This is a question that has been bothering for a while. I don't know if my answer is correct or not, but anyone look over it?

Both cylinders have the same length L. The first cylinder with radius R1 has a charge Q1 uniformly distributed inside the cylinder. The second cylinder is a conductor with radius R2 and charge Q2 (negative) uniformly distributed into the area between the first and second cylinder.

Find the electric field when:

a) r < R1 ; b) R1< r< R2; c) r> R2

So, I did it like this.

a) r< R1

$∮ E.dA = E2πrL = Qi/ε₀ $

$ Qi= ρV= [Q1/(πLR1^2)]*πLr^2 $

$E= rQ1/(2πLε₀R1^2) $

b) R1< r < R2

$∮ E.dA = E2πrL = Qi/ε₀ $

$E= Q1/ (2πrLε₀)$

c) r> R2

$ E1= Q1/ (2πrLε₀) and E2= -Q2/ (2πrLε₀)$ (because Q2 is negative I put the minus in front of it)

$ Eout = E1 + E2 = Q1/ (2πrLε₀) -Q2/ (2πrLε₀) $