Intermediate · 12 min
Wire resistance and resistivity
Separate a wire's resistance from the material property that helps determine it.
Material and geometry both matter
For a uniform wire, resistance is R = ρ × L ÷ A, where ρ is the material's resistivity, L is length, and A is cross-sectional area. Resistivity is measured in Ω·m and depends on temperature as well as material. At fixed temperature, doubling length doubles resistance; doubling area halves it. The symbol ρ here means resistivity, not mass density.
Worked example
A model material has resistivity 2 × 10⁻⁸ Ω·m. A wire 1 m long with area 1 × 10⁻⁶ m² has resistance 0.02 Ω.
Area is not diameter
A circular wire's cross-sectional area is proportional to the square of its diameter. Doubling diameter multiplies area by four, so resistance becomes one quarter for the same material, length, and temperature. Keep units consistent. An area of 1 mm² equals 10⁻⁶ m². Do not substitute a diameter directly where the formula requires an area.
Worked example
A wire has resistance 12 Ω. A second wire of the same material and length at the same temperature has twice its diameter. Second resistance = 12 ÷ 4 = 3 Ω
Compare wire designs
- For a wire with resistance 10 Ω, predict the resistance of an otherwise identical wire with twice the length.
- Predict the resistance if both length and cross-sectional area are doubled instead. Explain which geometric changes cancel in R = ρ × L ÷ A.