Resistivity of materials and its temperature dependence

By resistivity, materials fall into conductors (metals, low ρ), semiconductors (intermediate ρ) and insulators (very high ρ).

For metals over a limited range, ρ_T = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient of resistivity; α is positive for metals.

Metals become more resistive when heated mainly because the relaxation time τ falls (more frequent collisions); n hardly changes.

Alloys such as nichrome, manganin and constantan have high resistivity and very small temperature coefficient, so they are used in heating elements and standard resistors.

For semiconductors resistivity decreases with rising temperature (negative α), because the number density of charge carriers n increases with temperature.

For insulators and semiconductors, the rise in n with temperature outweighs the fall in τ, which is why their behaviour is opposite to metals.

In problems, ignore the change in the wire's dimensions on heating unless told otherwise; then R changes in the same ratio as ρ.