BC's for fields REVIEWED

This is the Title PhL 3.26.05 This has the field continuity conditions in perp and parallel notation. First part is incorporated into Ch 1 of lines doc. I decided on a t and n notation instead. If the two media are conducting dielectrics with Ohm's law Jc = σE, we can apply continuity (**) to our tiny box to find that div Jc = - ∂tρfree -∂t[∫V ρfree dV] = ∫S Jc dA (1.1.16) so that -∂tnfree = [Jn1- Jn2] = σ1En1 - σ2En2 or for monochrome time dependence -jω nfree = σ1En1 - σ2En2 . Recall now from *** that nfree = ε1En1- ε2En2 Adding the last equation to 1/jω times the previous equation gives 0 = [ε1 + σ1/jω] En1 - [ε2 + σ2/jω]En2 In terms of the complex dielectric constants ξi this says that 0 = ξ1En1 - ξ2En2 so that ξ1En1 = ξ2En2 In the limit that say medium 2 becomes a perfect conductor, ξ2 ≈ σ2/jω → ∞ and En2 → 0, but the product is maintained equal to ξ1En1 . But there is no free charge in such dielectrics, nor any at the boundary between them unless supplied by a very contrived apparatus, because any such free charge would leak away into the dielectrics. If one of the dielectrics is a "conductor", There will be polarization charge npol at the boundary due to "differential displacement" of the two dielectrics there. ********************** saving things here ************ E||(1) = E||(2) or (1/ε1)D||(1) = (1/ε2)D||(2) (1.1.20) H||(1) = H||(2) or (1/μ1)B||(1) = (1/μ2)B||(2) (1.1.21) [D(1) - D(2)] = nfree or [ε1E(1) - ε2E(2)] = nfree (1.1.22) B(1) = B(2) or μ1H(1) = μ2H(2) (1.1.23) Rules for continuity of parallel and normal fields at a boundary: (1.2.24) E||(1) = E||(2) or (1/ε1)D||(1) = (1/ε2)D||(2) (1.1.20) H||(1) = H||(2) or (1/μ1)B||(1) = (1/μ2)B||(2) (1.1.21) [D(1) - D(2)] = n or [ε1E(1) - ε2E(2)] = n (1.1.22) μ1H(1) = μ2H(2) or B(1) = B(2) (1.1.23)