Earnshaw's theorem and 'free space'

In summary, the conversation discusses the proof for Earnshaw's theorem which states that a group of point charges cannot be in a stable equilibrium only through electrostatic interactions. The use of Gauss's Law is questioned as it assumes a charge density of zero, while point charges have infinite density. This leads to the conclusion that there are no defined minima or maxima for the field potential, making stable equilibrium impossible.
  • #1
vetinari
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I am trying to understand the proof for Earnshaw's theorem. Though the theorem states

> that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges ([Wikipedia][1])

In the proof, Gauss's Law in free space is being used (namely that the charge density $\rho$ is zero). How is that correct if we're looking at a collection of point charges? I feel I am being wrong on a very fundamental level. [1]: http://en.wikipedia.org/wiki/Earnshaw%27s_theorem "Wikipedia"
 
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  • #2
Hi. I think the argument is based on the fact that point-charges are represented by delta-functions, so that Gauss' law returns zero everywhere except at these sources where it actually diverges. Hence, there are no well defined minima or maxima of the field potential and no stable equilibrium...
 

1. What is Earnshaw's theorem?

Earnshaw's theorem is a principle in physics that states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This means that it is impossible to have a stable arrangement of charges in free space that does not collapse or disperse.

2. What does 'free space' refer to in Earnshaw's theorem?

In this context, 'free space' refers to a region of space that is not influenced by any external forces, such as gravity or external electric fields. In other words, it is a vacuum in which only the charges themselves and their interactions are present.

3. How does Earnshaw's theorem relate to the stability of atoms and molecules?

Earnshaw's theorem applies to any collection of point charges, including the electrons and protons in atoms and molecules. This means that atoms and molecules are also subject to the principle that they cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of their constituent charges.

4. Are there any exceptions to Earnshaw's theorem?

In certain cases, it is possible to overcome Earnshaw's theorem by introducing additional forces, such as magnetic or gravitational forces. For example, the stability of atoms and molecules is maintained by the additional forces from the strong nuclear force and the Pauli exclusion principle. However, in the absence of these forces, Earnshaw's theorem holds true.

5. What practical applications does Earnshaw's theorem have?

Earnshaw's theorem has been used to explain the behavior of charged particles in various systems, such as ions in a Penning trap and electrons in a cathode ray tube. It also has implications for the design of systems that rely on the stability of charged particles, such as particle accelerators and ion traps.

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