ECE 307: Electricity and Magnetism Fall 2012

Transcription

1 C 7: lecticity n Mgnetism Fll Instuct: J.. Willims, Assistnt Pfess lecticl n Cmpute ngineeing Uniesity f Alm in Huntsille 6 Optics Builing, Huntsille, Al 5899 Pne: ( , emil: jn.willims@u.eu Cuse mteil pste n UAH Angel cuse mngement wesite Textk: M.N.O. iku, lements f lectmgnetics 5 t e. Oxf Uniesity Pess, 9. Optinl eing: H.M. ey, i G Cul n ll tt: n infml text n ect clculus, t e. Ntn Pess, 5. All figues tken fm pimy textk unless tewise cite.

2 lectsttic Fiels Applictins f electsttics lectic Pwe Tnsmissin X-y mcines igting Ptectin li tte lectnics Cpcits, esists, Fiel ffect Tnsists Tuc ps Keys Cte y Tues lectctigms Pticle septin Aspects f lectcemisty lectsttic cge tnsfe in liing gnisms 8/7/

3 Tpics Cee Cpte : lectsttic Fiels Culm s w n Fiel Intensity lectic Fiels ue t Cntinuus Cge istiutins lectic Flux ensity Guss s w Applictins f Guss s w lectic Ptentil eltinsip Between n An lectic iple n Flux ines lectic ensity in lectsttic Fiels Hmewk:,9,,7,9,5,5,5,5,5, 9,5,55 8/7/ All figues tken fm pimy textk unless tewise cite.

4 Funmentl Plem f lectmgnetic Fiel Tey: If ne s sme nume f suce cges (,,, ten wt fce tey exet n nte (test cge? In genel te psitins f te suce e gien s functin f time. Als, s genel ule, t suce n test cges e in mtin. F suc cses, u gl is t clc. te tjecty f te test cge. lutins t tese plems ely eily n te pinciple f supepsitin: Te intectin etween cges is cmpletely inepenent f te pesence f te neiging cges. Tus, we cn etemine te fce cting n test cge y summing ec fce pplie t te test cge y neiging cges iniiully. un esy? It s nt!!! In genel, te fce cting n cge is nt simply epenent n te psitin ect,, ut ls n te elcities n cceletins f te tw cges. As yu migt imgine, suc slutins e ey cmplex. Hwee we cn euce te plem t smeting me mngele y exmining nly cses in wic cges e fixe in spce e te time intel pse y te plem uc plems e efee t s lectsttic Plems n e te suject f te next tee cptes f tis k. 8/7/

5 Tw Mets Useful Mets f ling lectsttic Plems Culm s w Cn e use f ny sptil cnfigutin Will e use t stuy pint, line, sufce n lume cges Guss w F Often muc esie met f cses inling igly symmetic gemeties. Integl Fm iffeentil Fm 8/7/ 5

6 ummy igm f lectsttics Cge ensity lectic Fiel Ptentil l, lectic Fiel 8/7/ Figue is ecpie fm Giffits, Intuctin t lectynmics, e., Benjmin Cummings,

7 Pint Cge istiutins n Culm s w Te fce, F, etween tw pint cges n is: Alng te line jining tem iectly pptinl t te puct etween tem Inesely pptinl t te sque f te istnce etween tem k wee, F Te equtin e is esily clculte f test cge,, t te igin n suce cge,, t istnce wy. Te slutin is sligtly me cmplicte s we me u efeence fme wy fm te tw cge suc tt is efeence y te ect n is efeence y te ect. 8/7/ C / Nm

8 JAA Applet 8/7/ 8 ttp://

9 Pint Cge istiutins n Culm s w (cnt. Te slutin is sligtly me cmplicte s we me te efeence fme t sme lctin wy fm te tw cges Te lctin f is efeence y te ect Te lctin f is efeence y te ect Te ect etween n is Wee te nml ect is 8/7/ 9 ˆ

10 Pint Cge istiutins n Culm s w (cnt. F ( ˆ Tings t nte:. F ( F F F F ike cges epel ec te, ppsites ttct Te istnce etween n must e lge wit espect t te imete f te cges n must e sttic f tis slutin t e li Te signs f n must e tken int ccunt wen sling f te fce 8/7/

11 8/7/ Multiple lutins If, s stte efe, tee e me tn tw cges, ten ne cn sle f te sme fce cting n te test cge, s k k k k Tt F (... ( ( ( Inset xmple in clss

12 Multi-ple Fce xmple Tee equl pint cges e septe s swn. c f te cges is cnnecte y ey tin sting esigne t ek wen fce f.n is pplie in tensin. Clculte te cge equie t ek te stings, if = mm Wt is te electic fiel intensity t te cente f te sting AB? 8/7/ Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge,

13 lectic Fiel Te electic fiel intensity ect,, is te fce pe unit cge wen plce in n electic fiel Te electic fiel is inepenent f te test cge efine witin te spce It mkes n efeence t ll t test cge, tus, ne ccute unestning f te electic fiel is t sses te fce cting n pint, P, n ten iie ut te test cge use t sum ec f te cmpnent fces ( k k k k 8/7/

14 Cntinuus Cge istiutins ine, sufce, n lume cge istiutins e custmily sle s cntinuus cge mei. As suc, tese istiutins f cge cn e elute y integtin (f iffeentitin f te cge ensity in spce. ine ufce l s s lume eing t fiel slutins: l s 8/7/ Nte: yu text uses f ensity ee inste f. nt let yuself get cnfuse etween sufce cge ensity n te cylinicl xis unit ect

15 ine Cges Cnsie unifm cge ensity fm A t B lng te Z xis l l l 8/7/ 5 Nte: tis fm wul e ifficult t sle f B A (x,y, is fun y integtin et, l ( x, y, (,, [ x, y, ] Ten, An x y ( ( ( ( /

16 8/7/ 6 ine Cges (cnt. sec cs ( / ( ( sec tn OT Finlly A B yp cs j / Using O T sec sin cs sec ( cs ( sin ( sin cs sec sin cs Next,

17 8/7/ 7 ine Cges (cnt. Cnsie unifm cge ensity fm A t B lng te Z xis cs cs sin sin sin cs sec sin cs sec pecil Cse: Infinite line cge

18 8/7/ 8 ine Cge xmple Fin te electic fiel istnce e te mipint f stigt line segment f lengt, wic cies unifm line cge ensity,. x l / cs P x O Nte: symmety pies fiel ppgting in te iectin n = cs x l l l cs cs cs x x x x / (

19 8/7/ 9 ine Cge xmple (cnt. Fin te electic fiel istnce e te mipint f stigt line segment f lengt, wic cies unifm line cge ensity,. P x O x x x x / ( Z >> Nte: symmety pies fiel ppgting in te iectin n = cs

20 8/7/ ufce Cge Cnsie seet f cge in te xy-plne wit unifm cge ensity, s. le f te electic fiel t pint P=(,, s / ( / s

21 8/7/ ufce Cge (cnt. Cnsie seet f cge in te xy-plne wit unifm cge ensity, s. le f te electic fiel t pint P=(,, ( ( / / / lectic Fiel is inepenent f te istnce wy fm te plne!!!!!! Cn yu clculte te fiel etween tw plnes? Wt is te pysicl significnce f tis questin? s / ( Nte: Te symmety f tis plem equies tt ll slutins f cncel

22 Me xmples f istiute Cge Plems Cicul ing f cge wit ius, cies unifm cge ensity, n is plce n te xy plne n centee n te xis Fin te electic fiel t P(,, Wt lues f gie te mx electic fiel intensity? If te ttl cge n te ing is, ppximte te fiel intensity s te ius ges t e. Fin te electic fiel t P(,, l e / / l ( / 8/7/

23 8/7/ ing f cicul cge (cnt. Wt lues f gie te mx electic fiel intensity? Te extemum f ny functin cn e fun y tking te iffeentil wit espect t te ile f inteest n setting it equl t e. If te ttl cge n te ing is, ppximte te fiel intensity s te ius ges t e. Unifm cge ensity equiement implies tt =, tus / / / e / / / / l

24 8/7/ Cicul isk f Cge A cicul isk f ius, n s centee t =. Fin t P(,, / ( s s ( ( / / /, ( / /

25 ectngul ufce Cge Cnsie te fllwing flux seet x y xy( x s s s y xy( x 5 y / nc / m Ttl enclse cge n te sufce 5 / xy 9 Te fiel intensity t P(,,5 s s ( wee (,,5 ( x, y, ( x, y,5 p ( x, y,5 C x y 5 / 8/7/ 5 y s Fce expesse y =-mc t P F q / xy( x y 5 9 F / m 6 x xy x y xy 9 (,,5 / m (.5,.5,.5 / m x Using te lue gien f pemittiity f fee spce n pge 6 f yu text ( x, y,5 y 9 F 6 / 5 / m xy 9 C

26 lutins f Multiple Cge ensities Wt if we wnt t lk t te fiel intensity ue t multiple cge ensities f iffeent spes t iffeent lctins? Well, tt i i f seies f iffeent cge ensities t: s = nc/m t x= s =5 nc/m t y=- l = nc/m lng x=, = Fin t P(,,- 9 s ( C / m ( x x 8 / m 9 x F / m 6 9 s (5 C / m ( y y 7 / m 9 y 8/7/ F / m 6 6

27 8/7/ 7 lutins f Multiple Cge ensities ( l = nc/m lng x=, = m m m F m C P tt x x x s x l /,5,7 6 ( / ( 8 ( ( / 6 / ( ( ( (,, (,, (,, 9 9

28 Cge lumes Assuming unifm cge ensity, te ttl cge n in spee f ius,, is: Wt is te electic fiel n te sufce? Wt ut te electic fiel t pint P (,, fm te fm te cente f spee wit ius, centee t te igin? wee ( cs sin 8/7/ 8

29 Cge lumes ( Te symmety f te cge istiutin equies tt te sum f te cntiutins fm x n y equl e. We e left nly wit tems in te slutin cs cs wee sin 8/7/ 9

30 Gemety f Olique Tingles Cnsie w f Csines Wee cs cs Te mst cnenient wy t expess te intel is in tems f n, cs cs sin Becuse te lengt f te ect fm te sufce t te pint ies s functin f 8/7/

31 8/7/ Cge lumes ( As ges fm t, ies fm (- t (+ if P is utsie te spee iect sustitutin yiels: 8 cs sin sin cs cs cs Nte tt tis is ienticl t te electic fiel ue t pint cge t istnce >>.

32 8/7/ Cge lumes ( As ges fm t, ies fm (- t (+ if P is utsie te spee iect sustitutin yiels: 8 Nte tt tis is ienticl t te electic fiel ue t pint cge t istnce >>.

33 lectic Flux ensity Te electic fiel intensity,, is use t quntify fiel in cuum wee te ielectic f te spce etween cges is si t e (equl t te pemittiity f fee spce. Tis lne is nt epesenttie f fiels in ius mei Tus we efine te electic flux ensity ( electic isplcement,, s = +P wee P is te plitin ue t te mei F te mment we will ssume tt n plitin ccus n tt P is equl t e. A etile iscussin f plie mei will e pesente in Cpte 5 Using te isplcement ect we cn fin te electic flux tug sufce s s xmples: Infinite seet cge lume cge istiutin s x s x 8/7/

34 Guss w Guss w: Te electic flux,, tug ny clse sufce is equl t te ttl cge enclse y tt sufce, tus = enc Integl Fm Applying te iegence teem, we e yieling te iffeentil fm s s Tis is te fist f te Mxwell qutins wic clely sttes tt te lume cge ensity is equl t te iegence f te electic flux ensity Mxwell s B qutins B H J 8/7/ In mtte t 5 t enc

35 Guss w( Guss lw is simply n ltentie sttement f Culm s lw. Guss lw pies n esy mens f fining f symmeticl cge istiutins Applictins: Pint Cge sin 8/7/ 6 enc enc enc s, sin sin

36 8/7/ 7 Guss w: Unifmly Cge pee s s enc s s enc, sin sin, sin sin Outsie pee Insie pee

37 8/7/ 8 Guss w: Unifmly Cge eet s enc s ttm s tp s s s s s enc A A A s s A s xy (

38 8/7/ 9 Guss w: Infinite ine Cge l l l enc l s enc,

39 Guss s w xmple: Tw Unifmly Cge ells ( Hllw Cnucting ell 8/7/ Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge, ˆ,

40 Guss s w xmple: Pint Cge Insie Tw Unifmly Cge ells 8/7/ ˆ ˆ T simplify te plem, sle ec step f iffeent il lue, i Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge,

41 lutins wee is ie in spce = cs C/m m P(,/, m Z= Use Guss s w iffeentil fm: cs C / m cs C / m cs C / m P(,, P,, cs C / m.5c / m Nw sle f te ttl lume cge: Met : cs enc 8/7/ C

42 P(,/, =cs C/m m m Z= lutins wee ies in spce ( Nw sle f te ttl lume cge: Met : i tp Use Guss s w iffeentil fm: 8/7/ i ttm i s stp sttm cs C C cs C C i C cs C / m cs C / m cs C / m P(,, P,, cs C / m ssie Nte:.5C / m

43 8/7/ Pt Inepenence f te lectic Fiel et us clculte using te simplest pssile cnfigutin pint cge t te igin Nw let us clculte te integl f tis fiel etween ny pints n Tis slutin is clely pt inepenent n epens nly n te istnces f n fm te pint cge. It cst nting t me in te n iectins, ecuse te fiel s n ngul epenence. Tus, if I sle f ny clse lp in wic =, ten iusly q q q q l l l sin l

44 Pt Inepenence f te lectic Fiel ( Next, ecll tkes Teem: l Tus te Cul f must ls equl e f ny single pint cge t te igin. Nte wee tt tese esults l n mtte wee te cge is lcte if we e mny iffeent cges. Wy? Becuse te fiels f iniiul cges e linely inepenent, llwing us t sum ny fiel in its entiety using te pinciple f supepsitin.... ( Als ecll tt If Cul =, ten te fiel is si t e n ittinl ptentil fiel. Tese fiels e escie using scls (mgnitue ut n iectin. Tis is wy we use electsttic ptentil,, s ften t escie systems in wic nly electsttic fiels e f cncen. 8/7/ 5... ( ( Mxwell s n equtin f electsttics...

45 lectic Ptentil We efine te electic ptentil ( te ittinl scl fiel,, esciing ny electsttic ect fiel,, s te mgnitue f te iffeence etween t tw pints n n sme stn (cmmn efeence pint. ( p l ( ( p l Nw, te Gient teem sttes tt l cs l l mx negtie y cnentin l 8/7/ 6 l Nte te cucil le tt inepenence f pt plys: If ws epenent n pt, ten te efinitin f wul e nnsense ecuse te pt wul lte te lue f (p

46 Cmments n te lectic Ptentil Te w ptentil is ieus misnme Ptentil is cnnecte t ptentil enegy lectsttic ptentil is NOT te sme s ptentil enegy Antges f te ptentil fmultin If yu knw, ten it is esy t get t ny pint in spce y simply tking te gient Tis is ecuse te tee cmpnents f e nt s inepenent s te tey lk. Inste, lectic fiel cmpnents e explicitly inteelte y te equiement tt x is e. Tus, ne cn simply euce te plem t scl ne n igne te fuss f ect itin wen s esie Te efeence pint, Nte tt te efeence pint csen f te ptentil is ity n tus te lue f te ptentil cn e ltee simply y ming te efeence pint ( te psitin f te line integl in spce t ny iffeent lctin. Becuse ny cnge in efeence is simply n itie cnstnt t te fiel, te gient es nt cnge, yieling te sme electic fiel fm t lctins. Tus te ptentil cntins n pysicl significnce ecuse we cn just its lue t ny gien pint witut lteing u peceptin f te fiel t ll. Te ptentil eys te pinciple f supepsitin Units f te ptentil: Jules/Culm = lt 8/7/ 7

47 8/7/ 8 ummy igm f lectsttics l Figue is ecpie fm Giffits, Intuctin t lectynmics, e., Benjmin Cummings, 999.

48 Ptentil fm Pint Cge in peicl Cnuct 8/7/ 9 ˆ l l, ˆ Nte: we e stting ckws fm n me in tw te igin Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge,

49 8/7/ 5 Using Guss w Wit ielectics A l A enc s t t s t s s Cncentic Cnucting pees wit ius, (> wit ielectic fill Tw flt cnuctie pltes f e, A, fille wit ielectic l If cpcitnce, C=/, ten wt is te lue f ec exmple?

50 8/7/ 5 Using Guss w Wit ielectics: lectic Fiel n ltge eywee f Cge ielectic Cylinicl ell e Gune Cyline,, lectic fiel n ltge eeywee f cge ielectic cylinicl sell e gune inne cnuct l ln ln( ˆ ˆ, Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge, et in te igm equl in u cinte system

51 Using Guss w Wit ielectics: lectic Fiel n ltge eywee f Cge ielectic Cylinicl ell e Gune Cyline lectic fiel n ltge eeywee f cge ielectic cylinicl sell e gune inne cnuct ˆ, l ln ln ˆ ln et = 8/7/ 5 Plem n Figue fm N. I, ngineeing lectmgnetics, e, pinge,

52 Wk ne y te lectsttic Fiel It is ften useful t ccteie ny system tt cn impse fces n n ject y te wk it es t tt ject uppse cge q is lcte ne nte cge, q. Te fce cting n q cuses wk t e ne y isplcing q istnce l. W F l q l Te negtie sign inictes tt te wk ne y n extenl gent, q. Tus te ttl wk ne, ptentil enegy equie t me q istnce l fm t is: W q l Ntice te useful eltin pesente: Te wk ne t me cge fm ne lctin t te next is te cge times te iffeence in ptentil equie t me fm lctin t lctin in te ect fiel. In tis cse, te scl is muc simple mens f clculting wk ne witut ing t sum iniiul ect cmpnents ( ( q 8/7/ 5

53 Wk n negy in lectsttic Fiels T etemine te enegy pesent in n ssemly f cges, we must fist etemine te munt f wk necessy t ssemle tem. uppse we psitin cges, q, q, n q in n initilly empty spce. Initilly tee is n wk ne t tnsfe cge q fm infinity t u wk spce, ecuse te spce is initilly cge fee wit n electic fiel in te egin. Hwee, tee is fiel pesent fm q wen we me q int psitin. Te wk ne y tnsfeing q int u wkspce is te puct f q times te ptentil iffeence etween q n q. Te sme is tue s we psitin cge,q, wit espect t cges q n q W W W W W q q( Nte tt if te cges wee psitine in eese e ten: W W W W Tus y ing ll f te wk pssily pefme we tin W q q( W q q W q n k 8/7/ 5 k q k q ( q q q (

54 negy n negy ensity in lectsttic Fiels Nte tt tee is n esn peenting us fm eluting istiute cges in te sme mnne. Teefe ne cn wite tt te enegy, W, pesent in n electsttic fiel is: W l W W An since we cn sw y Guss w tt W l W W ( ( ( W 8/7/ 55 negy s negy ensity fist integl s ecmes lge W W w

55 8/7/ 56 lectic Fiel iples An electic iple cete wen tw pint cges f equl mgnitue ut ppsite sign e septe y smll istnce ( q q q l q l p p p q cs q ( p p Using te gemetic symmety pesent in te system, ne cn fin te ptentil t pint P s Futeme, we cn efine te iple mment s te cge times te isplcement ect = cs, suc tt te ptentil cn nw e witten s

56 8/7/ 57 lectic Fiel iples ( Te electic fiel ue t te iple centee t te igin my nw e etemine exctly y clculting te gient f te ptentil fiel sin (cs sin (cs cs ( p q q p p