A charge density of ohm Coulombs per square meter is uniformly distributed over a thin and very long cylindrical shell of radius a. You want to know the electric flux density field D and the electric field E generated by this charge at all points in space. a. Draw the Gaussian surface that you would use to calculate this field. b. Write Gauss's Law in differential form (i.e. in terms of divergence). c. Going step-by-step, justifying each step, transform this to the integral form of Gauss's Law, expressed in terms of total enclosed charge. d. State the symmetry argument by which you can determine the direction of the electric flux density D in this problem. What is the direction of D? e. Does symmetry allow D to be zero? f. Does symmetry allow D to point in the direction of the cylinder axis? Why not? g. The shell also has gravitational mass, by the way. What does symmetry say about the direction of the gravitational field? h. On what spatial variables does Dr , the radial component of the D vector, depend? How do you know? i. Solve for the D field in all regions of space. j. What is the electric field E in all regions of space?