= Solution
In a linear isotropic <dielectric> of <relative permittivity> $\epsilon$, the <electric displacement field> and <electric field> satisfy
$$
\boxed{\mathbf D=\epsilon_0\epsilon\mathbf E,\qquad
\nabla\cdot\mathbf D=\rho_f,\qquad\nabla\times\mathbf E=0.}
$$
Here $\rho_f$ denotes the free <charge density> called $\rho$ in the question. For spatially constant $\epsilon$, these also give $\nabla\cdot\mathbf E=\rho_f/(\epsilon_0\epsilon)$. Equivalently, $\mathbf E=-\nabla\varphi$ and $\nabla^2\varphi=-\rho_f/(\epsilon_0\epsilon)$. The <electric displacement field> has units $\mathrm{C\,m^{-2}}$ and the <electric field> has units $\mathrm{V\,m^{-1}}$.
The local equations determine the fields once their geometry and <boundary conditions> are specified; a density value alone does not determine a unique vector field without that information.
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