B Electric Field Created by a Finite Plate

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The discussion revolves around calculating the electric field created by a finite plate at a specific point in a Cartesian coordinate system. It questions the applicability of Gauss's law for this scenario, noting that only the z-components of the electric field contribute due to symmetry, as the x and y components cancel out. A cylindrical Gaussian surface is proposed for the calculation, despite the lack of symmetry. Advanced calculus is referenced, leading to a specific equation for the electric field at the point in question. The conversation highlights the complexities involved in applying Gauss's law to non-uniform charge distributions.
Heisenberg7
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Let's assume that we have a finite plate which is at the center of a cartesian coordinate system. Now let's define a point ##r## with coordinates ##(0, 0, z)##. My question is, can we use Gauss's law to find the electric field at this point? The direction of the electric field is going to be up so all the ##x## and ##y## components will cancel out leaving us with only ##z## components of all the electric fields created by the infinitesimal points on the plane. Now, would it be possible for us to create a gaussian surface like a cylinder and calculate the electric field at the point ##r##?

I've seen a video about this and the guy explaining it uses some pretty advanced calculus to find the electric field at the point r. In the end, he gets this equation: $$\vec{E}(\vec{r}) = \frac {Q}{4\pi\epsilon_oab} \arctan(\frac {ab}{2z \sqrt{a^2+b^2+4z^2}})$$
 
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I guess this is a dumb question because there is basically little to no symmetry (only along the z axis is the electric field pointed upwards).
 
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