Why are electromagnetic fields zero inside perfect conductors? I do not understand why we can make these assumptions in electrodynamics.
SOLUTION:
Firstly, we are going to analize the electric field.
[![Electric field inside perfect conductor][1]][1] [1]: https://i.stack.imgur.com/5HeUE.jpg
In the first conductor we can appreciate that the total charge distribution is electrically compensated ($\rho=0$). In consequence, the electric field will be completely null ($∇⋅\vec{D}=\rho=0$).
Later on we apply an external electric field that will distribute the charges inside the conductor almost instantaneously ($\sigma=\infty$). This redistribution will cause another electric field that will cancel the previously applied field.
We can conclude that the elctric field is zero unless we vary the magnetic field ($∇×\vec{E}=-\frac{∂\vec{B}}{∂t}$). The magnetic fields are defined by the following equations:
$∇⋅\vec{B}=0$
$∇×\vec{H}=\vec{J}+\frac{∂\vec{D}}{∂t}=\vec{J}$ext$+\sigma \vec{E}+\frac{∂\vec{D}}{∂t}$
We know that $\vec{D}=0$, so the induced current densities ($\vec{J}$ind$=\sigma\vec{E}$) will be zero as well. In addition, we will assume that the external current densities ($\vec{J}$ext) applied in the conductor are also zero. Finally we know that $\frac{∂\vec{D}}{∂t}=0$. So we can rewrite the Maxwell equations into the following ones:
$∇⋅\vec{B}=0$
$∇×\vec{H}=0$
Thanks to these equations we can conclude that the magnetic field is constant inside a perfect conductor. So the electric field will be allways zero ($∇×\vec{E}=-\frac{∂\vec{B}}{∂t}=0$).
As we have demonstrated before, we know that there can be a constant magnetic field within a perfect conductor. This field will be non-zero as long as it has been present since before the material transitioned to an infinite conductivity state. However adding some magnetic field would be an unnecessary complication. Therefore, the $\vec{B}=0$ condition is reasonable.