Optics, electromagnetic theory Electrostatics Conjecture

Maxwell's conjecture on point charges

Builds on Electric charge Electric field Electric potential

Statement

Let $n \ge 2$ , and let $q_1, \dots, q_n$ be nonzero real numbers, point charges fixed at distinct points $x_1, \dots, x_n \in \R^3$ . Let $E$ be their electric field on $\R^3 \setminus \set{x_1, \dots, x_n}$ . Then the number of isolated points $x$ with $E(x) = 0$ is at most $(n - 1)^2$ .

Remarks

The zeros of $E$ are the critical points of the electric potential $\varphi$ , so they are the equilibrium points of a test charge. Maxwell stated the bound in his Treatise on Electricity and Magnetism, with a heuristic argument but no proof. The conjecture is open in general. It was brought to wide attention by Gabrielov, Novikov and Shapiro in 2007, who studied it with methods of real algebraic geometry. Example: two equal charges $q$ at $(\pm 1, 0, 0)$ give a single zero of $E$ , at the midpoint, and $(2 - 1)^2 = 1$ .

Reading path

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Electric charge concept
Electric field concept
Electric potential concept
Maxwell's conjecture on point charges conjecture

Tag	`9ZGVBTS931`
URI	`m2://9ZGVBTS931`
Kind	conjecture
Writing	W0 stub
MSC	78A30, 31B05
Status	This copy of the site has no build data. Run `m2 status 9ZGVBTS931`.
About	Electric field
Uses	Electric charge, Electric field, Electric potential
Changed	3 November 2026

Maxwell's conjecture on point charges

Statement

Remarks

Reading path

Further reading

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