Earnshaw's theorem

Earnshaw's theorem states that point charges cannot remain in stable static equilibrium under electrostatic forces alone in free space. It follows from the fact that electrostatic potential has no local minima in charge-free regions.

Electrostatic potential: The potential energy per unit charge at a point in an electric field. Its curvature determines whether a charge can sit at a stable equilibrium.
Paul trap: An ion trap that confines charged particles using oscillating electric fields. Its time-varying fields evade the theorem’s restriction on static equilibrium.
Stable equilibrium: An equilibrium state in which small disturbances produce forces that restore the system toward its original state. The theorem rules out this kind of equilibrium for point charges under its stated conditions.
Coulomb's law: The inverse-square law giving the electrostatic force between point charges. It defines the force whose static equilibrium Earnshaw’s theorem constrains.
Laplace's equation: A partial differential equation stating that a function’s Laplacian is zero in a source-free region. Electrostatic potential satisfies it wherever there is no charge, forbidding a local minimum.
Penning trap: A device that confines charged particles with a static magnetic field and an electric quadrupole field. Magnetic forces supply confinement that static electrostatic forces alone cannot provide.
Thomson's theorem: A result that electrostatic equilibrium in a conductor minimizes electrostatic energy for fixed charges. It concerns conductor charge distributions, not stable trapping of a point charge in free space.
Electric field: A vector field describing the electric force per unit positive charge at each point. A stable charge position would require a restoring pattern in this field.
Poisson's equation: A partial differential equation relating a potential’s Laplacian to its source density. It generalizes the charge-free equation that supplies Earnshaw’s key constraint.
Magnetic levitation: Suspension of an object against gravity using magnetic forces. Magnetic forces can stabilize configurations that electrostatic forces cannot stabilize alone.

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