The electromagnetic self-force of a uniformly charged spherical ball

G. Vaman

arxiv: 1906.10999 · v3 · pith:TD6JWDLXnew · submitted 2019-06-26 · ⚛️ physics.class-ph

The electromagnetic self-force of a uniformly charged spherical ball

G. Vaman This is my paper

Pith reviewed 2026-05-25 15:14 UTC · model grok-4.3

classification ⚛️ physics.class-ph

keywords electromagnetic self-forceuniformly charged sphererectilinear motionLorentz contractionclassical electrodynamicsradiation reaction

0 comments

The pith

The electromagnetic self-force on a uniformly charged spherical ball is calculated for rectilinear motion by neglecting Lorentz contraction.

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

This paper computes the electromagnetic self-force that acts back on a uniformly charged spherical ball traveling in a straight line. The calculation proceeds by treating the ball as having its rest shape and neglecting any Lorentz contraction from its velocity. A reader would care because the self-force determines how a charged object responds to its own electromagnetic field, a longstanding issue in classical electrodynamics that connects to radiation reaction and the stability of charged matter.

Core claim

The electromagnetic self-force of a uniformly charged spherical ball moving on a rectilinear trajectory is calculated, neglecting the Lorentz contraction.

What carries the argument

Direct integration of the electromagnetic fields generated by the moving uniform charge distribution to obtain the net Lorentz force on the ball itself.

If this is right

The derived self-force supplies an explicit expression for the radiation-reaction term in the equation of motion of the ball.
The result applies directly to any rectilinear trajectory under the stated approximation.
The same method yields the self-force component parallel to the velocity for this symmetric charge distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approximation becomes increasingly accurate at lower speeds where contraction effects shrink.
The calculation could be compared against the known self-force for a point charge or a shell to check consistency in limiting cases.
Extending the same integration approach to curved trajectories would require reintroducing contraction or using a different frame.

Load-bearing premise

Neglecting Lorentz contraction does not change the validity of the self-force result for this trajectory and charge distribution.

What would settle it

A laboratory measurement of the force on a charged sphere moving at speeds where length contraction becomes measurable that deviates from the no-contraction prediction.

read the original abstract

We calculate the electromagnetic self-force of a uniformly charged spherical ball moving on a rectilinear trajectory, neglecting the Lorentz contraction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper calculates the self-force on a uniformly charged spherical ball in rectilinear motion while keeping the ball spherical and dropping Lorentz contraction.

read the letter

The core output is an explicit expression for the electromagnetic self-force on a uniformly charged ball whose center follows a straight world-line, with the charge distribution held spherical. The calculation is classical and restricted to that geometry and approximation. It adds one more analytic case to the existing literature on extended-charge self-forces, but does not introduce new methods or resolve open questions in the field. The derivation itself appears to follow standard integration of the Liénard-Wiechert fields or equivalent, applied to the uniform ball. That is the main concrete contribution. The approximation of no Lorentz contraction is stated up front, so the result is intended for regimes where v ≪ c or where the error from the spherical shape is accepted. The stress-test concern is therefore on target: for any appreciable fraction of c the lab-frame distribution is no longer spherical, which changes the fields and the integrated force. The paper would need to bound that error or restrict the velocity range clearly for the result to be used outside the non-relativistic limit. No independent numerical check or comparison to the contracted case is mentioned in the available material. The work is narrow in scope and does not claim broader implications. It is the sort of targeted calculation that can be useful to specialists who already know the classical self-force literature and want one more explicit integral evaluated. A referee could check the algebra and the handling of the approximation, but the paper does not look like it would change how the community treats the problem. I would send it to review if the derivation is fully written out and the error discussion is added, but would not cite it myself.

Referee Report

1 major / 1 minor

Summary. The manuscript calculates the electromagnetic self-force of a uniformly charged spherical ball moving on a rectilinear trajectory, explicitly neglecting the Lorentz contraction of the charge distribution.

Significance. If the spherical approximation can be justified with a controlled error bound, the result would supply an explicit self-force expression for a finite-size charge in a simplified relativistic setting, allowing direct comparison with point-particle radiation-reaction formulas and potentially serving as a benchmark for numerical work on extended charges.

major comments (1)

[Abstract] Abstract: the calculation holds the charge distribution spherical while the center follows a rectilinear world-line at arbitrary speed. For v comparable to c this distribution is no longer spherical in the lab frame; the resulting fields and integrated self-force therefore differ from the true relativistic case. The manuscript must supply a quantitative estimate of the error incurred by the approximation (or the velocity range in which the error remains negligible), because this assumption is load-bearing for the claimed self-force expression.

minor comments (1)

The abstract is extremely terse and supplies neither the final expression for the self-force nor the velocity regime considered; a one-sentence statement of the principal result would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the approximation. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the calculation holds the charge distribution spherical while the center follows a rectilinear world-line at arbitrary speed. For v comparable to c this distribution is no longer spherical in the lab frame; the resulting fields and integrated self-force therefore differ from the true relativistic case. The manuscript must supply a quantitative estimate of the error incurred by the approximation (or the velocity range in which the error remains negligible), because this assumption is load-bearing for the claimed self-force expression.

Authors: We agree that maintaining a spherical charge distribution while allowing arbitrary speed constitutes an approximation that neglects Lorentz contraction, as the manuscript states explicitly in the title and abstract. The calculation is performed for this specific model of a uniformly charged spherical ball. In response, the revised manuscript will add a dedicated paragraph clarifying the approximation's domain of applicability. We will state that the results are expected to remain accurate for non-relativistic velocities (v ≪ c), where the leading error from the neglected contraction enters at order (v/c)^2, and that the expression should be applied with increasing caution as v approaches c. A precise numerical error bound would require an independent calculation that incorporates the Lorentz-contracted distribution; this lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation claim is a direct calculation with no self-referential reductions visible

full rationale

The provided abstract and context describe a calculation of electromagnetic self-force under an explicit approximation (neglecting Lorentz contraction for a spherical ball on rectilinear trajectory). No equations, fitted parameters, self-citations, or uniqueness theorems are quoted that would reduce the output to the input by construction. The approximation is a modeling choice whose validity is external to the derivation chain itself; it does not create self-definitional or fitted-input circularity. This matches the default expectation of a self-contained computation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; Maxwell's equations and the uniform-charge assumption are implicit background.

axioms (2)

standard math Maxwell's equations govern the electromagnetic fields of the moving charge distribution.
Standard background invoked by any electromagnetic self-force calculation.
domain assumption The charge distribution remains spherically symmetric and uniform while moving.
Stated in the abstract as the model under consideration.

pith-pipeline@v0.9.0 · 5518 in / 1189 out tokens · 16507 ms · 2026-05-25T15:14:46.838590+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We calculate the electromagnetic self-force of a uniformly charged spherical ball moving on a rectilinear trajectory, neglecting the Lorentz contraction.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ρ(r) = 3e/(4πa³) θ(a−r) … multipole expansion … Fourier transforms … spherical Bessel j₁(ka)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Electromagnetic self-force of a point charge from the rate of change of the momentum of its retarded self-field

V. Hnizdo, G. Vaman, ”Electromagnetic self-force of a po int charge from the rate of change of the momentum of its retarded self-ﬁeld”, https:// arxiv.org/abs/1902.04488

work page internal anchor Pith review Pith/arXiv arXiv 1902
[2]

E. G. P. Rowe, ”Spherical delta functions and multipole e xpansions”, J. Math. Phys. 19, 1962 (1978)

work page 1962
[3]

V. I. Fabrikant, ”Elementary exact evaluation of inﬁnit e integrals of the product of several spherical Bessel functions, power and exponential ”, Quart. Appl. Math. LXXI, 573 (2012)

work page 2012
[4]

I. S. Gradshteyn and I. M. Ryzhic, ”Table of integrals, se ries and products”, 4th edition, Fizmatgiz, Moscow, (1963) (in russian). 7

work page 1963

[1] [1]

Electromagnetic self-force of a point charge from the rate of change of the momentum of its retarded self-field

V. Hnizdo, G. Vaman, ”Electromagnetic self-force of a po int charge from the rate of change of the momentum of its retarded self-ﬁeld”, https:// arxiv.org/abs/1902.04488

work page internal anchor Pith review Pith/arXiv arXiv 1902

[2] [2]

E. G. P. Rowe, ”Spherical delta functions and multipole e xpansions”, J. Math. Phys. 19, 1962 (1978)

work page 1962

[3] [3]

V. I. Fabrikant, ”Elementary exact evaluation of inﬁnit e integrals of the product of several spherical Bessel functions, power and exponential ”, Quart. Appl. Math. LXXI, 573 (2012)

work page 2012

[4] [4]

I. S. Gradshteyn and I. M. Ryzhic, ”Table of integrals, se ries and products”, 4th edition, Fizmatgiz, Moscow, (1963) (in russian). 7

work page 1963