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My question is of two parts:

(i) I know electric field due to an infinite plane sheet of charge at a point (finite distance $d$ away) is $2 \pi\ k\ \sigma$. From this, by reducing the linear dimensions of our system, can we conclude that electric field due to a finite plane sheet of charge at a point infinitely close to it will remain $2 \pi\ k\ \sigma$ ? (because $2, \pi, k, \sigma$ are all independent of the size of our system).

(ii) Is there a way to show that electric field due to any surface charge at a point infinitely close to it will still be $2 \pi\ k\ \sigma$ ?

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can we conclude that electric field due to a finite plane sheet of charge at a point infinitely close to it will remain 2π k σ

Yes. The distance plays no role in this scenario.

Is there a way to show that electric field due to any surface charge at a point infinitely close to it will still be 2π k σ?

Heuristically, if you take an arbitrary surface and zoom in infinitely close then to first order it is a plane. Similar to how you can decompose any curve into a polynomial using Taylor series expansion you can do the same thing with a surface. The “first order” term is a plane and so the field due to the first order term is the above field. As you move away from the surface second and higher order terms become important.

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  • $\begingroup$ So do you mean electric field at a point infinitely close to any arbitrary shaped surface charge of varying charge density is always $2 \pi k \sigma$? $\endgroup$ – N.G.Tyson Dec 24 '18 at 4:43
  • $\begingroup$ Why is there no contribution due to the rest of the points on our surface charge which are at a finite distance from the field point. $\endgroup$ – N.G.Tyson Dec 24 '18 at 4:53
  • $\begingroup$ There is a contribution from other points, but it is a second or higher order contribution. It exists but until you go some finite distance away it is small compared to the influence of the nearby surface. $\endgroup$ – Dale Dec 24 '18 at 12:28

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