Retarded electric and magnetic Casimir interaction of a polarizable system and a dielectric permeable wall

Using an expansion of the electromagnetic field in terms of quantized Fresnel modes, we calculate the retarded Casimir interaction, in the asymptotic domain, of a polarizable system and a dielectric permeable wall, that is, a wall characterized by a dielectric constant ${\mathrm{\epsilon }}_2$ and a magnetic permeability ${\mathrm{\mu }}_2$. We obtain explicit analytic results for the atom-wall and for the electron-wall interactions. When magnetic effects are ignored, the results reduce to those obtained by Lifshitz [Sov. Phys. 2, 73 (1956)] for the atom-wall interaction and by the present authors for the electron-wall interaction. The results simplify greatly for an ideal metallic wall and for a wall made of material such as liquid helium, for which the dielectric constant and the permeability are very close to unity. Other than in the determination of the amplitudes of the Fresnel modes, the calculations are entirely classical.