I thought that since a conductor as a whole, an electrically neutral medium, $\vec{\nabla}\cdot\vec{E}=0$ inside a conductor. But while reading Ashcroft and Mermin's Solid state physics, I found out that at equation $1.31$ they assumed $\vec{\nabla}\cdot\vec{E}=0$ but at $1.43$, they assumed $\vec{\nabla}\cdot\vec{E}\neq 0$. I cannot understand this. Please help.

First, they derive the expression for complex frequency-dependent effective permittivity $\epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$ assuming $\vec{\nabla}\cdot\vec{E}=0$ (page $17$-$18$). Then on page $19$, they assumed an equation $\vec{\nabla}\cdot\vec{E}(\omega)=4\pi\rho(\omega)$ to show the onset of plasma oscillations.

  • 4
    $\begingroup$ Could you give those of us who don't have the book a little more context? $\endgroup$
    – Philip
    Jun 6, 2020 at 18:39
  • $\begingroup$ In Maxwell's equations the RHS of the divergence of the electric field is equal to the charge density, so I imagine in the first case there is none, while in the second there is a non-zero charge density? $\endgroup$
    – ProfM
    Jun 6, 2020 at 19:22
  • $\begingroup$ Footnote on page 17: «We are considering here an electromagnetic wave, in which the induced charge density $\rho$ vanishes. Below we examine the possibility of oscillations in the charge density.» $\endgroup$
    – Felipe
    Jun 6, 2020 at 20:18
  • $\begingroup$ @Felipe But when is the induced charge density is zero? When is it nonzero? Those are not explained. $\endgroup$ Jun 6, 2020 at 20:44
  • $\begingroup$ @ProfM But why? We have metal, in both cases. $\endgroup$ Jun 6, 2020 at 20:49

1 Answer 1


They assume electrical neutrality in 1.31 as an approximation. An applied AC electric field will cause positive charges in some places and negative charges in others, but this effect is usually pretty small. The charge density wave (plasmon) they describe later (Eqs. 1.42-45) are an exception where the earlier approximation fails. However, these plasmons only occur at specific frequencies. If you apply a field at any other frequency, you don't get a plasmon and the approximation of zero charge density will probably work.

Sometimes you do need more accuracy than the approximation can provide. In that case, you need a more advanced theory, such as the one that produces Eq. 17.60. However, this type of thing gets complicated very quickly, and as Ashcroft and Mermin note, the more advanced theory reduces to the simple theory in the right limit. So, don't make your life complicated if you don't have to.


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