PERHAPS the most unexpected finding of the Ulysses mission so far has been the detection of quasi-periodic streams of high-velocity, submicrometre-sized dust particles during the spacecraft's encounter with Jupiter1. The impact geometry clearly shows that these small grains originate in the jovian system, but it is surprising that any dust can escape Jupiter's gravitational influence. Here we show how the Ulysses dust events could result from the acceleration and subsequent ejection of small grains by Jupiter's magnetosphere. Dust grains entering the plasma environment of the magnetosphere become charged, with the result that their motion is then determined by both electromagnetic and gravitational forces. We have modelled this process and find that only those particles in a certain size range gain sufficient energy to escape the jovian system. Moreover, if Io is assumed to be the source of the dust grains, its location in geographic and geomagnetic coordinates determines the exit direction of the escaping particles, providing a possible explanation for the observed periodicities. The calculated mass and velocity ranges of the escaping dust grains are consistent with the Ulysses findings.