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The phenomenon is shown for example in this video where a thin water jet is formed by letting water fall from a small hole punched under a plastic cup, and the jet is bent when a charged rod is moved close to it. Why does this happen?

What I found:

The people in the video claim it is due to the polar nature of water: the little dipoles align with the electric field, and if this is non-uniform it will cause a net force on the jet. This explanation is supported by Vemulapalli & Kukolich (1996), who also claim that it might also occur in non-polar liquids due to the induced dipole moment. In fact Brindle & Tomlison (1975) report that the phenomenon also occurs in non-polar liquids under appropriate conditions. On the other hand Ziaei-Moayyed, Goodman & Williams (2000) report (among other interesting things) that the effect still occurs in a uniform electric field, which seems to contradict the dipole explanation. They propose that it has to do with charged ions. However it's unclear (at least to me) that their setup actually produces a uniform field (or to what extent). For what it's worth, one can find a lot of things on Youtube, for instance supporting the ion explanation or arguing against it.

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I have a theory. The theory the second by Anne Elk (50th anniversary of Monty Python, you know). I wanted to submit this as a comment, but it's too long. I will delete upon complaint if it doth offend.

Right at the surface of a body of water each water molecule has only half the number of nearest neighbors as it would in the bulk, away from the surface. The surface molecules respond by clinging more tightly to those nearest neighbors that they do have, which is why we have surface tension in fluids that possess slightly attractive forces between molecules, the way water does. the surface molecules will have an incentive to line themselves up in a manner which minimizes the energy associated with the free surface, which might result in the water molecules rotating into position with their hydrogens all facing inward. If so, then you have a natural charge dipole at the surface which the charged probe rod can then act upon. I think this idea is testable...

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