ELECTROMAGNETIC FIELDS

Transcription

1 Pf. ng. an Macháč Dc. lcmagnic Filds LCTROMAGNTC FLD yllabus f Lcus Pf. ng. an Macháč Dc. - -

2 Pf. ng. an Macháč Dc. lcmagnic Filds nducin nvinmn: hmgnus - nnhmgnus ispic - anispic lina - nnlina nndispsiv - dispsiv Ln s fc F q ( v B) vlum chag dnsiy q ρ [C/m 3 ] lim sufac chag dnsiy q σ [C/m ] V lim V lin chag dnsiy q τ [C/m] l lim l cun dnsiy lim i [A/m ] sufac cun dnsiy K i l lim [A/m] l cninuiy quain (cnsvain f chag) d d d ρdv ρ y y - -

3 Pf. ng. an Macháč Dc. lcmagnic Filds diffnial pas: ϕ gadϕ A diva A A cula Ohm s law σ Mawll s quains: dψ d d d H dl d ε dφ dl d d d ε d Q µ H d H ε H µ ( ε ) ρ ( µ H ) µ H ( ρ dv ) d d maial quains σ D ε ε ε B µ H µ µ H lcsaic fild sucs - immbil lcic chags nd Mawll s quain dl 3d Mawll s quain (Gauss law) ε d Q ( ε ) ρ culf pnial fild - 3 -

4 Pf. ng. an Macháč Dc. lcmagnic Filds ϕ ϕ scala pnial vlag bwn pins A and B AB B A dl ϕ ϕ A B wk f chag anspain A Q AB Q ( ϕ ϕ ) quain f lcic fild lins capaciy y Q C dy d ngy f a chagd capaci A W Q C B dnsiy f lcic fild ngy w D ε D ε sufac dnsiy f fc n a chagd bdy sufac df d aaciv fc f capaci lcds w F dc d lcic fild sluin mhds using h Gauss s hm: cndiins: d n cns n - 4 -

5 Pf. ng. an Macháč Dc. lcmagnic Filds hn fm d Q ε w g Q ε suppsiin: ρdv 4πε V σd 4πε τdl 4πε c ϕ ϕ ϕ 4πε 4πε V c ρdv σd τdl 4πε mhd f imags ubsanc (nncnducing) in lcic fild dilcic maial is plaid - plaiain (cupld) chag is cad - lcic dipls wih dipl mmn: p qdl vc f lcic plaiain (spac dnsiy f dipl mmn): P n a sufac f dilcic p i i lim [C/m ] V V P σ v in a vlum f dilcic P ρ v σ v ρ v - sufac and spac dnsiy f cupld chag lcic inducin P ε κ D ε P D ρ nly f chag is h suc f his vc κ - lcic suscpibiliy D ε ( κ ) ε ε - 5 -

6 Pf. ng. an Macháč Dc. lcmagnic Filds Bunday cndiins a an infac f w dilcic n ( ε ε ) σ n n ( D D ) σ ( P P ) σ v D n D n σ ( ) n facin f lcic fild lins g gα α ε ε Pissn s quain: Laplac s quain: ρ ϕ ε ϕ dic ingain f h Pissn s (Laplac s) quain assuming ϕ ϕ( ) ( ) d ϕ ρ ϕ d ε dϕ d ϕ ( ) ρ d f ( ) k ε ( ) f d k k Mhd f spaain f vaiabls: sluin f h quain ϕ assumpin: ϕ ( ) X ( ). Y ( y) X X Y Y X m m X Y ± m Y numical mhds - mhd f fini diffncs (FD) ϕ ϕ ρ y ε ( ) ρ h 4ϕ ϕ ϕ ϕ3 ϕ4 ε - 6 -

7 Pf. ng. an Macháč Dc. lcmagnic Filds h disanc f nds This quain is win f all nds w g a sysm f quains f unknwn valus f pnial. laain (suplaain) mhd liminas h nd slv h sysm f quains 4 h ρ R ϕi 4 ϕ ε i R ϕ ϕ α α Valus f pnial a subsqunly cmpud in all nds in vy iain. mhd f fini lmns (F) Aa in which pnial is slvd is dividd lmnay pas (lmns). Th sluin f diffnial quain is appimad v hs lmns by simpl funcins - lina plynmials. Th iginal quain is ansfmd a sysm f lina algbaic quains using his appimain. Valus f pnial in all nds a dmind by slving h sysm f quains. cninuiy quain sainay fild Fild f sainay cun d d dq d d d Ohm s law in a diffnial fm σ dl ϕ ϕ bunday cndiins - 7 -

8 Pf. ng. an Macháč Dc. lcmagnic Filds ( ) n n n σ σ n n ( ) n Analgy wih lcsaic fild lcsaic fild cun fild D d Q D ε d σ ϕ ϕ ϕ ϕ D D n n n n ϕ D Q C ε ϕ G σ cmpuain f cun fild: cninuiy quain (analgy wih h Gass s law) suppsiin mhd f imags cmpuain f cnduciviy sisiviy cnduciviy G / l dl R R σ σ G σ σd G l l Kichhff s Laws s i k - spcial fm f h cninuiy quain nd k - spcial fm f h quain dl - 8 -

9 Pf. ng. an Macháč Dc. lcmagnic Filds wk (ngy) in cun fild A.. pw da P. d dnsiy f pw lsss p σ σ sucs - lcic cun - pman magn Bi-ava law Basic quains: ainay magnic fild µ B 4π B B µ c dl 4π V µ dv K d 4π s Mawll s quain (Amp s law) c B dl µ B µ 4h Mawll s quain B d B inducin flu φ d B - 9 -

10 Pf. ng. an Macháč Dc. lcmagnic Filds magnic fild cmpuain by h Amp s law assumpins: B dl n c B cns n c B µ l c B dl µ vc pnial: B nnunambiguusnss f chsing A A A A ' A ψ w g h Pissn s quain A µ is gnaal sluin (applicain f which is asi han h Bi-ava law): µ A dv 4π µ A 4π c dl µ K A d 4π magnic flu φ B d c Adl scala pnial H ϕ m ϕ m mhd f imags in magnic fild Calculain f inducancs saic dfiniin (u inducanc) - -

11 Pf. ng. an Macháč Dc. lcmagnic Filds L φc φ χ cil flu φ φ c i i ( Nφ) ngic dfiniin (inn inducanc) W L W L muual inducanc φ φ M Numann s fmula µ dl dl M π 4 c c Magniain f subsanc (subsanc in magnic fild) in vacuum B H µ [T H/m A/m] magnic fild sucs - cuns flwing in cnducs pmann magn - cas h sam fild as cun flwing hugh cnduc cun mus flw in a subsanc - cupld cun ( K ) v v v ams - lmnay cun lps - magnic mmn dm d in a pa f vlum m dm d magniain vc - vlum dnsiy f m - -

12 Pf. ng. an Macháč Dc. lcmagnic Filds M lim V i m V i sucs f M - cupld cuns n h sufac n ( M M ) Kv in h vlum M magnic fild vc H (innsiy f magnic fild) - nly f cun is suc f his vc H B M analgy D P µ M m H κ B µ B ( κ ) H µ µ H m ε µ ( H M ) ( D P) magniain - inn fild ffcs cnfmally wih h u fild amplificain diamagnics µ paamagnics µ fmagnics µ >> plaiain - inn fild ffcs agains h u fild anuain ε v hyssis cuv f fmagnics magnic sf maials magnic had maials bunday cndiins in cas n ( H H ) K K H H n ( B B ) µ H µ B n Bn n Hn g gα α µ µ in cas µ α Magnic fild ngy - -

13 Pf. ng. an Macháč Dc. lcmagnic Filds cun flwing hugh a cil W L dnsiy f magnic fild ngy w BH sysm f N cils W W i i i L f cils W L L ± M ngy ndd magni a subsanc - cain f magnic fild in uni vlum (hmgnus fild) vall ngy dw HdB i i in a lina subsanc W B p HdB W H p B p ubsanc wih hyssis - ngy is cnsumpd by vcming ficin fcs causing hyssis W hyssis cuv aa Fcs in magnic fild cnducing lp ( K Bd BdV ) df dl B F dl B c fc ffcing bwn w lp cnducs w paalll cnducs µ F 4π 3 ( dl R ) R dl / cc - 3 -

14 Pf. ng. an Macháč Dc. lcmagnic Filds F l µ πa fc calculad fm a chang f inducanc L F sufac dnsiy f fc df d w Magnic cicuis a slvd analgically wih lcic cicuis Hpkinsn s law N R m φ magnic sisanc (lucanc) l R m µ pmann magn clsd lp magn H magn wih an ai sli - dmagniain ffc f h sli causs B dcass dminain f a wking pin B µ l H i i quain f a lin i lv A wking pin is dmind by an inscin f his lin wih hyssis cuv. ngy and pw in lcmagnic fild Lins f vc H a idnical wih lins wh A cns s ha hy a idnical wih quipnials. Cnsqunly - 4 -

15 Pf. ng. an Macháč Dc. lcmagnic Filds H pw caid by a lin P H d Pyning s vc H Balanc f lcmagnic fild ngy d P P d ( W Wm ) PR pw supplid fm an u suc P dv ul lsss dv σ dv dv σ P changs f lcic and magnic ngy d d d m dv d ( W W ) ε µ H Pw adiad in suunding spac P R ( H ) d Pw is anspd alng cnducs is in h cas f lsslss cnducs (σ ) paalll wih hi sufac. n h cas f lssy cnducs (σ ) n pa f h Pyning s vc is dicd in cnduc and cvs lsss in i. Tim vaiabl lcmagnic fild is ncssay slv h cmpl Mawll s quains - 5 -

16 Pf. ng. an Macháč Dc. lcmagnic Filds H ε H µ ( ε ) ρ ( µ H ) lw changs - quasisainay fild - h m d is nglcd. Tim changs f magnic fild ca lcic fild C dφ dl d lcmiv fc inducd in a lp u vlag u i dφ dl d d u φ d di L d in h cas f a cil wih N lps dφ un d mving cnduc - lcic fild is inducd p v B iducd vlag u p p dl ( v B) dl Hamnic im vaiabl filds cmpl symblic mhd - phass scala: - 6 -

17 Pf. ng. an Macháč Dc. lcmagnic Filds ( ϕ ) invs ansfm in im dmain vc vaiabls M u m sin ϕ ˆ m u ( ) { m ). } ( ) M sin( ϕ ) y M sin( ϕ ) M sin( ϕ ) m ˆ M Mˆ Mˆ y y Mˆ ym y m wh Mˆ M m ϕ invs ansfm Mˆ ( ) ˆ m M in h cas f linaly plaid vc wh ϕ ϕ y ϕ ϕ is M M m sin ( ϕ ) M ˆ M m ϕ Mawll s quains ˆ ˆ H ( σ ε ) ˆ ˆ µ H ˆ ε ˆ ρ ˆ µ H ˆ ˆ ρ N: symbl ^ dning phass msly will n b usd hughu h fllwing. Pyning s vc insananus valu f sufac dnsiy f anspd pw H - 7 -

18 Pf. ng. an Macháč Dc. lcmagnic Filds im avagd valu (sufac dnsiy f aciv pw) av T T ( H ) d R{ H * } cmpl Pyning s vc ˆ ˆ ˆ H * av Q aciv pw sufac dnsiy Q m H { * } Balanc f lcmagnic fild ngy - hamnic vaying fild P P P av R ( Wmav Wav ) QR Q aciv pw supplid by a suc aciv pv supplid by a suc P * R{ R}dV Q V * m{ R}dV V ul lsss P av V σ dv avagd valus f lcic and magnic fild ngy dnsiis w mav µ H wav ε 4 4 aciv adiad pw P R R * { H } d - 8 -

19 Pf. ng. an Macháč Dc. lcmagnic Filds aciv adiad pw Q R m * { H } d snanc P R Q R in h cas W mav W av is Q Displacmn cun D D dψ d ε d d d ˆ ˆ ε D d cninuiy f lcic cun in dilcic wll cnducing mdium - >> σ>>ε D Wav quain By liminaing H fm Mawll s quains w g H k H k ρ ε µ suclss aa k H k k µε µσ H Plan lcmagnic wav ˆ ˆ k - 9 -

20 Pf. ng. an Macháč Dc. lcmagnic Filds - - k H H ˆ ˆ n n k k - dicin f ppagain β α α k - phas cnsan β - anuain Φ ˆ Φ α β ˆ ( ) ( ) Φ α β sin H H Φ α β ˆ ( ) ( ) H H Φ α β sin sufac f cnsan phas cns α - plan ppndicula α sufac f cnsan ampliud cns β - plan ppndicula β unifm wav β α nnunifm wav α β hav diffn dicins phas vlciy α f v gup vlciy α α d d d d v g v c g wavlngh f v f α π λ lain bwn vcs and H n n H - dicin f ppagain n H ϕ ε σ µ chaacisic impdanc

21 Pf. ng. an Macháč Dc. lcmagnic Filds in h cas k k w hav y H y H ansmid pw av R * { H } β cs( ϕ ) wav in idal dilcic k µε α µε β wav is n anuad sin α µ ε ( Φ ) H H sin α ( Φ ) y wav in cnducing mdia (σ>>ε) k µσ α β µσ ( ) H H µ σ β sin β sin π ϕ 4 ( α Φ ) ( α Φ ϕ ) y pnain dph δ β µσ lssy dilcic σ<<ε σ µ µ k µε σ α µε β σ ε ε µ ε - -

22 Pf. ng. an Macháč Dc. lcmagnic Filds vcs ( ) Wav plaiain H k a ppndicula in h plan wav Plaiain dmins h way in which an nd pin f mvs in a plan ppndicula h ppagain dicin. sin y ( α Φ ) sin( α Φ ) m ym y in h pin Φ α y m sin ( ) sin ( Φ) wh Φ Φ y Φ ym liminaing im w g y y ym m ym csφ m sin Φ - quain f llips gnal valus ym Φ - gnal llipic plaiain m lina plaiain Φ nπ π f Φ ( n ) - llipic plaiain spcial cas f m ym and ( ) π Φ n - cicula plaiain lf-handd plaiain Φ ( π ) vc as lf igh-handd plaiain Φ ( π π ) vc as igh (i is dmind in h dicin f wav ppagain) kin ffc Pw is ansmid fm a suc a lad by lcmagnic fild via dilcic suunding cnducs. Pw ls in cnducs ns hm hugh a sufac f cnducs - nmal cmpnn f h Pyning s vc. n h cas f nnsainay fild is his wav highly anuad - skin ffc - - -

23 Pf. ng. an Macháč Dc. lcmagnic Filds nnhmgnus disibuin f ( B ) cnduc (fmagnics) in a css-scin f Wav ppagaing fm h cnduc sufac in is vlum H H β α ( ) p β α ( ) H p ( ) p ( ) H σ p µ α β pnain dph δ δ µσ cnduc impdanc (cylind adius >> δ) p l H p l π l ( ) l σ π δ R L f i ( ) R f ffciv sisiviy R f l σ π δ Cnduc ffciv sisiviy f ac. is qual a dc. sisiviy f a lay bllw h sufac a dph δ (und an assumpin δ << cnducs ansvsal dimnsins). F a cnduc f abiay css-scin: R f - pim f a cnduc l δσ cylindical cnduc k ( ) p ( k) ( k ) µσ cun flwing hugh cnduc Φ ( ) ( ) ( k ) k q k π k p ( k ) ( k ) p k π ( k ) ( k ) - 3 -

24 Pf. ng. an Macháč Dc. lcmagnic Filds ( ) k π ( k) ( ) k mpdanc f a cnduc l l ( k ) ( k ) ( Φ Φ ) p k q 45 π σ µσ π σ q Fmagnics f cylindical shap B ( ) B p ( k) ( k ) πb φ Bd k p ( k ) ( k ) kin ffc in lngiudinally magnid sh B ( ) B p csh csh [( ) α] [( ) αa] α µσ magnic flu a abbp φ b B( ) d gh αa a ( ) [( ) αa] cmpl pmabiliy µ c Bav µ H αa gh a p ( ) [( ) α ] in h cas α a [( ) ] ( ) αa csh αa & sinh & [( ) ] ( ) αa αa [( ) a] & gh α lsss du ddy cuns - 4 -

25 Pf. ng. an Macháč Dc. lcmagnic Filds ( ) αa sinh[ ( ) α] µ csh[ ( ) αa] B µ lsss vlum dnsiy ls pw * p σ P lb a a p ( ) d Cun skin ffc in Casian cdinas impdanc ( ) a b a p csh csh ( ) d [( ) α] [( ) αa] p gh αa ( ) [( ) αa] l ( ) αa c gh a σ [( ) α ] TM wav H TM wav n ansmissin lin ansvsal fild is dscibd by T T T H T Tansvsal fild disibuin cspnds a sainay disibuin h lin mus cnsis a las f w spaa cnducs. vlag u( ) dl T cns. cun i( ) H dl using hm T cns. c - 5 -

26 Pf. ng. an Macháč Dc. lcmagnic Filds ( ) ( ) ( ) ( ) l d y u y y T T T ( ) ( ) ( ) ( ) l d y h i y h y H c T T T ( ) ( ) ( ) i L i R u ( ) ( ) ( ) u C u G i hamnic sady sa ( ) L R ( ) C G ( )( ) C G L R ( )( ) C G L R C L R G - valus laiv lngh uni C is dmind using chniqus knwn fm saic lcic filds L is dmind using chniqus knwn fm sainay magnic fild innal inducanc is nglcd skin ffc mus b akn in accun. G is dmind using analgy wih lcic fild R skin ffc mus b akn in accun ppagain cnsan ( )( ) C G L R γ lsslss lin LC γ wav impdanc C G L R lsslss lin C L

27 Pf. ng. an Macháč Dc. lcmagnic Filds phas vlciy v f α LC wav lngh v f λ f π α luin f h wav quain γ γ γ γ impdanc γ γ γ γ infinily lng lin α α lin f a fini lngh (L) nd f a lin: L (L) (L) ( γs) sinh( s) csh γ sl- csh( γs) sinh( γs) cdina masud fm h lin nd impdanc ( s) ( s) ( s) L L L gh gh ( γs) ( γs) lsslss lin ( αs) sin( αs) cs cs ( αs) sin( αs) - 7 -

28 Pf. ng. an Macháč Dc. lcmagnic Filds ( s) ( s) ( s) L L L g g ( αs) ( αs) pn-nd minad lin cs ( s) ( s) ( αs) sin( αs) ( s) c g α ( s) sh-nd minad lin ( αs) cs( s) sin α ( s) ( s) ( s) g α ( s) - 8 -