Derivation of Electrical Conductivity (σ)
In metals, the electrical conductivity depends on the number of charge carriers (free electrons) present in the material.
Let us consider a solid metal (S) of length 'l' and area of crosssection 'A'. Let 'n' be the number of electrons per unit volume. Then,
Total number of electrons in the metal N = nAl (1)
We know,
Total charge Q = Total number of electrons in the metal x charge of one electron
∴Total charge Q = N(e) (2)
The negative sign indicates that the charge of the electron is negative.
Substituting equation (1) in (2) we get,
Total charge present in the solid Q = nAl(e)
Now, when we apply a voltage 'V' to the metal, then the electrons in it starts to move with an average velocity called drift velocity (V_{d}) from one end to another end (through a distance l), giving rise to current conduction in the metal.
∴The current through the metal
Substituting equation (3) in (4) we get,
I = 
 ........ (5)  
J = 
 ........ (6) 
Substituting equation (5) in equation (6), we get
J = 

J = 
 ........ (7) 
We know, drift velocity
V_{d} = 

V_{d} = 
 ........ (8) 
Due to the applied electric field, the electrons gain the acceleration 'a'
Acceleration (a) = 

(or)
V_{d }= aτ .....(10)
If E is the field intensity and m is the mass of the electron, then the force experienced by the electron is,
F = eE .......(11)
From Newton's second law,
The force experienced by the electron is, F = ma ..........(12)
Comparing equation (10) and (11) we have
eE = ma
(or)
a = 
 ........ (13) 
V_{d } = 
 τ ........ (14) 
Substituting equation (14) in (9) we have
J = n(e) 
 τ 
J = 
 ........ (15) 
Here the number of electrons flowing per second through unit area (the current density), depends on the applied field. Thus if the electric field (E) applied is more, current density (J) will also be more.
So, we can write J ᾳ E
or J = σE .....(16)
Comparing equation (15) and (16) we can write
σE = 

σ =
ne^{2}τ m
Definition for Cefficient of Electrical Conductivity (σ)
It is defined as the quantity of electricity flowing per unit area per unit time maintained at unit potential gradient.
Unit : Ω^{1}m^{1}
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