electrostatics

We have a charge distribution in which all the charges are in equilibrium due to electrostatic forces . Can we prove that none of these charges will be in stable equilibrium ?
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 Recognitions: Homework Help Science Advisor Yes, since the potential due to the other charges satisfies $\nabla^2 V=0$. V is harmonic with has the property that is has no local maxima or minima.
 Recognitions: Homework Help Science Advisor It is called Thomson's theorem.

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 Yes, since the potential due to the other charges satisfies $\nabla^2 V=0$is being generated. Reload this page in a moment.. V is harmonic with has the property that is has no local maxima or minima. thats what i wanted to ask . how can we prove that $\nabla^2 V=0$
 Recognitions: Homework Help Science Advisor It's just Maxwell's (first) equation: $\vec \nabla \cdot \vec E =\rho/\epsilon_0$, or $\nabla^2 V=-\rho/\epsilon_0$. In the region where there is no charge density you have $\nabla^2 V=0$.