Resistivity and Temperature Dependence of Resistance
The resistance of a uniform conductor is R = ρL ÷ A, where ρ is the resistivity of the material (in Ω·m), L is the length and A the cross-sectional area; in metals resistance increases with temperature, R = R0[1 + α(T − T0)], while in semiconductors such as thermistors it usually decreases. In brief: doubling the length doubles R, and doubling the diameter reduces R to a quarter. Copper has a resistivity of 1.68 × 10−8 Ω·m and is a good conductor; glass has about 1010 to 1014 Ω·m. Some materials become superconductors, with zero resistance, below a critical temperature.
The choice of wire for a power cable, the filament of a lamp and the working of a temperature sensor all depend on resistivity and its change with temperature. This guide gives the formula for the resistance of a wire, explains the temperature effects in metals, semiconductors and superconductors, compares materials in a table, works through numerical examples and corrects common mistakes.
What is resistivity?
Resistance depends on the size of a component and on the material. For a uniform wire, resistance is proportional to its length L (more collisions with the lattice as the charge carriers travel farther) and inversely proportional to its cross-sectional area A (a wider path for charges): R = ρL ÷ A. The resistivity ρ is a property of the material at a given temperature, measured in ohm-meters (Ω·m). Its reciprocal is the conductivity σ = 1 ÷ ρ. For a wire of diameter d, A = πd2 ÷ 4, so R is inversely proportional to the square of the diameter. This is why thick cables are used for large currents (a lower resistance gives less heating and voltage drop), and why the filament of a lamp is a long, thin wire (high resistance).
How does temperature affect resistance?
- Metals: the resistance rises with temperature, because the ions of the lattice vibrate more and collide more often with the free electrons. For moderate changes, R = R0[1 + α(T − T0)], where α is the temperature coefficient of resistance (for copper, about 0.0039 K−1). The resistance of a tungsten filament at 2,500 °C is about 12 times its resistance at room temperature, which is why the V–I graph of a filament lamp curves, and why a bulb is most likely to fail at switch-on, when the cold filament has a low resistance and the current surge is highest.
- Semiconductors and insulators: the resistance falls as temperature rises, because more charge carriers are freed. An NTC thermistor (negative temperature coefficient) is used in temperature sensors and in thermostats; its resistance can fall from several kΩ to hundreds of Ω over a modest range of temperature.
- Superconductors: below a critical temperature (for example 9.2 K for niobium, 92 K for YBCO ceramics), the resistance drops to zero, allowing currents without loss, used in MRI magnets and particle accelerators.
- Light-dependent resistors (LDR) have a resistance that falls with illumination.
Resistivities of materials compared
| Material | Resistivity (Ω·m) | Type |
|---|---|---|
| Silver | 1.59 × 10−8 | Conductor |
| Copper | 1.68 × 10−8 | Conductor |
| Aluminum | 2.65 × 10−8 | Conductor |
| Nichrome | 1.1 × 10−6 | Alloy for heaters |
| Silicon (pure) | about 2,300 | Semiconductor |
| Glass | 1010 to 1014 | Insulator |
Worked example: wire resistance and heating
- Copper wire. L = 10 m, diameter 1.0 mm. A = π(0.50 × 10−3)2 = 7.85 × 10−7 m2. R = ρL ÷ A = 1.68 × 10−8 × 10 ÷ 7.85 × 10−7 = 0.214 Ω.
- Changing dimensions. Doubling the length: 0.43 Ω. Doubling the diameter: area is four times, so R = 0.054 Ω.
- Heating element. A nichrome wire of diameter 0.50 mm (A = 1.96 × 10−7 m2) and length 5.0 m: R = 1.1 × 10−6 × 5.0 ÷ 1.96 × 10−7 = 28 Ω. On 230 V, the current is 8.2 A and the power is 1.9 kW.
- Temperature dependence. A copper coil has R = 100 Ω at 20 °C. At 70 °C: R = 100[1 + 0.0039 × 50] = 119.5 Ω, an increase of 19.5 percent. The change of resistance is used to find the temperature of a motor winding.
- Voltage drop. The 10 m of copper wire in item 1 carries 5.0 A: V = IR = 5.0 × 0.214 = 1.07 V lost in the wire, and the power loss is I2R = 25 × 0.214 = 5.4 W.
How were these effects discovered?
Georg Ohm found in 1827 that the resistance of a wire depends on its length and thickness. Heike Kamerlingh Onnes discovered superconductivity in mercury in 1911 by cooling it to 4.2 K with liquid helium. The thermistor was developed at Bell Labs in the 1940s. The 1986 discovery of high-temperature superconductors by Georg Bednorz and Alex Müller led to materials that superconduct above the boiling point of liquid nitrogen (77 K).
Which mistakes are common?
Mistake 1: Confusing resistance and resistivity
Resistivity is a property of the material. Resistance depends on the material and on the shape.
Mistake 2: Forgetting that area depends on the square of the diameter
Doubling the diameter divides the resistance by four, not by two.
Mistake 3: Mixing mm and m
Convert the diameter to meters before calculating the area in m2.
Mistake 4: Assuming that resistance always rises with temperature
It falls in semiconductors (NTC thermistors) and vanishes in superconductors.