# What is the value of $\vec{\nabla}\cdot\vec{E}$ inside a conductor?

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.

• Could you give those of us who don't have the book a little more context? Commented Jun 6, 2020 at 18:39
• 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? Commented Jun 6, 2020 at 19:22
• 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.» Commented Jun 6, 2020 at 20:18
• @Felipe But when is the induced charge density is zero? When is it nonzero? Those are not explained. Commented Jun 6, 2020 at 20:44
• @ProfM But why? We have metal, in both cases. Commented Jun 6, 2020 at 20:49