Maxwell coils are solenoids of spheroidal shape which produce a consta
nt magnetic field inside the coil; at large distances the field of a M
axwell coil becomes that of a magnetic dipole. The current in systems
consisting of several confocal Maxwell coils can be adjusted in such a
way that the magnetic field inside the smallest coil is constant and
the back flux is confined to the interior of the largest coil. Quantum
-mechanical scattering of charged particles by Maxwell coils, or syste
ms thereof, is first formulated as a three-dimensional problem and the
n treated within a two-dimensional approximation. The evolution in tim
e of wave packets, which are Gaussian wave packets in the beginning, i
s calculated numerically. In all cases an Aharonov-Bohm effect is obse
rved in the angular distribution of the scattered wave packets. The ba
ck flux results only in an additional deflection of the packets, which
corresponds to the classical deflection caused by the Lorentz force.