Defects in the traditional analogy between the dipolar structure of a circular current and a simple electric dipole

Transcription

1 Dfct in th titionl nlog twn th iol tuctu of cicul cunt n il lctic iol M Chiw hic Dtnt cult of Scinc ngining n Tchnolog Wlt Siulu Univit /Bg X Nlon Mnl Div Mthth 57 South Afic. -il: chiw@wu.c. Atct. It i hown tht whn cicul cunt i olv into g itiution of itinct Ctin n coonnt lin cunt lnt ch itiution i colt gntic iol tht lctivl ct lik Ctin coonnt of th gntic toqu n vcto otntil lu onl th gntic fil oth Ctin coonnt. All th il in t of itiution own gntic iol ont which i titionll ttiut to th whol cicul cunt. In contt il lctic iol lign on th - i ct it n lctic toqu coonnt it full clinicll tic lctic fil n th lctic cl otntil ll of which il in t of th ol lctic iol ont. ch gntic o lctic Ctin toqu coonnt i il co ouct of itiution iol ont n on Ctin fil coonnt lll to n cluiv Ctin ln nicull icting th utull lll int-iol ilcnt whil th itiution cooning otntil vnih in tht ln. Un uch cil conition tition co on uviving Ctin coonnt of th gntic toqu o of th gntic vcto otntil to ctivl th lctic iol coin n toqu coonnt o th whol cl otntil. Singl fo thi n th qulit of th gntic iol ont of th two coonnt itiution of th clinicll tic cicul cunt tition incoctl fin ith of th gntic iol ont tht of th nti cicul cunt.. Intouction A follow u on th li [] w how tht th titionl nlog of th gntic iol tuctu of cicul cunt to tht of il lctic iol coniting of t lctic cl chg of inticl i ut ooit ign h n hot coing. Thi i on vluting fo th gntic ( cicul cunt) n lctic (il lin cl chg) itiution of iol thi iol ont toqu in n tnl fil iol gntic vcto n lctic cl otntil n iol gntic n lctic fil on th i tht n iol i cointion of ooit qul-i oitl itnt onool. n lntl lctic cunt ultili th f c ilit i not n ict n lntl gntic vcto chg in ntul contt to n lntl lctic cl chg n it titionl nlog of gntic cl chg. In thi ticl it i ov tht th contting vcto n cl ntu of th lntl gntic vcto n lctic cl chg wll th goti of th itiution th i oigin of th n intting iffnc n iiliti twn thi gntic n lctic oti o of which titionll ovlook o iintt. Th vl in thi t t coion.

2 . Mont of n toqu on Ctin gntic n lctic iol n cicl of iu ling in th -ln n cnt t th oigin in figu () n iuthl lin lntl gntic vcto chg t oint of il oition vcto in th th qunt i I φ I in co I () ( ) wh φ I i it lin gntic vcto chg nit n i t n ngl to th -i. nc th Ctin lntl gntic vcto chg coonnt I in () I co. () I φi ŝ ŷ ŷ q ŷ ŝ ẑ ŝ ŷ ẑ ŷ igu. iing t qul-i lntl ntiti of ooit ign into iol: () Ctin gntic vcto chg on n -cicl n () lctic cl chg on -i. hiit th inict ngul nnt il lin gntic vcto chg niti. Whn Ctin coonnt t i with th tching ut ooitl ict Ctin coonnt t n th contitut th n Ctin lntl gntic iol ctivl. Whil in figu () t th oint n of il oition vcto n on th -i th lin lntl lctic cl chg q q. (c) Th fo n lntl lctic iol. lctic cl chg lin niti. Th two gntic lntl iol n th on lctic lntl iol hv iol ont of ( ) () ŷ ŝ () q ()

3 ) ( () q q q ) )( ( (c) wh thi Ctin int iol ilcnt o Ctin iol ointtion vcto in co. () Th (u) ucit n in () to (c) n in uqunt icuion low ignif ntiti fo o u to ooit lntl gntic vcto o lctic cl chg tht i ctiv onool. Intgting () n () fo to n (c) fo to il th ont I I. (5) ch on i quivlnt to chg nit i ultili n vcto of l o unl ufc. Yt th cicul cunt gntic iol ont i twic th (Knnll) titionl vlu [8] :. I (6) In tnl gntic n lctic fil th Ctin lntl gntic n lctic iol chcti i gntic foc n lctic foc. A coul ont of th foc cting on th lntl iol th gntic n lctic toqu : (7) (7). q q q q (7c) ch til vcto ouct in (7) n (7) il on Ctin gntic toqu coonnt whil th uo vcto ouct in (7c) il two Ctin lctic toqu coonnt. Th toqu coonnt q q q q (8) how tch fot n inquliti. Uing (5) n (6) th totl gntic n lctic toqu (9). () Th contt in (7) to () nullif th titionl nlog twn gntic n lctic toqu n cill th tking of o [8] th cicul cunt totl gntic iol ont. Th con quivlnc in (9) n () highlight th uo gntic n lon lctic iol itiution.. Diol gntic vcto n lctic cl otntil n ocit fil Whn th gntic vcto n lctic cl chg itiution in figu th ouc of gntic vcto n lctic cl otntil wll th ocit gntic n lctic fil t fil oint i ol ignif th chg n thi oition. Thu in figu th fil oint in - o -ln i ilc fo oigin n fo lntl ouc Ctin gntic vcto chg t oint on cicl of iu in th - o -ln f () wh in n th goticl fcto f function of. Whil in figu ilcnt of fo lctic cl chg q n q t n on th -i in co f θ ()

4 φ θ φ I N igu : A fil oint in -ln n gntic ouc oint on -cicl. wh with th goticl fcto f ing nnt onl on. Blow ch nuing fit o oition fo o i ft inoil nion of f n o n wh n fo otntil n n fo fil n liction of qution () to (c). Thn th two Ctin lntl gntic iol on th cicl in figu n th lctic iol on th -i in figu will gnt t th gntic vcto n lctic cl otntil A in in A A f f in in () A co in A A f f co in () q q V V V f f co co. (c) f ch of A A n V h Ctin ln wh it lon i ctl o. Thi i titionll [] iug nc of lt gntic iol ont which wongl ili tht th ltt n on oint. Th i no uch infnc fo th lctic iol ont! Intgting n thn chnging to hicl unit vcto how tht A n A hv ll th hicl coonnt: in A in co in co θ φin ()

5 in in θco in co φ co A () co V. (c) θ igu : A fil oint in - o -ln n lctic ouc oint n on th i. Du to ltion in (5) th totl gntic vcto otntil i intchngl cluivl in t of ith o n thu cqui clinicl t: in in A A A φ φ. (5) Siill th fil t oint u to th two itinct Ctin lntl gntic iol in figu n th lctic iol in figu f f (6) in co in in in f f (6) co co in co in N

6 co q f f θin. q (6c) Intgting (6) to (6c) n chnging fo Ctin to hicl unit vcto how tht whil o not v with ch of n h ll th hicl coonnt n vi with : co θ in in φco in co in (7) co θco in φco in co in (7) ( co θin ). (7c) Alo u to (5) whn cluivl in t of ith o th cicul cunt totl gntic fil cqui clinicl t iil to tht of in (7c): co co θin θin. (8) Cll thi iilit cnnot utif th tition [8] of tking ith o th cicul cunt onl gntic iol ont. Agin th titionl nlog fil. In fct qution (7) to (8) cn lo otin inictl fo qution () to (5) ling th ltion A n V. (9). Concluion It h n hown tht titionl nlogi twn th tuctu n toqu of lctic n gntic iol ctivl onou. A cicul cunt i olvl into g itiution of itinct Ctin n coonnt lin cunt lnt; ch itiution ct lik Ctin coonnt of th iuthl gntic toqu n vcto otntil lu onl th gntic fil oth Ctin coonnt ll of which il in t of th itiution own gntic iol ont. qulit llow intchngl u of th gntic iol ont of th two colnt itiution n thu giv clinicl t to th cicul cunt gntic oti o tht tition incoctl ign ith of th gntic iol ont to th whol cicul cunt. In contt il lctic iol lign on th -i ct it n lctic toqu coonnt it full clinicll tic lctic fil n th lt lctic cl otntil ll of which il in t of th ol lctic iol ont. fnc [] Chiw M Mgntic vcto chg in th t of cicul cunt i of itinct Ctin lntl gntic iol ocing of SAI 56 th Annul Conf. of th South Aficn Intitut of hic (toi South Afic: 5 Jul ) I Bon n A Both (Univit of South Afic) nlin t htt:\\ [] Jckon J D 999 Clicl lctonic (Nw Yok: John Wil & Son) 79 [] Loin Coon D n Loin 987 lctogntic il n Wv (Nw Yok: W n & Co) [] ucll M 985 lcticit n Mgnti n (Lonon: McGw-ill) 59 [5] Bo T 988 Th foc on gntic iol A. J. h. 56(8) [6] Vin L 99 Toqu n foc on gntic iol A. J. h. 58() [7] Giffith D J 999 Intouction to lctonic (Lonon: nticll) 58 [8] Dil J 8 Diol in gntic fil wok n quntu in h. v