Earnshaw's Theorem and Plum Pudding I'm trying to understand what Thompson's motivation for his Plum-pudding model was. He chose the nucleus to be a sphere of uniform charge because it was mathematically nice, and chose the charge to be positive to circumvent Earnhsaw's theorem. It's this latter part I'm not understanding so much: the divergence $\nabla\cdot E=\frac{\rho}{\epsilon_0}>0$ (for positive test charge), so any point in the atom would have a negative divergence for a negative test charge (electron). This may be a dumb question, but why was it important to sidestep Earnshaw? What is physically wrong with having electrons in an atom be unstably configured? 
 A: 
What is physically wrong with having electrons in an atom be
  unstably configured?

Think of it like this. Imagine that you have a theory that describes how the atoms work, and where you use the best of you knowledge of electromagnetism and Newtonian mechanics. SO you place some positively charged points particles and some negatively charged points particles in a volume that represent your atom. So you have a model of an atom, yey! The only problem with your model is that the configuration is unstable. That is, your atom will fall part instantly, because of Earnhsaw's theorem. This is bad.
Let's try to fix it. So you postulate that the point charges are held in place by some force. The problem here is that there is know physical ways of dosing this. Thus, if you insist on you point particles you have to invoke "magic" to keep them stable.
The plum pudding model is (in this setting) a revision of the original model where you create a stable configuration (this is good), but you pay the price that some of you point particles are now tuned into a continuous charge density instead. 
