Conductors in Electrostatic Equilibrium

[oneplusone]

A long straight metal rod has a radius of 5 cm and a charge per unit length of 30 nC/m. Find the electric field x cm away where distance is perpendicular to the rod.The solution to this uses $\int E\cdot dA = \dfrac{q_{encl}}{\epsilon_0}$. My question is, why can you use this? I thought that is only when the gaussian surface is closed.

I understand the rest of the solution, but this theoretical part is confusing me…

thanks

 

[ModusPwnd]

My first thought; The rod is "long" which is a buzz word for infinite. An infinite road is enclosed by an area, the area of the gaussian cylinder that surrounds it. There are no endcaps because the rod is infinite so the area is closed.

 

[oneplusone]

Hmmm, so suppose that x < 5 (find field inside rod)…
does that mean you construct a gaussian cylinder that has radius x? And then use the equation above?

And since it is infinite, am I correct in saying that we can pull $E$ out of the integral (it is constant), and $A$ would just be $2\pi x l$ (we can ignore the "caps" of the cylinder since it's infinite?)