Final answer:
When the length of a wire increases, the resistance of the wire increases as well. This means that to maintain the same current, the potential difference across the wire must increase. Hence, the answer to the question is option 3: It increases.
Step-by-step explanation:
The question is asking about the relationship between the length of wire and the potential difference (voltage) across it. In general, the potential difference across a wire depends on the resistivity, length, and cross-sectional area of the wire, according to Ohm's law. However, if only the length of the wire is increased and all other factors remain constant, the potential difference required to drive the same current increases. This is because a longer wire has greater resistance, assuming the wire's material and temperature do not change.
Specifically, the resistance of a wire is directly proportional to its length. If a wire is stretched but its volume remains constant, the cross-sectional area decreases. Given that resistance is calculated with the formula R = ρ(L/A), where ρ is the resistivity, L is the length, and A is the cross-sectional area, the resistance increases as the length increases and area decreases.
So, if you were to stretch a wire to three times its original length, its resistance would increase by a factor of nine, provided that the wire's volume remains unchanged. Thus, to drive the same amount of current through the wire after it has been stretched, a greater potential difference would be required. Therefore, when the length of a wire is increased, the potential difference across it must also increase, option 3, to maintain the same current flow.