The Temperature Dependence of Resistance

Resistance is fundamentally the ability of the material to restrict the passage of electric current. Thus, just as material properties such as density, size, magnetization etc. change with temperature, so does resistance. Resistance is really a bulk property; the resistivity on the other hand is a material property. We shall talk about the Temperature dependence of resistivity in this article.

Resistivity of metals

The resistivity (ρ) of metals is dependent on the relaxation time (τ) of the free electrons in the metal as

ρ ∝ \( \frac {1}{τ}\)

(See our article on drift velocity)

The relaxation time is dependent on the mean free path (λ) and the root mean square speed \( (v_{rms})\) of the electron as:

τ = \( \frac{v_{rms}}{λ}\)

ρ ∝ \( \frac {1}{τ}\) ∝ \( \frac{λ}{v_rms} \)

Since the root mean square speed increases with temperature and mean free path, the resistivity and hence the resistance of the metal also rises.

This is mostly linear and is given as:

\( ρ_t \) = \( ρ_0 (1~+~αt)\)

Where, \( ρ_t \)  is the resistivity at t degrees centigrade and \( ρ_0 \)  is the resistivity at 0 degrees Celsius.

graph

Graph of Resistivity Vs Temperature of metals

Resistivity of Semiconductors

The number of charge carriers of semiconductors like Germanium and Silicon is lesser than those of metals but more than in insulators. These carriers are generated by thermal breaking of bonds. Thus as the temperature rises, more number of covalent bonds break, releasing more electrons which lowers the resistivity rapidly. This Temperature dependence is usually exponential which makes semiconductors very useful in electronic circuits detecting small changes in temperature.

Temperature dependence

Graph of Resistivity Vs Temperature of Semiconductors

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On a given winter night, Himanshu has two options: either to use a single layer blanket or a double layered blanket. You will advise him to use: