Resistivity, geometry, and temperature

In a long straight conductor carrying a steady current the field is uniform along the length, so V = EL, and the current spreads evenly across the section, so I = JA. Put both into the point form E = ρJ: V/L = ρI/A, which rearranges to V = (ρL/A)I. The bracket is the resistance, R = ρL/A, and that split is the whole point of the topic — ρ carries everything the material contributes, L/A everything the shape contributes. Double the length and R doubles, because the same current is pushed through twice as much scattering; double the area and R halves, because the charge has two lanes instead of one. A 25 m run of 1.50 mm² copper gives R = (1.68 × 10⁻⁸ Ω m)(25 m)/(1.50 × 10⁻⁶ m²) = 0.28 Ω; the same copper drawn out to 0.10 mm² over the same 25 m gives 4.2 Ω. Nothing about the copper itself changed.

R = ρL/A assumes A is the same everywhere along the current path. When it is not, cut the conductor into slices of thickness dx perpendicular to the current. Each slice is a resistor of value ρ dx/A(x), and slices in series add, so R = ∫₀^L ρ dx/A(x). For a cone whose radius tapers from a to b over a length L the integral gives R = ρL/(π a b): the effective area is the geometric mean of the two end areas, never their average, because 1/A weights the narrow end most heavily. The same integral settles a practical question. Draw a wire out to twice its length and its volume is unchanged, so A halves and R rises by a factor of four, not two. Stretch it by 1.0% and R rises by 2.0%, which is how a strain gauge turns a deformation into a measurable resistance change.

Two numbers, and only two, are open to a material: how many carriers it has, and how long each one survives between collisions. Between collisions an electron accelerates at a = eE/m, and each collision wipes out what it gained, so the drift it keeps is v_d = eEτ/m with τ the mean free time; with n carriers per cubic metre, J = n e v_d = (n e² τ/m)E, and matching that against E = ρJ gives ρ = m/(n e² τ). Copper has n = 8.5 × 10²⁸ m⁻³, roughly one conduction electron per atom, so ρ = 1.68 × 10⁻⁸ Ω m demands τ = m/(n e² ρ) = 2.5 × 10⁻¹⁴ s. Its electrons move at about 1.6 × 10⁶ m s⁻¹, a speed set by quantum statistics rather than by temperature, so the mean free path λ = v̄τ is about 40 nm — over a hundred atomic spacings through a lattice packed with ions, because a perfectly periodic, motionless lattice does not scatter at all. Which of the two numbers dominates a comparison? Pure silicon has n ≈ 10¹⁶ m⁻³, 8.5 × 10¹² times fewer than copper, and a resistivity 1.4 × 10¹¹ times larger: almost the whole gap is carrier count.

In a metal n is fixed: each atom donates its conduction electron whatever the temperature. All of the temperature dependence therefore lives in τ, and τ is set by whatever spoils the perfect periodicity of the lattice — thermal vibration, plus impurities and defects. The mean square displacement of a vibrating ion grows in proportion to T over the range that matters here, and the scattering rate grows with it, so 1/τ ∝ T and ρ climbs roughly in proportion to absolute temperature. Impurities are indifferent to T. The two scattering rates simply add, which is Matthiessen's rule, ρ(T) = ρᵣₑₛ + ρ_lattice(T). Cooling removes only the second term, so ρ flattens onto a residual floor fixed by how dirty the sample is rather than falling to zero. The ratio ρ(293 K)/ρᵣₑₛ is the standard purity measure: near 50 for commercial copper, past 1000 for a pure single crystal.

Define α as the fractional change of resistivity per kelvin at a reference temperature: α = (1/ρ₀)(dρ/dT) at T₀. Keeping only the first two terms of the Taylor series about T₀ gives ρ = ρ₀[1 + α(T − T₀)], and because the geometry factor L/A barely moves, the same bracket multiplies the resistance: R = R₀[1 + α(T − T₀)]. Copper has α = 3.9 × 10⁻³ K⁻¹ at 20 °C, so a winding that warms by 60 K gains 23% in resistance. The microscopic picture even predicts the size of α: if ρ were exactly proportional to absolute temperature then α would be exactly 1/T, which at 293 K is 3.4 × 10⁻³ K⁻¹ — close enough to the measured 3.9 × 10⁻³ K⁻¹ to confirm the mechanism, and different enough to show why α must be quoted with its reference temperature. Stretch the range and the truncation tells: tungsten with α = 4.5 × 10⁻³ K⁻¹ predicts 12.6 times the cold resistance at 2600 °C, where a real filament measures about 16.

ρ = m/(n e² τ) allows n to vary as well. In a semiconductor the carriers must be excited across an energy gap, so n climbs steeply with temperature — roughly as e^{(−E_g/(2k_B T))} — and that increase swamps the falling τ. Silicon's resistivity drops by orders of magnitude between room temperature and 200 °C, α is negative and large, and no single α describes it, because the fractional change per kelvin is itself strongly temperature dependent; an NTC thermistor sells exactly that behaviour as a sensor. Even for a metal, R = ρL/A is a steady-state, uniform-current model: it assumes the current is spread evenly over A, which holds down a long DC wire but fails near a contact and fails at high frequency, where the current crowds into a thin surface layer. Thermal expansion does change L and A, but at 1.7 × 10⁻⁵ K⁻¹ it is some 200 times weaker than α, so over 100 K it moves R by 0.17% where ρ moves it by 39%.