Boundary condition

Solution
Inside this medium H¯¯¯¯¯�¯ = 0 and J¯¯¯�¯ = 0 because otherwise infinite energy densities, μ|H|2/2�|H|2/2, are required; static E¯¯¯¯�¯ and B¯¯¯¯�¯ are unconstrained, however. Since ∇×H¯¯¯¯=0=J¯¯¯+∂D¯¯¯¯/∂t∇×H¯=0=J¯+∂D¯/∂t inside, dynamic E¯¯¯¯�¯ and D¯¯¯¯�¯ = 0 there too. Since H¯¯¯¯//H¯// and B¯¯¯¯⊥B¯⊥ are continuous across the boundary, H¯¯¯¯//=0H¯//=0 and H¯¯¯¯⊥H¯⊥ can be anything at the boundary. Since E¯¯¯¯//E¯// and D¯¯¯¯⊥D¯⊥ are continuous (let’s assume ρ_s = 0 if J¯¯¯�¯ = 0 ), static E¯¯¯¯�¯ and D¯¯¯¯�¯ are unconstrained at the boundary while dynamic E¯¯¯¯=D¯¯¯¯=0E¯=D¯=0 there because there is no dynamic electric field inside and no dynamic surface charge. Since only H¯¯¯¯⊥≠0H¯⊥≠0 at the boundary, this is non-physical and such media don’t exist. For example, there is no way to match boundary conditions for an incoming plane wave. This impasse would be avoided if σ ≠ 0, for then dynamic H¯¯¯¯//H¯// and E¯¯¯¯⊥E¯⊥ could be non-zero.
This page titled 2.6: Boundary conditions for electromagnetic fields is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David H. Staelin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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