Question

derive expression for resistivity in terms of its material parameters .

Answer 1

Answer:

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)

Hope it helps you

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