Question

Two wires are identical, except that one is aluminum and one is tungsten. the aluminum wire has a resistance of 0.22 ω. what is the resistance of the tungsten wire?

Answer 1

The resistance R of a piece of 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 its cross-sectional area.

Using this formula, and labeling with A the aluminum and with T the tungsten wire, we can write the ratio between $R_T$ (the resistance of the tungsten wire) and $R_A$ (the resistance of the aluminum wire):
$\frac{R_T}{R_A}= \frac{\rho_T \frac{L}{A} }{\rho_A \frac{L}{A} }$

the two wires are identical, so L and A are the same for the two wires and simplify in the ratio, and we get:
$R_T = \frac{\rho_T}{\rho_A} R_A$

By using the resistivity of the aluminum: $\rho_A=2.65 \cdot 10^{-8} \Omega m$ and the resistivity of the tungsten: $\rho_T = 5.6 \cdot 10^{-8} \Omega m$m we can get the resistance of the tungsten wire:
$R_T = \frac{\rho_T}{\rho_A} R_A = \frac{ 5.6 \cdot 10^{-8} \Omega m}{2.65 \cdot 10^{-8} \Omega m} (0.22 \Omega) = 0.46 \Omega$