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The dielectric constant is related to the electronic susceptance in an isotropic material. The susceptance is basically the ratio of polarization to applied electric field. You can think about a conductor as having "bound" electrons in that they cannot leave the entire material, but are free to polarize across the entire length of a conductor. When you apply an external electric field to a conductor, you polarize the entire conductor, such that the polarization causes the electric field inside the conductor to be zero (electrostatic equilibrium). In a normal dielectric, the bound electrons cannot move as far as in a conductor, they have a much smaller polarization. The definition of a dipole moment is charge separation times separation distance. In a conductor, the induced dipoles have distances of the magnitude of the size of the macroscopic object, which is much much larger than the dipole distances of a dielectric. Hence, the polarization vectors in a conductor are nearly infinite compared to the polarization vectors of a dielectric (whose order of magnitude we are accustomed to dealing with). The susceptance is therefore near infinite, and so is the relative permittivity.
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