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I know that the skin depth derived for AC current goes as $$\delta \propto \sqrt{\frac{1}{\omega \sigma}}$$ where $\omega$ is the angular frequency of the field and $\sigma$ is the conductivty.

Now: for a conductor with infinite conductivity but with DC current , we find a $0\times\infty $ in the denominator and I do not know how to treat it.

That said:

1) In a conductor with infinite conductivity and DC or AC current, does the current only flow on the surface?

2) What is keeping charge inside a real conductor? Shouldn't they always try to eliminate the electric field inside?

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1) A conductor with infinite conductivity is a superconductor and they definitively carry DC current in the bulk. The classical description of the skin effect breaks down for superconductors.

2) A conductor in general consists of atoms which have a positively charged nucleus. While in conductors the 'outermost' electrons have a relatively low binding energy (it takes little force to move them away from the atom and voltage is force per unit charge). However, if you were able to remove all of these quasi free electrons from the conductor, you would end up with a very positively charged (removing one electron of each atom of a mole corresponds to about 100'000 coulomb) conductor which very attract negative charges very strongly (due to the nuclei left in the conductor).

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  • $\begingroup$ Should note that an actual superconductor has AC resistivity even at zero temperature iirc. This is an important distinction as to whether you mean "a boundary where the fields vanish" or "a physical superconductor". $\endgroup$ – webb May 12 '14 at 15:33

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