Physics equations/Current and current density

Electric current

A simple electric circuit, where current is represented by the letter i. The relationship between the voltage (V), resistance (R), and current (I) is V=IR; this is known as Ohm's Law.

The SI unit for measuring an electric current is the ampere, which is the flow of electric charges through a surface at the rate of one coulomb per second. Electric current can be measured using an ammeter.More generally, electric current can be represented as the rate at which charge flows through a given surface as:

$I={\frac {\mathrm {d} Q}{\mathrm {d} t}}\,.$

The conventional symbol for current is $I$, which originates from the French phrase intensité de courant, The $I$ symbol was used by André-Marie Ampère, after whom the unit of electric current is named.

The electrons, the charge carriers in an electrical circuit, flow in the opposite direction of the conventional electric current.

The symbol for a battery in a circuit diagram.

The direction of the current is defined so the arrow would indicate the flow of positive charges. In metals, which make up the wires and other conductors in most electrical circuits, the positive charges are immobile, and the charge carriers are electrons. Because the electrons carry negative charge, their motion in a metal conductor is in the direction opposite to that of conventional (or electric) current.

Ohm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance,

$I={\frac {V}{R}}$

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

AC and DC

The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

Direct current (DC) is the unidirectional flow of electric charge. Direct current is produced by sources such as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric charge flows in a constant direction, distinguishing it from alternating current (AC). In alternating current (AC, also ac), the movement of electric charge periodically reverses direction. In direct current (DC, also dc), the flow of electric charge is only in one direction.

AC is the form in which electric power is delivered to businesses and residences. The usual waveform of an AC power circuit is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves. Audio and radio signals carried on electrical wires are also examples of alternating current. In these applications, an important goal is often the recovery of information encoded (or modulated) onto the AC signal.

Current density and Ohm's law

Main resource: Current density

Current density is a measure of the density of an electric current. It is defined as a vector whose magnitude is the electric current per cross-sectional area. In SI units, the current density is measured in amperes per square metre.

$I=\int {\vec {J}}\cdot d{\vec {A}}$

where $I$ is current in the conductor, ${\vec {J}}$ is the current density, and $d{\vec {A}}$ is the differential cross-sectional area vector.

The current density (current per unit area) ${\vec {J}}$ in materials with finite resistance is directly proportional to the electric field ${\vec {E}}$ in the medium. The proportionality constant is called the conductivity $\sigma$ of the material, whose value depends on the material concerned and, in general, is dependent on the temperature of the material:

${\vec {J}}=\sigma {\vec {E}}\,$

The reciprocal of the conductivity $\sigma$ of the material is called the electrical resistivity $\rho$ of the material and the above equation, when written in terms of resistivity becomes:

${\vec {J}}={\frac {\vec {E}}{\rho }}$ or

${\vec {E}}=\rho {\vec {J}}$

Conduction in semiconductor devices may occur by a combination of drift and diffusion, which is proportional to diffusion constant $D$ and charge density $\alpha _{q}$. The current density is then:

$J=\sigma E+Dq\nabla n,$

with $q$ being the elementary charge and $n$ the electron density. The carriers move in the direction of decreasing concentration, so for electrons a positive current results for a positive density gradient. If the carriers are holes, replace electron density $n$ by the negative of the hole density $p$.

In linear anisotropic materials, σ, ρ and D are tensors.

In linear materials such as metals, and under low frequencies, the current density across the conductor surface is uniform. In such conditions, Ohm's law states that the current is directly proportional to the potential difference between two ends (across) of that metal (ideal) resistor (or other ohmic device):

$I={V \over R}\,,$

where $I$ is the current, measured in amperes; $V$ is the potential difference, measured in volts; and $R$ is the resistance, measured in ohms. For alternating currents, especially at higher frequencies, skin effect causes the current to spread unevenly across the conductor cross-section, with higher density near the surface, thus increasing the apparent resistance.

Drift speed

The mobile charged particles within a conductor move constantly in random directions, like the particles of a gas. In order for there to be a net flow of charge, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the opposite direction of the electric field. The speed at which they drift can be calculated from the equation:

$I=nAvQ\,,$

where

$I$ is the electric current

$n$ is number of charged particles per unit volume (or charge carrier density)

$A$ is the cross-sectional area of the conductor

$v$ is the drift velocity, and

$Q$ is the charge on each particle.

Typically, electric charges in solids flow slowly. For example, in a copper wire of cross-section 0.5 mm², carrying a current of 5 A, the drift velocity of the electrons is on the order of a millimetre per second. To take a different example, in the near-vacuum inside a cathode ray tube, the electrons travel in near-straight lines at about a tenth of the speed of light.