Microscopic model of current: Consider a conductor with area of cross-section A and an electric field E 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 \(\vec v_d\).
The drift velocity of the electrons = Vd
The electrons move through a distance dx within a small interval of dt
v =\(\frac{dx}{dt}\) ; dx = vd dt .......(1)
Since A is the area of cross section of the conductor, the electrons available in the volume or length dx is
= volume x number per unit volume
= A dX x n ............(2)
Substituting for dx from equation (1) in (2)
= (A vd dt)n
Total charge in volume element dQ = (charge) x (number of electrons in the volume element)
dQ= (e)(A vd dt)n
Hence the current, I = \(\frac {dQ}{dt}\) = \(\frac {neAv_ddt}{dt}\)
I = ne A Vd .......... (3)
Current denshy (J):
The current density (J) is defined as the current per unit area of cross section of the conductor
J = \(\frac{I}{A}\)
The S.I unit of current density, \(\frac{A}{m^2}\) (or) Am-2
J = \(\frac {neAv_d}{A}\) (from equation 3)
J = nevd .............(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
\(\vec J\) = ne\(\vec v_d\)
Substituting i from equation \(\vec v_d\) = \(\frac {-er}{m}\) \(\vec E\)
But conventionally, we take the direction of (conventional) current density as the direction of electric field. So the above equation becomes
\(\vec J\)= σ\(\vec E\)............... (6)
where σ = \(\frac{n.e^2r}{m}\) is called condictivity.
The equation 6 is called microscopic form of ohm’s law.
