It's an engineering problem. Sure there is physics involved, plasma fields and Jacobians and all that stuff. But the source will keep pumping charged particles forever for all practical purposes. However, the source is sporadic. Sometimes there is a lot, sometimes there isn't. Once the particles are trapped in the Van Allen Belt as it is today, they stay for a long time. If they were drained quickly, then the level would drop and stay low until the next solar outburst. So, overall, over time, the density of charged particles will be reduced. It's still a sci-fi idea, but a space tether ought to be launched just to see what it does. Who knows, it might work or it might just scare people into learning a little about the universe besides what is on cable tonight.
My gut tells me "Not too long", but then I suppose I'm supposed to ruminate, not... oh damn, there's another epigram that got away.
Anyway, my trusty, though somewhat dusty "Asimov's New Guide to Science" relates the tale of a Mr. Christofilos, who in 1957 postulated that charged particles would be trapped along the magnetic lines of force in the atmosphere.
In 1958, Project ARGUS took place, and confirmed the Chrisofilos effect, and noted that the radiation from the nuclear bombs detonated in the magnetosphere was persistent, with Explorer IV detecting it "during several hundred of it's trips around the earth". (Don't let the quotation marks fool you, the info in this post is pretty much all plagiarism up to this point.)
Since, Explorer IV's period was 110 minutes, we're talking persistence measuring into days weeks here (why do I always find this stuff at the end of writing a post?).
Uh, so I guess the tether idea could work, as the particles are somewhat persistent.