Question

state and explain the boundary condition for the displacement vector D at the boundary separating two dielectric Media

Answer 1

Boundary conditions at dielectric surfaces state how the electric vectors \mathbf E and \mathbf D change at the interface between two different media (e.g. vacuum and a dielectric, two different dielectrics etc).
Consider a plane interface between two media 1 and 2 (these can be vacuum, or insulators). They have dielectric constants k1 and k2 (permittivity ε1 and ε2 respectively). For generality it is assumed that there is a free charge density σf due to charge q at the interface. \mathbf E_1 and \mathbf E_2 are electric field intensity vectors making angles θ1 and θ2 with the normal to the interface. Corresponding displacement vectors \mathbf D_1 and \mathbf D_2 will make angles θ1 and θ2 with normal (shown in the figure 2.6).