Temperature Dependence of Resistivity
Temperature Dependence of Resistivity :- The resistivity of almost all materials depends on temperature, but not all materials show the same temperature dependence. In a limited temperature range (about 100 ºC or less), the resistivity of a conductor is expressed by the following relation :-
ρt =resistivity of the conductor at T ºC
ρ0 = resistivity of the conductor at reference temperature T0 (i.e., room temperature 293K or 20ºC)
α = temperature coefficient of resistivity
ΔT = (T – T0) = change in temperature
Temperature coefficient of resistivity(α)
(Temperature Dependence of Resistivity)
From equation (1)
Therefore, the temperature coefficient of resistivity (α) can be defined as the change in resistivity (Δρ) per unit initial resistivity (ρ0) and per unit change in temperature (ΔT).
Unit of α :- C-1
Dimensions of α = [M0L0T0θ-1]
(i) Temperature Dependence of Resistivity of metals
The resistivity of a material is given by the following formula :-
m = mass of electron
n = number of free electrons in a unit volume
e = electron charge
= average relaxation time
In equation (2) there is no effect of temperature increase on m and e, but the value of decreases with increasing temperature in conductors, so it is clear from equation (2) that resistivity of conductors increases with increase in temperature.
The increase in resistivity of copper with increase in temperature is shown in the following graph :-
(ii) Temperature Dependence of Resistivity of alloys
The temperature coefficient of resistivity(α) of alloys is very low as compared to pure metals, that is, the resistivity of alloys increases very little with increase in temperature. Some alloys (constantan, manganin, nichrome etc.) have negligible temperature coefficient of resistivity (α) i.e. the effect of temperature on resistivity is very small. That is why they are used in laboratory in meter bridge, potentiometer wire, resistance box etc.
The increase in resistivity of nichrome with increase in temperature is shown in the following graph :-
(iii) Temperature Dependence of Resistivity of semiconductors and insulators
The temperature coefficient of resistivity(α) of semiconductor materials like carbon, silicon, germanium etc. is negative, that is, their resistivity decreases with increase in temperature.
In semiconductor materials, the number density of free electrons(n) increases with increasing temperature and the mean relaxation time () decreases, but here the increase in n is much greater than the decrease in . Hence, according to Equation (2), the net effect of increase in temperature is that the value of ρ decreases.
The change in resistivity of a typical semiconductor with increase in temperature is shown in the following graph :-
The temperature dependence of the resistivity of semiconductors and insulators is given by the following formula :-
Eg = Energy gap between conduction band and valence band
kB = Boltzmann constant
T = temperature of matter in Kelvin
The resistivity of insulators increases exponentially as the temperature decreases. At absolute zero the resistivity of insulators becomes infinite, that is, at 0K the conductivity of insulators becomes almost zero.
For semiconductors, Eg ≅ 1eV, so their resistivity is not very high, but for insulators Eg ≥1eV, hence their resistivity is very high.
(iv) Temperature Dependence of Resistivity of electrolytes
With increasing temperature, the viscosity of electrolytes decreases, as a result of which the ions inside them move more freely, which leads to an increase in conductivity. Therefore, the resistivity of electrolytes decreases with increase in temperature and their resistivity temperature coefficient (α) is negative.
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