# Calculating the electric field of a cylinder

I'm trying to calculate the electric field due to a hollow cylinder. The cylinder is "semi infinite" with one end at x=0 and the other at - infinity, and radius R. I'm trying to calculate the field at a point x along the axis (not above the end of the cylinder). The cylinder has charge density sigma (the problem specifically states to use surface charge density instead of line charge).
Am I correct in assuming that the radial field cancels? I can't figure out how to set this problem up.
I'm also supposed to calculate the field at x=0, at the "surface" of cylinder. Is this point inside the cylinder with E =0, or no?

Any help would be much appreciated, thanks!

Well, first you are right that the radial force outward from the axis of the cylinder should cancel on the axis of the cylinder. One way to see this is through a basic "symmetry argument." Imagine there were a radial force away from one spot on the axis.

Now imagine we rotated the problem around the axis of the cylinder. The cylinder looks exactly the same, but the radial component is pointing in another direction! We have considered the same problem but gotten different answers about the radial component, so "by symmetry" it must cancel. Another way to see this is that any radial field from one part of the cylinder at a point on the axis would be cancelled out by the the radial field from the opposing spot on the cylinder (i.e. the point that is directly across the axis).

Now: how to do this problem? Do you know how to calculate the electric field due to a ring of charge at a point sitting on the axis going perpendicularly through the center of the ring? Use the superposition principle to "build" the cylinder out of infinitely many rings.

So, I should calculate the field at point X due to a ring of charge with radius R, and then assume that I have infinitely many rings? Wouldn't that just lead to the field being infinite? I'm also under the impression that the closer the charge gets to the center of the ring (the second part of the problem) the larger the field gets, which would also lead to an infinite field.
I've been trying to finish this problem all day, and I'm not convinced that this is the proper way to do it...