Problem $3)(10$ points$)$ As shown in the figure below, the space between two conducting cylinders of length $L=25cm$ is half field with dielectric having relative dielectric constant $lo{n}_T=8.$ The cylinders have radius $0\mathrm{.}5$ cm and $2$ cm and are connected to a $100$ V battery.

$($a$)$ Find the fields vec$(E)$ and vec$(D)$ in the air and the dielectric.

$($b$)$ Find the surface charge induced on the inner conductor at points adjacent to the air, and at point adjacent to the dielectric.

$($c$)$ Find the capacitance of the system.