Field due to current in a wire Suppose a current flows in a straight cylindrical wire so that an electric field $\textbf{E}$ is maintained in the wire.
Will there be an electric field just outside the wire..?
 A: From your setup, it sounds like you have an E field directed along the wire that is driving the current (i.e. the wire has some finite conductivity).  If that is the case, then just outside the wire there must also be an E field, because of Faraday's Law.  The curl of E must be finite, and if you had a discontinuity of the tangential E field at the surface of the wire then the curl would be infinite.  
A: The voltage difference in a steady state current is independent of path, and deforming the path out of the wire, you can see that the electric field must be continuous. The electric field not only extends outside the wire, in conjunction with the magnetic field surrounding the wire, it is carrying the bulk of the momentum of the current.
A: When a steady current is flowing through the wire, the overall wire is charge neutral. In the presence of an external electric field $\textbf{E}$ the electrons are simply moving from one end of the wire to the other. The electron density at end of the wire from which the electrons are leaving gets replenished by (say) the battery. An equal flux of electrons is obviously coming out of the other end of the wire and going into the battery. In each region of the wire the free (or conducting) electron density matches that of the positive ion cores. As a result, the wire is charge neutral. Therefore there should not be any electric field outside the wire (Gauss's law).
Also, because you said that "$\textbf{E}$ is maintained," the magnetic field produced by the wire (Biot-Savart law) will be time independent. Therefore any electric field outside the wire, due to induction, will also not be possible (Lenz's law).
