(a) Find the vector potential inside and outside a solenoid that generates a magnetic field B = Bẑ inside an infinite cylinder of radius R. Work in the Coulomb gauge. 

(b) The A haronov-Bohm effect occurs because the magnetic flux is non-zero when the integration circuit is, say, the rim of a disk of radius ρ > R which lies perpendicular to the solenoid axis. Show that (φ is the angle in cylindrical coordinates) leads to identically zero vector potential outside the solenoid.

(c) The result in part (b) implies that we could eliminate the Aharonov-Bohm effect by a gauge transformation. Show, however, that the new magnetic field corresponds to a different physical problem, where

= ds . A