Field from non-conducting plate? 
For a non-conducting sheet, the electric field is given by:
$$E = \frac{\sigma}{2\epsilon_0}$$
  where $\sigma$ is the surface charge density.
This equation holds well for a finite nonconducting sheet as long as we are dealing with points close to the sheet and not too near its edges.

Why does the equation hold better with points closer to the sheet? I understand why the approximation worsens near the edges (because symmetry fails and causes fringe effects) but why is the approximation better near the sheet?
 A: Consider a square sheet with edges located at $(a,0)$, $(-a,0)$, $(0,a)$ and $(0,-a)$. Suppose, we wish to find the electric field at a point $(0,0,z)$. By symmetry, this electric field will point solely in the $z$-direction. To find the electric field, consider a small element on the sheet located at $(x,y)$ of area $dx dy$. The charge of this element is $\sigma dx dy$. The magnitude of the electric field at $(0,0,z)$ due to this element is then (treating the element as a point charge)
$$
dE = \frac{1}{4\pi \epsilon_0} \frac{\sigma dx dy}{x^2 + y^2 + z^2 }
$$
The $z$-component of this electric field is
$$
dE_z =  \frac{1}{4\pi \epsilon_0} \frac{\sigma z dx dy}{\left( x^2 + y^2 + z^2 \right)^{3/2}}
$$
Integrating this over the sheet, we find the total electric field at $(0,0,z)$ as
$$
E_z =  \frac{\sigma}{ \pi \epsilon_0}  \tan^{-1} \left[ \frac{a^2}{z \sqrt{ 2a^2 + z^2 } } \right]
$$
Let us now take the limit of small $z$. However, $z$ is a dimensionfull quantity, and you can't discuss the largeness or smallness of dimensionfull quantities, only dimensionless numbers. The only dimensionless number that I can construct using $z$ is $\frac{z}{a}$. So, when I say, $z$ is small, I really mean $\frac{z}{a}$ is small.
In this limit, we find
$$
E_z =  \frac{\sigma}{ \pi \epsilon_0}  \tan^{-1} \left[ \frac{1}{(z/a)\sqrt{ 2 + (z/a)^2 } } \right] = \frac{\sigma}{2\epsilon_0} + {\cal O}(z/a)
$$
Thus, when we are sitting close to the sheet, the field takes the form you described above.
But, here's the important thing. We didn't really care if $z$ itself is small (that sentence doesn't even make sense). What we really care about is if $z/a$ is small. Now, there are two ways to make this small - 


*

*Make $z$ small compared to $a$, i.e. move in very close to the sheet.

*Make $a$ large compared to $z$, i.e. make the sheet very very large. 
Both the statements above are completely equivalent. 
