fbpx
Wikipedia

Electrical resistivity and conductivity

Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-metre (Ω⋅m).[1][2][3] For example, if a 1 m3 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.

Resistivity
Common symbols
ρ
SI unitohm metre (Ω⋅m)
In SI base unitskg⋅m3⋅s−3⋅A−2
Derivations from
other quantities
Dimension
Conductivity
Common symbols
σ, κ, γii
SI unitsiemens per metre (S/m)
Other units
z
Derivations from
other quantities
Dimension

Electrical conductivity (or specific conductance) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) and γ (gamma) are sometimes used. The SI unit of electrical conductivity is siemens per metre (S/m). Resistivity and conductivity are intensive properties of materials, giving the opposition of a standard cube of material to current. Electrical resistance and conductance are corresponding extensive properties that give the opposition of a specific object to electric current.

Definition edit

Ideal case edit

 
A piece of resistive material with electrical contacts on both ends.

In an ideal case, cross-section and physical composition of the examined material are uniform across the sample, and the electric field and current density are both parallel and constant everywhere. Many resistors and conductors do in fact have a uniform cross section with a uniform flow of electric current, and are made of a single material, so that this is a good model. (See the adjacent diagram.) When this is the case, the resistance of the conductor is directly proportional to its length and inversely proportional to its cross-sectional area, where the electrical resistivity ρ (Greek: rho) is the constant of proportionality. This is written as:

 
 

where

The resistivity can be expressed using the SI unit ohm metre (Ω⋅m) — i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length).

Both resistance and resistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an intrinsic property and does not depend on geometric properties of a material. This means that all pure copper (Cu) wires (which have not been subjected to distortion of their crystalline structure etc.), irrespective of their shape and size, have 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 has a far larger resistivity than copper.

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. Resistance, however, 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 have lower resistance than narrow or long pipes.

The above equation can be transposed to get Pouillet's law (named after Claude Pouillet):

 
The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, if A = 1 m2,   = 1 m (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 Ω⋅m.

Conductivity, σ, is the inverse of resistivity:

 

Conductivity has SI units of siemens per metre (S/m).

General scalar quantities edit

For less ideal cases, such as more complicated geometry, or when the current and electric field vary in different parts of the material, it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the electric field to the density of the current it creates at that point:

 

where

  •   is the resistivity of the conductor material,
  •   is the magnitude of the electric field,
  •   is the magnitude of the current density,

in which   and   are inside the conductor.

Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by:

 

For example, rubber is a material with large ρ and small σ — because even a very large electric field in rubber makes almost no current 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.

As shown below, this expression simplifies to a single number when the electric field and current density are constant in the material.

Tensor resistivity edit

When the resistivity of a material has a directional component, the most general definition of resistivity must be used. It starts from the tensor-vector form of Ohm's law, which relates the electric field inside a material to the electric current flow. This equation is completely general, meaning it is valid in all cases, including those mentioned above. However, this definition is the most complicated, so it is only directly used in anisotropic cases, where the more simple definitions cannot be applied. If the material is not anisotropic, it is safe to ignore the tensor-vector definition, and use a simpler expression instead.

Here, anisotropic means that the material has 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 much less easily from one sheet to the adjacent one.[4] In such cases, the current does not flow in exactly the same direction as the electric field. Thus, the appropriate equations are generalized to the three-dimensional tensor form:[5][6]

 

where the conductivity σ and resistivity ρ are rank-2 tensors, and electric field E and current density J are vectors. These tensors can be represented by 3×3 matrices, the vectors with 3×1 matrices, with matrix multiplication used on the right side of these equations. In matrix form, the resistivity relation is given by:

 

where

  •   is the electric field vector, with components (Ex, Ey, Ez);
  •   is the resistivity tensor, in general a three by three matrix;
  •   is the electric current density vector, with components (Jx, Jy, Jz).

Equivalently, resistivity can be given in the more compact Einstein notation:

 

In either case, the resulting expression for each electric field component is:

 

Since the choice of the coordinate system is free, the usual convention is to simplify the expression by choosing an x-axis parallel to the current direction, so Jy = Jz = 0. This leaves:

 

Conductivity is defined similarly:[7]

 

or

 

both resulting in:

 

Looking at the two expressions,   and   are the matrix inverse of each other. However, in the most general case, the individual matrix elements are not necessarily reciprocals of one another; for example, σxx may not be equal to 1/ρxx. This can be seen in the Hall effect, where   is nonzero. In the Hall effect, due to rotational invariance about the z-axis,   and  , so the relation between resistivity and conductivity simplifies to:[8]

 

If the electric field is parallel to the applied current,   and   are zero. When they are zero, one number,  , is enough to describe the electrical resistivity. It is then written as simply  , and this reduces to the simpler expression.

Conductivity and current carriers edit

Relation between current density and electric current velocity edit

Electric current is the ordered movement of electric charges.[2]

Causes of conductivity edit

Band theory simplified edit

 
Filling of the electronic states in various types of materials at equilibrium. Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution (black: all states filled, white: no state filled). In metals and semimetals the Fermi level EF lies inside at least one band.
In insulators and semiconductors the Fermi level is inside a band gap; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes.

According to elementary quantum mechanics, an electron in an atom or crystal can only have certain precise energy levels; energies between these levels are impossible. When a large number of such allowed levels have close-spaced energy values – i.e. have energies that differ only minutely – those close energy levels in combination are called an "energy band". There can be many such energy bands in a material, depending on the atomic number of the constituent atoms[a] and their distribution within the crystal.[b]

The material's electrons seek to minimize the total energy in the material by settling into low energy states; however, the Pauli exclusion principle means that only one can exist in each such state. So the electrons "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 or above the Fermi level are free to move within the broader material structure, since the electrons can easily jump among the partially occupied states in that region. In contrast, the low energy states are completely filled with a fixed limit on the number of electrons at all times, and the high energy states are empty of electrons at all times.

Electric current consists of a flow of electrons. In metals there are many electron energy levels near the Fermi level, so there are many electrons available to move. This is what causes the high electronic conductivity of metals.

An important part of band theory is that there may be forbidden bands of energy: energy intervals that contain no energy levels. In insulators and semiconductors, the number of electrons is 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.

In metals edit

Like balls in a Newton's cradle, electrons in a metal quickly transfer energy from one terminal to another, despite their own negligible movement.

A metal consists of a lattice of atoms, each with an outer shell of electrons that freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice.[9] 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 drift towards the positive terminal. The actual drift velocity of electrons is typically small, on the order of magnitude of metres per hour. However, due to the sheer number of moving electrons, even a slow drift velocity results in a large current density.[10] The mechanism is similar to transfer of momentum of balls in a Newton's cradle[11] but the rapid propagation of an electric energy along a wire is not due to the mechanical forces, but the propagation of an energy-carrying electromagnetic field guided by the wire.

Most metals have electrical resistance. In simpler models (non quantum mechanical models) this can be explained by replacing electrons and the crystal lattice by a wave-like structure. When the electron wave travels through the lattice, the waves interfere, which causes resistance. The more regular the lattice is, the less disturbance happens and thus the less resistance. The amount of resistance is thus mainly caused by two factors. First, it is caused by the temperature and thus amount of vibration of the crystal lattice. Higher temperatures cause bigger vibrations, which act as irregularities in the lattice. Second, the purity of the metal is relevant as a mixture of different ions is also an irregularity.[12][13] The small decrease in conductivity on melting of pure metals is due to the loss of long range crystalline order. The short range order remains and strong correlation between positions of ions results in coherence between waves diffracted by adjacent ions.[14]

In semiconductors and insulators edit

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, about halfway between the conduction band minimum (the bottom of the first band of unfilled electron energy levels) and the valence band maximum (the top of the band below the conduction band, of filled electron energy levels). That applies for intrinsic (undoped) semiconductors. This means that at absolute zero temperature, there would be no free conduction electrons, and the resistance is infinite. However, the resistance decreases as the charge carrier density (i.e., without introducing further complications, the density of electrons) 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 producing holes in the valence band. (A "hole" is a position where an electron is missing; such holes can behave in a similar way to electrons.) For both types of donor or acceptor atoms, increasing dopant density reduces resistance. Hence, highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers dominates over the contribution from dopant atoms, and the resistance decreases exponentially with temperature.

In ionic liquids/electrolytes edit

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 solutions (electrolytes) varies tremendously with concentration – while distilled water is almost an insulator, salt water is a reasonable electrical conductor. Conduction in ionic liquids is also controlled by the movement of ions, but here we are talking about molten salts rather than solvated ions. In biological membranes, currents are carried by ionic salts. Small holes in cell membranes, called ion channels, are selective to specific ions and determine the membrane resistance.

The concentration of ions in a liquid (e.g., in an aqueous solution) depends on the degree of dissociation of the dissolved substance, characterized by a dissociation coefficient  , which is the ratio of the concentration of ions   to the concentration of molecules of the dissolved substance  :

 

The specific electrical conductivity ( ) of a solution is equal to:

 

where  : module of the ion charge,   and  : mobility of positively and negatively charged ions,  : concentration of molecules of the dissolved substance,  : the coefficient of dissociation.

Superconductivity edit

 
Original data from the 1911 experiment by Heike Kamerlingh Onnes showing the resistance of a mercury wire as a function of temperature. The abrupt drop in resistance is the superconducting transition.

The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In normal (that is, non-superconducting) 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. In a normal conductor, the current is driven by a voltage gradient, whereas in a superconductor, there is no voltage gradient and the current is instead related to the phase gradient of the superconducting order parameter.[15] A consequence of this is that an electric current flowing in a loop of superconducting wire can persist indefinitely with no power source.[16]

In a class of superconductors known as type II superconductors, including all known high-temperature superconductors, an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of magnetic vortices in the electronic superfluid, which dissipates some of the energy carried by the current. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen so that the resistance of the material becomes truly zero.

Plasma edit

 
Lightning is an example of plasma present at Earth's surface. Typically, lightning discharges 30,000 amperes at up to 100 million volts, and emits light, radio waves, and X-rays.[17] Plasma temperatures in lightning might approach 30,000 kelvin (29,727 °C) (53,540 °F), or five times hotter than the temperature at the sun surface, and electron densities may exceed 1024 m−3.

Plasmas are very good conductors and electric potentials play an important role.

The potential as it exists on average in the space between charged particles, independent of the question of how it can be measured, is called the plasma potential, or space potential. If an electrode is inserted into a plasma, its potential generally lies considerably below the plasma potential, due to what is termed a Debye sheath. The good electrical conductivity of plasmas makes their electric fields very small. This results in the important concept of quasineutrality, which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma (ne = ⟨Z⟩ > ni), but on the scale of the Debye length there can be charge imbalance. In the special case that double layers are formed, the charge separation can extend some tens of Debye lengths.

The magnitude of the potentials and electric fields must be determined by means other than simply finding the net charge density. A common example is to assume that the electrons satisfy the Boltzmann relation:

 

Differentiating this relation provides a means to calculate the electric field from the density:

 

(∇ is the vector gradient operator; see nabla symbol and gradient for more information.)

It is possible to produce a plasma that is not quasineutral. An electron beam, for example, has only negative charges. The density of a non-neutral plasma must generally be very low, or it must be very small. Otherwise, the repulsive electrostatic force dissipates it.

In astrophysical plasmas, Debye screening prevents electric fields from directly affecting the plasma over large distances, i.e., greater than the Debye length. However, the existence of charged particles causes the plasma to generate, and be affected by, magnetic fields. This can and does cause extremely complex behavior, such as the generation of plasma double layers, an object that separates charge over a few tens of Debye lengths. The dynamics of plasmas interacting with external and self-generated magnetic fields are studied in the academic discipline of magnetohydrodynamics.

Plasma is often called the fourth state of matter after solid, liquids and gases.[18][19] It is distinct from these and other lower-energy states of matter. Although it is closely related to the gas phase in that it also has no definite form or volume, it differs in a number of ways, including the following:

Property Gas Plasma
Electrical conductivity Very low: air is an excellent insulator until it breaks down into plasma at electric field strengths above 30 kilovolts per centimetre.[20] Usually very high: for many purposes, the conductivity of a plasma may be treated as infinite.
Independently acting species One: all gas particles behave in a similar way, influenced by gravity and by collisions with one another. Two or three: electrons, ions, protons and neutrons can be distinguished by the sign and value of their charge so that they behave independently in many circumstances, with different bulk velocities and temperatures, allowing phenomena such as new types of waves and instabilities.
Velocity distribution Maxwellian: collisions usually lead to a Maxwellian velocity distribution of all gas particles, with very few relatively fast particles. Often non-Maxwellian: collisional interactions are often weak in hot plasmas and external forcing can drive the plasma far from local equilibrium and lead to a significant population of unusually fast particles.
Interactions Binary: two-particle collisions are the rule, three-body collisions extremely rare. Collective: waves, or organized motion of plasma, are very important because the particles can interact at long ranges through the electric and magnetic forces.

Resistivity and conductivity of various materials edit

  • 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 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.

The degree of semiconductors doping makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a water/aqueous solution is highly dependent on its concentration of dissolved salts, and other chemical species that 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 25 °C. An EC meter is normally used to measure conductivity in a solution. A rough summary is as follows:

Resistivity of classes of materials
Material Resistivity, ρ (Ω·m)
Superconductors 0
Metals 10−8
Semiconductors Variable
Electrolytes Variable
Insulators 1016
Superinsulators

This table shows the resistivity (ρ), conductivity and temperature coefficient of various materials at 20 °C (68 °F; 293 K).

Resistivity, conductivity, and temperature coefficient for several materials
Material Resistivity, ρ,
at 20 °C (Ω·m)
Conductivity, σ,
at 20 °C (S/m)
Temperature
coefficient[c] (K−1)
Reference
Silver[d] 1.59×10−8 6.30×107 3.80×10−3 [21][22]
Copper[e] 1.68×10−8 5.96×107 4.04×10−3 [23][24]
Annealed copper[f] 1.72×10−8 5.80×107 3.93×10−3 [25]
Gold[g] 2.44×10−8 4.11×107 3.40×10−3 [21]
Aluminium[h] 2.65×10−8 3.77×107 3.90×10−3 [21]
Brass (5% Zn) 3.00×10−8 3.34×107 [26]
Calcium 3.36×10−8 2.98×107 4.10×10−3
Rhodium 4.33×10−8 2.31×107
Tungsten 5.60×10−8 1.79×107 4.50×10−3 [21]
Zinc 5.90×10−8 1.69×107 3.70×10−3 [27]
Brass (30% Zn) 5.99×10−8 1.67×107 [28]
Cobalt[i] 6.24×10−8 1.60×107 7.00×10−3[30]
[unreliable source?]
Nickel 6.99×10−8 1.43×107 6.00×10−3
Ruthenium[i] 7.10×10−8 1.41×107
Lithium 9.28×10−8 1.08×107 6.00×10−3
Iron 9.70×10−8 1.03×107 5.00×10−3 [21]
Platinum 10.6×10−8 9.43×106 3.92×10−3 [21]
Tin 10.9×10−8 9.17×106 4.50×10−3
Phosphor Bronze (0.2% P / 5% Sn) 11.2×10−8 8.94×106 [31]
Gallium 14.0×10−8 7.10×106 4.00×10−3
Niobium 14.0×10−8 7.00×106 [32]
Carbon steel (1010) 14.3×10−8 6.99×106 [33]
Lead 22.0×10−8 4.55×106 3.90×10−3 [21]
Galinstan 28.9×10−8 3.46×106 [34]
Titanium 42.0×10−8 2.38×106 3.80×10−3
Grain oriented electrical steel 46.0×10−8 2.17×106 [35]
Manganin 48.2×10−8 2.07×106 0.002×10−3 [36]
Constantan 49.0×10−8 2.04×106 0.008×10−3 [37]
Stainless steel[j] 69.0×10−8 1.45×106 0.94×10−3 [38]
Mercury 98.0×10−8 1.02×106 0.90×10−3 [36]
Bismuth 129×10−8 7.75×105
Manganese 144×10−8 6.94×105
Plutonium[39] (0 °C) 146×10−8 6.85×105
Nichrome[k] 110×10−8 6.70×105
[citation needed]
0.40×10−3 [21]
Carbon (graphite)
parallel to basal plane[l]
250×10−8 to 500×10−8 2×105 to 3×105
[citation needed]
[4]
Carbon (amorphous) 0.5×10−3 to 0.8×10−3 1.25×103 to 2.00×103 −0.50×10−3 [21][40]
Carbon (graphite)
perpendicular to basal plane
3.0×10−3 3.3×102 [4]
GaAs 10−3 to 108
[clarification needed]
10−8 to 103
[dubious ]
[41]
Germanium[m] 4.6×10−1 2.17 −48.0×10−3 [21][22]
Sea water[n] 2.1×10−1 4.8 [42]
Swimming pool water[o] 3.3×10−1 to 4.0×10−1 0.25 to 0.30 [43]
Drinking water[p] 2×101 to 2×103 5×10−4 to 5×10−2 [citation needed]
Silicon[m] 2.3×103 4.35×10−4 −75.0×10−3 [44][21]
Wood (damp) 103 to 104 10−4 to 10−3 [45]
Deionized water[q] 1.8×105 4.2×10−5 [46]
Ultrapure water 1.82×109 5.49×10−10 [47][48]
Glass 1011 to 1015 10−15 to 10−11 [21][22]
Carbon (diamond) 1012 ~10−13 [49]
Hard rubber 1013 10−14 [21]
Air 109 to 1015 ~10−15 to 10−9 [50][51]
Wood (oven dry) 1014 to 1016 10−16 to 10−14 [45]
Sulfur 1015 10−16 [21]
Fused quartz 7.5×1017 1.3×10−18 [21]
PET 1021 10−21
PTFE (teflon) 1023 to 1025 10−25 to 10−23

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 0 °C.[52]

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 popular science 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 1010 worse insulator than oven-dry.[45] In any case, a sufficiently high voltage – such as that in lightning strikes or some high-tension power lines – can lead to insulation breakdown and electrocution risk even with apparently dry wood.[citation needed]

Temperature dependence edit

Linear approximation edit

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

 

where   is called the temperature coefficient of resistivity,   is a fixed reference temperature (usually room temperature), and   is the resistivity at temperature  . The parameter   is an empirical parameter fitted from measurement data, equal to 1/ [clarify]. Because the linear approximation is only an approximation,   is different for different reference 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.[53] 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 edit

In general, electrical resistivity of metals 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 can be approximated through the Bloch–Grüneisen formula:[54]

 

where   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.   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 phonons (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.

The Bloch–Grüneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the first Brillouin zone and a Debye phonon spectrum.[55]

If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated by Augustus Matthiessen in the 1860s)[56][57] states 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's experiments that led in 1911 to discovery of superconductivity. For details see History of superconductivity.

Wiedemann–Franz law edit

The Wiedemann–Franz law states that for materials where heat and charge transport is dominated by electrons, the ratio of thermal to electrical conductivity is proportional to the temperature:

 

where   is the thermal conductivity,   is the Boltzmann constant,   is the electron charge,   is temperature, and   is the electric conductivity. The ratio on the rhs is called the Lorenz number.

Semiconductors edit

In general, intrinsic semiconductor resistivity decreases with increasing temperature. The electrons are bumped to the conduction energy band by thermal energy, where they flow freely, and in doing so leave behind holes in the valence band, which also flow freely. The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with temperature following an Arrhenius model:

 

An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the 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 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, they behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.[58]

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

 

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

Complex resistivity and conductivity edit

When analyzing the response of materials to alternating electric fields (dielectric spectroscopy),[59] in applications such as electrical impedance tomography,[60] it is convenient to replace resistivity with a complex quantity called impedivity (in analogy to electrical impedance). Impedivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity (in analogy to reactance). The magnitude of impedivity 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 admittivity. Admittivity is the sum of a real component 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 Mathematical descriptions of opacity.

Resistance versus resistivity in complicated geometries edit

Even 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   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

 

must be replaced with

 

where E and J are now vector fields. This equation, along with the continuity equation for J and the Poisson's equation for E, form a set of partial differential equations. 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 analysis may be required.

Resistivity-density product edit

In some applications where the weight of an item is very important, the product of resistivity and density is 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 conductivity-to-density ratio) is desirable. For example, for long-distance overhead power lines, aluminium is frequently used rather than copper (Cu) because it is lighter for the same conductance.

Silver, although it is the least resistive metal known, has a high density and performs similarly to copper by this measure, but is much more expensive. 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 (and lack of physical strength). Aluminium is far more stable. Toxicity excludes the choice of beryllium.[61] (Pure beryllium is also brittle.) Thus, aluminium is usually the metal of choice when the weight or cost of a conductor is the driving consideration.

Resistivity, density, and resistivity-density products of selected materials
Material Resistivity
(nΩ·m)
Density
(g/cm3)
Resistivity × density Resistivity relative to Cu, ie cross-sectional area required to give same conductance Approx. price, at
9 December 2018
[dubious ]
(g·mΩ/m2) Relative
to Cu
(USD
per kg)
Relative
to Cu
Sodium 47.7 0.97 46 31% 2.843
Lithium 92.8 0.53 49 33% 5.531
Calcium 33.6 1.55 52 35% 2.002
Potassium 72.0 0.89 64 43% 4.291
Beryllium 35.6 1.85 66 44% 2.122
Aluminium 26.50 2.70 72 48% 1.579 2.0 0.16
Magnesium 43.90 1.74 76 51% 2.616
Copper 16.78 8.96 150 100% 1 6.0 1
Silver 15.87 10.49 166 111% 0.946 456 84
Gold 22.14 19.30 427 285% 1.319 39,000 19,000
Iron 96.1 7.874 757 505% 5.727

See also edit

Notes edit

  1. ^ The atomic number is the count of electrons in an atom that is electrically neutral – has no net electric charge.
  2. ^ Other relevant factors that are specifically not considered are the size of the whole crystal and external factors of the surrounding environment that modify the energy bands, such as imposed electric or magnetic fields.
  3. ^ 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 ρ0 where ρ0 is the resistivity at 20 °C (in this case) and α is the temperature coefficient.
  4. ^ The conductivity of metallic silver is not significantly better than metallic copper for most practical purposes – the difference between the two can be easily compensated for by thickening the copper wire by only 3%. However silver is preferred for exposed electrical contact points because corroded silver is a tolerable conductor, but corroded copper is a fairly good insulator, like most corroded metals.
  5. ^ Copper is widely used in electrical equipment, building wiring, and telecommunication cables.
  6. ^ 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 method. Generally used for temper and alloy verification of aluminium.
  7. ^ Despite being less conductive than copper, gold is commonly used in electrical contacts because it does not easily corrode.
  8. ^ Commonly used for overhead power line with steel reinforced (ACSR)
  9. ^ a b Cobalt and ruthenium are considered to replace copper in integrated circuits fabricated in advanced nodes[29]
  10. ^ 18% chromium and 8% nickel austenitic stainless steel
  11. ^ Nickel-iron-chromium alloy commonly used in heating elements.
  12. ^ Graphite is strongly anisotropic.
  13. ^ a b The resistivity of semiconductors depends strongly on the presence of impurities in the material.
  14. ^ Corresponds to an average salinity of 35 g/kg at 20 °C.
  15. ^ The pH should be around 8.4 and the conductivity in the range of 2.5–3 mS/cm. The lower value is appropriate for freshly prepared water. The conductivity is used for the determination of TDS (total dissolved particles).
  16. ^ This value range is typical of high quality drinking water and not an indicator of water quality
  17. ^ Conductivity is lowest with monatomic gases present; changes to 12×10−5 upon complete de-gassing, or to 7.5×10−5 upon equilibration to the atmosphere due to dissolved CO2

References edit

  1. ^ Lowrie, William (2007). Fundamentals of Geophysics. Cambridge University Press. pp. 254–55. ISBN 978-05-2185-902-8. Retrieved March 24, 2019.
  2. ^ a b Kumar, Narinder (2003). Comprehensive Physics for Class XII. New Delhi: Laxmi Publications. pp. 280–84. ISBN 978-81-7008-592-8. Retrieved March 24, 2019.
  3. ^ Bogatin, Eric (2004). Signal Integrity: Simplified. Prentice Hall Professional. p. 114. ISBN 978-0-13-066946-9. Retrieved March 24, 2019.
  4. ^ a b c Hugh O. Pierson, Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, and applications, p. 61, William Andrew, 1993 ISBN 0-8155-1339-9.
  5. ^ J.R. Tyldesley (1975) An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
  6. ^ G. Woan (2010) The Cambridge Handbook of Physics Formulas, Cambridge University Press, ISBN 978-0-521-57507-2
  7. ^ Josef Pek, Tomas Verner (3 Apr 2007). "Finite‐difference modelling of magnetotelluric fields in two‐dimensional anisotropic media". Geophysical Journal International. 128 (3): 505–521. doi:10.1111/j.1365-246X.1997.tb05314.x.
  8. ^ David Tong (Jan 2016). "The Quantum Hall Effect: TIFR Infosys Lectures" (PDF). Retrieved 14 Sep 2018.
  9. ^ . ibchem.com
  10. ^ "Current versus Drift Speed". The physics classroom. Retrieved 20 August 2014.
  11. ^ Lowe, Doug (2012). Electronics All-in-One For Dummies. John Wiley & Sons. ISBN 978-0-470-14704-7.
  12. ^ Keith Welch. "Questions & Answers – How do you explain electrical resistance?". Thomas Jefferson National Accelerator Facility. Retrieved 28 April 2017.
  13. ^ "Electromigration : What is electromigration?". Middle East Technical University. Retrieved 31 July 2017. When electrons are conducted through a metal, they interact with imperfections in the lattice and scatter. […] Thermal energy produces scattering by causing atoms to vibrate. This is the source of resistance of metals.
  14. ^ Faber, T.E. (1972). Introduction to the Theory of Liquid Metals. Cambridge University Press. ISBN 9780521154499.
  15. ^ "The Feynman Lectures in Physics, Vol. III, Chapter 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity". Retrieved 26 December 2021.
  16. ^ John C. Gallop (1990). SQUIDS, the Josephson Effects and Superconducting Electronics. CRC Press. pp. 3, 20. ISBN 978-0-7503-0051-3.
  17. ^ See Flashes in the Sky: Earth's Gamma-Ray Bursts Triggered by Lightning
  18. ^ Yaffa Eliezer, Shalom Eliezer, The Fourth State of Matter: An Introduction to the Physics of Plasma, Publisher: Adam Hilger, 1989, ISBN 978-0-85274-164-1, 226 pages, page 5
  19. ^ Bittencourt, J.A. (2004). Fundamentals of Plasma Physics. Springer. p. 1. ISBN 9780387209753.
  20. ^ Hong, Alice (2000). "Dielectric Strength of Air". The Physics Factbook.
  21. ^ a b c d e f g h i j k l m n o Raymond A. Serway (1998). Principles of Physics (2nd ed.). Fort Worth, Texas; London: Saunders College Pub. p. 602. ISBN 978-0-03-020457-9.
  22. ^ a b c David Griffiths (1999) [1981]. "7 Electrodynamics". In Alison Reeves (ed.). Introduction to Electrodynamics (3rd ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 286. ISBN 978-0-13-805326-0. OCLC 40251748.
  23. ^ Matula, R.A. (1979). "Electrical resistivity of copper, gold, palladium, and silver". Journal of Physical and Chemical Reference Data. 8 (4): 1147. Bibcode:1979JPCRD...8.1147M. doi:10.1063/1.555614. S2CID 95005999.
  24. ^ Douglas Giancoli (2009) [1984]. "25 Electric Currents and Resistance". In Jocelyn Phillips (ed.). Physics for Scientists and Engineers with Modern Physics (4th ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 658. ISBN 978-0-13-149508-1.
  25. ^ "Copper wire tables". United States National Bureau of Standards. Retrieved 3 February 2014 – via Internet Archive - archive.org (archived 2001-03-10).
  26. ^ [1]. (Calculated as "56% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
  27. ^ Physical constants. (PDF format; see page 2, table in the right lower corner). Retrieved on 2011-12-17.
  28. ^ [2]. (Calculated as "28% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
  29. ^ IITC – Imec Presents Copper, Cobalt and Ruthenium Interconnect Results
  30. ^ "Temperature Coefficient of Resistance | Electronics Notes".
  31. ^ [3]. (Calculated as "15% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
  32. ^ Material properties of niobium.
  33. ^ AISI 1010 Steel, cold drawn. Matweb
  34. ^ Karcher, Ch.; Kocourek, V. (December 2007). "Free-surface instabilities during electromagnetic shaping of liquid metals". Proceedings in Applied Mathematics and Mechanics. 7 (1): 4140009–4140010. doi:10.1002/pamm.200700645. ISSN 1617-7061.
  35. ^ "JFE steel" (PDF). Retrieved 2012-10-20.
  36. ^ a b Douglas C. Giancoli (1995). Physics: Principles with Applications (4th ed.). London: Prentice Hall. ISBN 978-0-13-102153-2.
    (see also Table of Resistivity. hyperphysics.phy-astr.gsu.edu)
  37. ^ 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
  38. ^ Glenn Elert (ed.), "Resistivity of steel", The Physics Factbook, retrieved and 16 June 2011.
  39. ^ Probably, the metal with highest value of electrical resistivity.
  40. ^ 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.
  41. ^ Milton Ohring (1995). Engineering materials science, Volume 1 (3rd ed.). Academic Press. p. 561. ISBN 978-0125249959.
  42. ^ Physical properties of sea water 2018-01-18 at the Wayback Machine. Kayelaby.npl.co.uk. Retrieved on 2011-12-17.
  43. ^ [4]. chemistry.stackexchange.com
  44. ^ Eranna, Golla (2014). Crystal Growth and Evaluation of Silicon for VLSI and ULSI. CRC Press. p. 7. ISBN 978-1-4822-3281-3.
  45. ^ a b c Transmission Lines data. Transmission-line.net. Retrieved on 2014-02-03.
  46. ^ R. M. Pashley; M. Rzechowicz; L. R. Pashley; M. J. Francis (2005). "De-Gassed Water is a Better Cleaning Agent". The Journal of Physical Chemistry B. 109 (3): 1231–8. doi:10.1021/jp045975a. PMID 16851085.
  47. ^ ASTM D1125 Standard Test Methods for Electrical Conductivity and Resistivity of Water
  48. ^ ASTM D5391 Standard Test Method for Electrical Conductivity and Resistivity of a Flowing High Purity Water Sample
  49. ^ Lawrence S. Pan, Don R. Kania, Diamond: electronic properties and applications, p. 140, Springer, 1994 ISBN 0-7923-9524-7.
  50. ^ S. D. Pawar; P. Murugavel; D. M. Lal (2009). "Effect of relative humidity and sea level pressure on electrical conductivity of air over Indian Ocean". Journal of Geophysical Research. 114 (D2): D02205. Bibcode:2009JGRD..114.2205P. doi:10.1029/2007JD009716.
  51. ^ E. Seran; M. Godefroy; E. Pili (2016). "What we can learn from measurements of air electric conductivity in 222Rn ‐ rich atmosphere". Earth and Space Science. 4 (2): 91–106. Bibcode:2017E&SS....4...91S. doi:10.1002/2016EA000241.
  52. ^ Copper Wire Tables 2010-08-21 at the Wayback Machine. US Dep. of Commerce. National Bureau of Standards Handbook. February 21, 1966
  53. ^ Ward, Malcolm R. (1971). Electrical engineering science. McGraw-Hill technical education. Maidenhead, UK: McGraw-Hill. pp. 36–40. ISBN 9780070942554.
  54. ^ Grüneisen, E. (1933). "Die Abhängigkeit des elektrischen Widerstandes reiner Metalle von der Temperatur". Annalen der Physik. 408 (5): 530–540. Bibcode:1933AnP...408..530G. doi:10.1002/andp.19334080504. ISSN 1521-3889.
  55. ^ Quantum theory of real materials. James R. Chelikowsky, Steven G. Louie. Boston: Kluwer Academic Publishers. 1996. pp. 219–250. ISBN 0-7923-9666-9. OCLC 33335083.{{cite book}}: CS1 maint: others (link)
  56. ^ A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)
  57. ^ A. Matthiessen, Progg. Anallen, 122, 47 (1864)
  58. ^ J. Seymour (1972) Physical Electronics, chapter 2, Pitman
  59. ^ Stephenson, C.; Hubler, A. (2015). "Stability and conductivity of self-assembled wires in a transverse electric field". Sci. Rep. 5: 15044. Bibcode:2015NatSR...515044S. doi:10.1038/srep15044. PMC 4604515. PMID 26463476.
  60. ^ Otto H. Schmitt, University of Minnesota . otto-schmitt.org. Retrieved on 2011-12-17.
  61. ^ "Berryllium (Be) - Chemical properties, Health and Environmental effects".

Further reading edit

  • Paul Tipler (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 978-0-7167-0810-0.
  • Measuring Electrical Resistivity and Conductivity

External links edit

  • "Electrical Conductivity". Sixty Symbols. Brady Haran for the University of Nottingham. 2010.
  • Comparison of the electrical conductivity of various elements in WolframAlpha
  • Partial and total conductivity. "Electrical conductivity" (PDF).

electrical, resistivity, conductivity, this, article, about, electrical, conductivity, general, other, types, conductivity, conductivity, specific, applications, electrical, elements, electrical, resistance, conductance, electrical, resistivity, also, called, . This article is about electrical conductivity in general For other types of conductivity see Conductivity For specific applications in electrical elements see Electrical resistance and conductance Electrical resistivity also called volume resistivity or specific electrical resistance is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current A low resistivity indicates a material that readily allows electric current Resistivity is commonly represented by the Greek letter r rho The SI unit of electrical resistivity is the ohm metre W m 1 2 3 For example if a 1 m3 solid cube of material has sheet contacts on two opposite faces and the resistance between these contacts is 1 W then the resistivity of the material is 1 W m ResistivityCommon symbolsrSI unitohm metre W m In SI base unitskg m3 s 3 A 2Derivations fromother quantitiesr R A ℓ displaystyle rho R frac A ell DimensionM L 3 T 3 I 2 displaystyle mathsf M mathsf L 3 mathsf T 3 mathsf I 2 ConductivityCommon symbolss k g iiSI unitsiemens per metre S m Other unitszDerivations fromother quantitiess 1 r displaystyle sigma frac 1 rho DimensionM 1 L 3 T 3 I 2 displaystyle mathsf M 1 mathsf L 3 mathsf T 3 mathsf I 2 Electrical conductivity or specific conductance is the reciprocal of electrical resistivity It represents a material s ability to conduct electric current It is commonly signified by the Greek letter s sigma but k kappa especially in electrical engineering and g gamma are sometimes used The SI unit of electrical conductivity is siemens per metre S m Resistivity and conductivity are intensive properties of materials giving the opposition of a standard cube of material to current Electrical resistance and conductance are corresponding extensive properties that give the opposition of a specific object to electric current Contents 1 Definition 1 1 Ideal case 1 2 General scalar quantities 1 3 Tensor resistivity 2 Conductivity and current carriers 2 1 Relation between current density and electric current velocity 3 Causes of conductivity 3 1 Band theory simplified 3 2 In metals 3 3 In semiconductors and insulators 3 4 In ionic liquids electrolytes 3 5 Superconductivity 3 6 Plasma 4 Resistivity and conductivity of various materials 5 Temperature dependence 5 1 Linear approximation 5 2 Metals 5 2 1 Wiedemann Franz law 5 3 Semiconductors 6 Complex resistivity and conductivity 7 Resistance versus resistivity in complicated geometries 8 Resistivity density product 9 See also 10 Notes 11 References 12 Further reading 13 External linksDefinition editIdeal case edit nbsp A piece of resistive material with electrical contacts on both ends In an ideal case cross section and physical composition of the examined material are uniform across the sample and the electric field and current density are both parallel and constant everywhere Many resistors and conductors do in fact have a uniform cross section with a uniform flow of electric current and are made of a single material so that this is a good model See the adjacent diagram When this is the case the resistance of the conductor is directly proportional to its length and inversely proportional to its cross sectional area where the electrical resistivity r Greek rho is the constant of proportionality This is written as R ℓ A displaystyle R propto frac ell A nbsp R r ℓ A r R A ℓ displaystyle begin aligned R amp rho frac ell A 3pt Leftrightarrow rho amp R frac A ell end aligned nbsp where R displaystyle R nbsp is the electrical resistance of a uniform specimen of the material ℓ displaystyle ell nbsp is the length of the specimen A displaystyle A nbsp is the cross sectional area of the specimen The resistivity can be expressed using the SI unit ohm metre W m i e ohms multiplied by square metres for the cross sectional area then divided by metres for the length Both resistance and resistivity describe how difficult it is to make electrical current flow through a material but unlike resistance resistivity is an intrinsic property and does not depend on geometric properties of a material This means that all pure copper Cu wires which have not been subjected to distortion of their crystalline structure etc irrespective of their shape and size have 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 has a far larger resistivity than copper 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 Resistance however 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 have lower resistance than narrow or long pipes The above equation can be transposed to get Pouillet s law named after Claude Pouillet R r ℓ A displaystyle R rho frac ell A nbsp The resistance of a given element is proportional to the length but inversely proportional to the cross sectional area For example if A 1 m2 ℓ displaystyle ell nbsp 1 m 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 W m Conductivity s is the inverse of resistivity s 1 r displaystyle sigma frac 1 rho nbsp Conductivity has SI units of siemens per metre S m General scalar quantities edit For less ideal cases such as more complicated geometry or when the current and electric field vary in different parts of the material it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the electric field to the density of the current it creates at that point r E J displaystyle rho frac E J nbsp where r displaystyle rho nbsp is the resistivity of the conductor material E displaystyle E nbsp is the magnitude of the electric field J displaystyle J nbsp is the magnitude of the current density in which E displaystyle E nbsp and J displaystyle J nbsp are inside the conductor Conductivity is the inverse reciprocal of resistivity Here it is given by s 1 r J E displaystyle sigma frac 1 rho frac J E nbsp For example rubber is a material with large r and small s because even a very large electric field in rubber makes almost no current flow through it On the other hand copper is a material with small r and large s because even a small electric field pulls a lot of current through it As shown below this expression simplifies to a single number when the electric field and current density are constant in the material Derivation from general definition of resistivityThere are three equations to be combined here The first is the resistivity for parallel current and electric field r E J displaystyle rho frac E J nbsp If the electric field is constant the electric field is given by the total voltage V across the conductor divided by length ℓ of the conductor E V ℓ displaystyle E frac V ell nbsp If the current density is constant it is equal to the total current divided by the cross sectional area J I A displaystyle J frac I A nbsp Plugging in the values of E and J into the first expression we obtain r V A I ℓ displaystyle rho frac VA I ell nbsp Finally we apply Ohm s law V I R r R A ℓ displaystyle rho R frac A ell nbsp Tensor resistivity edit When the resistivity of a material has a directional component the most general definition of resistivity must be used It starts from the tensor vector form of Ohm s law which relates the electric field inside a material to the electric current flow This equation is completely general meaning it is valid in all cases including those mentioned above However this definition is the most complicated so it is only directly used in anisotropic cases where the more simple definitions cannot be applied If the material is not anisotropic it is safe to ignore the tensor vector definition and use a simpler expression instead Here anisotropic means that the material has 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 much less easily from one sheet to the adjacent one 4 In such cases the current does not flow in exactly the same direction as the electric field Thus the appropriate equations are generalized to the three dimensional tensor form 5 6 J s E E r J displaystyle mathbf J boldsymbol sigma mathbf E rightleftharpoons mathbf E boldsymbol rho mathbf J nbsp where the conductivity s and resistivity r are rank 2 tensors and electric field E and current density J are vectors These tensors can be represented by 3 3 matrices the vectors with 3 1 matrices with matrix multiplication used on the right side of these equations In matrix form the resistivity relation is given by E x E y E z r x x r x y r x z r y x r y y r y z r z x r z y r z z J x J y J z displaystyle begin bmatrix E x E y E z end bmatrix begin bmatrix rho xx amp rho xy amp rho xz rho yx amp rho yy amp rho yz rho zx amp rho zy amp rho zz end bmatrix begin bmatrix J x J y J z end bmatrix nbsp where E displaystyle mathbf E nbsp is the electric field vector with components Ex Ey Ez r displaystyle boldsymbol rho nbsp is the resistivity tensor in general a three by three matrix J displaystyle mathbf J nbsp is the electric current density vector with components Jx Jy Jz Equivalently resistivity can be given in the more compact Einstein notation E i r i j J j displaystyle mathbf E i boldsymbol rho ij mathbf J j nbsp In either case the resulting expression for each electric field component is E x r x x J x r x y J y r x z J z E y r y x J x r y y J y r y z J z E z r z x J x r z y J y r z z J z displaystyle begin aligned E x amp rho xx J x rho xy J y rho xz J z E y amp rho yx J x rho yy J y rho yz J z E z amp rho zx J x rho zy J y rho zz J z end aligned nbsp Since the choice of the coordinate system is free the usual convention is to simplify the expression by choosing an x axis parallel to the current direction so Jy Jz 0 This leaves r x x E x J x r y x E y J x and r z x E z J x displaystyle rho xx frac E x J x quad rho yx frac E y J x text and rho zx frac E z J x nbsp Conductivity is defined similarly 7 J x J y J z s x x s x y s x z s y x s y y s y z s z x s z y s z z E x E y E z displaystyle begin bmatrix J x J y J z end bmatrix begin bmatrix sigma xx amp sigma xy amp sigma xz sigma yx amp sigma yy amp sigma yz sigma zx amp sigma zy amp sigma zz end bmatrix begin bmatrix E x E y E z end bmatrix nbsp orJ i s i j E j displaystyle mathbf J i boldsymbol sigma ij mathbf E j nbsp both resulting in J x s x x E x s x y E y s x z E z J y s y x E x s y y E y s y z E z J z s z x E x s z y E y s z z E z displaystyle begin aligned J x amp sigma xx E x sigma xy E y sigma xz E z J y amp sigma yx E x sigma yy E y sigma yz E z J z amp sigma zx E x sigma zy E y sigma zz E z end aligned nbsp Looking at the two expressions r displaystyle boldsymbol rho nbsp and s displaystyle boldsymbol sigma nbsp are the matrix inverse of each other However in the most general case the individual matrix elements are not necessarily reciprocals of one another for example sxx may not be equal to 1 rxx This can be seen in the Hall effect where r x y displaystyle rho xy nbsp is nonzero In the Hall effect due to rotational invariance about the z axis r y y r x x displaystyle rho yy rho xx nbsp and r y x r x y displaystyle rho yx rho xy nbsp so the relation between resistivity and conductivity simplifies to 8 s x x r x x r x x 2 r x y 2 s x y r x y r x x 2 r x y 2 displaystyle sigma xx frac rho xx rho xx 2 rho xy 2 quad sigma xy frac rho xy rho xx 2 rho xy 2 nbsp If the electric field is parallel to the applied current r x y displaystyle rho xy nbsp and r x z displaystyle rho xz nbsp are zero When they are zero one number r x x displaystyle rho xx nbsp is enough to describe the electrical resistivity It is then written as simply r displaystyle rho nbsp and this reduces to the simpler expression Conductivity and current carriers editRelation between current density and electric current velocity edit Electric current is the ordered movement of electric charges 2 Causes of conductivity editBand theory simplified edit See also Band theory nbsp Filling of the electronic states in various types of materials at equilibrium Here height is energy while width is the density of available states for a certain energy in the material listed The shade follows the Fermi Dirac distribution black all states filled white no state filled In metals and semimetals the Fermi level EF lies inside at least one band In insulators and semiconductors the Fermi level is inside a band gap however in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes editAccording to elementary quantum mechanics an electron in an atom or crystal can only have certain precise energy levels energies between these levels are impossible When a large number of such allowed levels have close spaced energy values i e have energies that differ only minutely those close energy levels in combination are called an energy band There can be many such energy bands in a material depending on the atomic number of the constituent atoms a and their distribution within the crystal b The material s electrons seek to minimize the total energy in the material by settling into low energy states however the Pauli exclusion principle means that only one can exist in each such state So the electrons 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 or above the Fermi level are free to move within the broader material structure since the electrons can easily jump among the partially occupied states in that region In contrast the low energy states are completely filled with a fixed limit on the number of electrons at all times and the high energy states are empty of electrons at all times Electric current consists of a flow of electrons In metals there are many electron energy levels near the Fermi level so there are many electrons available to move This is what causes the high electronic conductivity of metals An important part of band theory is that there may be forbidden bands of energy energy intervals that contain no energy levels In insulators and semiconductors the number of electrons is 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 In metals edit Main article Free electron model source source source source source source Like balls in a Newton s cradle electrons in a metal quickly transfer energy from one terminal to another despite their own negligible movement A metal consists of a lattice of atoms each with an outer shell of electrons that freely dissociate from their parent atoms and travel through the lattice This is also known as a positive ionic lattice 9 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 drift towards the positive terminal The actual drift velocity of electrons is typically small on the order of magnitude of metres per hour However due to the sheer number of moving electrons even a slow drift velocity results in a large current density 10 The mechanism is similar to transfer of momentum of balls in a Newton s cradle 11 but the rapid propagation of an electric energy along a wire is not due to the mechanical forces but the propagation of an energy carrying electromagnetic field guided by the wire Most metals have electrical resistance In simpler models non quantum mechanical models this can be explained by replacing electrons and the crystal lattice by a wave like structure When the electron wave travels through the lattice the waves interfere which causes resistance The more regular the lattice is the less disturbance happens and thus the less resistance The amount of resistance is thus mainly caused by two factors First it is caused by the temperature and thus amount of vibration of the crystal lattice Higher temperatures cause bigger vibrations which act as irregularities in the lattice Second the purity of the metal is relevant as a mixture of different ions is also an irregularity 12 13 The small decrease in conductivity on melting of pure metals is due to the loss of long range crystalline order The short range order remains and strong correlation between positions of ions results in coherence between waves diffracted by adjacent ions 14 In semiconductors and insulators edit Main articles Semiconductor Insulator electricity and Charge carrier density 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 about halfway between the conduction band minimum the bottom of the first band of unfilled electron energy levels and the valence band maximum the top of the band below the conduction band of filled electron energy levels That applies for intrinsic undoped semiconductors This means that at absolute zero temperature there would be no free conduction electrons and the resistance is infinite However the resistance decreases as the charge carrier density i e without introducing further complications the density of electrons 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 producing holes in the valence band A hole is a position where an electron is missing such holes can behave in a similar way to electrons For both types of donor or acceptor atoms increasing dopant density reduces resistance Hence highly doped semiconductors behave metallically At very high temperatures the contribution of thermally generated carriers dominates over the contribution from dopant atoms and the resistance decreases exponentially with temperature In ionic liquids electrolytes edit 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 solutions electrolytes varies tremendously with concentration while distilled water is almost an insulator salt water is a reasonable electrical conductor Conduction in ionic liquids is also controlled by the movement of ions but here we are talking about molten salts rather than solvated ions In biological membranes currents are carried by ionic salts Small holes in cell membranes called ion channels are selective to specific ions and determine the membrane resistance The concentration of ions in a liquid e g in an aqueous solution depends on the degree of dissociation of the dissolved substance characterized by a dissociation coefficient a displaystyle alpha nbsp which is the ratio of the concentration of ions N displaystyle N nbsp to the concentration of molecules of the dissolved substance N 0 displaystyle N 0 nbsp N a N 0 displaystyle N alpha N 0 nbsp The specific electrical conductivity s displaystyle sigma nbsp of a solution is equal to s q b b a N 0 displaystyle sigma q left b b right alpha N 0 nbsp where q displaystyle q nbsp module of the ion charge b displaystyle b nbsp and b displaystyle b nbsp mobility of positively and negatively charged ions N 0 displaystyle N 0 nbsp concentration of molecules of the dissolved substance a displaystyle alpha nbsp the coefficient of dissociation Superconductivity edit Main article Superconductivity nbsp Original data from the 1911 experiment by Heike Kamerlingh Onnes showing the resistance of a mercury wire as a function of temperature The abrupt drop in resistance is the superconducting transition The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered In normal that is non superconducting 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 In a normal conductor the current is driven by a voltage gradient whereas in a superconductor there is no voltage gradient and the current is instead related to the phase gradient of the superconducting order parameter 15 A consequence of this is that an electric current flowing in a loop of superconducting wire can persist indefinitely with no power source 16 In a class of superconductors known as type II superconductors including all known high temperature superconductors an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field which may be caused by the electric current This is due to the motion of magnetic vortices in the electronic superfluid which dissipates some of the energy carried by the current The resistance due to this effect is tiny compared with that of non superconducting materials but must be taken into account in sensitive experiments However as the temperature decreases far enough below the nominal superconducting transition these vortices can become frozen so that the resistance of the material becomes truly zero Plasma edit Main article Plasma physics nbsp Lightning is an example of plasma present at Earth s surface Typically lightning discharges 30 000 amperes at up to 100 million volts and emits light radio waves and X rays 17 Plasma temperatures in lightning might approach 30 000 kelvin 29 727 C 53 540 F or five times hotter than the temperature at the sun surface and electron densities may exceed 1024 m 3 Plasmas are very good conductors and electric potentials play an important role The potential as it exists on average in the space between charged particles independent of the question of how it can be measured is called the plasma potential or space potential If an electrode is inserted into a plasma its potential generally lies considerably below the plasma potential due to what is termed a Debye sheath The good electrical conductivity of plasmas makes their electric fields very small This results in the important concept of quasineutrality which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma ne Z gt ni but on the scale of the Debye length there can be charge imbalance In the special case that double layers are formed the charge separation can extend some tens of Debye lengths The magnitude of the potentials and electric fields must be determined by means other than simply finding the net charge density A common example is to assume that the electrons satisfy the Boltzmann relation n e exp e F k B T e displaystyle n text e propto exp left e Phi k text B T text e right nbsp Differentiating this relation provides a means to calculate the electric field from the density E k B T e e n e n e displaystyle mathbf E frac k text B T text e e frac nabla n text e n text e nbsp is the vector gradient operator see nabla symbol and gradient for more information It is possible to produce a plasma that is not quasineutral An electron beam for example has only negative charges The density of a non neutral plasma must generally be very low or it must be very small Otherwise the repulsive electrostatic force dissipates it In astrophysical plasmas Debye screening prevents electric fields from directly affecting the plasma over large distances i e greater than the Debye length However the existence of charged particles causes the plasma to generate and be affected by magnetic fields This can and does cause extremely complex behavior such as the generation of plasma double layers an object that separates charge over a few tens of Debye lengths The dynamics of plasmas interacting with external and self generated magnetic fields are studied in the academic discipline of magnetohydrodynamics Plasma is often called the fourth state of matter after solid liquids and gases 18 19 It is distinct from these and other lower energy states of matter Although it is closely related to the gas phase in that it also has no definite form or volume it differs in a number of ways including the following Property Gas PlasmaElectrical conductivity Very low air is an excellent insulator until it breaks down into plasma at electric field strengths above 30 kilovolts per centimetre 20 Usually very high for many purposes the conductivity of a plasma may be treated as infinite Independently acting species One all gas particles behave in a similar way influenced by gravity and by collisions with one another Two or three electrons ions protons and neutrons can be distinguished by the sign and value of their charge so that they behave independently in many circumstances with different bulk velocities and temperatures allowing phenomena such as new types of waves and instabilities Velocity distribution Maxwellian collisions usually lead to a Maxwellian velocity distribution of all gas particles with very few relatively fast particles Often non Maxwellian collisional interactions are often weak in hot plasmas and external forcing can drive the plasma far from local equilibrium and lead to a significant population of unusually fast particles Interactions Binary two particle collisions are the rule three body collisions extremely rare Collective waves or organized motion of plasma are very important because the particles can interact at long ranges through the electric and magnetic forces Resistivity and conductivity of various materials editMain 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 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 The degree of semiconductors doping makes a large difference in conductivity To a point more doping leads to higher conductivity The conductivity of a water aqueous solution is highly dependent on its concentration of dissolved salts and other chemical species that 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 25 C An EC meter is normally used to measure conductivity in a solution A rough summary is as follows Resistivity of classes of materials Material Resistivity r W m Superconductors 0Metals 10 8Semiconductors VariableElectrolytes VariableInsulators 1016Superinsulators This table shows the resistivity r conductivity and temperature coefficient of various materials at 20 C 68 F 293 K Resistivity conductivity and temperature coefficient for several materials Material Resistivity r at 20 C W m Conductivity s at 20 C S m Temperature coefficient c K 1 ReferenceSilver d 1 59 10 8 6 30 107 3 80 10 3 21 22 Copper e 1 68 10 8 5 96 107 4 04 10 3 23 24 Annealed copper f 1 72 10 8 5 80 107 3 93 10 3 25 Gold g 2 44 10 8 4 11 107 3 40 10 3 21 Aluminium h 2 65 10 8 3 77 107 3 90 10 3 21 Brass 5 Zn 3 00 10 8 3 34 107 26 Calcium 3 36 10 8 2 98 107 4 10 10 3Rhodium 4 33 10 8 2 31 107Tungsten 5 60 10 8 1 79 107 4 50 10 3 21 Zinc 5 90 10 8 1 69 107 3 70 10 3 27 Brass 30 Zn 5 99 10 8 1 67 107 28 Cobalt i 6 24 10 8 1 60 107 7 00 10 3 30 unreliable source Nickel 6 99 10 8 1 43 107 6 00 10 3Ruthenium i 7 10 10 8 1 41 107Lithium 9 28 10 8 1 08 107 6 00 10 3Iron 9 70 10 8 1 03 107 5 00 10 3 21 Platinum 10 6 10 8 9 43 106 3 92 10 3 21 Tin 10 9 10 8 9 17 106 4 50 10 3Phosphor Bronze 0 2 P 5 Sn 11 2 10 8 8 94 106 31 Gallium 14 0 10 8 7 10 106 4 00 10 3Niobium 14 0 10 8 7 00 106 32 Carbon steel 1010 14 3 10 8 6 99 106 33 Lead 22 0 10 8 4 55 106 3 90 10 3 21 Galinstan 28 9 10 8 3 46 106 34 Titanium 42 0 10 8 2 38 106 3 80 10 3Grain oriented electrical steel 46 0 10 8 2 17 106 35 Manganin 48 2 10 8 2 07 106 0 002 10 3 36 Constantan 49 0 10 8 2 04 106 0 008 10 3 37 Stainless steel j 69 0 10 8 1 45 106 0 94 10 3 38 Mercury 98 0 10 8 1 02 106 0 90 10 3 36 Bismuth 129 10 8 7 75 105Manganese 144 10 8 6 94 105Plutonium 39 0 C 146 10 8 6 85 105Nichrome k 110 10 8 6 70 105 citation needed 0 40 10 3 21 Carbon graphite parallel to basal plane l 250 10 8 to 500 10 8 2 105 to 3 105 citation needed 4 Carbon amorphous 0 5 10 3 to 0 8 10 3 1 25 103 to 2 00 103 0 50 10 3 21 40 Carbon graphite perpendicular to basal plane 3 0 10 3 3 3 102 4 GaAs 10 3 to 108 clarification needed 10 8 to 103 dubious discuss 41 Germanium m 4 6 10 1 2 17 48 0 10 3 21 22 Sea water n 2 1 10 1 4 8 42 Swimming pool water o 3 3 10 1 to 4 0 10 1 0 25 to 0 30 43 Drinking water p 2 101 to 2 103 5 10 4 to 5 10 2 citation needed Silicon m 2 3 103 4 35 10 4 75 0 10 3 44 21 Wood damp 103 to 104 10 4 to 10 3 45 Deionized water q 1 8 105 4 2 10 5 46 Ultrapure water 1 82 109 5 49 10 10 47 48 Glass 1011 to 1015 10 15 to 10 11 21 22 Carbon diamond 1012 10 13 49 Hard rubber 1013 10 14 21 Air 109 to 1015 10 15 to 10 9 50 51 Wood oven dry 1014 to 1016 10 16 to 10 14 45 Sulfur 1015 10 16 21 Fused quartz 7 5 1017 1 3 10 18 21 PET 1021 10 21PTFE teflon 1023 to 1025 10 25 to 10 23The 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 0 C 52 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 popular science 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 1010 worse insulator than oven dry 45 In any case a sufficiently high voltage such as that in lightning strikes or some high tension power lines can lead to insulation breakdown and electrocution risk even with apparently dry wood citation needed Temperature dependence editLinear approximation edit The electrical resistivity of most materials changes with temperature If the temperature T does not vary too much a linear approximation is typically used r T r 0 1 a T T 0 displaystyle rho T rho 0 1 alpha T T 0 nbsp where a displaystyle alpha nbsp is called the temperature coefficient of resistivity T 0 displaystyle T 0 nbsp is a fixed reference temperature usually room temperature and r 0 displaystyle rho 0 nbsp is the resistivity at temperature T 0 displaystyle T 0 nbsp The parameter a displaystyle alpha nbsp is an empirical parameter fitted from measurement data equal to 1 k displaystyle kappa nbsp clarify Because the linear approximation is only an approximation a displaystyle alpha nbsp is different for different reference temperatures For this reason it is usual to specify the temperature that a displaystyle alpha nbsp was measured at with a suffix such as a 15 displaystyle alpha 15 nbsp and the relationship only holds in a range of temperatures around the reference 53 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 edit See also Bloch Gruneisen temperature and Free electron model Mean free dependence of the resistivity of gold copper and silver In general electrical resistivity of metals 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 r of a metal can be approximated through the Bloch Gruneisen formula 54 r T r 0 A T 8 R n 0 8 R T x n e x 1 1 e x d x displaystyle rho T rho 0 A left frac T Theta R right n int 0 Theta R T frac x n e x 1 1 e x dx nbsp where r 0 displaystyle rho 0 nbsp 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 8 R displaystyle Theta R nbsp 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 phonons 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 The Bloch Gruneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the first Brillouin zone and a Debye phonon spectrum 55 If more than one source of scattering is simultaneously present Matthiessen s rule first formulated by Augustus Matthiessen in the 1860s 56 57 states 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 s experiments that led in 1911 to discovery of superconductivity For details see History of superconductivity Wiedemann Franz law edit The Wiedemann Franz law states that for materials where heat and charge transport is dominated by electrons the ratio of thermal to electrical conductivity is proportional to the temperature k s p 2 3 k e 2 T displaystyle kappa over sigma pi 2 over 3 left frac k e right 2 T nbsp where k displaystyle kappa nbsp is the thermal conductivity k displaystyle k nbsp is the Boltzmann constant e displaystyle e nbsp is the electron charge T displaystyle T nbsp is temperature and s displaystyle sigma nbsp is the electric conductivity The ratio on the rhs is called the Lorenz number Semiconductors edit See also Kondo effect and Kondo semiconductor In general intrinsic semiconductor resistivity decreases with increasing temperature The electrons are bumped to the conduction energy band by thermal energy where they flow freely and in doing so leave behind holes in the valence band which also flow freely The electric resistance of a typical intrinsic non doped semiconductor decreases exponentially with temperature following an Arrhenius model r r 0 e E A k B T displaystyle rho rho 0 e frac E A k B T nbsp An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart Hart equation 1 T A B ln r C ln r 3 displaystyle frac 1 T A B ln rho C ln rho 3 nbsp 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 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 they behave like intrinsic semiconductors as the carriers from the donors acceptors become insignificant compared to the thermally generated carriers 58 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 ofr A exp T 1 n displaystyle rho A exp left T 1 n right nbsp where n 2 3 4 depending on the dimensionality of the system Complex resistivity and conductivity editWhen analyzing the response of materials to alternating electric fields dielectric spectroscopy 59 in applications such as electrical impedance tomography 60 it is convenient to replace resistivity with a complex quantity called impedivity in analogy to electrical impedance Impedivity is the sum of a real component the resistivity and an imaginary component the reactivity in analogy to reactance The magnitude of impedivity 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 admittivity Admittivity is the sum of a real component 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 Mathematical descriptions of opacity Resistance versus resistivity in complicated geometries editEven 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 R r ℓ A displaystyle R rho ell A nbsp 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 formulasJ s E E r J displaystyle J sigma E rightleftharpoons E rho J nbsp must be replaced withJ r s r E r E r r r J r displaystyle mathbf J mathbf r sigma mathbf r mathbf E mathbf r rightleftharpoons mathbf E mathbf r rho mathbf r mathbf J mathbf r nbsp where E and J are now vector fields This equation along with the continuity equation for J and the Poisson s equation for E form a set of partial differential equations 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 analysis may be required Resistivity density product editIn some applications where the weight of an item is very important the product of resistivity and density is 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 conductivity to density ratio is desirable For example for long distance overhead power lines aluminium is frequently used rather than copper Cu because it is lighter for the same conductance Silver although it is the least resistive metal known has a high density and performs similarly to copper by this measure but is much more expensive 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 and lack of physical strength Aluminium is far more stable Toxicity excludes the choice of beryllium 61 Pure beryllium is also brittle Thus aluminium is usually the metal of choice when the weight or cost of a conductor is the driving consideration Resistivity density and resistivity density products of selected materials Material Resistivity nW m Density g cm3 Resistivity density Resistivity relative to Cu ie cross sectional area required to give same conductance Approx price at 9 December 2018 dubious discuss g mW m2 Relative to Cu USDper kg Relativeto CuSodium 47 7 0 97 46 31 2 843Lithium 92 8 0 53 49 33 5 531Calcium 33 6 1 55 52 35 2 002Potassium 72 0 0 89 64 43 4 291Beryllium 35 6 1 85 66 44 2 122Aluminium 26 50 2 70 72 48 1 579 2 0 0 16Magnesium 43 90 1 74 76 51 2 616Copper 16 78 8 96 150 100 1 6 0 1Silver 15 87 10 49 166 111 0 946 456 84Gold 22 14 19 30 427 285 1 319 39 000 19 000Iron 96 1 7 874 757 505 5 727See also editCharge transport mechanisms Chemiresistor Classification of materials based on permittivity Conductivity near the percolation threshold Contact resistance Electrical resistivities of the elements data page Electrical resistivity tomography Sheet resistance SI electromagnetism units Skin effect Spitzer resistivity Dielectric strengthNotes edit The atomic number is the count of electrons in an atom that is electrically neutral has no net electric charge Other relevant factors that are specifically not considered are the size of the whole crystal and external factors of the surrounding environment that modify the energy bands such as imposed electric or magnetic fields 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 Dr a DT r0 where r0 is the resistivity at 20 C in this case and a is the temperature coefficient The conductivity of metallic silver is not significantly better than metallic copper for most practical purposes the difference between the two can be easily compensated for by thickening the copper wire by only 3 However silver is preferred for exposed electrical contact points because corroded silver is a tolerable conductor but corroded copper is a fairly good insulator like most corroded metals Copper is widely used in electrical equipment building wiring and telecommunication cables 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 method Generally used for temper and alloy verification of aluminium Despite being less conductive than copper gold is commonly used in electrical contacts because it does not easily corrode Commonly used for overhead power line with steel reinforced ACSR a b Cobalt and ruthenium are considered to replace copper in integrated circuits fabricated in advanced nodes 29 18 chromium and 8 nickel austenitic stainless steel Nickel iron chromium alloy commonly used in heating elements Graphite is strongly anisotropic a b The resistivity of semiconductors depends strongly on the presence of impurities in the material Corresponds to an average salinity of 35 g kg at 20 C The pH should be around 8 4 and the conductivity in the range of 2 5 3 mS cm The lower value is appropriate for freshly prepared water The conductivity is used for the determination of TDS total dissolved particles This value range is typical of high quality drinking water and not an indicator of water quality Conductivity is lowest with monatomic gases present changes to 12 10 5 upon complete de gassing or to 7 5 10 5 upon equilibration to the atmosphere due to dissolved CO2References edit Lowrie William 2007 Fundamentals of Geophysics Cambridge University Press pp 254 55 ISBN 978 05 2185 902 8 Retrieved March 24 2019 a b Kumar Narinder 2003 Comprehensive Physics for Class XII New Delhi Laxmi Publications pp 280 84 ISBN 978 81 7008 592 8 Retrieved March 24 2019 Bogatin Eric 2004 Signal Integrity Simplified Prentice Hall Professional p 114 ISBN 978 0 13 066946 9 Retrieved March 24 2019 a b c Hugh O Pierson Handbook of carbon graphite diamond and fullerenes properties processing and applications p 61 William Andrew 1993 ISBN 0 8155 1339 9 J R Tyldesley 1975 An introduction to Tensor Analysis For Engineers and Applied Scientists Longman ISBN 0 582 44355 5 G Woan 2010 The Cambridge Handbook of Physics Formulas Cambridge University Press ISBN 978 0 521 57507 2 Josef Pek Tomas Verner 3 Apr 2007 Finite difference modelling of magnetotelluric fields in two dimensional anisotropic media Geophysical Journal International 128 3 505 521 doi 10 1111 j 1365 246X 1997 tb05314 x David Tong Jan 2016 The Quantum Hall Effect TIFR Infosys Lectures PDF Retrieved 14 Sep 2018 Bonding sl ibchem com Current versus Drift Speed The physics classroom Retrieved 20 August 2014 Lowe Doug 2012 Electronics All in One For Dummies John Wiley amp Sons ISBN 978 0 470 14704 7 Keith Welch Questions amp Answers How do you explain electrical resistance Thomas Jefferson National Accelerator Facility Retrieved 28 April 2017 Electromigration What is electromigration Middle East Technical University Retrieved 31 July 2017 When electrons are conducted through a metal they interact with imperfections in the lattice and scatter Thermal energy produces scattering by causing atoms to vibrate This is the source of resistance of metals Faber T E 1972 Introduction to the Theory of Liquid Metals Cambridge University Press ISBN 9780521154499 The Feynman Lectures in Physics Vol III Chapter 21 The Schrodinger Equation in a Classical Context A Seminar on Superconductivity Retrieved 26 December 2021 John C Gallop 1990 SQUIDS the Josephson Effects and Superconducting Electronics CRC Press pp 3 20 ISBN 978 0 7503 0051 3 See Flashes in the Sky Earth s Gamma Ray Bursts Triggered by Lightning Yaffa Eliezer Shalom Eliezer The Fourth State of Matter An Introduction to the Physics of Plasma Publisher Adam Hilger 1989 ISBN 978 0 85274 164 1 226 pages page 5 Bittencourt J A 2004 Fundamentals of Plasma Physics Springer p 1 ISBN 9780387209753 Hong Alice 2000 Dielectric Strength of Air The Physics Factbook a b c d e f g h i j k l m n o Raymond A Serway 1998 Principles of Physics 2nd ed Fort Worth Texas London Saunders College Pub p 602 ISBN 978 0 03 020457 9 a b c David Griffiths 1999 1981 7 Electrodynamics In Alison Reeves ed Introduction to Electrodynamics 3rd ed Upper Saddle River New Jersey Prentice Hall p 286 ISBN 978 0 13 805326 0 OCLC 40251748 Matula R A 1979 Electrical resistivity of copper gold palladium and silver Journal of Physical and Chemical Reference Data 8 4 1147 Bibcode 1979JPCRD 8 1147M doi 10 1063 1 555614 S2CID 95005999 Douglas Giancoli 2009 1984 25 Electric Currents and Resistance In Jocelyn Phillips ed Physics for Scientists and Engineers with Modern Physics 4th ed Upper Saddle River New Jersey Prentice Hall p 658 ISBN 978 0 13 149508 1 Copper wire tables United States National Bureau of Standards Retrieved 3 February 2014 via Internet Archive archive org archived 2001 03 10 1 Calculated as 56 conductivity of pure copper 5 96E 7 Retrieved on 2023 1 12 Physical constants PDF format see page 2 table in the right lower corner Retrieved on 2011 12 17 2 Calculated as 28 conductivity of pure copper 5 96E 7 Retrieved on 2023 1 12 IITC Imec Presents Copper Cobalt and Ruthenium Interconnect Results Temperature Coefficient of Resistance Electronics Notes 3 Calculated as 15 conductivity of pure copper 5 96E 7 Retrieved on 2023 1 12 Material properties of niobium AISI 1010 Steel cold drawn Matweb Karcher Ch Kocourek V December 2007 Free surface instabilities during electromagnetic shaping of liquid metals Proceedings in Applied Mathematics and Mechanics 7 1 4140009 4140010 doi 10 1002 pamm 200700645 ISSN 1617 7061 JFE steel PDF Retrieved 2012 10 20 a b Douglas C Giancoli 1995 Physics Principles with Applications 4th ed London Prentice Hall ISBN 978 0 13 102153 2 see also Table of Resistivity hyperphysics phy astr gsu edu 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 Glenn Elert ed Resistivity of steel The Physics Factbook retrieved and archived 16 June 2011 Probably the metal with highest value of electrical resistivity Y Pauleau Peter B Barna P B Barna 1997 Protective coatings and thin films synthesis characterization and applications p 215 Springer ISBN 0 7923 4380 8 Milton Ohring 1995 Engineering materials science Volume 1 3rd ed Academic Press p 561 ISBN 978 0125249959 Physical properties of sea water Archived 2018 01 18 at the Wayback Machine Kayelaby npl co uk Retrieved on 2011 12 17 4 chemistry stackexchange com Eranna Golla 2014 Crystal Growth and Evaluation of Silicon for VLSI and ULSI CRC Press p 7 ISBN 978 1 4822 3281 3 a b c Transmission Lines data Transmission line net Retrieved on 2014 02 03 R M Pashley M Rzechowicz L R Pashley M J Francis 2005 De Gassed Water is a Better Cleaning Agent The Journal of Physical Chemistry B 109 3 1231 8 doi 10 1021 jp045975a PMID 16851085 ASTM D1125 Standard Test Methods for Electrical Conductivity and Resistivity of Water ASTM D5391 Standard Test Method for Electrical Conductivity and Resistivity of a Flowing High Purity Water Sample Lawrence S Pan Don R Kania Diamond electronic properties and applications p 140 Springer 1994 ISBN 0 7923 9524 7 S D Pawar P Murugavel D M Lal 2009 Effect of relative humidity and sea level pressure on electrical conductivity of air over Indian Ocean Journal of Geophysical Research 114 D2 D02205 Bibcode 2009JGRD 114 2205P doi 10 1029 2007JD009716 E Seran M Godefroy E Pili 2016 What we can learn from measurements of air electric conductivity in 222Rn rich atmosphere Earth and Space Science 4 2 91 106 Bibcode 2017E amp SS 4 91S doi 10 1002 2016EA000241 Copper Wire Tables Archived 2010 08 21 at the Wayback Machine US Dep of Commerce National Bureau of Standards Handbook February 21 1966 Ward Malcolm R 1971 Electrical engineering science McGraw Hill technical education Maidenhead UK McGraw Hill pp 36 40 ISBN 9780070942554 Gruneisen E 1933 Die Abhangigkeit des elektrischen Widerstandes reiner Metalle von der Temperatur Annalen der Physik 408 5 530 540 Bibcode 1933AnP 408 530G doi 10 1002 andp 19334080504 ISSN 1521 3889 Quantum theory of real materials James R Chelikowsky Steven G Louie Boston Kluwer Academic Publishers 1996 pp 219 250 ISBN 0 7923 9666 9 OCLC 33335083 a href Template Cite book html title Template Cite book cite book a CS1 maint others link A Matthiessen Rep Brit Ass 32 144 1862 A Matthiessen Progg Anallen 122 47 1864 J Seymour 1972 Physical Electronics chapter 2 Pitman Stephenson C Hubler A 2015 Stability and conductivity of self assembled wires in a transverse electric field Sci Rep 5 15044 Bibcode 2015NatSR 515044S doi 10 1038 srep15044 PMC 4604515 PMID 26463476 Otto H Schmitt University of Minnesota 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 Berryllium Be Chemical properties Health and Environmental effects Further reading editPaul Tipler 2004 Physics for Scientists and Engineers Electricity Magnetism Light and Elementary Modern Physics 5th ed W H Freeman ISBN 978 0 7167 0810 0 Measuring Electrical Resistivity and ConductivityExternal links edit nbsp Wikibooks has a book on the topic of A level Physics Advancing Physics Resistivity and Conductivity Electrical Conductivity Sixty Symbols Brady Haran for the University of Nottingham 2010 Comparison of the electrical conductivity of various elements in WolframAlpha Partial and total conductivity Electrical conductivity PDF Retrieved from https en wikipedia org w index php title Electrical resistivity and conductivity amp oldid 1184657287, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.