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When you rub a glass rod against silk cloth, the glass rod becomes positively charged and the cloth becomes negatively charged. So they now can attract each other. However, after this 'attraction' takes place, the effect is lost after the objects neutralize or nullify each others' charges.

What happens when a comb rubbed against hair is brought near paper bits? Here, since the paper is neutral, I'm assuming the same 'nullification' of charge cannot quite take place.

Also, is this 'nullification' completely precise? Does the silk provide the exact number of electrons the (positive) glass rod requires?

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Charge is conserved and so if the initially uncharged glass rod has a deficit of electrons (positively charged) then the cloth will have a equal surplus of electrons.

The comb becomes charged and when it is brought to some bits of paper there is a separation of charge within the paper. These are called induced charges. If the comb is negatively charged then the positive induce charge on the paper will be attracted (and closer than the negative induced charges) to the negative charges on the comb, so that paper is attracted to the comb.
If the paper and comb touch then the net result is that some of the negative charge on the comb is transferred to the paper.
If the paper obtains sufficient negative charge from the comb it will then be repelled from the comb as like charges repel.

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