Describe the microscopic model of current and obtain general from of Ohm's law.

Explanation

Microscopic model of current: Consider a conductor with area of cross-section A and an electric field `vecE` applied from right to left. Suppose there are n electrons per unit volume in the conductor and assume that all the electrons move with the same drift velocity `vecv_(d)`.
The drift velocity of the electrons `=v_(d)`
The electrons move through a distance dx within a small interval of dt
`v_(d)=(dx)/(dt), dx=v_(d)dt " "... (1)`
Since A is the area of cross section of the conductor, the electrons available in the volume or length dx is
= volume `xx` number per unit volume
`=adx xx n " " ...(2)`
Substituting for dx from equation (1) in (2)
`=(A v_(d)dt)n`
Total charge in volume element `dQ=` (charge) `xx` (number of electrons in the volume element)
`dQ=(e ) (A v_(d)dt)n`
Hence the current, `I=(dQ)/(dt)=(n eAv_(d)dt)/(dt)`
`I=n eAv_(d)" " ...(3)`
Current density (J): The current density (J) is defined as the current per unit area of cross section of the conductor
`J=(1)/(A)`
The S.I. unit of current density, `(A)/(m^(2))(or) Am^(-2)`
`J=(n eAv_(d))/(A)` (from equation 3)
`J=n ev_(d)" " ...(4)`
the above expression is valid only when the direction of the current is perpendicular to the area A. In general, the current density is a vector quantity and it is given by
`vecJ= n e vecv_(d)`
Substituting `vecv_(d)` from equation `vecv_(d)=(-e tau)/(m) vecE `
`vecJ= -(n.e^(2)tau)/(m)vecE " " ...(5)`
`vecJ= -sigma vecE`
But conventionally, we take the direction of (conventional) current density as the direction of electric field. So the above equation becomes
`vecJ=sigma vecE`
where `sigma (n e^(2)tau)/(m)` is called conductivity.
The equation 6 is called microscopic form of ohm.s law.