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# Can a 2.5-mm-diameter copper wire have the same resistance as a tungsten wire of the same length? Give numerical details.

Solution

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In order to determine the resistance we use the equation $\textbf{(18-3)}$:

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

where $\rho$ represents the resistivity of a certain material, $A=\pi r^2=\pi (d/2)^2=4\pi d^2$ is a cross sectional area where $d$ is diameter.

In our problem we have a $d_c=2.5mm$ copper wire with a length $L$ and a tungsten wire with a same length $L$. Let's denote with $R_c$ and $R_t$ the resistances of the copper and tungsten wire respectively. If the resistances are the same then:

$$$R_c=R_t\Rightarrow \rho_c\frac{L}{A_c}=\rho_t\frac{L}{A_t}\Rightarrow \rho_c A_t=\rho_t A_c,$$$

where $\rho_c=1.68\times 10^{-8}\Omega m$ and $\rho_t=5.6\times 10^{-8}\Omega m$ (table 18-1). Therefore from (1):

$\rho_c \frac{\pi d_t^2}{4}=\rho_t \frac{\pi d_c^2}{4}\Rightarrow \rho_c d_t^2=\rho_t d_c^2.$

Solve for $d_t$

$d_t=d_c\sqrt{\dfrac{\rho_t}{\rho_c}}=(2.5mm)\sqrt{\dfrac{5.6\times 10^{-8}\Omega m}{1.68\times 10^{-8}\Omega m}},$

$d_t=4.56mm.$

The is answer is: It can if the diameter of the tungsten wire is $d_t=4.56mm.$

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