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I found this video about Van De Graaff generator: https://www.youtube.com/watch?v=QarKXkXox6M. At 8:14 of the video, we can see that if we put the cupcake paper upside down, the paper flies away. Otherwise, the paper gets attracted to the dome of the generator.

I'm a newbie in physics and I'm even not good at English so I don't quite understand the explain in the video. What I'm guessing is: when put upside down, electrons will gather at the edge of the paper. So, the edges of the paper will contain a lot of charges and they have the same sign as the charge of the dome (positive charge), so they repel each other and the paper flies away. But I'm not sure.

Can someone explain for me why the cupcake paper flies away when put it upside down; and gets attracted to the dome when put it straight up?

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If the electric field is high enough the air becomes a conductor.
This called electrical breakdown and for air the breakdown electric field (potential gradient) is about $3\,\rm kV\, mm^{-1}$.

The surface charge density is larger the more curved a surface is and this in turn means that the electric field strength at sharp edges is larger which in turn means that air is more likely to breakdown (become a conductor) at an edge or point.

If you look at a cupcake case you will see that it is more pointed along the top rim as compared with the bottom rim.

After running the VdG generator foe a period of time the voltage of the dome becomes constant when the rate at which charges are deposited on the dome by the belt is equal to the rate at which charges leave the dome due to the air (and the VdG dome supports) being a conductor.
If the object on the dome is more pointed (sharper) [cupcake case flat side down] then the voltage of the dome is lower that than if the object on the dome is less pointed [cupcake case flat side up].
This means that the amount of charge on the dome and cupcake case is less when the cupcake case is flat side down and so is the force of repulsion between them so the cupcake case stays on the dome.
When the cupcake case is flat side up much more charge accumulates on the dome and cupcake case and the force of repulsion between them is sufficient to move the cupcake case away from the dome. The difference in the voltage (and hence charge stored) is shown by the lack of sparking between large and small domes when the cupcake case is flat side down.

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