Question

Suppose you wish to fabricate a uniform wire from a mass m of a metal with density rhom and resistivity rho. If the wire is to have a resistance of R and all the metal is to be used, what must be the length and the diameter of this wire? (Use any variable or symbol stated above as necessary.

Answer 1

Answer:

Explanation:

Resistivity and resistance are proportional and depends of the length and the cross-sectional area of the wire:

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

furthermore, the density is the mass divided by the volume, and the volume can be written as the area multiplyed by the length:

$\rho_m=\frac{m}{A*L}$

Now you have tw equations and two variables, so you can solve for each of them.

first, solve for A in both equations and replace them:

$\frac{L}{R}\rho=\frac{m}{\rho_mL}$

$L^2=\frac{mR}{\rho_m \rho} \\L=$

now replace this into any of the previous equiations:

$R=\frac{\sqrt{\frac{mR}{\rho_m \rho} } \rho}{A} \\A=\sqrt{\frac{mR\rho^2}{\rho_m \rho R^2} }\\A=\sqrt{\frac{m\rho}{\rho_m R} }$

If you assume the wire has circular cross-sectional area, then the area is:

$A=\pi(\frac{d}{2} )^2$

solving for d:

$d=2\sqrt{\frac{A}{\pi} }$

replacing A and simplifying:

$d=2 \sqrt[4]{\frac{m\rho}{\rho_m R \pi^2} }$