TheInfoList Ohm's law states that the
current Currents or The Current may refer to: Science and technology * Current (fluid) A current in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...
through a
conductor Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble like, for example, an orchestra. * Conductor (album), ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of ...
between two points is directly
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
to the
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ... across the two points. Introducing the constant of proportionality, the
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
, one arrives at the usual mathematical equation that describes this relationship: :$I = \frac,$ where is the current through the conductor in units of
ampere The ampere (, ; symbol: A), often shortened to "amp",SI supports only the use of symbols and deprecates the use of abbreviations for units. is the base unit of electric current An electric current is a stream of charged particles, such as ele ... s, ''V'' is the voltage measured ''across'' the conductor in units of
volt The volt is the derived unit for 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 ... s, and ''R'' is the
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
of the conductor in units of
ohm The ohm (symbol: Ω) is the SI derived unit SI derived units are units of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, ... s. 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 Georg Simon Ohm (, ; 16 March 1789 – 6 July 1854) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of inte ...
, 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 Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
form of the law used in
electromagnetics Electromagnetism is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics ...
and material science: :$\mathbf = \sigma \mathbf,$ where J is the
current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... at a given location in a resistive material, E is the electric field at that location, and ''σ'' (
sigma Sigma (uppercase Letter case is the distinction between the Letter (alphabet), letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written represent ... ) 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 coined ...
.

# History In January 1781, before
Georg Ohm Georg Simon Ohm (, ; 16 March 1789 – 6 July 1854) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of inte ...
's work,
Henry Cavendish Henry Cavendish FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resourc ...
experimented with
Leyden jar A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component An electronic component is any basic discrete device or physical entity in an electronic system Electronic may refer to: *Electronics, the scien ... 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 wor ...
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 –zinc Zinc is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the sa ... s, but later used a
thermocouple A thermocouple is an electrical device consisting of two dissimilar electrical conductors In physics and electrical engineering, a conductor is an object or type of material that allows the flow of Electric charge, charge (electrical current) ... 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 In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and b ... , ''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 A practical electrical power Electric power is the rate, per unit time, at which electrical energy Electrical energy is energy derived from electric potential energy or kinetic energy. When used loosely, ''electrical energy'' refers to ener ... 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 German 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 Samuel Finley Breese Morse (April 27, 1791 – April 2, 1872) was an American inventor and painter. After having established his reputation as a portrait painter, in his middle age Morse contributed to the of a single-wire system based on Eu ...
in 1855. The
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ... was discovered in 1897 by J. J. Thomson, and it was quickly realized that it is the particle (
charge carrier In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
) that carries electric currents in electric circuits. In 1900 the first (
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
) model of electrical conduction, the
Drude model The Drude model of electrical conduction Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that quantifies how strongly it resists electric current. Its inverse, ...
, was proposed by
Paul Drude Paul Karl Ludwig Drude (; 12 July 1863 – 5 July 1906) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of in ... , which finally gave a scientific explanation for Ohm's law. In this model, a solid conductor consists of a stationary lattice of
atom An atom is the smallest unit of ordinary 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 ato ... s (
ion An ion () is an atom An atom is the smallest unit of ordinary 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 ...
s), with
conduction electron In solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state ...
s moving randomly in it. A voltage across a conductor causes an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ... , 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ... (
thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concepts, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is ...
). 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 theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
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 German people, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students f ...
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 simple model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the Classical physics, classical Drude model with ...
. A year later,
Felix Bloch Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss Swiss may refer to: * the adjectival form of Switzerland *Swiss people Places *Swiss, Missouri *Swiss, North Carolina *Swiss, West Virginia *Swiss, Wisconsin Other uses *Swiss-sy ...
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 In solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state ph ...
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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
s, some
semiconductor A semiconductor material has an 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 curre ...
s, and some
insulators Insulator may refer to: * Insulator (electricity) An electrical insulator is a material in which the electron does not flow freely or the atom of the insulator have tightly bound electrons whose internal electric charge Electric charge is th ...
. While the old term for electrical conductance, the
mho The siemens (symbol: S) is the SI derived unit, derived unit of electric conductance, susceptance, electric susceptance, and admittance, electric admittance in the International System of Units (SI). Conductance, susceptance, and admittance are t ...
(the inverse of the resistance unit ohm), is still used, a new name, the
siemens Siemens AG ( ) is a German multinational Multinational may refer to: * Multinational corporation, a corporate organization operating in multiple countries * Multinational force, a military body from multiple countries * Multinational state, ...
, was adopted in 1971, honoring
Ernst Werner von Siemens Ernst Werner Siemens (von The term ''von'' is used in German language The German language (, ) is a West Germanic language mainly spoken in Central Europe. It is the most widely spoken and official or co-official language in Germany, ... . 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 Ensemble may refer to: Art * Musical ensemble A musical ensemble, also known as a music group or musical group, is a group of people who perform instrumental or vocal music, with the ensemble typically known by a distinct name. Some mus ...
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 are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
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 Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
, 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 are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
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 size of the atomic nucleus, and is related to the chemical bonds which bind atoms together. In solid mate ...
, 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 lustre, 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 Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon classical mechanics using the
Drude model The Drude model of electrical conduction Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that quantifies how strongly it resists electric current. Its inverse, ...
developed by
Paul Drude Paul Karl Ludwig Drude (; 12 July 1863 – 5 July 1906) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of in ... in 1900. The Drude model treats
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ... s (or other charge carriers) like pinballs bouncing among the
ion An ion () is an atom An atom is the smallest unit of ordinary 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 ...
s 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 velocityIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...
then determines the electric
current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... 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, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ... , −''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 The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current An electric current is a stream of cha ...
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. Water flow rate, as in
liter The litre (British English 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 cub ...
s per second, is the analog of current, as in
coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ... 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. 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 In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in mo ... region, Poiseuille's law describes the hydraulic resistance of a pipe, but in the
turbulent flow In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases ... 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 A network, in the context of electrical engineering and electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many te ...
, 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, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ... ) is placed on the top section, the ''I'' (
current Currents or The Current may refer to: Science and technology * Current (fluid) A current in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...
) is placed to the left section, and the ''R'' (
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
) 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 passive Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive (disambiguation), Pseudopassive * Passive language, a language from which an interpreter works * Passivity (b ... s are circuit elements that impede the passage of
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
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 Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
'' or in ''
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
'' 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 function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ... , 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 A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time- ...
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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s and the current and voltage waveforms are
complex exponential Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...
s. 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 system (or "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 ...
, 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 or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathemat ... , 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 passive Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive (disambiguation), Pseudopassive * Passive language, a language from which an interpreter works * Passivity (b ... 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 flow Flow may refer to: Science and technology * Flow (fluid) or fluid dynamics, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (math ...
) of either positive or negative polarity or AC (
alternating current Alternating current (AC) is an electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'natu ...
). 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... 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 Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the ...
(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 An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical condu ...
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 through a conductor produces heat. Joule's first law, also known as the Joule–Lenz law, :(''Energy dissipated pe ...
, 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ... 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'' (the driving "force") and ''flux, flux of heat'' (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having ''electric potential'' (the driving "force") and ''electric current'' (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. 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. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar (mathematics), 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 physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ... 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 Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... 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 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 of a uniform conductor is given in terms of resistivity by: :$= \rho \frac$ where ''l'' is the length of the conductor in International System of Units, 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,Seymour J, ''Physical Electronics'', pp. 48–49, Pitman, 1972 but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scattering, scatter 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, which studies the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of Green–Kubo relations, 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 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. 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. 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 (physics), plasma, 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 on the charged particles giving rise to an electric current $\mathbf$. The equation of motion for the electron gas, with a number density $n_e$, 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 Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... , and also put $\sigma=$ which is the electrical conductivity. This equation can also be equivalently written as : $\mathbf+\mathbf\times \mathbf=\rho\mathbf,$ where $\rho=\sigma^$ is the electrical resistivity. 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 * Magnetic circuit#Hopkinson's law: the magnetic analogy to Ohm's law, Hopkinson's law ("Ohm's law for magnetics") * Maximum power transfer theorem * Norton's theorem * Sheet resistance * Superposition theorem * Thermal noise * Thévenin's theorem