B.5.12—Resistance and resistivity

Resistance of a uniform conductor

$$R=\rho\frac{L}{A}$$

where ρ is resistivity, L is length and A is cross-sectional area. Resistivity is a material property at the stated conditions.

Read the scaling

At fixed material, doubling length doubles R. Doubling diameter makes area four times larger and reduces R to one quarter. A longer, thinner wire has greater resistance.

Units

Resistivity has SI unit Ω m. Use $$A=\pi r^2$$ for a circular wire and convert radius/diameter to metres before calculating.

Worked example from the mapped local textbook

Nichrome has $\rho=1.1\times10^{-6}\,\Omega\,\mathrm m$. A wire has $L=1.96\,\mathrm m$ and radius $r=0.21\,\mathrm{mm}=0.21\times10^{-3}\,\mathrm m$.

$$A=\pi r^2=1.39\times10^{-7}\,\mathrm{m^2}$$

$$R=\frac{\rho L}{A}=\frac{(1.1\times10^{-6})(1.96)}{1.39\times10^{-7}}=16\,\Omega$$

Converting the radius before squaring prevents a factor-of-$10^6$ error.

Common trap

Do not treat resistivity as the resistance of every sample of a material. Geometry changes resistance even when ρ is unchanged.