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**1. Homework Statement**

A very long cylinder has charge density L/m and radius a. Find the electric field at a distance r, with r > a.

**3. The Attempt at a Solution**

I construct a gaussian cylinder around it with radius r, and then from Gauss' law the field at any point on it's side surface is E= Q/Ae, where A is the area of the side of the cylinder, e = vacuum permittivity. So, Q = L*length, A = 2*pi*r. This gets me the right answer in the text book (L/(2*pi*e)), but there's a sticking point in my approach - this assumes all of the flux is concentrated through the sides, whereas there should also be some through the top and bottom of the cylinder, right? The best explanation I can think of is that because the cylinder is "very long", SA of the sides >>> SA of the top and bottom and so flux through these is negligible. This reasoning is a bit too heuristic for me though and it would be great if someone can give a better/more quantitative explanation.