Q $3(20$ points$)$

An infinite conductor cylindrical shell of inner radius $\backslash (\backslash$boldsymbol$\{$A$\}\backslash )$ and outer radius $\backslash (\backslash$boldsymbol$\{$B$\}\backslash )$ is placed with its axis aligned with the $\backslash (\backslash$mathbf$\{$Z$\}\backslash )-$axis with an infinite linear charge $\backslash (\backslash$boldsymbol$\{\backslash$rho$\}\_\{\backslash$boldsymbol$\{$l$\}\}\backslash )$ placed at the axis as shown in figure. If the electrical potential of the outer surface of the conductor cylinder is $\backslash (\backslash$boldsymbol$\{$V$\}\_\{\backslash$boldsymbol$\{$o$\}\}\backslash ).$ Find:

a$)$ The surface charge density on the inner and outer surfaces of the cylinder.

b$)$ The electrical field $\backslash (\backslash$boldsymbol$\{$E$\}\backslash )$ everywhere.

c$)$ The electrical potential everywhere inside the outer surface of the cylinder $($i$.$e$.\backslash (\backslash$rho