ECE 307: Electricity and Magnetism Fall 2012
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1 ECE 7: Electct nd Mgnetsm Fll Instucto: J.D. Wllms, ssstnt Pofesso Electcl nd Compute Engneeng Unvest of lbm n Huntsvlle 46 Optcs uldng, Huntsvlle, l 5899 Pone: (56) , eml: jon.wllms@u.edu Couse mtel posted on UH ngel couse mngement webste Tetboo: M.N.O. Sdu, Elements of Electomgnetcs 5 t ed. Ofod Unvest Pess, 9. Optonl Redng: H.M. Se, Dv Gd Cul nd ll tt: n nfoml tet on vecto clculus, 4 t ed. Noton Pess, 5. ll fgues ten fom pm tetboo unless otewse cted.
2 Couse Mtel Cptes -. (Revew Mtel) ectos lgeb, Coodnte Tnsfomtons, ecto Clculus Cpte 4. Coulomb s Lw, Electc Feld Intenst, Cge Dstbuton, Electc Flu Denst, Guss Lw, Electc Potentl, Eneg Cptes 5-6. Popetes of Mtels, Cuents, Contnut Equton, Posson s Equton, Lplce s Equton, Resstnce, Cpctnce, Imge Teo (Opt.) Cptes 7-8. ot-svt Lw, mpee s Lw, Mgnetc Flu Denst, Mwell s Equtons (Sttc), Mgnetc ecto Potentls, Mgnetc Foces, Mgnetc Mtels, ound Condtons Cpte 9. (Ptl) Fd s Lw, Mwell s Equtons (Tme ng), Optonl: Tme-Hmonc Felds, Plne Wves Pontng ectos Gdng: 8 -%, 7-8%, 6-7% C. Homewo (%): Tuned n weel. Gded on ttempted effot. Ems (5%): pe Semeste. Em scoes stocll ncese ove te semeste Em : Cptes 4, 5 nd 6 Em : Cptes 7, 8 nd 9 Fnl Compeensve Em (%): Includes mtel fom Cptes 4 toug 9
3 Mwell s Tme Dependent Equtons It ws Jmes Cl Mwell tt put ll of ts togete nd educed electomgnetc feld teo to 4 smple equtons. It ws onl toug ts clfcton tt te dscove of electomgnetc wves wee dscoveed nd te teo of lgt ws developed. Te equtons Mwell s cedted wt to completel descbe n electomgnetc feld (ete sttcll o dnmcll) e wtten s: Dffeentl Fom Integl Fom Rems D E t H v J D t DdS L S S ds E dl H dl Guss s Lw Nonestence of te Mgnetc Monopole Fd s Lw mpee s Ccut Lw L S 8/7/ dv v t J S ds D ds t
4 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto nlss Scls nd ectos Unt ecto ecto ddton nd Subtcton Poston nd Dstnce ectos ecto Multplcton Components of ecto Cpte : Coodnte Sstems nd Tnsfomtons Ctesn Coodntes Ccul Clndcl Coodntes Specl Coodntes Constnt Coodnte Sufces Cpte : ecto Clculus Dffeentl Lengt, e, nd olume Lne, Sufce, nd olume Integls Del Opeto Gdent of Scl Dvegence of ecto nd te Dvegence Teoem Cul of ecto nd Stoes s Teoem Lplcn of Scl Clssfcton of ecto Felds Homewo C. : 5c, 6, 8, 8b,, 6 C. : 7, 8,, 5, 7 C. : b, b, c, b, 6,, b,, Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste
5 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto nlss Scls nd ectos Unt ecto ecto ddton nd Subtcton Poston nd Dstnce ectos ecto Multplcton Components of ecto Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste
6 ectos nd Scls Scl: quntt defned onl b ts mgntude speed: 4 m/s Cge: Coulombs Cpctnce: 5 fds ecto: quntt defned b bot ts mgntude nd decton n spce Foce: F ( 5 4 ) N Feld: Functon tt specfes ptcul quntt evewee wtn sptl domn. Electc feld (vecto feld): oltge (scl feld): q E N / C q
7 Unt ecto vecto s bot mgntude nd decton Te mgntude of s scl wtten s II. unt vecto long s defned s vecto long te decton of wt mgntude of. s suc, te vecto m be defned s nd tus Emple:
8 ecto ddton nd Subtcton Two vectos, nd, cn be dded togete to genete td vecto, C Me sue to bde b te followng bsc lws of lgeb s ppled to vectos ddton Multplcton C C l l Commuttve ssoctve Dstbutve C C
9 Dstnce ectos ectos cn lso be used to defne te dstnce between two ponts n coodnte sstem, o between lne nd plne wtn coodnte sstem If gven two ponts, P nd Q, one cn fnd te dstnce between tem s te vecto,,,,, ),, ( ),, ( Q P P Q PQ Q P
10 ecto Multplcton: Dot Poduct Two vectos, nd, cn be multpled togete to genete td vecto, C Scl Poduct ecto Poduct Scl Tple Poduct ecto Tple Poduct Me sue to bde b te followng bsc lws of lgeb s ppled to dot poducts D C l C C C C Commuttve ssoctve cos Note: otogonl vecto dot poducts multpl to cosne vlue of eo pllel vecto dot poducts multpl to cosne vlue of
11 ecto Multplcton: Coss Poduct Me sue to bde b te followng bsc lws of lgeb s ppled to coss poducts C C C C nt-commuttve Not ssoctve Dstbutve n sn Note: otogonl vecto dot poducts multpl to sne vlue of C C C C C C C C C C Scl Tple Poduct ecto Tple Poduct
12 Components of ecto Gven two vectos, nd, one cn dectl fnd te scl component of long s cos Ts scl poduct s nown s te pojecton of long te decton. Te vecto component of long s smpl scl component multpled b te unt vecto long cos One cn lso fnd te ngle between nd usng te coss nd dot poduct of te two cos sn
13 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : Coodnte Sstems nd Tnsfomtons Ctesn Coodntes Ccul Clndcl Coodntes Specl Coodntes Constnt Coodnte Sufces Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste
14 Ctesn Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos n stt lnes tt ntesect t sngle pont (te ogn). Te vecto n ts coodnte sstem cn be wtten s
15 Clndcl Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos Te vecto n ts coodnte sstem cn be wtten s Wee te followng equtons cn be used to convet between clndcl nd ctesn coodnte sstems tn cos sn
16 Mt Tnsfomtons: Ct. nd Cl. cos sn sn cos cos sn sn cos
17 Specl Coodntes Coodnte sstem epesented b (,,) tt e tee otogonl vectos emntng fom o evolvng ound te ogn Te vecto n ts coodnte sstem cn be wtten s Wee te followng equtons cn be used to convet between specl nd Ctesn coodnte sstems tn tn sn cos sn sn cos
18 Mt Tnsfomtons: Ct. nd Sp. sn cos cos sn cos sn sn sn cos cos cos sn cos sn sn sn cos cos cos cos sn sn cos sn
19 ECE 7: Electct nd Mgnetsm Cptes -: Revew of Mtemtcl Essentls Cpte : ecto Clculus Dffeentl Lengt, e, nd olume Lne, Sufce, nd olume Integls Del Opeto Gdent of Scl Dvegence of ecto nd te Dvegence Teoem Cul of ecto nd Stoes s Teoem Lplcn of Scl Clssfcton of ecto Felds Note: Students wll not be tested specfcll on Cptes -. Howeve, te nfomton contned wtn tem wll be used n lmost eve spect of te couse. Knowledge nd slled pctce of tese concepts wll be equed to complete ll omewo, nd emntons tougout te semeste
20 Dffeentl Lengt, e, nd olume Ctesn Coodntes Dffeentl Dsplcement (dl) dl d Dffeentl Sufce e (ds) d ds dd ds dd ds dd d olume Dffeentl (dv) dv ddd Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl
21 Dffeentl Lengt, e, nd olume Clndcl Coodntes Dffeentl Dsplcement (dl) dl d d d Dffeentl Sufce e (ds) ds dd ds dd ds dd olume Dffeentl (dv) dv ddd Note: tem s used s multple to complete unts ssocted wt c dns of d
22 Dffeentl Lengt, e, nd olume Specl Coodntes Dffeentl Dsplcement (dl) dl d d snd Dffeentl Sufce e (ds) ds sn d d ds sn dd ds dd olume Dffeentl (dv) dv sn dd d Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl
23 Lne Integls Te lne ntegl s te ntegl of te tngentl component of long te cuve L Reques L be smoot, contnuous cuve. Te vecto, m be vecto feld component Lne ntegls e sd to be pt ndependent f te soluton of te tngentl component of s ndependent to te pt L ten wtn te feld. Te most common emple of pt ndependent ntegls used e wo (eneg) solutons ntegtng te foce ove te pt lengt, L. Pt ndependence of occus f = (te cul of s equl to eo) L dl dl b cosdl Fo contou of lengt, L Gves te cculton of ound te contou, L
24 Emple of Lne Integl Gven te followng equton fo F = [,- ] Fnd te lne ntegl of F long te followng pts between (,) nd (,) Pt : stt lne Pt : pbol d d d d d W d d / d d d d W d d d d d F W d d d F d d d,, Pt Dependent!!! Sow (on ou own) tt te cul does not equl
25 Note: Te sufce ntegl wll become te bss fo flu of te electc feld toug Gussn sufce n Cpte 4 Sufce Integls Te sufce ntegl s te ntegl of te vecto feld,, ove te closed contou, S povdes te net outwd flu of toug te sufce, S ds n S ds ds ds b cosds noml _ to _ te _ sufce S n ds
26 Sufce Integls: Fndng te e of te Sufce Sufce ntegls e double ntegls tt clculte te e of closed sufce Emples Ccle Clnde ( two ccles wt tubul sufce between tem L d L d d d d ds ds ds ds ds L S S S S S d d d d d ds S S S L
27 Sufce Integls (cont.) Unt convesons llow one to smplf poblems to ese clcultons Emple: Fnd te e cut fom te uppe lf of spee b te clnde Ts s te sum of te e of te spee wc pojects onto te ds n te (,) plne. Tus, we wnt to ntegte te e of te ds. Te geomet pesented sows te e tt wll be ntegted. Integton e Te desed e cn ten be clculted s: S ds / wee, d sn e dd d / cos dd dd / sn dd
28 e nd olume dv v v ddd S ds S S dd
29 olume Integls Used to clculte te volume, mss, centod, cge, etc of sold vdv v v dv v v s volumetc denst functon olume of spee v dv v snddd sndd d 4 olume of clnde v dv v olume ddd ( e L dd L _ of _ te _ ccle )( egt )
30 Del Opeto Del s vecto dffeentl opeto. Te del opeto wll be used n 4 dffeentl opetons tougout n couse on feld teo. Te followng equton s te del opeto fo dffeent coodnte sstems Gdent of Scl, s wtten s vecto Te dvegence of vecto,, s wtten s scl Te cul of vecto,, s wtten s vecto Te Lplcn of scl,, s wtten s scl sn,,
31 Gdent of Scl Te gdent of scl feld,, s vecto tt epesents bot te mgntude nd te decton of te mmum spce te of ncese of. To elp vsule ts concept, te fo emple topogpcl mp. Lnes on te mp epesent equl mgntudes of te scl feld. Te gdent vecto cosses mp t te locton wee te lnes pced nto te most dense spce nd pependcul (o noml) to tem. Te oentton (up o down) of te gdent vecto s suc tt te feld s ncesed n mgntude long tt decton. Emple,,,4, 4
32 Gdent of Scl () Fundmentl popetes of te gdent of scl feld Te mgntude of gdent equls te mmum te of cnge n pe unt dstnce Gdent ponts n te decton of te mmum te of cnge n Gdent t n pont s pependcul to te constnt sufce tt psses toug tt pont Te pojecton of te gdent n te decton of te unt vecto, s nd s clled te dectonl devtve of long. Ts s te te of cnge of n te decton of. If s te gdent of, ten s sd to be te scl potentl of Esl Poven Mtemtcl Reltons U U U U U U U U U n n
33 Dvegence of ecto Te dvegence of vecto,, t n gven pont P s te outwd flu pe unt volume s volume sns bout P. Te followng eltons cn be esl deved fo te dvegence of feld v ds dv s v sn sn sn lm Imge fom
34 Dvegence Teoem Te dvegence teoem sttes tt te totl outwd flu of vecto feld,, toug te closed sufce, S, s te sme s te volume ntegl of te dvegence of. Ts teoem s esl sown fom te equton fo te dvegence of vecto feld. ds s dv lm v v dv ds v poof s : ds s s ds s ds v v Dvegence of vecto (ed) out of specl sufce (geen) ttp:// dv
35 Dvegence Teoem Emples: d d d d d d d d d d ds S S s Suppose we wnted to fnd te flu of toug clnde of dus nd egt
36 Cul of ecto Te cul of vecto, s n l vecto wose mgntude s te mmum cculton of pe unt e s te e tends to eo nd wose decton s te noml decton of te e wen te e s oented to me te cculton mmum. cul lm S L dl S m Te followng eltons cn be esl deved fo te dvegence of feld n Inset: Feld lnes of (n blc) defned toug te closed loop dl (geen)
37 Cul of ecto () Cul of vecto n ec of te tee pm coodnte sstems dscussed n ts couse sn sn sn sn sn sn Ctesn Clndcl Specl
38 Cul of ecto (Emple) cot 6 cos 6 sn sn sn sn
39 Stoes Teoem Stoes teoem sttes tt te cculton of vecto feld, ound closed pt, L s equl to te sufce ntegl of te cul of ove te open sufce S bounded b L. Ts teoem s been poven to old s long s nd te cul of e contnuous long te closed sufce S of closed pt L Ts teoem s esl sown fom te equton fo te cul of vecto feld. cul lm S dl ds L poof L : dl s L dl L dl S L m n dl S S ds n ds Fo emple, see lne ntegl
40 Lplcn of Scl Te Lplcn dffeentl opeto s combnton of gdent nd dvegence opetos. Te Lplcn of scl feld s te dvegence of te gdent of n monc feld Lplce s Eqn. Feld wt ntenl dvegence Poson s Eqn. Te followng vecto dentt olds tue fo ll Lplcn felds govened b vecto, sn sn sn Lplcn
41 Clssfcton of ecto Felds ecto felds e unquel defned b te dvegence nd cul. Howeve nete te dvegence o te cul lone of vecto feld s suffcent to completel descbe te feld. ot flu nd otton must be consdeed concuentl. Tus we use te fou followng equtons to mtemtcll descbe nd pedct te effects of vecto feld () (b) (c) (d),,,,
42 Clssfcton of ecto Felds Te vecto feld,, s sd to be dvegenceless ( o solenodl) f Suc felds ve no souce o sn of flu, tus ll te vecto feld lnes enteng n enclosed sufce, S, must lso leve t. Emples nclude mgnetc felds, conducton cuent denst unde sted stte, nd mcompessble fluds Te followng equtons e commonl utled to solve dvegenceless feld poblems Te vecto feld,, s sd to be potentl (o ottonl) f Suc felds e sd to be consevtve. Emples nclude gvt, nd electosttc felds. Te followng equtons e commonl used to solve potentl feld poblems ds dv S v F dl L S ds
43 Clssfcton of ecto Felds () Retun gn to ou ntl sttement. vecto feld s defned b bot ts dvegence nd ts cul. Tus, we cn descbe n feld b evlutng te followng two equtons. v wee s denst component of te feld, v epesents volume, nd S epesents sufce e. We efe to te volume denst component s te souce denst, nd te e component s te cculton denst Hemolt s Teoem sttes tt n feld stsfng te condtons bove wose denst components vns t nfnt cn be completel descbed s te sum of two vectos. One of te vectos s ottonl, nd te ote s solenodl. In ou cse, te ottonl component s te electc feld, nd te solenodl component s te mgnetc feld. s suc one m completel descbe n electomgnetc feld usng te followng eltons F q E v v S S
44 (Optonl Mtel) ltentve ppoc to ecto Devtves Usng Scle Fctos (Optonl Mtel)
45 Devtves n Spce Scl Feld: scl functon, S, of poston coodntes n fnte dmensonl vecto spce. Fo most of ou puposes, te fnte vecto dmensonl vecto spce wll be lmted to dmensons (o pscll el spce). Defnng coodnte sstem {,, }wtn, ten llows fo dffeentton to be wtten s: ds d Nme of te coodnte sstem d d o ds Ctesn Clndcl Specl ds d d d sn 45
46 Devtves n Spce: Emple Deve te scle fctos fo te specl coodnte sstem ds d d d sn cos sn sn cos d d d d d d d d d d d d d d d d d d sn Hee we te onl te postve oots 46
47 Dffeentl Lengt, e, nd olume Ctesn Coodntes Dffeentl Dsplcement (dl) dl d d d d d d d Dffeentl Sufce e (ds) ds j ddê d d j ds ds ds dd dd dd olume Dffeentl (dv) dv j j d ddd d d d j d d Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl 47
48 Dffeentl Lengt, e, nd olume Clndcl Coodntes Dffeentl Dsplcement (dl) dl d d Dffeentl Sufce e (ds) ds ds ds olume Dffeentl (dv) d d d d d d d d d e e e d dv ddd ddd dd dd dd Note: tem s used s multple to complete unts ssocted wt c dns of d d 48
49 Dffeentl Lengt, e, nd olume Specl Coodntes Dffeentl Dsplcement (dl) Dffeentl Sufce e (ds) olume Dffeentl (dv) d dd d dd dv sn d d d d d d d dl sn dd dd ds dd dd ds d d d d ds sn sn Note: Dffeentl lengt nd sufce es e vectos. Dffeentl volume s scl 49
50 Del Opeto Del s vecto dffeentl opeto. Te del opeto wll be used n 4 dffeentl opetons tougout n couse on feld teo. Te followng equton s te del opeto fo dffeent coodnte sstems Gdent of Scl, s wtten s vecto Te dvegence of vecto,, s wtten s scl Te cul of vecto,, s wtten s vecto Te Lplcn of scl,, s wtten s scl sn,, 5 e e
51 Devtves n Spce: Te Gdent Defnton: Te gdent s vecto feld geneted b te poduct of te dffeentl opeto,, on scl feld,. gd ds ds d s suc, scl s smpl te poduct of two vectos. d ds cos ds Te gdent s te mmum te of cnge of te scl feld on te pont t wc te devtve s ten. It follows tt s te mmum te of cnge of d ds Fnll, s lws pependcul to equpotentl lnes of te sufce Te genel fom of te t component of te gdent vecto follows fom te defnton of te gdent lm ds d ds Yeldng te genel equton gd e 5
52 Devtves n Spce: Dvegence olume of te sufce: ddd Te dvegence of te scl feld ove te sufce s wtten s: dv j fo j j 5
53 Genel Equtons fo te cul n n otogonl cuvlne coodnte sstem cn be etended fom ou clcultons of te vecto coss poduct: Devtves n Spce: Te Cul 5 e e e e cul ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
54 Devtves n Spce: Lplcn Defnton: Te Lplcn of scl feld s defned s te dvegence of te gdent tt scl feld,. Geneled fom of te Lplcn of vecto feld,, n otogonl cuvlne coodntes 54 Lplcn
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