Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Earnshaw's theorem

The idea of particle instability in an electrostatic field originated with Samuel Earnshaw in 1839  and was formalized by James Clerk Maxwell  in 1874 who gave it the title "Earnshaw's theorem" and proved it with the Laplace equation. Earnshaw's theorem explains why a system of electrons is not stable and was invoked by Niels Bohr in his atom model of 1913 when criticizing J. J. Thomson's atom.

Earnshaw's theorem holds that a charged particle suspended in an electrostatic field is unstable, because the forces of attraction and repulsion vary at an equal rate that is proportional to the inverse square law and remain in balance wherever a particle moves. Since the forces remain in balance, there is no inequality to provide a restoring force; and the particle remains unstable and can freely move without restriction.

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