RESISTIVITY AND ITS DEPENDENCE UPON TEMPERATURE

It has been experimentally seen that the resistance R of a wire is directly proportional to its length L and inversely proportional to its cross sectional area A. Expressing mathematically we have

R ∝ L 

And 

R ∝ 1/A 

Or 

R ∝ L/A 

R = ρ L/A ....... (1)

where ρ is a constant of proportionality known as resistivity of specific resistance of the material of the wire. It may be noted that resistance is the characteristic of a particular wire whereas the resistivity is the property of the material of which the wire is made. From Eq. 1 we have

ρ = RA/L

The above equation gives the definition of resistivity as the resistance of a metre cube of a material. The SI unit of resistivity is (Ωm).

Conductance is another quantity used to describe the electrical properties of materials. In fact conductance is the reciprocal of resistance i.e.,

Conductance = 1 / resistance (R)

The SI unit of conductance is mho or siemen.

Likewise conductivity, σ is the reciprocal of resistivity i.e., 

σ = 1/ρ

The SI unit of conductivity is ohm^-1m^-1 or mho m^-1. Resistivity of various materials are given in Table.

It may be noted from Table that silver and copper are two best conductors. That is the reason that most electric wires are made of copper.

The resistivity of a substance depends upon the temperature also. It can be explained by recalling that the resistance offered by a conductor to the flow of electric current is due to collisions, which the free electrons encounter with atoms of the lattice. As the temperature of the conductor rises, the amplitude of vibration of the atoms in the lattice increases and hence, the probability of their collision with free electrons also increases. One may say that the atoms then offer a bigger target, that is, the collision cross-section of the atoms increases with temperature. This makes the collisions between free electrons and the atoms in the lattice more frequent and hence, the resistance of the conductor increases.

Experimentally the change in resistance of a metallic conductor with temperature is found to be nearly linear over a considerable range of temperature above and below 0 °C. Over such a range the fractional change in resistance per kelvin is known as the temperature coefficient of resistance i.e..

 α = (R_t - R₀)/R₀t

where R₀, and R_t, are resistances at temperature 0 °C and t°C. As resistivity ρ depends upon the temperature, Eq. gives 

R=ρ₀ L/A and R=ρ_t L/A

Substituting the values of R₀, and R_t, in Eq. we get

α = (ρ_t - ρ₀)/ρ₀t

where ρ₀ is the resistivity of a conductor at 0 °C and ρ_t is the resistivity at t °C. 

There are some substances like germanium, silicon etc; whose resistance decreases with increase in temperature. i.e;  these substances have negative temperature coefficients.