How a charge conerving alternative to Maxwell diplacement current entail a Darwin-like approximation to the olution of Maxwell equation 12 ab Alan M Wolky 1 Argonne National Laboratory 9700 South Ca Ave Argonne IL 60439 2 5461 Hillcret Ave Downer Grove IL 60515 a E- mail: AWolky@ANLgov b E- mail: AWolky@ATTnet Abtract Though ufficient for local conervation of charge Maxwell diplacement current i not neceary An alternative to the Ampere-Maxwell equation i exhibited and the alternative electric and magnetic field and calar and vector potential are expreed in term of the charge and current denitie The magnetic field i hown to atify the Biot-Savart Law The electric field i hown to be the um of the gradient of a calar potential and the time derivative of a vector potential which i different from but jut a tractable a the implet vector potential that yield the Biot-Savart Law The alternative decribe a theory in which action i intantaneou and o may provide a good approximation to Maxwell equation where and when the finite peed of light can be neglected The reult i reminicent of the Darwin approximation which aroe from the tudy claical charged point particle to order v/c 2 in the Lagrangian Unlike Darwin thi approach doe not depend on the contitution of the electric current Intead thi approach grow from a traightforward reviion of the Ampere Equation that enforce the local conervation of charge PACS 4120-q 4120 Cv 4120 Gz
I Introduction Though ufficient for local conervation of charge Maxwell addition of hi diplacement current to Ampere Law i not neceary Other addition could have been made with the ame effect a decribed elewhere[1] Here we how that one uch alternative to Maxwell diplacement current entail an expreion for the magnetic field that i given by Biot- Savart Law The ame alternative entail an expreion for the electric field that i reminicent of Darwin approximation of the olution of Maxwell Equation an approximation meant to be ued when the ource are relatively low moving claical point charge[2] Unlike Darwin Approximation our approach doe not rely on the contitutive equation for the current Rather we preent a macrocopic theory that i characterized by it alternative to Maxwell diplacement current The ret of thi paper i organized a follow Section II decribe the alternative to the diplacement current The magnetic field and vector potential that derive from the alternative are dicued in Section III Section IV i devoted to the concomitant electric field It calculation in term of it ource charge and current denity appear difficult but after the work decribed herein the final reult i not qualitatively more difficult to evaluate than the Biot-Savart Law Section V ummarize and conclude A preliminary remark may be helpful When dicuing the olution to Maxwell equation attention i focued on olution that decribe field that tend toward zero a the point of obervation recede farther and farther from the field ource Further the field ource charge and current are aumed to lie in a bounded region or to diminih fat enough at infinity to permit one to neglect urface integral at infinity The ame aumption will be made here II An alternative to Maxwell diplacement current For the reader ready reference thi ection recall ome obervation made in an earlier paper which preent a more thorough and omewhat more leiurely dicuion of them[1] In term of the field the four pre-maxwell Equation are a hown below [ ] = ρ ε 0 II1 [ ] = 0 II2 [ ] = t B II3 [ ] = µ 0 J II4 div E div B curl E curl B where ρ denote the total free and polarization charge denity and J denote the total free and magnetization current denity And we mut not forget EqII5 the fifth pre-maxwell Equation; it expree the local conervation of charge a requirement that cannot be compromied div J [ ] = t ρ II5 Indeed EqII5 doe more than aert current conervation It relate the current denity that i the ource of the tatic magnetic field to the charge denity that i a ource of the tatic electric field
The firt four pre-maxwell Equation decribe ituation in which the divergence of the current denity J i zero that i to ay ituation in which the charge denity i time independent Of coure thi exclude conideration of very important ituation for example ituation in which a capacitor plate are connected to an alternating current Another example i provided by an antenna excited by an ocillating current Thu one want to know what happen where and when the charge denity i changing A i widely appreciated the pre- Maxwell Equation cannot provide an anwer The divergence of the left hand ide of Ampere Equation EqII4 i alway zero and o only current with zero divergence can appear on the right Thu we come to the quetion mentioned in every textbook How hould Ampere Equation be revied to decribe all ituation? Maxwell anwer wa to add the term t ε 0 E and thereby arrive at what i often now often called the Ampere-Maxwell Equation curl B II6 [ ] = µ 0 J t ε 0 E It doe the job becaue we believe in the local conervation of charge a expreed by EqII5 And o the vanihing divergence of the left-hand ide of EqII6 i matched to the vanihing divergence of the right-hand ide Alternatively one could ay that EqII6 entail the local conervation of charge omething we are predipoed to believe In ummary the Ampere- Maxwell Equation i ufficient to guarantee local charge conervation But i it neceary? I there a different reviion to the Ampere Equation that would achieve the ame reult? Well ye there i That different reviion come to mind when one recall that the divergence of the gradient i the Laplacian and o one ha EqII7 % 1 div x ' x &' 4π x y = Δ 1 x 4π x y = δ 3 x y [ ] II7 Thu EqII8 preent a different generalization of Ampere Equation curl[ B[ tx] ] = µ 0 J[ tx] x d 3 y tρ[ ty] 4π x y - II8 After taking the divergence of both ide one confirm that EqII8 entail the local conervation of charge and thu our addition preent an alternative to Maxwell diplacement current Maxwell called the um of the conduction current and hi diplacement current the true current and denoted it by C[3] We will denote the analogou quantity in EqII8 by J ATM where ATM tand for Alternative to Maxwell III How the atm magnetic field depend on current The unuually attentive reader of either Panofky and Philip or Jackon might remember the olution of EqII2 and II8 [456] It i the familiar Law of Biot-Savart B ATM [ tr] = [ ] r r d 3 r µ 0 J tr III1 3 4π r r
TherighthandideofEqIII1canbere6expreedathecurlofavectorahown ineqii10 & B ATM d [ tr] 3 r = curl µ 0 J[ tr ] r 4π r r 1444 2444 3 ATM ' A B S In thi form the argument of the curl which will be denoted by A ATM B S bring to mind the vector potential that often appear in dicuion of magnetotatic In that ituation it divergence i zero and o it atifie the definition of the Coulomb gauge In general it doe not Indeed traightforward manipulation how that the divergence of the Biot-Savart vector potential A ATM B S i proportional to the divergence of the current denity J div r ATM d [ A B S ] 3 r div r 4π & ' [ ] µ 0 J tr r r [ [ ]] III2 = d 3 r div µ 0 J tr III3 4π r r Of coure the Biot-Savart vector potential i not the only one that give the ATM magnetic field One could add the gradient of a calar For example one could add the econd term on the right hand ide of EqIII4 and thereby obtain a divergence-free vector potential The reult i ATM Coulomb gauge vector potential A C ATM which i defined by div A A C ATM [ tr] = A ATM B S div µ 0 J tr [ ] = 0 [ ] [ ] d 3 r d [ tr] 3 z 4π r z r 4π r z 144444 42 444444 3 Γ B S C [ [ ]] III4a d 3 r div µ 0 J tr A ATM d C [ tr] 3 r = µ 0 J[ tr ] d 3 z 4π r 4π r r r z 4π r z III4b We draw attention to the Coulomb Gauge becaue it i often ued in tandard Maxwell electrodynamic And we will ue it to expre the electric field in term of it ource
IV$How the ATM electric field depend on charge and current$ Until now our focu ha been on the magnetic field But what of the electric field? Jut a with Maxwell Equation Faraday Law curl E [ ] = t B and the abence of magnetic pole div[ B] = 0 together entail that curl[ E t A] = 0 or E = ϕ t A Thu jut a i Maxwell electrodynamic ATM i gauge invariant We chooe the Coulomb gauge becaue in that gauge the equation div[ E] = ρ ε 0 entail the familiar Poion Equation 2 ϕ = ρ ε 0 having a it olution the familiar Coulomb potential Thu ATM electric field i given by EqIV1 E ATM = ϕ C t A ATM C = ϕ C t A ATM B S t Γ B S >C While both the Coulomb calar potential and the Biot-Savart vector potential are familiar and more or le tranparent the complex expreion denoted by Γ B S C i not Indeed it appear unfamiliar and difficult to evaluate becaue of it ixfold integration which i a convolution of two three dimenional integral Depite thi the following will how that it i not a bad a it look In fact the following will how it i no wore than the Biot-Savart vector potential We wih to implify Γ B S C Γ B S C [ [ ]] IV1 d 3 r div µ 0 J tr d 3 z 4π r z 4π r z IV2 In particular we wih to clearly ee the relation between the current denity at the ource point r and the vector potential at the field point r Remarkably thi can actually be arranged To do o we re-expre the integrand div J [ ] r z a div r J/ r z [ ] J r 1 r z and note that the urface integral concomitant with the total divergence i zero becaue of the current denity bounded upport Thu we have Γ B S C [ tr] d 3 r d 3 z 4π J [ tr 1 ] r r z -- 4π r z IV3 We recat the above by evaluating the gradient with repect to r the ource point and then interchanging the order of r and z integration The reult i hown in EqIV4 Γ B S C [ tr] = d 3 ' r 4π µ 0 J [ tr d ] 3 ' z 1 z r IV4 3 4π r z z r The above preent an integral over z for each ource point and each field point Thi i a purely geometric problem If the integral could be evaluated in term of elementary function the reult
would diplay what i now implicit the effect of the current denity J at the ource point r on the value of Γ B S C at the field point r Happily thi can be done To do it we will pick a generic ource point r imagine it to be the origin and then integrate over z - r Having done that for each ource point it will remain only to integrate over all the ource point To prepare the way we introduce ome notation The difference between field and ource point r r will be denoted by Δ rr that i Δ rr r r and the difference between z and r will be denoted by ρ that i ρ z r Uing thi we rewrite the integral over z a follow d 3 $ ' z & 1 z r d 3 $ $ '' ρ = & 1 & ˆ ρ 3 4π & % r z z r 4π & Δ rr ρ & % ρ 2 % I[ Δ rr ] IV5 We will evaluate I[ Δ rr ] by uing pherical polar coordinate In particular we put the origin at ρ = 0 and we orient thee coordinate o that the line from the origin to the north pole lie on Δ rr Denoting the coine of the polar angle θ by χ ie χ Co θ [ ] = dρρ 2 dχ I Δ rr [ ] we mut evaluate 1 2π dϕ ρsin[ θ]co[ ϕ] e ˆ 1 ρsin[ θ]sin[ ϕ] e ˆ 2 ρco[ θ] Δ ˆ rr IV6 4π ρ 3 ρ 2 2 0 1 0 Δ rr 2ρ Δrr χ where ˆ e 1 and ˆ e 2 are orthogonal to each other and to ˆ Δ rr Upon integrating over the azimuthal angle ϕ one find that the only non-zero contribution i proportional to ˆ Δ rr # Δ ˆ & 1 I[ Δ rr ] = rr χ % $ 2 d ρ dχ IV7 ' 0 1 ρ 2 1 2 ρ χ Thi integral can actually be done analytically though not without an irritation[7] The form of the integral over χ depend on whether ρ i more or le than one a hown below 1 χ 0 ρ 1 dχ = 2 ρ 2 1 2 ρ χ 3 ρ IV8a Uing both reult to integrate multiple integral i 1 1 ρ 1 1 dχ χ = 2 3 ρ 2 IV8b 1 ρ 2 1 2 ρ χ ρ from 0 to one find that the numerical value of the 1 χ d ρ dχ = 1 IV9 ρ 2 1 2 ρ χ 0 1
And o one ee that that the integral I[ Δ rr ] i imply half the unit vector pointing from r to r I[ Δ rr ] = 1 Δ ˆ rr IV10 2 Thu Γ B S C the quantity that manifet the time dependence of the charge denity can be written a hown next Γ B S C [ tr] = and it gradient appear below d 3 r 4π µ 0J[ tr ] 1 ˆ 2 Δ - IV11 rr r Γ B S C [ tr] = 1 2 ˆ d 3 r µ 0 J[ tr ] µ 0 J [ tr ] Δ ˆ rr Δ rr IV12 4π r r r r - And o the coulomb gauge vector potential i a hown in EqIV13 A ATM C = A ATM B S Γ B S >C A C ATM = 1 2 ˆ IV13a d 3 r µ 0 J[ tr ] µ 0 J [ tr ] Δ ˆ rr Δ rr - IV13b 4π r r r r - Here it may be helpful to recall where we tarted the relation between the electric field and the potential and thu the relation between the electric field and it ource E ATM = ϕ C t A ATM C = ϕ C t A ATM B S t Γ B S >C IV1 Note that we have found that while the magnetic field i given by it Biot-Savart vector potential thi vector potential i not ufficient to manifet the electric field induced by a changing magnetic field when the charge denity i alo changing A already tated the changing charge denity i manifet in Γ B-S->C V Summary and concluion A few point deerve emphai The firt i traightforward In the ATM electric field and magnetic field are determined by charge and current denitie and their time derivative at the ame time a the field i meaured In hort the ATM propagate influence intantaneouly Second unlike magneto-quaitatic approximation the ATM enable local charge conervation while including the poibility of a time dependent
charge denity Thu the ATM may approximate Maxwell theory when and where retardation may be neglected Third The ATM calar and vector potential depend on their ource charge and current in the ame way that Darwin Approximation of the Coulomb gauge olution of Maxwell Equation Thi i remarkable Darwin got hi approximation by auming hi charge carrier were claical point particle interacting with the field and then expanding that ytem Lagrangian in invere power of the peed of light The ATM i macrocopic theory that arrive at the ame dependence by tarting with an alternative to Maxwell Diplacement Current Fourth the Coulomb gauge vector potential of ATM doe not appear qualitatively more difficult to evaluate that the Biot-Savart vector potential Thu ATM provide a comprehenive including both electro and magneto quaitatic and tractable approximation to Maxwell equation Reference$ [1]Alan M Wolky On a charge conerving alternative to Maxwell diplacement current arxiv:14110642 Claical Phyic phyiccla-ph 2 November 2014 pp 14 [2] Charle G Darwin The dynamical motion of charged particle Phil Mag er 6 39 pp537-551 1920 [3] Jame C Maxwell A Treatie on Electricity and Magnetim unabridged Third Edition of 1891 vol2 Dover Publication Inc New York 1954 p 253 [4] Wolfgang Panofky and Melba Phillip Claical Electricity and Magnetim econd edition Addion-Weley Reading Ma 1962 p 128 [5] John D Jackon Claical Electrodynamic third edition Wiley New York 1999 p 178 [6] The author admit that he i not among the unuually attentive and that hi own path to EqIII1 wa roundabout - involving both Helmholtz Theorem and everal integration by part [7] A table of anti-derivative eg L Rade and B Wetergren Mathematic Handbook for Science and Engineering 4 th edition Springer New York 1999 page 151 #33 i helpful but not unambiguou becaue expreion of the form 1 r 2 appear and o one i left to wonder which branch i appropriate Fortunately it i eay to integrate numerically and o identify the correct branch