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A resistor is connected across a battery...

A resistor is connected across a battery and current flows through it. Compare the potential energy of an electron before entering the resistor be that after leaving the resistor.

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Current always flows from higher potential to lower potential and the direction of flow of electrons is opposite to the direction of current .

So, we can say that electrons will flow from lower potential to higher potential. The potential energy of electrons, given by U=eV, will increase.
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