A Dash of Maxwell s. A Maxwell s Equations Primer. Chapter V Radiation from a Small Wire Element

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1 Dash of Maxwll s Maxwll s quaios Pim Chap Radiaio fom a Small Wi lm By Gl Dash, mpyx LLC, GlDash a alum.mi.du Copyigh, 5 mpyx LLC ou las hap, w divd ou hid fom of Maxwll s quaios, whih w alld h ompuaioal fom: d ( d B N ρ N J v a (a (b ( (d Wh: li fild i /m B Magi flux dsiy, BH H Magi fild i mps/m olag o poial ρ Chag dsiy i Coulombs/m of a paiula hag lm, Disa fom a giv hag o u lm,, o h loaio of is v olum of a paiula hag lm, l Lgh of a paiula u lm, a a of a paiula u lm, J Toal u dsiy (boh oduiv ad displam i amps/ m of a paiula u lm, ε, Pmiiviy ad pmabiliy spivly Th magi of hs quaios lis i hi suiabiliy fo ompuaioal us. To solv Maxwll s quaios fo a giv assmblag of wis ad sous, all w d o kow is h disibuio of u ad hag. quaios ( ad (d allow us o ompu h volags ad vo poial

2 ov a volum of is. quaios (a ad (b h allow us o ompu h f spa li ad magi filds a ay poi i ha volum by simpl summaio. is im o pu hs quaios o wok by ompuig h adiaio fom a simpl suu, a sho wi lm. W hoos fo ou lm h o show i Figu. is a sho pi of wi wih h followig popis: << λ d << os( R[ ], a ay giv isa, is osa alog h lgh of Wh: l lgh of wi i ms fquy i adias f λ wavlgh i ms d diam of wi i ms u o h wi i amps h lm No ha his wi lm has osa u alog is i fii lgh. Si h u has o go somwh wo plas a povidd, o a ah d. Thy fom a apaio ad sv as svois of hag.

3 Figu : small wi lm ais a u. Ou ask is o div h magi ad li filds a ay giv obsvaio poi. Th lgh of h wi lm is l. W will b ug wo oodia sysms, Casia (x,y,z ad sphial (,,. W will sa ou aalysis by ompuig h vo poial. is always aligd wih h us ha podu i. Si w oly hav us i h z diio, will oly poi i h z diio. is simply: z N J a J a N ( ( z Wh: J u dsiy o a wi lm i amps/m a aa of wi lm i m

4 u o a wi lm i amps Howv, hs suls a o ompl. W hav o aou fo h fa ha h vo poial popagas as a wav hough spa. Si ou hypohial wi lm is suspdd i f spa, his wav popagas away fom h wi lm wih h spd of ligh,. To aou fo his popagaio, w adus h soluio i by addig a phas m: os( τ Wh: τ Tim o h obsvaio poi i sods Fquy i adias p sod τ Toal phas hag i adias Th m τ aous fo h fa ha h vo poial a h obsvaio poi is a fuio of somhig ha happd ali, amly h u a h sou a im τ. Th im i aks fo h fild o popaga o h obsvaio poi is qual o h disa dividd by h spd of ligh: τ /. Thfo: Noig ha : R[ R τ d : fλ ad z ( τ ( ad f, ] R[, λ f τ τ is kow as h add u. Th us of add us ad add poials a ommo i lomagis. s abov, hi pupos is o aou fo h fii popagaio spd of lomagi wavs as hy mov hough spa. h as of ou wi lm, h vo poial is plod i Figu. ]

5 Figu : Th vo poial is plod. Th small u lm as a vo poial whih falls off lialy wih disa. vss i phas vy half wavlgh as i popagas ouwad. Fom ou soluio fo h vo poial w a ompu h magi flux dsiy B ug quaio (b. No ha h magi flux dsiy, ad h h magi fild, is a fuio oly of, ad h oly a fuio of h us. Compuig h ul is somwha omplx mahmaially, bu w a g a iuiiv fl fom Figu. s pviously dsibd haps, w a us a imagiay paddlwhlyp dvi o s fo h xis of ul i a fild. Poi i Figu, h vo poial o h igh of h axis of h paddlwhl is ga ha ha o h lf ad i a oppog diio. This auss h paddlwhl o u, dmosaig ha h is ul a ha poi. Th ul of a vo fild is a vo i islf whos diio is dmid by h igh had ul. Th figs of h igh had poi i h diio of h paddlwhl spi ad h humb givs us h diio of h ul. Th ul of h vo poial a poi, whih is qual o h magi flux dsiy, pois owad h ad (ouwad fom h pag. Poi, h opposi is u. Poi, h paddlwhl dos o spi. Th is o ul a all. Wih a lil bi of imagiaio w a dis ha:. Th is o ul i h z diio.

6 . Th ul of h vo poial pois oly i h diio.. v i h diio, h is o ul alog h z axis. Figu : Th vo poial is usd o alula h magi flux dsiy, B, ad h magi fild, H. Th magi flux dsiy is qual o h ul of h vo poial. W a g a iuiiv fl fo h magiud ad diio of h ul by ug a imagiay paddlwhl, show i h upp lf had o. sd io h fild, i will spi if h vos o o sid of h paddlwhl a diff ha o h oh. Poi, h is ul i h oulokwis diio ad a Poi, h lokwis diio. Th is o ul a Poi. Th diio of h magi fild is dmid by h igh had ul. Th figs of h igh had poi i h diio of h ul. Thfo, h magi fild a Poi pois ouwad ad a Poi, iwad. Havig alulad h vo poial ad sudid i a las a iuiiv way h fom of h magi fild, ou x sp is o ompu h sala poial. To do his, w d o kow h disibuio of h hag a ay giv poi i im. Th hag is lad o h u o h wi by: dq d q d C

7 W a igo h osa C (sai hag ad ompu q as follows: R[ ] q d R[ q ( Fo bviy, i h aalysis ha follows w will assum ha h las mahmaial sp is always o ak h al pa of h soluio, ad simpl sa ha: q W assumd abov ha h u was osa ov h lgh of h wi, bu w do o mak h sam assumpio fo h hag q. Rah, w assum us h opposi, ha h hag q ds o b oad o h plas a h ds of h wi. ]

8 Figu : Th small wi lm is assumd o hav is hag oad o h plas a is ds. Th volag a a obsvaio poi is alulad fom h li fild. Som simplifyig gomi assumpios a usd. Th volag a a obsvaio poi a b ompud kowig h disibuio of hag (quaio (. ρ ρ w q q q q v v N O agai, w will aou fo popagaio im by ug add us.

9 By assumig ha >> l, l >>d, (l/ os, (l/ os ad λ >> l, w a show ha his quaio is qual o h followig (s ppdix fo divaio: ( os W a almos ady o ompu h magi ad li filds. Howv, w will fid i ovi o us sphial oodias isad of Casia oodias. Th asfomaio bw oodia sysms is illusad i Figu 5. Figu 5: Covsio fom Casia o sphial oodias i h x, z pla is illusad. xpsg h vo poial i sphial oodias w hav:

10 z z z os To fid B, ad h h magi fild HB/, w ak h ul of. a pvious hap, w divd h ul opaio i Casia oodias. W will disps wih a simila divaio i sphial oodias ad us sa h fomula fo ul i sphial oodias h. Wh, as i his as,, / ad /: B Coodias i Sphial of Cul ( Solvig fo h m ( /: ( ( Solvig fo h m /: Th ul of is hfo: H B B Tha solvs fo h magi fild. To fid h li fild, w us quaio (a. d d

11 s wih h ul opaio, w iodud h gadi opaio i a ali hap ad divd i i Casia oodias. s abov, w will disps wih h divaio h ad us sa h fomula fo h gadi i sphial oodias. Wh, as h, /, h gadi of h volag xpssd i sphial oodias is: Solvig fo h li fild i h diio: os d d ( ppdix B w show ha his is qual o: os Likwis: d d d d ( os os os Bu w kow ha / ε : ε So:

12 λ λ f f Si d d os os Thfo: os os os Fo : ε ε d d d d os d d d d d d ( d d Thfo: W ow a dfiiivly sa h soluio o Maxwll s quaios fo h sho u lm i Figu :

13 H os Ths h quaios may sm a umbl, bu hy a b dissd adily o val h udlyig physis of adiaio fom a wi lm. Tak h xpssio fo h magi fild: H ( ( ( osa u mom (add adiaio pa Fa Fild ompo Na Fild ompo Fou fudamal lms mak up h xpssio: a osa, a u lm adusd fo popagaio (ha is, add, a pa m, ad wo ms whih do h fall off of h fild wih disa. O of hs ms is popoioal o /, h oh o /. Th fis dos h fa fild ompo of h magi fild, ad h la h a fild ompo. W dfi h fa fild as follows: >> >> fλ f λ >> a disa muh ga ha λ/ (fa fild, h magi fild a b xpssd as: H ( h a fild, wh << λ/:

14 ( H Th li fa fild is dfid ug h sam iia as h magi fa fild, ha is h fa fild is dfid as xisig wh >>λ/. dd, i h fa fild h adial li fild,, a b igod ad h li fild osidd qual o: h a li fild: os Muh of ou is will fous o h fa filds. O agai, hs a: H No h followig:. Th magi ad li filds a oid 9 dgs fom ah oh i spa, ad. Th filds a i im phas. W hav s his ombiaio of magi ad li filds bfo. Ths quaios dsib a pla wav. Th diio of movm is dmid by h osspodu of h wo filds: H P Th vo P is kow as h Poyig vo. Th li fild is i uis of /m, ad h magi fild H i /m. Thi podu is i uis of W/m, psig h gy p ui aa big aid ouwad by h wav.

15 Th aio of wo filds is i uis of ohms ad is qual o: Ω ε ε ε ε 77 H Th valu 77 ohms is kow as h f spa impda. h a fild: H os Th a li ad magi filds a o i im phas. Fo xampl, a 9 dgs, ad Wh H λ Fis, w o ha h popagaio m a b igod i h a fild. Th, xpadig : ( ] R[ ( ] R[ H ( ( H os os os

16 Th wo filds a ou of phas i im, us as ad a ou of phas i a aiv iui. No pow is dissipad io spa hough h aio of h a filds. gy is us mpoaily sod i h magi ad li a filds us as gy is mpoaily sod i h apaios ad iduos of a aiv iui. ou x hap, w will apply ou soluios fo h sho wi lm o al wold aas suh as half wav dipols. Fom h o, higs will g asi as w l ou ompus do mos of h wok.

17 Rfs:. Kaus, J., lomagis, Fouh diio, MGaw Hill, 99.

18 ppdix W sa wih hs fomulas: q q ( W o ha: q ( By subsiuio: ( ( Fom Figu w o ha wh >>l ad λ>>: os ad os So h volag is qual o: ( os os ( ( os os L: os By subsiuio, ad oig ha >> l, h las m is qual o:

19 ( os ( os ( ( os os ( ( ( ( ( ( W a fuh simplify his xpssio by oig ha: os os ad >>l: ( ( ( os Howv, /λ: os os λ ad λ >>, <<, so : os W a sa ha h volag is appoximaly qual o:

20 ,so : ( ( ( ( ( ( ( os os os os os os

21 ppdix B W sa wih h xpssio fo h adial li fild, : os ( This paial divaiv is qual o: os W o ha: ( (

22 Pluggig his sul i yilds: ( os os