4. Consider an innitely long cylinder with a circular cross section of radius R. Let its symmetric axis be the z axis and set up the usual cylindrical coordinate system. Surface currents run parallel to the z axis on the cylinder surface, which is described by an azimuthal-angle dependent surface current density K() = (1/2 R) cos 35%. (a) (1 point) Show that the vector potential A everywhere only has the axial component (i.e. the 2 component), whose value does not depend on z, i.e. A = AZ (3, 45) i. (b) (1 point) Inside (5 R) the cylinder, there is no current, so the vector potential satises the equation VZA = 0. This leads to the Laplace's equation on the two-dimensional plane: 2 13(3Az) 13AZ=OI (2) $65 .9 as 4-52 61:52 Employ the method of separation of variables in cylindrical coordinates to solve for A: (s, 46) for both .9 R. You can use the results for the general form of the solutions from a previous homework problem. (c) (1 point) Calculate the magnetic eld B from the vector potential A. Describe the eld's qualitative behavior