Are $A$ and $\phi$ always zero inside a Faraday cage like $E$ and $B$?

Question

Are $A$ and $\phi$ always zero inside a Faraday cage, like $E$ and $B$ are?

If not, can its design be modified to accomplish that? If not, is there an analogous mechanism that'd always have $\vec{\mathbf{A}}=\vec{\mathbf{0}}$ and $\phi=0$ inside?

Answer 1

Within a closed metal surface the electrostatic field $\bf{E}$ zero, hence the potential $\phi$ whose gradient $\bf E=-\rm{grad}\phi$ the electric field is constant and can be set to any value as long as it is the same as is at its surface. This is because the surface of the metal must be an equipotential surface otherwise the free charges would be moved by the external field. Since there is no "magnetic conductor" as there is electric one no such requirement exists for a magnetizable material embedded in an external field. The closest analogy to a metallic conductor would be a a sphere made of high permeability soft magnet or the so-called $\mu$-metal (https://en.wikipedia.org/wiki/Mu-metal) but even that has some small stray fields inside. The superficial confinement that such magnetic material creates is nothing compared to something made out of copper in an external electric field.