Earnshaw's Theorem and electrostatics

[arishorts]

Homework Statement

14: a: It is impossible to have a stable equilibrium in electrostatics. This idea is known as Earnshaw’s Theorem. Let’s prove this fact. Assume that at a particular point P that a charge Q is in a stable equilibrium. Think about the direction of E⃗ necessary for the equilibrium. Now use Gauss’ Law with a spherical gaussian surface centered on P. Show that this leads to a contradiction.
b: Imagine a square in the xy plane with a point charge Q fixed at each corner. Now put a test charge q in the exact center of the square. What direction(s) can we move q in for which the equilibrium is stable? For which it is not stable? Explain.

Homework Equations

Gauss's equation: (E*A)/(Qenclosed*(Permittivity of free space))

The Attempt at a Solution

I've figured out that the direction of E has to be inward, but i don't understand why. With some research I've found the proof where the divergence comes out to zero, but wouldn't that mean it achieves stable equilibrium? Seems a bit contradictory. Also, if you use a spherical versus a circular surface, you get a difference in answer by a factor of 1/3... That's the best i can do. Please help.

 

[haruspex]

the divergence comes out to zero, but wouldn't that mean it achieves stable equilibrium?

No, that could be metastable, like a ball on a surface that is fully horizontal.

 

[arishorts]

I like your analogy, a lot actually. But how can i prove that using Gauss' theorem leads to a contradiction?