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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.