Temperature Dependence of Resistivity Explained Simply

Heat a copper coil and its resistance rises. Heat a semiconductor and its resistance falls, often sharply. Resistivity always responds to temperature, but the sign of the response reveals the material class. Far from a nuisance, this effect is the working principle of resistance thermometers and thermistors used everywhere.

The physics is a contest. In metals the carrier count is fixed, but atoms vibrate harder when hot, so electrons collide more and drift less. In semiconductors, heating frees far more carriers, and their increase overwhelms the extra collisions, so conduction improves.

Key Concepts

• Coefficient alpha: Fractional change of resistivity per degree. Unit per degree Celsius. Copper about 0.00393, aluminium about 0.00429, constantan near 0.00002 (hence standard resistors).
• Metals (positive alpha): Resistivity rises roughly linearly over moderate ranges due to lattice vibration scattering.
• Semiconductors (negative alpha): Resistivity falls steeply, roughly exponentially, as thermal energy creates carriers.
• Superconductivity: Below a critical temperature resistivity vanishes entirely, e.g. mercury at 4.2 K, found by Kamerlingh Onnes in 1911.

Formulas

• Metal (linear approx): R at T equals R0 [1 plus alpha (T minus T0)], likewise for rho. R0 is the value at reference T0, often 20 degrees C. Valid for moderate ranges only.
• Finding alpha: Alpha equals (R minus R0) divided by [R0 (T minus T0)] from two measurements.

Worked Example

A copper coil is 10.0 ohms at 20 degrees C, alpha equals 0.00393 per degree C. Find R at 70 degrees C, and the temperature when R reads 15.71 ohms.

1. Delta T equals 50. R equals 10.0 x [1 plus 0.00393 x 50] equals 10.0 x 1.1965 equals 11.97 ohms, about 12.0 ohms.
2. Reverse: 15.71 equals 10.0 [1 plus 0.00393 delta T], so 0.571 equals 0.00393 delta T.
3. Delta T equals 0.571 divided by 0.00393 equals 145 degrees, so T equals 165 degrees C. This is exactly how platinum thermometers infer temperature.

Common Misconceptions

• Misconception: All resistances increase with temperature. Correction: Only metals do. Semiconductors, carbon, insulators and electrolytes decrease.
• Misconception: Metals resist less when hot because electrons move faster. Correction: Random speed does not aid drift; more vibration means more collisions and higher resistivity.
• Misconception: The linear formula holds everywhere. Correction: At very low temperatures resistivity flattens to a residual value, and superconductors drop to zero, breaking the line completely.

Real-World Applications

• RTD thermometers: Platinum sensors use stable alpha for precise industrial temperature measurement.
• Thermistors: Semiconductor beads with large negative alpha sense temperature in AC units, chargers and medical probes.
• Filaments: A tungsten bulbs hot resistance is many times its cold value, limiting switch-on surge.
• Power grids: Hot conductors resist and lose more; summer load planning accounts for this.

Metals resist more when hot because collisions win; semiconductors resist less because new carriers win. R equals R0 [1 plus alpha delta T] captures the metal side and turns a plain coil into a thermometer.