Question

Using the concept of free electrons in a conductor, derive the expression for the conductivity of a wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field E.

Answer 1

The drift velocity of electrons in a conductor is given as

V_d = `(eEr)/m`  ...(1)

where, 
E = electric field set up across a conductor
m = mass of the electron
τ" data-mce-style="display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #212121; font-family: Roboto, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-style: initial; text-decoration-color: initial; position: relative;" data-mce-tabindex="0">ττ = average relaxation time
Now the current flowing in the conductor can be derived as

Electron drift to a small distance in a time Δt = V_dΔt

Amount of charge passing through the area A in time Δt, q = IΔt

IΔt = neAv_dΔt

or I = neAv_d   ...(2)

Where,

n → Number of free electrons per unit volume or number density.

Now from equation (1) and (2), we get

`I = ("ne"^2ArE)/m` ....3

Since the resistivity of a conductor is given as

`rho = m/("ne"^2r)`

Now, we know that conductivity of a conductor is mathematically defined as the reciprocal of resistivity of the conductor. Thus

`rho = 1/sigma` ....(4)

where `sigma = `conductivity of the conductor. Thus, from equation (3) and (4), we get

`sigma = ("ne"^2r)/m` ...(5)

Now, from equation (3) and (5), we have

`I/A = sigmaE .....(6)`

and the current  density is given as

`J = I/A`

Thus `J =sigmaE`