David L. Bergan 1 Inertial Mass Inertial Mass f Charged Eleentary Particles David L. Bergan Cn Sense Science P.O. Bx 1013 Kennesaw, GA 30144-8013 Inertial ass and its prperty f entu are derived fr electrdynaic effects n the spinning charged ring del f eleentary particles. By this result, a fundatin is laid fr Newtn's laws f echanics withut assuing intrinsic ass r nn-causal actins at a distance. Internal structure f eleentary particles is deduced, and the easured ass values fr eleentary particles require the charge distributed near the surface t have a finite thickness apprxiately equal t 0.174 percent f the particle's thickness diensin. INERTIAL MASS An iprtant prperty f atter is its tendency t aintain entu, and a bdy in tin aintains its velcity unless an external frce is applied. External frces can accelerate r decelerate atter in accrdance with Newtn's secnd law f tin: F where represents inertial ass. = a () 1 Gravitatinal ass and inertial ass are distinct prperties f atter that riginate by fundaentally different echaniss and relate t different frces. The frce f gravity is an attractin f tw bdies fr each ther. One bdy with ass exerts an external frce upn a secnd bdy in prprtin t the prduct f the tw asses. This paper is cncerned with inertial ass nly and des nt deal with gravitatinal ass. Inertial ass is nt an intrinsic prperty f atter but arises fr electrstatic charge and current that exist in the eleentary particles. All atter is cpsed f eleentary particles and receives its prperty f entu fr the charges and currents f its eleentary particles. The external structure f the eleentary particles has been specified by the spinning, charged ring del where electrstatic charge f ne quantu rtates with ri velcity equal t the speed f light. Earlier papers 1, shwed that electragnetic effects accunt fr the fundaental prperties f the charged eleentary particles, including size, agnetic ent, spin, and ass-equivalent energy. All these prperties are related t external electrstatic and agnetstatic fields. The effect f inertia, hwever, depends upn the interir charge distributin and crrespnding fields inside the trid. Inertial ass is the result f a reactin frce between charge (including currents, r ving charge) and the electragnetic fields induced by the tin f acceleratin. This eans that the inertial reactin takes place where the charge is lcated, i.e, within the trid. Thus, inertial ass f an eleentary particle is directly related t its internal structure (specifically, the distributin f its charge).
David L. Bergan Inertial Mass INTERNAL STRUCTURE OF CHARGED PARTICLES Due t the Culb frce f repulsin, charge in the ring is pushed utward t the surface f the ring, leaving the interir hllw. The exact distributin f charge is unknwn, but t a first apprxiatin, the charge resides in a thin shell f finite thickness t just belw the surface f a trid, as illustrated in figure 1. Fr purpses f calculatins, the charge within the thin shell is cnsidered t be unifrly distributed. At every pint where charge resides, the utwardly directed Culb frce f the charge is balanced by an inwardly directed agnetic pinch frce n the ving charge. The balanced cnfiguratin ccurs when t/r = 0.174 percent. This cncentratin f charge prduces the easured values f inertial ass f the free electrn (and als the easured inertial asses f the ther charged eleentary particles, i.e., the prtn, psitrn and antiprtn). R t r a Figure 1. Thickness f the Surface Charge n a Spinning Ring Charge is unifrly distributed in a thin shell at the surface f the trid. INERTIAL MASS OF CURRENT Huphreys 3 has shwn that a current resists any frce attepting t laterally accelerate the ving charge, thus revealing that electric current has inertial ass. His analysis shws that by Maxwell's First Equatin, a ving current induces an electric field. Then, fr Maxwell's Secnd Equatin, an accelerated electric field prduces an induced agnetic field within the regin f the flwing current. The induced agnetic field interacts with the ving charge t prduce the reactin frce f Newtn's secnd law; this frce f reactin is, f curse, the phenenn f inertia. Using this apprach, Huphreys fund, fr the case f unifr current density in a lng circular cylinder, that the inertial ass l fr agnetic effects per unit length is l I = 3 µ 16π c ( ) where I is the electric current, c is the speed f light and µ is the pereability f free space.
David L. Bergan 3 Inertial Mass INERTIAL MASS OF CURRENT IN A SPINNING RING Althugh Huphreys derived inertial ass fr current flwing in a cylinder, his slutin can be adapted t a spinning charged ring. Fr an eleentary particle, the ring is very thin r << R, and the curvature f a trid is negligible ver any shrt sectin f the ring. When the calculatins invlve nly the trid's surface and interir, then the results btained fr an infinitely lng cylinder can be applied ver a length πr t prvide an expressin f inertial ass fr a thin ring f radius R. Inertial ass fr the agnetic field f the spinning ring is btained by ultiplying the circuference and ass per unit length: = π R = l I R 8c () 3 Fr balanced frces at the tridal surface, charge near the surface f free electrns (and ther charged eleentary particles) ust rtate at a rate ω = c/r. 1 Using I = ωe/π, the agnetic inertial ass f a spinning ring is e = 3 µ 3π R ( 4) where e is the electrn charge. Equatin (4) applies t the case f unifr current density thrughut the interir f the ring; but, actually, the current f an eleentary particle is flwing near the surface f a ring with n current in the ring's interir (the ring is hllw). The effect f charge cncentratin in a thin shell beneath the ring surface is a larger inertial ass. In the next sectin, an apprpriate adjustent is deterined t accunt fr the hllw interir. INERTIAL MASS OF CURRENT IN HOLLOW RING The principle f superpsitin is used, alng with equatin (4), t deterine the cntributin f current t the inertial ass f a hllw, spinning ring. Cnsider tw cplanar and caxial rings, each with radius R. One ring has a thickness radius r that specifies the uter surface f the trid; the ther ring is slightly thinner, with thickness radius equal t a = r - t. Each ring has the sae unifr current density j but the currents are in ppsite directins. Then, the superpsitin f these tw rings prvides the hllw ring illustrated in figure 1. This cnfiguratin has a unifr charge and current density in the trid shell charge layer f thickness t. Let k represent the rati f the inner and uter radii f the shell as illustrated by the crss sectin f the ring illustrated in figure 1. k a / r ( 5) The current density is related t the ttal ring current I by
David L. Bergan 4 Inertial Mass j = r I ( 1 k ) π ( 6) The cpnents f current in the tw rings f thickness radii a and r are, respectively I = Ik a π a j = k ( 1 ) I = I a r j = k π ( 1 ) ( 7 a) ( 7 b) Equatin (3) is used t btain the net inertial ass fr current by suatin f the tw current cntributins: = a + r ( 8a) ( Ir Ia ) R = 8c ( 8 b) Inserting equatins (7) int (8b) gives I R 1 = 8c 1 ( + k ) ( k ) ( 9) Cparisn f equatins (3) and (9) reveals that the inertial ass is greater when the current is cncentrated in a shell at the surface f the trid. This is because ptential thery and the law f cnservatin f energy require re ptential energy t accuulate when charge eleents f like sign and repulsive frce are cpressed int saller vlue. With the cnditin fr a balance f Culb and Apère frces n the ving charge eleents f a ring ω = c/r) and the relatinship between current and spinning charge (I = ωe/π), equatin (9) can be written as INERTIAL MASS OF CHARGE e 1 = 3π 1 ( + k ) ( k ) ( 10) An eleentary particle with distributed electrstatic charge als pssesses inertial ass e. Unlike the case fr current, the cntributin fr charge is unrelated t the tangential velcity f the charge. It can be shwn that charge and current cntribute equal aunts f inertial ass when the current cnsists f charge ving with velcity equal t the speed f light.
David L. Bergan 5 Inertial Mass An equatin fr the inertial ass f accelerated electrstatic charge can be derived, fr Maxwell's first and secnd laws fllwing the sae apprach used by Huphreys fr current. In the spinning charged ring, where charge rtates with tangential velcity at the speed f light, charge and current cntribute equal aunts t the inertial ass f an eleentary particle. Fr equatin (10), this inertial ass is = + ( 11a) e e 1 = 16π R 1 ( + k ) ( k ) ( 11b) Mass and radius are inversely related in the eleentary particles; accrding t equatin (11b), a prtn (with ass greater than an electrn's ass) will be fund t be saller than an electrn. The sae relatinship was previusly denstrated by equatin (6) f reference []. THICKNESS OF THE CHARGE LAYER Equatin (11b) can be rewritten t deterine k, the rati f shell radii that define the thickness f the charge layer. k = 16π R e 16π R + e ( 1) where the prduct f radius and ass has a cnstant value (see equatin (7) f reference []) with given by standard references and R is btained fr the easured value f agnetic ent µ, R = µ/ec. Substituting this value f R int equatin (1) gives the value f k fr all the charged eleentary particles: k =. 99861 ( 13) The thickness t f the charged layer at the surface f the eleentary particles is given by t r ( k) = 100 1 = 0174. % ( 14) Since the thickness rati r/r is identical fr all the charged eleentary particles, and since the rati t/r is als identical in all the particles, the shapes f the eleentary particles are identical in every respect. SUMMARY The phenenn f entu has been theretically derived fr lateral acceleratin in the axial directin. Acceleratin in ther directins is re cplicated and nt analyzed here. The ring del specifies the shape and size f charged eleentary particles. In additin, the structure f charged eleentary particles has been described; an eleentary particle cnsists f
David L. Bergan 6 Inertial Mass its cnstituent aterial (electrstatic charge) and its structure (a thin shell at the surface f a trid). Evidently, the essential features f an eleentary particle are its charge (ne unit f psitive r negative sign) and energy (sall as in electrns and psitrns, r large as in prtns and anti-prtns). This descriptin f atter is useful t distinguish atter fr light, which has energy but n charge. Charge lcated in the thin tridal shell was assued t be unifrly distributed. Use f this apprxiatin eans that the structure described abve fr an eleentary particle is an apprxiatin t the actual structure f an eleentary particle. Anther wrk fund in these prceedings 4 shws that Maxwell's First Equatin (placing Faraday's Law f Magnetic Inductin int differential fr) is incrrect when a significant inductin effect is invlved r when high relative velcity between tw bjects is invlved. The apprxiatins ade in Maxwell First Equatin and its applicatin here t an accelerated charged bdy prduce a crrespnding apprxiatin in the result btained fr electrical inertia; the apprxiatin is sall fr sall acceleratins. And, the inertial effect is undubtedly real because the apprach taken is fundaentally based n the fundatinal laws f Culb, Apère, and Faraday. CONCLUSIONS The ring del and well established laws f electricity and agnetis predict the physical prperty f entu expressed by Newtn's secnd law. Inertial ass is a prperty f charges and currents. In the eleentary particles, charge and current prvide equal cntributins t the inertial ass f each particle. N assuptin f inherent ass is required t accunt fr the prperty f inertia bserved in all aterial bjects. Rather, using fundaental laws f electricity and agnetis, the inertial ass f an bject and its prperty f entu have been derived fr first principles. Matter and its prperty f inertia are fund wherever charge is fund. By eans f field thery and a few laws f electricity and agnetis, Maxwell was able t unify theries f light, electricity, and agnetis and explain a wide variety f bserved natural phenena particularly thse related t frces and radiant energy. Other wrks 1, have even denstrated the electrical character f the fundaental attributes f atter. Even inertial ass is shwn here t be a derived feature f electrical charges and a feedback effect f induced fields that exert a retarding frce n any accelerated charge. A fundaental apprach based n first principles has prduced a significant cntributin t the scientific gal f a unified field and frce thery. REFERENCES [1] Bergan, D. L., and Wesley, J. P., Spinning Charged Ring Mdel f Electrn Yielding Analus Magnetic Ment, Galilean Electrdynaics 1, 63-67 (Septeber/Octber 1990). [] Bergan, D. L., Spinning Charged Ring Mdel f Eleentary Particles, Galilean Electrdynaics, 30-3 (March/April 1991). [3] Huphreys, R., Appendix II, Inertial Mass f an Electric Current, Physics f the Future (by T. G. Barnes), pp. 195-0, Institute fr Creatin Research, El Cajn, CA (1983). [4] Bergan, D. L., Frces n Mving Objects, t be published in Hadrnic Press Suppleent.