlss (or,.. h currn flows on h surfac. 5 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Surfac currn K ff on a good conducor : If, h "surfac" currn Kff s no acly on h surfac, I s concnrad ovr a dph of on sn dph. K ff (un: A/ m s an ngrad valu of J (un: A/ m ovr h pnraon dph. K ff Jd K d d ff H (7 ( H H (9 (9 hr s (8.4 n Jacson; " " n (9 s " n" n (8.4. (9 shows ha h surfac currn K ff on a good conducor dpnds only on h H on s surfac. Physcally, K ff s h rspons of h conducor n ordr o shld s nsd from H (Faraday s law. Hnc, K ff s drmnd nrly y H. 5 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Tm - avragd powr loss on h surfac of a good conducor: dp loss powr gong no conducor ( da un ara of conducor surfac S S R ( ( H (7, (8 ( ( ( (3 dp loss (7, (8 ( H ( (3 Su. (3 no (3 H ( usful form o plan da nducon hang 4 H ( (8. (9 H ( K ff (8.5 No: If hr s rflcon, H ( H ( H ( ncdn rflcd 5

14 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud dp loss/ da n (8., oand y h Poynng vcor mhod, can shown o acly h Ohmc powr dsspad nsd h conducor. P ohmc powr n h conducor/un volum rssv dp da loss H R J H = H ( = (7 (8 P rssvd H d H 4 P rssv (5.69 sam as (8. Qusons:. Why dos a mcrowav ovn sav nrgy?. How would you dsgn an nducon coor? hgh and 53 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Dfnons : surfac mpdanc Z s, surfac rssanc R s, and surfac racanc X of mal s ( : rao of ( 7 9 Z ff ( ( s K Z ( o K s ff whr Zs Jacson p. 356, oom s calld h surfac Zs Rs Xs mpdanc. W may wr, (3 whr Rs Xs surfac rssanc surfac racanc ampl: Rs of coppr.6 a GH Th surfac mpdanc Z s s an nrnsc propry (rahr han surfac propry of mal. I s n fac h mpdanc of a good conducor: Z s. (mal 54 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas 3: Wavs a opcal frquncs and yond [Sc. 7.5 (Par D] Cas 3.: >> u < for all or som of h ound lcrons [a sucas of Sc. 7.5 (Par D, pp. 33-4, oal rflcon of lgh off h mrror and ulravol ransparncy of mals f N f N (7.5 m (ound ( m N f In gnral,, s p. 3. m In gnral, R( Im ( Th fr lcron rm s prdomnanly magnary whn <<. Bu, as shown aov, whn >>, coms prdomnanly ral, a qualav dparur from Cas. Ths radcally changs h mal rspons o M wavs. ampls ar gvn low and n Cas 3.. Quson: Wha s h physcal rason for h fr lcron rm o com prdomnanly ral whn >>? 55 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud L n Nf h fr lcron dnsy n h conducor ( f,.. ach aom n h conducor conans on avrag appromaly on fr lcron, s p.3, w oan from (7.5 p whr s h plasma frquncy of h conducon lcrons p n p S oom of p.33. m * and w hav rplacd m n (7.5 wh h ffcv mass m* of h conducon lcrons o accoun for h ffcs of ndng. For smplcy, w assum o ral y nglcng h wa dampng ffcs of ound lcrons. 56

15 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud p Su. / no, w oan p ( Hnc, s hr ral (propagaon whou anuaon or purly magnary (vanscn flds dpndng on h wav frquncy. p Whn p, and (. Thn, (33 H y (4 (34 57 n (33 and H n (34 ar vanscn flds whch fall off ponnally nsd h conducor. Thy do no consu a propagang wav. Ths s caus and H ar 9 ou of phas. Hnc, R[ H*] = No powr flow no h conducor. Thus, an ncdn wav wll oally rflcd from h conducor surfac, wh (33 and (34 rprsnng h shallow frng flds nsd h conducor. Ths s h prncpl of lgh rflcon off h mrror. By comparson, for mcrowav rflcon off a good conducor (Cas, and H ar 45 ou of phas n h conducor Som powr flows no h conducor. p p A hghr frquncs ( /, /. nc, ( coms ral. Th wav can hn propaga frly. Ths s h prncpl of ulravol ransparncy of mals. Quson: Why can h wav propaga whou anuaon n a conducor? (s dscusson a h nd of Cas 3.. y 58 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas 3.: 3 >> and >> for all lcrons n h mdum [a sucas of Sc. 7.5 (Par D, p. 33, applcal o X-ray frquncs and dyond] Undr h condons >> (ncludng and >>,wmay nglc and n (7.5, N f N f m (ound ( m (7.5 NZ (us f Z p m all, (7.59 whr s h dnsy of all NZ NZ lcrons p m (ound and fr n h mdum. ( p / c p ( Su. no and assum, w oan c p Alhough (7.6 prdcs vanscn flds for < p, h valdy of (7.6 rqurs >> and for all h lcrons n h mdum. Ths n urnrqurs >> p. Hnc, s always ral and h wav s always a propagang wav n h mdumm undr h valdy condon for (7.6. Th aov ramn for Cas 3. appls o oh dlcrc and conducng mda. (7.6 p 6

16 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Dscusson: To amn h physcal rason why w may nglc collsons and ndng forcs n (7.5 undr h condons >> and >>, w go ac o h quaon of moon for h lcrons: m( (, (7.49 By assumng (, (, w oan [s q. (] ( ( ( ( m m Thus, whn and, w hav / and /. Ths mpls ha, for h sam (, h collsonal dampng forc ( m / and h ndng forc ( m / dcras wh ncrasng and com nglgl a a suffcnly larg. rcs : plan " m / " and " m / " qualavly from h smpl cas of consan acclraon av : aand a. 6 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Cas 4: Wavs n plasmas [a sucas of Sc. 7.5 (Par D, p. 33] Th plasma s a parally ond (.g. onosphr or fully ond (.g. fuson plasmas gas. In gnral, ffcs of nural gas (f prsn and collsons can oh nglcd. Ion moon can also nglcd a suffcnly hgh frquncs. Thn, N m (ound m p f N f ( nglgl N f ( m (7.5 sam quaon as (7.59 u (35 wh a much smallr p whr p s h plasma frquncy dfnd as n n Nf plasma lcron dnsy, normally p m (36 much smallr han h dnsy of solds. 6 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud p Su. no, w oan / c p ( ( for plasmas sam quaon as (7.6 u c p (37 wh a much smallr p (37 s h wll nown dsprson rlaon for lcromagnc wavs n a plasma n h asnc of an rnally p appld sac magnc fld. (Sc. 7.6 consdrs h dsprson rlaon l for a magnd plasma. Whn ω s rmly larg (such as h gamma ray, all marals hav a dsprson rlaon gvn y (37 (Cas 3.. Bu for h plasma, (37 s vald for all frquncs (.g. MH. 63 III. Proprs of Plan Wavs n Dlcrcs and Conducors (connud Rwr c p (37 For < p, s purly magnary ( = and hnc and H ar vanscn flds gvn y (33 and (34: ; y H As n h cas of lgh rflcon off h mrror, an ncdn wav wav wll oally rflcd [Shorwav roadcasng plos h rflcon of rado wavs (~ MH off h onosphr]. For > p, s ral. Hnc, h wav wll propaga n h plasma, u wh a phas vlocy grar han h spd of lgh [as can sn from (37]. Ths mpls ha h plasma has an nd of rfracon ( n lss han. From (35, w hav p <. Thus, wh, w hav n <, as pcd. p 64

17 7.3 Rflcon and Rfracon of lcromagnc Wavs a a Plan Inrfac Bwn Dlcrcs c cs Modl: B' '' ', ' [ n' ], [ n ] B ncdn wav (a gvn lnarly polard homognous plan wav c rfracd wav (assumd B rflcd wav (assumd No : In Cas. of Par III, n v R. Hr, n. Knmac proprs: p rlaons wn angls of ncdnc, rflcon, and rfracon 65 Dynamc proprs: nnsy, phas, and polaraon rlaons 7.3 Rflcon and Rfracon (connud Knmac Proprs : Boundary condons for h flds a y y y hav h form: X y Y y Z y a any and y, whr X, Y, and Z ar funcons of h flds [s (7.37]. Snc,, ar lnarly l ndpndn, d w mus hav. Ohrws, w wll hav h rval condon X Y Z. For h sam rason, y y y. Hnc,,, and l nhsamplan plan. Whou loss of gnraly, w choos a convnn coordna sysm n whch y y y. Thn,,, and all l n h - plan, whch w call h plan of ncdnc. 66 Rflcon and Rfracon (connud Assum,,, and (hnc n and n ar all n sn cos '' r ', ' [ n' ] ral numrs. L sn r cos r, [ n ] sn cos r r r n /, c c (6 n n /, n / c r (angl of ncdnc angl of rflcon sn (Snll's law (7.36 snr n A no on Jacson (7.33: and In gnral, can compl, u s * always ral and posv. Thus, Jacson's formula n (7.33 s vald only whn s ral. 67 Rflcon and Rfracon (connud Dynamc Proprs : Informaon concrnng h nnsy, phas, and polaraon s conand n h compl,, and. Th nnsy, phas, and polaraon of rflcd and rfrac d wavs wh rspc o hos of h ncdn wav can oand from h oundary condons a : D connuous [ ] (39 B connuous [ ] (4 connuous [ ] (4 H connuous [ ] (4 No: ( Hr,,,, and (hnc n and n ar n gnral compl numrs (s frs paragraph of Jacson, p. 36. W assum ha (or s h gnrald lcrc prmvy. Hnc, h rsuls drvd low apply o any mda (ncludng mal. ( For a compl n (or n, h phas vlocy s h spd of lgh dvdd y R[ n]. [S lcur nos, h quaon for (5]. 68

18 Rflcon and Rfracon (connud Cas : plan of fncdnc d (h - plan, y, y y (39 s auomacally sasfd. (4 (43 (4 also gvs (43. Rflcon and Rfracon (connud (4 cos ncos c cos r n cos r n cos n c cos r (44 ncos n cos n n sn (43 (7.39 (44 ncos n n sn ncos n n sn 69 7 Rflcon and Rfracon (connud Cas : plan of ncdnc sn cos sn r cos r (45, sn cos, cos sn cos r sn r (46 cos sn Su. (45 and (46 no (39-(4 ylds nncos n cos n n n sn (7.4 n cosn n n sn n cos n n n sn 7 Rflcon and Rfracon (connud For normal ncdnc ( r, (7.39 rducs o n n n, (47 n n, rfrnc n n polaraon for (7.39 and ( 7.4 rducs o n n n,, (7.4 n n rfrnc n n polaraon for (7.4 Ths wo lmng rsuls ar dncal and show ha, f n n, 7 hr s a phas rvrsal of h rflcd wav a h nrfac.

19 slf-sudy Rflcon and Rfracon (connud Th rsuls for normal ncdnc ( r can prssd n rms of h mpdanc of h wo mda [Th mpdanc of h mdum s dfnd on p. 97 and n h lcur nos followng (7.]: Z (lowr mdum Z (uppr mdum Thus, (7.39 rducs o,, Z Z Z rfrnc polaraon Z Z for (7.39 Z Z If h lowr mdum s vacuum and h uppr mdum s coppr, Z Z w [lcur nos followng (7.] hav Z Zs (.6.6 for coppr a GH [(3] Thus, /,.. almos all of h ncdn wav wll 73 rflcd wh a phas rvrsal of h rflcd wav a h nrfac. slf-sudy 7.3 Rflcon and Rfracon (connud Dscusson: : Sourcs of lcromagnc flds n dlcrcs Th sourc-fr macroscopc Mawll quaons [(7.] can convrd no h mcroscopc form as follows: B B B Jacson p.56 and B lcur nos Ch. 4 D P D P pol J pol [lcur H B M nos, Ch. 4] D H P B M J M [(5.79] JM Jpol 74 slf-sudy 7.3 Rflcon and Rfracon (connud W s from h mcroscopc Mawll quaons ha, upon acon y h lcromagnc flds, ound lcrons of aoms/molculs n a dlcrc (, wll produc polaraon charg and currn dnss ( pol and J pol and magnaon currn dnsy (J M, hrough whch h dlcrc wll gnra s own flds. In h macroscopc Mawll quaons, pol, J pol,andj M ar hddn n D and H, u h flds hy gnra wll appar n h soluons. For ampl, as a wav s ncdn from a vacuum no an mdum, wll nduc pol and J pol ( pol = nsd a unform mdum, whras J pol salways prsn. pol and J pol ar h sourcs whch h gnra h rflcd wav and caus rfracon of h ransmd wav. Smlarly, n h cas of a chargd parcl ravlng n a dlcrc mdum a a spd grar han h spd of lgh n ha mdum, h pol and J pol nducd y h flds of h chargd parcl wll gnra h Chrnov radaon (rad n Jacson, Sc Polaraon y Rflcon and Toal Inrnal Rflcon Brwsr's Angl B: (for plan of ncdnc nn cos n cosn n n sn R wr (7.4 n cos n n n sn n cosn n n sn Assum,,, and (hnc n and n ar all ral numrs. L. W s ha, f, whr sasfs cos B sn B n n n n B hn,.. hr wll no rflcd wav. Consqunly, upon rflcon a h ncdn d angl B, wavs wh md polaraon com lnarly polard wh plan of ncdnc. B 76

20 7.4. Polaraon y Rflcon and Toal Inrnal Rflcon (connud Calculaon of B : B B 4 cos B sn B Rwr n cos n n n sn n n n n 4 sn B 4 sn n 4 n 4 sn B n n n n n n n n sn B n n n n an B n B (7.43 n B Polaraon y Rflcon and Toal Inrnal Rflcon (connud Toal Inrnal Rflcon: (occurs only whn n n Assum,,, and (hnc n and n ar all ral and n n. sn cos L n sn r cos r r Snll's law, sn [(7.36], can,n r snr n,,,n wrn: sn r sn, sn o whr sn n [ 9, nn]. Thus, f w hav S p. 7, / / sn r sn cos r [ sn ] [( sn ] sn r sn Th propagaon fac or ( of h rfracd wav havs as sn / sn [ ] ( sn cos ( sn r r sn (7.46 surfac wav 78 slf-sudy 7.4. Polaraon y Rflcon and Toal Inrnal Rflcon (connud Wav vcor and flds of h rfracd wav: : sn / [ ( ] sn sn sn Rwr (7.46: W s ha (of h rfracd wav may prssd as / sn sn sn sn whr, [ ] and oh and ar ral and posv quans drmnd y h ncdn angl. No ha (48 sasfs (6,... (48 Consdr h cas wh plan of ncdnc and wr (49 [(7] (5 Thn, [(9] B B y (5 (48-(5 gv h surfac wav soluon dscussd arlr n (. 79 slf-sudy 7.4. Polaraon y Rflcon and Toal Inrnal Rflcon (connud Poynng vcor : (Consdr plan of ncdnc as an ampl Rwr : S R ( ( + (5 (53 Su. (5, (53,, and no = from (5 / [ ] [ ] [ ] S [ ] (54 Powr flows along h -drcon. Thr s no powr flowng from h rgon no h rgon oal rflcon as pcd. 8

21 7.8 Suprposon of Wavs n On Dmnson; Group Vlocy Suprposon of Wavs : Consdr wavs (Fg., cos( and cos, n a dsprsv mdum characrd y. Assum and, hn gvs h approma phas vlocy ( v ph of h suprposd wav (Fg.. Th dffrnc n wavlnghs rsuls n alrnang rgons of consrucv/dsrucv nrfrncs, or spaal modulaons of h suprposd wav (Fg.. In addon, caus of h dffrnc n phas vlocs, rgons of consrucv nrfrnc, whch carry h fld nrgy, wll a dffrn posons a dffrn ms, movng a h group vlocy ( v g. cos cos( cos( cos( v g consrucv nrfrnc v ph Fg. nrfrnc dsrucv Fg Suprposon of Wavs n On Dmnson; Group Vlocy (connud Th aov qualav pcur can analyd as follows. cos cos cos ( cos ( v consrucv g nrfrnc nrfrnc dsrucv cos( cos, (A (B v ph Fg. whr and. Facor (A s h nvlop funcon of h modulad wav (Fg., whch dvds h wav no pacs, ach propagang a h spd g d vg (group vlocy d Facor (B gvs h phas spd of h wav whn ach pac, v ph (phas vlocy Suprposon of Wavs n On Dmnson; Group Vlocy (connud Suprposon of an Infn Numr of Wavs: Whn an nfn numr of wavs (cnrd around, wh a sprad, s Fg. 4 ar suprposd, nrfrncs can rsul n cancllaon vrywhr cp for a rgon of lngh (Fg. 3, whr h wavs ar consrucvly suprposd no a wav pac. n rr c c Phas vlocy: vp (7.88 n d c Group vlocy: vg (7.89 d n ( dn d L dl d Group dlay: g vg d d Can a wav pac propaga a h group vlocy fasr han h spd of lgh? Suprposon of Wavs n On Dmnson; Group Vlocy (connud Dscusson : ( Th puls shap gv y (7.85 s undsord n m. Howvr, f hgh ordr rms (.g. d arncl udd n h panson of d [(7.83], h puls wll roadn wh m. dv g Rason: v g vg vg d d d If d, hr s a sprad n v d g ( A shorr wav pac has a grar sprad n (and v g. Hnc, roadns fasr han a longr plus. ( Wav pacs n vacuum rman undsord ( c d. d Th followng scon gvs a mor rgorous ramn of h wav pac ncludng puls roadnng. 84

22 7.9 Illusraon of h Spradng of a Puls as I Propagas n a Dsprsv Mdum Rgorously, h ral quany u (,, whch w prssd n (7.8 as d, should wrn*: ( A u (, A dcc.. (7.9 Assum (, ar oh ral,.. no dsspaon. ( Th mdum s soropc, hn c ( (. u (, A d A*( d (, u A d ( A* d * No: In (7.9, A s no h Fourr ransform of u (,. Hnc, h "ral y condon" A A *( [s Sc. 8.8 of flcu r nos] s no applcal Illusraon of h Spradng of a Puls (connud u (, A d A*( d u (, A d ( A ( d (, u d A dd A* dd y d ( y A A* (56 [ ] u(, d A A (57 (56 (57 y assumpon ( A u (, (, u d ( Illusraon of h Spradng of a Puls (connud u (, p( cos (7.9 L nal condons ampl : u(, (7.93 dv [ a g d a ] d d (7.95 pc spradng of puls. A [ u (, u (,] d /L cos d L L L p [( ] p [( ] ( Illusraon of h Spradng of a Puls (connud ( u (, A dcc.. L L a ( ( ( L R [ ] d ( a a p[ ] p[ ( ] ( a ( a R L (7.98 L L a wav pac propagang forward ( a wav pac propagang acward whr L s a funcon of gvn y (7.99: a L L ( [ L ( ] 88

23 Suprlumnal ffc prmnal rsuls 89 9 Homwor of Chap. 7 Prolms:, 3, 4, 6, 3, 4, 9,,, 8 Oponal:37,, 3, 7, 9