An infinitely long, uniform line charge of density rho_L (C/m) is parallel to, and a distance d above the planar interface between two homogeneous dielectic media. The upper half-space (where the line charge resides) is characterized by permittivity epsilon_1, and the lower half-space has permittivity epsilon_2.
Find expression(s) for the scalar electric potential V(x,y) everywhere.
The problem is 2-dimensional. Orient the line charge in the z-direction. The line charge is at (x,y)=(0,d). The upper region (1) is y>0 and the lower region is y<0.