# Explanation for $E~$ not falling off at $1/r^2$ for infinite line and sheet charges?

For an infinite line charge, $E$ falls off with $1/r$; for an infinite sheet of charge it's independent of r! The infinitesimal contributions to $E$ fall off with $1/r^2$, so why doesn't the total $E$ fall off the same way for the infinite line and sheet charges?

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For any case where you are interested in obtaining the electric field using Gauss's Law $\displaystyle \frac{ q_{enc}}{\epsilon_o} = \Phi= \int \bf{dA} \cdot {E} ~$ you will end up picking a gaussian surface such that the electric field is constant or zero over the Gaussian surface so that it just evaluates to $|\bf{E}| \int |d\bf{A}_{||}|$ where the surviving integral is just over portions of the area parallel to the electric field. For example, for a point charge you will draw a concentric sphere. You can see from the above the further away you get from the point charge the bigger the area gets and so the electric field must get smaller in order to keep the flux constant. For a line charge we would draw a concentric cylinder and only the portion of the area that wraps around will survive. Again the flux must be constant but the area that wraps around a cylinder that grows much more slowly as the radius grows than that of the surface area of a sphere as the sphere's radius grows. Finally for a sheet we would draw a cylinder so that only the ends of the cylinder that are parallel with the sheet survive. Now as the length of the cylinder increases the surface area of the ends doesn't grow at all, so the electric field doesn't fall off at all.
The true explanation is, of course, the math. But I'm guessing you're familiar with the calculation. For an intuitive understanding, I'd put it like this: when you're very close to an infinite sheet of charge, the contributions to $\vec{E}$ of the pieces of charge far away from you mostly cancel out, because e.g. the electric field of a bit of charge far to your left is nearly antiparallel to the field of a bit far to your right. When you're further away from the sheet, then those contributions to the field are not so close to antiparallel, and they add up to a larger net electric field. That balances out the reduced electric field from the bits of charge closest to you.