Current density at a point in a conductor is defined as the amount of current flowing per unit area of the conductor around that point provided the area is held in a direction normal to the current.

Let I be the current distributed uniformly across a conductor of cross-sectional area A. The magnitude of the current density for all points on that cross-section of the conductor is

J=I/A

Current density is a vector quantity. Its direction is the direction of motion of positive charge.

As      I=Anev_d (refer article of relation between current and drift velocity)

Therefore                                     J=I/A=Anev_d/A

Or                                                 J=nev_d

The unit of current density is:

Ampere(metre)^-2 or Am^-2

For a particular surface of conductor, the current is the flux of J over the surface S and is given by

I=∫ J.ds

Where ds is elementary surface area vctor of an element taken over the particular surface S and integral is taken over the surface in question.

Conductance(G). The inverse of resistance (R) is called conductance of a conductor, that is,

Conductance, G=1/R

The unit of conductance is mho or siemen(symbol S). Its unit is inverse of Resistance that we have discussed in Ohm’s law.

Electrical conductivity. The inverse of resistivity (ρ) of a conductor is called its electrical conductivity(σ) i.e.

σ =1/ ρ

the unit of electrical conductivity is mho m^-1 or siemens/m.