Ohm's Law

 Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship: 

I = V / R

where I is the current through the conductor, V is the voltage measured across the conductor, and R is the resistance of the conductor. More specifically, Ohm's law states that the R in this relation is constant, independent of the current. If the resistance is not constant, the previous equation cannot be called Ohm's law, but it can still be used as a definition of static / DC resistance. Ohm's law is an emperical relation which accurately describes the conductivity of the vast majority of electically conducive materials over many orders of magnitude of current. However, some materials do not obey Ohm's law; these are called non-ohmic.

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wires. Ohm explained his experimental results by a slightly more complex equation than the modern form above.

In physics, the term Ohm's law is also used to refer to various generalizations of the law; for example, the vector form of the law used in electromagnetics and materials science:

J = σE

where J is the current density at a given location in a resistive material, E is the electric field at the location, and σ (sigma) is a material-dependent parameter called the conductivity. This reformulation of Ohm's law is due to Gustav Kirchhoff.

Ohm's law is an empirical law, a generalization from many experiments that have shown that current is approximately proportional to electric fields for most materials. It is less fundamental than Maxwell's equations and is not always obeyed. Any given material will break down under a strong-enough electric field, and some materials of interest in electrical engineering are "non-ohmic" under weak fields.

Ohm's law has been observed on a wide range of length scales. In the early 20th century, it was thought that Ohm's law would fail at the atomic scale, but experiments have not borne out this expectation. As of 2012, researchers have demonstrated that Ohm's law works for silicon wires as small as four atoms wide and one atom high.

The dependence of the current density on the applied electric field is essentially quantum mechanical. A qualitative description leading to Ohm's law can be based upon classical mechanics using the Drude model developed by Paul Drude in 1900.

The Drude model treats electrons (or other charge carriers) like pinballs bouncing among the ions that make up the structure of the material. Electrons will be accelerated in the opposite direction to the electric field by the average electric field at their location. With each collision, though, the electron is deflected in a random direction with a velocity that is much larger than the velocity gained by the electric field. The net result is that electrons take a zigzag path due to the collisions, but generally drift in a direction opposing the electric field.

The drift velocity then determines the electric current density and its relationship to E and is independent of collisions. Drude calculated the average drift velocity from p = -eEτ where p is the average momentum, -e is the charge of the electron and τ is the average time between the collisions. Since both the momentum and the current density are proportional to the drift velocity, the current density becomes proportional to the applied electric field; this leads to Ohm's law.

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