Bound charge inside and outside a dielectric

[E&M]

Homework Statement

A conducting wire of length a and charge density lambda is embedded inside a dielectric cylinder of radius b. To Show: a) Bound charge on the outer surface is equal in magnitude to the bound charge inside the surface.
b) volume density of bound charge is 0 in the dielectric.
c) what is the net charge along the axis?

Homework Equations

The Attempt at a Solution

I calculated D and E outside the dielectric which came out to be
D = lambda/(2*pi*s) along axial (s) direction
E = lambda/(2*epsilon_0*pi*s) along axial (s) direction
Using Gauss's Law I found out free charge enclosed and total charge enclosed using D and E. Then subtracting free charge from the total charge, I got the total bound charge enclosed, which came out to be 0. I don't know if it make sense or not. I need help in understanding the question.

Thanks.

 

[E&M]

I think I know what I am supposed to do conceptually but still I don't know how to write down everything in a systematic and scientific way. So, the bound charge enclosed came out to be 0 outside the cylinder. To satisfy this case, the bound charge inside the surface should be equal to the bound charge outside the surface.

For the third part of the question, E inside the dielectric will be lambda/(2*epsilon_0*epsilon_r*pi*s) along axial (s) direction
So, if we use Gauss's law, then we will get q enclosed to be lambda * l/epsilon_r, therefore charge per unit length will be lambda/epsilon_ri don't know about the third part of the question. would anyone please help me with this problem? thanks