The general rule is resistivity increases with increasing temperature in conductors and decreases with increasing temperature in insulators. Unfortunately there is no simple mathematical function to describe these relationships.
The temperature dependence of resistivity (or its reciprocal, conductivity) can only be truly understood with quantum mechanics. In the same way that matter is an assembly of microscopic particles called atoms and a beam of light is a stream of microscopic particles called photons, thermal vibrations in a solid are a swarm of microscopic particles called phonons. The electrons are trying to drift toward the positive terminal of the battery, but the phonons keep crashing into them. The random direction of these collisions disturbs the attempted organized motion of the electrons against the electric field. The deflection or scattering of electrons with phonons is one source of resistance. As temperature rises, the number of phonons increases and with it the likelihood that the electrons and phonons will collide. Thus when temperature goes up, resistance goes up.
For some materials, resistivity is a linear function of temperature.
ρ = ρ0(1 + α(T − T0))
The resistivity of a conductor increases with temperature. In the case of copper, the relationship between resistivity and temperature is approximately linear over a wide range of temperatures.
For other materials, a power relationship works better.
ρ = ρ0(T/T0)μ
The resistivity of a conductor increases with temperature. In the case of tungsten, the relationship between resistivity and temperature is best described by a power relationship.
see also: superconductivity
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