Electrostatic Fields in Matter

9 Boundary Conditions.What happens at the junction of two dielectrics?In the figure, the dielectric above the horizontal line has permittivityɛ 1 , while below the line it is ɛ 2 .ε1ε2Dn1Dn2Et1Et2Take the Gaussian pill-box on the left in the figure and imagineits height goes to zero. If there is no free charge at the boundary,there is no net flux of ⃗ D out of the pill-box. Hence the boundarycondition for ⃗ D is:D n1 = D n2 (24)We also know that the line integral of the electric field aroundthe rectangle on the right must be zero (since ∇ × ⃗ E = 0). As weimagine the rectangle shrinking to an infinitesimal size, we see that:E t1 = E t2 (25)i.e. across the boundary of 2 dielectrics, the normal component of⃗D is continuous and the tangential component of ⃗ E is continuous.13