Temperature dependence of resistance
In metals, resistivity rises with temperature as collisions get more frequent. In semiconductors it falls, because many more free charge carriers appear.
Why heating changes resistivity
Use ρ = m/(ne²τ). Heating can change two things: the relaxation time τ and the number density of free carriers n. In a metal, heating makes the ions vibrate more strongly. Electrons collide more often, so τ decreases. The number of free electrons hardly changes, so ρ increases. Over a limited range of temperature, ρ_T = ρ₀[1 + α(T − T₀)], with α positive. At very low temperatures the graph for copper is curved, not straight. In semiconductors and insulators, heating frees many more charge carriers, so n increases. This rise outweighs the fall in τ, so ρ decreases and α is negative. Some alloys, such as manganin and constantan, have a resistivity that changes very little with temperature. They are used for standard resistors.
Effect of heating on different materials
| Material | Examples | On heating, ρ ... | Sign of α |
|---|---|---|---|
| Metals | Copper, aluminium, tungsten | Increases | Positive |
| Alloys | Nichrome, manganin, constantan | Changes very little | Very small |
| Semiconductors | Silicon, germanium | Decreases sharply | Negative |
| Insulators | Glass, rubber | Decreases | Negative |
How to draw the ρ against T graphs
- Draw ρ (Ω m) on the y-axis and T (K) on the x-axis.
- Copper: starts just above the T-axis near 0 K, curves upwards, then becomes nearly a straight line at higher temperatures. Label it 'copper'.
- Nichrome: a nearly straight line well above the T-axis, rising only gently. Label it 'nichrome'.
- Semiconductor: a curve that falls steeply at low T and then levels off towards the T-axis. Label it 'silicon'.
- Draw each material on its own small graph, as their ρ values differ by many powers of ten.
Worked example: resistance of a heated coil
A coil of wire has a resistance of 20 Ω at 20 °C. The temperature coefficient of its metal is 4.0 × 10⁻³ °C⁻¹. Find its resistance at 70 °C. (Ignore the small change in its length and area.)
- Given: R₀ = 20 Ω at T₀ = 20 °C, T = 70 °C, α = 4.0 × 10⁻³ °C⁻¹.
- For small changes in size, R follows the same law as ρ: R_T = R₀[1 + α(T − T₀)].
- α(T − T₀) = 4.0 × 10⁻³ °C⁻¹ × 50 °C = 0.20.
- R_T = 20 Ω × (1 + 0.20) = 20 Ω × 1.20 = 24 Ω.
24 Ω
Worked example: finding α from two readings
A metal wire used as a resistance thermometer has a resistance of 5.0 Ω in melting ice (0 °C) and 7.0 Ω in boiling water (100 °C). Find the temperature coefficient of resistance of the metal. What is its resistance at 50 °C?
- Formula: R_T = R₀[1 + α(T − T₀)], so α = (R_T − R₀)/(R₀(T − T₀)).
- α = (7.0 Ω − 5.0 Ω) ÷ (5.0 Ω × 100 °C) = 2.0 Ω ÷ 500 Ω °C = 4.0 × 10⁻³ °C⁻¹.
- At 50 °C: R = 5.0 Ω × (1 + 4.0 × 10⁻³ × 50) = 5.0 Ω × 1.2 = 6.0 Ω.
α = 4.0 × 10⁻³ °C⁻¹; R = 6.0 Ω at 50 °C
A tungsten filament glows at over 2000 °C, so its hot resistance is many times its cold resistance. Digital thermometers use a tiny sensor whose resistance changes with body temperature, while manganin keeps lab standard resistors steady through an Indian summer.
2 marks: “Explain why the resistivity of a metal increases with temperature, while that of a semiconductor decreases.” ρ = m/(ne²τ). Metal: n is nearly fixed; heating raises ion vibrations, so τ falls and ρ rises. Semiconductor: heating frees many carriers; the rise in n outweighs the fall in τ, so ρ falls.
1 marks: “Why is manganin used to make standard resistance coils?” Its resistance changes very little with temperature (very small α), and its resistivity is fairly high.
3 marks: “A wire has a resistance of 10 Ω at 20 °C and 12 Ω at 120 °C. Find α for its metal and its resistance at 220 °C.” α = (12 − 10)/(10 × 100) = 2.0 × 10⁻³ °C⁻¹. At 220 °C: R = 10[1 + 2.0 × 10⁻³ × 200] = 10 × 1.4 = 14 Ω.
Metals: hotter means harder (τ falls, ρ rises). Semiconductors: hotter means easier (n rises, ρ falls). Manganin hardly cares.