Consider planewave refraction from a non-conducting medium ( , ) into a conducting medium ( ,

Consider planewave refraction from a non-conducting medium (ε , μ) into a conducting medium (ε , μ , σ). Ohmic loss requires that the refracted wave vector k₂ be complex. The figure below shows a proposed refraction geometry where k₂ = q + iκ.

(a) Explain why κ points in the +z-direction and why the angle of incidence still equals the angle of reflection. Derive the generalization of Snell’s law for this problem.
(b) Use the dispersion relation in the conductor to prove that θ₂ ≤ θ₁.
(c) Derive a quadratic equation for the variable κ². If δ(ω) is the skin depth, find Λ (ω) such that

(d) Show that θ_2V 

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