A wire of resistivity ρ must be replaced in a circuit by a wire four times as long. If, however, the total resistance is to remain as before, the diameter of the new wire must
A) remain unchanged.
B) be two times smaller.
C) be four times smaller.
D) be two times larger.
E) be four times larger.

Question

Quizplus · Accepted Answer

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area. If we replace a wire with a new wire that is four times as long, its resistance will increase by a factor of 4, since resistance is proportional to length. To keep the total resistance the same, we need to decrease the cross-sectional area of the new wire by a factor of 4.

The cross-sectional area of a wire is proportional to the square of its diameter. Therefore, we need to decrease the diameter of the new wire by a factor of 2 (since 2² = 4), which means that the new wire should have a diameter that is two times larger than the original wire. Hence, option D is the correct answer.

A Wire of Resistivity ρ Must Be Replaced in a Circuit