# (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.

a) The conductivity of a metallic wire is defined as it is the ability of the wire to allow the electric flow of electric charges as heat through it.
Numerically, the conductivity of a wire is defined as the reciprocal of resistivity.
That is, conductivity

SI unit is, ohm-1 m-1 or mho m-1
b) Consider the electric field is present then the acceleration of electrons is

where e = charge of the electron
m = mass of the electron
From (1) the drift velocity of the electron is given by
Average drift velocity

where Average force between successive collision which is known as relaxation time
Let 'n' is the number of free electrons per unit volume of a conductor the magnitude of the current is given by

substitute the value of V2 from (2) get,

we know the current density

from eqn (4) we get,

But
so, By comparing

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