FIELDS AND RADIATION FROM A MOVING ELECTRIC CHARGE

Transcription

1 FIELDS AND RADIATION FROM A MOING ELECTRIC CHARGE Musa D. Abdullahi, U.M.Y. University P.M.B. 18, Katsina, Katsina State, Nigeria E-ail: usadab@utlk.c, Tel: Abstract The paper assued that the charge and ass f a particle are independent f its velcity relative t an bserver. A particle f charge Q and ass, ving at tie t with velcity v is assciated with electrstatic field E at an angle θ t the directin f velcity. A agnetic field H is generated due t velcity and a reactive electric field E a is prduced due t acceleratin. As a result f aberratin f electric field, E beces a dynaic electric field E v, shifted by aberratin angle α fr the statinary psitin. The ving particle eits electragnetic radiatin if it is accelerated. The agnetic field H carries n energy. The energy f E v cntains the electrstatic energy f the charge and the kinetic energy f the particle. The reactive electric field E a acts n the sae charge Q prducing it t create a reactive frce QE a equal and ppsite t the accelerating frce (dv/dt), thereby causing inertia. It is als shwn that E ½ Mc is the su f intrinsic energies f the electric charges cnstituting a bdy f ass M. The ttal energy beces E ½ M(c + v ) fr a ving bdy, in cntrast t the relativistic frula E Mc. Keywrds: Aberratin, acceleratin, charge, energy, fields, ass, ptential, radiatin, spherical shell, velcity 1. Intrductin 1.1 Cnfiguratin f a Statinary Electric Charge If a statinary electric charge is t assue any cnfiguratin, it is st likely t be a spherical shell f charge Q, with centre O, radius a and unifr surface charge σ as a bundary shwn in Figure 1. Such a cnfiguratin has a cnstant ptential U and a balanced Figure 1 A statinary particle f charge Q in the fr f spherical shell f radius a setting up electrstatic field f intensity E and ptential ϕ at a distance r utside the shell. 1

2 (zer) electric field inside the charge and electrstatic field f intensity E and ptential ϕ at a distance r utside the charge. An inward pressure at the uter surface balances an utward pressure at the inner bundary. Equilibriu is reached at a certain fratin and a stable particle is btained. Such particles shuld ccur in pairs f ppsite charges as a dublet. If there is a Gd-particle, such a pair shuld be the ne. A particle f charge Q and ass, ving at tie t with velcity v, has an electrstatic field f intensity E at an angle θ t the velcity. The electrstatic field f the ving charge generates a agnetic field H. As a result f aberratin f electric field, the electrstatic field beces a dynaic electric E v displaced by aberratin angle α fr the statinary psitin. A reactive electric field E a is prduced due t acceleratin. The ving charge exhibits a reactive frce r inertia QE a dv/dt. The ving charge eits electragnetic radiatin if it underges acceleratin. The ptential ϕ and the fields E, H and E a are well knwn in electrdynaics but nt E v. Deducing expressins fr the fields H, E v and E a and explaining the surce f electragnetic radiatin and the cause f inertia f a bdy are the purpses f this paper. The reactive electric field E a acts n the sae charge Q prducing it t cause the inertial frce r reactive frce QE a, equal and ppsite t the accelerating frce (dv/dt). This explains the cause f inertia as a reactin t accelerating frce. In the prcess, an equatin r frula E ½ c is derived as the intrinsic energy f a charged particle, in ters f ass that is independent f velcity. Fr a bdy f ass M, a cpsite f nuerus charged particles, the electrstatic energy, assued t be the intrinsic energy, beces E ½ Mc, in cntrast t the relativistic equatin E Mc [1, ], c being the speed f light is given by: 1 c (1) µε where µ is the pereability and ε the perittivity f free space r vacuu, as shwn by J.C. Maxwell [3]. 1. Kinetic Energy f a Mving Charged Particle Newtn s secnd law f tin gives the ipressed frce f n a particle f ass ving with velcity v at tie t, in ters f its acceleratin dv/dt, as vectr equatin: dv f () dt Fr tin thrugh a distance s in a straight line, equatin () beces a scalar expressin: dv ds dv dv f v dt dt ds ds Wrk dne by an external agent in increasing the speed f the particle fr 0 t v is the kinetic energy k f the particle, given by the integral: k 1 f ds v dv v s v ( ) ( ) (3) In additin, the agent, such as an electric field, ay d wrk against a frictinal frce, which appears as heat r light radiatin [4]. The energy f agnetic field H f a ving electric charge ces t zer. The energy f the dynaic electric field E v is equal t the su f energy f the electrstatic field

3 E and the kinetic energy ½ v f the ving particle. While the well-knwn latitudinal agnetic field H plays a dinant part in the generatin f the reactive electric field E a and eissin f electragnetic radiatin, the less-knwn dynaic electric field E v is nt invlved in radiatin. The dynaic field E v serves as a strage ediu fr the electrstatic energy and kinetic energy f a charged particle ving with velcity v relative t an bserver. In the statinary state, there is n kinetic energy, nly electrstatic energy. 1.3 Ttal Energy f a Mving Charged Particle It is shwn in this paper that the ttal energy f a ving charged particle is the su f the electrstatic energy r intrinsic energy E and the kinetic energy k as given by: 1 1 E c + k ( c + v ) (4) Fr a bdy f ass M cpsed f equal nubers r equal aunts f psitive and negative electric charges, ving at speed v with kinetic energy K ½ Mv, the ttal energy E is: 1 1 E Mc K M c v ( ) + + (5) This is in cntrast t the relativistic ass-velcity frula, which akes E Mc [1, ].. Electrstatic Field, Ptential and Energy f a Spherical Charge A statinary charge Q shwn in Figure 1 is a spherical shell f radius a. The figure has intrinsic ptential U and intrinsic energy E ½ QU. It has electrstatic field E and ptential ϕ at a pint utside the sphere. Fr pints utside the shell, it is as if the charge Q is lcated at the centre f the sphere..1 Electrstatic Field f a Spherical Charge Gauss law [4, 5] gives the electrstatic field intensity at a pint distance r utside the spherical shell aking the charge Q, as: Q ϕ E û û ϕ (6) 4πε r r where ε is the perittivity f vacuu, ϕ is the electric ptential at a pint distance r fr the centre f the charge, û is a unit vectr in the radial directin and ϕ dentes the gradient f a scalar ϕ. A clsed surface inside the shell des nt enclse any charge. S Gauss law akes the electric field inside the spherical shell zer and the ptential there beces a cnstant U.. Intrinsic Ptential f a Spherical Charge In Figure 1 the space inside the spherical shell is a regin f cnstant ptential U. But the ptential ϕ varies inversely as the distance utside the sphere, thus: r Q ( ) Q ϕ dr (7) 4πε r 4πε r The electric ptential is the wrk dne in bringing a unit psitive charge fr infinity t a pint distance r fr the charge. The intrinsic ptential U at a pint n the surface f a spherical shell f charge Q and radius a, as well as inside the shell, is given by: Q U πε a (8) 4 3

4 .3 Intrinsic Energy f a Spherical Charge The electrstatic energy r intrinsic energy E is wrk dne, against the electric field, in cpsing the agnitude f the electric charge fr 0 t Q, at cnstant radius a, by infinitesial aunts dq, thus: Q q Q 1 E ( dq) QU 0 4πε a 8πε a (9) Equatin (9) ay als be btained by integrating the energy density thrughut vlue f space ccupied by an electrstatic field E, t give the electrstatic energy as: 1 Q 1 E ε ( ) E d QU (10) 8πε a 3. Electric Field and Energy f a Mving Electric Charge Figure shws a particle f charge Q at a pint O, with its electrstatic field E, ving at tie t at velcity v and acceleratin dv/dt. The ving charge is assciated with a agnetic field H, ut f the page n the left-hand side and int the page n the right. If the charge underges acceleratin a reactive electric field E a is prduced. As a result f aberratin f electric field, the field E alng OP in the directin f velcity f light c, at an angle θ t the velcity, appears displaced fr the statinary psitin t E v, alng ON at angle (θ-α) t the velcity v, as shwn in Figure. The dynaic electric field E v is put as: E ( ) E c v cv θ E v v Ev c+ v ± ô + + cs ± ô 1+ + csθ (11) c c c c where c is the speed f light and θ is the angle between the vectrs v and c. Figure An electric charge Q, and electrstatic field E, ving at tie t with velcity v and acceleratin dv/dt generating a agnetic field H and reactive electric field E a due t acceleratin. Due t aberratin f electric field E, at an angle θ t v, beces dynaic field E v, displaced thrugh angle α fr the statinary psitin. Fr θ 0, equatin (11) gives: E v v v ô E 1+ E 1+ c â c (1) 4

5 Fr θ π radians, equatin (11) gives: E v v v ô E 1 E 1 c â c (13) Fr θ π/ radians, equatin (11) gives: v Ev ± ô E 1+ (14) c 3.1 Energy f a Mving Electric Charge Equatin (11) gives Energy E n f the dynaic field E v as vlue integral: 1 1 v v E ε ( ) 1 cs ( ) n E d ε E θ d v + + c c The last ter in the abve equatin is: 1 0 v 1 v ε cs ( ) cs ( ) cs sin ( ) 0 E θ d ε E θ πr θ θ dθ c π c The energy E n beces: 1 v En ε E 1 ( d) + c (15) Equatin (15) cnsists f intrinsic energy f the charged particle, as: 1 E ε E ( d) (16) and kinetic energy K f the particle as: 1 v 1 K ε E ( d) v c The intrinsic energy E is the btained, fr the kinetic energy K and equatin (16), as: 1 1 E ε E( d) c (17) 3. Energy f a Rtating Electric Charge Fr a particle f charge Q and ass in the fr f spherical shell f radius a, rtating with angular velcity ω there is n tin f the centre f charge r centre f ass. As such the speed v, in equatin (11), is 0. Only the energy f the electrstatic field (equatin 16) reains. The radial electrstatic field E is perpendicular t the directin f latitudinal velcity, given by vectr prduct ω r êωr(sinθ) in spherical crdinates (r, θ, φ) with unit vectrs (û, î, ê). There is a lngitudinal agnetic field H as vectr prduct, given by: H êεwr( sinθ) E î εwr( sinθ) E Magnetic field at the surface f the spherical charge f radius r a beces: îwq( sinθ) îwq( sinθ) H 4πr 4πa A agnetic field des n wrk. Hwever, rtating electric charges, such as electrns in atter, can aintain a agnetic field in a peranent agnet. 4. Reactive Electric Field due t Acceleratin In Figure 1, the agnetic flux intensity B µ H due t field E fr an electric charge Q ving with velcity v, and setting a ptential ϕ, is given by the vectr prduct: 5

6 B µε v E µε v ϕ µε ϕv A (18) A µεϕv (19) where µ is the pereability and ε the perittivity f space, dentes the gradient f a scalar, dentes the curl f a vectr and A µ ε ϕv is the agnetic vectr ptential in the directin f velcity. A agnetic field, changing with tie t, sets an electric field E a, given by Faraday s law and equatin (19), as: B A E a (0) Equatin (19) and (0) give the reactive electric field E a, in ters f acceleratin, as: A dv E a µεϕ (1) dt The cardinal prpsal here is that the reactive field E a, given by equatin (1), acts n the sae charge Q prducing it t create a reactive frce equal and ppsite t the accelerating frce, s that: dv dv E aq µεϕ Q () dt dt The prduct ϕq is zer everywhere in space except at pints cntaining electric charges. If ϕ is the final ptential and Q the final charge, the prduct gives twice f the wrk dne in assebling the charge fr zer. This is equal t the electrstatic energy E f the charge. Equatin () then beces: µε E 1 1 E c (3) µε 4. Radiatin fr an Oscillating Electric Charge Fr a charged particle scillating with angular frequency ω and velcity v v sinωτ f aplitude v, the agnetic vectr ptential A (equatin 19), easured by an bserver at a pint distance r fr the charge and at present tie t and retarded tie τ t r/c, is: A µ ε ϕv sinωτ µ Qv 4πr sinω t r c (4) where c is the speed f light and r/c is the tie taken fr an effect at the lcatin f the charge t be felt at a pint distance r fr the charge. The retardatin f vectr ptential in equatin (4), resulting fr finite speed f light, is respnsible fr radiatin. Equatins (18) gives B and E a as: A µ Qv 1 r ω r B A û û sin ω t + cs ω t (5) r 4π r r c c c A ωµ Qv r E a csω t (6) 4π r c Taking the vectr (crss) prduct with E a and H gives the Pynting vectr: Q r 1 r ω r Ea H ωµ v ( û v ) csω t sinω t + csω t (7) 4π r c r c c c 6

7 Equatin (7) gives the pwer flw acrss unit area. This is ade up f pwer P s stred in the kinetic energy f the charge and pwer P r dissipated as electragnetic radiatin int space. The ttal pwer flw P, acrss surface area S surrunding the charge, is: Q r 1 r ω r P ωµ csω t sinω t cs ω t ( ).( d ) S + 4π r c r c c c v û v S Qv ωµ r µ ω r P sin ω t cs ω t sin θ( ds) S + 4π r r c c c v û. v (d S ) v sin θ ( ds ) where: ( ) ( ) ωµ µ ω sin ω 1 cs ω sin θ( ) Qv r r P t t ds S + + 4π r r c c c (8) In equatin (8) the ters with sinω and csω give zer average pwer ver ne cycle. The real r steady ter indicates dissipatin f pwer. Average pwer radiated, is: µ 0 ωqv c π 4π r µ ωqv c S 4π r µ 0 ωqv c π 4π r 3 π r sin θ( dθ) ( Qv ) sin θ ( ds) 3 π r sin θ( dθ) Qv 3 3 µ 0 ωqv 3 πsin θ( dθ) c π 4π µ ω µ r sin ( d )( d ) Qv c S θ ϕ θ ω 4πr 3πc µ ω (9) 1π c 0 where: sin θ( dθ ) π 3 The energy radiated, in tie T, is TP r. Equatin (9) shws that whenever an electrically charged particle is sehw disturbed r shaken, it eits electragnetic radiatin f pwer prprtinal t the square f prduct f aplitude f disturbance and frequency f scillatin. S, electric charges scillating in the electric fields f an at als eit crrespnding electragnetic radiatins at the icrwave frequencies f scillatin. This is in additin t eissin f heat and light radiatin, as shwn by the authr [4], due t frictinal tin f charged particles, in the electric field f the nucleus f an at r in the electric fields between the charged particles. This ay explain the surce f icrwave backgrund radiatin [7] Inertia f a Mving Electric Charge An electric charge f agnitude Q in the fr f spherical shell f radius a, under acceleratin, prduces a reactive electric field E a as given by equatin (1). This field acts n the sae charge Q prducing it t create a reactive frce F equal and ppsite t the accelerating frce, in accrdance with Newtn s secnd law f tin, thus: Q µε a Q ϕ v v F E (30) Equatin (30) explains the rigin f inertia. ( ) 7

8 At the lcatin f the charge, the ptential ϕ U and zer everywhere else, and, therefre, equatin (30) beces: UQ v v µε (31) µε UQ enµε (3) where E is the intrinsic energy f the charge as given by equatin (10). Substituting fr U fr equatin (8) we btain equatin (3) as: µ Q E Eµε (33) 4π a c Equatin (33), expressing ass in ters f electrical quantities, gives: 1 E c (34) 6. Cnclusin Equatins (1), (13), (14), (15), (17) and (9) are what this paper set ut t derive. The dynaic electric field E v is issing in electrdynaics. The kinetic energy f a ving charged particle is cntained in the dynaic electric field E v. Electragnetic radiatin by an scillating charged particle, with pwer as given by equatin (9), is in additin t eissin f heat and light radiatin by scillating atic particles. This ay be a pssible explanatin f icrwave backgrund radiatin detected in radi astrny. The paper als succeeded in explaining the cause f inertia as due t the reactive electric field E a, generated by an accelerated charged particle, acting n the sae charge t prduce the inertial frce equal and ppsite t the accelerating frce. Fr a bdy f ass M cpsed f electrically charged particles ving with velcity v at tie t, the reactive electric fields act at the lcatin f the respective charges t prduce the su f the inertial frces equal and ppsite the accelerating frce M(dv/dt). In this paper, the ass f a charged particle is expressed in ters f electrical quantities and the ass-energy equivalence law fr a bdy f ass M ving with speed v is derived as E ½ M(c + v ), where ½ Mc is the electrstatic energy f the electric charges cpsing the bdy. The ass-energy equivalence law, derived here fr basic electrical principles, differing fr the relativistic equatin by a factr f 1/, is ntewrthy. 7. References [1] M. Jaer; Cncept f Mass in Classical and Mdern Physics, Cabridge, MA, Harvard University ess (1961) [] M. Lange; The st faus equatin, Jurnal f Philsphy (001), 98, [3] J. C. Maxwell; A Treatise in Electricity and Magnetis, Oxfrd, 3 rd ed. (189) Part 1 [4] M. D. Abdullahi An Alternative Electrdynaics t the Thery f Special Relativity Online at: [5] D. J. Griffith; Intrductin t Electragnetis, entice-hall, Englewd Cliff, New Jersey (1996) [6] I. S. Grant & W. R. Phillips; Electragnetis, Jhn Wiley & Sns, N, Yrk (008) [7] en.wikipedia.rg/wiki/csic_icrwave_backgrund 8