A wire of resistivity ρ must be replaced in a circuit by a wire of the same material but 4 times as long.If,however,the resistance of the new wire is to be the same as the resistance of the original wire,the diameter of the new wire must be
A) the same as the diameter of the original wire.
B) 1/2 the diameter of the original wire.
C) 1/4 the diameter of the original wire.
D) 2 times the diameter of the original wire.
E) 4 times the diameter of the original wire.

Question

Quizplus · Accepted Answer

The resistance $R$ of a wire is given by $R = \rho \frac{L}{A}$, where $\rho$ is the resistivity, $L$ is the length, and $A$ is the cross-sectional area. If the length of the wire is increased by a factor of 4, to keep the resistance the same, the area must also be increased by a factor of 4. Since the area $A = \pi r^2 = \frac{\pi d^2}{4}$ for a wire (where $r$ is the radius and $d$ is the diameter), increasing the area by a factor of 4 means the diameter must be increased by a factor of $\sqrt{4} = 2$.

A Wire of Resistivity ρ Must Be Replaced in a Circuit