The vector potential of a steady azimuthal current density. Once again

Antonio O. Bouzas

The vector potential of a steady azimuthal current density. Once again

T0 review · 0 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For steady azimuthal currents, the vector potential is purely azimuthal, and a circular loop's field is captured by a Legendre function.

desk verdict A clean, correct pedagogical note that reproves a known result and repackages a standard integral-table formula; nothing new physically, but the presentation is genuinely useful and deserves referee time if the venue wants teaching notes. read the letter →

arxiv 2510.11663 v1 pith:GXPDRQZ3 submitted 2025-10-13 physics.class-ph

Antonio O. Bouzas This is my paper · ORCID

classification physics.class-ph

keywords vectorpotentialazimuthalcurrentdensitymagnetostaticscircularloopLegendrefunctionsCoulombgaugeintegraltransformsBessel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a clean structural fact: for a steady, localized current density that points in the azimuthal direction (with no azimuthal variation), the Coulomb-gauge vector potential also points azimuthally, and its magnitude is given by the integral in Eq. (6). The proof is an elementary shift-and-parity argument, shorter than earlier derivations and valid in spherical, cylindrical, and any orthogonal coordinate system that contains the azimuthal angle. Applying this to a circular current loop recovers the standard series expansions in spherical and cylindrical coordinates, and yields a new closed form for the vector potential in terms of a Legendre function of the second kind rather than complete elliptic integrals. The same Legendre form gives the magnetic induction field in Eq. (37), reproducing known on-axis results.

What carries the argument

The key identity is Eq. (6), the integral representation for Aφ, which turns the vector Poisson equation into a single scalar integral. For the circular loop, the pivotal mechanism is the Laplace-transform evaluation (from a standard table) of ∫₀^∞ dk J₁(ka)J₁(kρ)e^{−k|z|} as (1/π)√(a/ρ)Q_{1/2}(ξ); this converts the Bessel integral (30) into the closed Legendre form (33). The derivative relation dQ_ν/dz = (z²−1)^{−1/2}Q^{1}_ν then yields the field components (37).

What would settle it

Evaluate the Bessel integral (30) numerically at (ρ/a=0.5, z/a=0.3) using independent adaptive integration, and compare with (33) using a different Legendre-function implementation; any discrepancy beyond roundoff would disprove the closed form. Similarly, compare (37) with a direct Biot–Savart integration at a few off-axis points.

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Extended reading notes

Core claim

The central claim is that any steady azimuthal current configuration generates a vector potential with only an azimuthal component, A=Aφ(r)φ̂, where Aφ is the integral (6) with cos(φ′−φ)/|r−r′|. For the circular loop, this reduces to the Bessel integral (30), which the paper evaluates in closed form as (33), Aφ=(μ0I/2π)√(a/ρ)Q_{1/2}(ξ) with ξ=(ρ²+z²+a²)/(2aρ). The magnetic induction then follows from the curl as (37), expressed through Q_{1/2} and its derivative Q^{1}_{1/2}. The paper also notes the equality of the elliptic-integral and Legendre forms, which yields the identity (35).

Load-bearing premise

The closed-form expressions for the circular loop depend on a textbook Laplace-transform identity that the paper quotes without proof; if that identity were misapplied, equations (33) and (37) would not follow, although equation (6) itself would still hold.

Editorial extensions

If this is right

With (6), the vector potential of any azimuthally symmetric steady current can be written down as a single scalar integral without solving vector Poisson equations.
The closed form (33) gives the loop's vector potential and field with Legendre functions, an alternative to elliptic integrals that may be more convenient for asymptotics and numerical work.
Equating the elliptic and Legendre forms produces the relation (35) between Q_{1/2} and complete elliptic integrals.
The proof of (6) carries over to any orthogonal coordinate system containing φ, so the result applies to non-standard geometries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Legendre-function form (33) suggests that other azimuthal magnetostatic problems whose potentials involve similar Bessel-Laplace integrals could also be simplified, without needing to pass through elliptic integrals.
Because the proof of (6) rests only on periodicity and parity, the same technique likely works for any vector field whose source is a localized, divergence-free azimuthal vector field—for instance, in other gauges or in analogous diffusion problems.
The reliance on an external integral-table identity means the derivation is not self-contained; a motivated reader might prove the identity by contour integration or by verifying the ODE satisfied by Q_{1/2}.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this is a straightforward, honest teaching note, not a research advance. The main formula (6) — that a steady azimuthal current density gives an azimuthal vector potential with that explicit integral — was already proved by Vasilyev in spherical coordinates and Zangwill in cylindrical ones. What the paper adds is a simpler, uniform proof that works for both, and the claim that it works in any orthogonal coordinate system containing the azimuthal angle. I checked the derivation; the angular integration trick is clean and correct. No hidden assumptions, no circularity, no fitted parameters. It does what it says.

The compact closed forms (33) and (37) are the strongest selling point. They are indeed simpler than the textbook elliptic-integral forms, and the paper is honest enough to credit the Laplace-transform identity to Erdélyi's table. The equivalence to the elliptic-integral result is checked numerically at a few points and via the on-axis limit of the B-field, which is an independent check. That is fine for a note of this kind. The extra relation (35) between Q_{1/2} and complete elliptic integrals is a nice byproduct.

The soft spots are modest. First, the "any orthogonal coordinate system" generalization in Section 2 is asserted, not proved. The proof as written uses explicit spherical and cylindrical forms for the unit vector decomposition (14); extending it to a general orthogonal system would need a metric-based argument that is absent. That is a scope overclaim, but it does not affect the spherical and cylindrical results. Second, calling (33) "new" is generous — it is a known integral-table identity in new clothes. The novelty is presentational. Third, Table 1 checks four points; a skeptical reader might want a few more, but the on-axis limit and the exact equivalence with (31) are convincing enough.

Bottom line: this is a good classroom note for graduate or advanced undergraduate students. The proof of (6) is genuinely simpler than the literature, and the Legendre-function closed forms are convenient. I would not cite it as a research result, but I would happily hand it to a student who asks why the vector potential of a loop is azimuthal. For a journal that publishes pedagogical or expository physics, it deserves a serious referee — instructed to ask for a proof or a clear caveat on the orthogonal-coordinate claim, and a softer statement about novelty.

For peer review: yes, send it out. It is correct, clearly written, and modest in scope.

Referee Report

0 major / 5 minor

Summary. The paper presents a short derivation of an integral expression, Eq. (6), for the vector potential of a time-independent, localized azimuthal current density J=Jφ(r)φhat. The derivation uses a change of azimuthal variables and a parity argument to show that A is azimuthal and proportional to φhat. The result is then applied to a circular current loop: Section 3.1 recovers the spherical-harmonic expansion, Section 3.2 recovers the cylindrical Bessel-transform form, and Section 3.3 gives two closed forms, Eqs. (31) and (33), the latter in terms of the Legendre function of the second kind Q_{1/2}. Section 3.4 gives the corresponding magnetic induction components in Eq. (37). The paper is explicitly intended as a pedagogical supplement to standard textbooks.

Significance. If correct, this is a useful and clean contribution to the pedagogical literature. The proof of Eq. (6) is elementary and more transparent than the earlier derivations cited in [2,3]. The closed form (33) is compact, and the paper provides a non-trivial identity, Eq. (35), connecting Q_{1/2} with complete elliptic integrals. The numerical cross-checks in Table 1, the recovery of the spherical-harmonic and Bessel expansions, and the independent on-axis limit (39) all support the correctness of the central results. The paper has no free parameters and does not assume the target result. Its main weaknesses are presentational: the claimed generality to 'any orthogonal system' is not formally established, and the key integral identity (30) is quoted from a table rather than derived; neither issue affects the correctness of the spherical/cylindrical results or the circular-loop formulas.

minor comments (5)

[Introduction and §2] The paper claims that the proof of (6) is valid 'in any orthogonal system of curvilinear coordinates containing the azimuthal angle φ.' The written proof, however, only defines R(α) explicitly for spherical and cylindrical coordinates. The generalization is plausible (for an axisymmetric orthogonal system the distance depends on φ−φ′ through a term 2ρρ′ cos(φ−φ′), so R is even and 2π-periodic), but it is not stated or proved in this form. Please either supply this short argument or restrict the claim to the two coordinate systems used in the applications.
[Table 1] The last row (ρ/a=10^{-6}, z/a=1.74) shows a relative difference of about 2×10^{-3} between Eq. (33) and Eqs. (30)/(31), although the absolute difference is 4×10^{-10}. The text says 'perfect equality of the three values to six decimals ... except for a minor numerical difference of order 10^{-10}.' This is misleading: the difference is not merely a round-off at the displayed precision, but likely reflects numerical cancellation in evaluating Q_{1/2} at a large argument. Please report the discrepancy in relative terms, or compute Eq. (33) with higher precision and state the error tolerance.
[§3.4] In the Biot–Savart formula displayed at the end of Section 3.4, the denominator is written as |r−r′|^{3/2}; it should be |r−r′|^3. Also, the integration variable is written as j(r′) in that formula but as J(r′) elsewhere; the notation should be unified.
[References] References [5], [7], and [8] contain the typo 'Trascendental' instead of 'Transcendental.' The author affiliation line also appears to read 'CINVESTA V-IPN'; if this is a typo for CINVESTAV-IPN it should be corrected.
[§3.3] The evaluation of the Laplace transform in Eq. (30) is quoted from Eq. (13) of §4.14 in [7] and is not derived. For a pedagogical note, this is acceptable, but it would be helpful to state the quoted identity explicitly (with the prefactor 1/π√(aρ) Q_{1/2}(ξ)) and to mention that the branch of Q_{1/2} is the one with the cut from −∞ to 1, so that the reader can verify the application. A parenthetical derivation via contour integration or recurrence would make the note more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and independently benchmarked.

full rationale

The central formula (6) is derived in Section 2 directly from the standard Coulomb-gauge Green's-function integral (7), using only the 2π-periodicity and parity of the denominator (10), the trigonometric identity (14), and the vanishing integral of an odd integrand. None of these steps assumes (6) or any target result. The circular-loop forms (25), (26), (30) are obtained by substituting the standard delta-function current densities (21), (27) into (6); they reproduce known textbook results [1,2]. The closed expressions (31) and (33) are independent evaluations of the same Laplace transform (30), one in elliptic integrals and one via a quoted standard identity from the Bateman tables [7]; their equivalence is checked numerically in Table 1, and the on-axis limit (39) independently recovers the Amperian result. There is no fitted parameter, no post-hoc selection, no author self-citation, and no load-bearing appeal to the author's own prior work. The only unsupported claim is the unproved extension to 'any orthogonal system,' but that is a scope assertion, not a circular step, and it does not feed back into the derived results. Thus no circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted and no entities are invented. The paper imports standard magnetostatic equations, standard expansions from textbooks, and one integral-table identity; the last is external but numerically cross-checked.

assumptions (6)

domain assumption Magnetostatic vector potential satisfies ∇²A=−μ0J in Coulomb gauge with A→0 at infinity and Green-function solution (7).
Section 2, Eq. (7); standard physical premise, not proved in the paper.
standard math Uniqueness for the Laplace equation: a function harmonic in all space and vanishing at infinity is identically zero (used to discard A⊥J for fixed-direction currents).
Section 1, after Eq. (3); standard uniqueness theorem.
standard math Spherical-harmonic expansion of 1/|r−r′| (Jackson Eq. 3.70 / Zangwill Eq. 4.83).
Section 3.1, Eq. (23); standard expansion taken from [1,2].
standard math Bessel-Laplace expansion of 1/|r−r′| (Jackson Problem 3.16), Eq. (29).
Section 3.2, Eq. (29); cited from [1].
standard math Integral-table identity (Erdélyi et al., Tables of Integral Transforms, Vol. I, §4.14 eq. 13) converting the Laplace transform (30) to Q_{1/2}(ξ).
Section 3.3, Eq. (33); external table entry, verified numerically in Table 1.
standard math Legendre-function derivative relation dQν/dz=(z²−1)^{−1/2}Q¹ν(z) (Eq. 36, from [8]).
Section 3.4; used to take the curl of (33).

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Cite this review

Pith. "Pith review of The vector potential of a steady azimuthal current density. Once again." pith.science (2026). https://pith.science/paper/GXPDRQZ3

@misc{pith2026251011663,
  author       = {Pith},
  title        = {Pith review of: The vector potential of a steady azimuthal current density. Once again},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GXPDRQZ3}},
  note         = {Machine review of arXiv:2510.11663}
}

read the original abstract

We give an integral expression for the vector potential of a time-independent, steady azimuthal current density. Our derivation is substantially simpler and somewhat more general than others given in the literature. As an illustration, we recover the results for the vector potential of a circular current loop as an orthogonal expansion in spherical and cylindrical coordinates. Additionally, we obtain closed analytical expressions for the vector potential and the magnetic induction of a circular current loop in terms of Legendre functions of the second kind, that are simpler than the results in terms of complete elliptic integrals given in textbooks.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 1 linked inside Pith

[1]

Classical Electrodynamics,

J. D. Jackson,“Classical Electrodynamics,” Third Edition, John Wiley & Sons, New York, N.Y. (1999)

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[2]

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A. Zangwill, “Modern Electrodynamics,” Cambridge Univ. Press, Cambridge, UK (2012)

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[3]

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A. Vasilyev, “Vector Potential and Magnetic Field of Axially Symmetric Currents,” [arXiv:1208.4983 [physics.class-ph]]

arXiv
[4]

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Wolfram Research Inc., “Mathematica,” Version 14.2, Champaign, IL (2025)

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A. Erdelyi et al., “Tables of Integral Transforms,” Vol. I, McGraw-Hill, New York, NY, (1954)

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Classical Electrodynamics,

J. D. Jackson,“Classical Electrodynamics,” Third Edition, John Wiley & Sons, New York, N.Y. (1999)

1999

[2] [2]

Modern Electrodynamics,

A. Zangwill, “Modern Electrodynamics,” Cambridge Univ. Press, Cambridge, UK (2012)

2012

[3] [3]

Vector Potential and Magnetic Field of Axially Symmetric Currents,

A. Vasilyev, “Vector Potential and Magnetic Field of Axially Symmetric Currents,” [arXiv:1208.4983 [physics.class-ph]]

arXiv

[4] [4]

Mathematica,

Wolfram Research Inc., “Mathematica,” Version 14.2, Champaign, IL (2025)

2025

[5] [5]

Higher Trascendental Functions,

A. Erdelyi et al., “Higher Trascendental Functions,” Vol. II, McGraw-Hill, New York, NY, (1953)

1953

[6] [6]

Elliptic integral,

Wikipedia contributors, “Elliptic integral,” in “Wikipedia, The Free Encyclopedia,” (2025). https://en.wikipedia.org/w/index.php?title=Elliptic integral&oldid=1303198271

2025

[7] [7]

Tables of Integral Transforms,

A. Erdelyi et al., “Tables of Integral Transforms,” Vol. I, McGraw-Hill, New York, NY, (1954)

1954

[8] [8]

Higher Trascendental Functions,

A. Erdelyi et al., “Higher Trascendental Functions,” Vol. I, McGraw-Hill, New York, NY, (1953). 6

1953