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Let's assume an infinitely long, neutral conducting wire, with charges flowing through it. Let it be connected to an external battery providing the necessary voltage for the flow of these charges. Now, at any given time, in any element of the wire, the amount of charge entering that element will be equal to the amount of charge leaving that element (because the wire is neutral). This should imply that the overall neutrality of the wire results in a net 0 electric field outside the wire.

But the charges are flowing BECAUSE the wire is connected to a battery, and that there is some potential difference between any two points of the wire. So wont this potential difference result in an electric field outside the conducting wire?

Initially I was convinced by the first argument till I read the second explanation in another thread. In short, does a infinitely long, neutral conducting wire produce an electric field outside it?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – rob
    Commented Jul 2, 2021 at 18:34

1 Answer 1

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Current flow in a conductor requires an (E) field. A uniform current (in a long wire) requires a uniform field. This can only come from a gradient in the charge density. A power source takes electrons from one end of the wire and puts them into the other. One end of the wire has a high positive charge density which gradually becomes negative as you approach the other end. There would be an (E) field leaving the + end and going into the – end of the wire. With a variable charge density, Gauss's law does require this flux through the surface of the wire. I have read on this site (but not verified) that a uniform field in the wire requires that the excess charge must reside on the surface of the wire.

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    $\begingroup$ If the wire has uniform resistivity, then yes, the charges will be on the surface. If the wire has nonuniform resistivity, charges may be present wherever the gradient of resistivity is not zero, or the gradient is not well defined, as at the boundary between two different materials. $\endgroup$ Commented Jun 30, 2021 at 23:22
  • $\begingroup$ @MathKeepsMeBusy The by far largest change in conductivity will be at the surface, so unless you consider a conductor with dielectric or vacuum cavities - which would make the conduction unnecessarily complicated in this context - the charge is on the surface. $\endgroup$
    – my2cts
    Commented Jul 1, 2021 at 9:51
  • $\begingroup$ @my2cts Or at the junction of a typical resistor. Or if the wire has a temperature gradient. $\endgroup$ Commented Jul 1, 2021 at 11:50
  • $\begingroup$ @MathKeepsMeBusy That is all true but here we are talking about a much simpler case. Making it more complex may obsure the actual question. $\endgroup$
    – my2cts
    Commented Jul 1, 2021 at 14:51
  • $\begingroup$ @my2cts the person to whom I was responding wrote: "I have read on this site (but not verified) that a uniform field in the wire requires that the excess charge must reside on the surface of the wire." I gave that person the general rule. $\endgroup$ Commented Jul 1, 2021 at 15:27

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