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Your Question : Assuming that voltage remains constant, what happens to the current in a wire if the length of the wire increase?

Assuming that voltage remains constant, what happens to the current in a wire if the length of the wire increase?

Answer:

1. You can think of every inch of wire as a resistor (with a very low resistance). 
2. When you make the wire longer, you're just adding more inches in series, so the resistance is increasing. That means the current is decreasing, if the voltage between the ends of the wire is constant.

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