Electric field due to a current flowing through an infinitely long wire

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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– rob
Commented Jul 2, 2021 at 18:34