What causes the DNA fragments to stop moving in gel electrophoresis? I'm currently studying VCE BioChemistry, and we're studying the separation of DNA strings of different lengths via gel electrophoresis.  

(This involves having 'clumps' of DNA at one end of a gel medium and applying an electric current, pushing the DNA strands across the gel. Different sized strands move at different speeds through the gel)

At some point, the strands stop moving such that there is a distribution of strands across the gel. What causes the strands to stop?
Surely they'd continue until the end of the gel.

If suppose the electric current could be stopped, thus causing the strands to stop moving, but how would one know when to stop the current?
There are electrophoretic methods where the molecules stop automatically like Isoelectric Focusing, but in the common agarose gel electrophoresis of DNA you mention this is not the case.

The DNA is negatively charged, it will move along the electric field you set up until you stop applying any current. It will happily move out of your gel if you let it run for too long.

To see the progress of your gel some dyes are usually added to the sample like bromophenol blue or xylene cyanol. They don't interact with the DNA, they just provide you a reference on how far your gel has run. You still need to have a rough idea how your samples run compared to those dyes if you want to use the full length of your gel.