  We calculate the average polarisability of two dimensional mesoscopic rings
in the presence of an Aharonov-Bohm flux. The screening is taken into account
self-consistently within a mean-field approximation. We investigate the effects
of statistical ensemble, finite frequency and disorder. We emphasize
geometrical effects which make the observation of field dependent
polarisability much more favourable on rings than on disks or spheres of
comparable radius. The ratio of the flux dependent to the flux independent part
is estimated for typical GaAs rings.
