Maxwell's Equations PDF

Hoxcell Eauckions Euoon of ontinuil forepretnta the tinme vareng Equa ion of onknu ej resens he las o A onsorakon of choug 7ha is he chogg fowing sote thirough a closed sunfoa in out Ci-e Cuanend) thenake of dereQse of honge yolume is equal to within he volumei...

Hoxcell Eauckions Euoon of ontinuil forepretnta the tinme vareng Equa ion of onknu ej resens he las o A onsorakon of choug 7ha is he chogg fowing sote thirough a closed sunfoa in out Ci-e Cuanend) thenake of dereQse of honge yolume is equal to within he volumei a closed susoce I Cum ont Pleuain out hrout 193 where in a Noleum. oíthin he -dQ RoBe of decnerse dt hVolume Cunent densi y and J- Qondchion whene J= value of I ond Q in Szslihuting the loo -( df dv Gau sss bie gonte Theorem to Li. Now to Of chave eepaion to |CT T)dv =-Jdv co teoo. Volune integta hein ineyondo Gyual af this s eLakien Airne vay Eqaion of ontinuby or steael coninuiy fRetd. Cuvnts cloe, not, Yary corth Hrne foy s tealy cuntns. that is his e s h e e of  Moxwell's equaions on Gauss's law in Elecostali YMOell's finst egpaion skodes Hhat he totl elechic tlhx PE passin1 closed hypo th elrcal salace is a througttines sunface :. the netchangi encloscd qthe or jo.ds -6E - bisaconnent vecker. where b over a volume V and p chaye be distei buted Let the choge he be the vleme ohounge den sitymub 4henelore Maoell's a The total elec dis placemen thmough toa closed hypotneal suací enclos i21 the totaf charge Cnclostal oithin the lene Gcuuyy divenance heorem at he l HS As ho tulo yoleme thein Gouss's Qoo fon agnaksny , slcdes thak the total, mogne Hc fluxhm Omergia xo orouh closed sunface s xeno he nt magnalc fluy eneng'ghregh any clgsed sunface bho alales ha he magnal honopale does noth eist. Gouss's iogunce As B.ds o Diferontia Porem  3) Maraell's4hid eqaidh of Elecomgnehe tnduchon. magneie magnelic Pl fl inkost hat, oheneser coheneuen ele ctom ofive por (e Cemy Hstates a) chcuit changes hen indlucad he Cintletf- Jhia incuced emf emf mayrelke f nlmees. set up in Qants so o inucod emf is magnikule of inked oit the Cincuát BThe B) of magrolic fly -dm of induced where d8.ds atofe whichtaley which that the bechuge of Lenz's law ao olrechon is such e sjgn em sek epa cuwnent io it oppos es the incueal eMect preduted Hho yhe mage it c defciy is he closed b Cause pro emfates that emt dlbe definahion of consertvalve eleclric,Pield generata of the non- Ohe bateny f6. dl-.ds Ma el4 he emf anoun a closedl poth is Capal to the time 0f, he magnate osplaement thug ay dexi yatve boundled b he path. sukoce we stoke's Heem 4o LHs of e ad.ds Jx¬ ).ds when hein Cue foo suttace dB TXE hus. iReia Qren of js the Moelyy hico ef.  ime fot S)Qnd ¼7--3f coninuyD} of e to Toking bot On - (CTxR)di S heorten tstoke Acortolng total Hmes magnalic indegealof klo to ant clgron indu felAmpene' d s Actorclin 4o Proof the Cincuttal o oliffetenial for line anaitLauoiy Cineuital Moxals Dwhere For me dependen eld e nel vecor C The e mugt add gome fe that Mooell suggostd Cond On R s -bisplacomentcuieal d bofh sido) of a@) dlesiy) (J-Conluchio S Jakny dvesgnte D Oing 6) in ) we -6. - fo Slokes hereem 4HdL C)ds ( The mag Une euvehve the Condachion eleon pley she ime eledic dsplacnad thromyh ay bowndaol he pa SemmUy CA ) H- J T F X - J+e V CIr Dds Mar coell eofy fon 9tatc ell, ularcwes o fiee Spare In stae eld vecte b,E, g a elo not vay iHh tine e Consueon Ceenen SO Sa D 68.ds o B.ds =o B ds  Displaterjent Cuseat and R-c nekoorh coith a volloae oet the Cunnek htough Ondl cdv thnough the apaci tor, nat flow prackte chinbnt doesMows one elechrocle / out of Cuero,t thef into the Bat,' he Cuont hot flows the packon pals 1he net efec is a i! Ahe thene is a Ghene olecyode. conlinis Copactot. the pats Cuant Moalon hnagh he resjs br. Bact Cluun hroLgl the Curont Plocoing Rom obve resutt, Conouckion Cunent - Hence esistor i ,known oy obey chm's Ooud. aohile he Cuot Plocalh dislacomont cuont. apocky commo ekokic Reld in side each elemet he eleckic elomont dl ylded b. lathenako prvof; As'v' aOnoss the Equals he vollaaA EY o bohere ArCross - Sechona Ortea r Conduchvihy of eajslonte elemeat Capacitone of a A C=  A Cv-d) A OE A 6,de o - G Elecic ohee ad se, lohenee moves that diploonet efechrie Aield E ie choy in neke pootiaf Jhere coill be of elechromg A Cund te Polenkialy hSoalor One that Moxel's feld eas we know V-0-O t of cand of a veckon is civengence twe knou that Ke10 i e davognc yCTxA ) =0 oloo Conpoioy nece where is kooun ay veckor pofenlil magneie in ) (yxa') )+a -0 + Vx CO)-0 4)  )-¼ Cy¢) Ond in (y we (l0 dt cwe (10) U 6 in VxCXA) dt Prom Now csin the valee a 2 from - in o salar poenial. aad subha chns AEao adoiy  tronsformoion Geye ie gvXROitN scalan öt Consider traasfoemalon Using in Oeget ) TC91#) Iransforaliny  Lortente Gtaug expresed to Reld Cgolions kooo that Marwel: of elechonagnic potendiab to)ond 4 One gen tom 2 842 Prom eg? O ond oe note, the Oceatren te of both poBenial and ? in ench, equalony tie ty one inkeedocled thy intocloclyj NOnishes providedl This condifion is called orentz C Conolthon an eohi bids iede indorralaionshlp of elechvo maqnake petetaly a thi's Cofal Hon 3).the gauge s koown a Lonontz Gaupe Sing in O an @ we Nosv phOse veocy of ave phose (?- )R'- - 5 ’ whexe J operaton  46 Gouag Acond ) in we od Lonedzondition ieiovL. aeo ie when Coulomb on Thonsvence Gouao ouaions eypressel thot 14axaeht feld feerns of cala Co) Onl veeton polenthal CP) one pojoniol Ond Couloms Giouas condhian q.A:0 in v 042 9ypened os R C1): Cooxdin ales of Souyte point ) R cliotonte ber sad f dy Ak trau element at s  3 dvl Sueh that No iy vector calcuusewe have R du' from 4C7) © CUs Solve jst fean of RHs ofa dv! dv R Coulonh 6GOugk CvA=ots hereino indofaee bekoeo difloret mel Boundoy ondition ah boundany coniions sPed b Hrael's e ie hihnterlaa eponalih. lebtc ond magroic felds al hehe elechiomaqnalic eechomazpaic Red el be obained eo media cah Qd salsR gome yecho B, , tes mecl interloce bet? at the Dislacam erd vechoC Boundaay Concikon for elechic ConoHon that the elkci'e displaconst Nediun) n, The inteface seperati8 Co) must saksß af at the Intere the boundar oniion Sy o obtoin for B, fel us imagine a pill box Medium beas Che Surfare at the inteatbceHaq: 2 mecio ay shoun in box is Conpesed or 4 seuafaces s,,s, S3S a a, The pll NS,- Toprmost Cyper) sunfore of pl boy S = Lowermost sunfoce o pperpil boy Suaface o cyincleeSe S Quloe ol es, Cuaed Sutface o! lawer oalinder Cyide dd B Cad be he eleckic obsplaonat veclena in meblilm and imeolium(repe chvelyosibo fhom Maxwel's 1s e f : above eq ove the wolne erclosed pil box  to Giayes's dwne Jheo Acrodigj ( S a coe Conpainny Sg cotll be no ontai buhon Aho,then thene Now i Ond Sy The Jilime inlegalSurfoce ond hente RASCanges to Snfece integal olme, chan daitcs) ole Jods : ) Dn melien where Conngonent op 6in mediun Qy D=Nonal romolonposesls of eleckeie disalacomèns vechoy C5) Thus noruno! chanoes not Contneoes sis 00t aonoss the inenface eqAal to the cnd chag Cun omount wich jo Boundo ConiHon formoghetc Peld inclucioh c Hon hat the magnehie feld incuchia, incluc The ines must ks a he conoikon for is Ocaled boundar meolia bocundoy corci ion or &, laf ces To obtoin at the inken rce bol9 pil box ihe sudfhed Consiclex a ttaleaface is Coy2ose ol 4 The pil box Cuppe«) sufoce of pil box S, Tep most Scunfoce of pill box S,lowermost Ban of Sg- Cuaned surface lowe Sy Cuaned Sutoce of let B ond B be theneolemmcgnake o and meium reapeckvely Aeld indu cHon in Prtom Maxeoel's eg bn Magake Qeld inolchon C®) cwe hawe pill box Qboye e over the volee enclosed by toe hawe heorany we hawe, eondiry to Gouos ivegae, Now S S2 be no conri bukon Nowif Ah, the, thene toill sunfoe Cn S, cppreaches, Sunlaces Sa Qnd Ond entine Suface fakes Suylace he ehape entine su Pace 52  A A B Bin B okin ol maghdie Retd.inda Honce. normal compenent acioss he inte fbce bes' is ontnuous lon elechr'e Reld cE) Boundory Cond'ion ConelHon that E must saf Mediin ) The theintenfoce seponaliyy the ute at bounday Concl ion 2 meoia is calld fon do onden to get bounda conclion En Console, a omal neckanglan nelangdan fon E, ACCD A at indeface closed loop Melium the $untace uea of et s be 1 is clean Closed loop.Ae coA foun fgune SE be he elechrie eld in meli meoliu,  ee haue Ecl ag both sidlas aver the sealbte of he oap coe gas xE) ds -(e6ar S he0Mem cwe houe No Congpoiiy Co fA dt ColxAh) -OB Colx nA)SU we hawe nusfanglal Conpon nds or oe conhnud us acros the nder Pate bei? meda  S 3 c+y. dr coso' 2h Hn + QA DQ A PC d ABCA 4 have oe kes-theoram we sto Acconcing to ABCDA closed bowndad bySufaces the ove sicle, jntegnafiny both given is intoity fleld Por Moraets eg A medi o un inkoniky in Aeld MediunO mognelic mognelic he be Hand Hiolot on for ion ond boundoy obtain toonden Jo &lesepo on Hon condibouncdoy callgd, aler is he otatisfy s CH)inensiy Relmagneli' d e thatcondton The inteosit feld eke mggn CondiHon fot Boundary  then face indeqra on R:S chong La Nouto if bh ’o delihc in ewe will se Omoan Cunat Bne integal and toe sfoce Cirtnb cogik e place of Hence iDnalal Conponot of mge acHoss he 1he difforant Con i nuo media. theonem and vecho Pr coaNes gprocad in ohen elecroaqreic then Some Bromhein Sowtce a is 4anserned from the Boe to enetg this negion The nate of Plou ofenogg pe by a vecor Qed alled polat is qlven Poynlirg vecer a a elatel oiH elec-dc Aold CE) od raznoie Aeldi point y follouos at hat point y Pning veclor C'): Plo of enengg PosQe Time XArea 3I unit o Toule Sec xm2 Dimengiana fomala of yM'e3  ace) Congoidy JcExn)dv Acondsg renMt fheo V Talay sidey both intkeyal on yolme -CR Subtnachng.o e fhom on prodack,o Jalung. do sigides both Moreel's4e to Tabi 8ides bot, withproduck H oot Taknq 34 Maxeoel's Ho gunfaceS closed bound a by Voleme v medibm region centotn a of Consicen a entéring ane waves em which negi 'non he inReld and Pield elecle in andchages displacing he neCats o eneogg nelates JE the buolonp onh wth CF') veetor 7heoreny loynting  )dv -6cx®)dFce)dv Jte1 E)+ The above expression epeera n theorrem dv= mal Cnea XSmall dv= dsd 4h *Gas)da. dt hawe kleqno into, sommafio Jn 4he lout step b couse in COse of di'sete Should se Sunmakon. elechik and 7he lonealz force eperiented hy ckaige 4 in agnehc freld amount oforkdone Hhis fomcein  do aË+4cxB)] di xo') *Edt + 4cdx ) -de adt qE-d? clo dt dw dt for atl the do chog dt we hawe Coapoiin do dt Thu s 1s ten on RHs of represents he mate ot which onh is done ewithin volume v Pield in the cag 5d feem on RHS of q, Sntepnetahion of Thus 2 team on RHs of eq ) repteses tmeo ate of inCreane oof eneng4tored in elecic and ielc coiHhln vot manelic Helda volume.V Jnterpnelakon o P the the tern on LHs o Plow iinte volune Surface s.