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.

Explanation:

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

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

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:

now replace this into any of the previous equiations:

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

solving for d:

replacing A and simplifying:

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