Resistivity & Temperature
Metals resist more when hot; semiconductors less; superconductors not at all.
Lesson at a glanceShow the key points
- ρ(T) = ρ(T₀)[1 + α(T − T₀)]; R follows the same rule.
- Resistivities: metals ~10⁻⁸ Ω m, insulators up to ~10¹⁷ Ω m, semiconductors between.
- Semiconductors have negative α: thermistors measure ~10⁻³ °C changes.
- Superconductors: zero resistance below a critical temperature (mercury 4.2 K, 1911).
Warm a metal and it resists more. Warm a semiconductor and it resists less. Cool certain materials far enough and resistance vanishes altogether.
Metals get worse when hot
Heating makes the lattice vibrate harder, so electrons collide more often. τ falls, the drift speed falls, and the resistivity rises. Over modest temperature ranges the rise is close to linear:
The length and area of a wire barely change with temperature, so the resistance follows the same rule: R(T) = R(T₀)[1 + α(T − T₀)].
| Material | ρ at room temperature (Ω m) | α (K⁻¹) |
|---|---|---|
| Silver | 1.47 × 10⁻⁸ | 0.0038 |
| Copper | 1.72 × 10⁻⁸ | 0.0039 |
| Aluminium | 2.63 × 10⁻⁸ | 0.0039 |
| Tungsten | 5.51 × 10⁻⁸ | 0.0045 |
| Iron | 9.71 × 10⁻⁸ | 0.0050 |
| Nichrome | 100 × 10⁻⁸ | 0.0004 |
| Germanium | 0.46 | −0.048 |
| Silicon | ≈ 10³ | −0.075 |
| Fused quartz | 7.5 × 10¹⁷ | – |
Resistivities span more than 25 powers of ten: metals around 10⁻⁸ Ω m are good conductors, materials like fused quartz are insulators, and silicon and germanium sit in between: semiconductors.
A copper coil has a resistance of 20.0 Ω at 20 °C. What is its resistance at 80 °C?
A tungsten filament has 30 Ω at 20 °C and 360 Ω when glowing. Estimate its temperature using the linear rule.
Thermistors
In semiconductors α is negative and often large: warming them frees more charge carriers, and the resistivity drops. (The simple free-electron picture doesn’t apply here.) A small bead of semiconducting metal oxide (of nickel, iron, cobalt, copper…) sealed in epoxy is a thermistor. Put it in a circuit with a cell: as the temperature rises its resistance falls and the current climbs. Thermistors can detect changes of about 10⁻³ °C.
Superconductors
Some materials lose their resistance completely below a critical temperature. Kamerlingh Onnes found this in mercury in 1911 (Tc = 4.2 K). Above Tc the material behaves like a normal metal. A current started in a superconducting loop keeps flowing for years with no emf.
- 1Metals: ρ(T) = ρ(T₀)[1 + α(T − T₀)], α > 0; R follows the same law.
- 2Conductors ~10⁻⁸ Ω m, insulators up to ~10¹⁷ Ω m, semiconductors in between.
- 3Semiconductors: α < 0, large → thermistors.
- 4Superconductors: ρ = 0 below T_c (mercury 4.2 K).
Check yourself
Keep exploring
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