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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.

====================================

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

===

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

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

I = charge crossing a cross section in time t / time t

So, I = e (n A v t) / t = n e A v

Substituting in (2) we get,

J / (n e) = v = e J ρ τ / m

=>

Or,

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.

====================================

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

===

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_i + a τ = 0 + e E τ /m =***v***--- (1)***e J ρ τ / m*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 =**--- (2)**e J ρ τ / m**.*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)*