Rigorously prove that electric field is zero in a perfect conductor I have ran into a problem while trying to prove that the electric field is zero in a perfect conductor
My argument went something like this:

We know that: $$\vec J = \sigma \vec E$$
In a perfect conductor $\sigma = \infty$
Therefore to maintain a constant current, $\vec E$ must be zero.

While this $\infty \times 0$ is a constant argument is not never seen before, I feel ike this can be made much rigorous. 
Can someone help provide me with an argument why the electric field must be zero in a perfect conductor?
 A: Suppose we impose a current density $\newcommand{\j}{\mathbf{J}}\j$, then the resulting electric field $\newcommand{\e}{\mathbf{E}}\e$ is given by $\e = \rho \j$, where $\rho$ is the resistivity. In a perfect conductor, $\rho=0$. So in a perfect conductor with some fixed current $\j$, the electric field satisfies $\e = \rho \j = 0 \j =  \mathbf{0}$. I don't know if this is any more satisfying, but it doesn't use infinity.
A: The electrical field $\mathbf{E}$ is an external field, which "drags" the conductor electrons through the conductor "lattice". The conductivity $\sigma$ describes the resistance of the "lattice". When the resistance is zero, a non zero current can exist in the conductor without necessity to support it with an external field.
If $\mathbf{E}$ is non zero, the current will be growing (or varying), just like a solution to $m\mathbf{a}=\mathbf{F}_{ext}$. 
A: 
Can someone help provide me with an argument why the electric field
  must be zero in a perfect conductor?

It's not clear exactly what you're looking for.  In a sense, any argument attempting to prove that the electric field must be zero in a perfect conductor will beg the question.
For example, here's an excerpt from "Electromagnetics for High-Speed Analog and Digital Communication Circuits":

We could in fact define a perfect conductor as a material with zero
  electric field inside the material.  This is an alternative way to
  define a perfect conductor without making any assumptions about
  conductivity.

From this starting point, one reasons that
(1) if there's an electric field inside, it's not a perfect conductor
(2) if the material has a finite conductivity and there is a steady current through, there is an electric field inside proportional to the current density  
(3) thus, if the material has finite conductivity, it's not a perfect conductor
Again, it's not clear to me precisely what you're looking for.  If the above fails to address your question, please update and clarify your question.
A: Consider a metal sheet placed in a uniform electrical field normal to the sheet surface. The electrons will be "dragged" by the electrical field and form an excess of negative charges on one side of the metal sheet and an excess of positive charges on the other side of the metal sheet.
The excess charges produce an electrical field that cancels the applied field in the interior of the metal sheet.
The same consideration applies for a metallic half space.
