The charged particle scattering amplitude on the magnetic field of a toroidal solenoid is obtained in the first Born and high-energy approximations. It is shown that multiconnectedness of the accessible space, non-triviality vector potentials in it and the single-valuedness of the wavefunctions used are not enough for the existence of the Aharonov-Bohm effect. Criteria are given for this. Surrounding the toroidal solenoid with barriers of different geometrical forms, the author observes that the magnetic field contribution to the scattering amplitude depends crucially both on the barrier height and its form. The latter could hardly be explained by particle penetration into the H not=0 region. The author proposes an addendum to Tonomura's experiments which probably could clear up any doubts about the existence of the AB effect. The gedanken experiment is discussed which shows that the effect of inaccessible fields could be observed even in a simply connected space.