Comment on ‘Vector potential of the Coulomb gauge’*

The expression for the Coulomb-gauge vector potential in terms of the ‘instantaneous’ magnetic field derived by Stewart (2003 Eur. J. Phys. 24 519) 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 AC in terms of the ‘instantaneous’ magnetic field B AC(r, t) = ∇ 4π × ∫ dr ′ 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 AC 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], an arbitrary-gauge vector potential A, as any 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) = 0 ∇ ·A⊥(r, t) = 0. (2) The longitudinal and transverse parts in (2) are given explicitly by A‖(r, t) = − ∇ 4π ∫ dr ′ ∇′ ·A(r′, t) |r − r′| A⊥(r, t) = ∇ 4π × ∫ dr ′ ∇′ ×A(r′, t) |r − r′| . (3) * 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. 0143-0807/04/020021+02$30.00 © 2004 IOP Publishing Ltd Printed in the UK L21 L22 Letters and Comments Let us now decompose the vector potential A in terms of the Coulomb-gauge vector potential AC as follows: A(r, t) = [A(r, t)−AC(r, t)] + AC(r, t). (4) If the curl of [A−AC] vanishes, then, according to equation (2) and the fact that the Coulombgauge vector potential is by definition divergenceless, the Coulomb-gauge vector potentialAC 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−AC] does vanish ∇× [A(r, t)−AC(r, t)] = ∇×A(r, t) −∇×AC(r, t) = B(r, t)−B(r, t) = 0. (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 AC(r, t) = A⊥(r, t) = ∇ 4π × ∫ dr ′ ∇′ ×A(r′, t) |r − r′| = ∇ 4π × ∫ dr ′ 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 VC in terms of the ‘instantaneous’ electric field E that is analogous to expression (6) for the Coulomb-gauge vector potential VC(r, t) = 1 4π ∫ dr ′ ∇′ ·E(r′, t) |r − r′| . (7) This follows directly from the definition VC(r, t) = ∫ d3r ′ 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 AC and VC according to E = −∇VC − ∂AC c∂ t B = ∇×AC. (8) 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 VC and AC as E‖ = −∇VC and E⊥ = −∂AC/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]).