Physics

The Maxwell Conjecture is False

Philip Arathoon, Gavin Ball, Matthew D. Kvalheim

Featured August 13, 2026

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AI-generated analysis — This is SciGrove's AI interpretation of the paper, not peer-reviewed content.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

Simply

By carefully placing five tiny electric charges, the authors found a way to create at least 24 stable spots where a test charge would feel no push or pull, proving an old idea about how many such spots are possible was wrong.

In depth

The paper presents a counterexample to Maxwell's conjecture, demonstrating a configuration of five point charges that produces at least 24 non-degenerate critical points in its electrostatic potential, exceeding the conjectured maximum of 16 for five charges. This is achieved by starting with three charges in an equilateral triangle and carefully adding two small axial charges, causing a central equilibrium to bifurcate into many new critical points.

Key Takeaways

1
The paper provides a definitive counterexample to Maxwell's 1873 conjecture regarding the maximum number of non-degenerate critical points for $n$ point charges.
2
A specific configuration of five point charges is shown to generate at least 24 non-degenerate critical points, disproving the $(n - 1)^{2}$ upper bound for $n = 5$ .
3
The authors introduce an iterative construction that can generate arbitrarily many critical points, with a ratio of 10 critical points per added charge pair, significantly exceeding previous bounds.

Conceptual Flow

HIGH LEVEL

Methodology: Creating Many Stable Spots

They started with a simple setup of three charges, then added two tiny charges very carefully to make the central 'no-push-pull' spot split into many new ones.

Start with 3 Charges

Add 2 Tiny Charges

Observe New Spots

Results: Disproving an Old Rule

Instead of the expected maximum of 16 stable spots for five charges, their new setup created at least 24, showing the old rule was incorrect.

Old Rule (Max 16 Spots)

New Setup (24+ Spots)

Old Rule Disproven