The Relationship Between Loss, Conductivity, and Dielectric Constant

Transcription

1 Th Rlatonhp Btwn Lo, Conuctvty, an Dlctrc Contant Gnral xpron Th quton ha bn ak how lo, conuctvty, an lctrc contant ar ntrrlat. Anwrng th quton rqur a farly xtnv rvw of bac lctroagntc. Frt, au that on ha a pc of arbtrary atral. Th atral a of ato, olcul, on. thn th atral xt lctron, thr boun to nual ato fr to ov about. An lctrc fl appl acro th obct. Th lctron wll naturally want to ov bcau of th lctrc fl. Th conucton lctrc currnt nty (a aur of th flow of lctron) var rctly wth th trngth of th lctrc fl. Thu J c () whr a contant of proptonalty, an t call conuctvty. Th conuctvty prov a aur of how fat an lctron can flow through a atral. It fn a () qµ whr q th charg an µ th lctrc oblty (not th prablty) of th u. Lkw, th lctrc flux nty var lnarly wth th applcaton of th lctrc fl o that D ε (3) r, ε th contant of proptonalty, an t call prttvty. Th t-haronc vron of Maxwll quaton tat that J + ω D () J th lctrc currnt nty, an t ha two part. Th frt part th pr lctrc currnt nty, J (that, J an xctaton to th yt by an out ourc), an th con part th afnton conucton lctrc currnt nty, J, cau by th applcaton of an xtrnal lctrc fl. Thu, w hav c J + J + ω D (5) c J + + ω D (6) In ot atral thr xt at lat on of thr typ of lctrc pol. Any kn of pol xhbt a polarty; that, on of th pol can b crb a bng Chr Bhop /3/

2 ngatvly charg, an th othr can b crb a bng potvly charg. Th thr typ of pol ar a follow.. Molcul arrang n uch a way a to xhbt an balanc of charg. F ntanc, watr boun n uch a way that th two ngatv hyrogn ato ar on on of th olcul, an a potv oxygn ato on th othr. nc, watr ha a nt lctrc polarty.. Ion hav nhrntly oppotly charg part. F ntanc, tabl alt, NaCl, ha a potv ou ato (Na+) an a ngatv chln ato (Cl-). 3. Mot ato hav a clou of lctron urrounng th nuclu. nc th a of an lctron uch l than th a of th nuclu, th applcaton of an lctrc fl cau th lctron to ract an ov uch quckly than th nuclu can ract. Th rult that th lctron clou hft t poton an no longr cntr about th nuclu. nc, th ato n wth th potvly charg nuclu on on an th ngatvly charg lctron clou on th othr. hn an xtrnal lctrc fl appl, th pol algn wth th fl. Th acton cau a tr to b a to th lctrc flux nty that ha th a vct rcton a th appl fl. Th rlatonhp can b athatcally crb a D ε χ (7) ε + Th tr χ known a th lctrc ucptblty an rv a a proptonalty contant btwn th lctrc fl an th pton of th lctrc flux nty cau by th prnc of th lctrc. On can rwrt th quaton a D ε ( + χ ) (8) D ε ε (9) r whr ε r known a th rlatv prttvty of th u. ε r n gnral a coplx quantty. To unrtan why, conr an altrnatng lctrc fl appl to a pol. hn th fl frt trk th pol, th pol rotat to algn tlf wth th fl. A t pa, th lctrc fl rvr t rcton, an th pol ut rotat agan to ran algn wth th crct polarty. A t rotat, nrgy lot through th gnraton of hat (frcton) a wll a th acclraton an clraton of th rotatonal oton of th pol. Th gr to whch th pol out of pha wth th ncnt lctrc fl an th lo that nu trn how larg th agnary part of th prttvty a a functon of atral an frquncy. Th largr th agnary part, th nrgy bng pat through oton, an th l Chr Bhop /3/

3 3 nrgy avalabl to propagat pat th pol. Thu, th agnary part of th rlatv prttvty rctly rlat to lo n th yt. To rprnt th ral an agnary part of th abolut prttvty, th followng convnton u. ε ε ε ε r () Rturnng to Maxwll quaton (6), w now hav that J + + ω( ε ε ) () ( + ωε ) + ωε J + () J + + ωε (3) In th lat tp, w hav fn an ffctv conuctvty, + ωε () Th ffctv conuctvty th valu that uually pcf n ata ht, although t ght b labl a rly conuctvty. Th frt tr on th rght-han of th abov quaton th tatc conuctvty, an w can fn th lat tr to b conuctvty u to an altrnatng fl. Thu + (5) a Agan rturnng to Maxwll quaton (3), w hav now J + ωε (6) ωε J + ωε ( tanδ ) (7) r, w hav fn th lo tangnt, tan δ a tan δ (8) ωε can alo xpan Maxwll quaton (6) a J + ε ωε (9) ωε ε Chr Bhop /3/

4 Th lat quaton hghlght th fact that two tr contrbut to th lo tangnt. Th frt tr,, crb lo u to collon of lctron wth othr lctron an ωε 7 ato. F ntanc, f th tatc conuctvty hgh (coppr ha 5.8x / ), thn charg flow vry aly wthout any collon. At frt glanc t trang that a tr that approach nfnty n th nurat crb a low lo tructur, but t ut b rbr that nfnt conuctvty pl ro lctrc fl (an fnt currnt nty). That, nta of vwng th currnt nty a a functon of th lctrc fl, J c () vw th lctrc fl a a functon of th currnt nty, J c () Now nfnt conuctvty ak n. A on ght xpct, n conuct, th tr of ε (9),, onat th othr tr of (9),. ωε ε ε Th tr of (9) crb how uch nrgy uppl by an xtrnal lctrc ε fl pat a oton an hat. In lctrc, th tr uually onat th frt tr. In tal, th ral part of th prttvty uually qual to th prttvty of fr pac, an th agnary part uually ro. conuct antan a rlatv balanc btwn th two tr. Thu, whn an ffctv conuctvty pcf on a ata ht, t uful to rbr that t ar fro two ourc. F a tal, th ffctv conuctvty u alot ntrly to th collon of lctron, an th polaraton pnnt tr ropp. Maxwll quaton (9) ruc to J + ( ωε + ) () F a lctrc, th ffctv conuctvty u alot ntrly to polaraton lo (pol oton), an th frt tr ropp fro th calculaton. Maxwll quaton (9) bco ε J + ωε (3) ε turn now to calculatng th powr abb an trantt by a u. can bgn wth Maxwll quaton an fn th followng rlatonhp. Th Chr Bhop /3/

5 5 quaton ar rv aung that pha rprnt th fl, an a pnnc of t ω uppr. Not that f th quaton wr rv ung t rvatv, th rult woul b ffrnt. That, on can not rly rplac by ω n th rult t bcau of th non-lnar natur of th quaton (prouct of fl). F th raon, th oyntng vct, uually rprnt a, hr ha th f. If on u th uual (non-pha) rvaton, thn th rult ut b avrag ovr t to obtan th rult hr. ( ) + + ω + () ( M + J ) ( ) (5) (6) J J (7) µ (8) ε (9) In th quaton, rprnt th coplx uppl powr, coplx xtng (trantt) powr, rprnt th rprnt th ral pat powr, an an rprnt t nrg. Uually th lat two tr ar trctly agnary, but f µ ε ar coplx, thn thr both tr ay b coplx. Th ral part of th tr can b xtract an a to th pat powr. r w gn th poblty of coplx prablty an rwrt th quaton t a follow. ( ) + + ω + (3) ( M + J ) (3) Chr Bhop /3/

6 6 ( ) (3) ( + ωε ) (33) µ (3) ε (35) ar concrn hr prarly wth th pat powr tr,. A nton bf, t pulng to that th pat powr var rctly wth th tatc conuctvty. Rwrtng n tr of th currnt nty hlp th ntuton. + ωε J (36) now wh to copar th pat powr wth th xtng powr an th t powr. A an xapl, w choo a block of lctrc atral that ha a plan wav propagatng n. Th wav gnat fro out th block, o th pr ourc, J an M, ar ro. Th plan wav ntr at an xt at. Thu, rprnt both th ntrng an th xtng powr pnng on th urfac at whch t valuat. th aount of powr pat n th u, an an ar th t nrg. Bf w proc, w not that non-ro but fnt tatc conuctvty pl that charg ar prnt wthn th u. Th fact tru bcau non-ro fnt tatc conuctvty pl that charg tak a fnt t to travl through th u, o that th t-avrag charg non-ro. Zro tatc conuctvty pl that fr charg nvr ntr th u. Infnt tatc conuctvty pl that charg progr ntantanouly through th u; thu, th t-avrag charg agan ro. Oftn book talk about a ourc-fr loy u. By ourc-fr th book an, n part, that th tatc conuctvty thr ro nfnt. That, th t-avrag charg ro. By loy th a book an that thr polaraton lo prnt n th atral. That, th prttvty ha a non-ro agnary part. hn conrng a atral wth a fnt non-ro conuctvty, t woul that cuon about a ourc-fr u coul not apply. owvr, bcau th charg can b xpr n tr of th lctrc fl ( J c ), th probl can b trat by th a tchnqu u to trat ourc-fr a. Th fact vry Chr Bhop /3/

7 7 ftunat, othrw atral wth non-ro tatc conuctvt coul only b trat nurcally. A t, w wll now rv th clo-f oluton to a plan wav travlng n a u wth fnt non-ro tatc conuctvty an wth polaraton lo. Th wav quaton that w ut olv + k ω µ (37) + Rcallng that ( k µ ) ω (38) k ( ε ε ) ω µ ε ω µ (39) on obtan whr ε + ω µ ε + () ε ω ε + ω µ ε () ω ε + ( tanδ ) ω µ ε () + (3) ω µ ε ( tan ) δ () A oluton to th wav quaton x (5) can b foun a follow. Chr Bhop /3/

8 8 ( tan ) ω µ ε δ (6) δ δ ( + tan δ ) co n ω µ ε (7) δ δ ω µ ε co n (8) coδ ω µ ε + coδ coδ coδ coδ (9) Th oluton to th wav quaton thn Lttng ω coδ coδ x (5) coδ ω µ ε + coδ µ ε coδ coδ ω µ ε (5) an β + coδ coδ ω µ ε (5) thn β (53) β x (5) Notc that f an ε thn δ o that (55) β k (56) an Chr Bhop /3/

9 9 x k (57) whch th uual xpron f a plan wav n lol pac. Th agntc fl can b foun fro µ ω (58) y ω µ (59) y ω µ (6) Th xpron, (6), only qual to y y η µ ε (6) whn, a conton that not gnrally tru. now uar th quaton of th fl. x β (6) y β ω µ β (63) Th nxt tp to coput th r powr. ( ) + ω µ β (6) (65) δ ε µ co (66) Chr Bhop /3/

10 ε ε (67) F a block of atral havng lngth, valuat a follow., an cro-ctonal ara, A, th ntgral Th powr ntrng th atral at : n β + A β + ω µ ω µ (68) Th powr xtng th atral at : out β + A β + ω µ ω µ (69) Th powr pat n th atral a hat: ( ) A (7) Th powr t n agntc fl: ( ) (7) ε A ε coδ 8 co δ Th powr t n lctrc fl: ( ) A ε ε 8 (7) Not that n an out appar on th a of quaton (3). In (68) th gn of n rvr whn t fn to ak th u of th output, pat, an t powr qual to th nput powr. Takng th rato of to th ral part of, on obtan n out ω µ ω µ R{ } n out β co δ ω µ ε coδ ω ε tanδ (73) Chr Bhop /3/

11 Thu, th pat (ral) powr qual to th ffrnc btwn th nput an output ral powr. Lkw, on can how that ω I ( + ) { } n out (7) to prov that th t (agnary) powr qual th ffrnc n th nput an output ractv powr. Th rato of th pat powr to ral nput powr foun a R R { } n { } n ( ) A (75) A β ω µ ( ) ω µ (76) β If thn th gnral quaton (8), (5), (5), an (6) through (67) pcal a ε ε tan δ + (77) ε ω ε ε δ k n (78) δ β k co (79) β x (8) ( β + θ ) y (8) η whr µ θ η η (8) ε ( ) η θ (83) Chr Bhop /3/

12 ωε (8) µ ε (85) ε ε (86) A, an β bco nfnt, an th fl go to ro. Th aupton that an ε o not yl any ructon of th quaton. rary ourc: Balan, Aanc ngnrng lctroagntc. Chr Bhop /3/