Consider a potential problem in the half-space defined by z ≥ 0, with Dirchlet boundary conditions on the plane z = 0 (and at infinity).
(a) Write down the appropriate Green function G(x, x')
(b) If the potential on the plane z = 0 is specific to be Φ= V inside a circle of a radius a centered at the origin, and Φ= 0 outside that circle, find an integral expression for the potential at the point P specified in terms of cylindrical coordinates (ρ , Φ, z).
(c) Show that, along the axis of the circle (ρ = 0), the potential is given by:
Φ = V(1-z/√a2+z2)
(d) Show that at rage distances (ρ^2 + z^2 >> a^2) the potential can be expanded in a power series in (ρ^2 + z^2)^-1, and that the leading terms are:
Φ = (Va2/2)(z/(ρ2+z2)3/2)[1- 3a2/4(ρ2+z2) + 5(3ρ2a2+a4/8(ρ2+z2)2)+.....]
Verify that the result of parts c and d are consistent with each other in their common range of validity.