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So, this has been bothering me for a few days and I'm having trouble understanding where the fault is. If we consider a uniform charge density ##\rho## extending through all space, then by symmetry, I would argue that ##\mathbf{E}=0## in all space. However, this does not agree with what a naive application of Gauss's Law would predict since ##\nabla\cdot\mathbf{E}=0\ne\rho/\epsilon##. So where exactly is the argument breaking down? Is there something unusual...