4. Method of Green's function: The semi-infinite region z > 0 is free of charge. Its boundary on the left, the z = 0 plane, is kept at a fixed electrostatic potential using some experimental apparatus installed in the region z 0 and a > 0): Vo ( 1 - x 2 + y 2 V (x, y, z = 0) = a 2 if x2+ yz a2. The zero point of the potential is chosen to be at infinity z - too to the right. See Figure 2. free of charges at z > 0 y V(x = 0, y = 0, z> 0)? Z V(x, y, z = 0) given 2 = 0 Figure 2: Electrostatic potential in semi-infinite free space(a) (1 point) To find the potential V(x, y, z) everywhere for z > 0, we would like to use the method of Green's function. Define a suitable Green's function G(r; r') by writing down the Poisson's equation it satisfies and imposing some appropriate boundary conditions. Explain that the Green's function you seek is unique.(b) (0.5 point) Use the method of images to nd 60'; r'). (c) (1 point) Derive an integral solution for V(x, y, z) at z > 0, in terms of 60'; r') and the given potential V(x, y, z = 0) on the z = 0 plane. (d) (1 point) Use the explicit expression for V(x, y, z = 0) given in Eq. (3) to nd the potential along the positive z axis, ie. V(x = 0,y = 0, z) for all z > 0