Consider the interface or the boundary between two dielectrics with different properties, L H }1 }o}r1 DQG }2 }o}r2 are the permittivities of the two media 1 and 2. Assume that there are no free charges at the interface between the two media. If we construct a pillbox with infinitesimally small thickness and area.
if D->1 and D->2 are the electric flux densities or the displacement densities in the media 1and 2 respectively, then since there are no charges enclosed, it follows from Gauss’s law that the surface integral of D over the pill box surface is zero. Thus, D->1.n1^ûV D->2 .n^2ûV ZKHUH ûs is the pillbox surface (top and bottom) and n^1 and n^2 are the unit outward normals respectively. Thus, (D->1 - D->2).n^1 =0. Since n^1 is the unit normal to the interface, D->1 - D->2 = 0, where D->1 and D->2 are the normal components of the electric flux densities in the respective media. Thus D->1 = D->2. The normal components of displacement densities are equal at the boundary between two dielectric media. In other words, the normal component of the electric flux density across the charge free interface between the two media is continuous, meaning that the number of lines of displacement flux entering one face is the same as the number leaving the other face.
The above relation is useful in evaluation of the flux density and hence field intensity- components in one medium due to those in the adjacent medium.
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