Vector potential of the Coulomb gauge

Transcription

1 INSTITUTE OF PHYSICSPUBLISHING Eur. J. Phys. 24 (23) EUROPEANJOURNAL OF PHYSICS PII: S143-87(3) Vector potential of the Coulomb gauge AMStewart Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT 2, Australia Received 23 April 23 Published 22 July 23 Online at stacks.iop.org/ejp/24/519 Abstract The vector decomposition theorem of Helmholtz leads to a form of the Coulomb gauge in which the potentials are expressed in a form that is totally instantaneous. The scalar potential is expressed in terms of the instantaneous charge density, the vector potential in terms of the instantaneous magnetic field. 1. Introduction Despite its apparent generality, the vector decomposition theorem of Helmholtz does not seem to have found a central place in classical electromagnetic theory. This may be due to amisconception that it can be applied only to vector fields that do not change with time. In this paper we show that the Helmholtz theorem leads directly to a dynamical vector potential for electromagnetism and that this vector potential is the Coulomb gauge vector potential, but in a form that is instantaneous in time. The scalar potential of the Coulomb gauge is expressed in terms of the instantaneous charge density, as usual; the vector potential is expressed in terms of an integral over the instantaneous magnetic field. Section 2 of this paper summarizes recent advances in the understanding of the Coulomb gauge. Section 3 shows how Helmholtz s theorem suggests a form for the vector potential and section 4 shows how this form is consistent with the dynamics of the Coulomb gauge. Section 5 demonstrates the mathematical equivalence between the new form of the vector potential and the one obtained recently by Jackson [1. Section 6 gives a simple example of the use of the new form of the potential by using it to obtain the vector potential of the idealized Bohm Aharonov solenoid and section 7 summarizes the results of the paper. 2. Potentials and fields The physically observable electromagnetic fields E(x, t) and B(x, t) that result from source densities of electric charge ρ(x, t) and current J(x, t) may be expressed as the spatial and temporal gradients of the scalar and vector potentials ϕ(x, t) and A(x, t) [2 E = ϕ A/ ct and B = A. (2.1) In classical electromagnetism, the potentials are often useful as an aid to calculating the fields but in quantum theory they appear to be essential in describing properly the interaction of /3/5519+6$3. 23 IOP Publishing Ltd Printed in the UK 519

2 52 AMStewart charge with the electromagnetic field [3 6. For this purpose, it is necessary to obtain the potentials that correspond to a given configuration of fields or sources. It has long been understood that the potentials are not unique [7. If they are changed to A = A + χ and ϕ = ϕ χ/ ct, (2.2) where the gauge function χ(x, t) is any single-valued, continuously differentiable function of x and t that vanishes at infinity, then the fields E and B remain unchanged. Therefore, many different sets of potentials ϕ and A (gauges) correspond to a given set of fields E and B which result from the presence of ρ and J. Somegauges may be more useful than others in a particular calculation, and the transformation from one gauge to another is made by choosing an appropriate gauge function. The matter has recently been discussed by Jackson [1, who has given many examples of gauge transformations. Onegauge that is used frequently in introductoryelectrodynamics [2, 8 and has many uses in non-relativistic quantum electrodynamics [9 is the Coulomb gauge [1 12 with potentials ϕ c and A c.thisgauge has the feature that A c =. (2.3) The scalar potential of the Coulomb gauge is readily found to be [1, 2 ϕ c (x, t) = d 3 x ρ(x, t) R, (2.4) where R = x x.itismoredifficult to calculate the vector potential of this gauge and an expression for it in closed form in terms of the sources has been obtained only recently [1, R/c A c (x, t) = c d 3 x dτ τj(x, t τ) ˆR/R 2. (2.5) By substituting (2.4) and (2.5) into (2.1) the expressions for the fields in terms of the sources [13 are obtained [1 [ ˆR E(x, t) = d 3 x R 2 ρ(x, t ˆR ) + cr t ρ(x, t ) 1 c 2 R t J(x, t ) (2.6) and B(x, t) = 1 [ d 3 x J(x, t ) + R c c t J(x, t ) ˆR /R 2, (2.7) with the retarded time, t,givenbyt = t R /c,wherer = x x,thevariablex being used for later convenience. Thescalar potential of the Coulomb gauge (2.4), when expressed in terms of the sources, has the interesting featureofbeinginstantaneous, the potential at time t being given by the sources at the same time t; the vector potential (2.5) is, though, non-local in time. The instantaneous nature of the scalar potential illustrates the non-physical nature of gauge potentials. Any physically measurable quantity, such as the fields E and B (2.6), (2.7), must be retarded to take account of the finite velocity of light, but a non-physically-measurable entity, such as a potential, need not be. 3. The Helmholtz theorem The vector decomposition theorem of Helmholtz [14 states that any three-vector field that vanishes at spatial infinity can be expressed as the sum of two terms, the longitudinal A l (x) and transverse A t (x) components, which have the properties A l (x) = and A t (x) =. (3.1) These components are given explicitly as A l (x) = d 3 x A(x ) (3.2) 4π x x

3 Vector potential of Coulomb gauge 521 and A t (x) = d 3 x A(x ) 4π x x. (3.3) The question we ask in this paper, and answer in the affirmative, is whether (3.2) and (3.3) can represent a vector potential consistent with the Maxwell equations. For example, is it valid to take, inthe Coulomb gauge, the longitudinal component (3.2) to be zero, which it must be from (2.3), and the transverse component from (2.1) to be A t (x, t) = 4π d 3 x B(x, t) x x, (3.4) where we have displayed the time component explicitly to signify that this is an equation that changes with time? 4. The transverse Helmholtz component as a gauge potential We consider E and B to be the Maxwell fields generated by the sources ρ and J. If we differentiate equation (3.4) with respect to time, apply the Maxwell equation c E = B/ t and make use of the vector identity E(x ) x x = E(x ) x x E(x ) 1 x x, (4.1) we get 1 A t (x, t) = c t 4π d 3 x E(x, t) 1 x x, (4.2) noting that the volume integral of the left-hand side of (4.1) vanishes for fields that vanish at infinity. Using the identity for the curl of a cross-product and the property that f (x x ) = f (x x ) this becomes 1 A t (x, t) = 1 { } d 3 x E(x ) 2 1 c t 4π x x +[E(x ) 1. x x (4.3) The first term on the right-hand side of (4.3) contains a delta function and gives E(x). Inthe second term, the on the right is taken outside the integral. Next, using the identity similar to (4.1) but with a divergence instead of a curl we get 1 A t (x, t) = E(x) + 1 d 3 x E(x ). c t 4π x x (4.4) If we identify E,through a Maxwell equation, as 4π times the charge density ρ and define ascalar potential ϕ t to be ϕ t (x, t) = d 3 x ρ(x, t) x x, (4.5) equation (4.4) becomes A t / ct = E + ϕ t. (4.6) Thisexpression, which is similar to the first partof (2.1) relatinge to the electromagnetic potentials, suggests that the transverse potential of equation (3.4) can be identified with the vector potential of the Coulomb gauge. In the next section of the paper, it is shown that equation (2.5) is, in fact, identical to the expression found recently for the vector potential of the Coulomb gauge, equation (3.4).

4 522 AMStewart 5. The transverse Helmholtz component as a Coulomb gauge potential If we substitute (2.7) into (3.4) we get A t (x, t) = 4πc d 3 x d 3 x x x [J(x, t ) + R c t J(x, t ) ˆR /R 2. (5.1) We need to demonstrate that this leads to (2.5). From the relations R = x x and R = x x we get x x = R R.Weconverttheintegral over x to one over R and note that with t fixed / t = c / R so A t (x, t) = 4πc d 3 x d 3 R [ R R J(x, t ) R R J(x, t ) ˆR /R 2. (5.2) We take the spherical coordinates of R to be (r,θ,φ),with the reference axis of these coordinates along the direction of the vector R, sothatθ is the angle between R and R. The corresponding Cartesian coordinates of R with respect to R we denote as (x, y and z ). The magnitude R R is seen to be independent of φ. The two terms in (5.2) containing J, which we denote collectively as G, do not depend on φ either; only the vector cross-product G ˆR does. This vector cross-product has Cartesian x component [G y cos(θ) G z sin(θ) sin(φ), y component [G z sin(θ) cos(φ) G x cos(θ)and z component [G x sin(θ) sin(φ) G y sin(θ) cos(φ). When we integrate these over φ from to 2π we are left only with the terms that do not contain sin(φ) or cos(φ), so 2π Accordingly (5.2) becomes A t (x, t) = 2c d 3 x dφ G ˆR = 2π cos(θ)g ˆR. (5.3) dr π [ sin(θ) cos(θ) dθ R R J(x, t ) R R J(x, t ) ˆR. (5.4) To carry out the integral over θ we express the inverse of R R in terms of Legendre polynomials P l (x) 1 R R = r< l P r l= > l+1 l (cos θ) (5.5) where the variable r > or r < is the larger or smaller of R and R.Because of the orthogonality of the polynomials, the only non-vanishing integral over θ is that for l = 1, as P 1 (cos θ) = cos θ so π sin(θ) cos 2 (θ) dθ = 2/3, (5.6) giving A t (x, t) = 3c d 3 x dr r [ < r> 2 The domain of the integral over r is J(x, t ) R R J(x, t ) ˆR. (5.7) dr r < r> 2 If the variable of integration is changed to τ = r/c the integral becomes dr r < r> 2 G ˆR = [ c 2 R 2 = 1 R 2 R R/c τ dτ + R c dr r dr + R R r. (5.8) 2 R/c dτ τ 2 [ J(t τ) τ J(t τ) τ ˆR. (5.9)

5 Vector potential of Coulomb gauge 523 The integrationoverτ is readily performedby parts to give dr r R/c < r> 2 G ˆR = 3c2 τ dτ J(t τ) ˆR, (5.1) R 2 which gives finally R/c A t (x, t) = c d 3 x dτ τj(x, t τ) ˆR/R 2, (5.11) which shows that (2.5) and (3.4) are identical. This confirms the suggestion that (3.4) gives an alternative form of the vector potential of the Coulomb gauge. In addition, the scalar component (4.5) is the same as the scalar potential of the Coulomb gauge (2.4), demonstrating the complete equivalence of the two sets of potentials. 6. The Bohm Aharonov vector potential As a simple illustration of the use of equation (3.4) we calculate the vector potential in the Coulomb gauge for the field B(r) = ẑfδ(x)δ(y) in a long thin solenoid along the z-axis where F is the magnetic flux in the solenoid. This is the geometry that gives rise to the Bohm Aharonov effect [3. Operating with the curl on (3.4) provides an alternative form of this equation, A t (r, t) = 1 4π d 3 r B(r, t) (r r ) r r 3, (6.1) which, for this particular situation, gives A t (r, t) = F dx dy dz δ(x )δ(y )ẑ (r r ). (6.2) 4π r r 3 Noting that r =ẑz we get A t (r, t) = F 4π (ŷx ˆxy) dz r r. (6.3) 3 The numerator of the integrand is a vector in the φ direction of magnitude (x 2 + y 2 ) 1/2,using cylindrical coordinates (r,φ,z); thedenominator is [x 2 + y 2 + (z z ) 2 3/2.Theintegral then comes to A t (r, t) = ˆφF(x 2 + y 2 ) 1/2 4π dz [x 2 + y 2 + (z z ) 2 3/2 (6.4) and, carrying out this standard integration, we finally get A t (r, t) = ˆφF/(2π r ). Thisisthe standard result that is usually obtained in a simpler way with Stokes s theorem by equating the lineintegral of the vector potential to the flux enclosed in a circular path around the solenoid. Since the potential is instantaneous, it applicable to any arbitrary time dependence of the magnetic flux. 7. Conclusion The vector decomposition theorem of Helmholtz leads to a form of the vector potential of the Coulomb gauge that, like the scalar potential, is instantaneous. It has been confirmed that this new form is mathematically identical to an expression for the vector potential derived recently [1. The gauge can then be expressed in a way that is totally instantaneous. The scalar potential is expressed in terms of the instantaneous charge density (2.4), the vector potential in terms of the instantaneous magnetic field (3.4). Of course, the scalar potential can be expressed in terms of the instantaneous electric field by using an inhomogeneous Maxwell equation.

6 524 AMStewart The gauge condition A c = ofthecoulomb gauge can be maintained by making any gauge transformation with 2 χ =. The fundamental solution of this equation that is regular at infinity is χ = a/r +b. However,ifthe gauge function is required to be differentiable and to vanish at infinity [15 then a and b must both be zero. Accordingly,the gauge function vanishes, so the potentials of the Coulomb gauge are unique. This gauge is said to be completely fixed. Totally instantaneous gauges are not unprecedented. The Poincaréormultipolar gauge ϕ(r, t) = r 1 du E(ur, t) A(r, t) = r 1 duub(ur, t) (7.1) is a case in point [5, Of course, this gauge can be expressed in a form that is manifestly non-local in time by expressing the fields in the integrands in terms of the sources by means of (2.6) and (2.7). References [1 Jackson J D 22 Am. J. Phys [2 Jackson J D 1999 Classical Electrodynamics (New York: Wiley) [3 Aharonov Y and Bohm D 1959 Phys. Rev [4 Stewart A M 2 J. Phys. A: Math. Gen [5 Stewart A M 2 Aust. J. Phys [6 Stewart A M 23 J. Mol. Struct. (Theochem) [7 Jackson J D and Okun L B 21 Rev. Mod. Phys [8 Griffiths D J 1999 Introduction to Electrodynamics (Upper Saddle River, NJ:Prentice-Hall) [9 Craig D P and Thirunamachandran T 1998 Molecular Quantum Electrodynamics: an Introduction to Radiation Molecule Interactions (New York: Dover) [1 Brill O L and Goodman B 1967 Am. J. Phys [11 Rohrlich F 22 Am. J. Phys [12 Jefimenko O D 22 Am. J. Phys [13 Jefimenko O D 1989 Electricity and Magnetism (Star City: Electret) [14 Arfken G 1995 Mathematical Methods for Physicists (San Diego, CA: Academic) [15 Scharf G 1989 Finite Quantum Electrodynamics (Berlin: Springer) [16 Valatin J G 1954 Proc. R. Soc. A [17 Woolley R G 1971 Proc. R. Soc. A [18 Stewart A M 1999 J. Phys. A: Math. Gen

7 INSTITUTE OF PHYSICSPUBLISHING Eur. J. Phys. 25 (24) L21 L22 EUROPEANJOURNAL OF PHYSICS PII: S143-87(4) LETTERS AND COMMENTS Comment on Vector potential of the Coulomb gauge * VHnizdo National Institute for Occupational Safety and Health, 195 Willowdale Road, Morgantown, WV 2655, USA vbh5@cdc.gov Received 16 September 23 Published 16 February 24 Online at stacks.iop.org/ejp/25/l21 (DOI: 1.188/143-87/25/2/L4) Abstract The expression for the Coulomb-gauge vector potential in terms of the instantaneous magnetic field derived by Stewart (23 Eur. J. Phys ) by employing Jefimenko s equation for the magnetic field and Jackson s formula for the Coulomb-gauge vector potential can be proven much more simply. In a recent paper [1, Stewart has derived the following expression for the Coulomb-gauge vector potential A C in terms of the instantaneous magnetic field B A C (r, t) = 4π d 3 r B(r, t) r r. (1) Stewart starts with expression (1) as an ansatz suggested by the Helmholtz theorem, and then proceeds to prove it by substituting in (1) Jefimenko s expression for the magnetic field in terms of the retarded current density and its partial time derivative [2 and obtaining, after some non-trivial algebra, an expression for A C in terms of the current density derived recently by Jackson [3. In this comment, we give a more simple proof of formula (1) using only the Helmholtz theorem. According to the Helmholtz theorem [4, anarbitrary-gauge vector potential A, asany non-constant three-dimensional vector field that vanishes sufficiently rapidly at infinity, can be decomposed uniquely into a longitudinal part A,whose curl vanishes, and a transverse part A,whose divergence vanishes A(r, t) = A (r, t) + A (r, t) A (r, t) = A (r, t) =. (2) The longitudinal and transverse parts in (2) are given explicitly by A (r, t) = 4π d 3 r A(r, t) r r A (r, t) = 4π d 3 r A(r, t). (3) r r * This comment is written by V Hnizdo in his private capacity. No support or endorsement by the Centers for Disease Control and Prevention is intended or should be inferred /4/221+2$3. 24 IOP Publishing Ltd Printed in the UK L21

8 L22 Letters and Comments Let us now decompose the vector potential A in terms of the Coulomb-gauge vector potential A C as follows: A(r, t) = [A(r, t) A C (r, t)+a C (r, t). (4) If the curl of [A A C vanishes, then, according to equation(2) and the fact that the Coulombgauge vector potential is by definition divergenceless, the Coulomb-gauge vector potential A C is the transverse part A of the vector potential A. But because the two vector potentials must yield the same magnetic field, the curl of [A A C does vanish [A(r, t) A C (r, t) = A(r, t) A C (r, t) = B(r, t) B(r, t) =. (5) Thus the Coulomb-gauge vector potential is indeed the transverse part of the vector potential A of any gauge. Therefore, it can be expressed according to the second part of (3) and the fact that A = B as A C (r, t) = A (r, t) = 4π d 3 r A(r, t) = r r 4π d 3 r B(r, t) r r. (6) The right-hand side of (6) is expression (1) derived by Stewart. In closing, we note that there is an expression for the Coulomb-gauge scalar potential V C in terms of the instantaneous electric field E that is analogous to expression (6) for the Coulomb-gauge vector potential V C (r, t) = 1 4π d 3 r E(r, t). (7) r r This follows directly from the definition V C (r, t) = d 3 r ρ(r, t)/ r r of the Coulombgauge scalar potential and the Maxwell equation E = 4πρ. Expressions (6) and (7) may be regarded as a totally instantaneous gauge, but it would seem more appropriate to view them as the solution to a problem that is inverse to that of calculating the electric and magnetic fields from given Coulomb-gauge potentials A C and V C according to E = V C A C B = A C. (8) c t The first equation of (8) gives directly the longitudinal part E and transverse part E of an electric field E in terms of the Coulomb-gauge potentials V C and A C as E = V C and E = A C /c t (the apparent paradox that the longitudinal part E of a retarded electric field E is thus a truly instantaneous field has been discussed recently in [5). References [1 Stewart A M 23 Vector potential of the Coulomb gauge Eur. J. Phys [2 Jefimenko O D 1989 Electricity and Magnetism 2nd edn (Star City, WV: Electret Scientific) Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley) [3 Jackson J D 22 From Lorenz to Coulomb and other explicit gauge transformations Am. J. Phys [4 Arfken G 1995 Mathematical Methods for Physicists (San Diego, CA: Academic) [5 Rohrlich F 22 Causality, the Coulomb field, and Newton s law of gravitation Am. J. Phys Jefimenko O D 22 Comment on Causality, the Coulomb field, and Newton s law of gravitation Am. J. Phys Rohrlich F 22 Reply to comment on Causality, the Coulomb field, and Newton s law of gravitation Am. J. Phys

9 INSTITUTE OF PHYSICSPUBLISHING Eur. J. Phys. 25 (24) L23 L27 EUROPEANJOURNAL OF PHYSICS PII: S143-87(4) LETTERS AND COMMENTS On vector potential of the Coulomb gauge Valery P Dmitriyev Lomonosov University, PO Box 16, Moscow , Russia dmitr@cc.nifhi.ac.ru Received 2 August 23 Published 16 February 24 Online at stacks.iop.org/ejp/25/l23 (DOI: 1.188/143-87/25/2/L5) Abstract The question of an instantaneous action (Stewart 23 Eur. J. Phys ) can be approached in a systematic way applying the Helmholtz vector decomposition theorem to a two-parameter Lorenz-like gauge. We thus show that only the scalar potential may act instantaneously. 1. Introduction The role of the gauge condition in classical electrodynamics was recently highlighted [1. This is because of probable asymmetry between different gauges. The distinct feature of the Coulomb gauge is that it implies an instantaneous action of the scalar potential [2 4. The question of simultaneous co-existence of instantaneous and retarded interactions has been mostly debated [5. Paper [2 concludes that the vector decomposition theorem of Helmholtz leads to a form of the vector potential of the Coulomb gauge that, like the scalar potential, is instantaneous. This conclusion was arrived at considering the retarded integrals for electrodynamic potentials. Constructing within the same theorem wave equations the author of [3 finds that the scalar potential propagates at infinite speed while the vector potential propagates at speed c in free space. In order to resolve the discrepancy between [2 and [3 the latter technique will be developed below in a more systematic way. Recently the two-parameter generalization of the Lorenz gauge was considered [1, 4: A + c ϕ =, (1) cg 2 t where c g is a constant that may differ from c. We will construct wave equations applying the vector decomposition theorem to Maxwell s equations with (1). Thus simultaneous coexistence of instantaneous and retarded actions will be substantiated. 2. Maxwell s equations in the Kelvin Helmholtz representation Maxwell s equations in terms of electromagnetic potentials A and ϕ read as 1 A + E + ϕ = (2) c t /4/223+5$3. 24 IOP Publishing Ltd Printed in the UK L23

10 L24 Letters and Comments E c ( A) +4πj = (3) t E = 4πρ. (4) The Helmholtz theorem says that a vector field u that vanishes at infinity can be expanded into asumof its solenoidal u r and irrotational u g components. We have for the electric field E = E r + E g, (5) where E r = (6) E g =. (7) The similar expansion for the vector potential can be written as A = A r + c c g A g, (8) where A r = (9) A g =. (1) If we substitute equations (5) and (8) into (2), we obtain 1 c A r t + E r + 1 c g A g t + E g + ϕ =. (11) By taking the curl of equation (11), we obtain, using equations (7) and (1), ( ) 1 A r + E r =. (12) c t On the other hand, from equations (6) and (9), we have ( ) 1 A r + E r =. (13) c t If the divergence and curl of a field are zero everywhere, then that field must vanish. Hence, equations (12) and (13) imply that 1 A r + E r =. (14) c t We subtract equation (14) from (11) and obtain 1 A g + E g + ϕ =. (15) c g t Similarly, if we expressthecurrent density as j = j r + j g, (16) where j r = (17) j g =, (18) equation (3) can be written as two equations E r c ( A r ) +4πj r = (19) t E g +4πj g =. (2) t From equations (5) and (6), equation (4) can be expressed as E g = 4πρ. (21)

11 Letters and Comments L25 3. Wave equations for the two-speed extension of electrodynamics We will derive from equations (14), (15), (19), (2), and (21) the wave equations for the solenoidal (transverse) and irrotational (longitudinal) components of the fields. In what follows we will usethegeneral vector relation ( u) = 2 u + ( u). (22) The wave equation for A r can now be found. We differentiate equation (14) with respect to time: 1 2 A r + E r =. (23) c t 2 t We next substitute equation (19) into (23) and use equations (22) and (9) to obtain 2 A r c 2 t 2 2 A r = 4πcj r. (24) The wave equation for E r can be found as follows. We differentiate equation (19) with respect to time: 2 ( E r t 2 c A ) r +4π j r =, (25) t t and substitute equation (14) into (25). By using equations (22) and (6), we obtain 2 E r c 2 t 2 2 E r = 4π j r t. (26) In the absence of the electric current, equations (24) and (26) are wave equations for the solenoidal fields A r and E r. To find wave equations for the irrotational fields, we need a gauge relation. Substituting (8) into equation (1) we get the longitudinal gauge A g + 1 ϕ =. (27) c g t The solenoidal part of the vector potential automatically satisfies the Coulomb gauge, equation (9). The wave equation for A g can be found as follows. We first differentiate equation (15) with respect to time: 1 2 A g + E g + ϕ =. (28) c g t 2 t t We then take the gradientofequation (27), ( A g ) + 1 ϕ =, (29) c g t and combine equations (28), (29) and (2). If we use equations (22) and (1), we obtain 2 A g c 2 t 2 g 2 A g = 4πc g j g. (3) Next, we will find the wave equation for ϕ. Wetakethedivergence of equation (15), 1 A g + E g + 2 ϕ =, (31) c g t and combine equations (31), (27), and (21): 2 ϕ t 2 c2 g 2 ϕ = 4πcg 2 ρ. (32) Equations (32) and (3) give wave equations for ϕ and A g.

12 L26 Letters and Comments We may try to find a wave equation for E g using equation (15) in (32) and (3). However, in the absence of the charge, we have from equation (21) E g =. (33) Hence, by equations (33) and (7), we have E g =. (34) We see that Maxwell s equations (2) (4) with the longitudinalgauge (27) imply that the solenoidal and irrotational components of the fields propagate with different velocities. The solenoidal components A r and E r propagate with the speed c of light, and the irrotational component A g of the magnetic vector potential and the scalar potential ϕ propagate with the speed c g. 4. Single-parameter electrodynamics In reality, electrodynamics has only one parameter, the speed of light, c. Then, to construct from the above the classical theory, we have to choose among two variants: two waves with equal speeds or a single wave. If we let c g = c, (35) the two-parameter form (1) becomes the familiar Lorenz gauge A + 1 ϕ =. (36) c t Another possible choice is c g c. (37) The condition (37) turns equation (1) into the Coulomb gauge A =. (38) Substituting c g = (39) into the dynamic equation (32), we get 2 ϕ = 4πρ. (4) The validity of equation (4) for the case when ϕ and ρ may be functions of time t means that the scalar potential ϕ acts instantaneously. Substituting (39) into equation (27) we get for the irrotational part of the vector potential: A g =. (41) Insofar as the divergence (41) and the curl (1) of A g are vanishing, we have A g =. (42) So, onfirst sight by (39) the irrotational component A g of the vector potential propagates instantaneously. However, according to relation (42), with (39) A g vanishes. Putting (42) into equation (3) we also obtain j g =. (43) Putting (42) into (8) and (43) into (16) we get A = A r and j = j r. (44) Substituting (44) into equation (24) gives 2 A t 2 c2 2 A = 4πcj. (45) Equation (45) indicates that in the Coulomb gauge (38) the vector potential A propagates at speed c.

13 Letters and Comments L27 5. Mechanical interpretation Recently, we have shown [6 that in the Coulombgauge electrodynamics is isomorphic to the elastic medium that is stiff to compression yet liable to shear deformations. In this analogy the vector potential corresponds to the velocity and the scalar potential to the pressure of the medium. Clearly, in an incompressible medium there are no longitudinal waves, the pressure acts instantaneously,and the transversewave spreads at finite velocity. This mechanicalpicture provides an intuitive support to the electrodynamic relations (38), (4) and (45) just obtained. 6. Conclusion By using a two-parameter Lorenz-like gauge, we extended electrodynamics to a two-speed theory. Turning the longitudinal speed parameter to infinity we come to electrodynamics in the Coulomb gauge. In this way we show that the scalar potential acts instantaneously while the vector potential propagates at the speed of light. Acknowledgment Iwould like to express my gratitude to Dr I P Makarchenko for valuable comments concerning the non-existence of longitudinal waves of the electric field and longitudinal waves in the Coulomb gauge. References [1 Jackson J D 22 From Lorenz to Coulomb and other explicit gauge transformations Am. J. Phys [2 Stewart A M 23 Vector potential of the Coulomb gauge Eur. J. Phys [3 Drury D M 22 Irrotational and solenoidal components of Maxwell s equations Galilean Electrodyn [4 Chubykalo A E and Onoochin V V 22 On the theoretical possibility of the electromagnetic scalar potential wave spreading with an arbitrary velocity in vacuum Hadronic J [5 Jackson J D 22 Criticism of Necessity of simultaneous co-existence of instantaneous and retarded interactions in classical electrodynamics by Chubykalo A E and Vlaev S J Int. J. Mod. Phys. A (Preprint hep-ph/2376) [6 Dmitriyev V P 23 Electrodynamics and elasticity Am. J. Phys

14 INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 25 (24) L29 L3 EUROPEAN JOURNAL OF PHYSICS PII: S143-87(4) LETTERS AND COMMENTS Reply to Comments on Vector potential of the Coulomb gauge AMStewart Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT 2, Australia Received 4 December 23 Published 16 February 24 Online at stacks.iop.org/ejp/25/l29 (DOI: 1.188/143-87/25/2/L6) Abstract The instantaneous nature of the potentials of the Coulomb gauge is clarified and a concise derivation is given of the vector potential expressed in terms of the instantaneous magnetic field. The comment by Dmitriyev [1 on the paper of Stewart [2 about the potentials of the Coulomb gauge asserts that only the scalar potential of that gauge may act instantaneously. The question to be asked here is: instantaneously with respect to what? In the Lorenz gauge, all components of the potential that originate from the charge and current sources propagate at the speed of light [3. In the Coulomb gauge [3 the scalar potential acts instantaneously with respect to the charge density. However, from the expression for the vector potential obtained by Jackson [3 R/c A(x,t)= c x d 3 y dττj(y,t τ) ˆR/R 2, (1) it can be seen that, with respect to the current source J, the vector potential propagates with different time delays, from τ =, instantaneously, to τ = R/c, where R is the distance between x and y, corresponding to the speed of light. If the vector potential of the Coulomb gauge is expressed in an equivalent form, in terms of the fields rather than the sources [2, A(x,t)= x 1 4π d 3 y B(y,t) x y, (2) an expression that may be obtained either from the Helmholtz theorem [2 or by analogy with the Biot Savart law, then the vector potential propagates instanteously with respect to the magnetic field. However, this causes no conceptual difficulty because the fields themselves propagate at the speed of light. What counts is the velocity of propagation of the fields and this must always be the speed of light. The sole criterion for the validity of a given set of potentials is that they reproduce the correct fields; the velocity of propagation of the potentials is of no physical consequence. To confirm that equation (2) does indeed reproduce the correct fields it is necessary to show that it satisfies the equations B = A A = B =. (3) /4/229+2$3. c 24 IOP Publishing Ltd Printed in the UK L29

15 L3 Letters and Comments The second of these conditions is satisfied by a standard vector identity. From the curl of (2) we get x A(x,t)= 1 [ 4π d 3 1 y x x x y B(y,t), (4) and, by expanding the triple vector product, x A(x,t)= 1 { } 4π d 3 y x 2 1 x y B(y,t)+[B(y,t) 1 x x. (5) x y The first term gives a delta function, leading to B(x,t). The second term becomes x d 3 1 y[b(y,t) y 4π x y, (6) and, using the vector identity for the divergence of the product of a vector and a scalar, y B(y,t) x y = B(y,t) 1 y x y + y B(y,t) (7) x y becomes zero because B = and the volume integral of the divergence gives rise to a surface integral that vanishes. Hence it is proved that for the potential of (2) B(x,t) = A(x,t). Hnizdo [4 also gives a proof. References [1 Dmitriyev V P 24 On vector potential of the Coulomb gauge Eur. J. Phys. 25 L23 [2 Stewart A M 23 Vector potential of the Coulomb gauge Eur. J. Phys [3 Jackson J D 22 From Lorenz to Coulomb and other explicit gauge transformations Am.J.Phys [4 Hnizdo V 24 Comment on Vector potential of the Coulomb gauge Eur. J. Phys. 25 L21