Is there a current flowing through a potentiometer wire when the balance point is achieved?

To have a potential difference (voltage) between two points on a resistive wire, there should be current flowing through it. If there is no current flowing through the wire, there would be no potential drop, V=IR. If I is 0, V will be 0.

So, even when your potentiometer is balanced, there will be current in the potentiometer wire for a potential to balance the EMF that you are trying to measure.

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