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Im following Ncert textbook for physics and I was learning about Charge due to infinitely planar sheet. In this they say that the electric field due to the infinitely long planar sheet to be the same as the electric field enclosed within the gaussian surface

Title of subtopic: Field due to uniformly charged infinite plane sheet

Then : Therefore the net flux through the Gaussian surface is $2 EA$. The charge enclosed by the closed surface is $\sigma A$. Therefore by Gauss’s law, $E=\frac{\sigma}{2\epsilon_{0}}$

My doubt is how can this be the charge for all of the charge in the plane sheet, For problems also they use this same formula. Problems like this:

Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2 . What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?

Use the same formula. How can this be possible because If we want to find the whole charge of the plate we need to integrate right but here we are not integrating and using formula. Have I misunderstood something

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  • $\begingroup$ The total charge of an infinite plane with a nonzero surface charge density is infinite. $\endgroup$
    – Ghoster
    Apr 16, 2023 at 5:16
  • $\begingroup$ then shouldnt the topic be changed to something else ? @Ghoster $\endgroup$
    – Razz
    Apr 16, 2023 at 5:59
  • $\begingroup$ The whole point of setting these problems is so that you learn a different way to look at those equations rather than always falling back to Coulomb's law and integrating. The correct future viewpoint is in terms of Maxwell's equations, not Coulomb's law. Of course, you can put in a lot of work and integrate to prove that that solution is correct, but that is not the correct way to look at the problem. $\endgroup$ Apr 16, 2023 at 8:55
  • $\begingroup$ Then shouldn’t the topic be changed to something else? What’s the problem with “Field due to uniformly charged infinite plane sheet”? It doesn’t even mention the total charge of the sheet. The total charge of the sheet is irrelevant. $\endgroup$
    – Ghoster
    Apr 16, 2023 at 16:50

1 Answer 1

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You are dealing with approximations of situations which occur in the real world. The approximations are made to make the analysis of a real situation easier whilst at the same getting a result which is not too far from that obtained by undertaking more complex analysis.

There is no such thing in the real world as an infinite charged plate.
If you have a charged plate whose dimensions are very large compared with the distance from the plate at which you are trying to find the electric field then the field from that plate of finite size approximates to that of an imaginary plate of infinite size.

If the two parallel plates have linear dimensions which are very much larger than the separation of the plates then most of the electric field between the two plates would approximately the same as that if the plates were infinite in extent.

In the example that you have given you are to assume as an approximation that the "fridge" field outside the parallel plate arrangement is zero.
In the real world that can never be so but the approximation gets better as the separation of the plates decrease and/or the dimensions of the plates increase.

Here are two images in the article Solving the Generalized Poisson Equation with the Finite-Difference Method which illustrates the true complexity of the parallel plate arrangement which you usually get around by assuming that the linear dimensions of the plates are much greater than their separation.

enter image description here

As has been pointed out the total charge on an "infinite" plate is infinite but that should not bother you because you will never come across such a plate except in your imagination.

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