Electric Field Inside matter

Right up close to any atomic nucleus the electric field is huge, really enormous. But at a distance of 1 nanometre or more from an isolated neutral atom, the field is almost zero because the contributions from electrons and protons cancel each other out.

In a solid material, similar statements apply, except that the distance between atoms is such that you never get right away from one atom before running into the next, so you never get to the place where the electrons of a single atom cancel the nuclear field of that atom. Now the field at any one spot is the total from all the nuclei and electrons around, and it can be quite big at one point, but it will vary in direction from one point to another, with the result that when you average over a region of size a few nanometres, the average field is very close to zero. It is this average field which people are referring to when they say there is no electric field in a conductor. For greater precision, allow the region you are averaging over to be larger still, say one micrometre diameter. Now the average field is extremely close to zero, and for a conductor there is a further consideration.

For a conductor, the situation is such that if there were a field on average, then the conduction electrons would move. So they keep moving until they take up positions throughout the conductor such as to provide a field which very precisely balances whatever average field there would otherwise be inside the material. But this balancing can only cancel the average field, not its spatial variation at the atomic scale.

I confess I have not bothered to add quantitative statements to say what the words "huge" and "extremely close to zero" mean here. But you can estimate them yourself. Use Coulomb's law for the field at the edge of an atomic nucleus. For a region of size 1 micrometre or so in an insulating solid, estimate how far from zero the net charge inside that region could plausibly be. (If the solid is polarised, it can be non-zero). For a conductor, the feedback mechanism provided by the movement of the conduction electrons means that once steady state has been achieved the field (strictly speaking, the chemical potential gradient, but let's not get into that ...) is zero at places where conduction electrons can move freely. Inside an atom they can't move freely. On distances that extend over many atoms, they can (but quantum theory is needed to understand this).

The length scale of interest for classical EM is much larger than Angstroms. We are interested in scales where matter can be approximated as a continuum. At those scales the fields of the electrons and protons cancel out quite precisely for most materials.

Indeed, although nuclear electric fields are screened by electrons, the screening is not perfect. At angstrom, subatomic scale there are variations in the electric field. At nanometer scale these are already largely ironed out. When discussing electric field strength, atomic style units should be used. For example, the normal definition of E is Newton per Coulomb. Newton per elementary charge is more reasonable, as is the Bohr radius as length unit. In such units the E field takes values of the order of unity. I am guessing and leave it to the op to check. It is also clear the a very strong macroscopic electric field has only a moderate impact at atomic scale.