Please solve complete problem. I need complete step by step solution. Course: Electromagnetic theory

(a) State the general form of Poisson's equation in electrostatics, defining any symbols you introduce. (2 marks) (b) A long metal cylinder with radius a, is coaxial with, and entirely inside, an equally long metal tube with internal radius 2a and external radius 3a. The space between the cylinder and the tube is filled with an LIH dielectric material with relative permittivity &. This space is also permeated by a uniform free charge distribution with density p. Outside the tube, the charge density is zero. Show that Poisson's equation for the electrostatic potential in the dielectric material in the region between the cylinder and the tube described above can be simplified to the following equation, explaining your reasoning and choice of coordinate system: 1 d Pf r dr EEO (5 marks) (c) Find the general solution of Poisson's equation in the dielectric material. (7 marks) Now assume that the metal cylinder and tube are both held at zero potential. (d) Using appropriate boundary conditions, show that the potential within the dielectric at distance r from the axis of symmetry is given by V(r) = Pf 3 In() for a