Consider two infinite, plane parallel, perfectly conducting plates at z = 0 and z = z0 > 0, which hold equal and opposite surface charge densities and are kept at potentials V = 0 and V = Vp > 0, respectively. The region between the plates is filled with two slabs of perfect dielectric materials having permittivities ε1 for 0 < z < d (region 1) and ε2 for d < z < z0 (region 2)

a. Find the general solution for the electric potentials, V= V(z), (in terms of Vp, d, zo, ε1, and ε2) in the two regions by solving Laplace's equation piecewise and enforcing the continuity of V(z) at z = d. Hint: you will also need to use the fact that there is no surface charge accumulatiorn at a boundary between two perfect dielectrics.
b. Given that Z0=4d = 2 m, ε1=3ε0, ε2=ε0, and E(0< z< d) = -5z V/m , what is the electrostatic potential Vp on the conductor plate at z=z0?
c. Given the parameters in part (b) above, what is the surface charge density p, on the plate at z= zo?