Electric field of a uniformly charged infinite cylinder (${\mathbf{E}}_{\mathrm{in}}=\rho \mathbf{r}/(2{ϵ}_0)$)

For an infinite insulating cylinder of radius $R$ and uniform fixed charge density $\rho$, Gauss's law gives interior electric field $\rho \mathbf{r}/(2{ϵ}_0)$ and exterior electric field $\rho R^2\mathbf{r}/(2{ϵ}_0|\mathbf{r}{|}^2)$, where $\mathbf{r}$ is transverse displacement from its axis. The electric field is continuous at $|\mathbf{r}|=R$ when there is no separate surface charge density. The interior linear dependence makes displaced-cylinder superposition particularly simple.

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.