Temperature Dependence of Resistivity

Resistivity-

 

Before you go through this article, make sure that you have gone through the previous article on Resistivity of Conductor.

 

We have learnt-

• Resistivity of a conductor is defined as the resistance of a unit cube of the material of the conductor.
• It depends upon the nature of material of the conductor and temperature of the conductor.

 

In this article, we will discuss the temperature dependence of resistivity.

 

Temperature Dependence of Resistivity-

 

The resistivity of any material depends upon the number density (n) of free electrons and the mean collision time () as-

 

 

For Metals (Conductors)-

 

• For metals, the number density (n) of free electrons is almost independent of temperature.
• As temperature increases, the thermal speed of free electrons increases and also the amplitude of vibration of the metal ions increases.
• Consequently, the free electrons collide more frequently with the metal ions.
• Thus, the mean collision time () decreases.

 

Hence,

On increasing the temperature, the resistivity of metals (conductors) increases.

 

For most of the metals, resistivity increases linearly with the rise in temperature.

In such cases, resistivity at any temperature T is given by-

 

 

where-

• ρ_T = Resistivity at temperature T
• ρ₀ = Resistivity at a reference temperature T₀
• α = Temperature coefficient of resistivity

 

Temperature Coefficient of Resistivity   The temperature coefficient of resistivity α may be defined as the increase in resistivity per unit resistivity per degree rise in temperature.     The unit of α is °C-1. The dimension of α is [T-1].

 

For metals, the value of α is positive meaning that resistivity increases with increasing temperature.

 

At low temperatures, the resistivity of metals increases as a higher power of temperature.

The following graph shows the variation of resistivity (ρ) of copper as a function of temperature T-

 

 

For Alloys-

 

• When two or more metals are mixed together to form an alloy, complex crystalline structure is formed.
• This is because of what alloys have very high resistivity than their constituent metals.

ρ_alloys > ρ_metals

 

Alloys have very weak dependence on temperature. For alloys, α → 0

 

Example-

The resistivity of nichrome has weak dependence as shown in the following graph while that of manganin is almost independent of temperature.

α_alloys < α_metals

 

For Semiconductors and Insulators-

 

In case of insulators and semiconductors, the relaxation time does not change with temperature but the number density (n) of free electrons increases exponentially with the rise in temperature.

 

Hence,

On increasing the temperature, the resistivity of semiconductors and insulators decreases. For semiconductors and insulators, α is negative.

 

 

For Electrolytes-

 

As the temperature increases, the inter-ionic attractions (solute-solute, solvent-solute and solvent-solvent types) decreases and also the viscous forces decreases and therefore the ions move more freely.

 

Hence,

As the temperature of electrolytic solution increases, the resistivity of electrolyte decreases.

 

Read the next article on-

Ohm’s Law

 

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