Uniform electric field in an off-axis cylindrical cavity ($\mathbf{E}=\rho \mathbf{a}/(2{ϵ}_0)$)

Remove an axial cylindrical hole, wholly contained in an infinite insulating cylinder with uniform fixed charge density $\rho$. Let $\mathbf{a}$ point from the original axis to the hole axis. The superposition principle replaces the removed material by a cylinder of charge density $-\rho$. At any point $\mathbf{r}$ inside the hole, the two interior electric fields add to $\rho [\mathbf{r}-(\mathbf{r}-\mathbf{a})]/(2{ϵ}_0)=\rho \mathbf{a}/(2{ϵ}_0)$. The result is uniform throughout the hole, even when the hole is not concentric; it does not apply to a hole in an electrical conductor whose electric charges can redistribute.