Current and Current Density

Electric Current. In an isolated metallic conductor, the free electrons present in it are in random motion like the molecules of a gas. So the net rate at which electrons passes through any hypothetical plane in zero. It the through any hypothetical plane is zero. If the ends of the conductor are connected to a battery, an electric field E will be set up at every point within the conductor. This field acts on the electrons and gives them a resultant motion in the direction – E. If a charge q passes through any cross-section of the conductor in time t, then the current i is defined by

                I = q/t

When q is in coulomb and t in second, then i is in amprere. If the rate of flow of charge with time is not constant the current varies with time. Then at any instant t current is defined as the net charge flowing across the area per unit time. Current is a scalar quantity. Conventionally, the direction of electric current is taken along the direction of motion of positive charges. When the current is opposite to the direction of electron flow.

Current density. Current is a macroscopic quantity. The related microscopic quantity is the current density J.

The current density at any point is defined as the quantity of charge passing per second through a unit area taken perpendicular to the direction of the flow of charge at that points on that cross section is given by

                J = q/t = i
                        A    A

The unit of current density is A/m².

Current i is related to the current density J by
                i = ∫ J. dS

Where dS is an element of area and the integral is taken over any surface cutting across the conductor.
 
From this equation we may define the current i as the flux of the current density vector J through a given area.