A Prelude to EE 4301

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1 A Prelude o EE 401 b Duncn L. McFrlne The Erik onsson School of Engineering nd Compuer Science The Universi of Tes Dlls Richrdson Tes 7508 dlm@udlls.edu Foreword This rher focused nd unfinished documen is wrien s he firs se of redings for UTD s undergrdue engineering elecromgneics course which is usull ken in suden s senior er. These wo chpers ken closel from m lecure noes inend o provide bridge beween he rher dvnced ebooks doped for he course nd he dispre bckgrounds of our sudens. We do mke cerin ssumpions bou our sudens bckgrounds; like ll ssumpions hese re no lws vlid. We ssume h he sudens hve some mhemicl muri gined from Signls nd Ssems course nd mulivrie clculus course. We lso ssume h he sudens lerned inroducor elecromgneics (o he poin of Mwell s equions in inegrl form) in sophomore phsics course; sndrd would be chpers -7 in Hllid nd Resnick s Fundmenls of Phsics ( rd ediion). Our gol in EE401 is o ech some obviousl useful elecromgneics o our engineers. To his end we concenre on he rnsfer of elecromgneic energ (nd herefore informion) from one poin o noher. Plne wve propgion rnsmission lines wveguides nd nenns hen re ll esil moived relevn opics included in his semeser course. To cover his mbiious schedule I hve found h necessr simplificion hs been o resric our choice of coordine ssems o Cresin () coordines wherever possible. (M discussion of nenns requires some mnipulions in sphericl world bu since hese come he end of he semeser nd re kep o minimum mos sudens bsorb hem wih onl he usul level of complins). This simplificion o reciliner coordines is I hink m bigges compromise in he course nd i should be considered conroversil. Drf une

2 Chper 1: Mhemicl Preceps Sclr Fields nd ecor Fields In mn of our elecricl engineering eperiences o de ou hve looked he emporl or ime vring behvior of properies or signls of ineres. For emple volge s funcion of ime m be wrien s or curren s funcion of ime m be wrien s I. These re emples of funcions of one vrible. In elecromgneics hese (s well s mn oher quniies of ineres) re lso funcions of spce. Here we review how o hndle funcions of hree dimensionl spce. Once we define convenion consising of coordine ssem nd n origin hree numbers re needed o deermine poin in spce. For emple longiude liude nd n liude will give he posiion of n irplne in fligh. These hree numbers m be hree disnces wo disnces nd n ngle one disnce nd wo ngles or hree ngles. In his course we will mosl consider he simples coordine ssem he Cresin coordine ssem for keeping rck of where poin in spce is. A simple coordine ssem is mos pproprie for he sud of compliced geomeries which chrcerie mn modern devices. Thus we m wrie for volge field (1) 1 The fc h his quni is now funcion of hree dimensionl spce mkes his field. The fc h his quni is sclr s opposed o vecor mkes his sclr field. To begin o visulie n emple of volge field hink bou consrucing lrge hree dimensionl circui. In describing he operion of his ssem i migh prove convenien o keep rck of he volge differen posiions in spce relive o some ground poin (origin). I hve seen srucure hese cpciors wired ogeher for high volge swiching pplicions. In such srucure hese cpciors were indeed b n code which suggesed heir locion in he room. This indeing scheme ws lso used in modeling he chnging volge in spce nd ime which occurred when swich ws closed. Oher emples of sclr fields include he emperure differen locions in he mosphere. A ho ir blloonis rveling from Plno Tes o Rockwll Tes migh cre bou emperures differen liudes differen poins long his roue. While we re dngling up in he ir we cn lso hink bi bou lighning. Cused b ir brekdown (ioniion) due o volge differenils beween clouds nd he ground he fc h lighning occurs in some plces nd no ohers suggess h volge in he mosphere lso vries wih spce. So fr we hve ried o moive wh we migh wn o keep rck if quni s funcion of hree dimensionl spce. Since vecor quniies re ofen of ineres (force veloci momenum...) I is resonble o wn o consider vecor fields () 1 M definiion here is rher mhemicl nd prgmic nd inherenl ignores he philosophicl view h field implies some sor of inermedir which replces he docrine of cion---disnce.

3 We hve picked he smbol becuse we re hinking bou he elecromgneic quni curren densi. Curren is vecor quni becuse i is he flow of chrged pricles nd boh he mgniue (number) nd he direcion of he curren is imporn. In Cresin coordines he vecor field decomposiion becomes (4) The hree componens of re hemselves sclr fields nd so we cn s h o describe he informion in vecor field we need o keep rck of hree sclr fields. nd re uni vecors (of lengh one) nd blnce he vecor nure of he lef hnd side of Eq. (4) o he righ hnd side of Eq. (4). Tking derivives of fields: The del operor he grdien he divergence nd he curl. You re ll well fmilir wih mnipuling funcions of one vrible nd hrdl need o be reminded of he uili of heir clculus. Derivives for emple led o opimiions which in urn m led o improved design. The lws of nure re ofen cs in differenil equions whose soluion m ield quniive undersnding of he phsicl nure of he ssem or device under sud. Here we look differeniing sclr nd vecor fields. The rules for king he ime derivive of sclr or vecor field re srigh forwrd emples of pril differeniion: d d (5) d [ (6) d Similrl king he derivive wih respec o priculr direcion in spce is lso srigh forwrd: ] = d d (7) d [ (8) d As prcicl mer when king he pril derivive wih respec o one vrible preend for he momen h he oher prmeers re consns. The following emples illusre he echnique of pril differeniion. Your choice of coordine ssem for he vecor decomposiion m be differen from our choice of coordines o describe he field s een. For emple in wriing ] = we hve used clindricl coordines r dependence of he sclr field. r r () for he vecor decomposiion bu Cresin coordines for he

4 Emple: Pril differeniion C cos( w)ep Cwsin( w)ep / w / w sin( w) 0 sinw w cosw We would rell like o ke some sor of firs derivive wih respec o spce nd hve his derivive be independen wih respec o he priculr coordine ssem used. In consrucing such derivive i mkes some sense o mi ogeher equl prs of derivives in ech direcion sin w The esies w o combine hese hree erms is o dd hem ogeher in some w. (Mulipling hem ogeher migh occur o us firs bu mong oher objecions he dimensions of he finl quni would be ou of line wih derivive.) If we dd hese hree erms ogeher in sclr mnner we migh lose rck of which erm cme from which derivive. Therefore i mkes he mos sense o dd hem up vecorill: (1) I is comforing consequence of he deep significnce of our mhemics h his quni hs rel phsicl mening: The direcion of his vecor field poins in he direcion of he lrges slope. The mgniude of his vecor field is he mgniude of his slope. This vecor field herefore is clled he grdien of he sclr field. grd (1) Consider for second he elevion of mounin in Colordo. This sclr field is funcion of (or longiude nd liude or b USGS convenion Norhing nd Esing ). If ou re hiking n poin on his mounin nd pour lile wer ou from our cneen he wer will flow in he direcion of he grdien of he elevion field evlued h poin ec opposie cull! (9) (10) (11) 4

5 nd is iniil ccelerion will be reled o he mgniude of he grdien of he elevion field evlued h poin. On of our difficul sks in elecromgneics is o r o visulie he vrious fields in spce. I is someimes convenien o hink bou he surfces formed b consn field vlues. To be specific here will be surfces upon which he volge will be consn. To be confusing we ll cll hese isopoenils. A surfce of mel would be n isopoenil since in sed se we would epec he volge o equlie ou upon he surfce. I is n ineresing nd imporn chrcerisic of he grdien h is lws perpendiculr o he surfces of consn vlue. Reurning o our mounin hike we pull ou of our conour mp nd find he locion where we re spilling wer from our cneen. The direcion where he wer is flowing is perpendiculr o he conour which runs hrough our locion. This gives us convenien w o clcule he uni vecor which is norml o surfce: I is well worh our ime o inroduce he del operor: grd nˆ (14) grd (15) We will rouinel use he del operor o signif some sor of firs derivive wih respec o spce. For emple he grdien of sclr field m be wrien grd (16) Emple: Tking he grdien C coswep C coswep / / w C cos w wep C cos / w 0 (17) wep / w 5

6 6 We now urn o king spil derivives of vecor fields nd we will mke considerble use of. Noice h is vecor. Couning nd vecor field we relie h we re working wih wo vecors. Two vecors m inerc ogeher b eiher he do produc or he cross produc. Therefore firs spil derivives of vecor fields come in wo flvors: del doed wih (18) And del crossed wih (19) Boh of hese hve useful phsicl menings nd re imporn in our sud of elecromgneics. Therefore le s lern he mechnics of evluing hem. is clled he divergence or div for shor. Evluing he do produc shows us he mhemicl mening div Noice h reurns sclr field. Noice h his new sclr field includes he digonl erms he derivive of he componen he derivive of he componen nd he derivive of he componen. is clled he curl. Evluing he cross produc shows us he mhemicl mening curl (1) Perhps he use of mri for king he cross produc of wo vecors is fmilir o ou. I find i convenien o remember where he minus signs go in he finl epression. Noice h de

7 reurns vecor field nd h his new vecor field conins he off-digonl erms which were no included in he divergence. To begin o ge n ppreciion for when o use he divergence nd when o use he curl noe wo rules of humb: 1. Use he divergence when ou wn he firs derivive o mch sclr quni. Use he curl when ou wn he firs derivive o mch vecor proper. (Bewre hough h ero m be eiher vecor or sclr).. Use he divergence when he prllel digonl erms re imporn o he problem. Use he curl when he orhogonl off-digonl erms re imporn. Emple: Tking he divergence. sin w 0 () Emple: Tking he curl. sinw sin w 0 0 () sin w w sinw sin 7

8 A phsicll meningful emple of curl s significnce cn be borrowed from fluid mechnics. Consider n rbirr flow of wer; he veloci of he wer pricles form veloci field v which is vecor field. If I hrow ino his flow smll wig or drinking srw here is ver good chnce h his wig will begin o roe. The is of his roion is in he direcion of v nd is 4 speed of roion will be proporionl o he mgniude of v. I is significn rrngemen if he curl of fluid s veloci field is ero; if v 0 hen he flow is clled irroionl. Also in fluid flow 5 he conservion of mss lw m be wrien s v where v is once gin he veloci field nd is he densi. Th he re of chnge of densi of he flow poin cn be wrien in erms of divergence suggess h phsicll he divergence represens flu. We will obin clerer picure of his when we consider one of he inegrl heorems below. Second derivives Second derivives of fields re priculrl imporn in elecromgneics becuse he propgion of elecromgneic energ nd herefore informion is governed b wve equion which s we will see is pril differenil equion which includes second derivive wih respec o ime nd second derivive wih respec o spce. So given he grdien he divergence nd he curl le us lis our choices for second derivives wih respec o spce. If I sr wih sclr field nd ke is grdien I cree vecor field. There re wo ws o ke he derivive wih respec o his new vecor field he divergence nd he curl. Thus here re ecl wo ws o ke generl second derivive of sclr field wih respec o spce: nd (4) (5) If I sr wih vecor field nd ke is divergence I cree sclr field. There is one w o ke he derivive of his new sclr field he grdien. On he oher hnd if I sr wih vecor field nd ke is curl I cree noher vecor field. There re wo ws o ke he derivive wih respec o his new vecor field he divergence nd he curl. Thus here re ecl hree ws o ke generl second derivive of vecor field wih respec o spce: (6) (7) 4 To be rigorous he vorici vecor w is 1 given b w v. The vorici is wice he insnneous re of rigid bod roion in flow field. 5 which is incompressible 8

9 9 nd (8) Le us eplore ech of hese five new fields o see which quniies migh be useful. Consider he firs quni he div(grd): (9) Which hs n esheicll nice inuiive form. Now consider he curl grd : de (0) = 0 Since he order of differeniion never mers. For ll sclr field hen 0. While his is useful vecor ideni i does no ield nd new field or quni which we cn pu o work for us. Therefore he unmbiguous nd highl useful choice for second spil derivive of sclr field is which we will cll he Lplcin of sclr field. Emple: Tking he Lplcin of sclr field. / ep cos w w C 1 / ep cos w w C 1 / ep cos w w C 0 / ep cos w w C

10 10 Th he div(curl) equls ero m be shown in mnner similr o Eq (0). = 0 () I urns ou h div(grd) is ver mess nd herefore of limied use ecep in problems where compliced meril forces us o mke use of i. Mos of he ime we r o drem up phsicl pproimions which jusif seing equl o ero. To be eplici: () Wh mkes his epression compliced is he presence of nd in he erm; he presence of nd in he erm nd he presence of nd in he erm. This coupling will limi our bili o nlicll solve differenil equions mde from. In his course we cn usull find reson o void i. Finll le us wrie ou he curl(curl): de (4)

11 11 Wh mkes his epression compliced is he presence of nd in he erm; he presence of nd in he erm nd he presence of nd in he erm. These re he ver sme objecionl erms s in Eq (). We cn combine Eqs () nd (4) in w which ges rid of hese coupling erms if we define he eminenl useful Lplcin of vecor (5) This vecor quni is ofen wh we men b second spil derivive of vecor field. Emple: Tking he Lplcin of vecor field. sin w w sin w sin 0 w sin 4 4 6

12 Two inegrl heorems Ne o king derivives i m be sid h doing inegrls is he mos eciing psime. Here we will show in Cresin coordines he vlidi of wo inegrl heorems of vecor clculus nmel Sokes heorem nd he divergence heorem. Their demonsrion will lso provide review emples of how o perform muliple inegrls line inegrls nd re inegrls. Sokes Theorem A rel ese mogul who bus New York Ci proper b he squre block wns o know how mn squre blocks here re in region defined b he sree corners sree 4 venue; 8 sree venue; 101 sree 7 venue; 54 sree 6 venue. There is surveor s formul which llows he quick clculion of his re wihou lo of hssle: Are The significnce of his emple is h i is nurl o rele he re enclosed b line o he coordines of h line. This formul is disn relive o Sokes heorem which eques line inegrl evlued round closed ph o n inegrl over he re enclosed b h ph: =88 dl nˆ da Le s demonsre his equli for ver smll squre of re. This squre is siing in some generl vecor field (8) whose cener is some poin. Our ph inegrl will begin he poin nd rvel in he direcion o he poin he lower righ hnd corner of he squre. From here he ph goes up o he poin hen bck over o before reurning o he sring poin. The four line vecors for he four ph segmens re dl respecivel. This couner-clockwise roue b he righ-hnd-rule gives us surfce norml vecor nˆ. The erms on he lef hnd side of Eq. (8) re hen ; ; ;. Noe h we hve mde use of he (7) 1

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14 14 smll nure of nd o evlue long he midpoin of ech line segmen. Upon compleing he do produc nd rrnging erms he lef hnd side of Eq. (8) becomes dl (9) Turning our enion o he righ hnd side of Eq. (8) n ˆ (40) If is sufficienl smll hen he quni m be ken s consn over he re nd pulled hrough he inegrl signs. The righ hnd side of Eq. (8) hen becomes da n ˆ (41) Coninuing our mnipulions of Eq. (9) dl (4) If we ke he limi s he lenghs go o ero we obin lim lim 0 0 (4) Which is now in he form of Eq. (41). Sokes heorem in Eq. (1) holds mcroscopicll for lrge closed curve nd n (!) surfce (fl concve conve ec.) bounded b h perimeer curve. Our demonsrion is microscopic nd even if we resric our discussion o Cresin coordines is onl building block of he finl proof. To his end consider wo djcen squres s shown in Fig. ech of which is similr o he one which srred in our bove nlsis. If we wrie down ll he line inegrls s we did in Eq. (4) we noice h he shred side is wrien wice once for dl posiive he oher for dl negive. The conribuion of his shred side hus vnishes due o he vecor nure of our heorem. Onl he si eerior line segmens mch up o he (now doubled) re. B eension we cn keep dding on smll squres unil we hve mpped ou lrge no necessril fl re. All shred sides will cncel nd he line inegrl bou he perimeer will ield re informion bou he enclosed surfce.

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16 Emple:Frd s lw in differenil form Frd s lw of elecromgneics m be wrien in inegrl form s: E dl B nˆ da (44) The closed line inegrl on he lef hnd side of he equion defines he limis of inegrion of he righ hnd side. Appling Sokes heorem o he lef hnd side we cn conver he line inegrl o n re inegrl. E dl ˆ B nˆ da (45) E nda The re in boh surfce inegrls mus be he sme so we m combine hem under he sme limis of inegrion Which will equl ero if B E nˆ da 0 (46) B E (47) This is Frd s lw of elecromgneics in differenil form. Divergence Theorem Our second inegrl heorem is he Divergence heorem. For n rbirr vecor field d nda ˆ (48) The surfce of he volume on he lef hnd side of Eq. (48) gives he re limis of inegrion on he righ hnd side of Eq. (48). We will demonsre his heorem in Cresin coordines in mnner nlogous o our demonsrion of Sokes heorem bove. Our building block is smll cube wih sides nd. The cener of his cube is he poin nd i is ligned nicel wih he -- is. This siuion is shown in Fig.. For his cube he lef hnd side of he heorem m be wrien d ddd (49) We hve ssumed h differenil volume. is smll enough so h does no significnl chnge over he 16

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18 18 Figure There re si erms on he lef hnd side of Eq. (48) one for ech fce of he cube. Evluing in he middle of ech fce nd grouping b he norml uni vecor of ech fce we hve da ˆn (50) Mulipling ppropriel b one nd king he limis gives da lim ˆ 0 n lim 0 (51) lim 0 Which is he sme s Eq. (49). The Divergence heorem in Eq. (48) holds mcroscopicll for lrge closed surfce nd is enclosed volume. Our demonsrion is microscopic nd even if we resric our discussion o Cresin coordines is onl building block of he finl proof. Our eension of our demonsrion o he mcroscopic world is nlogous o he rgumen presened for Sokes heorem. Consider wo djcen cubes shring one side. If we wrie down ll he surfce inegrls s we did in Eq. (51) we noice h he shred surfce ppers wice once for nˆ posiive he oher nˆ for negive. The conribuion of his shred surfce hus vnishes due o he vecor nure of our heorem. Onl he en eerior surfces mch up o he (now doubled) volume. B eension we cn keep dding on smll cubes unil we hve mpped ou lrge volume. All shred surfces will cncel nd he surfce inegrl bou he eerior will ield volumeric informion bou he inerior. Emple: Guss lw in differenil form Guss lw of mgneism m be wrien in inegrl form s: 0 ˆ da n B (5) Appling he divergence heorem o he lef hnd side we cn conver he re inegrl o n inegrl over he volume enclosed b he sme re. 0 da B (5)

19 Evluing his inegrl in ver smll region round poin gives B 0 (54) h poin. This is Guss lw for mgneism in differenil form. From our discussion of he divergence heorem i follows h he divergence of vecor field m be wrien s lim nˆ da 0 (55) In fc his is ofen he w mhemicins define he divergence. Phsicll we cn s h he efflu (ouflow) of per uni volume poin in spce. Clindricl nd sphericl coordine ssems This secion will evenull be wrien is Generl vecor ideniies Equions (0) () nd (5) re emples of vecor ideniies ruhs which will be used in derivions hroughou he elecromgneics course. You should become fmilir wih hese epressions bu don spend hours memoriing hem. The re lised here s he re in our book for es reference. f nd g re sclr fields; A nd B re vecor fields. f g f g (56) A B A B (57) A B A B (58) fg fg gf (59) A A f f A f (60) A B B A - A B (61) A f A f A f (6) A B A B - B A B A - A B (6) 19

20 f f (64) 0 A (65) f 0 (66) A A A (67) A B A B B A A B B A (68) A B C B C A C A B (69) A C CA B A B C B (70) Problems 1. Appl Sokes heorem o Ampere s Lw in inegrl form s per emple o show Ampere s lw in differenil form H dl D nˆ da (71) D H (7). Drw he nlog of fig. for he divergence heorem.. Appl he divergence heorem o Guss lw for he elecrici in inegrl form o show Guss lw for he elecrici in differenil form 4. erif h he vecor ideniies re rue in Cresin coordines. D n ˆ da (7) D (74) 0