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I just came across Earnshaw's Theorem which states:

As an example it said that equal and fixed point charges at the corners of a cube could not hold stationary a point charge at the center.

There was no proof provided, but after a little thought it occured to me that Poisson's Equation for the potential would reduce to Laplace's Equation:

[itex] \nabla^{2}V=-\frac{\rho}{\epsilon_{0}} \Longrightarrow \nabla^{2}V=0[/itex]

since there is no charge anywhere but at the corners.

Because Laplace's Equation does not allow solutions with local maxima or minima then it would be impossible to find a a stable configuration.

Mathematically this seems fine. However, visually it seems that if I were to lightly tap the charge at the center then it would feel a repulsion pushing it back to the center (causing it to then oscillate about the center). Clearly this is wrong. Does anyone know of a physical way to imagine the instability?

*A charged particle cannot be held in stable equilibrium by electrostatic forces alone.*As an example it said that equal and fixed point charges at the corners of a cube could not hold stationary a point charge at the center.

There was no proof provided, but after a little thought it occured to me that Poisson's Equation for the potential would reduce to Laplace's Equation:

[itex] \nabla^{2}V=-\frac{\rho}{\epsilon_{0}} \Longrightarrow \nabla^{2}V=0[/itex]

since there is no charge anywhere but at the corners.

Because Laplace's Equation does not allow solutions with local maxima or minima then it would be impossible to find a a stable configuration.

Mathematically this seems fine. However, visually it seems that if I were to lightly tap the charge at the center then it would feel a repulsion pushing it back to the center (causing it to then oscillate about the center). Clearly this is wrong. Does anyone know of a physical way to imagine the instability?

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