# Dilemma of classical physics: stationary particles that can't be in stable equilibrium

Statement from Electricity and Magnetism (Edward Purcell):

$$U = −0.8738Ne^2 /4π\epsilon_0 a$$ The negative sign shows that work would have to be done to take the crystal apart into ions. In other words, the electrical energy helps to explain the cohesion of the crystal. If this were the whole story, however, the crystal would collapse, for the potential energy of the charge distribution is obviously lowered by shrinking all the distances. We meet here again the familiar dilemma of classical – that is, nonquantum – physics. No system of stationary particles can be in stable equilibrium, according to classical laws, under the action of electrical forces alone.

This explanation was based on the crystal lattice of Nacl.

I'm very new to this. What is the dilemma?

In my guess the dilemma is that the ions in NaCl are in static equilibrium due to electrostatics forces alone, which contradicts with theory in the classical physics mentioned in the text. Am I right in thinking so?

• – Farcher Apr 1 '18 at 8:09
• @Fracher I had seen that as well, but unfortunately it doesn't answer my question. I'm not confused with the presence or absence of charge like asked in the question related to the link. My question is different. I'd be happy if you could help me get rid of my confusion. – suiz Apr 1 '18 at 8:38
• Hint: Earnshaw's theorem. – Qmechanic Apr 1 '18 at 10:10
• No, you have it backwards. The ions are not in static equilibrium under the classical model, as Purcell explains, and generally they never are in a classical model. The dilemma is the fact that real matter is stable anyway. Both of the links above directly answer your question. – knzhou Apr 1 '18 at 10:18
• @knzhou So, actual matters are stable with electrostatic force alone but the classical model explains that only electrostatic force cannot hold the ions in static equilibrium, which is the dilemma here. Did I get it right? – suiz Apr 1 '18 at 16:02