Current Distributions

In elementary electric circuit theory, current is the rate at which electric charge passes a particular point in a circuit. For example, 1 A is 1 C per second. In this view current is a scalar quantity, and there are only two possible directions because charge is constrained to flow along defined channels. Direction is identified by the sign of the current with respect to a reference direction, which is in turn defined with respect to a reference voltage polarity. The convention in electrical engineering defines positive current as the flow of positive charge into the positive voltage terminal of a passive device such as a resistor, capacitor, or inductor. For an active device such as a battery, positive current corresponds to the flow of positive charge out of the positive voltage terminal.

However, this kind of thinking only holds up when the systems being considered are well-described as “lumped” devices connected by infinitesimally thin, filament-like connections representing wires and circuit board traces. Many important problems in electrical engineering concern situations in which the flow of current is not limited in this way. Examples include wire- and pin-type interconnects at radio frequencies, circuit board and enclosure grounding, and physical phenomena such as lightning. To accommodate the more general class of problems, we must define current as a vector quantity. Furthermore, current in these problems can spread out over surfaces and within volumes, so we must also consider spatial distributions of current.

As noted above, if a current

is constrained to follow a particular path, then the only other consideration is direction. Thus, a line current is specified mathematically as

, where the direction

may vary with position along the path. For example, in a straight wire

is constant, whereas in a coil

varies with position along the coil.

In some cases, current may be distributed over a surface. For example, the radio-frequency current on a wire of radius

made from a metal with sufficiently high conductivity can be modeled as a uniform surface current existing on the wire surface. In this case, the current is best described as a surface current density

, which is the total current

on the wire divided by the circumference

of the wire:

Imagine that current is distributed within a volume. Let

be the current passing through a small open planar surface defined within this volume, and let

be the area of this surface. The volume current density

at any point in the volume is defined as

In general,

is a function of position within this volume. Subsequently the total current passing through a surface

is

In other words, volume current density integrated over a surface yields total current through that surface. You might recognize this as a calculation of flux, and it is (but it is uncommon to refer to net current as a flux).¹

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  However, it is not common to refer to net current as a flux; go figure!

• “Electric current” on Wikipedia.
• “Current density” on Wikipedia.