At the exact center of a finite wire (i.e. a distance, say $L/2$ from each end), why can I not apply Gauss's Law in integral form to find an EXACT solution for the electric field? At the center of the wire, $E$ is entirely radial, so it seems like I should be able to draw an infinitesimally $\epsilon$ thick cylinder as my Gaussian surface, pull $E$ out of the integral (it should be a constant in the limit $\epsilon\to 0$), and get an exact expression for the electric field. Obviously, I do not. Could someone explain in as mathematical a way as possible why this is the case? BTW, I have similar questions regarding Ampere's Law. It seems like I should be able to apply that to finite wires as well. I know that the reason that doesn't work is that it only works for loops of wire, but again, just from the math, I cannot explain that either. Thanks!