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I was rereading Purcell's Electricity and Magnetism as research for another question, and I found this passage:

In metals Ohm's law is obeyed exceedingly accurately up to current densities far higher than any that can be long maintained. No deviation has ever been clearly demonstrated experimentally. (Second edition, 1985, section 4.5, p143.)

Here all characteristics of the metal, including temperature, must be kept constant, and in such conditions Purcell states that no deviation from a linear relation between electric field and current density has been demonstrated.

Now, this book is relatively outdated (two pages back it touches briefly on superconductivity and mentions that the highest Tc on record is 21 K), so I wonder whether the fact above, admittedly less technologically crucial than high-Tc superconductivity, is still held to be experimentally true, or whether deviations from linearity have in fact been observed. Purcell goes on to state that

According to one theoretical prediction, departures on the order of 1 percent might be expected at a current density of 10$^9$ amps/cm$^2$

though I'm having some trouble guessing whether we have technology to reach such current densities.

To put things on a more concrete footing, then: have deviations from Ohm's law in metals been observed? Is this an active question in any field of research? Do confirmations of this follow from other, related research? Or, alternatively: has the prediction above been verified or falsified? Do we have technology to test those regimes? (if not, what ranges are experimentally viable?) Additionally, can someone supply a reference to the theoretical prediction?


Brief edit: I just got my hands on a sample of the new (2013) edition of Purcell (now in SI units!), revised by Morin. They have updated the superconductivity bit to include high-Tc SCs, but the Ohm's law passage remains much the same. They do mention that the current density above is over a million times stronger than normal, but don't specify what the state of the art is in current density, and don't give references to the 'theoretical prediction'. It's real nice to have that book brought back to SI life, though!

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That current density is exceedingly difficult to reach due to the fact that any conductor small enough to not experience the skin effect under such current is usually small enough to melt very quickly under same. –  Ignacio Vazquez-Abrams Oct 17 '12 at 22:19
    
As a theoretical consideration, recall that Ohm's law fundamentally occurs because of a balance between relaxation and driving EMF. Thus unless there is a dependence between the relaxation mechanism and the driving EMF, the relationship will be linear. I can't think of anything related to electric fields that fundamentally changes, but perhaps it is possible to generate enough magnetic field to get magneto-resistance effects? That might be where the 1 percent prediction comes from. –  genneth Oct 22 '12 at 23:48

1 Answer 1

I wish I could just leave a comment but not enough rep. :-(

Anyway, in superconductors, there's a critical external magnetic field where when exceeded the material is no longer superconducting. Something about the external field preventing the formation of cooper pairs (someone else might be able to explain this better). So if the current in a cable is so large that the magnetic field generated exceeds the critical field in the cable itself (some sort of self-interaction), then the resistance will increase even if the material is kept at the same temperature. I don't know if this is considered a violation of Ohm's law. The value of the critical external field increases for decreasing temperature. This is why at the LHC superfluid helium at 2K is used to cool the coils rather than standard boiling helium at 4.2K.

Just a thought...

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