In terms of drift velocity ,electric current flowing through a conducting wire of length L and uniform area of cross-section A
is
I=dQ/dt =neAv_{d}=(ne^{2}Aτ/ml) V
The above can be rearranged to give the ohm's law i.e,
V=IR
where R=(ml/ne^{2}Aτ)
Now R=ρl/A (9)
Where ρ is called the specific resistance or resistivity of the conductor
And ρ=m/ne^{2}τ (10)
From equation (9) ,we can see that resistance of the wire is proportional to its length and inversely proportional to its cross-sectional area.
Thus resistance of a long and thin wire will greater then the resistance of short and thick wire of the same material
Now from equation (9)
R=ρl/A (11)
And from ohm law R=V/I
Therefore
ρ=(V/I)(A/L)
=(V/L) / (I/A)
=E/J (12)
Where E=V/L is the electric field at any point inside the wire and J=I/A is current density at any point in the wire. Unit of resistivity is ohm-meter.
Thus from equation (12) ,electric resistivity can also be defined as the ratio of electric field intensity at any point in the conductor and the current density at that point.
The greater the resistivity of the material ,greater would be the field needed to establish a given current density
Perfect conductor have zero resistivities and for perfect insulators resistivity would be infinite
Metals and alloys have lowest resistivities and insulators have high resistivities and exceeds those of metals by a factor of 10^{22}
The reciprocal of resistivity is called conductivity and is represented by σ
Unit of conductivity is ohm^{-1}meter^{-1}(Ω^{-1}m^{-1}) and
σ is defined as
σ=1/ρ
Since ρ=E/J
or σ=J/E
or J=σE (13a)
The above relation can also be written in vector form as both J and E are vector quantities where vector J being directed towards E J=σE (13b)
Variation of resistivity with temperature
Resistance and hence resistivity of conductor depends on numbers of factors
One of the most important factors is dependence of resistance of metals on temperature
Resistivity of the metallic conductor increases with increase on temperature
when we increase the temperature of the metallic conductor,its constituent atoms vibrate with greater amplitudes then usual.This results to the more frequent collision between ions and electrons
As a result average time between the two successive collision decreases resulting the decrease in drift velocity
Thus increase collision with the increase in temperature results in increase resistivity
For small temperature variations ,resistivity of the most of the metals varies according to the following relations
ρ(T)=ρ(T_{0})[1 + α(T-T_{0})] (14)
Where ρ(T) and ρ(T_{0}) are the resistivities of the material at temperature T and T_{0} respectively and α is the constant for given material and is known as coefficient of resistivity.
Since resistance of a given conductors depends on the length and cross-sectional area of the conductor through the relation
R=ρl/A
Hence temperature variation of the resistance can be given as
R=R(T_{0})[1 + α(T-T_{0})] (15)
Resistivity of alloys also increase with temperature but this increase is much small as compared to metals
Resistivities of the non-metals decreases with increase in temperature .This is because at high temperature more electrons becomes available for conduction as they set themselves loose from atoms and hence temperature coefficient of resistivity is negative for non-metals
A similar behaviour occurs in case of semi-conductors .temperature coefficient of resistivity is negative for semi-conductors and its value is often large for a semi-conductor materials
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