Variation of Resistivity with Temperature

We have already discussed the concept of resistivity in detail in the earlier sections. The concepts of resistance and resistivity are quite interrelated. While resistance as the name suggests, measures how much a material obstructs the flow of current through it. Resistance is measured in ohms. Mathematically, it can be formulated by the formula E= I x R, where R is the resistance of the object, I is the electric current flowing through it and E is the amount of voltage applied across the object.

Resistivity is a slightly different concept from resistance. Resistivity is the amount of resistance present or exhibited by a material having specific or standard dimensions. Hence, we may define the resistivity of a material to be the resistance offered by a piece of material of length one meter and cross-sectional area of one square meter.       

Both resistance and resistivity depend on temperature, but the only difference is that the dependence is linear for relatively small temperature changes while for big changes, the dependence is non-linear.   

Variation of Resistivity with Temperature in Conductor

The attributes like shape or the cross-sectional area have an impact on the resistance of the conductor, but resistivity is not affected by it. Resistivity is only affected by the change in temperature. The relationship between resistivity and temperature is
Δρ = α * ΔT * ρ₀
Where:
Δρ : Change of the resistivity
α : Resistivity, temperature coefficient
ΔT : Change of temperature
ρ₀ : Original resistivity
For example, at 20 °C (293 K), the resistivity of Copper at 20 °C is 1.68 * 10^-8 , it's temperature coefficient is 0.0039 K^-1, its resistivity at 30 °C is 1.75E-8.

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Another Expression for Resistivity Variation with Temperature

As stated above, the resistivity ‘r’ of a conductor is unaffected by its shape and size but is affected by its temperature. Increase in temperature results in an increase in ‘r’ especially in the case of Ohmic conductors. If the given temperature is denoted by ‘T’ then the expression for ‘r’ is:

r_(t) = r₀ (1 + a DT), where r₀ = resistivity at 0^oC,

a = temperature coefficient of resistivity.

Also, a = (p – p₀) / p₀ΔT or 

a = 1 / p  dp / dT.

For conductors ‘a’ is a positive constant but for insulators and semiconductors it is negative.

We have tabulated below the values of resistivity of certain metals. It will provide you a clear idea of the effect of temperature on resistivity of metals.  

Besides metals, we list below certain insulators with their resistivity:

• P.V.C. 5.4 x 10¹⁵

• Glass 10⁴

• Quartz 10¹²

• P.T.F.E 10¹²

Here, P.T.F.E stands for polytetrafluoroethylene which is used for the insulation of high voltage cables. The values in the above table clearly show that the resistivity of insulators is much higher than that of conductors.   

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