Electrostatics Boundary Value Problems (Finding resistance of conductor): When is $\rho _v$ Zero? When do we assume in Boundary Value Problems (BVP) that $\rho _v$ is zero? I was solving a problem in Elements of Electromagnetics By Sadiku which says that we want to find the resistance of a bent iron bar.

In the solution they solved Laplace equation $\nabla^2 V=0 $ rather than poisson equation $\nabla^2 V= -\frac{\rho _v}{\epsilon} $ did they assume that $\rho_v =0$ since this is a conductor and hence $\vec E=0$ and then $Q=0$ .
 A: In order to calculate the dc resistance of a conductor with specific conductivity $\sigma$, you assume time independent (stationary) current densities and charge densities so that the current continuity equation becomes $$\nabla· \vec J=-\frac {\partial \rho}{\partial t}=0 \tag 1$$ where $\vec J$ is the current density and $\rho$ is the charge density.  The electric field is irrotational, i.e., it has an electric potential $\phi$ so that $$\vec E=-\nabla \phi \tag 2$$ Further the current density is related to the electric field by $$\vec J=\sigma \vec E=-\sigma \nabla \phi \tag 3$$ Inserting the current density of eq. (3) into eq. (1) results in the Laplace equation for the electrical potential $$ \Delta \phi=0 \tag 4$$ When eq. (4) is solved with the appropriate boundary conditions, the electric field distribution and thus also the current densities, total current and resistance is obtained. In the derivation of equation (4) it has only been assumed that the time derivative of the charge density is zero, not the charge density itself. Also a constant conductivity $\sigma=n·e·\mu$ ($n$ is the electron density and $\mu$ is the electron mobility of the metal) has been assumed, which implies that any possible net charge density $\rho$ contributing to conduction is negligible  compared to the mobile charge density (electrons in a metal) determining the conductivity, i.e., $\rho \ll n·e$.
Added note: Your assumption that the electric field inside the conductor is incorrect. This only holds for a conductor without an applied potential difference which is obviously not the case here where you are considering the electrical resistance.
When a constant $\sigma$ is assumed, equation (4) shows that there are no volume charges because it is equivalent to $\nabla \vec E=0$. However, to fulfill the boundary conditions at the surface there will be surface charges.
