If an electric current is flowing through an electric wire, can we consider that wire charged?
The answer is required with a proof. Can we consider the wire to be charged positively or negatively?
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Jerk_dadt is correct. Electric current is the flow of free electrons in the conductor. At any instant, the number of electrons leaving the wire is always equal to the number of electrons flowing from the battery into it. Hence, the net charge on the wire is zero.
If you say the current carrying conductor is charged, it will violate the Kirchoff junction rule, which is based on the fact that in an electric circuit, a point can neither act as a source of charge, nor can the charge accumulate at that point.
No, not necessarily. Current is the just the movement of electrons already in the wire (that is neutral). The electric field (or voltage) applied causes them to have a net movement in one particular direction (i.e. opposite direction of conventional current).
So copper has 29 protons and 29 electrons per atom. A copper wire would have a net zero charge. It conducts electricity very well because its outer electrons can move "freely" through the wire since they are loosely bound. There is no net charge involved here (for either positive or negative charge).
See charge current continuity equation.
Del on Current Density = change in charge per unit of volume.... Current Density = conductivity * Electric Field
What this means is that any time we have a divergence in the electric field, we will have a violation of Kirchhoff's Junction Law, or in other words, we will have a local accumulation of charge. Note, charge is STILL GLOBALLY conserved, however not locally. So where do we observe divergence in the electric field? We observe divergence in the electric field if we have currents running through materials of non-uniform composition, currents running across interfaces of different materials, conductors with currents running through non uniform geometries and on wave fronts where there are voltage fluctuations propagating down the line.