a negatively charged conductor is connected to a neutral conductor. Which conductor is at higher potential? What is the direction of flow of current?

Dear Student ,

When two conductors of different potentials are joined together then charge continues to flow from conductor of high potential to that of low potential until potentials of both conductors become neutral.The direction of flow of electric  current is taken opposite to the direction of flow of electron,and electron flow from low potential to high potential (i am considering you know this fact), so the flow of current is from high potential to low potential.

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It sounds like you're asking "If two conductive materials are brought in contact, and one of them is electrically neutral, and one is positively charged, which direction will charge flow?"

If that is indeed your question, then the answer is that negative charges (electrons) will flow from the neutral object to the positive object until they are at the same electrical potential (both bodies will end up positively charged).

The reason is this: the positively charged object will exert an attractive force on each electron in the neutral object, and a repulsive force on each proton in the neutral object. However, the protons are unable to move, being bound to the nucleus, which is stuck in a lattice of chemical bonds. The electrons in the highest energy levels, on the other hand, are free to move in a conductor, so they react to the attractive force by flowing towards the positively charged object.

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