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Electrostatic self-energy of a uniformly charged regular polygon: Exact reduction, closed-form expressions, and convergence to the disk

J. Batle1,* and O. Ciftja2,†

  • 1CRISP—Centre de Recerca Independent de sa Pobla, E-07420 sa Pobla, Mallorca, Spain
  • 2Department of Physics, Prairie View A&M University, Prairie View, Texas 77446, USA

  • *Contact author: batlequantum@gmail.com
  • †Contact author: ogciftja@pvamu.edu
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APS Open Sci. 1, 000183 – Published 23 October, 2026

DOI: https://doi.org/10.1103/c72v-54h8

Abstract

We evaluate the electrostatic self-energy of a uniformly charged regular polygon, treated as a planar charge distribution embedded in three-dimensional space and interacting through the Coulomb potential. By exploiting the piecewise-linear nature of the boundary, the four-dimensional Coulomb integral can be reduced exactly to a finite sum of elementary sector contributions that is valid for any number of sides. This construction leads to closed-form expressions for the uniformly charged triangle, square, pentagon, and hexagon. The pentagon result, although involving a quartic number field, is remarkably compact. As the number of sides increases, the energy decreases monotonically toward the corresponding value for a uniformly charged disk. A boundary-perturbation analysis gives the leading relative correction, which decreases as the inverse square of the number of sides. The second variation of the energy produces a nonanalytic correction proportional to the inverse fourth power of the number of sides multiplied by its logarithm. This term provides a direct signature of the polygonal corners. The arithmetic structure of the exact results also explains why compact closed forms cease to be available beyond the uniformly charged hexagon. All results are validated by high-precision numerical calculations and provide useful benchmarks for studies of confined two-dimensional Coulomb systems.

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APS Open Sci. 1, 000183– Published 23 October, 2026
Vol. 1
  • Received 5 June 2026
  • Accepted 9 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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