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arxiv: 2604.22908 · v1 · submitted 2026-04-24 · ⚛️ physics.class-ph · cond-mat.mtrl-sci· cs.NA· math.NA· physics.app-ph· physics.comp-ph· physics.soc-ph

Recognition: unknown

Electric potential of insulated conducting objects in presence of electric charges -- some exact and approximate results

Hrvoje \v{S}tefan\v{c}i\'c, Karlo Filipan

Authors on Pith no claims yet

Pith reviewed 2026-05-08 08:43 UTC · model grok-4.3

classification ⚛️ physics.class-ph cond-mat.mtrl-scics.NAmath.NAphysics.app-phphysics.comp-phphysics.soc-ph
keywords electrostaticsconductorselectric potentialJ formalismcapacitancegeometrical approximation
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The pith

A new J formalism determines the electric potential of insulated conducting objects exactly or approximately from their geometry alone.

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

The paper introduces the J formalism to reformulate how to find the electric potential on an insulated conducting object near external charges. This reformulation lets one calculate the potential exactly for spherical objects and approximate it for others by using only the object's geometrical properties. No calculation of the surface charge distribution or the potential in the space around the object is needed. The method is tested numerically with the Robin Hood method to check its accuracy and efficiency. It is proposed as a way to compute capacitances of conducting objects more directly.

Core claim

The J formalism reformulates the electrostatic boundary-value problem for insulated conductors so that their electric potential in the presence of external charges follows directly from geometrical properties, giving exact values for spheres and practical approximations for general shapes without determining surface charges or the exterior potential.

What carries the argument

The J formalism, a new way to express the potential of an insulated conductor through its geometrical features rather than solving the full boundary value problem.

If this is right

  • The potential on a spherical conductor is obtained exactly from its radius and position relative to charges.
  • Non-spherical objects receive good approximations based on their shape and size metrics alone.
  • Capacitance calculations for conducting objects become feasible without full electrostatic field solutions.
  • The computational effort focuses only on the object geometry instead of volume or surface integrals over charges.

Where Pith is reading between the lines

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

  • This could enable quicker estimates in engineering applications involving conductors in electric fields.
  • Similar geometry-based approaches might apply to other boundary problems in electrostatics or heat conduction.
  • Extensions to multiple conductors or dynamic cases could be explored using the same formalism.

Load-bearing premise

The J formalism correctly reformulates the electrostatic boundary-value problem for insulated conductors and yields valid exact or approximate potentials solely from geometrical properties.

What would settle it

Solving Laplace's equation numerically for an insulated sphere with a nearby point charge and comparing the surface potential to the value given by the J formalism.

Figures

Figures reproduced from arXiv: 2604.22908 by Hrvoje \v{S}tefan\v{c}i\'c, Karlo Filipan.

Figure 1
Figure 1. Figure 1: Distributions of J(⃗xi) for sphere of radius R = 10 for different mesh refinements: average approaches theoretical value of 4πR (for k = 1) for more refined meshes. The green triangle denotes the mean value and the orange line the median value. The orange dashed line corresponds to the theoretical value for the J value of a sphere of radius R. established for any surface geometry: J(⃗xi) = XN j=1 Aij∆Sj . … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the initial and final potential distributions for the view at source ↗
Figure 3
Figure 3. Figure 3: Average potential across iterations for different spheres centered at view at source ↗
Figure 4
Figure 4. Figure 4: Average and standard deviation of percentage differences: a) between view at source ↗
Figure 5
Figure 5. Figure 5: Distributions of J(⃗xi) for a cube of side 5 for different mesh refine￾ments. electric charges’ positions). To measure the spatial characteristics relevant for the two aforemen￾tioned tendencies, a number of dimensionless quantities could be used. We first introduce the barycenter of the insulated object surface located at R⃗ = 1 S Z S ⃗r dS ≈ P i P ⃗ri∆Si i ∆Si . (18) Here, ⃗r refers to the radius vector … view at source ↗
Figure 6
Figure 6. Figure 6: Distributions of J(⃗xi) for a torus with radii R = 10 and r = 10 3 for different mesh refinements. external charges to the insulated object is introduced as a dimensionless quantity cc = 1 N 2 dmax + dmin X i dq,i . (20) The value of cc is small for charges close to the insulated object and large for charges far from the insulated object. A systematic analysis of how the accuracy of the approximation depen… view at source ↗
Figure 7
Figure 7. Figure 7: Initial and final potential distributions for examples with a single unit view at source ↗
Figure 8
Figure 8. Figure 8: Average and standard deviations over all (main graph) and 100 first view at source ↗
Figure 9
Figure 9. Figure 9: The average and standard deviations of potential over all and 500 first view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of calculated integral value and the difference between view at source ↗
Figure 11
Figure 11. Figure 11: Histograms of integral expression in (9) for spheres of radius 10 and view at source ↗
Figure 12
Figure 12. Figure 12: Histograms of integral expression in (9) for: a) cylinder with the base view at source ↗
Figure 13
Figure 13. Figure 13: Histograms of integral expression in (9) for two objects shown in view at source ↗
read the original abstract

Determination of the electric potential of insulated conducting objects is an important problem both theoretically and practically. For an insulated conducting object in the presence of external charges or charges distributed on the object surface, the problem of potential determination is reformulated using a newly introduced $J$ formalism. Using the $J$ formalism, it is shown how the electric potential can be calculated exactly for spherical objects and efficiently approximated for other object geometries using geometrical properties of the insulated conducting object. This approach does not require calculation of the surface charge distribution at the object surface of the calculation of the electric potential in the surrounding space. Properties and the performance of the approach are investigated numerically using the Robin Hood method. Possible applications of the approach based on the $J$ formalism are outlined for calculation of capacitance of conducting objects.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a J formalism to reformulate the electrostatic boundary-value problem for an insulated conducting object in the presence of external charges. It claims that this allows exact determination of the constant potential on spherical objects and efficient approximations for other geometries using only geometrical properties of the object, without explicit computation of the surface charge distribution or the exterior potential field. Numerical properties are investigated via the Robin Hood method, and applications to capacitance calculations are outlined.

Significance. If the J formalism is shown to be mathematically equivalent to the standard BVP (Laplace equation outside, constant potential and fixed total charge on the conductor) while enabling direct geometry-based evaluation, the approach could provide a practical shortcut for potential and capacitance estimates on complex shapes. The exact spherical results would serve as a useful benchmark, but the absence of derivations, error analysis, or quantitative validation in the provided text limits assessment of its advantage over existing methods such as image charges or boundary-element techniques.

major comments (3)
  1. [J formalism section] The section introducing the J formalism: the reformulation must be shown explicitly to enforce constant potential on the conductor and fixed total charge without implicit reference to the surface charge density or the exterior solution. Provide the step-by-step equivalence proof to the standard boundary conditions, as the central bypass claim depends on this.
  2. [Spherical objects section] The section on spherical objects: the claimed exact results must be derived or verified against known analytic solutions (e.g., image-charge method for a point charge near an insulated sphere). Include the explicit expression for the potential in terms of J and demonstrate agreement with error bounds.
  3. [Numerical results] The numerical investigation section: the Robin Hood method tests are referenced but supply no data tables, error metrics, convergence rates, or comparisons to standard solvers. Add quantitative validation (e.g., relative error vs. geometry parameters) to support the approximation claims for non-spherical shapes.
minor comments (2)
  1. [Abstract] The abstract contains a minor grammatical issue: 'does not require calculation of the surface charge distribution at the object surface or the calculation of the electric potential' should read 'or calculation of the electric potential' for parallelism.
  2. Notation for the J quantity should be defined with a clear equation number on first use and distinguished from standard electrostatic quantities (e.g., current density or other J symbols) to avoid confusion.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to provide the requested explicit derivations, verifications, and quantitative validations.

read point-by-point responses

  1. Referee: [J formalism section] The section introducing the J formalism: the reformulation must be shown explicitly to enforce constant potential on the conductor and fixed total charge without implicit reference to the surface charge density or the exterior solution. Provide the step-by-step equivalence proof to the standard boundary conditions, as the central bypass claim depends on this.

    Authors: We agree that an explicit step-by-step equivalence proof is required to fully substantiate the central claim. The manuscript introduces the J formalism as a reformulation that bypasses explicit surface charge and exterior potential calculations by relying on geometrical properties, but we acknowledge the need for a clearer derivation showing direct enforcement of constant potential and fixed total charge. In the revised manuscript, we will add a dedicated subsection with the full equivalence proof to the standard BVP. revision: yes

  2. Referee: [Spherical objects section] The section on spherical objects: the claimed exact results must be derived or verified against known analytic solutions (e.g., image-charge method for a point charge near an insulated sphere). Include the explicit expression for the potential in terms of J and demonstrate agreement with error bounds.

    Authors: We accept this point. The manuscript states that exact results are obtained for spherical objects via the J formalism, but we will strengthen the section by deriving the explicit potential expression in terms of J, verifying it against the image-charge method for a point charge near an insulated sphere, and including quantitative agreement with error bounds. revision: yes

  3. Referee: [Numerical results] The numerical investigation section: the Robin Hood method tests are referenced but supply no data tables, error metrics, convergence rates, or comparisons to standard solvers. Add quantitative validation (e.g., relative error vs. geometry parameters) to support the approximation claims for non-spherical shapes.

    Authors: We agree that the numerical section requires more quantitative support. The manuscript investigates properties using the Robin Hood method but does not include detailed metrics. In the revision, we will add data tables, error metrics, convergence rates, and comparisons to standard solvers, with relative errors plotted against geometry parameters for non-spherical shapes. revision: yes

Circularity Check

0 steps flagged

J formalism is an independent reformulation with no reduction to inputs by construction

full rationale

The paper introduces a new J formalism as a reformulation of the electrostatic BVP for insulated conductors and derives exact results for spheres plus geometry-based approximations for other shapes, all without explicit surface charge or exterior potential. Numerical performance checks rely on the independent Robin Hood method rather than the target quantities themselves. No equations or steps are shown to define J in terms of the predicted potential, fit parameters to the output, or rely on self-citation chains for the central claims. The derivation therefore remains self-contained and does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the validity of the newly introduced J formalism as a correct reformulation of the electrostatic problem for insulated conductors.

axioms (1)
  • domain assumption The J formalism provides a valid reformulation of the electric potential problem for insulated conducting objects in the presence of charges.
    This is the foundational premise stated in the abstract for deriving exact and approximate results.
invented entities (1)
  • J formalism no independent evidence
    purpose: To reformulate the potential determination problem without requiring surface charge distribution or surrounding potential calculations.
    New mathematical construct introduced in the paper with no independent evidence or prior references mentioned.

pith-pipeline@v0.9.0 · 5467 in / 1193 out tokens · 35119 ms · 2026-05-08T08:43:39.043817+00:00 · methodology

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Reference graph

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