Length Gage

A 30m long extension cord made from two #19 Gage copper wire. A)resistance of wire at 20°C?

b) replace copper wire w/ aluminum wire of same length. what is minimum diameter so that the new wire has a resistance no greater than the old?
(Resistivity: copper=1.67e-8, aluminum= 2.65e-8)
Resistance of wire: R=ρ (L/A) ; ρ = density of wire
Temp dependance: R=Ro[1+α(T-To)]

Although it’s true that ρ usually denotes density, in the context of conductor resistance, in which density plays no role at all, is used to symbolize resistivity. For circular wires, A = (π/4) d²; in consequence,

R = ρ L / A = 4 ρ L / π d².

Furthermore, the effective length of a 30 m long extension cord actually is twice the length of the extension, i.e., 60 m; an extension cord consists of two wires, equal both in length and size (diameter), so the effective length of such a line is twice the physical lenght of the line. The formula becomes

R = 8 ρ L / π d².

On the other hand, diameter of a 19 AWG wire is 35.89 mil (thousandths of an inch), equivalent to 9.116E-4 m.

a) 1.673E-8 Ω·m is the resistivity of 99.999% pure copper at 20 °C. Using this value,

R = 8 × 1.673E-8 × 30 / (9.116E-4)² π = 1.538 Ω.

b) Solving for d in the last equation,

d = √ (8 ρ L / π R).

Likewise, 2.655E-8 Ω·m is the resistivity of 99.96% pure aluminum, at 20 °C. Thus,

d = √ ( 8 × 2.655E-8 × 30 / 1.538 π ) = 1.1484E-3 m = 1.1484 mm.

This nearly equals 45.21 mil, whose closest AWG equivalent is 17 ga. wire (45.26 mil dia.). Since this wire is a little larger than strictly necessary, we can be sure its resistance will not be greater than that of the old wire.

Additional notes: while this computations are correct from the mathematical standpoint, they are not very much realistic. Commercial (international standard-annealed) grade copper is only 99.91% pure, and its resistivity at 20 °C is 1.724E-8 Ω·m. Similarly, commercial grade aluminum (AIEE standard hard-drawn, the grade used to manufacture wire) is 99.5% pure, exhibiting a resistance at 20 °C of 2.828E-8 Ω·m. Both these are never used as pure metals for making wire, because their very poor tensile strength. Using real figures,

a) 1.585 Ω.

b) 1.0488 mm, or 41.29 mil; while geometrically the nearest size is 18 ga., this wire is only 40.30 mil dia., so, for ensuring a lower resistance than 1.585 Ω, 17 ga. aluminum wire should be used, as before.

A Tech Authority – HEIDENHAIN Certo Length Gauge Kit