I read the following problem: Prove that a dielectric medium for which $\varepsilon \to \infty$ behaves as a perfect conductor in the presence of static electric fields.
So, the easy part is that the normal component of an electric field is cancelled when entering the medium, as in a conductor. We also know that the tangent component of the electric field is continuous through the media, but how can we prove it vanishes for this case? This is usually done for a conductor by saying that any tangent field on the surface would produce a flowing motion of charges, hence leading with a non-static situation, impliying that this component (and the normal also) vanish for a conductor. But I cannot see how to prove this for an infinite constant dielectric without using the previous argument (assuming a priori that it is a conductor with free movable charges), this would be circular reasoning.
I could sum up the problem by asking: how do we prove that such infinite dielectric has an equipotential surface when exposed to static electric fields? or equivalently: how do we prove that the total field is actually normal to the surface of such dielectric?
Thanks, any help is appreciated.