PEP 332: Mathematical Methods for Physicists. Math Methods (Hassani 2009) Ch 15 Applied Vector Analysis. (1) E = ρ ϵ 0 ; (2) B =0; (3) E = B (1) ; (2)

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1 PEP 33: Mmticl Mthods fo Phsicts Mth Mthods (Hsni 9 Ch 5 pplid c nls doul dl options mgntic multipols plcin s ( E d Q ϵ ; ( d ; (3 C E d dφ m dt ; (4 C d µ I ( E ρ ϵ ; ( ; (3 E ; (4 µ J µ ϵ E

2 Φ( (53 ov how C R $ # wh Φ5 potntil om givn dimnsions Ch Doul Dl Options Th s Options 5 Doul Dl ( d d## d d (c how dimnsions s om om C R dq( W op itslf Φ( K vc (54 cn m smdiﬀnt comtions pplid c nls Ω Th s w om dimnsions pplid c nls dict diﬀntition cn il vif 46 Evlut ldimnsions tgl s om om (c how (53 conjunction (5 umticll phsics olvg s pdictd tncimplis lcmgntic (f (5 & % & % olvg s pdictd tnc lcmgntic sm othphsics lcsttic gvittionl filds consvtiv 3 ( pplid c nls wvs fct s wvs popgt t pplid spd light (86 H c nls wvs fct wvs popgt lighf(86 H (49 stts s consvtiv vc t fild spd gdint its poposd phnomnon light fo n lcmgntic phnomnon 46 Evlut l tgl fom oig pot ( : poposd phnomnon light fo n lcmgntic phnomnon potntil (5 s on o h if fild In simil fhion w cn dictl vif followg idntit: phsics olvg s pdictd tnc lcmgntic lft Kg s Collg ondon spg 865 tund PEP 33 Mmticl Mthods fo Phsicts & % & % ( long stight l jog pots; phsics olvg s pdictd tnc lcmgntic lft Kg s Collg ondon 865 s tund gdint function n it consvtiv W com (86 3tips spg cn ( s wvs fct wvs popgt t spd light H h cotth stt H md piodic Cmidg luctntl ( long pol psg though pots wll pot ( (55 wvs fct s wvstips popgt t spd light (86 H h cotth stt H md piodic Cmidg luctntl sttmnts on sg Homwo 5: Polms fom Ch 4 5 (Hsni 9 poposd phnomnon light fo n lcmgntic phnomnon ccptd oﬀ Cmidg fist Cvndh Psso Phsics ( fom n oig fom pot ( : ccptd n oﬀ fom Cmidg light fist Cvndh Psso Phsics poposd phnomnon fo n lcmgntic phnomnon du Wdnsd / 87 H dsignd Cvndh lo st it up lft Kg s Collg ondon hlpd spg 865 tund PEP 33 Mmticl Mthods fo Phsicts Empl 5 Mgntic filds cn lso wittn tms so-clld (c Is consvtiv? ( long stight l jog pots; 87 H dsignd Cvndh lo st it up865 tund lft Kg s Collg ondon hlpd spg h cotth stt md piodic tips itscmidg luctntl vc potntils To fd H pssion fo vc potntil w sustitut o 5 vc fild consvtiv (i cul vnhs if ( long pol psg though pots wll pot h cotth sttfild H( md piodic tips Cmidg luctntl cos s consvtiv? 47 Is vc (5 5:oﬀ mgntic fildgdint tgl: Homwo Polms fom Ch 45 (Hsni 9 ccptd n fom Cmidg fist Cvndh Psso onl if it cn wittn scl which c Phsics function ( ccptd n potntil oﬀ fom Cmidg fist Cvndh Psso Phsics If so fd its $ % du Wdnsd / 55 Polms 87 Hconsvtiv? dsignd lo hlpd st it up ##Is scl function Cvndh fild s potntil # # (c 55 Polms 87 H dsignd Cvndh lo hlpd st it up m dq( v( ( vc fild 3 givn m dq( v( 48 fild lw consvtiv? zo Ω Is Ω 5 how cul gdint function cos s 47 vc ( ' ( * Empl 5 Th lcsttic gvittionl filds which w dnot 5 how function lw zo Φ cul gdint (z/ z If so fditspotntil givn nqution fom gnicll W wnt t out tgl Howv coss poduct pvnts 5how cul vc lw zo 5 how dq( th cul vc lw zo dict pull out o w nd gt ound tg mnipultg 48 Φ vc fild ( givn wh constnts ( componnt K componnt 53 if (59 3 cn Usg scond ltion (4 w wit 5 how cul gdint function lw zo ( * Ω 53 if '(59 componnt componnt Polms Polms ( Dtm wh not 5 how gdint function lw zo Φocul consvtiv (z/ ssumg z dfd which it 53: gion gdint function contctl ( Fd potntil if consvtiv 54 Povid dtils Empl 5 how cul vc lw zo & '( 54 Povid dtils templ 53: zo i gion h gdint fit which no pot v( 5 how cul vc lw zo ( Comput th componnts vif (5 v v 49 Th componnts vc givn componnts fild ( Comput th vif (5 wh Φ constnts 53 if (59 ot componnt f f f componnt show ou pssions givn ( Clcult z 53 if (59 componnt componnt f f f show ou ot pssions givn ( Clcult ( Dtm wh o not consvtiv 3 3 z z z s v( z mpl ( Fd potntil ifitempl consvtiv mpl 54 Povid dtils 53:

3 Ch 5 Doul Dl Options Empl 5 Mgntic filds cn lso wittn tms so-clld vc potntils To fd pssion fo vc potntil w sustitut (5 mgntic fild tgl: m dq( v( ( { ( } Ω 3 m dq( v( Ω W wnt t out tgl Howv coss poduct pvnts dict pull out o w nd gt ound th mnipultg tg Usg scond ltion (4 w cn wit ( v( {}}{ ( v v ( {}} ( { ( v( ( (f f (f (4 ( ( W not v cus diffntits spct ( z which v( dpndnt ustitutgthltltionpssionfo w ot ( m dq( v( dq( ( v( m Ω (56 Ω

4 Ch 5 Doul Dl Options m Ω dq( ( v( ( m Ω dq( v( (56 wh w hv tn out tgl sc it diffntits spct pmts tgtion Ω sumd dpndnt ( z Th vc potntil dfd lt l which w wit dq( v( m (57 Ω If chgs confd on dimnsion so w hv cunt loop n dq( v( Id (57 ducs m I d (58 n impotnt consqunc s (56 (55 (59 c vc fild ltd dnsit its souc w conclud no mgntic chgs

5 Ch 5 Mgntic Multipols mgntic dipol momnt multipol pnsion mgntic vc potntil givs: ( m I ê d mi d mi }{{} ê d sustitutg qution 3 qution 58 ( d m I (58 Nt us omil pnsion (5 ϵ ϵê ê α Up scond od ϵ thilds { ( ϵ ϵê ê 3 8 ( ϵ ϵê ê } { ϵê ê ϵ [ 3 (ê ê ] } ê 3 [ 3 (ê ê ] (3

6 Ch 5 Mgntic Multipols }{{} mgntic dipol momnt To fcilitt clcultg scond tgl choos Ctsin coodts oint ou so ê -diction Dnot tgl Thn ê d ê d d (ê d ê d ê z dz W vlut ch componnt sptl d d ( nd gng cus gng nd pots loop cocid Now consid idntit ( d d d( ( nd gng (5

7 Ch 5 Mgntic Multipols mgntic dipol momnt n nlogous idntit volvg z Fo -componnt w hv d d d d d }{{} ( d d }{{} It follows (5 ( d d mi mi ( ( d Ths dd up nothg ( d z d d ( d ê z m µ ê z ê z wh w hv dfd mgntic dipol momnt µ µ I d (5 similclcultionwillild (

8 Ch 5 Mgntic Multipols mgntic dipol momnt similclcultionwillild z mi z mi ( d ê m µ ê Thfo ê ê z ê z m ( ê µ ê z ê z µ ê }{{} µ(ê ê z ccul Rcllg ê ê z ê ouchoicointtion ê ê wfllot mµ ê mµ 3 (5 Th stig smlnc twn vc potntil mgntic Empl 5 t us clcult mgntic dipol momnt cicul cunt dius Plcgcicl-pln its cnt t oig w hv µ I d I (ê ρ (dϕ ê ϕ I π dϕ ê z Iπ ê z o mgnitud mgntic dipol momnt cicul loop cunt poduct cunt loop Its diction ltd diction cunt ight-h ul

9 t t t t t t If so fd its potntil 55 Polms poposd phnomnon light fo nh lcmgntic 55 Polms (d how ( f ( (f ot (5 dfd phnomnon fom oig pot : poposd phnomnon light fo n lcmgntic phnomnon 48 vc fild givn lft Kg s Collg ondon spg 865 tund PEP 33sum Mmticl Mthods fo Phsicts pssions pis pnss Ch 5 Mgntic Multipols ( long stight l jog pots; 5 how cul Collg gdint function lwzo lft Kg s ondon spg 865 tund ' ( * 5 how cul gdint function lw zo Φ H md piodic (5 h cotth stt tips Cmidg luctntl (z/ ( long pol psg though pots wll pot h cotth stt H md piodic tips z Cmidg luctntl 5 how cul vc lw zo Homwo 5: Polms fom Ch 4 5 (Hsni 9 ccptd n oﬀ fom Cmidg fist Cvndh Psso Phsics 5 how cul vc lw zo sptl convnint coodt 55 tg ch componnt d ( ccptd n oﬀ fom Cmidg fist Cvndh Psso Phsics wh Φif / constnts du Wdnsd 87 H dsignd Cvndh lo hlpd st it up sstm show its tgl ound n closd loop vnhs 53 (59 componnt componnt (c Is consvtiv? 87 H dsignd Cvndh lo hlpd st it up 53 if componnt ( Dtm wh o(59 not componnt consvtiv 56 Rcll tl mgntic foc cunt loop givn ( Fd potntil if it consvtiv on 54 Povid dtils Empl 53: s consvtiv? 47 Is vc fild ( cos 54 Povid dtils Empl 53: (o Comput th componnts vif (5 49 Th componnts vc fild givn If fd its potntil ( Comput th componnts vif (5 FI d ou ot pssions givn ( Clcult f f z f show z f show givn ( Clcult 3 3 ou ot pssions f f z z s z 48 vc fild givn mpl mpl foc how tl on (cunt loop loctd homognous mgntic 5 how cul gdint function lw zo f givn (5 (c if ' * ( Dtm wh consvtiv o not 5 how cul function lw zo (5 (c if f Φ givn gdint (z/ fild zo (d Ifhow ( fd its ot dfd z H ff (f (5 ( it consvtiv potntil (d how ( (f ot (5 H dfdzo 5 how cul vc lw Div sum pssions pis qution pnss 5 how cul vc lw 57 diﬀntil fom s lt fom co- zo sum pssions pis pnss wh Φif constnts (5 tgl spondg fom 53 (59 componnt componnt (5 53 if (59 componnt componnt ( Dtm wh o not consvtiv convnint coodt 55 tg ch componnt d sptl 58 ttg s show mgntic fild stsptl convnint coodt 55 tg ch componnt d ( Fd potntil if it consvtiv 54 Povid dtils Empl 53: sstm show its tgl ound n closd loop vnhs it o fis sm wv qution lctic fild In pticul 54 Povid dtils Empl 53: sstm show its tgl ound n closd loop vnhs ( Comput th componnts vif (5 popgts sm spd 49 Th componnts vc fild givn ( Comput componnts vif (5 56 Rcll th tl mgntic foc on cunt loop givn 56 Rcll foc oni(ωt cunt givn f tl i(ωt f mgntic z f show ouloop ot pssions givn ( Clcult wh i E 59 Consid E E f f z f givn ( Clcult 3 3 show ou ot pssions z z s mpl z ω constnts Th E so dfd psnt pln wvs F I d mpl F I d f givn (5 (c if movg diction vc ( Dtm wh consvtiv o not (5 (c if f givn ( how stf s f if:(5 how tl foc on cunt loop loctd ot pc homognous mgntic (d how ( f (f H dfd ( If it consvtiv fd its potntil howhow tl foc on cunt loop loctd ot homognous mgntic (d ( f (f (5 H dfd fild zo sum ( pssions pis pnss fild zo ; ( ; E sum pssions pis pnss (5 ωqution fom co57 Div diﬀntil fom s lt (5 ω ; (4 qution E E fom 57 Div (3 diﬀntil s lt fom co c spondg tgl fom spondg tgl fom 55 tg ch componnt d sptl convnint coodt Polms Polms

10 Ch 53 plcin dfition plcin Th gdint n impotnt fquntl occug op clld plcin: (f f f f f z (53 govnd chödg qution quntum mchnics W dcuss on sitution which plcin occus ntull Th sult mpl ov Thom 34 cn comd ot n impotnt qution lcsttics gvit clld Poson qution: (Φ 4πKρ Q o Φ( 4πKρ Q ( (54 Th ptil diffntil qution whos solution dtms potntil t vious pots spc In mn situtions dnsit gion tst zo Thn RH vnhs w ot n impotnt spcil c ov qution clld plc s qution: Φ( (55

11 Ch 53 plcin fluid dnmics fo volum fluid oundd sufc outsid pssu p noml potd volum F pd pd Th shows p foc dnsit whosvolumtglgivsfoc If dnsit fluid ms lmnt h vlocit v if o focs ctg on fluid (givn foc dnsit f n tl foc givn : ρ(dv/dt d pd f d th holds fo n volum pticulfonfitsimlvolumfo which tgls com tg Hnc scond lw motion fo fluid ρ(dv/dt p f (54 Th tl tim divtiv vlocit

12 Ch 53 plcin fluid dnmics fo fluid ρ(dv/dt p f (54 Th tl tim divtiv vlocit Th tl tim divtiv vlocit dv dt v v d dt v d dt v z dz dt v ustitutg th (54 dividg ρ ilds v (v v p f ρ (v v (55 Th Eul s qution on fundmntl fluid dnmics Th foc dnsit f Eul s qution usull gvittionl foc c gvittionl foc on n lmnt ρ d gρ d whg gvittionl ccltion (o fild gvittionl foc dnsit ρg (55 coms v (v v p ρ g (56

13 Ch 53 plcin fluid dnmics v p (v v ρ g (56 Empl 53 In hdosttic situtions unifom gvittionl fild fluid not movg (56 coms p ρg if g ngtiv z-diction n p p p z ρg Thus pssu dpndnt dpndsonlonhightz W sum fluid (ll liquid compssil mng its dnsit dos not dpnd on pssu Thn tgtg z qution givs p ρgz C If liquid h f sufc t z h wh pssu p nc p ρgh p p ρg(h z

14 Ch 54 s s fou tgl fom ( E d Q ; ( ϵ d ; (3 C E d dφ m dt ; (4 C d µ I ( Guss lw: lctic flu though closd sufc tl chg volum ( Guss lw fo mgntm: cospondg flu fo mgntic fild zo (3 Fd s lw: t chng mgntic flu dφm/dt connctd E (4 mp s lw: souc mgntic fild cunt I fou diffntil fom ( E ρ ϵ ; ( ; (3 E ; (4 µ J

15 s Contiution # it plcs fo it tions impotnt cus pticul mph on tgs must qul Th povs thid qution dq( cus ρ Q function position wll c lt hold Φ( K pim ojcts Th diﬀntil fom Th J d Q (58 (57 cn ct diﬀntil fom wll diﬀntil 54 s Ω foch it tgs must qul Th povs thid qution s impotnt it plcs pticul mph hitdcus fou (58 onsttd pondg (58 (53 conjunction (5 ut ρhitd hich pim ojcts Th diﬀntil fom s contiution fo consvtion lctic chg out m 86s H noticd whil scond thid fou (58 sttd pondg o( llvolums volum E ; In pticul ( w cn oth ;m lcsttic gvittionl filds consvt ϵ noticd 86s contdiction out m 86s H noticd whil scond thid constnt o lcmgntm o Thn tgl will pcts ppoimtl tg ρ pcts constnt o lcmgntm o nc volum nonzo (ut smll wgumnt ld contdiction t us h (55 Jonl (58 (3 ( E E ; ; (4 µin ( tc ; simil fhion w cn dictl vif follo ld contdiction t us tc gumnt (55 o fo tg vnh ϵh lth qution (58 vnhs Thfo H lt qution (58 vnhs Thfo divd fist Thom 34 Equ tg qution (4 givs tg oth sids w gt J sc Th contdicts ( J (58 (3 E ; (4 µ tg oth sids w gt divtion J Th contdicts diﬀntil w div thid qution lv fom contuit qution (3 fo chgs which psss iﬀntil fom contuit qution diﬀntil fom contuit qution (3 fo chgs which psss howv contuit qution (q 35 givg Mgntic consvtion cn lctic chg : which v simil thid d Empl filds lso wittn i d divd fist Thom 34 Equ consvtion lctic chg cus fim stlhmnt consvtion lctic chg cus fim stlhmnt tuns H thid qution (57 ρ vc potntils To fd pssion fo vc H w thid qution lv ltg divtion Qdiv chg consvtion dcidd t fou chg consvtion dcidd t ltg fou JQ (3 (5 Th mgntic ution which v simil thid d comptil m m chg consvtion clu clu fist m m comptil chg consvtion Th fild tgl: fist om tuns w H qution (57 tim w ot H thid E d qution If diﬀntit qution spct ## # # qution If w diﬀntit qution ot diffntitd qution ( spct spct tim: tim w m dq( v( ( # # dq( v( # ρ # m 3 E ρ ρ E ρ E ρ ϵ E ρ Ω Ω H Eitslf not E d tivit ms consvd quntit ut ms ϵ 47 E ϵ ϵ ϵ W ϵt out tgl #wnt Howv c dφ d m suggstd if fou constnt Th ( ; n d qution dict d s Th pull out o wchg nd gt ound th mn modifid (4 constnt consvtion suggstd if fou constnt dt chg dt consvtion fouth qution hd modifid clud # scond Usg ltion (4 w cn wit chg consvtion fouth qution hd modifid clud With th modifiction fou (58 com ϵ E/ E dφm d (4 With d d µ (59 Jmodifiction µ ϵ th fou com ϵ E/ ssumd chng flu coms out soll du dt dt (58 & '( v( h mgntic fild Th ms it possil push tim v v nsid h phsics it mmtics whn tgl upon which coms ptil divtiv v sumd chng flucoms out soll du unction mgntic position wll c lt hold focs logic pu dduction fild Th ms it possil push tim h tgs must qul Th w povs pio thid qution lt qution uch momnts mmtics ion sid tgl upon which it coms ptil divtiv v( fochold logic cption Copnicus s lt h- function position duction wll c

E. Computation of Permanent Magnetic Fields

E. Computation of Permanent Magnetic Fields E. Computtion of Pmnnt Mgntic Filds Th following pssgs should giv n impssion, how pmnnt mgnts cn b clcultd in spct of thi fild distibution. This ovviw ctinl cnnot cov ll subjcts. It will ml intoduc th