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...Aka the Kelvin hydroelectric generator, the Kelvin electrostatic generator, or Lord Kelvin's thunderstorm.

Let's start with 2 metal cups, one above the other, in such a way that the top cup can pour its contents below it into the bottom cup. At this point, they are both uncharged. If we could magically make the top cup negatively charged, would the bottom cup become positively charged per the electrostatic induction effect? Or would something else happen? (Or nothing happens to the bottom cup?)

Secondly, if we pour the water from the negatively charged top cup into the bottom cup in such a way that a solid stream is prevented from connecting the two cups electrically, would the top cup and bottom cup switch electrical charge signs? I mean, would the top cup become positively charged and the bottom cup negatively charged?

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Your description of the setup sounds like we're dealing with only one side of the Kelvin water dropper. Where the inductor (top cup) and the collector (bottom cup) are electrically isolated. It's confusing when you say "top cup" because it cannot have a bottom in order to allow water to fall to bottom cup. I will refer to it as the top loop instead. Note that both the top loop and bottom cups are metal (just like in Kelvin dropper and thus cannot "hold" static charge like a balloon can. It can only have a charge if it is induced by a charged object which are the ions in the water in this case.)

Answer to first question:

Yes the bottom cup would become positively charged. The negatively charged top loop would repel any negative charges back up the stream while the remaining positive charged are attracted. And since the stream breaks into drops when the leaves the top cup (as you said) you are left with only positive charges in the bottom cup. However if the stream does not transition from a solid to broken stream within the top loop then a separation of charges could not occur resulting in a neutral bottom can. tldr; if the stream breaks into drops before enter bottom cup, the bottom cup will become positively charged

For this I am assuming that you pour water from a negatively charged cup (which is insulated from you and the environment) into a neutral bottom cup.

Answer to second question:

These cups are charged via electrostatic induction as by the ions in the water. This means that for the top cup to be negatively charged there is an excess amount of negatively charged ions. So if you pouted this water into the neutral bottom cup then it would become negatively charged. If you poured all the water then the top cup would become neutral as there are no more anions to induce a charge in it. I'm sure you can deduce what would happen if the bottom cup was positively charged (water contains excess positive ions). Hopefully you find this useful (:

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