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

Explanation

and obtain general form of Ohm.s law. Let us consider a conductor with area of cross section A and an electric field `vec E` applied from right to left. If there are n electrons per unit volume in the conductor. Then let us assume that all the electrons move with the same drift velocity `vec V _d`

Let the drift velocity of the electrons be `v_a` The electrons move through a distance du 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 with time dt is
= volume `xx` number per unit volume
` = Adx xx n " ...(2)"`
Substituting for dx form 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) = (ne A n_d dt)/(dt) `
` I = ne A v_d " ...(3)`
Current density (J)
The current density (J) is defined as the current per unit area of cross-section of the conductor,
`J = I / A`
The S.I unit of current density is `A/m_2`
(i.e.) A `m^(-2)`
` J= (ne Av_d)/A` ( from equation (3))
` J = nev_d" ...(4)"`
This expression holds only when the direction of the current is perpendicular to the area A. In general, the current density is a vector quantity. It is given by
`vecJ = ne vecv_d`
Substituting `vecv_d ` from equation `vecv_d = (e tau )/m vecE`
`vec J = (n* e^2 tau )/m vecE" ...(5) "`
`vec J = sigma vecE" ...(6)"`
where `sigma = (ne^2 tau )/m ` is is called conductivity. The equation (5) is called microscopic form of Ohm.s law.