UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 8 Prof. Steven Errede LECTURE NOTES 8

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1 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd LETURE NOTES 8 A Mr Sphisticatd Tratmnt EM Wav rpagatin in nducting Mdia In th prvius 436 Lctur Nts 7, w discussd th prpagatin EM wavs in r, t E r, t and th r, t r, t t in th cnductr. cnducting mdia (.g. mtals), taking int accunt Ohm s Law cntinuity quatin r Hwvr, this tratmnt cnducting mdia nglctd crtain inrtial cts assciatd with th init mass r / cnductin lctrns in th mtal cnductr w nd t cnsidr mr carully th actual mtin th r / cnductin lctrns in th cnductr, and thir rspns.g. t th applicatin a mnchrmatic plan EM wav angular rquncy ω. Sinc th r / cnductin lctrns in a mtal ar nt bund t individual atms in a cnductr, thr ar n lastic rstring rcs acting n th r / cnductin lctrns {i.. spring cnstant k 0 }, as thr was in th cas th plarizatin bund atmic lctrns,.g. in a disprsiv, linar nn-cnducting mdium. Thus, th dirntial quatin dscribing th mtin th r / cnductin lctrns in a mtal is th gnral rm: dv r, t m m v r t r t E r t F r t F r, t E r, t whr th driving rc ild Er, t, 0,, = th charg th lctrn tims th lctric th mnchrmatic plan EM wav prpagating in th cnducting mtal. Thus, w btain a irst-rdr inhmgnus dirntial quatin th rm: dv r, t m m v r t E r t,, n.b. hr again, w nglct th cts th magntic Lrntz rc trm, FLrntz rt, vrt, Brt, Ert, bcaus: v v vd vprp. Nt als that m = mass th r /cnductin lctrn in th mtal cnductr which, dpnding n th micrscpic/quantum mchanical ails th cnductr is {tn} nt qual 3 t th mass a {truly r, islatd} lctrn 9. 0 kg m ~ m. m, but {tn} Th quantity m is knwn as th rictinal / ul dissipatin cnstant, assciatd with lctrn nrgy / mmntum lsss i.. r /cnductin lctrn scattring lsss in th cnductr. Th damping cnstant {n.b. units angular rquncy (radians/sc)} is assciatd with dissipativ/ nrgy lss mchanism(s) th mtin r / cnductin lctrns in th mtal r / cnductin lctrns scattring ach thr, atms, lattic vibratins, crystal dcts and impuritis in ral mtals. Thus, w can als din a rlaatin tim cnstant assciatd with th r / cnductin lctrns prsnt in th mtal cnductr. rla rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

2 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Th r currnt dnsity in th mtal cnductr is r, tn vr, t r {Amps/m }, whr: n r / cnductin lctrn numbr dnsity in th mtal cnductr {#/unit vlum = #/m 3 }. q th lctric charg r / cnductin lctrns {ulmbs}. vr, t vd u {.g. v ~80m scin cppr.} Thus: D = th drit vlcity c r / cnductin lctrns in mtal at rt, dv r, t m m v r t E r t,, dv r, t nm nm vrt nert dv r, t m n m nv r t n E r t d m nv r, t m nv r, t n E r, t d r r, t m m r r t n E r t multiply thrugh by n :,,,,,, which is als a irst-rdr linar, inhmgnus dirntial quatin. r: {m/s}. d r r, t n r r t E r t m,, Rcall that th slutin t an inhmgnus dirntial quatin is th sum th slutin t th crrspnding hmgnus dirntial quatin, plus a particular slutin satisying th inhmgnus dirntial quatin. Thus, w irst nd t btain th slutin t th hmgnus E r, t 0. dirntial quatin, i.. whn n EM wavs ar prsnt in th cnducting matrial, vis. E r, t 0, th gnral slutin t th irst-rdr linar, hmgnus dirntial quatin: a.) I d r r, t r r, t 0 is th rm a dampd pnntial (bcaus γ > 0): t t rla r, t r r r r r Transint r currnt(s) will dcay t whr: sc thir initial t = 0 valu ( r r in a charactristic rlaatin tim t sc. rla rla ) rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

3 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd {prssing cnsrvatin Using th cntinuity quatin r r, t r r, t t r lctric charg} in th abv hmgnus quatin, with thus: n r r t r r t E r t m 0 d r rt, r rt, 0 t t r t,,, rt, rt, r t 0 t t rla r r, t r r r r r t. rt, r E r, t 0: rt, 0 But rm 436 Lctur Nts 7, p. w als btaind th rlatin Thus, cmparing ths tw quatins w s that:. b.) Fr a static nn-zr lctric ild Er E tim dpndnc, hnc d r, t 0 quatin bcms: n n r E r E cnstant r r m m rla r rt r. t, r, thn this static prblm can hav n plicit and thus th irst-rdr linar, inhmgnus n n r r E r E cnstant m m r: r r Th static / D cntinuity quatin r r charg is: r r 0 t r r cnstant cn r,{r t} and: r r cnt. with static slutin(s): Hwvr, Ohm s Law (r D / stady r lctric currnts) is: r Er th static cnductivity th mtal (Simns/m) r whr: / static rsistivity ( -m ). Thus, w s hr that: r Er Er r n m i.. th static cnductivity n m m r quivalntly, th static rsistivity:. {n.b. bth purly ral quantitis} n Th static / D rsistanc.g. a lng wir is thus: R Ohms {again, a purly ral quantity} m A n A rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 3

4 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd c.) Fr a harmnically tim-dpndnt {in gnral, cmpl} lctric ild (.g. assciatd with a mnchrmatic EM plan wav (r an A currnt, using Ohm s law), with angular rquncy : d i t, Er, t E r r r t n and: r r, t Er, t m W assum that th {in gnral, cmpl} r currnt dnsity slutin t this linar i t r, t r. inhmgnus irst-rdr dirntial quatin is th gnral rm: Thn th slutin t this linar, inhmgnus irst-rdr dirntial quatin is: n n n i r rt, Ert, Ert, Ert, sinc: m m m r rt, Ert, Ert, Ert, Ert, i i i/ Whr: Th A cnductivity: Th A rsistivity: n m i/ n.b. cmpl quantity i/ m i/ i/ cmpl quantity n Thus, w s that th {cmpl} A rsistanc (aka impdanc, Z).g. a lng wir is: m Z R A i/ i/ n.b. cmpl quantity A A n A n.b. an har th ct cmpl R A r r in {audiphil-typ} hi-i str systms by cmparing gd (thick) vs. bad (thin/chap) L & R ludspakr cabls! r rt, Th A cntinuity quatin is: r r, t with: Er, t t n m r r, t E r, t Er, t Er, t i/ i/ But: r Thus: r rt, n m rt, Ert, Ert, Ert, i/ i/ r rt, r rt, t i/ r: r rt, t i/ r rt, Gauss Law 4 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

5 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Th slutin t th abv dirntial quatin r th r lctric charg vlum dnsity is i t rt, r prvidd that: th rm: r r rt, r t i/ rt, r i i.. that: r but: i i/ Thus, w s that: i / r i i i/ Tchnically-spaking, this rlatin shuld b: n.b. implis r r n i / m i/ {cmpl!} r r i vacuum vacuum i vacuum i vacuum mtal mtal mtal mtal bcaus th vacuum prvads all spac th cnducting mdium and th vacuum cist in th sam rgin spac-tim, invking th suprpsitin principl, th tw lctric prmittivitis (n.b. bth scalar quantitis) ar additiv. Hwvr, it can b sn that bcaus th {D} cnductivity mtals is s high 7 {typically ~0 Simns/m}, that vn r {angular} rquncis in th ptical rgin 6 { ~ 0 radians/sc } th lctric prmittivity r spac, is dward by th scnd trm, i i i i/ Farads/ m, s tn th trm is nglctd/drppd in many ttbks... Fr tchnical crrctnss / cmpltnss sak, w will includ it hr. Thus, w s that vn r mtal cnductrs: i / i i i / i / r r r i / i i i/ i / Fr pur cppr mtal at lw rquncis, i.. 0, th static cnductivity is Simns/m; th numbr dnsity r lctrns in pur cppr mtal is using n m u u u u u n m n m u n 6 0 u /m and 3 40 sc, which crrspnds t a rlaatin tim r r / cnductin lctrns in pur cppr u 4.50 sc. rla u a cnnctin btwn cmpl A lctric prmittivity r () and cmpl A cnductivity () in a mtal / cnductr. 7 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 5

6 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Fr nducting Mtals with Fr Elctrns: r, t Er, t Nt that in th static limit (ω 0): i t E r, t E r E r static / cnstant r E i t r r, t r r r static / cnstant r r t it rt, r r static / cnstant r r r n i/ static / cnstant m i/ static / cnstant r singular!!! i i i i r / / In th prvius 436 Lctur Nts 7.5 n disprsin in nn-cnducting, nn-magntic linar / hmgnus / istrpic mdia, w shwd / drivd th cmpl lctric prmittivity (du t bund atmic lctrns) th dilctric mdium t b: bnd bnd bnd K b n bnd sc bnd m r n i with: and: bnd bnd n K I nly n / a singl rsnanc ists, this rlatin rducs t:. bnd b n i b n m with: 0 and: 0 3 m mparing this prssin t that r cnducting mdia with r / cnductin lctrns: n r with static cnductivity: / i m n n n i r 4 m / i m i m n 0 i r m with: 0 0 a 0 0! 0 r: k m b 6 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

7 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Nt that th sign in th abv rmula {rlativ t that r bund atmic lctrns} ariss bcaus act that r / cnductin lctrns in mtal ar nt bund (i.. phas shit = 0 Hz). Thn r disprsiv cnducting mdia cntaining r lctrns, Mawll s quatins bcm: D E r 5) Ohm s Law: ) r ) B 0 6) ntinuity Equatin: 3) B E t B D t 4) r r r E t r E i r t Assum disprsiv cnducting mdium is nn-magntic Again, w us th curl pratr n quatins 3) and 4) t btain wav quatin(s) r EM wavs prpagating in a disprsiv mtal / cnductr,.g. r quatin 3): r E E B r t t t E E E n hr, r nn-magntic E r and: r cnductrs t t c v r 0 in r t 0 sc psr charg-quilibratd cnductrs. Thus, r tims t 0 sc psw btain: E E E E v t t i t Using quatin 4) abv, an idntical wav quatin is btaind r th magntic ild B rt, B B B B v t t i t Slutins t abv wav quatin(s) r mnchrmatic EM plan wavs prpagating in th ẑ dirctin in a disprsiv cnducting mdia with r lctrns will b th rm.g. : E r, t E ikz t k ˆ B r t ke r t B and:,, ikz t. Nt that hr {again} th wavnumbr k is cmpl and rquncy-dpndnt: with k k and m k k k i {rlatd t absrptin/dissipatin th EM wav as it prpagats in th disprsiv cnductiv mdium.} rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 7

8 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd W can assciat th cmpl wavnumbr k with a cmpl ind rractin n n r: n k k c c whr th cmpl ind rractin: n ni thus: n n k and: m n lugging in th prssin Er, t : c c. ikz t E r mnchrmatic plan EM wavs prpagating in a disprsiv, nn-magntic cnducting mdium int th abv wav quatin, w btain th s-calld charactristic quatin r this situatin {Ercis - plicitly wrk this ut yurslvs!}: i with: k c i c and: n m r: n m c Thn: n k = (cmpl ind rractin) n i i Th Lw Frquncy Bhavir a Disprsiv, Nn-Magntic nducting Mdium: Dinitin lw rquncy: whr: n m and:. In this rgim, th charactristic/man/avrag tim btwn succssiv cllisins princd by th r / cnductin lctrns in th disprsiv, nn-magntic cnducting mdium cll is shrt in cmparisn t th scillatin prid EM wavs, i.. cll. Fr 0 (but nt 0 i!): k c i k i i kping nly trms linar in ω i Nw nt that: i {las wrk this ut/driv this yurslvs!} Thus r 0 i.. : k i k k i i i Frm which w s immdiatly that r mnchrmatic plan EM wavs prpagating in a disprsiv, nn-magntic cnducting mdium, that: k and thus hr {again} w s that th B -ild lags th E -ild by: k B E tan tan 45. k 4 8 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

9 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd c Similarly, r 0 i.. sinc: n k Thn: c c n k i i i. Using: i i i n n i i which again implis that: n Hr again, w can din a lw-rquncy skin dpth: sc. th charactristic distanc that th E and B ilds all t thir z = 0 valus. W din a lw-rquncy absrptin cicint: and crrspnding lw-rquncy tinctin cicint: sc Th charactristic distanc vr which th {tim-avragd} EM wav nrgy dnsity and intnsity EM I r z u r u all t thir initial (z = 0) valus, rspctivly ar: EM 0 I r I and 0 z c 3 u 7 Fr pur cppr mtal, r which u 4.0 sc and 60 Simns/ m and thus u / sc, thn r lw rquncis,.g. 0 Hz w s that pur cppr mtal is in th lw-rquncy rgim r vacuum wavlngths : 8 c m 300m c w/ u In pur cppr mtal, th skin dpth u 6 ur 0 m m k sc assciatd with mnchrmatic EM wavs rquncy ~ 0 Hz = THz = 0 6 MHz is: cu 8 sc THz, 300m 6.50 m 65 nm Thus at lw rquncis, w s that: 0 Hz 65 nm 0 Hz m 300m sc u Sinc bth ar prprtinal t sam as br, but with E rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 9

10 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd In pur cppr mtal, th skin dpth sc assciatd with mnchrmatic EM wavs rquncy ~ 0 0 Hz = 0 GHz = 0 4 MHz (.g. cll phns) {which crrspnds t a vacuum 8 0 wavlngth c m 3 cm} is: cu sc 0 Hz 6.50 m 650 nm Th wavlngth EM wavs with rquncy 0 0 Hz prpagating in th pur cppr mtal is: u 5 u 0 m 0m k Thus at lw rquncis, w {again} s that: Hz 650 nm 0 Hz 0m 3.0 cm sc u Th High Frquncy Bhavir a Disprsiv, Nn-Magntic nducting Mdium: Dinitin high rquncy: whr: n m and:. Th squar th cmpl ind rractin r a mtal is: n i i i i At high rquncis, th r / cnductin lctrns in a mtal will undrg a grat many scillatins br scattring i.. th prid scillatins is shrt cmpard t th man tim btwn scattrings, ssntially th rlaatin tim,. In th high-rquncy rgim, th vlum charg dnsity r lctrns and th lattic psitiv ins in mtal tgthr can b thught as a plasma whs lctrn charg dnsity scillats lngitudinally { i.. in th dirctin prpagatin th EM wav) at th {natural} angular rsnanc rquncy knwn as th plasma rquncy, dind as: Th plasma rquncy in a mtal/cnductr: n m rla. (radians/sc) and: Hz 6 Fr pur cppr: rad / sc u Th crrspnding plasma wavlngth in cppr is: u u u Hz c m = 4.7 nm u Oprativ in th ptical UV prtin th EM spctrum typical r many mtals!!! 0 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

11 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd W can rwrit th (cmpl ind rractin) in trms th plasma rquncy : i i m i m i i i n n n usd: i i i Thus: n n n n whr: and: i m m m Nw: n ni ni n in with: n n Equating th ral and imaginary prtins n n in and: mn n us th llwing trick : i i i 4 i i i i i i n i n in i n n n Nw din: and: n and:. Thn: n and: n Thus: n n r: n n 4n n 0 4 Din: a b c n, thn: 0 which is a quadratic quatin th rm: 0 with: a =, b = and c =. must slct +v rt n physical grunds! Th slutin is: b b 4ac a rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

12 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Th mpl Ind Rractin: n ni n whr: n and: n whr: n and: m n m Typical plasma rquncis r r lctrns in mtals ar: 6 0 rad / sc. 3 4 Typical γ-actrs r r lctrns in mtals ar: rla 0 0 rad / sc. Hnc, in a typical mtal/cnductr w s that: rla Thus, at high rquncis : Thn: and: 3 ~0-0 i.. at high rquncis.. And: Thn: n And: {Using r } at high rquncis. n At high rquncis, r th cmpl ind rractin n is prdminantly ral, i.. th imaginary part ral part n at high rquncis. Mtals ar transparnt t γ-rays and -rays r!!! rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

13 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Sm {alkali} mtals ar transparnt / bgin t b transparnt in th UV rgin th EM spctrum! Alkali Mtals Mtal c c 7 3 Li 55.0 nm 3 Na 09.0 nm 39 9 K 87.0 nm 63 9u 4.7 nm Rb 3.0 nm UV n m Wll abv th plasma rquncy, thn: k k B k tan 0 : r B is {narly} in-phas with E r EM wavs prpagating in mtals/cnductrs with rquncy wll abv th plasma rquncy. Wll abv th plasma rquncy, w als hav, such that: dv r, t m m v r, ter, tma r, ter, t Fr, t ikz t Bcaus Er, t E, and a r, t rt r t t w s that r that th r lctrns in a mtal scillat {narly} in-phas with th driving E th mnchrmatic EM wav. Wll abv th plasma rquncy, a mtal s cmpl ind rractin is ral (i.. absrptin is small), th cnductr bcms incrasingly transparnt as ω is incrasd bynd. Nt als that n r {i.. in th s-calld anmalus disprsin rgin}!!! n r ( ) and: r ( ) rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 3

14 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd r r Sinc n r r nn-magntic cnductrs, r and r {als} prdminantly ral at high rquncis ( ), i.. K r c c Th skin dpth r p is: sc Blw th plasma rquncy, th ind rractin is cmpl, i.. thrr nn-ngligibl; Mtals ar thus paqu r. Wll blw th plasma rquncy : k and k tan tan 45 k B lags E by 45 r 4 ar r p. is larg and Th r / cnductin lctrns in a disprsiv cnductr als lag E -ild by sam phas lag. Fr, sinc th incidnt EM wav is nt transmittd, much th EM wav is rlctd. Mtals hav a high rlctin cicint R r. In crtain situatins, such as EM wavs prpagating in th Earth s insphr, r.g. in a tnuus lctrnic plasma in th labratry, th lctrns ar {truly} r, hnc damping is ngligibl ( 0 ) in such situatins. Thn r situatins with ngligibl damping, i.. 0 rlatins simply smwhat: n K n r r r and th abv, thn n n n with: n m Th abv rlatin hlds {i.. is valid} vr a wid rang rquncis, including. Fr situatins with 0 n,, th imaginary part cmpl mn 0 bcaus 0. In this rgim, whr 0, th wavnumbr k is: which is smtims prssd as: ck k c disprsin rlatin r k. {n.b. In disprsiv cnducting mdia, whr γ is nt small, th abv rmula apply nly r.} Nt that in a tnuus plasma with : k c is purly imaginary!!! 4 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

15 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd EM wavs with incidnt n such a plasma ar rlctd r, bcaus th EM kz ilds insid tnuus plasma ar pnntially attnuatd by th actr. It is prcisly this act that nabls.g. shrt-wav radi cmmunicatin arund th glb th shrt-wav radi wavs rlct th plasma in th arth s insphr! Fr tnuus lctrn plasmas in th labratry: n lctrns/m n radians/sc m m k c at ω = 0. At ω = 0, th static attnuatin cnstant is: Skin dpth (at ω = 0): sc c cm cm r static (r lw-rquncy) EM ilds Epulsin EM ilds within a plasma is wll-knwn phnmna,.g. in cntrlld thrmnuclar prcsss, and can als b plitd.g. in attmpts at cninmnt a ht plasma. Elctrmagntic Wav rpagatin in a Linar, Hmgnus, Istrpic Disprsiv nductiv Mdium In a cnducting mdium, mtals in particular, th ptical / EM prprtis many mtals ar dminatd by th r lctrns in th mtal, rsulting in high rlctanc R ~ 85-95%, narly indpndnt th rquncy / wavlngth th incidnt light, at last in th visibl light rgin th EM spctrum (~ 350 nm < λ < 780 nm). Ths r lctrns in such mtals as aluminum, tin, sdium, ptassium, csium, vanadium, nibium, gadlinium, hlmium, yttrium, scandium, smium, and vn irn hav a silvry-gray apparanc {du t visibl light phtns scattring th r lctrns in th mtal} ssntially ths mtals ar clrlss bcaus thir wavlngth-indpndnt rlctanc, but rmmbr / raliz that th physical clr an bct illuminatd.g. by whit light (= lat distributin in rquncy) is rlctd light hnc th illuminatd bct is a pr absrbr light at that wavlngth indd, th bct prrntially absrbs light at thr rquncis / wavlngths, and nt s much at th rquncy / wavlngth assciatd with th clr w prciv it t b, rm rlctd light s igur blw: rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 5

16 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Sm cnducting mtals, such as gld and cppr, as wll as varius allys.g. brnz and brass (zinc, tin and cppr) d nt hav a silvry-gray apparanc, but hav a yllw / rang / rd ting t thm. This is bcaus ths mtals prrntially absrb (i.. transmit) in th grn-blu rgin and thus rlct mr-s in th yllw / rang / rd rgin th visibl light spctrum, giving ths mtals thir charactristic hus / clrs. Gld-mtalizd saty glasss with a vry thin layr gld dpsitd n thm hav transmittanc T() pakd ~ in th grn prtin th visibl spctrum. Gggls/saty glasss catd with a thin gld layr ar usd by ppl wh wrk arund high-tmpratur urnacs. Visibl light rm th urnac {pakd ~ in th grn} is transmittd, but inrard light (i.. hat) is strngly rlctd by gld! Bcaus r / cnductin lctrns in a mtal hav n rsnancs (cpt th n at ω = 0) bcaus thr ar n rstring rcs acting n thm. In such mtals as gld and cppr, bund lctrns.g. in th s-calld valnc bands th mtal als play a nn-ngligibl rl.g. in th ptical prprtis th mtal i.. in th visibl light prtin th EM spctrum! Sinc (i.. w assum) th rspns th r lctrns in th cnductin band is sparabl / indpndnt th rspns th bund lctrns in th valnc band(s) t incidnt EM wavs (valid as lng as th amplituds E/ B ar nt t larg). Thus w {again} us th principl linar r suprpsitin r th cmpl lctric prmittivity: whr:, n.b. vac tt vac bnd b n bund n i sc bnd with: m n bund sc n b = # dnsity (= #/m 3 ) bund atmic lctrns in th disprsiv cnducting mdium r and whr: and: i whr: 0 0 n 0 0 i 0 4 m n = # dnsity (#/m 3 ) r cnductin-band lctrns in th disprsiv cnducting mdium. Th plasma rquncy r lctrns in th disprsiv cnducting mdium is: n m 6 rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd.

17 UIU hysics 436 EM Filds & Surcs II Fall Smstr, 05 Lct. Nts 8 r. Stvn Errd Thn: b n bund tt n i i 0 sc tt 4 m 0 tt bnd r b n bund tt bnd r n i i sc 0 4 m 0 I th disprsiv cnducting mdium is nn-magntic, th cmpl ind rractin n ni is rlatd t th cmpl prmittivity i n by: i b n bund tt tt n i sc 0 4 m 0 with: n sc, bund n m and with: n ninin in by th rlatins: mpl n ni is rlatd t cmpl kki n r: n k with: n k k c c c c and: rssr Stvn Errd, Dpartmnt hysics, Univrsity Illinis at Urbana-hampaign, Illinis All Rights Rsrvd. 7