Temperature dependence of resistivity

The variation of resistivity of a metallic conductor, an alloy, a semiconductor, or an insulator with an increase in temperature is not the same in all cases.

Temperature dependence of resistivity of a metallic conductor

In terms of relaxation time, the resistivity of the material of a metallic conductor is given by: ρ = (m)/(ne2τ)

Here, Here, m = mass of an electron, n = electron density or number of electrons per unit volume, e = charge of an electron, and τ = average relaxation time of the free electrons.

This equation of resistivity shows that the resistivity of a metallic conductor is inversely proportional to the average relaxation time τ of the free electrons in the conductor. This means, ρ α (1/τ)

If the temperature of the metallic conductor increases, the amplitude of the vibrations of the positive ions in the conductor also increases. Due to this, the free electrons collide more frequently with the vibrating ions and as a result, the average relaxation time τ of the free electrons also decreases.

Since, ρ α (1/τ), the resistivity of the material of the metallic conductor increases with an increase in temperature.

For pure conductors such as copper, the resistivity increases linearly with temperature in the temperature range around and above the room temperature.

If ρ0 and ρt are resistivities of the material of the metallic conductor at 0° centigrade and t° centigrade respectively, then it is found that ρt = ρ0 ( 1 + α t), where α is the temperature coefficient of resistivity.

As the temperature of the metallic conductor increases, the average relaxation time τ of the free electrons in the conductor decreases. As resistivity ρ is inversely proportional to the average relaxation time τ, hence, resistivity increases with the rise in temperature in case of metallic conductor.

Temperature coefficient of resistance

Since resistance R is directly proportional to the resistivity ρ, hence we can find that:

Rt = R0 ( 1 + α t), where α is the temperature coefficient of resistance of the conductor. Rt and R0 represent the resistance of the conductor at 0° centigrade and t° centigrade respectively.

Temperature coefficient of resistance of the conductor α = (RtR0) / (R0 t)

For metals, the value of the Temperature coefficient of resistance α is positive and lies between 10-2 to 10-4 °C-1.

Graph of resistivity vs Temperature change for a metallic conductor

The resistivity of a metal conductor increases linearly with an increase in temperature and if a graph is plotted between resistivity (ρ) and Temperature difference, then the graph will be a straight line.

However, at a temperature much below 0° C, the graph deviates considerably from the straight-line graph.

Temperature dependence of resistivity of a semiconductor

The resistivity of a semiconductor decreases exponentially with temperature.

Summary

The resistivity of conductors increases with an increase in temperature.
And the resistivity of semiconductors decreases with an increase in temperature.

Temperature dependence of resistivity
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