TheInfoList

OR:

Ohm's law states that the current through a conductor between two points is directly proportional to the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship: :$I = \frac,$ where 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 empirical relation which accurately describes the conductivity of the vast majority of electrically conductive 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 wire. Ohm explained his experimental results by a slightly more complex equation than the modern form above (see ' below). 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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
and material science: :$\mathbf = \sigma \mathbf,$ where J is the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector (geometric), vector whose magnitude is the electric current per ...
at a given location in a resistive material, E is the electric field at that location, and ''σ'' ( sigma) is a material-dependent parameter called the conductivity. This reformulation of Ohm's law is due to
Gustav Kirchhoff Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coine ...
.

# History

In January 1781, before Georg Ohm's work,
Henry Cavendish Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
experimented with
Leyden jar A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typi ...
s and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of electrification" (voltage). He did not communicate his results to other scientists at the time, and his results were unknown until Maxwell published them in 1879.
Francis Ronalds Sir Francis Ronalds Fellow of the Royal Society, FRS (21 February 17888 August 1873) was an English scientist and inventor, and arguably the first History of electrical engineering, electrical engineer. He was knighted for creating the first wo ...
delineated "intensity" (voltage) and "quantity" (current) for the dry pile—a high voltage source—in 1814 using a gold-leaf electrometer. He found for a dry pile that the relationship between the two parameters was not proportional under certain meteorological conditions. Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book ''Die galvanische Kette, mathematisch bearbeitet'' ("The galvanic circuit investigated mathematically"). He drew considerable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used
voltaic pile file:Voltaic pile.svg, upright=1.2, Schematic diagram of a copper–zinc voltaic pile. The copper and zinc discs were separated by cardboard or felt spacers soaked in salt water (the electrolyte). Volta's original piles contained an additional zin ...
s, but later used a
thermocouple A thermocouple, also known as a "thermoelectrical thermometer", is an electrical device consisting of two dissimilar electrical conductors forming an electrical junction. A thermocouple produces a temperature-dependent voltage as a result of the ...
as this provided a more stable voltage source in terms of internal resistance and constant voltage. He used a galvanometer to measure current, and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation :$x = \frac,$ where ''x'' was the reading from the
galvanometer A galvanometer is an electromechanical measuring instrument for electric current. Early galvanometers were uncalibrated, but improved versions, called ammeters, were calibrated and could measure the flow of current more precisely. A galvanom ...
, ''l'' was the length of the test conductor, ''a'' depended on the thermocouple junction temperature, and ''b'' was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results. In modern notation we would write, :$I = \frac ,$ where $\mathcal E$ is the open-circuit emf of the thermocouple, $r$ is the internal resistance of the thermocouple and $R$ is the resistance of the test wire. In terms of the length of the wire this becomes, :$I = \frac ,$ where $\mathcal R$ is the resistance of the test wire per unit length. Thus, Ohm's coefficients are, :$a = \frac , \quad b = \frac .$ Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies" and the Minister of Education proclaimed that "a professor who preached such heresies was unworthy to teach science." The prevailing scientific philosophy in Germany at the time asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone. Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s. However, Ohm received recognition for his contributions to science well before he died. In the 1850s, Ohm's law was widely known and considered proved. Alternatives such as " Barlow's law", were discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855. The
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
was discovered in 1897 by J. J. Thomson, and it was quickly realized that it is the particle (
charge carrier In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and electron hole, holes. Th ...
) that carries electric currents in electric circuits. In 1900 the first ( classical) model of electrical conduction, the Drude model, was proposed by Paul Drude, which finally gave a scientific explanation for Ohm's law. In this model, a solid conductor consists of a stationary lattice of
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
s ( ions), with conduction electrons moving randomly in it. A voltage across a conductor causes an
electric field An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...
, which accelerates the electrons in the direction of the electric field, causing a drift of electrons which is the electric current. However the electrons collide with atoms which causes them to scatter and randomizes their motion, thus converting kinetic energy to
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
(
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
). Using statistical distributions, it can be shown that the average drift velocity of the electrons, and thus the current, is proportional to the electric field, and thus the voltage, over a wide range of voltages. The development of
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
in the 1920s modified this picture somewhat, but in modern theories the average drift velocity of electrons can still be shown to be proportional to the electric field, thus deriving Ohm's law. In 1927
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a Germans, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students for th ...
applied the quantum Fermi-Dirac distribution of electron energies to the Drude model, resulting in the
free electron model In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with ...
. A year later, Felix Bloch showed that electrons move in waves ( Bloch electrons) through a solid crystal lattice, so scattering off the lattice atoms as postulated in the Drude model is not a major process; the electrons scatter off impurity atoms and defects in the material. The final successor, the modern quantum band theory of solids, showed that the electrons in a solid cannot take on any energy as assumed in the Drude model but are restricted to energy bands, with gaps between them of energies that electrons are forbidden to have. The size of the band gap is a characteristic of a particular substance which has a great deal to do with its electrical resistivity, explaining why some substances are
electrical conductor In physics and electrical engineering, a conductor is an object or type of material that allows the flow of Electric charge, charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric ...
s, some
semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
s, and some insulators. While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the
siemens Siemens AG ( ) is a German Multinational corporation, multinational Conglomerate (company), conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad. The principal ...
, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers. In the 1920s, it was discovered that the current through a practical resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for the average current, in the case of ordinary resistive materials. Ohm's work long preceded
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
and any understanding of frequency-dependent effects in AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

# Scope

Ohm's law is an empirical law, a generalization from many experiments that have shown that current is approximately proportional to electric field for most materials. It is less fundamental than
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
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 Atomic spacing refers to the distance between the atomic nucleus, nuclei of atoms in a material. This space is extremely large compared to the nuclear size, size of the atomic nucleus, and is related to the chemical bonds which bind atoms together ...
, but experiments have not borne out this expectation. As of 2012, researchers have demonstrated that Ohm's law works for
silicon Silicon is a chemical element with the Symbol (chemistry), symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a Tetravalence, tetravalent metalloid and semiconductor. It is a member ...
wires as small as four atoms wide and one atom high.

# Microscopic origins

The dependence of the current density on the applied electric field is essentially quantum mechanical in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...
using the Drude model developed by Paul Drude in 1900. The Drude model treats
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
s (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 In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector (geometric), vector whose magnitude is the electric current per ...
and its relationship to ''E'' and is independent of the collisions. Drude calculated the average drift velocity from ''p'' = −''eEτ'' where ''p'' is the average
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
, −''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.

# Hydraulic analogy

A hydraulic analogy is sometimes used to describe Ohm's law. Water pressure, measured by pascals (or PSI), is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes water to flow. The water volume flow rate, as in
liter The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric units, metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubi ...
s per second, is the analog of current, as in
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In 2019 redefinition of the SI base units, the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant curre ...
s per second. Finally, flow restrictors—such as apertures placed in pipes between points where the water pressure is measured—are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge, that is, the electric current, through an electrical resistor is proportional to the difference in voltage measured across the resistor. More generally, the
hydraulic head Hydraulic head or piezometric head is a specific measurement of Fluid pressure#Hydrostatic pressure, liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22. It is usually measured as a liquid surface eleva ...
may be taken as the analog of voltage, and Ohm's law is then analogous to
Darcy's law Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of e ...
which relates hydraulic head to the volume flow rate via the
hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and Rock (geology), rocks, that describes the ease with which a fluid (usually water) can move through the porosity, pore space, or fractures ...
. Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohm analogy. The method can be applied to both steady and transient flow situations. In the linear
laminar flow In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
region, Poiseuille's law describes the hydraulic resistance of a pipe, but in the
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by Chaos theory, chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disrup ...
region the pressure–flow relations become nonlinear. The hydraulic analogy to Ohm's law has been used, for example, to approximate blood flow through the circulatory system.

# Circuit analysis

In circuit analysis, three equivalent expressions of Ohm's law are used interchangeably: :$I = \frac \quad \text\quad V = IR \quad \text \quad R = \frac.$ Each equation is quoted by some sources as the defining relationship of Ohm's law, or all three are quoted, or derived from a proportional form, or even just the two that do not correspond to Ohm's original statement may sometimes be given. The interchangeability of the equation may be represented by a triangle, where ''V'' (
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
) is placed on the top section, the ''I'' ( current) is placed to the left section, and the ''R'' ( resistance) is placed to the right. The divider between the top and bottom sections indicates division (hence the division bar).

## Resistive circuits

Resistor A resistor is a passivity (engineering), passive terminal (electronics), two-terminal electronic component, electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce c ...
s are circuit elements that impede the passage of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electron ...
in agreement with Ohm's law, and are designed to have a specific resistance value ''R''. In schematic diagrams, a resistor is shown as a long rectangle or zig-zag symbol. An element (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as an ''ohmic device'' (or an ''ohmic resistor'') because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range. Ohm's law holds for circuits containing only resistive elements (no capacitances or inductances) for all forms of driving voltage or current, regardless of whether the driving voltage or current is constant ( DC) or time-varying such as AC. At any instant of time Ohm's law is valid for such circuits. Resistors which are in '' series'' or in '' parallel'' may be grouped together into a single "equivalent resistance" in order to apply Ohm's law in analyzing the circuit.

## Reactive circuits with time-varying signals

When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
, so Ohm's law (as defined above) does not directly apply since that form contains only resistances having value ''R'', not complex impedances which may contain capacitance (''C'') or inductance (''L''). Equations for time-invariant AC circuits take the same form as Ohm's law. However, the variables are generalized to
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s and the current and voltage waveforms are complex exponentials. In this approach, a voltage or current waveform takes the form ''Ae'', where ''t'' is time, ''s'' is a complex parameter, and ''A'' is a complex scalar. In any
linear time-invariant system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant system, ...
, all of the currents and voltages can be expressed with the same ''s'' parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms. The complex generalization of resistance is impedance, usually denoted ''Z''; it can be shown that for an inductor, :$Z = sL$ and for a capacitor, :$Z = \frac.$ We can now write, :$V = Z\,I$ where ''V'' and ''I'' are the complex scalars in the voltage and current respectively and ''Z'' is the complex impedance. This form of Ohm's law, with ''Z'' taking the place of ''R'', generalizes the simpler form. When ''Z'' is complex, only the real part is responsible for dissipating heat. In a general AC circuit, ''Z'' varies strongly with the frequency parameter ''s'', and so also will the relationship between voltage and current. For the common case of a steady
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
, the ''s'' parameter is taken to be $j\omega$, corresponding to a complex sinusoid $Ae^$. The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and voltages in a circuit, which can be in different phases due to the different complex scalars.

## Linear approximations

Ohm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductors and circuit components (
resistor A resistor is a passivity (engineering), passive terminal (electronics), two-terminal electronic component, electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce c ...
s) specifically made for this behaviour. Both are ubiquitous in electrical engineering. Materials and components that obey Ohm's law are described as "ohmic" which means they produce the same value for resistance (''R'' = ''V''/''I'') regardless of the value of ''V'' or ''I'' which is applied and whether the applied voltage or current is DC (
direct current Direct current (DC) is one-directional electric current, flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor (material), conductor such as a wire, but can also flow throug ...
) of either positive or negative polarity or AC (
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
). In a true ohmic device, the same value of resistance will be calculated from ''R'' = ''V''/''I'' regardless of the value of the applied voltage ''V''. That is, the ratio of ''V''/''I'' is constant, and when current is plotted as a function of voltage the curve is ''linear'' (a straight line). If voltage is forced to some value ''V'', then that voltage ''V'' divided by measured current ''I'' will equal ''R''. Or if the current is forced to some value ''I'', then the measured voltage ''V'' divided by that current ''I'' is also ''R''. Since the plot of ''I'' versus ''V'' is a straight line, then it is also true that for any set of two different voltages ''V''1 and ''V''2 applied across a given device of resistance ''R'', producing currents ''I''1 = ''V''1/''R'' and ''I''2 = ''V''2/''R'', that the ratio (''V''1 − ''V''2)/(''I''1 − ''I''2) is also a constant equal to ''R''. The operator "delta" (Δ) is used to represent a difference in a quantity, so we can write Δ''V'' = ''V''1 − ''V''2 and Δ''I'' = ''I''1 − ''I''2. Summarizing, for any truly ohmic device having resistance ''R'', ''V''/''I'' = Δ''V''/Δ''I'' = ''R'' for any applied voltage or current or for the difference between any set of applied voltages or currents. There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their ''I''–''V'' curve) is ''nonlinear'' (or non-ohmic). An example is the p–n junction diode (curve at right). As seen in the figure, the current does not increase linearly with applied voltage for a diode. One can determine a value of current (''I'') for a given value of applied voltage (''V'') from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio ''V''/''I'' for some point along the nonlinear curve is sometimes called the ''static'', or ''chordal'', or DC, resistance, but as seen in the figure the value of total ''V'' over total ''I'' varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude Δ''V'' volts or Δ''I'' amps centered at that same point along the curve and measuring Δ''V''/Δ''I''. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the ''dynamic'', ''small-signal'', or ''incremental'' resistance, defined as the one over the slope of the ''V''–''I'' curve at the average value (DC operating point) of the voltage (that is, one over the
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the ''V''–''I'' curve at the DC operating point.

# Temperature effects

Ohm's law has sometimes been stated as, "for a conductor in a given state, the electromotive force is proportional to the current produced." That is, that the resistance, the ratio of the applied
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''Transd ...
(or voltage) to the current, "does not vary with the current strength ." The qualifier "in a given state" is usually interpreted as meaning "at a constant temperature," since the resistivity of materials is usually temperature dependent. Because the conduction of current is related to
Joule heating Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current through a conductor (material), conductor produces heat. Joule's first law (also just Joule's law), also known in c ...
of the conducting body, according to
Joule's first law Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical con ...
, the temperature of a conducting body may change when it carries a current. The dependence of resistance on temperature therefore makes resistance depend upon the current in a typical experimental setup, making the law in this form difficult to directly verify. Maxwell and others worked out several methods to test the law experimentally in 1876, controlling for heating effects.

# Relation to heat conductions

Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
in heat conductors when subjected to the influence of temperature differences. The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases. Specifically, solving a heat conduction (Fourier) problem with ''
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
'' (the driving "force") and '' flux of heat'' (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous
electrical conduction Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
(Ohm) problem having ''
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a unit of electric charge from a reference point to the sp ...
'' (the driving "force") and ''
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
'' (the rate of flow of the driven "quantity", i.e. charge) variables. The basis of Fourier's work was his clear conception and definition of
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials ...
. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavior will be lost when real materials (e.g. ones having a thermal conductivity that is a function of temperature) are subjected to large temperature gradients. A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

# Other versions

Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an
electrical circuit An electrical network is an interconnection of electronic component, electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of e ...
. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law: :$\mathbf = \rho \mathbf$ where "E" is the
electric field An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...
vector with units of volts per meter (analogous to "V" of Ohm's law which has units of volts), "J" is the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector (geometric), vector whose magnitude is the electric current per ...
vector with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "ρ" (Greek "rho") is the
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
with units of ohm·meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written as J = $\sigma$E where "σ" (Greek "sigma") is the conductivity which is the reciprocal of ρ. The voltage between two points is defined as: :$= -\int$ with $d \mathbf l$ the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation: :$V = \ \ \text \ \ E = \frac.$ Since the E field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density J will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:Lerner L, ''Physics for scientists and engineers'', Jones & Bartlett, 1997
pp. 732–733
/ref> :$J = \frac.$ Substituting the above 2 results (for ''E'' and ''J'' respectively) into the continuum form shown at the beginning of this section: :$\frac = \frac\rho \qquad \text \qquad V = I \rho \frac.$ The
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its Multiplicative inverse, reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares s ...
of a uniform conductor is given in terms of
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
by: :$= \rho \frac$ where ''l'' is the length of the conductor in SI units of meters, ''a'' is the cross-sectional area (for a round wire ''a'' = ''πr''2 if ''r'' is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters. After substitution of ''R'' from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form: :$=. \$ A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
,Seymour J, ''Physical Electronics'', pp. 48–49, Pitman, 1972 but a real metal has
crystallographic defect A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in Crystal, crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the Crysta ...
s, impurities, multiple
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbe ...
s, and thermal motion of the atoms. Electrons
scatter Scatter may refer to: * Scattering, in physics, the study of collisions * Statistical dispersion or scatter * Scatter (modeling), a substance used in the building of dioramas and model railways * Scatter, in computer programming, a parameter in Br ...
from all of these, resulting in resistance to their flow. The more complex generalized forms of Ohm's law are important to
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases which arise from electromagnetic forces between atoms. More ge ...
, which studies the properties of
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
and, in particular, its electronic structure. In broad terms, they fall under the topic of
constitutive equations In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially Kinetics (physics), kinetic quantities as related to Kinematics, kinematic quantities) that is specific to a ma ...
and the theory of transport coefficients.

## Magnetic effects

If an external B-field is present and the conductor is not at rest but moving at velocity v, then an extra term must be added to account for the current induced by the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
on the charge carriers. :$\mathbf = \sigma \left(\mathbf + \mathbf\times\mathbf\right)$ In the rest frame of the moving conductor this term drops out because v= 0. There is no contradiction because the electric field in the rest frame differs from the E-field in the lab frame: E′ = E + v×B. Electric and magnetic fields are relative, see
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
. If the current J is alternating because the applied voltage or E-field varies in time, then reactance must be added to resistance to account for self-inductance, see
electrical impedance In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of Electrical_resistance, resistance and Electrical_reactance, reactance in a electrical circuit, circuit. Quantitatively, the impedan ...
. The reactance may be strong if the frequency is high or the conductor is coiled.

## Conductive fluids

In a conductive fluid, such as a
plasma Plasma or plasm may refer to: Science * Plasma (physics) Plasma () 1, where \nu_ is the electron gyrofrequency and \nu_ is the electron collision rate. It is often the case that the electrons are magnetized while the ions are not. Magnetized ...
, there is a similar effect. Consider a fluid moving with the velocity $\mathbf$ in a magnetic field $\mathbf$. The relative motion induces an electric field $\mathbf$ which exerts
electric force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between char ...
on the charged particles giving rise to an
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
$\mathbf$. The equation of motion for the electron gas, with a
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration In chemistry, concentration is the Abundance (chemistry), abundance of a constituent divided by the total volume of a mixture. S ...
$n_e$, is written as :$m_en_e=-n_e e \mathbf+ n_e m_e \nu \left(\mathbf_i-\mathbf_e\right)-en_e\mathbf_e\times \mathbf,$ where $e$, $m_e$ and $\mathbf_e$ are the charge, mass and velocity of the electrons, respectively. Also, $\nu$ is the frequency of collisions of the electrons with ions which have a velocity field $\mathbf_i$. Since, the electron has a very small mass compared with that of ions, we can ignore the left hand side of the above equation to write :$\sigma\left(\mathbf+\mathbf\times \mathbf\right)=\mathbf,$ where we have used the definition of the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector (geometric), vector whose magnitude is the electric current per ...
, and also put $\sigma=$ which is the
electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
. This equation can also be equivalently written as : $\mathbf+\mathbf\times \mathbf=\rho\mathbf,$ where $\rho=\sigma^$ is the
electrical resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
. It is also common to write $\eta$ instead of $\rho$ which can be confusing since it is the same notation used for the magnetic diffusivity defined as $\eta=1/\mu_0\sigma$.

* Fick's law of diffusion * Hopkinson's law ("Ohm's law for magnetics") * Maximum power transfer theorem * Norton's theorem *
Sheet resistance Sheet resistance, is a measure of Electrical resistance, resistance of thin films that are uniform in thickness. It is commonly used to characterize materials made by semiconductor doping, metal deposition, resistive paste printing, and Insulated ...
* Superposition theorem *
Thermal noise A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
* Thévenin's theorem

# References

''Ohm's Law''
chapter fro

book an
series
* John C. Shedd and Mayo D. Hershe
"The History of Ohm's Law"
''
Popular Science ''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, inclu ...
'', December 1913, pp. 599–614, Bonnier Corporation , gives the history of Ohm's investigations, prior work, Ohm's false equation in the first paper, illustration of Ohm's experimental apparatus. * Explores the conceptual change underlying Ohm's experimental work. * Kenneth L. Caneva
"Ohm, Georg Simon."
'' Complete Dictionary of Scientific Biography''. 2008 * s:Scientific Memoirs/2/The Galvanic Circuit investigated Mathematically, a translation of Ohm's original paper. {{DEFAULTSORT:Ohm's Law Electronic engineering Circuit theorems Empirical laws Eponyms Electrical resistance and conductance Voltage
Law Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. ...