Jupiter has a very large magnetic field, the largest of any planet. Contributing factors to this field are Jupiter's fast rotation (<10 hours) and its conducting core (ultradense hydrogen is a conductor!). The magnetic field around the planet forms a shield against the solar wind (charged particles, or plasma) coming from the sun, and creates its own plasma environment, or magnetosphere that extends to 130 Jupiter Radii and beyond -- extending all the way to Saturn. If the entire magnetosphere were visible from Earth, it would appear in the sky bigger than our moon!

As jupiter rotates, it sweeps its plasma around with it at very large speeds, probably exceeding the rotational speed of the moons. The moons are bombarded by this magnetosphere plasma, and therefore radiation is a problem for spacecraft (and one day people?) in the Jovian system. The density of the plasma increases as one nears Jupiter, for example rising from 1 particle per cubic centimeter at 14 Jupiter Radii to 100 particles per cubic centimeter at 3 Jupiter Radii. Some locations are denser (more radiation) than others, such as the Io plasma torus. Much more info on Jupiter's magnetosphere can be found here.