Expression for Current density

Consider a wire of length L and area of cross section A in which there is an electric field E towards the left. Let n be the number of free electrons per metre3. The free electrons move towards the right with a constant drift velocity vd.

Number of conduction electrons in the wire = n (AL).
Let e be the charge on each electron.
Magnitude of charge q, passing out of the wire in time t is
        q = (n AL)e
where t is given by    t = L/vd

The current in the wire is
        i = q/t = nALe/ L/v_d = n Aev_d

But i/A is the current density J. Therefore
        J = nev_d
In vector form,     J = (ne) v_d.              …. (1)

(ne)  is the volume charge     density p of the current carriers. For positive carriers ne is positive and Eq. (1) predicts that J and V_d point in the same direction. For electrons J and v_d point in opposite direction.