Are magnetic hysteresis losses relevant to alternating currents flowing in a wire? Say we have an AC in a magnetically lossy material, like iron. Because of iron's relatively high permeability, skin effect will be more pronounced than it is in say, copper, so this iron wire isn't so great a conductor, practically speaking.
However, skin effect arises due to eddy currents, which are themselves due to the time varying magnetic field due to the time varying current. So, this suggests there is a magnetic field at work, and there's the potential for magnetic hysteresis to be an additional source of loss in the material.
Yet, I'm not sure if this is true or not, if somehow the geometry of the conductor and the fields around it make this a non-issue. I would think that if a time-varying magnetic field exists inside the iron, then there would be hysteresis losses. Is this true? Does such a field exist?
To be clear: the issue is with iron as an electric conductor, not as part of a magnetic circuit such as a transformer core as is the more common application.
 A: The skin effect tells us that the AC current in a conductor with high(ish) magnetic permeability will be confined to a region near the surface with characteristic thickness
$$\delta = \sqrt{\frac{2\rho}{\omega \mu}}$$
Since the current is effectively moving in a cylindrical sheet, there will be a region just inside that current sheet that will experience a tangential B-field (strongest close to the surface) that changes direction as the current changes direction.
Such a region will exhibit magnetic hysteresis, and this will increase the power dissipation due to the DC current. You would have to do the volume integral to see whether, for a particular scenario, this is a significant effect - but theoretically it is certainly possible.
A: In my opinion the contribution of hysteresis losses in an iron wire can be important when comparing with conductive losses, but this is an issue strongly dependent of the iron alloy used according to hysteresis loop, conductivity and permeability. The analysis of the induced current distribution in conducting wires subjected to a harmonic axial voltage is important in designing many electrical devices such as transformers and transmission lines. The azimuthal magnetic field induces axial electric currents and therefore the impedance of the wire depends on the excitation frequency. The current density, and the EM fields are increasingly confined to a thin layer at the boundary of the wire as the frequency increases in such a manner that the internal core is completely electromagnetically screened. To minimize this effect at higher frequencies it is necessary to enhance the surface-to-volume ratio by using thin high-conductivity wires. A correct evaluación of the two losses contributions can be made analitically or by numerical simmulations.
Here http://scitation.aip.org/content/aapt/journal/ajp/77/11/10.1119/1.3160663 you can see the study for axial magnetic field and azimuthal currents. The problem you are interested is the reciprocal one: axial currents and azimuthal field, but the way you must approach this issue is practically the same.
