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In the modern texts of electromagnetism in the presence of stationary currents the electric field is assumed conservative $\nabla \times E =0 $. Using this we get $E_{||}^{out}=E_{||}^{in}$ which means we have the same amount of electric field just outside of the wire! Is this correct? Is there any experimental proof?

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Outside a current carrying conductor, there is, in fact, an electric field. This is discussed for example, in "Surface charges on circuit wires and resistors play three roles" by J. D. Jackson, in American Journal of Physics -- July 1996 -- Volume 64, Issue 7, pp. 855 . To quote Norris W. Preyer quoting Jackson, "Jackson describes the three roles of surface charges in circuits: "(1) to maintain the potential around the circuit, (2) to provide the electric field in the space around the circuit, (3) and to assure the confined flow of current.""

Experimental verification was provided by Jefimenko several decades ago. A modern experimental demonstration is provided by Rebecca Jacobs, Alex de Salazar, and Antonio Nassar, in their article "New experimental method of visualizing the electric field due to surface charges on circuit elements", in American Journal of Physics -- December 2010 -- Volume 78, Issue 12, pp. 1432 .

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That equation is true for electrostatics. Inside an electrically neutral current-carrying wire, the electric parallel to the wire is zero. So outside the wire it's also zero.

More importantly, Gauss's law will tell you that the components perpendicular to the wire must also be zero.

So the electric field is zero everywhere for an electrically neutral current-carrying wire.

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then what is the $E$ in the equation $J=\sigma E$ always we have inside the current which by the way has zero flux through every closed surface inside the wire!! –  richard Apr 22 '13 at 15:53
    
For a short wire for length $L$, $EL=V$. But for a long wire, $L \rightarrow \infty$ so $E \rightarrow 0$. For a short wire the electric field isn't zero everywhere. –  santa claus Apr 22 '13 at 17:24
    
Alec, for long wires, one would use higher voltage to maintain the electric field, and hence the current, the same. Your answer is incorrect; there is electric field both inside and outside the wire. –  Ján Lalinský Oct 31 '13 at 14:17
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Your problem is clearly and comprehensively treated by Hans De Vries in:

http://chip-architect.com/physics/Magnetism_from_ElectroStatics_and_SR.pdf

The quintessence is that a current carrying wire appears electrostatically charged to an observer in relative motion to that wire, even when the same current carrying wire appears uncharged to an observer at rest relative to that wire. The observed electric field is identical to that given by the Lorentz-formula E = v X B. For simplicity, disregard the resistance of the wire and assume a superconducting wire.

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Link-only answers are bad answer. Stack Exchange sites seek to be repositories of good questions with good answers, not link farms. –  dmckee Apr 8 at 15:49
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