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| {{About|electrical conductivity in general|other types of conductivity|Conductivity (disambiguation){{!}}Conductivity|specific applications in electrical elements|Electrical resistance and conductance}}
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| '''Electrical resistivity''' (also known as '''resistivity''', '''specific electrical resistance''', or '''volume resistivity''') quantifies how strongly a given material opposes the flow of [[electric current]]. A low resistivity indicates a material that readily allows the movement of [[electric charge]]. Resistivity is commonly represented by the [[Greek alphabet|Greek letter]] ρ ([[Rho (letter)|rho]]). The [[SI]] unit of electrical resistivity is the [[ohm]]⋅[[metre]] (Ω⋅m)<ref>{{cite book|author=Lowrie |title=Fundamentals of Geophysics |url=http://books.google.com/books?id=h2-NjUg4RtEC&pg=PA254 |publisher=Cambridge University Press |isbn=978-1-139-46595-3 |pages=254–}}</ref><ref>{{cite book|author=Narinder Kumar |title=Comprehensive Physics XII |url=http://books.google.com/books?id=IryMtwHHngIC&pg=PA282 |year=2003 |publisher=Laxmi Publications |isbn=978-81-7008-592-8 |pages=282–}}</ref><ref>{{cite book|author=Eric Bogatin |title=Signal Integrity: Simplified |url=http://books.google.com/books?id=_IiONSphoB4C&pg=PA114 |year=2004 |publisher=Prentice Hall Professional |isbn=978-0-13-066946-9 |pages=114–}}</ref> although other units like [[ohm]]⋅[[centimetre]] (Ω⋅cm) are also in use. As an example, if a {{nowrap|1 × 1 × 1}} m solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.
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| '''Electrical conductivity''' or '''specific conductance''' is the reciprocal of electrical resistivity, and measures a material's ability to conduct an [[electric current]]. It is commonly represented by the Greek letter σ ([[Sigma (letter)|sigma]]), but κ ([[kappa]]) (especially in electrical engineering) or γ ([[gamma]]) are also occasionally used. Its SI unit is [[Siemens (unit)|siemens]] per [[metre]] (S/m) and [[Electrostatic units|CGSE unit]] is reciprocal [[second]] (s<sup>−1</sup>).
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| == Definition ==
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| === Resistors or conductors with uniform cross-section ===
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| [[Image:Resistivity geometry.png|thumb|A piece of resistive material with electrical contacts on both ends.]]
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| Many [[resistor]]s and [[electrical conductor|conductors]] have a uniform cross section with a uniform flow of electric current and are made of one material. (See the diagram to the right.) In this case, the electrical resistivity ''ρ'' (Greek: [[Rho (letter)|rho]]) is defined as:
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| :<math>\rho = R \frac{A}{\ell}, \,\!</math>
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| where
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| :''R'' is the [[electrical resistance]] of a uniform specimen of the material (measured in [[ohm]]s, Ω)
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| :''<math>\ell</math>'' is the length of the piece of material (measured in [[metre]]s, m)
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| :''A'' is the cross-sectional area of the specimen (measured in square metres, m<sup>2</sup>).
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| The reason resistivity is defined this way is that it makes resistivity an ''[[intrinsic property]]'', unlike resistance. All copper wires, irrespective of their shape and size, have approximately the same ''resistivity'', but a long, thin copper wire has a much larger ''resistance'' than a thick, short copper wire. Every material has its own characteristic resistivity – for example, rubber's resistivity is far larger than copper's.
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| In a [[hydraulic analogy]], passing current through a high-resistivity material is like pushing water through a pipe full of sand, while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. But resistance is not ''solely'' determined by the presence or absence of sand; it also depends on the length and width of the pipe: short or wide pipes will have lower resistance than narrow or long pipes.
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| The above equation can be transposed to get '''Pouillet's law''' (named after [[Claude Pouillet]]):
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| :<math>R = \rho \frac{\ell}{A}. \,\!</math>
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| The resistance of a given material will increase with the length, but decrease with increasing cross-sectional area. From the above equations, resistivity has [[SI]] units of [[ohm]]⋅[[metre]]. Other units like ohm⋅cm or ohm⋅inch are also sometimes used.
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| The formula <math>R = \rho \ell / A</math> can be used to intuitively understand the meaning of a resistivity value. For example, if <math>A=1\text{m}^2</math> and <math>\ell=1\text{m}</math> (forming a cube with perfectly-conductive contacts on opposite faces), then the resistance of this element in ohms is numerically equal to the resistivity of the material it is made of in ohm-meters. Likewise, a 1 ohm⋅cm material would have a resistance of 1 ohm if contacted on opposite faces of a 1 cm×1 cm×1 cm cube.
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| Conductivity ''σ'' (Greek: [[Sigma (letter)|sigma]]) is defined as the inverse of resistivity:
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| :<math>\sigma=\frac{1}{\rho}. \,\!</math>
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| Conductivity has SI units of [[Siemens (unit)|siemens]] per meter (S/m).
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| === General definition ===
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| The above definition was specific to resistors or conductors with a uniform cross-section, where current flows uniformly through them. A more basic and general definition starts from the fact that if there is [[electric field]] inside a material, it will cause [[electric current]] to flow. The electrical resistivity ''ρ'' is defined as the ratio of the electric field to the density of the current it creates: | |
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| :<math>\rho=\frac{E}{J}, \,\!</math>
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| where
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| :''ρ'' is the resistivity of the conductor material (measured in ohm⋅metres, Ω⋅m),
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| :''E'' is the [[Magnitude (mathematics)|magnitude]] of the [[electric field]] (in [[volt]]s per [[metre]], V⋅m<sup>−1</sup>),
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| :''J'' is the magnitude of the [[current density]] (in [[ampere]]s per [[square metre]], A⋅m<sup>−2</sup>),
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| in which ''E'' and ''J'' are inside the conductor.
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| Conductivity is the inverse:
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| :<math>\sigma=\frac{1}{\rho} = \frac{J}{E}. \,\!</math>
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| For example, rubber is a material with large ''ρ'' and small ''σ'', because even a very large electric field in rubber will cause almost no current to flow through it. On the other hand, copper is a material with small ''ρ'' and large ''σ'', because even a small electric field pulls a lot of current through it.
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| ==Causes of conductivity==
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| ===Band theory simplified===
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| {{see also|Band theory}}
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| {{Band structure filling diagram}}
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| [[Quantum mechanics]] states that electrons in an atom cannot take on any arbitrary energy value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible. When a large number of such allowed energy levels are spaced close together (in energy-space) i.e. have similar (minutely differing energies) then we can talk about these energy levels together as an "energy band". There can be many such energy bands in a material, depending on the atomic number (number of electrons) and their distribution (besides external factors like environment modifying the energy bands).
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| The material's electrons seek to minimize the total energy in the material by going to low energy states, however the [[Pauli exclusion principle]] means that they cannot all go to the lowest state. The electrons instead "fill up" the band structure starting from the bottom. The characteristic energy level up to which the electrons have filled is called the [[Fermi level]]. The position of the Fermi level with respect to the band structure is very important for electrical conduction: only electrons in energy levels near the [[Fermi level]] are free to move around since the electrons can easily jump among the partially occupied states in that region. In contrast, the low energy states are rigidly filled with a fixed number of electrons at all times, and the high energy states are empty of electrons at all times.
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| In metals there are many energy levels near the Fermi level, meaning that there are many electrons available to move. This is what causes the high electronic conductivity in metals.
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| An important part of band theory is that there may be forbidden bands in energy: energy intervals which do not contain any energy levels. In insulators and semiconductors, the number of electrons happens to be just the right amount to fill a certain integer number of low energy bands, exactly to the boundary. In this case, the Fermi level falls within a band gap. Since there are no available states near the Fermi level, and the electrons are not freely movable, the electronic conductivity is very low.
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| ===In metals===
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| A [[metal]] consists of a [[lattice]] of [[atom]]s, each with an outer shell of electrons which freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice.<ref>[http://web.archive.org/web/20120416024619/http://ibchem.com/IB/ibnotes/brief/bon-sl.htm Bonding (sl)]. ibchem.com</ref> This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (a [[voltage]]) is applied across the metal, the resulting [[electric field]] causes electrons to move from one end of the conductor to the other.
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| Near [[room temperature]]s, metals have resistance. The primary cause of this resistance is the thermal motion of ions. This acts to scatter electrons (due to destructive interference of free electron waves on non-correlating potentials of ions) .{{citation needed|date=June 2012}} Also contributing to resistance in metals with impurities are the resulting imperfections in the lattice. In pure metals this source is negligible .{{citation needed|date=June 2012}}
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| The larger the cross-sectional area of the conductor, the more electrons per unit length are available to carry the current. As a result, the resistance is lower in larger cross-section conductors. The number of scattering events encountered by an electron passing through a material is proportional to the length of the conductor. The longer the conductor, therefore, the higher the resistance. Different materials also affect the resistance.<ref>Suresh V Vettoor [http://www.ias.ac.in/resonance/Sept2003/pdf/Sept2003p41-48.pdf Electrical Conduction and Superconductivity]. ias.ac.in. September 2003</ref>
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| ===In semiconductors and insulators===
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| {{main|Semiconductor|Insulator (electricity)}}
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| In metals, the [[Fermi level]] lies in the conduction band (see Band Theory, above) giving rise to free conduction electrons. However, in [[semiconductors]] the position of the Fermi level is within the band gap, approximately half-way between the conduction band minimum and valence band maximum for intrinsic (undoped) semiconductors. This means that at 0 kelvin, there are no free conduction electrons and the resistance is infinite. However, the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, [[dopant]] atoms increase the majority charge carrier concentration by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance, hence highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.
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| ===In ionic liquids/electrolytes===
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| {{main|Conductivity (electrolytic)}}
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| In [[electrolyte]]s, electrical conduction happens not by band electrons or holes, but by full atomic species ([[ion]]s) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the concentration – while distilled water is almost an insulator, [[salt water]] is a very efficient electrical conductor. In [[cell membrane|biological membranes]], currents are carried by ionic salts. Small holes in the membranes, called [[ion channel]]s, are selective to specific ions and determine the membrane resistance.
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| ===Superconductivity===
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| {{main|Superconductivity}}
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| The electrical resistivity of a metallic [[electrical conductor|conductor]] decreases gradually as temperature is lowered. In ordinary [[Electrical conductor|conductors]], such as [[copper]] or [[silver]], this decrease is limited by impurities and other defects. Even near [[absolute zero]], a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An [[electric current]] flowing in a loop of [[superconducting wire]] can persist indefinitely with no power source.<ref name="Gallop">
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| {{cite book
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| |author=John C. Gallop
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| |year=1990
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| |title=SQUIDS, the Josephson Effects and Superconducting Electronics
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| |url=http://books.google.com/?id=ad8_JsfCdKQC&printsec=frontcover
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| |publisher=[[CRC Press]]
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| |pages=3, 20
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| |isbn=0-7503-0051-5
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| }}</ref>
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| In 1986, it was discovered that some [[cuprate]]-[[perovskite (structure)|perovskite]] [[ceramic]] materials have a critical temperature above {{convert|90|K|°C|abbr=on|0}}. Such a high transition temperature is theoretically impossible for a [[conventional superconductor]], leading the materials to be termed [[high-temperature superconductors]]. [[Liquid nitrogen]] boils at 77 K, facilitating many experiments and applications that are less practical at lower temperatures. In conventional superconductors, electrons are held together in pairs by an attraction mediated by lattice [[phonon]]s. The best available model of high-temperature superconductivity is still somewhat crude. There is a hypothesis that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons.<ref>{{Cite book
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| | author = D. Pines
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| | contribution = The Spin Fluctuation Model for High Temperature Superconductivity: Progress and Prospects
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| | year = 2002
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| | title = The Gap Symmetry and Fluctuations in High-Tc Superconductors
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| | pages = 111–142
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| | place = New York
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| | publisher = Kluwer Academic
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| | isbn = 0-306-45934-5
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| | doi = 10.1007/0-306-47081-0_7
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| | series = NATO Science Series: B:
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| | volume = 371
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| }}</ref>
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| ==Resistivity of various materials==<!-- [[Temperature coefficient]] links here -->
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| {{Main|Electrical resistivities of the elements (data page)}}
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| * A [[Electrical conductor|conductor]] such as a [[metal]] has high conductivity and a low resistivity.
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| * An [[Electrical insulation|insulator]] like [[glass]] has low conductivity and a high resistivity.
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| * The conductivity of a [[semiconductor]] is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of [[light]], and, most important, with [[temperature]] and composition of the semiconductor material.
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| The degree of [[Doping (semiconductor)|doping]] in semiconductors makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a [[Solution (chemistry)|solution]] of [[Water (molecule)|water]] is highly dependent on its [[concentration]] of dissolved [[salts]], and other chemical species that [[Ionization|ionize]] in the solution. Electrical conductivity of water samples is used as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower the conductivity (the higher the resistivity). Conductivity measurements in water are often reported as ''specific conductance'', relative to the conductivity of pure water at {{val|25|u=°C}}. An [[EC meter]] is normally used to measure conductivity in a solution. A rough summary is as follows:
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| {|class="wikitable"
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| |-
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| |'''Material'''
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| |'''Resistivity''' <br /> ρ (Ω·m)
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| |-
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| |[[Superconductors]]
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| | 0
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| |-
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| |[[Metal]]s
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| | 10<sup>−8</sup>
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| |-
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| |[[Semiconductor]]s
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| | variable
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| |-
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| |[[Electrolyte]]s
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| | variable
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| |-
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| |[[Electrical insulation|Insulators]]
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| | 10<sup>16</sup>
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| |}
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| This table shows the resistivity, conductivity and [[temperature coefficient]] of various materials at 20 [[Celsius|°C]] (68 [[Fahrenheit|°F]], 293 [[Kelvin|K]])
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| {| class="wikitable sortable"
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| |-
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| ! Material
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| ! class="unsortable" | ρ (Ω·m) at {{val|20|u=°C}}
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| ! class="unsortable" | σ (S/m) at {{val|20|u=°C}}
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| ! class="unsortable" | Temperature<br /> coefficient<ref group="note">The numbers in this column increase or decrease the [[significand]] portion of the resistivity. For example, at {{convert|30|C|K|abbr=on}}, the resistivity of silver is {{val|1.65|e=-8}}. This is calculated as {{nowrap|Δρ = α ΔT ρ<sub>o</sub>}} where ρ<sub>o</sub> is the resistivity at {{val|20|u=°C}} (in this case) and α is the temperature coefficient.</ref><br /> (K<sup>−1</sup>)
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| ! class="unsortable"| Reference
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| |[[Graphene|Carbon (graphene)]]||{{val|1|e=-8}}||-||-||<ref name="UMDnews">[https://newsdesk.umd.edu/scitech/release.cfm?ArticleID=1621 Physicists Show Electrons Can Travel More Than 100 Times Faster in Graphene]. Newsdesk.umd.edu (2008-03-24). Retrieved on 2014-02-03.</ref>
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| |[[Silver]]||{{val|1.59|e=-8}}||{{val|6.30|e=7}}||0.0038||<ref name="serway">{{cite book | author=Raymond A. Serway | title=Principles of Physics | edition=2nd| year=1998 | publisher=Saunders College Pub | location=Fort Worth, Texas; London | isbn=0-03-020457-7 | page=602}}</ref><ref name="Griffiths">{{cite book | author = David Griffiths| authorlink=David Griffiths (physicist)|editor =Alison Reeves| title = Introduction to Electrodynamics| origyear = 1981 | accessdate = 2006-01-29 | edition = 3rd | year = 1999 | publisher = [[Prentice Hall]] | location = Upper Saddle River, New Jersey | isbn = 0-13-805326-X | oclc = 40251748 | page = 286 | chapter = 7. Electrodynamics}}</ref>
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| |[[Copper]]||{{val|1.68|e=-8}}||{{val|5.96|e=7}}||0.003862||<ref name="Giancoli">{{cite book | author = Douglas Giancoli| editor =Jocelyn Phillips | title = Physics for Scientists and Engineers with Modern Physics| origyear = 1984| accessdate = 2013-03-04 | edition = 4th | year = 2009| publisher = [[Prentice Hall]] | location = Upper Saddle River, New Jersey | isbn = 0-13-149508-9 | page = 658 | chapter = 25. Electric Currents and Resistance}}</ref>
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| |[[Annealing (metallurgy)|Annealed]] [[copper]]<ref group="note">Referred to as 100% IACS or International Annealed Copper Standard. The unit for expressing the conductivity of nonmagnetic materials by testing using the [[Eddy current|eddy-current]] method. Generally used for temper and alloy verification of aluminium.</ref> ||{{val|1.72|e=-8}}||{{val|5.80|e=7}}||0.00393|| <ref>[http://archive.org/details/copperwiretables31unituoft Copper wire tables : United States. National Bureau of Standards : Free Download & Streaming : Internet Archive]. Archive.org (2001-03-10). Retrieved on 2014-02-03.</ref>
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| |[[Gold]]<ref group="note">Gold is commonly used in [[electrical contacts]] because it does not easily corrode.</ref>||{{val|2.44|e=-8}}|||{{val|4.10|e=7}}||0.0034||<ref name="serway"/>
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| |[[Aluminium]]<ref group="note">Commonly used for high voltage power lines</ref> ||{{val|2.82|e=-8}}||{{val|3.5|e=7}}||0.0039||<ref name="serway"/>
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| |-
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| |[[Calcium]]||{{val|3.36|e=-8}}||{{val|2.98|e=7}}||0.0041||
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| |-
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| |[[Tungsten]]||{{val|5.60|e=-8}}||{{val|1.79|e=7}}||0.0045||<ref name="serway"/>
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| |-
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| |[[Zinc]]||{{val|5.90|e=-8}}||{{val|1.69|e=7}}||0.0037||<ref>[http://physics.mipt.ru/S_III/t Physical constants]. (PDF format; see page 2, table in the right lower corner)]. Retrieved on 2011-12-17.</ref>
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| |[[Nickel]]||{{val|6.99|e=-8}}||{{val|1.43|e=7}}||0.006||
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| |-
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| |[[Lithium]]||{{val|9.28|e=-8}}||{{val|1.08|e=7}}||0.006||
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| |[[Iron]]||{{val|1.0|e=-7}}||{{val|1.00|e=7}}||0.005||<ref name="serway"/>
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| |[[Platinum]]||{{val|1.06|e=-7}}||{{val|9.43|e=6}}||0.00392||<ref name="serway"/>
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| |-
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| |[[Tin]]||{{val|1.09|e=-7}}||{{val|9.17|e=6}}||0.0045||
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| |[[Carbon steel]] (1010)||{{val|1.43|e=-7}}||{{val|6.99|e=6}}||||<ref>[http://www.matweb.com/search/DataSheet.aspx?MatGUID=025d4a04c2c640c9b0eaaef28318d761 AISI 1010 Steel, cold drawn]. Matweb</ref>
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| |-
| |
| |[[Lead]]||{{val|2.2|e=-7}}||{{val|4.55|e=6}}||0.0039||<ref name="serway"/>
| |
| |-
| |
| |[[Titanium]]||{{val|4.20|e=-7}}||{{val|2.38|e=6}}||X||
| |
| |-
| |
| |Grain oriented [[electrical steel]]||{{val|4.60|e=-7}}||{{val|2.17|e=6}}||||<ref>{{cite web|url=http://www.jfe-steel.co.jp/en/products/electrical/catalog/f1e-001.pdf |title=JFE steel |format=PDF |accessdate=2012-10-20}}</ref>
| |
| |-
| |
| |[[Manganin]]||{{val|4.82|e=-7}}||{{val|2.07|e=6}}||0.000002||<ref name="giancoli">{{cite book | author=Douglas C. Giancoli | title=Physics: Principles with Applications | edition=4th | year=1995 | publisher=Prentice Hall | location=London | isbn=0-13-102153-2}}<br>(see also [http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/rstiv.html Table of Resistivity]. hyperphysics.phy-astr.gsu.edu)
| |
| </ref>
| |
| |-
| |
| |[[Constantan]]||{{val|4.9|e=-7}}||{{val|2.04|e=6}}||0.000008||<ref>John O'Malley (1992) ''Schaum's outline of theory and problems of basic circuit analysis'', p. 19, McGraw-Hill Professional, ISBN 0-07-047824-4</ref>
| |
| |-
| |
| |[[Stainless steel]]<ref group="note">18% chromium/ 8% nickel austenitic stainless steel</ref>||{{val|6.9|e=-7}}||{{val|1.45|e=6}}|| || <ref>Glenn Elert (ed.), [http://hypertextbook.com/facts/2006/UmranUgur.shtml "Resistivity of steel"], ''The Physics Factbook'', retrieved and [http://www.webcitation.org/5zURdqzDl archived] 16 June 2011.</ref>
| |
| |-
| |
| |[[Mercury (element)|Mercury]]||{{val|9.8|e=-7}}||{{val|1.02|e=6}}||0.0009||<ref name="giancoli"/>
| |
| |-
| |
| |[[Nichrome]]<ref group="note">Nickel-iron-chromium alloy commonly used in heating elements.</ref>|| {{val|1.10|e=-6}}||{{val|9.09|e=5}}||{{val|0.0004}}||<ref name="serway"/>
| |
| |-
| |
| |[[GaAs]]||{{val|1|e=-3}} to {{val|1|e=8}}||{{val|1|e=-8}} to {{val|e=3}}|| ||<ref name="Ohring">{{cite book | author = Milton Ohring | title = Engineering materials science, Volume 1| edition = 3rd | year = 1995 | page = 561|isbn=0125249950|publisher=Academic Press}}</ref>
| |
| |-
| |
| |[[Amorphous carbon|Carbon (amorphous)]]||{{val|5|e=-4}} to {{val|8|e=-4}} <!-- 3.5×10<sup>−5</sup> Serway figure removed because unclear what form of carbon is being referenced-->||{{val|1.25|e=3}} to {{val|2|e=3}}|| −0.0005||<ref name="serway"/><ref>Y. Pauleau, Péter B. Barna, P. B. Barna (1997) ''Protective coatings and thin films: synthesis, characterization, and applications'', p. 215, Springer, ISBN 0-7923-4380-8.</ref>
| |
| |-
| |
| |[[Graphite|Carbon (graphite)]]<ref group="note">Graphite is strongly anisotropic.</ref>||{{val|2.5|e=-6}} to {{val|5.0|e=-6}} //[[basal plane]]<br />{{val|3.0|e=-3}} ⊥basal plane ||{{val|2|e=5}} to {{val|3|e=5}} //basal plane<br />{{val|3.3|e=2}} ⊥basal plane|| ||<ref name=Pierson>Hugh O. Pierson, ''Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, and applications'', p. 61, William Andrew, 1993 ISBN 0-8155-1339-9.</ref>
| |
| |-
| |
| |[[Diamond|Carbon (diamond)]]||data-sort-value="1E12"|{{val|1|e=12}}||~{{val|e=-13}}|| ||<ref>Lawrence S. Pan, Don R. Kania, ''Diamond: electronic properties and applications'', p. 140, Springer, 1994 ISBN 0-7923-9524-7.</ref>
| |
| |-
| |
| |[[Germanium]]<ref name="semi" group="note">The resistivity of [[semiconductor]]s depends strongly on the presence of [[Impurity|impurities]] in the material.</ref>||{{val|4.6|e=-1}}||2.17|| −0.048||<ref name="serway"/><ref name="Griffiths"/>
| |
| |-
| |
| |[[Sea water]]<ref group="note">Corresponds to an average salinity of 35 g/kg at {{val|20|u=°C}}.</ref>||{{val|2|e=-1}}||{{val|4.8}}|| ||<ref>[http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_9.html Physical properties of sea water]. Kayelaby.npl.co.uk. Retrieved on 2011-12-17.</ref>
| |
| |-
| |
| |[[Drinking water]]<ref group="note">This value range is typical of high quality drinking water and not an indicator of water quality</ref>||{{val|2|e=1}} to {{val|2|e=3}}||{{val|5|e=-4}} to {{val|5|e=-2}}|| ||{{Citation needed|date=January 2011}}
| |
| |-
| |
| |[[Silicon]]<ref name="semi" group="note"/>||{{val|6.40|e=2}}||{{val|1.56|e=-3}}||{{val|-0.075}}||<ref name="serway"/>
| |
| |-
| |
| |[[Wood|Wood (damp)]]||{{val|1|e=3}} to {{val|1|e=4}}||{{val|e=-4}} to {{val|e=-3}}||||<ref name="Transmission Lines data">[http://www.transmission-line.net/2011/07/electrical-properties-of-wood-poles.html Transmission Lines data]. Transmission-line.net. Retrieved on 2014-02-03.</ref>
| |
| |-
| |
| |[[Deionized water]]<ref group="note">Conductivity is lowest with monoatomic gases present; changes to {{val|1.2|e=-4}} upon complete de-gassing, or to {{val|7.5|e=-5}} upon equilibration to the atmosphere due to dissolved CO<sub>2</sub></ref> ||{{val|1.8|e=5}}||{{val|5.5|e=-6}}|| ||<ref>{{cite journal|doi=10.1021/jp045975a|title=De-Gassed Water is a Better Cleaning Agent|year=2005|author=R. M. Pashley , M. Rzechowicz, L. R. Pashley, and M. J. Francis|journal=The Journal of Physical Chemistry B|volume=109|pmid=16851085|issue=3|pages=1231–8}}</ref>
| |
| |-
| |
| |[[Glass]]||{{val|10|e=10}} to {{val|10|e=14}}||{{val|e=-11}} to {{val|e=-15}}||?||<ref name="serway"/><ref name="Griffiths"/>
| |
| |-
| |
| |[[Hard rubber]]||{{val|1|e=13}}||{{val|e=-14}}||?||<ref name="serway"/>
| |
| |-
| |
| |[[Wood|Wood (oven dry)]]||{{val|1|e=14}} to {{val|1|e=16}}||{{val|e=-16}} to {{val|e=-14}}||||<ref name="Transmission Lines data"/>
| |
| |-
| |
| |[[Sulfur]]||{{val|1|e=15}}||{{val|e=-16}}||?||<ref name="serway"/>
| |
| |-
| |
| |[[Air]]||{{val|1.3|e=16}} to {{val|3.3|e=16}}||{{val|3|e=-15}} to {{val|8|e=-15}}|| ||<ref>{{cite journal|doi=10.1029/2007JD009716|title=Effect of relative humidity and sea level pressure on electrical conductivity of air over Indian Ocean|year=2009|author= S. D. Pawar, P. Murugavel, D. M. Lal|journal=Journal of Geophysical Research|volume=114|pages=D02205|bibcode=2009JGRD..11402205P}}</ref>
| |
| |-
| |
| |[[PEDOT:PSS]]||{{val|1|e=-3}} to {{val|1|e=-1}}||{{val|1|e=1}} to {{val|1|e=3}}||?||
| |
| |-
| |
| |[[Fused quartz]]||{{val|7.5|e=17}}||{{val|1.3|e=-18}}||?||<ref name="serway"/>
| |
| |-
| |
| |[[Polyethylene terephthalate|PET]]||{{val|10|e=20}}||{{val|e=-21}}||?||
| |
| |-
| |
| |[[PTFE|Teflon]]||{{val|10|e=22}} to {{val|10|e=24}}||{{val|e=-25}} to {{val|e=-23}}||?||
| |
| |}
| |
| | |
| The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C value is only an approximation when used at other temperatures. For example, the coefficient becomes lower at higher temperatures for copper, and the value 0.00427 is commonly specified at {{val|0|u=°C}}.<ref>[http://library.bldrdoc.gov/docs/nbshb100.pdf Copper Wire Tables]. US Dep. of Commerce. National Bureau of Standards Handbook. February 21, 1966</ref>
| |
| | |
| The extremely low resistivity (high conductivity) of silver is characteristic of metals. [[George Gamow]] tidily summed up the nature of the metals' dealings with electrons in his science-popularizing book, ''One, Two, Three...Infinity'' (1947): "The metallic substances differ from all other materials by the fact that the outer shells of their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current." More technically, the [[free electron model]] gives a basic description of electron flow in metals.
| |
| | |
| Wood is widely regarded as an extremely good insulator, but its resistivity is sensitively dependent on moisture content, with damp wood being a factor of at least {{val|e=10}} worse insulator than oven-dry.<ref name="Transmission Lines data"/> In any case, a sufficiently high voltage – such as that in lightning strikes or some high-tension powerlines – can lead to insulation breakdown and electrocution risk even with apparently dry wood.
| |
| | |
| ==Temperature dependence==
| |
| | |
| ===Linear approximation===
| |
| The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a [[linear approximation]] is typically used:
| |
| :<math>\rho(T) = \rho_0[1+\alpha (T - T_0)]</math>
| |
| where <math>\alpha</math> is called the ''temperature coefficient of resistivity'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>\rho_0</math> is the resistivity 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>M.R. Ward (1971) ''Electrical Engineering Science'', pp. 36–40, McGraw-Hill.</ref> When the temperature varies over a large temperature range, the [[linear approximation]] is inadequate and a more detailed analysis and understanding should be used.
| |
| | |
| ===Metals===
| |
| In general, electrical resistivity of [[metal]]s increases with [[temperature]]. Electron–[[phonon]] interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity ρ of a metal is given by the Bloch–Grüneisen formula:
| |
| | |
| :<math>\rho(T)=\rho(0)+A\left(\frac{T}{\Theta_R}\right)^n\int_0^{\frac{\Theta_R}{T}}\frac{x^n}{(e^x-1)(1-e^{-x})}dx</math>
| |
| | |
| where <math>\rho(0)</math> is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the [[Fermi surface]], the [[Debye radius]] and the number density of electrons in the metal. <math>\Theta_R</math> is the [[Debye temperature]] as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:
| |
| | |
| # n=5 implies that the resistance is due to scattering of electrons by [[phonon]]s (as it is for simple metals)
| |
| # n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
| |
| # n=2 implies that the resistance is due to electron–electron interaction.
| |
| | |
| If more than one source of scattering is simultaneously present, Matthiessen's Rule
| |
| (first formulated by [[Augustus Matthiessen]] in the 1860s)
| |
| <ref>A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)</ref><ref>A. Matthiessen, Progg. Anallen, 122, 47 (1864)</ref> says that the total resistance can be approximated by adding up several different terms, each with the appropriate value of ''n''.
| |
| | |
| As the temperature of the metal is sufficiently reduced (so as to 'freeze' all the phonons), the resistivity usually reaches a
| |
| constant value, known as the '''residual resistivity'''. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as [[superconductivity]].
| |
| | |
| An investigation of the low-temperature resistivity of metals was the motivation to [[Heike Kamerlingh Onnes|Heike Kamerlingh Onnes's]] experiments that led in 1911 to discovery of [[superconductivity]]. For details see [[History of superconductivity]].
| |
| | |
| ===Semiconductors===
| |
| {{main|Semiconductor}}
| |
| In general, resistivity of [[intrinsic semiconductor]]s decreases with increasing temperature. The electrons are bumped to the [[conduction band|conduction energy band]] by thermal energy, where they flow freely and in doing so leave behind [[Electron hole|holes]] in the [[valence band]] which also flow freely. The electric resistance of a typical [[intrinsic semiconductor|intrinsic]] (non doped) [[semiconductor]] decreases [[exponential decay|exponentially]] with the temperature:
| |
| | |
| :<math>\rho= \rho_0 e^{-aT}\,</math>
| |
| | |
| An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the [[Steinhart–Hart equation]]:
| |
| | |
| :<math>1/T = A + B \ln(\rho) + C (\ln(\rho))^3 \,</math>
| |
| | |
| where ''A'', ''B'' and ''C'' are the so-called '''Steinhart–Hart coefficients'''.
| |
| | |
| This equation is used to calibrate [[thermistor]]s.
| |
| | |
| [[Extrinsic semiconductor|Extrinsic (doped) semiconductors]] have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.<ref>J. Seymour (1972) ''Physical Electronics'', chapter 2, Pitman</ref>
| |
| | |
| In non-crystalline semiconductors, conduction can occur by charges [[quantum tunnelling]] from one localised site to another. This is known as [[variable range hopping]] and has the characteristic form of
| |
| :<math>\rho = A\exp(T^{-1/n})</math>,
| |
| where ''n'' = 2, 3, 4, depending on the dimensionality of the system.
| |
| | |
| ==Complex resistivity and conductivity==
| |
| When analyzing the response of materials to alternating [[electric field]]s, in applications such as [[electrical impedance tomography]],<ref>Otto H. Schmitt, University of Minnesota [http://web.archive.org/web/20101013215436/http://www.otto-schmitt.org/OttoPagesFinalForm/Sounds/Speeches/MutualImpedivity.htm Mutual Impedivity Spectrometry and the Feasibility of its Incorporation into Tissue-Diagnostic Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements]. otto-schmitt.org. Retrieved on 2011-12-17.</ref> it is necessary to replace resistivity with a [[complex number|complex]] quantity called '''impeditivity''' (in analogy to [[electrical impedance]]). Impeditivity is the sum of a real component, the resistivity, and an imaginary component, the '''reactivity''' (in analogy to [[Reactance (electronics)|reactance]]). The magnitude of Impeditivity is the square root of sum of squares of magnitudes of resistivity and reactivity.
| |
| | |
| Conversely, in such cases the conductivity must be expressed as a [[complex number]] (or even as a matrix of complex numbers, in the case of [[anisotropic]] materials) called the ''[[Admittance|admittivity]]''. Admittivity is the sum of a real component called the conductivity and an imaginary component called the [[Susceptance|susceptivity]].
| |
| | |
| An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real [[permittivity]]. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more [[opacity (optics)|opaque]] the material is). For details, see [[Mathematical descriptions of opacity]].
| |
| | |
| ==Tensor equations for anisotropic materials==
| |
| Some materials are [[Anisotropy|anisotropic]], meaning they have different properties in different directions. For example, a [[crystal]] of [[graphite]] consists microscopically of a stack of sheets, and current flows very easily through each sheet, but moves much less easily from one sheet to the next.<ref name=Pierson/>
| |
| | |
| For an anisotropic material, it is not generally valid to use the scalar equations
| |
| :<math> J = \sigma E \,\, \rightleftharpoons \,\, E = \rho J . \,\!</math>
| |
| For example, the current may not flow in exactly the same direction as the electric field. Instead, the equations are generalized to the 3D tensor form<ref>J.R. Tyldesley (1975) ''An introduction to Tensor Analysis: For Engineers and Applied Scientists'', Longman, ISBN 0-582-44355-5</ref><ref>G. Woan (2010) ''The Cambridge Handbook of Physics Formulas'', Cambridge University Press, ISBN 978-0-521-57507-2</ref>
| |
| | |
| :<math> \mathbf{J} = \sigma \mathbf{E} \,\, \rightleftharpoons \,\, \mathbf{E} = \rho \mathbf{J} \,\!</math>
| |
| | |
| where the conductivity ''σ'' and resistivity ''ρ'' are rank-2 [[tensor]]s (in other words, 3×3 matrices). The equations are compactly illustrated in component form (using [[index notation]] and the [[Einstein notation|summation convention]]):<ref>K.F. Riley, M.P. Hobson, S.J. Bence (2010) ''Mathematical methods for physics and engineering'', Cambridge University Press, ISBN 978-0-521-86153-3</ref>
| |
| :<math> J_i = \sigma_{ij} E_j \,\, \rightleftharpoons \,\, E_i = \rho_{ij} J_j . \,\!</math>
| |
| The ''σ'' and ''ρ'' tensors are inverses (in the sense of a [[Invertible matrix|matrix inverse]]). The individual components are not necessarily inverses; for example, ''σ<sub>xx</sub>'' may not be equal to 1/''ρ<sub>xx</sub>''.
| |
| | |
| ==Resistance versus resistivity in complicated geometries==
| |
| If the material's resistivity is known, calculating the resistance of something made from it may, in some cases, be much more complicated than the formula <math>R = \rho \ell /A </math> above. One example is [[Spreading Resistance Profiling]], where the material is inhomogeneous (different resistivity in different places), and the exact paths of current flow are not obvious.
| |
| | |
| In cases like this, the formulas
| |
| :<math> J = \sigma E \,\, \rightleftharpoons \,\, E = \rho J \,\!</math>
| |
| need to be replaced with
| |
| :<math> \mathbf{J}(\mathbf{r}) = \sigma(\mathbf{r}) \mathbf{E}(\mathbf{r}) \,\, \rightleftharpoons \,\, \mathbf{E}(\mathbf{r}) = \rho(\mathbf{r}) \mathbf{J}(\mathbf{r}), \,\!</math>
| |
| where '''E''' and '''J''' are now [[vector field]]s. This equation, along with the [[continuity equation]] for '''J''' and the [[Poisson's equation]] for '''E''', form a set of [[partial differential equation]]s. In special cases, an exact or approximate solution to these equations can be worked out by hand, but for very accurate answers in complex cases, computer methods like [[Finite element method|finite element analysis]] may be required.
| |
| | |
| ==Resistivity density products==
| |
| In some applications where the weight of an item is very important resistivity density products are more important than absolute low resistivity – it is often possible to make the conductor thicker to make up for a higher resistivity; and then a low resistivity density product material (or equivalently a high conductance to density ratio) is desirable. For example, for long distance [[overhead power line]]s, aluminium is frequently used rather than copper because it is lighter for the same conductance.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Material
| |
| ! Resistivity (nΩ·m)
| |
| ! Density (g/cm<sup>3</sup>)
| |
| ! Resistivity-density <br />product (nΩ·m·g/cm<sup>3</sup>)
| |
| |-
| |
| | [[Sodium]]
| |
| | 47.7
| |
| | 0.97
| |
| | 46
| |
| |-
| |
| | [[Lithium]]
| |
| | 92.8
| |
| | 0.53
| |
| | 49
| |
| |-
| |
| | [[Calcium]]
| |
| | 33.6
| |
| | 1.55
| |
| | 52
| |
| |-
| |
| | [[Potassium]]
| |
| | 72.0
| |
| | 0.89
| |
| | 64
| |
| |-
| |
| | [[Beryllium]]
| |
| | 35.6
| |
| | 1.85
| |
| | 66
| |
| |-
| |
| | [[Aluminium]]
| |
| | 26.50
| |
| | 2.70
| |
| | 72
| |
| |-
| |
| | [[Magnesium]]
| |
| | 43.90
| |
| | 1.74
| |
| | 76.3
| |
| |-
| |
| | [[Copper]]
| |
| | 16.78
| |
| | 8.96
| |
| | 150
| |
| |-
| |
| | [[Silver]]
| |
| | 15.87
| |
| | 10.49
| |
| | 166
| |
| |-
| |
| | [[Gold]]
| |
| | 22.14
| |
| | 19.30
| |
| | 427
| |
| |-
| |
| | [[Iron]]
| |
| | 96.1
| |
| | 7.874
| |
| | 757
| |
| |}
| |
| | |
| Silver, although it is the least resistive metal known, has a high density and does poorly by this measure. Calcium and the alkali metals have the best resistivity-density products, but are rarely used for conductors due to their high reactivity with water and oxygen. Aluminium is far more stable. Two other important attributes, price and toxicity, exclude the (otherwise) best choice: Beryllium. Thus, aluminium is usually the metal of choice when the weight of some required conduction (and/or the cost of conduction) is the driving consideration.
| |
| | |
| ==See also==
| |
| {{colbegin}}
| |
| * [[Conductivity near the percolation threshold]]
| |
| * [[Contact resistance]]
| |
| * [[Electrical impedance]]
| |
| * [[Electrical resistivities of the elements (data page)]]
| |
| * [[Electrical resistivity tomography]]
| |
| * [[Ohm's law]]
| |
| * [[Sheet resistance]]
| |
| * [[SI electromagnetism units]]
| |
| * [[Skin effect]]
| |
| {{colend}}
| |
| | |
| ==Notes==
| |
| <references group="note" />
| |
| | |
| ==References==
| |
| {{Reflist|35em}}
| |
| | |
| ==Further reading==
| |
| * {{cite book | author= Paul Tipler| title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics |edition=5th | publisher=W. H. Freeman | year=2004 | isbn=0-7167-0810-8}}
| |
| | |
| ==External links==
| |
| * {{cite web|title=Electrical Conductivity|url=http://www.sixtysymbols.com/videos/conductivity.htm|work=Sixty Symbols|year=2010|publisher=[[Brady Haran]] for the [[University of Nottingham]]}}
| |
| | |
| {{DEFAULTSORT:Electrical Resistivity And Conductivity}}
| |
| [[Category:Electronics]]
| |
| [[Category:Physical quantities]]
| |
| [[Category:Materials science]]
| |