Question

We usually think of wires as perfect (R=0) conductors, which is not quite accurate. Real wires have resistance, and an effort is made to keep this resistance small – usually less than an ohm. a. Design a uniform cylindrical wire that has a resistance of 25 mΩ and uses $1 worth of copper. What are the length and diameter of your wire? Note: The resistivity (⍴) of copper is 1.72x10-8 Ωm, and you should assume that copper is currently $2.69 per pound. b. This wire is cut in half, and the two strands are used to power a toaster

Answer 1

Answer:

Length of copper wire is 5,22m and the diameter of copper wire is 2,14x10⁻³m.

Explanation:

The resistance of a wire is determined by:

R = ρL/A

Where R is resistance (25x10⁻³Ω); ρ is resistivity (1,72x10⁻⁸Ωm); R is length and A is cross sectional area.

Replacing you will obtain:

1,45x10⁶ m⁻¹ = L/A (1)

$1 of copper are:

$1×$\frac{1 lb}{2,69}$ = 0,37 lb. In kg:

0,37lb×$\frac{0,453592kg}{1lb}$ = 0,169 kg of copper.

As density of copper is 8960 kg/m³. The volume of this amount of copper is:

0,169kg×$\frac{1m^{3}}{8960kg}$ = 1,88x10⁻⁵ m³

The volume of a wire is L×A. Thus:

1,88x10⁻⁵ m³ = L×A (2)

Replacing (1) in (2)

1,88x10⁻⁵ m³ = 1,45x10⁶ m⁻¹A²

A = 3,6x10⁻⁶ m²

Thus, L = 5,22 m

Thus, length of copper wire is 5,22 m

As area = 3,6x10⁻⁶ m² = $\frac{pi}{4}d^{2}$ Where d is diameter.

The diameter of copper wire is 2,14x10⁻³m

I hope it helps!