9.2 Model of conduction in metals  (Page 4/7)

Current density

Although it is often convenient to attach a negative or positive sign to indicate the overall direction of motion of the charges, current is a scalar quantity, $I=\frac{dQ}{dt}$ . It is often necessary to discuss the details of the motion of the charge, instead of discussing the overall motion of the charges. In such cases, it is necessary to discuss the current density, $\overrightarrow{J}$ , a vector quantity. The current density    is the flow of charge through an infinitesimal area, divided by the area. The current density must take into account the local magnitude and direction of the charge flow, which varies from point to point. The unit of current density is ampere per meter squared, and the direction is defined as the direction of net flow of positive charges through the area.

The relationship between the current and the current density can be seen in [link] . The differential current flow through the area $d\overrightarrow{A}$ is found as

where $\theta$ is the angle between the area and the current density. The total current passing through area $d\overrightarrow{A}$ can be found by integrating over the area,

Consider the magnitude of the current density, which is the current divided by the area:

Thus, the current density is $\overrightarrow{J}=nq{\overrightarrow{v}}_{\text{d}}$ . If q is positive, ${\overrightarrow{v}}_{\text{d}}$ is in the same direction as the electrical field $\overrightarrow{E}$ . If q is negative, ${\overrightarrow{v}}_{\text{d}}$ is in the opposite direction of $\overrightarrow{E}$ . Either way, the direction of the current density $\overrightarrow{J}$ is in the direction of the electrical field $\overrightarrow{E}$ .