EL. Consider a magnetic eld created by a straight electric wire pointing along the z-axis carrying an electric current J in the positive direction. The so-called vector potential created by this current is equal to 251.036, 2:) = J log(r)z. [We omit the magnetic permeability {in and a factor 2a.) The magnetic: field E ruralogue of equation 13 = 'v in electrostatics: is [3' = x xi. We follow the same story line as in the previous problem but. now use cylindrical coordinates. Show that. o o .l l _ B=VX A=Jf?g V See page 527 in Boas for the curl in cylindrical coordinates, 5 . re _. 1 1r 0 :2 r _ _ a a a V X A r or no a: A,' r'z'lg A; ; . Ampere's law, V x E = f, states how the magnetic eld is related to electric currents. Similar to the point charge issue in problem 3, this naively fails for a wire of zero thickness. ShowLusing problem Lb, or using the above formula for the curl in cylindrical coordinates, that V x B = U for all r a i}. . Stokcs' theorem deals with such zero thickness wires more easily. Consider two nested circles in the z = 0 plane, one with radius R1 and the other with radius R2, with only electric current along 23 through their center. Evaluate the line integral in Stokes' theorem along both circles assuming that g : $60. The result is independent of R. Explain how this result is related to the absence of electric current pointing perpendicular through the 2;: Uplane inbetween the two circles according to Stokes' theorem and thus confirms that VXB=Uforallr>lL