I am following the step on this introduction to magnetic monopole. I understand that the vector potential produced by semi-infinite solenoid is exactly the vector potential of magnetic monopole. But we know that the magnetic field of infinite solenoid vanishes outside the solenoid while the magnetic field of monopole does not? How can I explain this discrepancy?
The key here is that the solenoid is semi-infinite, not infinite. If we view the Dirac string as an infinitely thin solenoid, the magnetic flux in the solenoid escapes through the open end, spreading out radially to give a monopole field.
The field of one end of a semi infinite solenoid cannot be exactly that of a monopole, as such objects do not exist. Only the field outside the solenoid resembles that of a monopole.