A current I starts at z = −∞and flows up the z-axis as a linear filament until its hits an origin-centered sphere of radius R. The current spreads out uniformly over the surface of the sphere and flows up lines of longitude from the south pole to the north pole. The recombined current flows thereafter as a linear filament up the z-axis to z = +∞.
(a) Find the current density on the sphere.
(b) Use explicitly stated symmetry arguments and Amp`ere’s law in integral form to find the magnetic field at every point in space.
(c) Check that your solution satisfies the magnetic field matching conditions at the surface of the sphere.