### General Expression for Conductivity

Consider a tube of metal with large number of free electrons as shown in the Fig.1.
Let A = Cross-sectional area in m2
L = Length in m
V = Voltage applied in volts
T = Time required by an electron to travel distance of 'L' m
v = Drift velocity of electron = L/T                             ............. (1)
E = V/L = electric field                                                 ........... (2)
v = μ E  where μ = Mobility of electrons                           ........(3)
Consider any cross-section as shown in the Fig. 2.
 Fig. 1  N electrons crossing in T
Let N be the number of electrons passing through area A in time T. So number of electrons crossing the area A in unit time is N/T.
If q = Charge on each electron = 1.6 x 10-19 C
Then the total charge crossing the cross-section area A in unit time is,
dq = Charge on each electron x (N/T) = (Nq/T) C
But charge passing per unit time through a cross-section is the current.
...     I = charge passing/ time = (Nq/T)/(1sec)
...     I = (Nq/T) A                                         ............(4)
The current density J for the bar is current per unit cross-sectional area of the conducting material.
...     J = I/A     A/m2                                      ............. (5)
...     J = (Nq/TA)  but T = L/v  from (1)
But   LA = Volume of the tube
...              n = Concentration of free electrons
= Number of electrons per unit volume
...              n = N/(LA)  /m3
...              J = nqv   but   v = μ E
...               J = nqE   A/m2                                      ................. (6)
This is the general expression for current density in a given material.
The current density is related to electric field E by the relation,
J = σ  E                                         ................. (7)
Where  σ  = Conductivity of the material in (Ω-m)-1
Note : The conductivity indicates the ease with which current can flow through the given material.
Comparing (6) and (7)
σ  = nqμ (Ω-m)-1     ........... (8)
This is the general expression for the conductivity of the given material
...              ρ  = Resistivity = 1/σ  (Ω-m)                        ........... (9)
Note : The resistivity is the reciprocal of the conductivity.

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