Electrical resistivity and conductivity 20 languages

Article Talk Tools

From Wikipedia, the free encyclopedia

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.

Resistivity
Common symbols ρ
SI unit ohm metre (Ω⋅m)
Other units s (Gaussian/ESU)
In SI base units kg⋅m3⋅s−3⋅A−2
Derivations from
other quantities

Dimension

Conductivity
Common symbols σ, κ, γ
SI unit siemens per metre (S/m)
Other units (Gaussian/ESU)
Derivations from
other quantities

Dimension

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.

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)[citation needed] and
γ (gamma)[citation needed] 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

is the electrical resistance of a uniform specimen of the material

is the length of the specimen
is the cross-sectional area of the specimen

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).

Conductivity, , is directly proportional to

Where: = electron concentration, = hole concentration, = electron mobility, = hole

mobility.

General scalar quantities ​[ edit ]

If the geometry is more complicated, or if the resistivity varies from point to point within the material,
the current and electric field will be functions of position. Then 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 at the point ,

is the electric field at the point ,
is the current density at the point .

The current density is parallel to the electric field by necessity.

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.

This expression simplifies to the formula given above under "ideal case" when the resistivity is
constant in the material and the geometry has a uniform cross-section. In this case, the electric
field and current density are constant and parallel.

Derivation of the constant case from the general case [show]

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]

The relation between current density and electric current velocity is governed by the equation

where

= current density (A/m²),

= charge of the carrier (C) — e.g., C for electrons,

= number of charge carriers per unit volume (1/m³),

= drift velocity (m/s) — the average velocity of charge carriers in the direction of the electric
field.

Which can be rearranged to show current velocity's inverse relationship to the number of charge
carriers at constant current density.

Causes of conductivity ​[ edit ]

Band theory simplified ​[ edit ]

See also: Band theory

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. "intrin." indicates intrinsic semiconductors.
edit

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 ]

Main article: Free electron model

An animation of Newton's cradle.

Like the balls in 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 , 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 ]

Main article: Superconductivity

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)

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), 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

Very low: air is an

excellent insulator until
it breaks down into
Electrical Usually very high: for many purposes, the
plasma at electric field
conductivity conductivity of a plasma may be treated as infinite.
strengths above 30
kilovolts per
centimetre.[20]

Two or three: electrons, ions, protons and neutrons

One: all gas particles
can be distinguished by the sign and value of their
Independently behave in a similar way,
charge so that they behave independently in many
acting influenced by gravity
circumstances, with different bulk velocities and
species and by collisions with
temperatures, allowing phenomena such as new
one another.
types of waves and instabilities.

Maxwellian: collisions
usually lead to a Often non-Maxwellian: collisional interactions are
Velocity Maxwellian velocity often weak in hot plasmas and external forcing can
distribution distribution of all gas drive the plasma far from local equilibrium and lead to
particles, with very few a significant population of unusually fast particles.
relatively fast particles.

Binary: two-particle
Collective: waves, or organized motion of plasma, are
collisions are the rule,
Interactions very important because the particles can interact at
three-body collisions
long ranges through the electric and magnetic forces.
extremely rare.

Resistivity and conductivity of various materials ​[ edit ]

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

A conductor such as a metal has high conductivity and a low resistivity.

An insulator such as 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

Conductivity, Temperature
Resistivity, ρ,
Material σ, coefficient[c] Reference
at 20 °C (Ω·m)
at 20 °C (S/m) (K−1)

Silver[d] 1.59 × 10−8 63.0 × 106 3.80 × 10−3 [21][22]

Copper[e] 1.68 × 10−8 59.6 × 106 4.04 × 10−3 [23][24]

Annealed copper[f] 1.72 × 10−8 58.0 × 106 3.93 × 10−3 [25]

Gold[g] 2.44 × 10−8 41.1 × 106 3.40 × 10−3 [21]

Aluminium[h] 2.65 × 10−8 37.7 × 106 3.90 × 10−3 [21]

Brass (5% Zn) 3.00 × 10−8 33.4 × 106 [26]

Calcium 3.36 × 10−8 29.8 × 106 4.10 × 10−3

Rhodium 4.33 × 10−8 23.1 × 106

Tungsten 5.60 × 10−8 17.9 × 106 4.50 × 10−3 [21]

 Zinc 5.90 × 10−8 16.9 × 106 3.70 × 10−3 [27]

Brass (30% Zn) 5.99 × 10−8 16.7 × 106 [28]

7.00 × 10−3[30]
Cobalt[i] 6.24 × 10−8 16.0 × 106 [unreliable source?]

Nickel 6.99 × 10−8 14.3 × 106 6.00 × 10−3

Ruthenium[i] 7.10 × 10−8 14.1 × 106

Lithium 9.28 × 10−8 10.8 × 106 6.00 × 10−3

Iron 9.70 × 10−8 10.3 × 106 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%

11.2 × 10−8 8.94 × 106 [31]
P / 5% Sn)

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

46.0 × 10−8 2.17 × 106 [35]
steel

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

6.70 × 105
Nichrome[k] 110 × 10−8 [citation needed]
0.40 × 10−3 [21]

Carbon (graphite) 250 × 10−8 to 2 × 105 to 3 × 105 [4]

parallel to basal plane[l] 500 × 10−8 [citation needed]

0.5 × 10−3 to 1.25 × 103 to

Carbon (amorphous) −0.50 × 10−3 [21][40]
0.8 × 10−3 2.00 × 103

Carbon (graphite)
perpendicular to basal 3.0 × 10−3 3.3 × 102 [4]

plane

10−3 to 108 10−8 to 103 [41]

GaAs [clarification needed] [dubious – discuss]

Germanium[m] 4.6 × 10−1 2.17 −48.0 × 10−3 [21][22]

Sea water[n] 2.1 × 10−1 4.8 [42]

3.3 × 10−1 to
Swimming pool water[o] 0.25 to 0.30 [43]
4.0 × 10−1

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

Bone 1.66 × 102 6 × 10−3 [44]

Silicon[m] 2.3 × 103 4.35 × 10−4 −75.0 × 10−3 [45][21]

Wood (damp) 103 to 104 10−4 to 10−3 [46]

Deionized water[q] 1.8 × 105 4.2 × 10−5 [47]

Ultrapure water 1.82 × 105 5.49 × 10−6 [48][49]

Glass 1011 to 1015 10−15 to 10−11 [21][22]

Carbon (diamond) 1012 ~10−13 [50]

Hard rubber 1013 10−14 [21]

Air 109 to 1015 ~10−15 to 10−9 [51][52]

Wood (oven dry) 1014 to 1016 10−16 to 10−14 [46]

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.[53]

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.[46] 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.[54] 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–Grüneisen 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 ρ of a metal can be approximated through the Bloch–Grüneisen formula:[55]

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.[56]

If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated
by Augustus Matthiessen in the 1860s)[57][58] 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.[59]

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.

Kondo insulators ​[ edit ]

Kondo insulators are materials where the resistivity follows the formula

where , , and are constant parameters, the residual resistivity, the Fermi liquid

contribution, a lattice vibrations term and the Kondo effect.

Complex resistivity and conductivity ​[ edit ]

When analyzing the response of materials to alternating electric fields (dielectric spectroscopy),[60]
in applications such as electrical impedance tomography,[61] 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 material with a low resistivity–density product
(or equivalently a high conductivity/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;[62] 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 product of selected materials, and relative

to copper (Cu)
Resistivity Density Resistivity × density
Material Relative Relative Relative
(nΩ·m) (g/cm3) (g·mΩ/m2)
to Cu to Cu to Cu

Sodium 47.7 2.843 0.97 0.108 46 0.31

Lithium 92.8 5.53 0.53 0.059 49 0.33

Calcium 33.6 2.002 1.55 0.173 52 0.35

Potassium 72.0 4.291 0.89 0.099 64 0.43

Beryllium 35.6 2.122 1.85 0.206 66 0.44

Aluminium 26.50 1.579 2.70 0.301 72 0.48

Magnesium 43.90 2.616 1.74 0.194 76 0.51

Copper 16.78 1 8.96 1 150 1

Silver 15.87 0.946 10.49 1.171 166 1.11

Gold 22.14 1.319 19.30 2.154 427 2.84

Iron 96.1 5.727 7.874 0.879 757 5.03

History ​[ edit ]

John Walsh and the conductivity of a vacuum ​[ edit ]

In a 1774 letter to Dutch-born British scientist Jan Ingenhousz, Benjamin Franklin relates an
experiment by another British scientist, John Walsh, that purportedly showed this astonishing fact:
Although rarified air conducts electricity better than common air, a vacuum does not conduct
electricity at all.[63]

Mr. Walsh ... has just made a curious Discovery in Electricity. You know we find that in
rarify’d Air it would pass more freely, and leap thro’ greater Spaces than in dense Air; and
thence it was concluded that in a perfect Vacuum it would pass any distance without the
least Obstruction. But having made a perfect Vacuum by means of boil’d Mercury in a long
Torricellian bent Tube, its Ends immers’d in Cups full of Mercury, he finds that the Vacuum
will not conduct at all, but resists the Passage of the Electric Fluid absolutely.

However, to this statement a note (based on modern knowledge) was added by the editors—at the
American Philosophical Society and Yale University—of the webpage hosting the letter:[63]

We can only assume that something was wrong with Walsh’s findings. ... Although the
conductivity of a gas, as it approaches a vacuum, increases up to a point and then
decreases, that point is far beyond what the technique described might have been
expected to reach. Boiling replaced the air with mercury vapor, which as it cooled created
a vacuum that could scarcely have been complete enough to decrease, let alone
eliminate, the vapor’s conductivity.

See also ​[ edit ]

Charge 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 strength
Physical crystallography before X-rays

Notes ​[ edit ]

a. ^ The atomic number is the count of electrons in an atom that is electrically neutral – has no net electric
charge.
b. ^ 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.
c. ^ 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.
 d. ^ 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.
e. ^ Copper is widely used in electrical equipment, building wiring, and telecommunication cables.
f. ^ 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.
g. ^ Despite being less conductive than copper, gold is commonly used in electrical contacts because it
does not easily corrode.
h. ^ Commonly used for overhead power line with steel reinforced (ACSR)
i. ^ a b Cobalt and ruthenium are considered to replace copper in integrated circuits fabricated in
advanced nodes[29]
j. ^ 18% chromium and 8% nickel austenitic stainless steel
k. ^ Nickel-iron-chromium alloy commonly used in heating elements.
l. ^ Graphite is strongly anisotropic.
m. ^ a b The resistivity of semiconductors depends strongly on the presence of impurities in the material.
n. ^ Corresponds to an average salinity of 35 g/kg at 20 °C.
o. ^ 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).
p. ^ This value range is typical of high quality drinking water and not an indicator of water quality
q. ^ 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. ^ Bonding (sl) . 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" (PDF). 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 Archived 2011-11-23 at the Wayback Machine. (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 archived 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 Archived 2018-01-18 at the Wayback Machine.
Kayelaby.npl.co.uk. Retrieved on 2011-12-17.
43. ^ [4] . chemistry.stackexchange.com
44. ^ Sovilj, S.; Magjarević, R.; Lovell, N. H.; Dokos, S. (2013). "Simplified 2D Bidomain Model of Whole
Heart Electrical Activity and ECG Generation" . Computational and Mathematical Methods in
Medicine. doi:10.1155/2013/134208 . PMC 3654639 . PMID 23710247 .
45. ^ Eranna, Golla (2014). Crystal Growth and Evaluation of Silicon for VLSI and ULSI . CRC Press.
p. 7. ISBN 978-1-4822-3281-3.
46. ^ a b c Transmission Lines data . Transmission-line.net. Retrieved on 2014-02-03.
47. ^ 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 .
 48. ^ ASTM D1125 Standard Test Methods for Electrical Conductivity and Resistivity of Water
49. ^ ASTM D5391 Standard Test Method for Electrical Conductivity and Resistivity of a Flowing High
Purity Water Sample
50. ^ Lawrence S. Pan, Don R. Kania, Diamond: electronic properties and applications, p. 140, Springer,
1994 ISBN 0-7923-9524-7.
51. ^ 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 .
52. ^ 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 .
53. ^ Copper Wire Tables Archived 2010-08-21 at the Wayback Machine. US Dep. of Commerce.
National Bureau of Standards Handbook. February 21, 1966
54. ^ Ward, Malcolm R. (1971). Electrical engineering science. McGraw-Hill technical education.
Maidenhead, UK: McGraw-Hill. pp. 36–40. ISBN 9780070942554.
55. ^ 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 .
56. ^ 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 .
57. ^ A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)
58. ^ A. Matthiessen, Progg. Anallen, 122, 47 (1864)
59. ^ J. Seymour (1972) Physical Electronics, chapter 2, Pitman
60. ^ 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 .
61. ^ 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.
62. ^ "Berryllium (Be) - Chemical properties, Health and Environmental effects" .
63. ^ a b Franklin, Benjamin (1978) [1774]. "From Benjamin Franklin to Jan Ingenhousz, 18 March 1774" .
In Willcox, William B. (ed.). The Papers of Benjamin Franklin. Vol. 21, January 1, 1774, through March
22, 1775. Yale University Press. pp. 147–149 – via Founders Online, National Archives.

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 ]

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).

https://edu-physics.com/2021/01/07/resistivity-of-the-material-of-a-wire-physics-practical/

Categories: Electrical resistance and conductance Physical quantities Materials science

This page was last edited on 24 May 2025, at 17:20 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you
agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit
organization.

Privacy policy About Wikipedia Disclaimers Contact Wikipedia Code of Conduct Developers Statistics

Cookie statement Mobile view