Electrical Resistivity and Conductivity - Wikipedia, The Free Encyclopedia

From Wikipedia, the free encyclopedia

Electrical resistivity (also known as resistivity, specific electrical resistance, or volume resistivity) is aproperty of a material; it quantifies how strongly the material opposes the flow of electric current. A lowresistivity indicates a material that readily allows the movement of electric charge. The SI unit of electricalresistivity is the ohm⋅metre (Ω⋅m). It is commonly represented by the Greek letter ρ (rho).

Electrical conductivity or specific conductance is the reciprocal quantity, and measures a material's ability toconduct an electric current. It is commonly represented by the Greek letter σ (sigma), but κ (kappa) (especially

in electrical engineering) or γ gamma are also occasionally used. Its SI unit is siemens per metre (S⋅m−1) and

CGSE unit is reciprocal second (s−1).

1 Definition1.1 Resistors or conductors with uniform cross-section1.2 General definition

2 Causes of conductivity2.1 Band theory simplified2.2 In metals2.3 In semiconductors and insulators2.4 In ionic liquids/electrolytes2.5 Superconductivity

3 Resistivity of various materials4 Temperature dependence

4.1 Linear approximation4.2 Metals4.3 Semiconductors

5 Complex resistivity and conductivity6 Tensor equations for anisotropic materials7 Resistance versus resistivity in complicated geometries8 Resistivity density products9 See also10 Notes11 References12 Further reading

Resistors or conductors with uniform cross-section

Many resistors and conductors have a uniform cross section with a uniform flow of electric current and aremade of one material. (See the diagram to the right.) In this case, the electrical resistivity ρ (Greek: rho) isdefined as:

Electrical resistivity and conductivity - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity

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A piece of resistive material with

electrical contacts on both ends.

where

R is the electrical resistance of a uniform specimen of the material(measured in ohms, Ω) is the length of the piece of material (measured in metres, m)

A is the cross-sectional area of the specimen (measured in squaremetres, m²).

The reason resistivity is defined this way is that it makes resistivity amaterial property, unlike resistance. All copper wires, no matter theirshape and size, have approximately the same resistivity, but a long, thincopper 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.

In a hydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipefull 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 notsolely determined by the presence or absence of sand; it also depends on how wide the pipe is (it is harder topush water through a skinny pipe than a wide one) and how long it is (it is harder to push water through a longpipe than a short one.)

The above equation can be transposed to get Pouillet's law:

The resistance of a given material will increase with the length, but decrease with greater cross-sectional area.From the above equations, resistivity has SI units of ohm⋅metre. Other units like ohm⋅cm or ohm⋅inch are alsosometimes used.

Conductivity is the inverse of resistivity:

Conductivity has SI units of Siemens per meter (S/m).

General definition

The above definition was specific to resistors or conductors with a uniform cross-section, where current flowsuniformly through them. A more basic and general definition starts from the fact that if there is electric fieldinside a material, it will cause electric current to flow. The electrical resistivity ρ is defined as the ratio of theelectric field to the density of the current it creates:

where


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Electron energy levels in an

insulator

ρ is the resistivity of the conductor material (measured in ohm⋅metres, Ω⋅m),

E is the magnitude of the electric field (in volts per metre, V⋅m−1),

J is the magnitude of the current density (in amperes per square metre, A⋅m−2),

in which E and J are inside the conductor.

Conductivity is the inverse:

For example, rubber is a material with large ρ and small σ, because even a very large electric field in rubber willcause 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.

Band theory simplified

Quantum mechanics states that electrons in an atom cannot take on any arbitraryenergy value. Rather, there are fixed energy levels which the electrons can occupy,

and values in between these levels are impossible[citation needed]. When a largenumber 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 energylevels together as an "energy band". There can be many such energy bands in amaterial, depending on the atomic number (number of electrons) and theirdistribution (besides external factors like environment modifying the energybands). Two such bands important in the discussion of conductivity of materialsare: the valence band and the conduction band (the latter is generally above the

former)[citation needed]. Electrons in the conduction band may move freelythroughout the material in the presence of an electrical field.

In insulators and semiconductors, the atoms in the substance influence each other so that between the valenceband and the conduction band there exists a forbidden band of energy levels, which the electrons cannotoccupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for itto leap across this forbidden gap and into the conduction band. Thus, even large voltages can yield relativelysmall currents.

In metals

A metal consists of a lattice of atoms, 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[citation needed]. This 'sea'of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (avoltage) is applied across the metal, the resulting electric field causes electrons to move from one end of theconductor to the other.

Near room temperatures, metals have resistance. The primary cause of this resistance is the thermal motion ofions. This acts to scatter electrons (due to destructive interference of free electron waves on non-correlating


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potentials of ions)[citation needed]. Also contributing to resistance in metals with impurities are the resulting

imperfections in the lattice. In pure metals this source is negligible[citation needed].

The larger the cross-sectional area of the conductor, the more electrons per unit length are available to carry thecurrent. As a result, the resistance is lower in larger cross-section conductors. The number of scattering eventsencountered 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.[1]

In semiconductors and insulators

Main articles: Semiconductor and Insulator (electricity)

In metals, the Fermi level lies in the conduction band (see Band Theory, above) giving rise to free conductionelectrons. However, in semiconductors the position of the Fermi level is within the band gap, approximatelyhalf-way between the conduction band minimum and valence band maximum for intrinsic (undoped)semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance isinfinite. However, the resistance will continue to decrease as the charge carrier density in the conduction bandincreases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier concentrationby donating electrons to the conduction band or accepting holes in the valence band. For both types of donor oracceptor atoms, increasing the dopant density leads to a reduction in the resistance, hence highly dopedsemiconductors behave metallically. At very high temperatures, the contribution of thermally generated carrierswill dominate over the contribution from dopant atoms and the resistance will decrease exponentially withtemperature.

In ionic liquids/electrolytes

Main article: Conductivity (electrolytic)

In electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions)traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by theconcentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. Inbiological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, areselective to specific ions and determine the membrane resistance.

Superconductivity

Main article: superconductivity

The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In ordinaryconductors, such as copper or silver, this decrease is limited by impurities and other defects. Even near absolutezero, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance dropsabruptly 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.[2]

In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above90 K (−183 °C). Such a high transition temperature is theoretically impossible for a conventionalsuperconductor, leading the materials to be termed high-temperature superconductors. Liquid nitrogen boils at77 K, facilitating many experiments and applications that are less practical at lower temperatures. Inconventional superconductors, electrons are held together in pairs by an attraction mediated by lattice phonons.


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The best available model of high-temperature superconductivity is still somewhat crude. There is a hypothesisthat electron pairing in high-temperature superconductors is mediated by short-range spin waves known as

paramagnons.[3]

Main article: Electrical resistivities of the elements (data page)

A conductor such as a metal has high conductivity and a low resistivity.An insulator like glass has low conductivity and a high resistivity.The conductivity of a semiconductor is generally intermediate, but varies widely under differentconditions, such as exposure of the material to electric fields or specific frequencies of light, and, mostimportant, with temperature and composition of the semiconductor material.

The degree of doping in semiconductors makes a large difference in conductivity. To a point, more doping leadsto higher conductivity. The conductivity of a solution of water is highly dependent on its concentration ofdissolved salts, and other chemical species that ionize in the solution. Electrical conductivity of water samples isused as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower theconductivity (the higher the resistivity). Conductivity measurements in water are often reported as specificconductance, relative to the conductivity of pure water at 25 °C. An EC meter is normally used to measureconductivity in a solution. A rough summary is as follows:

MaterialResistivityρ (Ω•m)

Metals 10−8

Semiconductors variable

Electrolytes variable

Insulators 1016

Superconductors 0

This table shows the resistivity, conductivity and temperature coefficient of various materials at 20 °C (68 °F)

Material ρ (Ω•m) at 20 °C σ (S/m) at 20 °CTemperature

coefficient[note 1]

(K−1)Reference

Silver 1.59×10−8 6.30×107 0.0038 [4][5]

Copper 1.68×10−8 5.96×107 0.0039 [5]

Annealed copper[note 2] 1.72×10−8 5.80×107 [citation needed]

Gold[note 3] 2.44×10−8 4.10×107 0.0034 [4]

Aluminium[note 4] 2.82×10−8 3.5×107 0.0039 [4]

Calcium 3.36×10−8 2.98×107 0.0041

Tungsten 5.60×10−8 1.79×107 0.0045 [4]


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coefficient[note 1]

(K−1)Reference

Zinc 5.90×10−8 1.69×107 0.0037 [6]

Nickel 6.99×10−8 1.43×107 0.006

Lithium 9.28×10−8 1.08×107 0.006

Iron 1.0×10−7 1.00×107 0.005 [4]

Platinum 1.06×10−7 9.43×106 0.00392 [4]

Tin 1.09×10−7 9.17×106 0.0045

Carbon steel (1010) 1.43×10−7 6.99×106 [7]

Lead 2.2×10−7 4.55×106 0.0039 [4]

Titanium 4.20×10−7 2.38×106 X

Grain oriented electricalsteel 4.60×10−7 2.17×106 [8]

Manganin 4.82×10−7 2.07×106 0.000002 [9]

Constantan 4.9×10−7 2.04×106 0.000008 [10]

Stainless steel[note 5] 6.9×10−7 1.45×106 [11]

Mercury 9.8×10−7 1.02×106 0.0009 [9]

Nichrome[note 6] 1.10×10−6 9.09×105 0.0004 [4]

GaAs 5×10−7 to 10×10−3 5×10−8 to 103 [12]

Carbon (amorphous) 5×10−4 to 8×10−4 1.25 to 2×103 −0.0005 [4][13]

Carbon (graphite)[note 7]2.5e×10−6 to 5.0×10−6

//basal plane

3.0×10−3 ⊥basal plane

2 to 3×105 //basalplane

3.3×102 ⊥basalplane

[14]

Carbon (diamond)[note 8] 1×1012 ~10−13 [15]

Germanium[note 8] 4.6×10−1 2.17 −0.048 [4][5]

Sea water[note 9] 2×10−1 4.8 [16]

Drinking water[note 10] 2×101 to 2×103 5×10−4 to 5×10−2 [citation needed]

Silicon[note 8] 6.40×102 1.56×10−3 −0.075 [4]

Deionized water[note 11] 1.8×105 5.5×10−6 [17]

Glass 10×1010 to 10×1014 10−11 to 10−15 ? [4][5]

Hard rubber 1×1013 10−14 ? [4]

Sulfur 1×1015 10−16 ? [4]


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coefficient[note 1]

(K−1)Reference

Air 1.3×1016 to 3.3×1016 3×10−15 to

8×10−15 [18]

Paraffin 1×1017 10−18 ?

Fused quartz 7.5×1017 1.3×10−18 ? [4]

PET 10×1020 10−21 ?

Teflon 10×1022 to 10×1024 10−25 to 10−23 ?

The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C valueis 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 0 °C.[19]

The extremely low resistivity (high conductivity) of silver is characteristic of metals. George Gamow tidilysummed 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 shellsof their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metalis filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displacedpersons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rushin the direction of the force, thus forming what we call an electric current." More technically, the free electronmodel gives a basic description of electron flow in metals.

Linear approximation

The electrical resistivity of most materials changes with temperature. If the temperature T does not vary toomuch, a linear approximation is typically used:

where is called the temperature coefficient of resistivity, is a fixed reference temperature (usually roomtemperature), and is the resistivity at temperature . The parameter is an empirical parameter fitted frommeasurement data. Because the linear approximation is only an approximation, is different for differentreference temperatures. For this reason it is usual to specify the temperature that was measured at with a

suffix, such as , and the relationship only holds in a range of temperatures around the reference.[20] Whenthe temperature varies over a large temperature range, the linear approximation is inadequate and a moredetailed analysis and understanding should be used.

Metals

In general, electrical resistivity of metals increases with temperature. Electron–phonon interactions can play akey role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperatureof 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


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formula:

where is the residual resistivity due to defect scattering, A is a constant that depends on the velocity ofelectrons at the Fermi surface, the Debye radius and the number density of electrons in the metal. is theDebye temperature as obtained from resistivity measurements and matches very closely with the values ofDebye temperature obtained from specific heat measurements. n is an integer that depends upon the nature ofinteraction:

n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)1.n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)2.n=2 implies that the resistance is due to electron–electron interaction.3.

If more than one source of scattering is simultaneously present, Matthiessen's Rule (first formulated by Augustus

Matthiessen in the 1860s) [21][22] says that the total resistance can be approximated by adding up severaldifferent 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 usuallyreaches 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 impurityconcentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effectknown as superconductivity.

An investigation of the low-temperature resistivity of metals was the motivation to Heike Kamerlingh Onnes'sexperiments that led in 1911 to discovery of superconductivity. For details see History of superconductivity.

Semiconductors

Main article: Semiconductor

In general, resistivity of intrinsic semiconductors decreases with increasing temperature. The electrons arebumped to the conduction energy band by thermal energy, where they flow freely and in doing so leave behindholes in the valence band which also flow freely. The electric resistance of a typical intrinsic (non doped)semiconductor decreases exponentially with the temperature:

An even better approximation of the temperature dependence of the resistivity of a semiconductor is given bythe Steinhart–Hart equation:

where A, B and C are the so-called Steinhart–Hart coefficients.

This equation is used to calibrate thermistors.

Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increasesstarting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors.


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After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly dueto the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsicsemiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally

generated carriers.[23]

In non-crystalline semiconductors, conduction can occur by charges quantum tunnelling from one localised siteto another. This is known as variable range hopping and has the characteristic form of

,

where n = 2, 3, 4, depending on the dimensionality of the system.

When analyzing the response of materials to alternating electric fields, in applications such as electrical

impedance tomography,[24] it is necessary to replace resistivity with a complex quantity called impeditivity (inanalogy to electrical impedance). Impeditivity is the sum of a real component, the resistivity, and an imaginarycomponent, the reactivity (in analogy to reactance). The magnitude of Impeditivity is the square root of sum ofsquares of magnitudes of resistivity and reactivity.

Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix ofcomplex numbers, in the case of anisotropic materials) called the admittivity. Admittivity is the sum of a realcomponent called the conductivity and an imaginary component called the 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 opaque the material is). For details, see Mathematicaldescriptions of opacity.

Some materials are anisotropic, meaning they have different properties in different directions. For example, acrystal 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.[14]

For an anisotropic material, it is not generally valid to use the scalar equations

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[25][26]

where the conductivity σ and resistivity ρ are rank-2 tensors (in other words, 3×3 matrices). The equations are

compactly illustrated in component form (using index notation and the summation convention) [27]:


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The σ and ρ tensors are inverses (in the sense of a matrix inverse). The individual components are notnecessarily inverses; for example σxx may not be equal to 1/ρxx.

If the material's resistivity is known, calculating the resistance of something made from it may, in some cases, bemuch more complicated than the formula above. One example is Spreading Resistance Profiling,where the material is inhomogeneous (different resistivity in different places), and the exact paths of currentflow are not obvious.

In cases like this, the formulas

need to be replaced with

where E and J are now vector fields. This equation, along with the continuity equation for J and the Poissonequation for E, form a set of partial differential equations. In special cases, an exact or approximate solution tothese equations can be worked out by hand, but for very accurate answers in complex cases, computer methodslike finite element analysis may be required.

In some applications where the weight of an item is very important resistivity density products are moreimportant than absolute low resistivity- it is often possible to make the conductor thicker to make up for a higherresistivity; and then a low resistivity density product material (or equivalently a high conductance to densityratio) is desirable. For example, for long distance overhead power lines— aluminium is frequently used ratherthan copper because it is lighter for the same conductance.

Material Resistivity (nΩ•m) Density (g/cm3)Resistivity-density

product (nΩ•m•g/cm3)

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


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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 products, but are rarely used for conductors due to their highreactivity with water and oxygen. Aluminium is far more stable. Two other important attributes, price andtoxicity, exclude the (otherwise) best choice: Beryllium. Thus, aluminium is usually the metal of choice whenthe weight of some required conduction (and/or the cost of conduction) is the driving consideration.

Conductivity near the percolation thresholdElectric effective resistanceElectrical resistivities of the elements (datapage)Electrical resistivity imaging

Ohm's lawSheet resistanceSI electromagnetism unitsSkin depth

^ The numbers in this column increase or decrease the significand portion of the resistivity. For example, at 30 °C(303 K), the resistivity of silver is 1.65×10−8. This is calculated as Δρ = α ΔT ρo where ρo is the resistivity at20 °C (in this case) and α is the temperature coefficient.

1.

^ Referred to as 100% IACS or International Annealed Copper Standard. The unit for expressing the conductivityof nonmagnetic materials by testing using the eddy-current method. Generally used for temper and alloy verificationof aluminium.

2.

^ Gold is commonly used in electrical contacts because it does not easily corrode.3.^ Commonly used for high voltage power lines4.^ 18% chromium/ 8% nickel austenitic stainles steel5.^ Nickel-iron-chromium alloy commonly used in heating elements.6.^ Graphite is strongly anisotropic.7.^ a b c The resistivity of semiconductors depends strongly on the presence of impurities in the material.8.^ Corresponds to an average salinity of 35 g/kg at 20 °C.9.^ This value range is typical of high quality drinking water and not an indicator of water quality10.^ Conductivity is lowest with monoatomic gases present; changes to 1.2×10−4 upon complete de-gassing, or to7.5×10−5 upon equilibration to the atmosphere due to dissolved CO2

11.

^ Suresh V Vettoor Electrical Conduction and Superconductivity (http://www.ias.ac.in/resonance/Sept2003/pdf/Sept2003p41-48.pdf) . ias.ac.in. September 2003

1.

^ John C. Gallop (1990). SQUIDS, the Josephson Effects and Superconducting Electronics(http://books.google.com/?id=ad8_JsfCdKQC&printsec=frontcover) . CRC Press. pp. 3, 20. ISBN 0-7503-0051-5.http://books.google.com/?id=ad8_JsfCdKQC&printsec=frontcover.

2.

^ Pines, D. (2002), "The Spin Fluctuation Model for High Temperature Superconductivity: Progress and Prospects",The Gap Symmetry and Fluctuations in High-Tc Superconductors, New York: Kluwer Academic, pp. 111-142,doi:10.1007/0-306-47081-0_7 (http://dx.doi.org/10.1007%2F0-306-47081-0_7) , ISBN 0-306-45934-5

3.

^ a b c d e f g h i j k l m n o Serway, Raymond A. (1998). Principles of Physics (2nd ed ed.). Fort Worth, Texas;London: Saunders College Pub. p. 602. ISBN 0-03-020457-7.

4.

^ a b c d Griffiths, David (1999) [1981]. "7. Electrodynamics". In Alison Reeves (ed.). Introduction toElectrodynamics (3rd edition ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 286. ISBN 0-13-805326-X.OCLC 40251748 (//www.worldcat.org/oclc/40251748) .

5.

^ Physical constants (http://physics.mipt.ru/S_III/t) . (PDF format; see page 2, table in the right lower corner)].Retrieved on 2011-12-17.

6.


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^ Matweb (http://www.matweb.com/search/DataSheet.aspx?MatGUID=025d4a04c2c640c9b0eaaef28318d761)7.^ JFE steel (http://www.jfe-steel.co.jp/en/products/electrical/catalog/f1e-001.pdf)8.^ a b Giancoli, Douglas C. (1995). Physics: Principles with Applications (4th ed ed.). London: Prentice Hall.ISBN 0-13-102153-2.(see also Table of Resistivity (http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/rstiv.html) )

9.

^ John O'Malley, Schaum's outline of theory and problems of basic circuit analysis, p. 19, McGraw-HillProfessional, 1992 ISBN 0-07-047824-4

10.

^ Glenn Elert (ed.), "Resistivity of steel" (http://hypertextbook.com/facts/2006/UmranUgur.shtml) , The PhysicsFactbook, retrieved and archived (http://www.webcitation.org/5zURdqzDl) 16 June 2011.

11.

^ Ohring, Milton (1995). Engineering materials science, Volume 1 (3rd edition ed.). p. 561.12.^ Y. Pauleau, Péter B. Barna, P. B. Barna, Protective coatings and thin films: synthesis, characterization, andapplications, p. 215, Springer, 1997 ISBN 0-7923-4380-8.

13.

^ a b Hugh O. Pierson, Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, andapplications, p. 61, William Andrew, 1993 ISBN 0-8155-1339-9.

14.

^ Lawrence S. Pan, Don R. Kania, Diamond: electronic properties and applications, p. 140, Springer, 1994 ISBN0-7923-9524-7.

15.

^ Physical properties of sea water (http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_9.html) .Kayelaby.npl.co.uk. Retrieved on 2011-12-17.

16.

^ Pashley, R. M.; Rzechowicz, M; Pashley, LR; Francis, MJ (2005). "De-Gassed Water is a Better Cleaning Agent".The Journal of Physical Chemistry B 109 (3): 1231. doi:10.1021/jp045975a (http://dx.doi.org/10.1021%2Fjp045975a) . PMID 16851085 (//www.ncbi.nlm.nih.gov/pubmed/16851085) .

17.

^ Pawar, S. D.; Murugavel, P.; Lal, D. M. (2009). "Effect of relative humidity and sea level pressure on electricalconductivity of air over Indian Ocean". Journal of Geophysical Research 114: D02205. Bibcode2009JGRD..11402205P (http://adsabs.harvard.edu/abs/2009JGRD..11402205P) . doi:10.1029/2007JD009716(http://dx.doi.org/10.1029%2F2007JD009716) .

18.

^ Copper Wire Tables (http://library.bldrdoc.gov/docs/nbshb100.pdf) . US Dep. of Commerce. National Bureau ofStandards Handbook. February 21, 1966

19.

^ Ward, MR, Electrical Engineering Science, pp. 36−40, McGraw-Hill, 1971.20.^ A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)21.^ A. Matthiessen, Progg. Anallen, 122, 47 (1864)22.^ Seymour J, Physical Electronics, chapter 2, Pitman, 197223.^ Otto H. Schmitt, University of Minnesota Mutual Impedivity Spectrometry and the Feasibility of its Incorporationinto Tissue-Diagnostic Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements(http://web.archive.org/web/20101013215436/http://www.otto-schmitt.org/OttoPagesFinalForm/Sounds/Speeches/MutualImpedivity.htm) . otto-schmitt.org. Retrieved on 2011-12-17.

24.

^ An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, ISBN0-582-44355-5

25.

^ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN978-0-521-57507-2

26.

^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge UniversityPress, 2010, ISBN 978-0-521-86153-3

27.

Paul Tipler (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and ElementaryModern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.

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