The pair-model of monopolar and dipolar moments of elemental electric scalar and magnetic vector charges

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1 The pa-model of opola dpola moments of elemental elet sala magnet veto hages Max Chwa hyss Depatment, Faulty of Sene, Engneeng Tehnology, Walte Ssulu Unvesty, Mthatha 57, Easten Cape, South Afa E-mal: Abstat. A sequel to epesentng elemental soues of magnet felds as elemental magnet veto hages s ealzng that elet magnet dpole moments ae dffeent lasses of moments. The dstnton between opola dpola moments beomes leae when any dstbuton of elet sala hage o magnet veto hage s depted as one o moe pas of hages wth equal magntudes. It s shown hee that sepaaton of the hages (elet sala o magnet veto) s essental fo the vey exstene othe attbutes of an elet o magnet dpole, but not fo a opole. These epesentatons ae makedly dffeent fom the tadtonal analogous epesentatons notons of elet magnet dpole moments as soues of oespondng felds o potentals.. Intoduton The eent onepton of elemental soues of magnet felds as elemental magnet veto hages [] has many spn-offs. Ths pape shows elet magnet dpola moments as pas of own opola moments of elet sala magnet veto hages, eveals faults n tadtonal [5] notons, despton temnologes.. epesentaton lassfaton of opola dpola moments dp dq (a) ˆ Fgue. (a) Monopola dp dq ˆ dq d p dp dq (b) d p (b) dpola d p ollnea moments fo paed sala hages. ˆ ŝ dq ˆ d p dp sdq

2 Fo elemental elet sala hages dq dq at ponts of postons, fgue (a) shows the opola moments d p d p. The jont moment s the opola moment dp dp dp dq dq dq () s ollnea wth the poston veto dq of the ente of ts net hage dq. If the hages wee negatve n sgn, eah moment would be opposte n deton to the poston veto. Fgue (b) shows the opola moments d p d p of dq dq, as well as the sum dp dp dp dq dq dq dq sdq () It s a dpola moment, whh s ollnea wth the nta dpola dsplaement s dawn fom dq to dq. Tadtonally an ntegal of any mxtue of () () would qualfy as an elet dpole moment [5]. Yet an ntegal of dm s smply alled a moment of the total mass m [6]. In stak ontast to tadton (Glbet model of 6 o Da magnet opole of 9) [5], fgue (a) shows paallelogam epesentatons of aeal (aea-lke) opola moments d m dm of dental elemental magnet veto hages d Q d Q at. The sum s dm dm dm dq dq dq d () Q dq dm s dq dm d Q ` d Q dq If the two magnet veto hages ae dq dq, that s, of same magntude but opposte detons, the jont moment s shown n fgue (b) as a fguatve paallelogam based on s, wth ts aea veto beng dm dm dm dq dq dq dq s dq (4) Toque, aea veto angula momentum ae aeal moments of the vetos foe, dsplaement lnea momentum espetvely. Howeve, note that when the opposte sdes of a paallelogam ae egaded as dsplaements n opposte detons, then ts aea veto beomes the aeal moment of ethe pa of opposte sdes. Natually the aeal moment of both pas s twe the aea veto [].. Geomet expanson of elet sala magnet veto potentals Fgue dsplays postons of two dental elet sala hages, wth the feld pont n fee spae (pemttvty ) dsplaed fom,, by ˆ. Smlaly, fgue 4 shows d m s ` dq d m dm dq d m (a) (b) Fgue. (a) Monopola d m (b) dpola d m aeal moments fo paed veto hages.

3 that fo elet sala hages of opposte sgn but same magntude. In these two stuatons the elet sala potentals at ae espetvely dv dq dq Fgue. oston veto of ente dsplaement of fom. dv dv dv dq 4 f f f dvo (5) dv dp dq s dq Fgue 4. Inta soue dsplaement s dsplaements, of fom. dv dp dv dv dq 4 f f fdpdvo (6)

4 whee dq dvo (7) 4 s the potental at due to a sngle postve elemental elet sala hage dq at the ogn, f (8) f f the opola geometal enhanement fato, fdp f f (9) the dpola geometal enhanement fato. As ˆ 5 ˆ ˆ, a bnomal expanson of gves ˆ ˆ f.... () Substtutng () nto (5) shows that the potental at due to a sngle dq at, an be eated by plang at that dq plus an nfnte sees of ts fatons, whose exteme values ae odeed as, dq dq, dq,.... () Thus, the tadtonal namng of suessve tems n suh expansons as opole, dpole, quadupole, otupole so on [5], s napt. Indeed, t s absud to speak of the opole tem, the dpole tem, the quadupole tem, the otupole tem so on, of a pont elet sala hage (a tadtonal pototype opole). Applyng the appoxmaton n () to equatons (8) (9) espetvely yelds f ˆ () sˆ fdp () The value of f vaes wth but stays lose to ± ts sgn s smply the om sgn of the elet sala hages, so that fom (5) (7), dv vanshes only at. The value of f dp vaes wth s but stays lose to ±, ts sgn s that of the podut s ˆ. Thus dv dp has a elatvely edued magntude. Exept fo ˆ s, the appoxmaton () s nvald fo all n the plane pependulaly bsetng s, whee dv dp s always exatly zeo. Ths fault s neve mentoned. Usng the tansfomatons dq dm,, dv d dq dq,, dv da, espetvely yeld hazed equatons (gavtatvty, pemeablty ) [7] fo the gavtatonal sala d magnet veto d A potentals. Note that although dq s the magntude of a ollnea moment, dq s nethe an aeal moment no ts magntude. Now let us examne two exteme onfguatons of opoles dpoles. Fom the above, when o, both the moment potental of a opole suvve; but these de out fo a dpole as s f dp. n the othe h, f o, both the moment potental of a dpole ae nonzeo quanttes as s f ; wheeas fo a opole, the moment vanshes but dp ts potental emans as f.

5 4. Conluson Any dpole (elet o magnet) has equal amounts of sepaated hages wth opposte sgns o detons, eates a poston-ndependent (ollnea o aeal) moment a elatvely edued magntude of the elet sala o magnet veto potental as the th ode tem s absent. The spatal sepaaton of hages s a equement of the exstene of a dpole. A opole s a pa of equal amounts of hages wth the same sgn o deton, has a ente of non-zeo net hage, a postondependent moment an enhaned potental whh s domnated by the th ode tem neve vanshes exept at. Faults n the tadtonal notons temnologes of elet magnet dpole moments have also been evealed. efeenes [] Chwa M 9 Mystey of a ula uent s two Catesan elemental magnet dpoles Bohue, 54 th Annual Confeene of the South Afan Insttute of hyss (Duban, 6 th th July 9) abstat # 5 A. [] Chwa M Status of elet magnet dpole moments n a omplete haatezaton of eletomagnet fst ode moments, o. nd Walte Ssulu Unvesty Intenatonal eseah Confeene (East London, 5 th 7 th August 9) pp , ISBN [] Gffths D J 999 Intoduton to eletodynams, d edton (London, entehall) pp , ISBN [4] ollak G L Stump D Eletomagnetsm (Cape Town, Addson-Wesley) pp , ISBN [5] Jakson J D 975 Classal Eletodynams, nd edton (New Yok, John Wley) pp. 7 8, ISBN 47 4 X. [6] Seway A Jewett J J W 4 hyss fo Sentsts Engnees, 6 th edton (London, Saundes College ublshng) p. 7, ISBN [7] Chwa M Zeo outwad magnet flux fom sufaes enlosng non-dpola magnet soues, o. 55 th Annual Confeene of the South Afan Insttute of hyss (etoa, 7 th Septembe st tobe ) SAI_45, ISBN