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 bsic diffntil qutions of mgntosttics, povid quick glnc onto th numicl mthod of Finit Elmnts nd will thn plin th most popul mthods to gt filds of mgnts nlticll. Som fomuls will b psntd, which bsd on th tho of vcto nlsis. So th mning of som smbols which in us will b dpictd in n ppndi t th nd of this chpt. 1. Gnl Th oot of ll fomuls fo th nlsis of mcoscopic mgntic sstms th Mwll qutions togth with som mtil lws. Fom ths Mwll qutions ptil diffntil qutions of lctomgntic potntils cn b divd. Thos cov ll filds of EM phnomn lik sttic nd tim dpndnt lctic nd mgntic filds, cunt distibutions, lctic cicuits o wv phnomn. H fist n qution fo sttic pmnnt mgnts togth with DC cunts will b divd. Eq. B.4 ws: B B H H E.1 0 This constitutiv ltion includs pmnnt mgnts with mnnc induction B nd gnl fild dpndnt but ltivl smll pmbilit µ. But it lso cn stnd fo soft mgntic mtils, wh B is ltivl smll nd µ cn b v lg. W now us th vcto potntil A nd in ddition M instd of B, i..: B A nd M B /0 E. Th us of A follows fom B=0, comp EA.8 in th ppndi. Appling th ottion opto t both sids of E.1 nd using E. povids: 1 M A 0 H E.3 Rmmbing q. A.4 w s, tht th fist tm t th ight sid is nothing ls thn th cunt dnsit j. 1 M A 0 j E.4 Known pmts fo this mgntosttic fomultion th DC cunt dnsit, th distibution of mnnt mgntition s wll s pmbilitis of hd o soft mgntic mtils. Th vcto potntil A is th unknown vibl. In ddition to ptil diffntil qutions s E.4, dditionl constints fo th vcto potntil hv to b dmndd. Ths ltions fo its bhvio t th sptil bods of th poblm, s wll s gug ltions, s blow. Th consid tht A is not dfind uniqul b onl th diffntil qution. Solutions of E.4 fo A nd so fo B, H tc. nlticll vilbl fo onl fw spcil css, which mns tht numicl mthods hv to b pplid thn. Th most popul on is nowds th Finit Elmnt Mthod FEM. 1
Th Finit Elmnt Mthod spts spc into smll lmnts, ssuming lin o polnomil bhvio of th componnts of A in ch lmnt. Fig. E1 shows such n lmnt distibution fo two dimnsionl mpl. Th bhvio of th potntil in n lmnt is pmtid with th hlp of its vlus on th nods o dgs of th lmnt. Und ths conditions E.4 cn b fomultd to sstm of lin qutions with th nod o dg vlus of A s unknowns. Th Finit Elmnt Mthod so povids n ppoimtion of lit which incss in qulit with gowing numb of lmnts. Disdvntgous in this mthod is, tht FEM softw pckgs v pnsiv on on hnd, which mns svl 10.000 US$ fo 3D pckgs. In ddition most of thm dmnd high gd of tining nd th nlss fil tim consuming in most css. Bsid th dict solution of diffntils qutions nd bsid th FEM mthod, svl oth numicl mthods lik FDM Finit Diffnc Mthod, BEM Bound Elmnt Mthod o FIT Finit Intgtion Tchniqu ist, but of lss populit o hv to suff fom fw disdvntgs compd to FEM. Fig. E1: Cunt coil nd unidictionl pmnnt mgnt suoundd b ion nd i, nld with FEM mthod. As noth ppoimtion mthod th so clld mthod of mgntic cicuits is mntiond in nl v book bout mgntism. This ws in tnsiv us in th pst bfo FEM togth with chp comput soucs bcm vilbl. This mthod is mostl usd to tt nl closd mgntic cicuit sstms, tht consist of mgnts nd cunt conductos which mbddd into soft mgntic mtils To gt good sults h th istnc of onl smll i gps hs to b dmndd. Pls f to littu fo futh infomtion. Aft this gnl intoduction, w will focus on th mthmticl ttmnt of th dimnsionl sstms tht consist onl of pmnnt mgnts.. Ttmnt of M Pmnnt Mgnts In bsnc of DC cunts nd soft mgntic mtils in E.4 th is j=0. µ is clos to on fo th most css of hd mgntic mtils. So E.4 ducs to th following fomul:
A 0 M Now w intoduc gug qution fo th vcto potntil, h th so clld Coulomb gug: A 0 E.5 Togth with th vcto idntit EA.9 fom th ppndi blow, th bov thn bcoms A 0 M E.6 This diffntil qution hs gnl solution b q. E.7: 0 ' M 0 M n' A dv' df' 4π - ' 4π - ' V F E.7 H is th loction w th fild hs to b clcultd nd is th vcto of th mgnts loctions. Th intgtions don ov th mgnts volum V s wll s ov th mgnts sufc F. Eq. E.7 is of lss populit thn th following fomultion, wh instd of th vcto potntil scl potntil is usd. Tking q. E.1 togth with stting µ =1 fo th mgnt lds togth with E. to: M H B 0 0 Tking into ccount tht B hs no soucs, i.. B=0 s q. A.3, this lds to H E.8 Sinc j=0. i.. H=0, vcto nlsis povids tht H cn b pssd b scl potntil, s q. EA.7 of th ppndi: H E.9 This povids togth with E.8 in th bsnc of cunts: M E.10 Th gnl solution of E.10 is givn b: 1 ' M 1 M n' dv' df' 4π - ' 4π - ' V Th intgtion is pfomd gin ov th mgnts volum s wll s ov its sufcs. M F E.11 Which of both fomultions i.. th vcto o scl potntil fomultion is tkn, oftn dpnds on th s of solving th spctiv intgls nd m b diffnt fo diffnt gomtis. In gnl, oftn th intgls cn not b pssd b plicit fomuls but hv to b ttd numicll. 3. Empl fo th Us of th Scl Potntil Fomultion In th following w will show n s mpl of th ppliction of mgntic scl potntil on homognousl mgntid clind o disc mgnt with il hight h nd dius. 3
Th fild shll b computd t distnc d fom th mgnts sufc, s th sktch blow. Th mgntition is ointd in il diction, i.. M=M *. Fig.E: Sktch of homognousl mgntid flt clind. Th fild shll b computd t point with distnc d fom its upp fc. Whn w tk th potntil solution q. E.11 w s tht insid th volum intgl th is M =0, so tht th volum intgl itslf vnishs. This is lws th cs with homognous mgntition, so th filds oigintd onl b th pol fcs. Fo simplifiction w combin th st of q. E.11 with q. E.9 wh th Nbl opto cn b tkn und th intgl. So w gt fo th fild: 1 1 H M n' df' 4π - ' F E.1 Th scl poduct of mgntition with th sufc s noml vcto is onl non o t th hd fcs of th clind, w th plus sign is vlid t th top nd minus t th bottom sid: M n E.13 Th sufc lmnt of hd fc is M df' 'd'd' E.14 Fom smmt it follows tht t th cnt point t distnc d, th cn onl b n il, i.. -componnt of th fild. So w onl nd th -pt of th Nbl opto und th intgl: 1 1 ' ' 1 ' ' =+/- h/ is th loction of th hd fcs, nd is th coodint of th point of intst. Both nd ltd to th coodint oigin, which is loctd t th cnt of th mgnt. Aft diffntition nd pssing both nd with th hlp of h nd d on gts: 1 d fo th upp fc E.15 ' 3 ' d 4
nd 1 d h fo th low fc E.15b ' 3 ' d h Now w summi E.1-E.15 nd gt fo th fild: Hd H with wh H H t H b E.16 M π d ' Ht d ' 0 d' E.16 4π 0 3 ' d nd M π d h ' Hb d ' 0 d' E.16b 4π 0 3 ' d h Th intgtion ov th ngl is lmnt nd th intgls ov th dius cn b tkn fom intgl mps. Doing this th finl sult cn b wittn to Hd H E.17 with wh M Hd gd h - gd E.17 w gw E.17b w E.g. mgnt with =5mm, h=3mm nd M=800kA/m =1T oigints fild of 171 ka/m t distnc of 1mm. Whs this mpl shows sult which cn b obtind quit sil, in gnl th intgls cn not b solvd b plicit pssions nd hv to b ttd numicll. In th cs of non homognous distibutions of mgntition dditionll th volum intgls of q. E.11 hv to b solvd, which dmnd dditionl ffots. Appndi In th bov som smbols of vcto nlsis lik th Nbl opto w usd nd shll b dpictd h. Th Nbl opto cn b intoducd in Ctsin coodints s vcto of singl componnt diffntil optos:,, EA.1 5
6 In Ctsin coodints this cn b pplid lik vcto in th fom of dot poduct nd coss poduct with oth vctos. With scl filds it cn b pplid b simpl multipliction. So ntitis lik divgnc, ottion nd gdint cn b fomd: div EA. ot EA.3 gd EA.4 Sinc w usd gd in clind coodints,, in th mpl bov, w will giv it h lso in this coodint sstm. Epssions fo div nd ot in oth coodint sstms cn b found in littu. 1 gd EA.5 Anoth opto following fom Nbl is th Lplc opto. In Ctsin coodints: Δ EA.6 Som impotnt chctistics of vcto filds in ltion with th bov optos s follows: 0 EA.7 In wods: Whn vcto fild is cul f it cn b pssd s th gdint of scl potntil A 0 EA.8 In wods: Whn vcto fild is souc f it cn b pssd s th cul ottion of vcto potntil On idntit btwn diffnt opto pssions of vcto fild which ws usd bov is th following: Δ EA.9 A lot of oth idntitis btwn th bov optos in fnc to scl nd vcto filds cn b found in littu. Also f to littu with spct to th vlution of sufc nd volum intgls s wll s to ltions btwn thm, lik th Stoks, Guss o Gns ltions.