In electrostatics, a charge distribution in equilibrium cannot achieve stable equilibrium due to the properties of the electric potential, which satisfies the equation ∇²V=0. This indicates that the potential is harmonic, lacking local maxima or minima, as stated in Earnshaw's theorem. The discussion clarifies that the relationship between electric field and potential is derived from Maxwell's equations, specifically highlighting that in regions without charge density, the potential also satisfies ∇²V=0. The conversation briefly corrects a misattribution of the theorem to Thomson instead of Earnshaw. Overall, the key takeaway is that stable equilibrium for charges is impossible in electrostatic systems.
