Question

Derive an expression for electrical conductivity on the basis of free electron theory

Answer 1

The assumptions of the Drude-Lorentz classical theory of free-electrons are the following. Metals contain free electrons that move through a lattice of positive ions. These free electrons are responsible for electrical conduction when an electric potential is maintained across the conductor.

Answer 2

Expression for electric conductivity on the idea of unfastened electron theory:

Explanation:

Let E be the implemented electric powered discipline, m be the mass of the electron and e be the rate at the electron. The pressure F because of implemented discipline can be-

F = eE

Also F = ma, in which a is the acceleration.

∴ a = eE/m

Because of collisions of electrons for the duration of motion, the electrons will now no longer get accelerated indefinitely. τ be the relaxation time (collision time) then the common digital pace called drift velocity is given -

vd = aτ = (eE/m) τ … (i)

Let I be the current carried with the aid of using a conductor on utility of electrical discipline E similar to flow pace vd. In time dt, the electrons will travel a distance vd dt and the quantity of electrons crossing any cross-sectional region A in time dt can be contained in volume Avd dt. Thus, if there are n electrons in line with unit volume of the conductor, the entire rate flowing thru the section in time dt is-

dQ = enAvd dt

or I = dQ/dt = en Avd

And current density J = I/A = en vd … (ii)

Using equation (i), we have,

J = en (eEτ/m) = ne2 τE/m … (iii)

For a specific material, the amount ne2τ/m in (iii) is consistent at specific temperature and is called electric conductivity ‘σ’ of the cloth.

∴ J = ne2τE/m = σE … (iv)

or σ = ne2τ/m … (v)

R = r l/A in which r is resistivity of material and l is the length of conductor. Also r = 1/σ, therefore, from equations (i) and (iv), we have,

I = JA = σEA = EA/ᵨ = El/R

Also E = V/l

∴ I = Vl/Rl = V/R

that is not anything however Ohm’s law. That is why equation (iv) and (v) are also called Ohm’s law.

Equation (v) also can be written as-

∴ σ = ne²τ/m = neμ   (vi)

Where, µ (= eτ/m) is the mobility obtained with the aid of using electrons in the presence of electrical field. Using equation (i), the mobility of electrons also can be expressed as-

μ = eτ/m = vd/E  (vii)

Let λ be the mean free direction and V’ be the root mean square velocity of electrons,

τ = λ / V'

V' = √three KT/m

τ = λ√ m / 3KT

Now the electric conductivity ‘σ’ may be expressed as-

σ = ne²τ /m

  = ne²λ /m

  = (ne²λ /m ) √(m/3KT)

σ = ne²λ/ √3mKT

Answer = σ = ne²λ/ √3mKT