(a) Let (,) be a solution of Laplaces equation in a cylindrical region < R. Show that the function (,) = (R 2 /,)

(a) Let Φ(ρ,Ф) be a solution of Laplace’s equation in a cylindrical region ρ < R. Show that the function Ψ(ρ,φ) = Φ(R²/ρ,φ) is a solution of Laplace’s equation in the region ρ > R.

(b) Show that a suitable linear combination of the functions Φ and Ψ in part (a) can be used to solve Poisson’s equation for a line charge located anywhere inside or outside a grounded conducting cylindrical shell.

(c) Show that a linear combination of the functions Φ and Ψ can be used to solve Poisson’s equation for a line charge located inside or outside a solid cylinder of radius R and dielectric constant κ₁ when the cylinder is embedded in a space with dielectric constant κ₂.

(d) Comment on the implications of this problem for the method of images.