Temperature Dependence of Resistivity

Resistance is not a fixed property of a material — it changes with temperature. In most metals, resistance rises as the material heats up. Think of atoms in a metal as a crowd of people standing in a corridor; electrons are trying to walk through them. At low temperatures the crowd is still and orderly, so electrons pass through easily. As temperature rises, the atoms vibrate more vigorously, constantly bumping into electrons and disrupting their flow. This increased scattering raises the material's resistivity, and therefore its resistance. The relationship between resistivity and temperature is approximately linear over moderate temperature ranges. The physical property that quantifies how strongly a material responds to temperature change is called the temperature coefficient of resistivity, denoted by α (alpha). Metals such as copper and aluminium have positive values of α — their resistance increases with temperature. This is why the filament in a traditional light bulb has a much higher resistance when hot than when cold; designers must account for this when specifying circuits. Not all materials behave this way. Semiconductors and insulators typically show the opposite behaviour — their resistivity decreases as temperature rises, giving them a negative temperature coefficient. Thermistors exploit this property to measure temperature electronically. Understanding how temperature affects resistance is essential for designing safe, reliable electrical circuits in everything from household wiring to aerospace electronics.

Key takeaways
  • Resistivity as a material property
  • The microscopic picture
  • The linear approximation
  • Positive vs negative temperature coefficients
  • Practical consequences in circuits

Resistivity as a material property

Resistivity (ρ) is an intrinsic property of a material that describes how strongly it opposes the flow of electric current, independent of shape or size. Resistance R of a sample depends on ρ, its length L, and cross-sectional area A via R = ρL/A. When we say resistivity changes with temperature, we mean the material itself becomes more or less conductive — every component made from that material is affected simultaneously.

The microscopic picture

In a metallic conductor, current is carried by free electrons drifting through a lattice of positive metal ions. As temperature increases, thermal energy causes the ions to vibrate with greater amplitude. These vibrating ions scatter conduction electrons more frequently, reducing the average time between collisions (the mean free time, τ). Since resistivity is inversely proportional to τ, more scattering means higher resistivity. This is the fundamental reason metals have positive temperature coefficients.

The linear approximation

For most metals over a temperature range of roughly 0 °C to several hundred degrees Celsius, resistivity varies linearly with temperature. If ρ₀ is the resistivity at reference temperature T₀ (commonly 20 °C), then at temperature T the resistivity is ρ = ρ₀[1 + α(T − T₀)]. The same formula applies to resistance directly: R = R₀[1 + α(T − T₀)], because R is simply proportional to ρ for a component with fixed geometry.

Positive vs negative temperature coefficients

Metals like copper (α ≈ +3.9 × 10⁻³ K⁻¹) have positive α — resistance rises with temperature. Semiconductors like silicon have negative α because rising temperature liberates more charge carriers, dramatically increasing conductivity. Carbon also has a slight negative α. NTC (Negative Temperature Coefficient) thermistors exploit this dramatically: a small rise in temperature causes a large drop in resistance, making them ideal sensors. PTC (Positive Temperature Coefficient) devices are used as self-resetting fuses — they become highly resistive when overheated, automatically limiting current.

Practical consequences in circuits

Temperature-dependent resistance has real engineering implications. Copper cables carrying large currents warm up, increasing their resistance and causing further heating — a potential runaway effect. Incandescent bulb filaments (tungsten, α ≈ +4.5 × 10⁻³ K⁻¹) have cold resistances roughly 10× lower than their operating resistance at ~2700 °C, causing a large inrush current when first switched on. Precision resistors are made from alloys such as Nichrome or Manganin, which are specifically chosen for their very low α, ensuring stable resistance across a wide temperature range.

The Key Equation

ρ=ρ0[1+α(T−T0)]\rho = \rho_0\left[1 + \alpha(T - T_0)\right]
  • \rho — Resistivity at temperature T (Ω·m): The resistivity of the material at the temperature of interest.
  • \rho_0 — Reference resistivity (Ω·m): The resistivity of the material at the reference temperature T₀ (often 20 °C).
  • \alpha — Temperature coefficient of resistivity (K⁻¹): A material constant indicating how strongly resistivity changes per kelvin of temperature change. Positive for metals, negative for semiconductors.
  • T — Temperature (°C or K): The temperature at which resistivity is being calculated.
  • T_0 — Reference temperature (°C or K): The temperature at which ρ₀ was measured, typically 20 °C. Note: T and T₀ must be in the same units.

Worked Example

Problem. A copper wire has a resistance of 4.00 Ω at 20 °C. The temperature coefficient of resistivity for copper is α = 3.9 × 10⁻³ °C⁻¹. Calculate the resistance of the wire when it reaches an operating temperature of 80 °C.

R=R0[1+αΔT]=4.00×[1+(3.9×10−3)(60)]≈4.94 ΩR = R_0[1 + \alpha\Delta T] = 4.00\times[1+(3.9\times10^{-3})(60)] \approx 4.94\,\Omega

Answer. The resistance of the copper wire at 80 °C is approximately 4.94 Ω, an increase of about 23% compared to its room-temperature value.

Figure
Diagram
Resistivity vs Temperature for Metals & SemiconductorsTemperature (°C)Resistivity (ρ)01230100200300400Metal (Cu)α > 0Semiconductorα < 0ρ₀ρ risesAt T₀ = reference tempboth cross ρ₀
Origins · 1 minGustav Kirchhoff and how this was discovered
GK
Gustav Kirchhoff
1824–1887 · German

Gustav Kirchhoff made foundational contributions to circuit theory, formulating his famous circuit laws (Kirchhoff's Current Law and Kirchhoff's Voltage Law) which remain central to electrical circuit analysis. Working alongside Robert Bunsen, he also pioneered spectroscopy. His early circuit work built directly on Ohm's empirical findings and helped formalise the understanding of how resistance and conductance behave in connected circuits, including the temperature-dependent behaviour studied by his contemporaries. Kirchhoff and Bunsen used their newly invented spectroscope to discover two new elements — caesium and rubidium — in 1860 and 1861, simply by observing the colours of light emitted by mineral samples heated in a flame.

  • Resistance & Resistivity — Temperature dependence is a direct extension of understanding what resistivity is and how it defines a material's opposition to current flow.
  • Ohm's Law — Ohm's Law (V = IR) is the framework used to apply resistance values in circuits; temperature changes resistance and therefore directly affect how Ohm's Law is used in practice.
  • Electrical Power & Heating — Current through a resistor generates heat (P = I²R), which in turn raises temperature and changes resistance — creating a feedback loop that is central to understanding heating elements and fuses.
  • Series & Parallel Circuits — In real circuits, components heat up during operation; knowing how temperature alters individual resistances is essential for accurately analysing series and parallel circuit behaviour.