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I think I am misunderstanding some point about conductors and fields within them. I have been told that within a conductor, the charge is null because the inside field is null, and all of the charges are on the surface.

Does it mean that the NET charge within an uncharged conductor is null, although there are electrically neutral atoms (with their electrons and protons cancelling) inside the uncharged conductor? If so, in this case, both within the conductor and outside it the net charge is null, isn't it?

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The statement, "all of the charge is on the surface" is only true in a static case of a charged conductor. There are other forces at work. For example, when there is current we have resistance--which tends to cause the EMF to push some of the carriers into the material. That's why the cross section is used in defining resistance rather than just the perimeter of the cross section. For an uncharged conductor, you can think of it as the two polarities of charges battling it out for the surface, and the result is no extra charge on the surface or within the conductor, either way. An uncharged conductor is neutral at any point in or on the conductor (unless there's an external field).

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To expand a bit on the previous answer, here's a qualitative example of a static charge distribution (not numerically exact) and giving the field lines near a conducting sphere placed in an external electric field.

enter image description here

(Image credit: from an older edition of a textbook by F. Ulaby, possibly Fundamentals of Applied Electromagnetism, published by Pearson).

The electrons reorganize so as to produce a final charge distribution that will exactly cancel the external field inside the conductor. There is no field anywhere inside and so, by Gauss' law, an closed surface will contain no net charge. In the example of the figure, the positive charges are really a deficit of negative charges, which have been displaced by the external field. Any volume inside the conductor (aside from small volumes of the order or smaller than atomic sizes) is overall neutral.

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