Electric field everywhere for a hollow cylindrical conductor?

[Vitani11]

Homework Statement

An infinitely long, hollow, conducting cylinder has a inner radius a and outer radius b and carries a linear charge density λ along its length. What is the electric field everywhere?

Homework Equations

∫E⋅dA = Q_enc/∈

Variables
∈ = permittivity constant
a = inner radius
b = outer radius
λ = linear charge density
E = electric field
r = distance to point of E field
Q_enc = enclosed charge

The Attempt at a Solution

For inside (r∠a) and in the shell (a∠r∠b) the electric field is zero. I don't know what to do for outside the shell. I think the charge is concentrated on the outer shell just as for a spherical conductor, is this true? Here is my attempt for outside.

 

[Dilemma]

Use Gauss's law:
∫E⋅dA = Q / ∈

E ⋅ (2πrL) = λ * L / ∈ where L denotes the imaginary Gaussian surface's length.

Therefore,

E = λ / (2π∈r)

I hope this was helpful.

 

[Vitani11]

That is exactly what I did in the picture. Great - thank you!