Two concentric cylindrical conducting shells of length L are separated by a vacuum. The inner shell has surface charge density +σ and radius ra. The outer shell has radius rb. Using Gauss’ Law, as a function of radius r find: The direction and magnitude of electric field inside and outside the shells. Be sure to clearly state the Gaussian surfaces that you are using!
Find an expression for the voltage between the shells.
Electric flux = ∫ EdA = qencl / ε0
σ = q/A
The Attempt at a Solution
I'm using a cylindrical Gaussian surface. I understand that when r < ra the field E = 0. Here's what I've gotten so far for when r is between ra and rb:
E = qencl / ε0⋅Agauss = qencl / ε0⋅2πrl
⇒ E = σ2πral / ε0⋅2πrl
⇒ E = σra / rε0
That doesn't seem very... right. I don't think there should be a dependence on ra for starters. Would the expression for the flux through a Gaussian surface with r > rb be the same also, as there isn't any non-induced charge on it? Also, for the last part of the question I don't understand why there would be a voltage between the shells when the charge on the outer shell is just induced by the inner cylinder, ie the net charge on the outer cylinder is 0. Isn't it? Sorry if I'm not making much sense, I'm almost through my exams and my brain is extra melty.