  The Aharonov-Bohm (AB) effect was convincingly demonstrated using a
micro-sized toroidal magnet but it is almost always explained using an
infinitely-long solenoid or an infinitely-long flux line. The main reason for
this is that the formal treatment of the AB effect considering a toroidal
configuration is too cumbersome. But if the micro-sized toroidal magnet is
modelled by a closed flux line of arbitrary shape and size then the formal
treatment of the AB effect is exact, considerably simplified, and
well-justified. Here we present such a treatment that covers in detail the
electromagnetic, topological, and quantum-mechanical aspects of this effect. We
demonstrate that the AB phase in a closed flux line is determined by a linking
number and has the same form as the AB phase in an infinitely-long flux line
which is determined by a winding number. We explicitly show that the two-slit
interference shift associated with the AB effect in a closed flux line is the
same as that associated with an infinitely-long flux line. We emphasise the
topological nature of the AB phase in a closed flux line by demonstrating that
this phase is invariant under deformations of the charge path, deformations of
the closed flux line, simultaneous deformations of the charge path and the
closed flux line, and the interchange between the charge path and the closed
flux line. We also discuss the local and nonlocal interpretations of the AB
effect in a closed flux line and introduce a non-singular gauge in which the
vector potential vanishes in all space except on the surface surrounded by the
closed flux line, implying that this potential is zero along the path of the
charged particle except on the crossing point where this path intersects the
surface bounded by the closed flux line, a result that questions the alleged
physical significance of the vector potential and the local interpretation of
the AB effect.
