Current density represents the current flowing per unit area of cross-section at a point. To relate current, a microscopic quantity, to the microscopic motion of the charges, let's examine a conductor of cross-sectional area A, as shown in given figure.
A microscopic picture of current flowing in a conductor
The total current through a surface can be written as
where is the current density (the SI unit of current density are A/m2). If q is the charge of each carrier, and n is the number of charge carriers per unit volume, the total amount of charge in this section Q = q(nAx). Suppose that the charge carriers move with a speed vd. then the displacement in a time interval t will be x = vdt, which implies
The speed vd at which the charge carriers are moving is known as the drift speed. Physically, vd is the average speed of the charge carriers inside a conductor when an external electric field is applied. Actually an electron inside the conductor does not travel in a straight line; instead, its path is rather erratic as shown.
Motion of an electron in a conductor
From the above equations, the current density can be written as
Thus, we see that and d point in the same direction for positive charge carriers, in opposite directions for negative charge carriers.