Gauss's Law Problem: Electric Field Inside a Charged Cylinder

[bcjochim07]

Homework Statement

A long, thin straight wire w/ linear charge density lambda runs down the center of a thin, hollow metal cylinder of radius R. The cylinder has net linear charge density of 2lambda. Assume lambda is positive. Find an expression for the electric field strength inside the cylinder.

I used a cylinder of length L for my gaussian surface:

the charge inside is lambda * L

E*2pir*L = lambda * L / permittivity constant

E= lambda/ (2pi*r*permittivity constant)

This is correct, but the thing I don't see is how the electric field from the metal cylinder cancels inside the cylinder. Wouldn't it only cancel at the very center?

Homework Equations

The Attempt at a Solution

 

[bcjochim07]

Oh... if I draw a gaussian surface inside, it only encloses the wire. I'm trying to fit my head around this.

 

[bcjochim07]

Could someone please give a good explanation for this? Thanks