# DC current in ideal conductor and skin effect?

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?