Finding Electric Field outside a Charged Cylinder

I'm trying to solve a problem that involves finding the electric field due to a uniformly cylinder of radius $r$, length $L$ and total charge $Q$. Well, my thought was: if I am to use Gauss' Law, I'll have to use a gaussian surface enclosing the cylinder.

Then here arises my doubt: My try was to enclose the cylinder with another cylinder with radius $R > r$, parametrize it and find it's normal $\hat{n}$ at each point. Then, I would need to calculate $\mathbf{E}\cdot \hat{n}$ and integrate. This would only be good to calculate $\mathbf{E}$ if it's magnitude is constant on the gaussian cylinder so that the magnitude would drop out of the integral. Also, I would need $\mathbf{E}$ parallel to the normal. It seems that both those things are true, but how can I see it and justify/prove it?

I just want suggestions, hints, in how can I see/prove that $\mathbf{E}$ has constant magnitude on the gaussian cylinder and how can I see/prove that $\mathbf{E}$ is parallel to the normal at each point. I also thought on some argument arround symmetry but I didn't find any consistent one.

Thanks in advance for your help!

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The field isn't constant on your Gaussian surface since you have a finite cylinder. The ends and the corners mess things up. You can see this by looking at very short distances away from the cylinder, so that it effectively looks like an infinite cylinder, plane or corner depending on where you are. But if you go very far away the finite cylinder effectively looks like a point charge. So the field is going to be some (relatively) complicated thing. So using Gauss's law isn't a good strategy. zakk's answer takes you from here. – Michael Brown Apr 22 '13 at 6:01

2 Answers

Symmetries! Your system system has radial symmetry. You can use cylindrical coordinates, with the $z$ axis being the charged cylinder axis; you will easily observe that, due to the symmetry of the system, every quantity depends only on $r=x^2+y^2$ and $z$, rather than $x$, $y$, $z$. The answers to your questions can be easily derived from this fact.

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For external points, cylinder behaves like a line of charge. So you just have to put the distance in the formula of electric field of a point charge as (radius + distance from the cylinders surface)

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