Spinning Charged Ring Model of Electron Yielding Anomalous Magnetic Moment

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1 Brgan & Wsly 1 Ring Mdl f Elctrn Spinning Chargd Ring Mdl f Elctrn Yilding Analus Magntic Mnt David L. Brgan J. Paul Wsly P.O. Bx 1013 Wihrdastrass 4 Knnsaw, GA USA 771 Blubrg, Grany Suary: A unifrly chargd spinning ring is prpsd as a dl fr th lctrn. Fur paratrs, th radius f th ring R, th half-thicknss r, th ttal charg, and th tangntial vlcity c ar chsn t yild th fur lctrn charactristics, th ass, th charg, th spin h /, and th agntic nt µ. Th dl is cpltly stabl undr lctragntic frcs aln. Th twic classical valu fr th gyragntic rati is xplaind. Th siz f th lctrn quals th ratinalizd Cptn wavlngth, and th frquncy f rtatin quals th Cptn frquncy. Th dl yilds t a highr rdr apprxiatin th analus agntic nt in agrnt with bsrvatin. INTRODUCTION Evr sinc discvry f th lctrn in th nintnth cntury, a ajr gal f physics has bn t find an lctrn dl that accunts fr th physical prprtis bsrvd and asurd with vr incrasing prcisin. Early dls prpsd by Abraha 1 and Lrntz prvidd a physical pictur f th lctrn but culd nt satisfactrily accunt fr s lctrical prprtis. Latr dls such as th Dirac pint dl 3 and th quantu chanical dl hav bn cnfrd t xprintal data; yt, ths athatical dls hav nly tnuus links t a chanical and physical structur that is th pursuit f physics (th study f th physical natur and prprtis f th univrs). Th nd fr a bttr lctrn dl is statd by Ivan Sllin wh wrt in 198 that...a gd thry f lctrn structur still is lacking... Thr is still n gnrally accptd xplanatin fr why lctrns d nt xpld undr th trndus Culb rpulsin frcs in an bjct f sall siz. Estiats f th aunt f nrgy rquird t assbl an lctrn ar vry larg indd. Elctrn structur is an unslvd ystry, but s is th structur f st thr lntary bjcts in natur, such as prtns [and] nutrns... 4 Early dls f th lctrn 1, wr nt ralistic, priarily bcaus thy did nt tak int accunt th spin and agntic nt f th lctrn. All f th dls prpsd s far hav had t assu ad hc nn-lctragntic frcs t hld th dl tgthr. Th dl prpsd hr is th first dl vr prpsd that is cpltly stabl undr lctragntic frcs aln; th dl ds nt radiat. * THE MODEL AND SOME OF ITS CONSEQUENCES An lctrn is assud t b a unifr surfac charg dnsity σ that frs a ring f radius R and half-thicknss r spinning abut th axis f sytry with th angular vlcity ω = c/r, as * Rcntly, w larnd that lctragntic ring dls hav bn prpsd by Parsn, Cptn, Iida, Jnnisn, and Bstick. Rfrncs t ths prpsals can b fund in thr paprs availabl fr Cn Sns Scinc.

2 Brgan & Wsly Ring Mdl f Elctrn shwn in Figur 1. Th unifr surfac charg dnsity vr th ring σ ust b chsn t yild th ttal charg ; thus, whr A = (π r)(πr) is th surfac ara whn r << R. = Aσ = 4π σrr () 1 ω = c/r R r Figur 1: Th Spinning Ring Mdl f th Elctrn Th ass f th lctrn is btaind fr th classical lctragntic nrgy f th chargd spinning ring and ass-nrgy quivalnc. Th ttal lctragntic nrgy f th spinning ring E t is givn by th lctrstatic nrgy, E C = whr C is th capacitanc f th spinning ring, plus th agntstatic nrgy, E LI = whr L is th slf-inductanc f th spinning ring, and th currnt I is ω c I = = π πr ( ) () 3 ( 4) Fr th thin ring whr r << R it ay b shwn 5,6,7 that th capacitanc and inductanc hav th valus R C = 4 π ε ln( 8 Rr / ) [ ] [ ] () 5 L = µ R ln( 8Rr / ) µ R ln( 8Rr / ) ( 6) Equatins () thrugh (6) yild quipartitin f lctric and agntic nrgis such that

3 Brgan & Wsly 3 Ring Mdl f Elctrn c Et = E E = ln( 8Rr / ) ( 7) 8 π ε R Th dl ds nt radiat as th surcs, th surfac charg dnsity σ and th surfac currnt dnsity σc/πr, ar cnstant and d nt chang with ti. Whn chargs ccupy th ntir circl cntinuusly, thn...th ttal radiatin fild will vanish...; this iplis that a cntinuus ring currnt will nt radiat... 8 Th tangntial spd c f th spinning ring ds nt vilat any principl. Th ring is nt a atrial ring, s n attr r ass is travling with th spd c. Actually th vlcity ay b rgardd as ncssarily c, bcaus th dl ay b viwd as cnsisting f lctric and agntic filds nly; and lctragntic filds ncssarily prpagat with th vlcity c in a vacuu. STABILITY OF THE MODEL AS A FUNCTION OF THE HALF-THICKNESS OF THE RING Th ssntial waknss f prir dls f th lctrn has bn that frcs f unknwn rigin hav t b pstulatd ad hc t hld th lctrn tgthr against lctrstatic rpulsin. In cntrast, th prsnt dl is cpltly stabl undr th actin f classical lctragntic frcs aln, and n frcs f unknwn rigin hav t b pstulatd. Th agntic frc n th thicknss f th ring prducs an inward dirctd prssur P r pinch frc, F = qvxb; thus, P = σ vb () 8 whr B is th agntic fild at th surfac f th thicknss du t th spinning f th chargd ring. Hr th surfac charg σ vs with th vlcity v=c, s th pinch prssur bcs P = σ cb ( 9) Intgrating th lin intgral f B arund th thicknss yilds, accrding t th inapprpriatly nad Apr intgral law, π rb = µ I ( 10) Using quatin (4), th agntic fild at th thicknss bcs B = ( 4 crr 11 ) π ε by Fr quatins (1), (9), and (11) th agntstatic prssur n th thicknss is thn givn P B = = 4 16πε rr µ ( 1) Th lattr rsult, varying as B /µ is th wll knwn inward dirctd agntstatic strss f th fild. Th lctrstatic rpulsiv frc F = qe n th thicknss f th ring prducs an utward prssur r tnsin P givn by P =+σ E ( 13)

4 Brgan & Wsly 4 Ring Mdl f Elctrn whr E is th lctric fild at th surfac f th thicknss. Fr Gauss s law and quatin (1), th lctric fild at th surfac f th thicknss is σ E = = = ε E ε 4π ε rr ( 14) Cbining quatins (13) and (14), th lctrstatic utward dirctd tnsin r prssur bcs P = = ε E ( 15) 4 16πεrR Th lattr rsult, varying as ε E is th wll knwn lctrstatic strss r tnsin f th fild. Cparing quatins (1) and (15), it is sn that th utward lctrstatic rpulsiv prssur is balancd xactly by th inward agntstatic prssur r pinch; thus, P = P ( 16) Th thicknss f th ring thus hlds itslf tgthr in quilibriu undr classical lctrical and agntic frcs aln. STABILITY OF THE MODEL AS A FUNCTION OF THE RADIUS OF THE RING In additin t th stability f th thicknss invlving r, th ring as a clsd currnt lp f radius R is als stabl undr classical lctric and agntic frcs aln. T discuss th stability f th ring as a functin f R, it is ncssary t us th fundantal Wbr 9 lctrdynaics as gnralizd by Wsly, 10 which (unlik th Maxwll thry) prscribs th frc btwn tw ving chargs fr th utst. In particular, th frc btwn tw chargs q at s and q' at s, sparatd by th distanc S = s - s' with rlativ vlcity V = v - v' is 4πε c qq S F= c 3 S ( V S) 3 + V S d V + S dt ( 17) Th utward frc f pr unit lngth f th ring Rdφ du t lctrstatic rpulsin (th first tr in th brackt n th right f quatin (17)) ay b btaind by cnsidring th chang in lctrstatic nrgy whn th ring is nlargd fr R t R + dr. Fr quatins () and (7), this lctrstatic rpulsin pr unit lngth bcs f [ q C] / R r = R ln( 8 / ) 8π ε R ( 18) Siilarly th frc f pr unit lngth f th ring f radius R du t agntstatic frcs ay b btaind by cnsidring th chang in agntstatic nrgy fr an xpansin f th ring fr R t R + dr. Fr quatins (3) and (7) this agntstatic rpulsiv frc pr unit lngth bcs f ( LI / ) ln ( 8Rr / ) = f R = 8π ε R ( 19)

5 Brgan & Wsly 5 Ring Mdl f Elctrn Th fact that th frc f is utward ay b sn fr th fact that th pinch frc n th thicknss is slightly gratr n th insid f th thicknss than n th utsid f th thicknss, th agntic fild bing slightly strngr insid thn utsid. Fr quatins (18) and (19), th nt utward frc n th ring pr unit lngth du t th lctrstatic and agntstatic ffcts is ln( 8R/ r) f = f + f 4πε R ( 0) Ths frcs aris fr th first, scnd and third trs in th brackt n th right sid f quatin (17). Th ffct f th last tr in th brackt n th right f quatin (17) ay b btaind by nting that fr r = 0 (an apprxiatin that is justifid, as r << R) th dl givs * d v d v S dv/ dt = ( s-s ) = R ω [ 1 cs( φ φ )] ( 1) dt dt whr φ is th angular psitin f q and φ is th angular psitin f q and bth chargs ar fixd at th sa radial distanc R with th sa angular vlcityω = φ = φ. Th situatin is sytric abut th axis f sytry f th ring, s avraging th angl (φ φ ) vr th ring yilds zr fr th csin part f quatin (1); and S dv/ dt = R ω c ( ) φ Substituting this rsult () int quatin (17) shws that th ffct f th acclratin tr is just twic that f th lctrstatic frc and is in th ppsit dirctin. Thus th acclratin frc pr unit lngth f a is dirctd inward and is givn by f a = f ( 3) Fr th quality f th lctrstatic and agntstatic frcs, it ay b sn that th ttal frc f lctragntic rigin n th ring fr quatins (0) and (3) is zr; thus, fa + f + f = 0 ( 4) Th ring is, thus, in quilibriu undr classical lctragntic frcs aln, fr th ring dinsin R, as wll as fr th thicknss dinsin r. Th spinning ring is cpltly stabl bcaus th frcs upn it ar in quilibriu, and its charg and currnt distributins d nt vary with ti. Accrding t Maxwll thry, n radiatin is pssibl; and withut any ans f lsing nrgy, th ring rtains its lctragntic nrgy and shap. THE NATURAL FREQUENCY AND SIZE OF THE RING Still anthr way t s that th spd f th ring c is actually rly th vlcity f an lctragntic fild is t cnsidr th spinning ring as an LC-circuit. Th rsnant frquncy f an LC-circuit with a standing lctragntic fild f vlcity c is ω LC π = ( 5) LC * Wbr s lctrdynaics includs a frc fr acclratin which is ntirly issing in th Maxwll thry.

6 Brgan & Wsly 6 Ring Mdl f Elctrn Fr quatins (5) and (6) this givs ω LC π c = = 4π R c R ( 6) Th Cptn frquncy is ftn assciatd with an lctrn. It is givn by th Planck frquncy cnditin whr th nrgy is takn as th rst nrgy f th lctrn; thus, ω C c = D ( 7) Equating th natural rsnant LC-frquncy f th dl ω LC, quatin (6), with th Cptn frquncy ω C, quatin (7), yilds ω LC = ωc R = D ( 8) c Th radius R f th ring quals th ratinalizd Cptn wavlngth, 3.86 x trs. Prvius stiats f th lctrn siz hav bn bth largr and sallr. * Quantu thry is gnrally assud t spcify a larg sard ut distributin f ass and charg which can b n th rdr f atic dinsins abut trs. A vastly sallr stiat f ~ 10 - trs was rcntly publishd; 11 this is vn sallr than th siz f a prtn. Yt th lctrn s agntic nt is knwn t xcd th prtn s, and quatin (9) rquirs a largr lctrn siz fr rasnabl stiats f charg rtatins. Mdrn atic dls assign sall prtns t th nuclus and largr lctrns t th vlu surrunding th nuclus. THE MAGNETIC MOMENT OF THE ELECTRON Th agntic nt f th dl t th first apprxiatin, using quatin (4) is givn by πr ω cr µ AI = = π ( 9) whr A' is th ara f th currnt lp A' = π R. Fr quatin (8) fr R as th Cptn wavlngth, th dl yilds th agntic nt t th first apprxiatin as D µ = µ B ( 30) whr µ B is th Bhr agntn. This fits th bsrvd valu t abut 3 r 4 placs. A clsr apprxiatin that includs th s-calld analus agntic nt is drivd blw. ELECTRON SPIN Evans stats 1 Epirically, it was ncssary t assu that ach lctrn psssss an intrinsic angular ntu, in additin t its usual rbital angular ntu, as thugh it * On th basis f scattring xprints, Cptn stiatd th radius f th ring lctrn is abut x10-1. [A. H. Cptn, Th Siz and Shap f th Elctrn, Phys. Rv. Sc. Sris 14, (1919).]

7 Brgan & Wsly 7 Ring Mdl f Elctrn wr a spinning rigid bdy. Th bsrvabl agnitud f this spin angular ntu is h /. This agnitud f th angular ntu f spin p s = D ( 31) shuld b idntifid with th fr lctrn and shuld nt b cnfusd with th spin f an lctrn bund in an at and cupld t thr atic particls. * Th angular ntu f a fr lctrn dpnds upn its agntstatic nrgy nly and nt upn its lctrstatic nrgy, sinc nly ving charg will prduc ntu. A chargd nn-spinning ring wuld hav lctrstatic nrgy, but n angular ntu. Th spinning ring has th tangntial vlcity c, s that th spin angular ntu is whr p s = cr ( 3) E Et = c c ( 33) is th ass quivalnt f th agntstatic nrgy, as givn by quatins (3) and (7). Th fact that nly agntstatic ass is invlvd and th agntstatic ass is nhalf th ttal ass f th lctrn ans that th angular ntu f th lctrn p s, as givn by quatin (3), is nly n-half th aunt that wuld b assciatd with an rdinary spinning acrscpic bdy. This spinning ring dl thn xplains why th lctrn has a gyragntic rati µ /p s twic as larg as th classical rati /, r fr quatins (30) and (31) µ p s = ( 34) Fr quatins (7), (8), (3), and (33), th spin f th lctrn accrding t th prpsd dl bcs If R p s D = ln ( 8Rr / ) = ( 35) 8πε c = D/ c is th Cptn wavlngth, thn th half-thicknss r is t b chsn such that whr α = / 4 πε c D is th fin structur cnstant, and ln( 8Rr / ) = π / α ( 36) r = 8R xp( π/ α) 0 ( 37) * Fr an atic lctrn, Th ttal angular ntu f a singl particl is th suatin f its spin nta, and is rprsntd by th intrinsically psitiv nubr j. Th agnitud f th angular ntu f th crrspnding tin is D j( j + 1 )...[and] j is rstrictd t half-intgr valus f j = 1/... 1 Thus, althugh quantu thry spcifis p = 3 D/ fr th cbind rbital and spin nta f atic lctrns, this valu shuld nt b takn as th spin ntu f th fr lctrn.

8 Brgan & Wsly 8 Ring Mdl f Elctrn Th valu f r is s sall that it can b ignrd xcpt whr a singularity ight thrwis aris. ANOMALOUS MAGNETIC MOMENT T th first apprxiatin, th spinning ring dl f th lctrn yilds n Bhr agntn fr th agntic nt f th lctrn, quatin (30), in agrnt with bsrvatins t 3 r 4 placs. A r accurat xprssin is btaind by using a bttr valu fr th slf inductanc L f th spinning ring. Instad f th scnd f quatins (6), a r accurat xprssin fr th slf inductanc is [ ] L = µ R ln( 8R / r ) µ R ln( 8R / r ) ( 1 α/ π) ( 38) whr th apprxiatin invlvs stting ln ( 8 R / r ) = π / α as givn by quatin (36) and α = / 4 πε cd 1 / 137, and whr th radius f th ring R' and th half-thicknss r ar t b dtrind anw. * Th r accurat xprssin (38) rvals that th agntstatic nrgy is slightly lss than th lctrstatic nrgy, lading t r accurat ring paratrs. Th rlvant rlatins that ust b satisfid by apprpriat chics f th paratrs R', r', and ω' ar: angular frquncy: ω = c / R ( 39) currnt: I = ω / π = c / π R ( 40) agntstatic nrgy: E = L I / ( πεr )( c c) ( R r ) ( απ) = / 8 / ln 8 / 1 / ( 41) lctrstatic nrgy: E = ( πεr ) ( R r ) / 8 ln 8 / ( 4) angular ntu: ps = R = E c R / c = D / ( 43) ttal nrgy: E = E + E = c t agntic nt: µ = πr I = ( cr )( c c) ( 44) / / ( 45) Fr Lnz law, it ay b assud that th physical syst will adjust s that changs in th paratrs fr th first apprxiatin will b as sall as pssibl. It ay b assud that th agnitud f th chang in agntstatic nrgy will qual th agnitud f th chang in th lctrstatic nrgy fr sytry; thus, E E = E E ( 46) * It is n lngr pssibl t hav bth quipartitin f nrgy btwn lctrstatic and agntstatic nrgy and t als hav th rquird tangntial vlcity f th ring c qual t c. Th LC-frquncy ω LC is n lngr applicabl, as it is drivd assuing quipartitn btwn agntstatic and lctrstatic nrgis. Th Cptn frquncy ω C, bing rlatd t n bsrvatin, cannt b usd. Th r accurat xprssin (38) fr th agntstatic nrgy is th iprtant prprty lading t r accurat ring paratrs.

9 Brgan & Wsly 9 Ring Mdl f Elctrn Using ths cnditins, th radius f th spinning ring R nd nt b changd fr th first apprxiatin; thus, D R = R = c Only th angular frquncy ω and th half-thicknss r nd t b chsn anw such that ( 1 / ) ( c / D)( 1 / ) ω = ω + α π = + α π ( 47) ln( 8Rr / ) = ln( 8Rr / ) ( 1+ α / π ) = π / α + 1/ ( 48) Th nw half-thicknss r thn bcs 8D π 1 r = xp 0 ( 49 ) c α Fr quatin (39), th nw tangntial vlcity f th ring c bcs gratr than th frr ri vlcity c thus, c = c( 1+ α / π ) ( 50) Th nw tangntial vlcity f th ring is xactly qual t th spd f light which ay b shwn by calculating th cnditin fr dinsinal stability f th ring using th xact quatin fr inductanc. Equatin (50) indicats that th first apprxiatin f tangntial vlcity (labld c) is actually slightly lss than th spd f light. Substituting quatins (47), (48) and (50) int (41) and (4), th nw agntstatic and lctrstatic nrgis bc E = E ( 1 α / π ) E = E ( 1+ α / π ) ( 51) which ar sn t cnsrv nrgy fr quatins (7) and (44) and als th sytry cnditin (46). Substituting th first f quatins (51) and (47) int (43), it is sn that th angular ntu f spin p s is cnsrvd as D/. Substituting quatins (47) and (50) int (45), th r xact agntic nt bcs µ = µ ( 1+ α / π ) = ( D/ )( 1+ α / π ) ( 5) Th analus agntic nt is thn givn by µ / µ 1 = α / π = ( 53) which agrs with th bsrvatins t 6 r 7 placs. Cnsidring th gyragntic rati µ /p s fr quatins (43) and (5), it is sn t pssss th sa analy α/π as th agntic nt. Highr rdr apprxiatins fr th agntic nt f th lctrn can b btaind by cnsidring still r accurat xprssins fr th capacitanc and slf-inductanc f th spinning ring. REFERENCES [1] Abraha, M., Ann. dr Phys., 10, 105 (1903). [] Lrntz, H. A., Th Thry f Elctrns, nd d., Dvr, Nw Yrk (195). [3] Dirac, P., Classical Thry f Radiating Elctrns, Prc. Ryal Scity, Vlu 167, p. 148 (1938).

10 Brgan & Wsly 10 Ring Mdl f Elctrn [4] Sllin, I., Atic Structur and Spctra, Mc-Graw Hill Encyclpdia f Scinc and Tchnlgy, Vlu 1, p. 857 (198). [5] Mn, P., and Spncr, D. E., Fild Thry fr Enginrs, p. 375, D. Van Nstran, Nw Yrk (1961). [6] Syth, W. R., Static and Dynaic Elctricity, nd Ed., p. 318, McGraw-Hill, Nw Yrk (1950). [7] Gray, A., Abslut Masurnts in Elctricity and Magntis, p. 73, MacMillan, Lndn (191). [8] Panfsky, W. K., and Phillips, M., Classical Elctricity and Magntis, p. 370, Addisn- Wsly, Rading, Massachustts (196). [9] Wbr, W. E., Abh. Libnizns Gs., Lipzig, 316 (1846); Ann. dr Phys., 73, 9 (1848); Wilhl Wbr s Wrk, Vlus 1-6, Julius Springr, Brlin (1893). [10] Wsly, J. P., Spc. Sci. Tch., 10, 47 (1987); Phys. Essays, May 1990; Bull. A. Phys. Sc., 8, 1310 (1983); Prgrss in Spac-Ti Physics 1987, d. Wsly, J. P., pp , Bnjain Wsly, 771 Blubrg, Wst Grany (1987); J. Phys. D,, 849 (1989). [11] Dhlt, H., Exprints n th Structur f an Individual Elntary Particl, Scinc 47, pp (Fb., 1990). [1] Evans, R. D., Th Atic Nuclus, pp. 140, 149, McGraw-Hill, Nw Yrk (1955).

Spinning Charged Ring Model of Electron Yielding Anomalous Magnetic Moment

Spinning Charged Ring Model of Electron Yielding Anomalous Magnetic Moment Brgan & Wsly 1 GE1, ing Mdl f Elctrn Spinning Chargd ing Mdl f Elctrn Yilding Analus Magntic Mnt David L. Brgan J. Paul Wsly P.O. Bx 767306 Wihrdastrass 4 swll, GA 30076-7306 USA 771 Blubrg, Grany Suary: