Consider a one-dimensional situation on a straight line (say, x-axis). Let a charge of magnitude q be located at x=x0, the potential satisfies the Poisson's equation d2Vdx2=−ρ(x)ϵ0=−qδ(x−x0)ϵ0. If q>0, V′′(x0)<0, and if q<0, V′′(x0)>0. Therefore, it appears that the potential V does have a minimum at x=x0, for q<0. Does this imply that x=x0 is a point of stable equilibrium? I must be missing something because this appears to violate Earnshaw's theorem (or it doesn't)?
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