Physics

The Maxwell Conjecture is False

Philip Arathoon, Gavin Ball, Matthew D. Kvalheim

Featured August 4, 2026

AI-generated analysis — This is SciGrove's AI interpretation of the paper, not peer-reviewed content. Always refer to the original paper.

Simply

By adding two tiny charges to a simple triangle of three charges, the authors found a way to create many more zero-electric-field spots than previously thought possible, proving an old idea wrong.

In depth

The paper presents a counterexample to Maxwell's conjecture, demonstrating a configuration of five point charges that yields at least 24 non-degenerate critical points of the electrostatic potential, exceeding the conjectured maximum of

(n - 1)^{2} = 16

for

n = 5

. This is achieved by starting with a symmetric three-charge configuration and carefully adding two small axial charges with a precisely chosen strength, which causes a central equilibrium to bifurcate into many new critical points.

Key Takeaways

1
The study provides a direct counterexample to Maxwell's 1873 conjecture, showing that the number of non-degenerate critical points of an electrostatic potential can exceed $(n - 1)^{2}$ .
2
The counterexample is constructed by perturbing a simple three-charge equilateral triangle configuration with two small axial charges of a specific strength, leading to a significant increase in critical points.
3
The authors introduce an iterative construction that can generate configurations with an arbitrarily large number of critical points, achieving an asymptotic critical point to charge ratio of 10.

Conceptual Flow

HIGH LEVEL

Methodology: Perturbing Symmetry to Create Equilibria

The authors start with a simple, symmetric setup, then add tiny charges in a special way to make existing 'balance points' split into many new ones.

Simple Charge Setup

Known Balance Points

Add Small Charges Carefully

Complex Charge Setup

Many New Balance Points

Results: Breaking the Old Limit

The old rule said there could only be 16 balance points for 5 charges, but the authors found at least 24, showing the old rule was incorrect.

Old Rule (Max 16 Points)

New Charge Configuration

Found 24+ Points

Old Rule Broken

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