Temperature Dependence of Resistivity - Notes | Study Physics Class 12 - NEET

Resistivity is the nature of a material that allows or resists the flow of electric current through a given element or material. What is surprising about resistivity is the temperature dependence of electrical resistance! It is hard to comprehend how the temperature of an element can affect the degree of conductance of such material but believe it or not, this is the world of science and it happens almost every day, all around us!

The Concept of Electrical Resistance

Resistivity is the phenomenon of specific electrical resistance of a material or volume resistivity of a material. It can also be defined as the intrinsic property of a material that displays how the material resists the flow of current in the material. The concept can also be defined as the resistance that is displayed by a conductor which has unit length and unit area of the given cross section.
So resistivity is not dependent upon the length and area of a cross-section of a given material. However, the resistance of a material depends upon the length and area of the cross-section of the material in question. The resistivity manifests as:
ρ = RA/L,
where R is the resistance in ohms, A is the area of cross-section in square meters and L is the length in meters. The unit of resistivity is universally accepted as ohm-meter.

The Concept of Temperature Resistivity

The resistivity of materials is dependent upon the temperature of the material.
ρt = ρ0 [1 + α (T – T₀)] 
is the equation that defines the connection between the temperature and the resistivity of a given material. In this equation ρ0 is the resistivity at an equilibrium temperature, ρt is the resistivity at t0 C, T₀ is referred to as the reference temperature and α is the temperature coefficient of resistivity.

Understanding the Equation

It is known that an electric current is the movement of free electrons from one atom to the other when there is a potential difference between the two. In the case of conductors, no gap is present between the conduction band and valence band of the electrons. In most cases, these bands overlap each other.
The valence electrons in a given atom are loosely bound to the nucleus in a conducting material. Quite often, metals or conductors have a low ionization energy and therefore, they tend to lose electrons very fluidly. When an electric current is applied, the electrons are free to move within the structure on their own. This happens in the case of the normal temperature of a material.
However, when the temperature increases gradually, the vibrations in the metal ions in the lattice structure also undergo an increase. In this case, the atoms begin to vibrate with a higher amplitude. Such vibrations, in turn, cause frequent collisions between the free electrons and the remaining electrons.
Each such collision drains out some degree of energy of the free moving electrons and renders them in a condition in which they are not able to move. Thus, it causes a restriction in the movement of the delocalized electrons.
In the case of metals or conductors, it is rightly said that they hold a positive temperature coefficient. The value α is positive. For most of the metals, the resistivity increases in a linear pattern with an increase in the temperature in a range of 500K.

What happens in Insulators?

In the case of insulators, the forbidden energy gap between the conduction band and the valence band is very high. The valence band is filled with the electrons of the atoms. Diamond is a unique example of an insulator. Here, all the valence electrons are involved in the covalent bond formation and as a result, conduction does not take place. The electrons are too tightly bound to the nucleus of the atom.

Resistance of Pure Metals

(i) We know that 

For a given conductor, l, A and n are constant, hence R  is directly proportional to (1/τ)

If λ represents the mean free path (Average distance covered between two successive collisions) of the electron and v_rms, the root-mean-square speed, then

 Hence R is directly proportional to  

Now,

(a) λ decreases with rise in temperature because the amplitude of vibrations of the +ve ions of the metal increases and they create more hindrance in the movement of electrons and,

(b) (i) v_rms increases because v_rms is directly proportional to  under root T. Therefore, Resistance of the metallic wire increases with rise in temperature. As ρ is directly proportional to  R and σ is directly proportional to (1/ρ), hence resistivity increases and conductivity decreases with rise in temperature of the metallic of the metallic wires.

(ii) If R₀ and R_τ represent the resistances of metallic wire at 0°C and t°C respectively then R_t is given by the following formula :

where α is called as the Temperature coefficient of resistance of the material of the wire.

α depends on material and temperature but generally it is taken as a constant for a particular material for small change.

R_t - R₀ = R₀ α t

for very small change in temperature dR = R₀ α dt

(c) Resistance of semiconductors

(i) There are certain substances whose conductivity lies in between that of insulators and conductors, higher than that of insulators but lower than that of conductors. These are called as semiconductors, e.g., silicon, germanium, carbon etc.

(ii) The resistivity of semiconductors decreases with increase in temperature i.e., a for semiconductors is -ve and high.

(iii) Though at ordinary temperature the value of n (no. of free electrons per unit volume) for these materials is very small as compared to metals, but increases very rapidly with rise in temperature (this happens due to breaking of covalent bonds). Though τ decreases but factor of n dominates. Therefore, the resistance

  goes on decreasing with increase in temperature.

Specific Resistance or Resistivity 

• R of a conductor is directly proportional to its length (l).
• R of a conductor is inversely proportional to the area of cross-section (A).

 Resistivity depends only upon the material of which the conductor is made. It is defined as the resistance of the conductor made of a given material having length of 1 meter and area of cross-section 1 m²

If P₁ and P₂ are resistivities of a material at temperatures T1

and T2 respectively, then :

Solved Examples for You

Question: State the properties and features of temperature resistivity in conductors and insulators.
Solution: The resistivity of a material is defined as the resistance offered by a conductor having a given unit length and unit area of cross-section. The unit of resistivity is ohm meter. The formula for deriving resistivity is ρ = RA/L. Here, R is the resistance in ohms, A is the area of cross-section in square meters and L is the length in meters.

• In the case of metals or conductors, when the temperature increases, the resistivity of the metal increases as a result. Thus, the flow of current in the metal decreases. This phenomenon reflects a positive temperature coefficient. The value α is positive in this case.
• In the case of insulators, the conductivity of the material generally increases, when the temperature is made to increase. When the conductivity of the material undergoes an increase, it is easy to decipher that the resistivity of the material decreases and the current flow of the material increases.