Question

Two wires have the same resistance and diameter. If the wires are made of platinum and gold with resistivities respectively of 11.0 ✕ 10−8 Ω · m and 2.44 ✕ 10−8 Ω · m, determine the ratio of their lengths.

Answer 1

Answer:

0.222

Explanation:

The resistance of a wire is given by

$R=\rho \frac{L}{A}$

where

$\rho$ is the resistivity

L is the length of the wire

A is the cross-sectional area

Here let's call $R_p$ the resistance of the platinum wire and $R_g$ the resistance of the gold wire. The two resistances are equal, so we can write

$R_p = R_g$

$\rho_P \frac{L_p}{A_p}=\rho_g \frac{L_g}{A_g}$

Where

$\rho_p = 11.0\cdot 10^{-8} \Omega m$ is the resistivity of platinum

$\rho_g = 2.44\cdot 10^{-8}\Omega m$ is the resistivity of gold

We know that the two wires also have same diameter, so same cross-sectional area, so

$A_p = A_g$

Therefore we can rewrite the equation as

$\frac{L_p}{L_g}=\frac{\rho_g}{\rho_p}$

And so the ratio of the lengths of the two wires is

$\frac{L_p}{L_g}=\frac{2.44\cdot 10^{-8}}{11.0\cdot 10^{-8}}=0.222$