Electromagnetic fields inside perfect conductors

Why are electromagnetic fields zero inside perfect conductors? I do not understand why we can make these assumptions in electrodynamics.

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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

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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$=sigmavec{E}$) 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.