To achieve a resistance of 9.0 Ω with 1.0 g of copper (density 8900 kg/m^3), what length should the wire have?

• A. 0.775 m
• B. 77.5 m
• C. 7.75 m
• D. 0.0775 m

Answer: (C)

Resistance of a wire depends on the material’s resistivity and its geometry. The resistance formula uses resistivity, not density: R = ρ_res L / A. The cross-sectional area A relates to the volume and length by V = A L. With a fixed mass, you get V from the density: V = m / ρ_density. Combine these to eliminate A: R = ρ_res L^2 / V, so L = sqrt(R V / ρ_res).

Plug in the numbers: mass is 1.0 g = 0.001 kg, density is 8900 kg/m^3, so the volume is V = 0.001 / 8900 ≈ 1.12 × 10^-7 m^3. The resistivity of copper is about 1.68 × 10^-8 Ω·m. With a target resistance of 9.0 Ω, L ≈ sqrt( (9.0 × 1.12 × 10^-7) / (1.68 × 10^-8) ) ≈ sqrt(60) ≈ 7.75 m.

So the wire should be about 7.75 meters long. If you mistakenly used density in place of resistivity, you’d get a vastly different length, illustrating why the resistivity value is essential here.