general theory so this concept can Weber electrodynamics extendedto standard science. Here the sec ' include radiation causality does. 0H0 1 J.P. Wesley Conclusion Weiherdammstrasse 24, 7712 B/umberg, FFiG beg/5:,Th and other, speculations are all very ‘ ' Received: June 1985 en matrices and elliptic functions web but for. realizatio ' Chitfas to be forged in detail. Sotam only a case oft a recrpe’ 1'n search ofa n the mH , or better, acordon bleu. ”b References 1 Bohm D t (1980)., ., V/iO/eness and [he2 Imp/(care‑ Order. Routledge & Kevan P1l Ld 2 Clendinnen ' 3‘ (u I 'LOHdOt (1981). 1.1., 'Phase termsI -‘ the iAXlOlTlv of~Choice', Specular. Sci chlznol 44 . Abstract Weber's law for the force between moving charges satisfies Newton’s Third Law, the conservation of energy, and yields Ampere‘s original empirical law for the force between current 4a gag/”10]”Clendinnen,) 6(5),l.l.,521‘Furt‐529her(1983)Remarks on Phase' versus the Axiom of Choice',I.Specular”“3me5n elements. Weber's law predicts the observed force between two closed current loops. It yields the observed temporal and motional electromagnetic induction, and the observed force on Ampere’s World,oten. PJ.7, tand Hersh. , R., 7'"\ll on‐Cantorian’ Set Tl " ' ' I aw is derived. This Weber field Bergm' pp. _12_220‘ W.H. Freeman. San Francisco “3:010 . .IWHI/ICHIHIICS in the Modem bridge. The electromagnetic field appropriate for the Weber force 1 is then extended to rapidly varying effects and radiation by introducing time retardation. An ann, P.G., Introduction to [I r _ . ( - )5).‑ additional electromagnetic wave, the Weber wave is obtained. Absolute space is included by York (1942). He [72me of Relamwy, Chap. 17. Prentice‐Hall. Net introducing time retardation using the phase velocity observed in a moving system as predicted by 6 Hancock ( ,H., Tl ‘I ’ ' the Voigt‐Doppler effect. 7 (1976).Well, A., EllipticwgiizgfIonsEllipticAccordingFunctions,to EisensteinDover Publicationsand KroneckerNewpYork89 Sprinogl(1953‘ 3“” . , , a , 2. 8 Cayle 9 Margenau,‘ Yr A-,H.,An TheEleme”WY. Treatise- ' on Elliptic Functions, Bell & Sons, London (1876). Introduction Nat - ,.~ . on 10 Edwards, A~W-F., FOIUIZZIIgeI'Of Eritrea! Rea/11% P422, McGraw-Hill, New York (1950). The electromagnetic theory of Weber,1 as first presented in 1846, is based (1977). [OILS ofMtJIhe/natical Genetics, p.21, Cambridae University Press D . the force between two moving point charges using the separation distance, the ll To”, .‑ plxms, R, and B' relative velocity, and the relative acceleration. His theory is the only electro‑ (1974). 1rd, C" The 566"” Life Of Plants, p.188, Penguin Books, London magnetic theory ever proposed th at satisfies both Newton’s Third Law and the Gordon R and y varying (1978). , ‘y Jacobs 0“: AG,«The Shaping‑ of Tissues in Embryos‘, Scicnr. Am, 84 conservation of energy. Weber’s original theory was limited to 810W! effects where time retardation and radiation were not involved. It is an ‘action‑ at-a-distance’ theory, no fields being necessary. It is now known that electro‑ magnetic waves exist and that a field theory is required to describe such waves. This paper, therefore, first expresses Weber’s original action-at‐a‐distance theory in terms of fields and then second introduces time retardation to yield electromagnetic waves. in this way, it is possible to obtain an electromagnetic theory compatible with all of the evidence which is consistent wrth Newton 5 Third Law and the Conservation of energy. ' .~ .‑ Maxwell’s theory, being based upon the Biot‐Savart Law, does not satisfy Newton’s Third Law. Maxwell’s theory cannot, therefore, predzict the obseryed fOrCe 0“ Ampere’s bridge. Because Maxwell’s radiation theory 15ag6“?{?”j"' tion from erroneously assumed slowly varying effects it fails to predict 116 correct radiation field. An additional wave, the ‘Weber wave’, asderived here. is needed. Speculations in Science and Technology, Vol. 10, No. 1, page 47 The WEber force 49 J.P, Wesley Weber. assumed th at the ‑ Weber Electrodynamics posmons r3 61mg), U Of two interact' .1 no,- and it changes sign when th e subscripts 1and 3are interchanged to give and ri moving with the v 1110 D lenient Iidsi due to IzdSZ. €l0citie , iii: force oncurrent e The Biot‐Savart Law is quite U = qiqill/r ‐ (v-r)3/2c3r3] S‘3 and V1is and ‘Il at different; it 15 where c is the ve [‐ (d82‘d51)r + dsl(dso_~r)]/r3 (8) locity of light and (i) CZFB =1311dsg.\f(dsl.\fr)/r3 = 1211 r: ._ (8) clearly does notsatisfy Newton’s Third Law: the force r2 Tl, v=v,_vl The Biot‐Savart Law he subscripts i and 2does not yield The first term on th isnot directed along r, and interchanging t hat these laws energy. 6 right of equati '7 the negative for the force on 1,dsi due to Izdsz. It may be noted t V O“(I)- issimply the Couloinb L} When the s Porenlial (7) and (8) refer to moving charges by using the replacements the rate y C I ~ - t. 5 (11"1 = Ildsls (12V: 2 Izdsz (9) 5". Chan (3S i CNN 2 - . The absurdity of the Biot‐Savart Law (or any law violating Newton’s Third df (Clzflivr/r-‘m + V2/C3 _ 3(v _ -2 - .2 r) /2c°i +(r-dv/diyc2] (3) Law) can be immediately demonstrated. The self force on a closed current loop4 may be considered. Ampere’s Law (7), which satisfies Newton’s Third ing from each othe r and U dec ‑ Law, clearly yields zero for the self force on a closed current loop. Mecha‑ 5 result (3) may be written in therf‘oarslif nically coupling current element [2d52 togetlierwith current element Ildsi,the Biot‐Savart Law (8) predicts a net nonvanishing force on the two coupled the WebF er force‘ on charge (12 (4) due to charge q, is given by current elements equal to (10) w=(q2q1r/r3)[1 + 1/2/62 CZFB = 12111" X (dsl ><d52)/l'3 ‐ 3(v-r)2/2c2r3 + (r-dv/dt)/c2] (5) This result (5) satisfies 2 of a closed Integrating this result (10) with respect to C132 over a portion mterc - N€Wton’ ‘ . loop Chanoggifimg‘ SUbSCriptg l and 7 331331;? harp; smce the force is along r' and current loop and with respect to dsi over the remaining portion 1 of the yields a net force on the entire closed current loop equal to b e S] ‐ aln ti i ‘. The Weber Egréf 11““, the magnitude remailieirlfgrflfeon Charge 4] due to ([2) Secs?ma multiplymg. NCWIOH’SSCSFVesa 50 con energy. Assuming thatsame.the charges al l' . nd Law for a . €COnd Law fOr “trt' l 9 9 50 MW ”F” 2 Fl i it ><(dsi ><(ism/r3 (11) p mole 1 by V1 and Caddinoc y‘15€21S 1C6 ‘ by V3 and Newwn's which need not vanish. By altering the way in which the loop is divided into WhereV2-d“1:11;;7/Cl vi.dPi/dr =(v2 ‐ “le2_dU/dr (6) portlons 1 and 2, which is merely a matter of labelling, the force predicted by equation (11) can be altered. Nothing need be changed physically. Interchan‑ Syéifffble-in ‘ ~‘ Thel aretotaltheegreqmenltja ofthe particles. This result (6) is immediatel! ging labels 1 antl 2 reverses the direction of the force. This net force on aclosed wn Straps dent of then depends Only guy + T> Where T is the kinetic ener I of thE current loop could be used in principle to lift a spaCe Ship by its 0 boot from the forces inside the space ship itself. And no energy h‘ilS {'0 be expended. thus , energythe Pathis coflServed.or the rate ofponChangethe 6ndof POints.the systemThe betweentotal enerthggIethlnplfifrii:g) d L The absurdity is clear. it is frequently claimed that only the net ponderomotive force between two closed current loops can be observed to test Proposed laws for the force between current elements or moving charges. Since the Blot‐Savfll'l' Law he absurdity' the Law of Biot‐Savart (equation (8)) and the Ampere Law (equation (7)) Yield Precisely the same “a he it appear that force between two closed current loopS; it is frequently claimed that it (1.065 not matter which law is used. To try to argue from an obviously false Premlsc, [he i -’ IS often nixulilgtfdefs Olilg‘m’ empirical law3 is satisfied the Blot‐Savart Law violating Newton’s Third Law, in order to obtain a‘coi'rect nipelilpal law. For examplgr force laWS that are not compatible conclusion is hardly satisfactory scientific thinking- Moreover, Observations are mem 172141” Ampere’s orioi mlany text books label the Blot‐Sa‑ not limited to the force between two closed current l00p5_ Three independent Ce 2 " 87 a curbrentna empmcal law forat r1theseparatedforce FAonby element [ids] Observations not limited to the force between two closed current loOPS are readily available to test the validity of any proposed force law between cuff-em _ '3 + elements or moving Charges: (1) the force on one portion of a closqdpcunent lflarl)’ satisf‘ ydszTMdSITer-l (7) 100p due to the remaining portion of the 1 bemeasurcd’thc mpclc 00p can ‘ the ‘ 168 N , ~ . harge mOVing in 'cuum ln ' eWton 8 Third Law: The force is directed brldge experlment.3 (2) The force on a C a va WeberE/ectrodynamics form any I . (Persoml COmniunic. ca tical without artifiCial conductor movino igtlfin) has shown that Ffhe 213136 Odbserved. (3} Mill To obtain an exact result in closed mathema _ . localized Tl - ‘3 1‘3 Presence Of “C6 BleCtr‘ “ «fl “noul'iritiesm C it is convenient.