# Field of an infinite plane sheet of charge

## Homework Statement

Use Gauss's law to find the electric field caused by a thin, flat, infinite sheet with a uniform positive surface charge density σ.

## Homework Equations

φ(flux) = ∫E*dA = qencl / ε0

## The Attempt at a Solution

I set up a diagram like this: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesht.html#c1

1) I'm sorry if this is a really vague question but I don't get why the cylinder is smaller than the sheet? I thought the cylinder would have to enclose the sheet? If we don't enclose all the charge of the plane, how can we use Gauss's Law?

then EA = qencl / ε0, because I assumed E and A are constants in this situation.

A = 2πrL, where L is the length of the gaussian cylinder.

then E = qencl / (ε0 * A)

E = qencl / ε02πrL

which is so wrong..

I read the answer in the book but these are what makes no sense to me:

1) The flux though the cylindrical part of the cylinder is zero because E * n = 0 everywhere.

2) The flux through each flat end of the surface is +EA because E * n = E everyone, so the total flux through both ends is +2EA.

If I assume these two things are true I understand what they do after which is

2EA = σA / ε0

E = σ/2ε0

Please help on how I can understand what the book is telling me in 1) and 2) and also how do they arrive at the diagram that they do?

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