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If there is a electric set up such that A and B are respectively positive and negative terminals of the cell and a resistor PQ is inserted in the circuit. My question is how the current flows between A and P yet there is no Potential difference between A and P?

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It is because when you insert a resistor, or when a conductor (resistors are conductors) is connected to a voltage source such as a battery, the negative terminal will present an opposing electrostatic force the conductor as well as have a ready supply of additional electrons. This field will repel electrons...
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It is because when you insert a resistor, or when a conductor (resistors are conductors) is connected to a voltage source such as a battery, the negative terminal will present an opposing electrostatic force the conductor as well as have a ready supply of additional electrons. This field will repel electrons in the conductor and exert a force on them and will attempt to inject fresh electrons into the conductor. At the opposite end, where the conductor is connected to the positive terminal of the voltage source, a force exists to pull electrons from the conductor. 

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