How are neutral conductors neutral even though they microscopically aren't?

Question

When dealing with electromagnetic waves inside conductors we take $$\nabla \cdot E = 0$$

and I believe we use the fact that conductors are neutral to do this. However, even though conductors on average are neutral, microscopically they really aren't and they have free charges moving around especially if there is an electromagnetic wave going through them. So how does this justification hold? Is it because the divergence operator really cares about the global neutrality of the domain and not the microscopic parts?

Answer 1

You are asking a good question, and it is sad that there are people downvoting.

The meaning of the Maxwell's equations in materials is actually an extremely delicate subject. To get the zero, we actually have to average the fields over a region at least as big as a unit cell of the material. It is not at all easy to define, and nobody actually knows a scheme that solves all the theoretical problems that an averaging process brings up.

That is, this topic is so difficult, that multiple people can spend their entire lives into this and still get nowhere. Beware.

Answer 2

The equation $$ \nabla \cdot \mathbf E = 0 $$

is from macroscopic EM theory, where material medium is a continuous region of space with different behaviour and relation between the fields $\mathbf E,\mathbf D,\mathbf B,\mathbf H$ than they have in the vacuum. The equation is not generally valid for all mediums, but it is for uniform, isotropic and uncharged material, such as glass or water. In conductors one can have non-zero total charge, but only if maintained by external forces, such as when the conductor rotates in magnetic field. Left free of external forces, initial charge distribution inside the conductor will quickly move to its surfaces and establish a neutral medium inside.

There are no macroscopic charges in these models inside and thus $\rho_{total}=0$.

Of course, we know conductors have charged particles in them, and the above equation would not be correct if by $\mathbf E$ we meant the microscopic electric field in between and inside the charged particles.

The justification for the above equation is that the field $\mathbf E$ there is not the microscopic electric field, but the macroscopic electric field from the macroscopic EM theory, which is a smooth function of position and does not manifest any microscopic variations, except possible jumps at the boundaries of different material media or the boundary with the vacuum. Also the fact that results of derivations where Maxwell's equations and this assumption are used, are in very good agreement with many experimental observations in optics and electrical engineering.

There are cases where the equation does not hold, such as in anisotropic materials with different permittivities in different directions of space, such as in some crystals. Then one has $\nabla \cdot \mathbf D = 0$ instead, and divergence of electric field does not generally vanish; a non-zero polarization charge can appear inside the crystal.