The length of a certain wire is doubled and at the same time its radius is reduced by a factor of 2. What is the new resistance of this wire?
A) It is 4 times as large.
B) It is $1 / 2$ as large.
C) It it 8 times as large.
D) It is 6 times as large.
E) It is 2 times as large.

Question

The length of a certain wire is doubled and at the same time its radius is reduced by a factor of 2. What is the new resistance of this wire? 
A) It is 4 times as large.
B) It is $1 / 2$ as large.
C) It it 8 times as large.
D) It is 6 times as large.
E) It is 2 times as large.

Quizplus · Accepted Answer

The resistance $R$ of a wire is given by $R = \rho \frac{L}{A}$, where $\rho$ is the resistivity of the material, $L$ is the length of the wire, and $A$ is the cross-sectional area. When the length $L$ is doubled, the resistance becomes $2R$. When the radius $r$ is halved, the cross-sectional area $A = \pi r^2$ becomes $\pi (\frac{r}{2})^2 = \frac{1}{4}\pi r^2$, which means the area is reduced to a quarter of its original size, making the resistance four times larger due to the inverse relationship between area and resistance. Therefore, the new resistance is $2 \times 4 = 8$ times as large as the original resistance.

The Length of a Certain Wire Is Doubled and at the Same