Boundary Conditions in Dielectrics
Boundary conditions (BC's) are probably the most important tool at the interface between mathematics and physics. Gauss' law tells us that the parallel components of the E-field above and below an interface between two media must equal. The difference between the vertical components of the electric field and the electric displacement are equal to some constant. Of course, there are some interesting consequences for the electric field in adjacent media as can be seen in the figures to the left. While light travels through two media which result in total internal reflection for certain angles, an electric field simply exists in the 2 media. There is no such thing as total internal reflection. The field eventually runs parallel to the interface when the angle of the optical normal approaches 90 degrees. A similar surprising result exists at normal "incidence". One field becomes very very small since parallel components must always equal. The relationship, similar to Gauss law can be derived from these boundary conditions: tan(theta2)/tan(theta1)=epsilon2/epsilon1.