Question

Find the ratio of the diameter of aluminium to copper wire, if they have the same

Answer 1

Answer:

1.24

Explanation:

The resistivity of copper$\rho_1=2.65\times 10^{-8}\ \Omega-m$

The resistivity of Aluminum,$\rho_2=1.72\times 10^{-8}\ \Omega-m$

The wires have same resistance per unit length.

The resistance of a wire is given by :

$R=\rho \dfrac{l}{A}\\\\R=\rho \dfrac{l}{\pi (\dfrac{d}{2})^2}\\\\\dfrac{R}{l}=\rho \dfrac{1}{\pi (\dfrac{d}{2})^2}$

According to given condition,

$\rho_1 \dfrac{1}{\pi (\dfrac{d_1}{2})^2}=\rho_2 \dfrac{1}{\pi (\dfrac{d_2}{2})^2}\\\\\rho_1 \dfrac{1}{{d_1}^2}=\rho_2 \dfrac{1}{{d_2}^2}\\\\(\dfrac{d_2}{d_1})^2=\dfrac{\rho_1}{\rho_2}\\\\\dfrac{d_2}{d_1}=\sqrt{\dfrac{\rho_1}{\rho_2}}\\\\\dfrac{d_2}{d_1}=\sqrt{\dfrac{2.65\times 10^{-8}}{1.72\times 10^{-8}}}\\\\=1.24$

So, the required ratio of the diameter of Aluminum to Copper wire is 1.24.