(a) Define the term ‘conductivity’ of a metallic wire. Write its SI unit.
(b) 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.

Question

Philoid · Accepted Answer

a) Conductivity of a wire is defined as the reciprocal of resistivity ρ . Its S.I units is mho/m.

b) As we know that resistance R of a conductor is defined as

Where ρ = resistivity of conductor

l = length of conductor

A = area of cross section of conductor

According to ohm’s law V = IR

Where V = voltage across conductor and I = current through conductor.

Substituting the value of R in ohm’s law

Now

So, voltage becomes

V = j ρ l ….(i)

We know that electric field E is given as

Therefore V = E l.

So, equation (i) becomes

E l = j ρ l

E = j ρ

Now we know that

Therefore j = σ E

The average drift velocity of electron is given as

Where e = charge of electron

E = electric field

m = mass of electron

τ = relaxation time

If n is the number of free electrons per unit volume, the current I is given by

I = neA |V_d|

Now