Resistivity of materials and temperature
When Ohm’s law fails, and what things are made of
Ohm’s law, V equals I R, works beautifully for a metal wire. But it is not a law of nature like Newton’s laws. Many devices simply do not obey it.
Here is the first kind of failure. Plot I against V for a filament bulb. At first the graph is straight, but then it bends towards the V axis. V stops being proportional to I.
The reason: the filament heats up as the current grows, and a hotter filament has a larger resistance. So V over I keeps rising.
The second kind: the relation depends on the sign of V. Connect a diode one way and a current flows; reverse the battery, keep the same size of V, and almost no current flows at all.
The third kind: the relation between V and I is not unique. For gallium arsenide, the curve rises, falls, then rises again. For one value of current, there can be more than one voltage.
Materials and devices like these are called non-ohmic. They are not mistakes; diodes and other semiconductor devices are useful precisely because they behave this way.
Now compare ohmic materials by their resistivity. Picture a ladder of resistivity on a scale that goes up by powers of ten.
At the bottom are the metals, conductors, with ρ from about 10⁻⁸ to 10⁻⁶ ohm metre. Silver and copper are the best, around 1.6 to 1.7 × 10⁻⁸.
At the top are insulators like glass, rubber and quartz: 10¹⁰ up to 10¹⁶ ohm metre and more. That is more than 10²² times the resistivity of copper.
In between are semiconductors such as germanium and silicon. Their resistivity falls as they get hotter, and it can be changed hugely by adding a tiny amount of impurity.
Resistivity and temperature
Heat a copper wire and measure its resistivity. Over a wide range of temperature, the graph of ρ against T is very nearly a straight line.
So we write ρ T equals ρ₀ times, 1 plus α, T minus T₀. Here ρ₀ is the resistivity at a reference temperature T₀, and α is the temperature coefficient of resistivity.
α is the fractional change in resistivity per degree. For copper it is about 0.0068 per degree Celsius, so 100 degrees of heating raises ρ by about 68 percent.
Why does a metal resist more when hot? Recall ρ equals m over n e² τ. The ions vibrate harder, the electrons collide more often, and the time τ between collisions gets shorter.
Alloys like nichrome, manganin and constantan behave differently. Their resistivity is high, and it hardly changes with temperature. Manganin’s α is only about 2 × 10⁻⁶ per degree.
That is why these alloys are used to make standard resistors: their value stays the same on a cold morning and a hot afternoon.
Semiconductors go the other way: their resistivity falls as T rises. In ρ equals m over n e² τ, heating sets many more charge carriers free, so n grows faster than τ shrinks.
Worked example. A toaster’s nichrome element has a resistance of 75.3 ohm at 27 degrees Celsius. On a 230 volt supply, the current settles at 2.68 ampere.
Hot resistance: R equals 230 over 2.68, which is 85.8 ohm. With α equal to 1.70 × 10⁻⁴ per degree, R T equals R₀, 1 plus α, T minus T₀.
So T minus 27 equals 85.8 minus 75.3, divided by 75.3 times 1.70 × 10⁻⁴. That is 10.5 over 0.0128, about 820 degrees. The element runs at about 847 degrees Celsius.
Try it: ohmic or not?
- Sweep the voltage from 0 to 6 V for the filament bulb. The curve bends towards the V axis, and the readout V/I keeps growing.
- Switch to the 10 Ω resistor: the curve is a straight line through the origin, and V/I never changes.
- Try the diode with negative V. Reverse the battery and almost no current flows.
- For the GaAs device, find two different voltages that give the same current.
Try it: heat it up
- Raise the temperature of copper and watch ρ/ρ₀ climb along a straight line: ρ_T = ρ₀[1 + α(T − T₀)].
- Overlay nichrome and manganin. Their lines are almost flat: good for standard resistors.
- Pick silicon. Its resistivity falls steeply as it gets hotter (the graph switches to a log scale).
- Open the resistivity ladder to see conductors, semiconductors and insulators across 24 powers of ten.
A platinum thermometer
A platinum resistance thermometer has a resistance of 5.00 Ω at the ice point (0 °C) and 5.23 Ω at the steam point (100 °C). In a hot bath its resistance is 5.795 Ω. What is the temperature of the bath?
- 79.5 °C
- 245.65 °C
- 345.65 °C
- 2520 °C
Answer
C. First α = (R₁₀₀ − R₀)/(R₀ × 100) = 0.23/(5 × 100) = 0.00046 °C⁻¹. Then T = (R_T − R₀)/(R₀ α) = 0.795/(5 × 0.00046) = 0.795/0.0023 ≈ 345.65 °C. Check: the rise of 0.795 Ω is 0.795/0.23 = 3.4565 times the 100-degree rise. 245.65 °C measures from the steam point and forgets the first 100 °C; 2520 °C uses R_T instead of R_T − R₀; 79.5 °C assumes 1 Ω per 100 °C.
Hotter: more or less resistance?
A copper wire and a piece of pure silicon are both heated from 20 °C to 100 °C. What happens to their resistivities?
- Both increase, because the ions vibrate more
- Copper’s increases; silicon’s decreases
- Copper’s decreases; silicon’s increases
- Neither changes noticeably
Answer
B. Use ρ = m/(ne²τ). In copper, n is fixed and τ falls as the ions vibrate harder, so ρ rises (α ≈ 0.0068 °C⁻¹). In silicon, heating frees many more charge carriers, so n grows much faster than τ falls, and ρ drops sharply. Almost no change is what alloys like manganin show, not copper.
Summary
- Temperature dependence:
ρ_T = ρ₀[1 + α(T − T₀)] - Temperature coefficient:
α = (ρ_T − ρ₀) / [ρ₀(T − T₀)] - Resistivity from collisions:
ρ = m / (n e² τ)
- Ohm’s law fails when V is not proportional to I (bulb), when the relation depends on the sign of V (diode), or when V is not a unique function of I (GaAs).
- Resistivity spans about 24 powers of ten: metals 10⁻⁸–10⁻⁶ Ω m, semiconductors in between, insulators 10¹⁰–10¹⁶ Ω m and more.
- In metals ρ rises with T because τ falls. In semiconductors ρ falls with T because n rises.
- Alloys like nichrome, manganin and constantan have high resistivity that hardly changes with temperature, so they are used for standard resistors and heating elements.