Poisson's equation for conductor with non-uniform conductivity

Question

In my introductions to electromagnetics course we have a computer lab where one of the preliminary question is as follows (my attempt at an answer is below the bold part):

Is my reasoning correct? Is my assumption that the charge density $rho$ is zero in a conductor with non uniform conductivity correct?

Even if my reasoning above is correct, it still bugs me that it is mostly a mathematical reasoning and that I can't formulate for myself the physical interpretation of this. Ok, so equation 1 is not valid in this case, but what does that mean?

Thorbearius · Accepted Answer

Ok, so it seems my reasoning above was correct as my teacher passed my lab report. If the conductor is divided into discrete regions of different (uniform) conductivity each region will have zero charge density, however at the continuity breaks in conductivity there will be a surface charge and thus the conductor will in fact have some charge density. I reason that the same must then be true for a non uniform conductivity as one could view it as infinitely small regions of different uniform conductivity.

After giving it some thought I guess the physical interpretation of this is that the electrostatic model does not work for conductors with non uniform conductivity.

Denis Cété · Answer

The case of non-uniform conductivity is found in semiconductors: the conductivity depends on carrier density, which depends on the potential (hence non-uniform conductivity, "band-bending" etc.).
Note that the current has 2 components: one is proportional to the electric field, and the other is proportional to the gradient of carrier concentration.
A good presentation: https://ecee.colorado.edu/~bart/book/book/title.htm

Poisson's equation for conductor with non-uniform conductivity

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