P1. Use the method of Green function to find an integral expression for the electrostatic potential in the half-space defined by z ≥ 0. The electrostatic potential Φ on the plane z = 0 is zero everywhere except inside a circle of radius R. In this area, the electrostatic potential is given by Φ(ρ, Φ, 0)= V₀ (R-ρ)²/R², where ρ is the distance to the origin.

(a) Find the potential along the axis of the circles (the z axis).

(b) Find the potential at large distance from the origin.

(c) Discuss what happens to these potentials when a point charge q is placed at (x₀, y₀, z₀), where z₀ is positive, and the potential on the plane z = 0 remains the same.

P2 Consider the region bounded by two concentric spherical surfaces. On the inner boundary (r = a) of this region, the electrostatic potential is specified by Φ (a, θ, Φ) = V₀ cos²(θ) in spherical coordinates, and on the outer boundary (r = b), the electrostatic potential is specified by Φ (b, θ, Φ) = 2V₀. Find the electrostatic potential in this region (a < r < b).

P3 A system consists of 4 point charges and a localized distribution of charge centered at the origin. The point charges are q, 2q, -q and -2q, located at (a, 0, 0), (-a, 0, 0), (0, a, 0) and (0, -a, 0), respectively. The charge density of the localized distribution of charge is given by

ρ(r) = q/a⁵.r²e^-r cos²(θ)sinΦ

Find the multipole moments of the system and make a multipole expansion of the potential at large distances (r >> a), up to the term (a/r)³.