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Earnshaw's Theorem and electrostatics

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data
    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.


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


    3. 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.
     
  2. jcsd
  3. Jan 29, 2013 #2

    haruspex

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    No, that could be metastable, like a ball on a surface that is fully horizontal.
     
  4. Jan 29, 2013 #3
    I like your analogy, a lot actually. But how can i prove that using Gauss' theorem leads to a contradiction?
     
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