View source for Electrical resistance and conductance

{{short description|Opposition to the passage of an electric current}}
{{About|specific applications of conductivity and resistivity in electrical elements|other types of conductivity|Conductivity (disambiguation){{!}}Conductivity|electrical conductivity in general|Electrical resistivity and conductivity}}
{{Redirect|Resistive|the term used when referring to touchscreens|Resistive touchscreen}}
{{use dmy dates|date=July 2020}}
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{{Infobox physical quantity
| name = Electric resistance
| unit = ohm (Ω)
| symbols = {{mvar|R}}
| baseunits = kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−2</sup>
| dimension = <math>\mathsf{M} \mathsf{L}^2 \mathsf{T}^{-3} \mathsf{I}^{-2}</math>
}}
{{Infobox physical quantity
| name = Electric conductance 
| unit = siemens (S)
| symbols = {{mvar|G}}
| dimension = <math>\mathsf{M}^{-1} \mathsf{L}^{-2} \mathsf{T}^3 \mathsf{I}^2</math>
|baseunits=kg<sup>−1</sup>⋅m<sup>−2</sup>⋅s<sup>3</sup>⋅A<sup>2</sup>}}

The '''electrical resistance''' of an object is a measure of its opposition to the flow of [[electric current]]. Its [[Multiplicative inverse|reciprocal]] quantity is '''{{vanchor|electrical conductance|CONDUCTANCE}}''', measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with mechanical [[friction]]. The [[International System of Units|SI]] unit of electrical resistance is the [[ohm]] ({{math|[[Omega|Ω]]}}), while electrical conductance is measured in [[siemens (unit)|siemens]] (S) (formerly called the 'mho' and then represented by {{math|℧}}).

The resistance of an object depends in large part on the material it is made of. Objects made of [[electrical insulator]]s like [[rubber]] tend to have very high resistance and low conductance, while objects made of [[electrical conductor]]s like metals tend to have very low resistance and high conductance. This relationship is quantified by [[electrical resistivity and conductivity|resistivity or conductivity]]. The nature of a material is not the only factor in resistance and conductance, however; it also depends on the size and shape of an object because these properties are [[Intensive and extensive properties|extensive rather than intensive]]. For example, a wire's resistance is higher if it is long and thin, and lower if it is short and thick. All objects resist electrical current, except for [[superconductor]]s, which have a resistance of zero.

The resistance {{mvar|R}} of an object is defined as the ratio of [[voltage]] {{mvar|V}} across it to [[electric current|current]] {{mvar|I}} through it, while the conductance {{mvar|G}} is the reciprocal:
<math display=block>R = \frac{V}{I}, \qquad G = \frac{I}{V} = \frac{1}{R}.</math>

For a wide variety of materials and conditions, {{mvar|V}} and {{mvar|I}}  are directly proportional to each other, and therefore {{mvar|R}} and {{mvar|G}} are [[Constant (mathematics)|constants]] (although they will depend on the size and shape of the object, the material it is made of, and other factors like temperature or [[Deformation (mechanics)|strain]]). This proportionality is called [[Ohm's law]], and materials that satisfy it are called ''ohmic'' materials.

In other cases, such as a [[transformer]], [[diode]] or [[battery (electricity)|battery]], {{mvar|V}} and {{mvar|I}} are not directly proportional. The ratio {{mvar|{{sfrac|V|I}}}} is sometimes still useful, and is referred to as a ''chordal resistance'' or ''static resistance'',<ref name="brown">{{cite book |last1=Brown |first1=Forbes T. |title=Engineering System Dynamics: A Unified Graph-Centered Approach |date=2006 |publisher=CRC Press |location=Boca Raton, Florida |isbn=978-0-8493-9648-9 |page=43 |edition=2nd |url=https://books.google.com/books?id=UzqX4j9VZWcC}}</ref><ref name=kaiser>{{cite book |title=Electromagnetic Compatibility Handbook |first=Kenneth L. |last=Kaiser |publisher=CRC Press |location=Boca Raton, Florida |year=2004 |isbn=978-0-8493-2087-3 |pages=13{{ndash}}52 |url=https://books.google.com/books?id=nZzOAsroBIEC}}</ref> since it corresponds to the inverse slope of a [[Chord (geometry)|chord]] between the origin and an [[current–voltage characteristic|{{mvar|I}}–{{mvar|V}} curve]]. In other situations, the [[derivative]] <math display="inline"> \frac{\mathrm{d}V}{\mathrm{d}I} </math> may be most useful; this is called the ''differential resistance''.

==Introduction==

[[File:ResistanceHydraulicAnalogy.svg|thumb|350px|alt=analogy of resistance |The [[hydraulic analogy]] compares electric current flowing through circuits to water flowing through pipes. When a pipe (left) is filled with hair (right), it takes a larger pressure to achieve the same flow of water. Pushing electric current through a large resistance is like pushing water through a pipe clogged with hair: It requires a larger push ([[electromotive force]]) to drive the same flow ([[electric current]]).]]

In the [[hydraulic analogy]], current flowing through a wire (or [[resistor]]) is like water flowing through a pipe, and the [[voltage drop]] across the wire is like the [[pressure drop]] that pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow. 

The [[voltage drop]] (i.e., difference between voltages on one side of the resistor and the other), not the [[voltage]] itself, provides the driving force pushing current through a resistor. In hydraulics, it is similar: the pressure difference between two sides of a pipe, not the pressure itself, determines the flow through it. For example, there may be a large water pressure above the pipe, which tries to push water down through the pipe. But there may be an equally large water pressure below the pipe, which tries to push water back up through the pipe. If these pressures are equal, no water flows. (In the image at right, the water pressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is mostly determined by two properties:
* geometry (shape), and
* material

Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe. In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.

Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the same shape and size. Similarly, [[electron]]s can flow freely and easily through a [[copper]] wire, but cannot flow as easily through a [[steel]] wire of the same shape and size, and they essentially cannot flow at all through an [[insulator (electrical)|insulator]] like [[rubber]], regardless of its shape. The difference between copper, steel, and rubber is related to their microscopic structure and [[electron configuration]], and is quantified by a property called [[resistivity]].

In addition to geometry and material, there are various other factors that influence resistance and conductance, such as temperature; see [[#Dependence on other conditions|below]].

==Conductors and resistors==
[[File:Metal film resistor.jpg|thumb|250px|A 75 Ω [[resistor]], as identified by its [[electronic color code]] (violet–green–black–gold–red). <!-- This resistor uses the 4 band variant of the colour code so the bands represent 7 - 5 - 0 - x0.1</sup> making 75 Ω ---> An [[ohmmeter]] could be used to verify this value.]]

Substances in which electricity can flow are called [[electrical conductor|conductors]]. A piece of conducting material of a particular resistance meant for use in a circuit is called a [[resistor]]. Conductors are made of high-[[Electrical resistivity and conductivity|conductivity]] materials such as metals, in particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs.

==Ohm's law==
{{main|Ohm's law}}
[[File:FourIVcurves.svg|thumb|500px|The [[current–voltage characteristic]]s of four devices: Two [[resistor]]s, a [[diode]], and a [[Battery (electricity)|battery]]. The horizontal axis is [[voltage drop]], the vertical axis is [[electric current|current]]. Ohm's law is satisfied when the graph is a straight line through the origin. Therefore, the two resistors are ''ohmic'', but the diode and battery are not.]]

For many materials, the current {{mvar|I}} through the material is proportional to the voltage {{mvar|V}} applied across it:
<math display=block>I \propto V</math>
over a wide range of voltages and currents.  Therefore, the resistance and conductance of objects or electronic components made of these materials is constant. This relationship is called [[Ohm's law]], and materials which obey it are called ''ohmic'' materials.  Examples of ohmic components are wires and [[resistor]]s. The [[current–voltage characteristic|current–voltage graph]] of an ohmic device consists of a straight line through the origin with positive [[slope (mathematics)|slope]].

Other components and materials used in electronics do not obey Ohm's law; the current is not proportional to the voltage, so the resistance varies with the voltage and current through them.  These are called ''nonlinear'' or ''non-ohmic''. Examples include [[diode]]s and [[fluorescent lamp]]s.

== Relation to resistivity and conductivity ==
{{main|Electrical resistivity and conductivity}}
[[File:Resistivity geometry.png|thumb|A piece of resistive material with electrical contacts on both ends.]]

The resistance of a given object depends primarily on two factors: what material it is made of, and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area; for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire. Also, for a given material, the resistance is proportional to the length; for example, a long copper wire has higher resistance than an otherwise-identical short copper wire. The resistance {{math|R}} and conductance {{math|G}} of a conductor of uniform cross section, therefore, can be computed as

<math display=block>\begin{align}
  R &= \rho \frac{\ell}{A}, \\[5pt]
  G &= \sigma \frac{A}{\ell} \,.
\end{align}</math>

where <math>\ell</math> is the length of the conductor, measured in [[metre]]s (m), {{mvar|A}} is the cross-sectional area of the conductor measured in [[square metre]]s (m<sup>2</sup>), {{mvar|σ}} ([[Sigma (letter)|sigma]]) is the [[electrical conductivity]] measured in [[Siemens (unit)|siemens]] per meter (S·m<sup>−1</sup>), and {{mvar|ρ}} ([[Rho (letter)|rho]]) is the [[electrical resistivity]] (also called ''specific electrical resistance'') of the material, measured in ohm-metres (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are [[multiplicative inverse|reciprocals]]: <math>\rho=1/\sigma</math>. Resistivity is a measure of the material's ability to oppose electric current.

This formula is not exact, as it assumes the [[current density]] is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires.

Another situation for which this formula is not exact is with [[alternating current]] (AC), because the [[skin effect]] inhibits current flow near the center of the conductor. For this reason, the ''geometrical'' cross-section is different from the ''effective'' cross-section in which current actually flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC current, their resistances increase due to the [[proximity effect (electromagnetism)|proximity effect]]. At [[utility frequency|commercial power frequency]], these effects are significant for large conductors carrying large currents, such as [[busbar]]s in an [[electrical substation]],<ref>{{cite journal |last1=Fink |last2=Beaty |name-list-style=amp |title=Standard Handbook for Electrical Engineers |journal=Nature |year=1923 |volume=111 |issue=2788 |edition=11th |pages=17–19 |doi=10.1038/111458a0 |bibcode=1923Natur.111..458R |hdl=2027/mdp.39015065357108 |s2cid=26358546 |hdl-access=free }}</ref> or large power cables carrying more than a few hundred amperes.

The resistivity of different materials varies by an enormous amount: For example, the conductivity of [[PTFE|teflon]] is about 10<sup>30</sup> times lower than the conductivity of copper. Loosely speaking, this is because metals have large numbers of "delocalized" electrons that are not stuck in any one place, so they are free to move across large distances. In an insulator, such as Teflon, each electron is tightly bound to a single molecule so a great force is required to pull it away. [[Semiconductor]]s lie between these two extremes. More details can be found in the article: [[Electrical resistivity and conductivity]]. For the case of [[electrolyte]] solutions, see the article: [[Conductivity (electrolytic)]].

Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed to light. See [[#Dependence on other conditions|below]].

==Measurement==
{{main|Ohmmeter}}
[[File:Sanwa SM-1005s.jpg|thumb|alt=photograph of an ohmmeter |An [[ohmmeter]]]]
An instrument for measuring resistance is called an [[ohmmeter]].  Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use [[four-terminal sensing]].

==Typical values==
{{see also|Electrical resistivities of the elements (data page)|Electrical resistivity and conductivity}}

{| class="wikitable"
|+ Typical resistance values for selected objects
|-
! scope="col" | Component
! scope="col" | Resistance (Ω)
|-
| 1 meter of copper wire with 1&nbsp;mm diameter
| 0.02{{efn|The resistivity of copper is about {{val|1.7e-8|u=Ω.m}}.<ref name="Cutnell">{{cite book |last1=Cutnell |first1=John D. |last2=Johnson |first2=Kenneth W. |author-link2=Kenneth Alan Johnson |title=Physics |date=1992 |publisher=Wiley |location=New York |isbn=978-0-471-52919-4 |page=559 |edition=2nd |url=https://archive.org/details/physics00cutn/mode/2up? |url-access=registration}}</ref>
}}
|-
| {{val|1|u=km}} [[overhead power line]] (''typical'')
| 0.03<ref>{{cite book |first=John D. |last=McDonald |title=Electric Power Substations Engineering |edition=2nd |date=2016 |publisher=CRC Press |location=Boca Raton, Florida |isbn=978-1-4200-0731-2 |pages=363ff}}</ref>
|-
| [[AA battery]] (''typical [[internal resistance]]'')
| 0.1{{efn|For a fresh Energizer E91 AA alkaline battery, the internal resistance varies from {{val|0.9|u=Ω}} at {{val|−40|u=°C}}, to {{val|0.1|u=Ω}} at {{val|+40|u=°C}}.<ref>{{cite report |url=http://data.energizer.com/PDFs/BatteryIR.pdf |title=Battery internal resistance |publisher=Energizer Corp. |access-date=13 December 2011 |archive-date=11 January 2012 |archive-url=https://web.archive.org/web/20120111122430/http://data.energizer.com/PDFs/BatteryIR.pdf |url-status=dead }}</ref>}}
|-
| [[Incandescent light bulb]] filament (''typical'')
| 200–1000<ref group=lower-alpha>A {{val|60|u=W}} light bulb (in the USA, with {{val|120|u=V}} [[mains electricity]]) draws RMS current {{sfrac|{{val|60|u=W}}|{{val|120|u=V}}}} = {{val|500|u=mA}}, so its resistance is {{sfrac|{{val|120|u=V}}|{{val|500|u=mA}}}} = {{val|240|u=Ω}}. The resistance of a {{val|60|u=W}} light bulb in Europe ({{val|230|u=V}} mains) is {{val|900|u=Ω}}. The resistance of a filament is temperature-dependent; these values are for when the filament is already heated up and the light is already glowing.</ref>
|-
| Human body
| 1000–100,000{{efn|{{val|100|u=kΩ}} for dry skin contact, {{val|1|u=kΩ}} for wet or broken skin contact. High voltage breaks down the skin, lowering resistance to {{val|500|u=Ω}}. Other factors and conditions are relevant as well. For more details, see the [[electric shock]] article, and NIOSH&nbsp;98-131.<ref>{{cite web |url=https://www.cdc.gov/niosh/docs/98-131/pdfs/98-131.pdf |access-date=2014-11-02 |id=Publication No. 98-131 |title=Worker Deaths by Electrocution |publisher=[[National Institute for Occupational Safety and Health]]}}</ref>}}
|}

==Static and differential resistance==
{{see also|Small-signal model}}
{{multiple image
| align     = right
| direction = horizontal
| width     = 200
| image1    = DifferentialChordalResistance.svg
| width1    = 110
| alt1      = Differential versus chordal resistance
| caption1  = The [[Current–voltage characteristic|current–voltage curve]] of a non-ohmic device (purple).  The '''static resistance''' at point ''A'' is the [[Multiplicative inverse|inverse]] [[slope]] of line ''B'' through the origin. The '''differential resistance''' at ''A'' is the inverse slope of [[tangent line]] ''C''.
| image2    = Negative_differential_resistance.svg
| width2    = 90
| alt2      = Negative differential resistance
| caption2  = The [[Current–voltage characteristic|current–voltage curve]] of a component with [[negative resistance|negative differential resistance]], an unusual phenomenon where the current–voltage curve is non-[[Monotonic function|monotonic]].}}

Many electrical elements, such as [[diode]]s and [[battery (electricity)|batteries]] do {{em|not}} satisfy [[Ohm's law]]. These are called ''non-ohmic'' or ''non-linear'', and their [[Current–voltage characteristic|current–voltage curves]] are {{em|not}} straight lines through the origin.

Resistance and conductance can still be defined for non-ohmic elements. However, unlike ohmic resistance, non-linear resistance is not constant but varies with the voltage or current through the device; i.e., its [[Biasing|operating point]]. There are two types of resistance:<ref name=brown/><ref name=kaiser/>

{{glossary start}}
{{term|Static resistance}}
{{ghat|Also called '''chordal''' or '''DC resistance'''}}
{{defn|This corresponds to the usual definition of resistance; the voltage divided by the current
<math display=block>R_\mathrm{static} = \frac{U}{I} \,.</math>

It is the slope of the line ([[chord (geometry)|chord]]) from the origin through the point on the curve. Static resistance determines the power dissipation in an electrical component. Points on the current–voltage curve located in the 2nd or 4th quadrants, for which the slope of the chordal line is negative, have ''negative static resistance''. [[Passivity (engineering)|Passive]] devices, which have no source of energy, cannot have negative static resistance. However active devices such as transistors or [[op-amp]]s can synthesize negative static resistance with feedback, and it is used in some circuits such as [[gyrator]]s.
}}
{{term|Differential resistance}}
{{ghat|Also called '''dynamic''', '''incremental''', or '''small-signal resistance'''}}
{{defn|[[Electrical resistance#Static and differential resistance|Differential resistance]] is the derivative of the voltage with respect to the current; the [[slope]] of the current–voltage curve at a point
<math display=block>R_\mathrm{diff} = \frac { {\mathrm d}U}{ {\mathrm d}I} \,.</math>

If the current–voltage curve is non[[monotonic]] (with peaks and troughs), the curve has a negative slope in some regions—so in these regions the device has ''[[negative differential resistance]]''. Devices with negative differential resistance can amplify a signal applied to them, and are used to make amplifiers and oscillators. These include [[tunnel diode]]s, [[Gunn diode]]s, [[IMPATT diode]]s, [[magnetron]] tubes, and [[unijunction transistor]]s.
}}
{{glossary end}}

==AC circuits==
===Impedance and admittance===
{{main|Electrical impedance|Admittance}}

[[File:VI phase.svg|thumb|right|300px|The voltage (red) and current (blue) versus time (horizontal axis) for a [[capacitor]] (top) and [[inductor]] (bottom). Since the [[amplitude]] of the current and voltage [[Sine wave|sinusoids]] are the same, the [[absolute value]] of [[electrical impedance|impedance]] is 1 for both the capacitor and the inductor (in whatever units the graph is using). On the other hand, the [[phase (waves)|phase difference]] between current and voltage is −90° for the capacitor; therefore, the [[argument (complex analysis)|complex phase]] of the [[electrical impedance|impedance]] of the capacitor is −90°. Similarly, the [[phase (waves)|phase difference]] between current and voltage is +90° for the inductor; therefore, the complex phase of the impedance of the inductor is +90°.]]

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their [[Phase (waves)|phases]]. For example, in an ideal [[resistor]], the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a [[capacitor]] or [[inductor]], the maximum current flow occurs as the voltage passes through zero and vice versa (current and voltage are oscillating 90° out of phase, see image below). [[Complex number]]s are used to keep track of both the phase and magnitude of current and voltage:

<math display=block>\begin{array}{cl}
u(t) &= \operatorname\mathcal{R_e} \left( U_0 \cdot e^{j\omega t}\right) \\
i(t) &= \operatorname\mathcal{R_e} \left( I_0 \cdot e^{j(\omega t + \varphi)}\right) \\
Z &= \frac{U}{\ I\ } \\
Y &= \frac{\ 1\ }{Z} = \frac{\ I\ }{U}
\end{array}</math>

where:
* {{mvar|t}} is time;
* {{math|''u''(''t'')}} and {{math|''i''(''t'')}} are the voltage and current as a function of time, respectively;
* {{math|''U''<sub>0</sub>}} and {{math|''I''<sub>0</sub>}} indicate the amplitude of the voltage and current, respectively;
* <math>\omega</math> is the [[angular frequency]] of the AC current;
* <math>\varphi</math> is the displacement angle;
* {{mvar|U}} and {{mvar|I}} are the complex-valued voltage and current, respectively;
* {{mvar|Z}} and {{mvar|Y}} are the complex [[electrical impedance|impedance]] and [[admittance]], respectively;
* <math>\mathcal{R_e}</math> indicates the [[real part]] of a [[complex number]]; and
* <math>j \equiv \sqrt{-1\ }</math> is the [[imaginary unit]].

The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts:
<math display=block>\begin{align}
Z &= R + jX \\
Y &= G + jB ~.
\end{align}</math>

where {{mvar|R}} is resistance, {{mvar|G}} is conductance, {{mvar|X}} is [[electrical reactance|reactance]], and {{mvar|B}} is [[susceptance]]. These lead to the [[complex number]] identities
<math display=block>\begin{align}
R &= \frac{G}{\ G^2 + B^2\ }\ ,  \qquad & X = \frac{-B~}{\ G^2 + B^2\ }\ , \\
G &= \frac{R}{\ R^2 + X^2\ }\ ,  \qquad & B = \frac{-X~}{\ R^2 + X^2\ }\ ,
\end{align}</math>
which are true in all cases, whereas <math>\ R = 1/G\ </math> is only true in the special cases of either DC or reactance-free current.

The [[argument (complex analysis)|complex angle]] <math>\ \theta = \arg(Z) = -\arg(Y)\ </math> is the phase difference between the voltage and current passing through a component with impedance {{mvar|Z}}. For [[capacitor]]s and [[inductor]]s, this angle is exactly -90° or +90°, respectively, and {{mvar|X}} and {{mvar|B}} are nonzero. Ideal resistors have an angle of 0°, since {{mvar|X}} is zero (and hence {{mvar|B}} also), and {{mvar|Z}} and {{mvar|Y}} reduce to {{mvar|R}} and {{mvar|G}} respectively. In general, AC systems are designed to keep the phase angle close to 0° as much as possible, since it reduces the [[AC_power#Reactive power|reactive power]], which does no useful work at a load. In a simple case with an inductive load (causing the phase to increase), a capacitor may be added for compensation at one frequency, since the capacitor's phase shift is negative, bringing the total impedance phase closer to 0° again.

{{mvar|Y}} is the reciprocal of {{mvar|Z}} (<math>\ Z = 1/Y\ </math>) for all circuits, just as <math>R = 1/G</math> for DC circuits containing only resistors, or AC circuits for which either the reactance or susceptance happens to be zero ({{mvar|X}} or {{math|''B'' {{=}} 0}}, respectively) (if one is zero, then for realistic systems both must be zero).

===Frequency dependence ===
A key feature of AC circuits is that the resistance and conductance can be frequency-dependent, a phenomenon known as the [[universal dielectric response]].<ref>{{cite journal |title=Stress-dependent electrical transport and its universal scaling in granular materials |journal=Extreme Mechanics Letters |volume=22 |pages=83–88 |doi=10.1016/j.eml.2018.05.005 |year=2018 |last1=Zhai |first1=Chongpu |last2=Gan |first2=Yixiang |last3=Hanaor |first3=Dorian |last4=Proust |first4=Gwénaëlle |arxiv=1712.05938 |s2cid=51912472 }}</ref> One reason, mentioned above is the [[skin effect]] (and the related [[proximity effect (electromagnetism)|proximity effect]]). Another reason is that the resistivity itself may depend on frequency (see [[Drude model]], [[deep-level trap]]s, [[resonant frequency]], [[Kramers–Kronig relations]], etc.)

==Energy dissipation and Joule heating==
{{main|Joule heating}}
[[File:Cartridge-heater-hot.jpg|thumb|Running current through a material with resistance creates heat, in a phenomenon called [[Joule heating]]. In this picture, a [[cartridge heater]], warmed by Joule heating, is [[Incandescence|glowing red hot]].]]

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, electrical energy is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called ''[[Joule heating]]'' (after [[James Prescott Joule]]), also called ''ohmic heating'' or ''resistive heating''.

The dissipation of electrical energy is often undesired, particularly in the case of [[electric power transmission|transmission losses]] in [[overhead power line|power lines]]. [[Electric power transmission|High voltage transmission]] helps reduce the losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, for example in [[electric stove]]s and other [[electric heating|electric heaters]] (also called ''resistive heaters''). As another example, [[incandescent lamp]]s rely on Joule heating: the filament is heated to such a high temperature that it glows "white hot" with [[thermal radiation]] (also called [[incandescence]]).

The formula for Joule heating is:
<math display=block>P=I^2R</math>
where {{mvar|P}} is the [[electric power|power]] (energy per unit time) converted from electrical energy to thermal energy, {{mvar|R}} is the resistance, and {{mvar|I}} is the current through the resistor.

== Dependence on other conditions ==

=== Temperature dependence ===
{{main|Electrical resistivity and conductivity#Temperature dependence}}
Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For the detailed behavior and explanation, see [[Electrical resistivity and conductivity]].

As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a [[resistance thermometer]] or [[thermistor]]. (A resistance thermometer is made of metal, usually platinum, while a thermistor is made of ceramic or polymer.)

Resistance thermometers and thermistors are generally used in two ways. First, they can be used as [[thermometer]]s: by measuring the resistance, the temperature of the environment can be inferred. Second, they can be used in conjunction with [[Joule heating]] (also called self-heating): if a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to [[fuse (electrical)|fuses]], or for [[feedback]] in circuits, or for many other purposes. In general, self-heating can turn a resistor into a [[nonlinear element|nonlinear]] and [[hysteresis|hysteretic]] circuit element. For more details see [[Thermistor#Self-heating effects]].

If the temperature {{mvar|T}} does not vary too much, a [[linear approximation]] is typically used:
<math display=block>R(T) = R_0[1+\alpha (T - T_0)]</math>
where <math>\alpha</math> is called the ''temperature coefficient of resistance'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>R_0</math> is the resistance at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference.<ref>{{cite book |last=Ward |first=M.R. |title=Electrical Engineering Science |pages=36–40 |publisher=McGraw-Hill |year=1971}}</ref>

The temperature coefficient <math>\alpha</math> is typically {{val|+3e-3|u=K−1}} to {{val|+6e-3|u=K−1}} for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.{{efn|See [[Electrical resistivity and conductivity]] for a table. The temperature coefficient of resistivity is similar but not identical to the temperature coefficient of resistance. The small difference is due to [[thermal expansion]] changing the dimensions of the resistor.}}

=== Strain dependence ===
{{main|Strain gauge}}
Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon [[strain (materials science)|strain]].<ref>{{Citation|last=Meyer|first=Sebastian|display-authors = etal|date=2022|pages=114712|language=en|doi=10.1016/j.scriptamat.2022.114712|title=Volume 215|series=Scripta Materialia|chapter=Characterization of the deformation state of magnesium by electrical resistance|volume=215 |s2cid=247959452 |doi-access=free}}</ref> By placing a conductor under [[tension (mechanics)|tension]] (a form of [[stress (physics)|stress]] that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under [[Compression (physics)|compression]] (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on [[strain gauge]]s for details about devices constructed to take advantage of this effect.

=== Light illumination dependence ===
{{main|Photoresistor|Photoconductivity}}
Some resistors, particularly those made from [[semiconductor]]s, exhibit ''[[photoconductivity]]'', meaning that their resistance changes when light is shining on them. Therefore, they are called ''[[photoresistor]]s'' (or ''light dependent resistors''). These are a common type of [[photodetector|light detector]].

==Superconductivity==
{{main|Superconductivity}}
[[Superconductor]]s are materials that have exactly zero resistance and infinite conductance, because they can have {{math|1=''V'' = 0}} and {{math|1=''I'' ≠ 0}}. This also means there is no [[joule heating]], or in other words no [[dissipation]] of electrical energy. Therefore, if superconductive wire is made into a closed loop, current flows around the loop forever. Superconductors require cooling to temperatures near {{val|4|u=K}} with [[liquid helium]] for most metallic superconductors like [[niobium–tin]] alloys, or cooling to temperatures near {{val|77|u=K}} with [[liquid nitrogen]] for the expensive, brittle and delicate ceramic [[high temperature superconductors]].
Nevertheless, there are many [[technological applications of superconductivity]], including [[superconducting magnet]]s.

==See also==
{{Portal|Electronics}}
* [[Conductance quantum]]
** [[Von Klitzing constant]] (its reciprocal)
* [[Electrical measurements]]
* [[Contact resistance]]
* [[Electrical resistivity and conductivity]] for more information about the physical mechanisms for conduction in materials.
* [[Johnson–Nyquist noise]]
* [[Quantum Hall effect]], a standard for high-accuracy resistance measurements.
* [[Resistor]]
* [[RKM code]]
* [[Series and parallel circuits]]
* [[Sheet resistance]]
* [[SI electromagnetism units]]
* [[Thermal resistance]]
* [[Voltage divider]]
* [[Voltage drop]]

==Footnotes==
{{notelist|1}}

==References==
{{reflist|25em}}

==External links==
{{Commonscat|Electrical resistance and conductance}}
*{{cite web |url=http://www.cvel.clemson.edu/emc/calculators/Resistance_Calculator/index.html |archive-url=https://web.archive.org/web/20100711132344/http://www.cvel.clemson.edu/emc/calculators/Resistance_Calculator/index.html |archive-date=2010-07-11 |department=Vehicular Electronics Laboratory |title=Resistance calculator |publisher=Clemson University}}
*{{cite web |url=http://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/ |title=Electron conductance models using maximal entropy random walks |series=Wolfram Demonstrantions Project |website=wolfram.com}}

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[[Category:Electrical resistance and conductance| ]]
[[Category:Electricity]]
[[Category:Electromagnetic quantities]]