Question

A very long, solid insulating cylinder with radius $R$ has a cylindrical hole with radius $a$ bored along its entire length. The axis of the hole is a distance $b$ from the axis of the cylinder, where $a<b<R$ (as shown in figure). The solid material of the cylinder has a uniform has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field inside the hole, and show that this is uniform over the entire hole.

Answer 1

${\overrightarrow{E}}_{whole}={\overrightarrow{E}}_{cylinder}-{\overrightarrow{E}}_{hole}$
${\overrightarrow{E}}_{whole}=\frac{\rho \overrightarrow{r}}{2{ϵ}_0}-\frac{\rho \overrightarrow{S}}{2{ϵ}_0}=\frac{\rho }{2{ϵ}_0}(\overrightarrow{r}-\overrightarrow{s})=\frac{\rho }{2{ϵ}_0}\overrightarrow{b}$
Note that ${\overrightarrow{E}}_{whole}$ is uniform within the hole.