Electric Field inside an ideal conductor

Question

I have some doubts about the electric field inside an ideal conductor (let's call it E). Precisely, I have read two different descriptions

1) On physics books I read that the electric field inside in a conductor in electrostatic equilibrium is equal to 0. The physical reason of this is the fact that all charges, at equilibrium, are distributed in the external surface of the conductor since only this distribution can reduce their repulsion forces. A mathematical view of this is given by the equation J = sigma * E. In fact, since sigma = infinite and J = 0 (since equilibrium means that charge do not move), necessarily we must have E = 0.

According to this explanation, E = 0 only in electrostatic equilibrium.

2) On Electromagnetic Fields books I read an explanation which is similar, but not identical. I read that, since J must have a finite value, and J = sigma * E and sigma = infinite, we get E = 0.

According to this explanation, there is not any mention of electrostatic equilibrium. It seems that inside a conductor in any condition we have E = 0. Also if there is a voltage source applied on it or something similar.

Now I have two questions:

• Which is the correct description?

• It is known that a metal is able to reflect EM waves. Is it due to the fact that E = 0 in its inner points?

Answer 1

The first description is true for all conductors while the second is only true for perfect conductors.

The reason the first works for all conductors is that in electrostatics, we get to say 'if $\textit{any}$ charge moves, our assumption of electrostatics is violated', so an electric field in even an imperfect conductor would violate the assumption and so is not allowed. On the other hand, if we allow a fully dynamical system, we can have moving charge in a conductor $\textit{if}$ it is not perfect. This should be obvious since we have charge moving through conductors all the time in the real world. So in this case, the argument only hold because the conductivity, $\sigma$, is infinite, which is only true for perfect conductors.

As for the second, the best answer I know to give is that we use boundary conditions to understand how the fields behave near surfaces, and when you carry out the calculations, you get reflection. If I recall correctly, $\textbf{E}=0$ is used to get the idealized perfect reflection in the idealized perfect conductor problem.