(a) Use completeness relations to represent (x x')(y y') and then the method of direct integration for the inhomogeneous differential equation which remains

(a) Use completeness relations to represent δ(x − x')δ(y − y') and then the method of direct integration for the inhomogeneous differential equation which remains to find the interior Dirichlet Green function for a cubical box with side walls at x = ±a, y = ±a, and z = ±a.

(b) Use the result of part (a) to find the charge density that must be glued onto the surfaces of an insulating box with sides walls at x = ±a, y = ±a, and z = ±a so that the electric field everywhere outside the box is identical to the field of a (fictitious) point charge Q located at the center of the box. It is sufficient to calculate σ(x, y) for the z = a face. Give a numerical value (accurate to 0.1%) for σ(0, 0).

(c) This problem was solved in the text by a different method. Check that both methods give the same (numerical) answer for σ(0, 0).