Nine identical wires, each of diameter d and length L, are connected in series. The combination has the same resistance as a single similar wire of length L but whose diameter is:
A) 3d
B) 9d
C) d/3
D) d/9
E) d/81

Question

Quizplus · Accepted Answer

Resistance of a wire is given by the formula R = (ρL)/A, where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.

For the combination of wires, the total resistance can be written as R = 9[(ρL)/A], since the wires are connected in series.

For the single wire, the resistance can be written as R' = (ρL)/A', where A' is the cross-sectional area of the single wire.

Since the wires are identical, their diameters are the same, and hence their cross-sectional areas are proportional to the square of their diameters. Therefore, we can write A = 9A', since there are 9 wires in the combination.

Substituting this in the expression for the resistance of the combination, we get R = 9[(ρL)/(9A')], which simplifies to R = (ρL)/A'.

Comparing this with the expression for R', we see that A' = 3^2A = 9A, which means that the diameter of the single wire is 3 times the diameter of each of the wires in the combination.

Therefore, the answer is C) d/3.

Nine Identical Wires, Each of Diameter D and Length L