Answer:
The value of the electric field is [tex]E_{net} = \dfrac{r \textbf{b}}{2\epsilon_{0}}[/tex]
Explanation:
We know that the electric field inside a solid cylinder at a distance [tex]\textbf{r}[/tex] from the centre is given by
[tex]E = \dfrac{\rho \textbf{r}}{2 \epsilon_{0}}[/tex]
Let's consider the cross-section of the cylinder as shown in the figure. Let `O' be the centre of the long solid insulating cylinder having radius 'R'. Also consider that [tex]O'[/tex] be the cetre of the hole of radius 'a' situated at a distance 'b' from 'O'. Given, the volume charge density of the material is 'r'. So, the volume charge density inside the hole will be '-r'. Let's consider 'P' be any arbitrary point inside the hole situated at a distance 's' from [tex]O'[/tex].
So, the electric field '[tex]E_{O}[/tex]' due to the long cylinder at point 'P' is given by
[tex]E_{O} = \dfrac{r \textbf{c}}{2 \epsilon_{0}}[/tex]
and the electric field '[tex]E_{O'}[/tex]'due to the hole at point 'P' is given by
[tex]E_{O'} = \dfrac{\rho \textbf{s}}{2 \epsilon_{0}}[/tex]
So the net electric field ([tex]E_{net}[/tex]) inside the hole is given by
[tex]E_{net} = E_{O} - E_{O'} = \dfrac{r}{2\epsilon_{0}}(\textbf{c - s}) = \dfrac{r \textbf{b}}{2\epsilon_{0}}[/tex]
