Question

Derive an expression for resistivity in terms of relaxation timeDerive an expression for resistivity in terms of relaxation timeDerive an expression for resistivity in terms of relaxation time

Answer 1

The following are the steps:

1.  First, we need to define symbols and their meanings. 
2.  Then we obtain an expression for average drift velocity of electrons in terms of resistivity (electric field) and current density. 
         v = a  t = e E τ /m  = e J ρ τ / m    ---   (1)
 
3.  Then we obtain an expression for average drift velocity in terms of current (charge flowing) across any cross section.
         v = J / (n e)
 
4.  Equate them both.  we get the answer.
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Let
Relaxation time = average time between two successive collisions of an electron =  τ
emf applied across a resistor/conductor = V
Resistance of the conductor = R = ρ L / A
Resistivity = ρ
conductivity = s = 1/r
Area of cross section of the resistance = A
Length of the resistance wire = L
mass of an electron = m
electrostatic charge on an electron = e
drift velocity of an electron = v
current flowing in the conductor = I = V /R
N = Avogadro number
f = number of free conducting electrons (in the outermost shell) in one atom
d = density of the conductor
M = molar mass of the conductor
n = electron volume density = number of electrons in unit volume of a conductor
total number of electrons  = Mass * N * f /Molar mass  = A L d N f / M
n = number / volume = d N f / M
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Electric field intensity = E = V / L, assuming that it is uniform along the length of the conductor wire.
Force on an electron in this electric field = F = e E
Acceleration = F / m = a = e E /m = e V / (m L)
current density = J =  I / A = V / (A R) = V A / (  r A L) = V /(r L)
         J = σ E = E / ρ
         Or,  E =   J ρ

Velocity gained in between collisions due to electric field E and force F  = v
       v = v_i +  a  τ = 0 + e E  τ /m  = e J ρ τ / m    ---   (1)

The average of velocities v_i of all electrons just after collisions is 0, as they get bounced in all random directions.  Hence the average velocity of an electron along the length of a resistor or conductor wire is equal to that gained due to electrostatic field E.
  So drift velocity = v = e J ρ τ / m            --- (2).


I = charge crossing a cross section in time t / time t
So, I = e (n A v t) / t = n e  A v
J = n e v


Substituting in (2)  we get,
     J / (n e) = v = e J  ρ τ / m
      =>   τ =  m / ne² ρ
   Or,    ρ = m / (n e² τ)