Temperature Dependence of Resistivity

The resistivity of a material is temperature dependent. The variation in resistivity for different materials is also different. For example, the resistivity of metals, semiconductors and superconductors varies as shown in the graphs below.

Let a conductor has resistance R₀, at the temperature T₀ = 0°C

Then the resistance of the conductor at temperature T°C be R_t

Then the change in temperature ΔT = T°C – T₀°C

Change in resistance ΔR = R_t – R₀

The change in resistance ΔR depends upon the following factors

Directly on the initial resistance
Directly on raise of temperature
On the nature of material by which the conductor is made up of.

R_t – R₀ ∝ R₀ × ΔT

R_t – R₀ = α R₀ΔT

α represents temperature coefficient of resistivity

R_t = R₀ + α × R₀ΔT

∴ R_t = R₀[1 + α(T - T₀)]

But R_t = ρ_t l/A and R₀ = ρ₀l/A

Substitute the R_t and R₀ in the R_t equation

∴ ρ _t l/A	=	ρ₀l/A[1 + α(T - T₀)]
ρ_t	=	ρ₀[1 + α(T - T₀)]