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, ''V'', drives an
Ohm's law states that, in an Electrical Circuit , the Current passing through a conductor between two points is Proportional to the Potential Difference (i.e. Voltage Drop or Voltage ) across the two points, and inversely proportional to the Resistance between them.
In mathematical terms, this is written as:
:I = rac VR
where ''I'' is the current in Ampere s, ''V'' is the potential difference in Volt s, and ''R'' is a constant, measured in Ohm s, called the Resistance . The potential difference is also known as the Voltage Drop , and is sometimes denoted by ''E'' or ''U'' instead of ''V''.

''Handbook of Chemistry and Physics'', Fortieth Edition, p.3112, 1958

The law was named after the physicist Georg Ohm , who, in a treatise published in 1827 , described measurements of applied voltage, and current passing through, simple electrical circuits containing various lengths of wire, and presented a slightly more complex equation than the one above to explain his experimental results.

The resistance of most resistive devices ( Resistors ) is constant over a large range of values of current and voltage. When a resistor is used under these conditions, the resistor is referred to as an ohmic device because a single value for the resistance suffices to describe the resistive behavior of the device over the range. When sufficiently high voltages are applied to a resistor, forcing a high current to flow through it, the device is no longer ohmic because its measured resistance is different (typically greater) from the value measured under standard conditions (see temperature effects, below).


ELEMENTARY DESCRIPTION AND USE


Electrical circuits consist of electrical devices connected by wires (or other suitable conductors). (See the article Electrical Circuits for some basic combinations.) The above diagram shows one of the simplest electrical circuits that can be constructed. One electrical device is shown as a circle with + and - terminals, which represents a voltage source such as a battery. The other device is illustrated by a zig-zag symbol and has an R beside it. This symbol represents a resistor, and the R designates its resistance. The + or positive terminal of the voltage source is connected to one of the terminals of the resistor using a wire of negligible resistance, and through this wire a current I is shown to be passing, in a specified direction illustrated by the arrow. The other terminal of the resistor is connected to the - or negative terminal of the voltage source by a second wire. This configuration forms a complete circuit because all the current that leaves one terminal of the voltage source must return to the other terminal of the voltage source. (While not shown, because electrical engineers assume that it exists, there is an implied current I, and an arrow pointing to the left, associated with the second wire.)

Voltage is the electrical force that moves (negatively charged) electrons through wires and electrical devices, current is the rate of electron flow, and resistance is the property of a resistor (or other device that obeys Ohm's law) that limits current to an amount proportional to the applied voltage. So, for a given resistance R (ohms), and a given voltage V (volts) established across the resistance, Ohm's law provides the equation (I=V/R) for calculating the current that must flow through the resistor (or device).

The 'conductor' mentioned by Ohm's law is a circuit element across which the voltage is measured. Resistors are conductors that slow down the passage of electric charge. A resistor with a high value of resistance, say greater than 10 megaohms, is a poor conductor, while a resistor with a low value, say less than 0.1 ohm, is a good conductor. (Insulators are materials that, for most practical purposes, do not allow a current to flow when a voltage is applied. However, all materials have electrical properties and, with enough applied voltage, break down and allow a current to flow.)

In a circuit diagram like the one above, the various components may be joined by connectors, contacts, welds or solder joints of various kinds, but for simplicity these connections are usually not shown.


PHYSICS

Physicists often use the continuum form of Ohm's Law:

\mathbf{J} = \sigma \cdot \mathbf{E}

where J is the Current Density (current per unit area), σ is the Conductivity (which can be a Tensor in anisotropic materials) and '''E''' is the Electric Field .

The common form V = I \cdot R used in circuit design is the macroscopic, averaged-out version.

The continuum form of the equation is only valid in the Reference Frame of the conducting material. If the material is moving at velocity v relative to a Magnetic Field '''B''', a term must be added as follows
:
\mathbf{J} = \sigma \cdot \left( \mathbf{E} + \mathbf{v} imes\mathbf{B} ight)

The analogy to the Lorentz Force is obvious, and in fact Ohm's law can be derived from the Lorentz force and the assumption that there is a drag on the charge carriers proportional to their velocity.

A perfect metal lattice would have no Resistivity , but a real metal has Crystallographic Defect s, impurities, multiple Isotope s, and thermal motion of the atoms. Electrons Scatter from all of these, resulting in resistance to their flow.

Ohm's law is sufficient to derive both Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL). Let us first examine only the right-hand side of the equation:


:\sigma E\,

and calculate the line integral around a closed contour:


:\int { \sigma E \cdot dl }

Applying Stokes's Theorem , we can write over the surface bounded by the countour:


:\int_S { \sigma
abla imes E \cdot dA }

but, since ''E'' is the gradient of a scalar potential, yielding:


:\int_S \sigma
abla imes \left(
abla (\phi) ight) \cdot dA

and gradients are irrotational, we have:

:\int_S \sigma imes ec{0} \cdot dA

thereby proving KCL. Returning to the original formulation of Ohm's law:

:J = \sigma E\,

and forming the closed line integrals again:


:\int J \cdot dl = \oint \sigma E \cdot dl

and recalling from Maxwell's equations that curl(H) = J:


:\int
abla imes (H) \cdot dl = \int \sigma E \cdot dl

we apply Stokes's theorem to obtain:


:\int_S H \cdot dA = \oint \sigma E \cdot dl

From our preceding derivation, we know that the right-hand side evaluates to zero:


:\int_S H \cdot dA = 0

thus proving that the net current flow through an open surface is zero, which restates KCL.


HOW ELECTRICAL AND ELECTRONIC ENGINEERS USE OHM'S LAW


Ohm's Law is one of the equations used in the analysis of electrical circuits, whether the analysis is done by engineers or computers. Even though, today, computers running electronic computer aided design and analysis programs do the bulk of the work predicting and optimizing the performance of electrical circuits (in particular, those circuits to be fabricated on silicon chips), most electrical engineers still use Ohm's Law every working day. Whether designing or debugging an electrical circuit, electrical engineers must have a working knowledge of the practical aspects of Ohm's law.

Virtually all electronic circuits have resistive elements which are almost always considered ideal ohmic devices, i.e. they obey Ohm's Law. From the engineer's point of view, resistors (devices that "resist" the flow of electrical current) develop a voltage across their terminal conductors (e.g. the two wires emerging from the device) proportional to the amount of current flowing through the device.

More specifically, the voltage measured across a resistor at a given instant is strictly proportional to the current passing through the resistor at that instant. When a functioning electrical circuit drives a current I, measured in amperes, through a resistor of resistance '''R''', the voltage that develops across the resistor is I '''R''', the value of '''R''' serving as the proportionality factor. (That current must have been supplied by a circuit element functioning as a current source and it must be passed on to a circuit element that serves as a current sink.) Thus resistors act like current-to-voltage converters (just as springs act like displacement-to-force converters).

Similarly, a circuit may incorporate a resistor (of resistance R) designed to function as a voltage-to-current converter. In such a circuit, a desired voltage '''V''' is established across the resistor in order to force a current '''I''' exactly equal to 1/R times '''V''' to flow through the resistor.

The DC resistance of a resistor is always a positive quantity, and the current flowing through a resistor generates (waste) heat in the resistor as it does in one of Ohm's wires. Voltages can be either positive or negative, and are always measured with respect to a reference point. When we say that a point in a circuit has a certain voltage, it is understood that this voltage is really a voltage difference (a two terminal measurement) and that there is an understood, or explicitly stated, reference point, often called ground or common. Currents can be either positive or negative, the sign of the current indicating the direction of current flow. Current flow in a wire consists of the slow drift of electrons due to the influence of a voltage established between two points on the wire.