Electrical resistivity (also called specific electrical resistance or volume resistivity) and its inverse, electrical conductivity, is a fundamental property of a material that quantifies how strongly it resists or conducts electric current. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek alphabet ρ (rho). The SI unit of electrical resistivity is the ohmmeter (Ω⋅m).
For example, if a 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) and γ (gamma) are sometimes used. The SI unit of electrical conductivity is siemens per metre (S/m).
where
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. This means that all pure copper 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 highresistivity material is like pushing water through a pipe full of sand — while passing current through a lowresistivity 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 material is proportional to the length, but inversely proportional to the crosssectional area. Thus resistivity can be expressed using the SI unit "ohm metre" (Ω⋅m) — i.e. ohms divided by metres (for the length) and then multiplied by square metres (for the crosssectional area).
For example, if A = , $\backslash ell$ = (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).
where
in which $E$ and $J$ 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.
! Derivation from general definition of resistivity 
There are three equations to be combined here. The first is the resistivity for parallel current and electric field:
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:
If the current density is constant, it is equal to the total current divided by the cross sectional area:
Plugging in the values of E and J into the first expression, we obtain:
Finally, we apply Ohm's law, V/ I = R.

Here, anisotropy 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. 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 threedimensional tensor form:J.R. Tyldesley (1975) An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, G. Woan (2010) The Cambridge Handbook of Physics Formulas, Cambridge University Press,
where the conductivity σ and resistivity ρ are rank2 , 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:
\begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix} = \begin{bmatrix} \rho_{xx} & \rho_{xy} & \rho_{xz} \\ \rho_{yx} & \rho_{yy} & \rho_{yz} \\ \rho_{zx} & \rho_{zy} & \rho_{zz} \end{bmatrix}\begin{bmatrix} J_x \\ J_y \\ J_z \end{bmatrix}
where
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 xaxis parallel to the current direction, so J_{ y} = J_{ z} = 0. This leaves:
Conductivity is defined similarly:
\begin{bmatrix} J_x \\ J_y \\ J_z \end{bmatrix} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}\begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix}
or
Both resulting in:
Looking at the two expressions, $\backslash boldsymbol\{\backslash rho\}$ and $\backslash boldsymbol\{\backslash sigma\}$ 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, σ_{xx} may not be equal to 1/ ρ_{xx}. This can be seen in the Hall effect, where $\backslash rho\_\{xy\}$ is nonzero. In the Hall effect, due to rotational invariance about the zaxis, $\backslash rho\_\{yy\}=\backslash rho\_\{xx\}$ and $\backslash rho\_\{yx\}=\backslash rho\_\{xy\}$, so the relation between resistivity and conductivity simplifies to:
If the electric field is parallel to the applied current, $\backslash rho\_\{xy\}$ and $\backslash rho\_\{xz\}$ are zero. When they are zero, one number, $\backslash rho\_\{xx\}$, is enough to describe the electrical resistivity. It is then written as simply $\backslash rho$, and this reduces to the simpler expression.
where 𝑛 is the density of charge carriers (the number of carriers in a unit volume), 𝑞 is the charge of one carrier, $\backslash vec\{\backslash upsilon\}\_a$ is the average speed of their movement. In the case where the current consists of many carriers
where $j\_i$ is the current density of the $i$th carrier.
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.
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 wavelike 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. 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.
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 $\backslash alpha$, which is the ratio of the concentration of ions $N$ to the concentration of molecules of the dissolved substance $N\_0$:
The specific electrical conductivity ($\backslash sigma$) of a solution is equal to:
where $q$: module of the ion charge, $b^+$ and $b^$: mobility of positively and negatively charged ions, $N\_0$: concentration of molecules of the dissolved substance, $\backslash alpha$: the coefficient of dissociation.
In 1986, researchers discovered that some cuprateperovskite ceramic materials have much higher critical temperatures, and in 1987 one was produced with a critical temperature above . Such a high transition temperature is theoretically impossible for a conventional superconductor, so the researchers named these conductors hightemperature superconductors. Liquid nitrogen boils at 77 K, cold enough to activate hightemperature superconductors, but not nearly cold enough for conventional superconductors. In conventional superconductors, electrons are held together in pairs by an attraction mediated by lattice . The best available model of hightemperature superconductivity is still somewhat crude. There is a hypothesis that electron pairing in hightemperature superconductors is mediated by shortrange spin waves known as .
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 ( n_{e} = ⟨Z⟩> n_{i}), 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 nonneutral 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, . 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 . The dynamics of plasmas interacting with external and selfgenerated magnetic fields are studied in the academic discipline of magnetohydrodynamics.
Plasma is often called the fourth state of matter after solid, liquids and gases.Yaffa Eliezer, Shalom Eliezer, The Fourth State of Matter: An Introduction to the Physics of Plasma, Publisher: Adam Hilger, 1989, , 226 pages, page 5
It is distinct from these and other lowerenergy 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:
The degree of semiconductors doping makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a solution of water is highly dependent on its concentration of dissolved salts, and other chemical species that Ionization in the solution. Electrical conductivity of water samples is used as an indicator of how saltfree, ionfree, or impurityfree 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 . An EC meter is normally used to measure conductivity in a solution. A rough summary is as follows:
Superconductors  0 
10^{−8}  
Variable  
Variable  
Insulators  10^{16} 
∞ 
This table shows the resistivity ( ρ), conductivity and temperature coefficient of various materials at 20 Celsius (68 Fahrenheit, 293 Kelvin)
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 . Copper Wire Tables . US Dep. of Commerce. National Bureau of Standards Handbook. February 21, 1966
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):
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 worse insulator than ovendry. In any case, a sufficiently high voltage – such as that in lightning strikes or some hightension power lines – can lead to insulation breakdown and electrocution risk even with apparently dry wood.
where $\backslash alpha$ is called the temperature coefficient of resistivity, $T\_0$ is a fixed reference temperature (usually room temperature), and $\backslash rho\_0$ is the resistivity at temperature $T\_0$. The parameter $\backslash alpha$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, $\backslash alpha$ is different for different reference temperatures. For this reason it is usual to specify the temperature that $\backslash alpha$ was measured at with a suffix, such as $\backslash alpha\_\{15\}$, and the relationship only holds in a range of temperatures around the reference.M.R. Ward (1971) Electrical Engineering Science, pp. 36–40, McGrawHill. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.
where $\backslash rho(0)$ 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. $\backslash Theta\_R$ 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:
If more than one source of scattering is simultaneously present, Matthiessen's Rule (first formulated by Augustus Matthiessen in the 1860s)A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)A. Matthiessen, Progg. Anallen, 122, 47 (1864) 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 lowtemperature 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.
At high metal temperatures, the WiedemannFranz law holds:
where $K$: thermal conductivity, $k$; Boltzmann constant, $e$: electron charge, $T$: temperature, $\backslash sigma$: electrical conductivity coefficient.
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 socalled Steinhart–Hart coefficients.
This equation is used to calibrate .
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.J. Seymour (1972) Physical Electronics, chapter 2, Pitman
In noncrystalline 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.
Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix of complex numbers, in the case of anisotropic materials) called the Admittance. Admittivity is the sum of a real component called the conductivity and an imaginary component called the Susceptance.
An alternative description of the response to alternating currents uses a real (but frequencydependent) conductivity, along with a real permittivity. The larger the conductivity is, the more quickly the alternatingcurrent signal is absorbed by the material (i.e., the more opaque the material is). For details, see Mathematical descriptions of opacity.
In cases like this, the formulas
must be replaced with
where E and J are now . 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.
/ref> (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.
Sodium  47.7  0.97  46  31%  2.843  0.31  
Lithium  92.8  0.53  49  33%  5.531  0.33  
Calcium  33.6  1.55  52  35%  2.002  0.35  
Potassium  72.0  0.89  64  43%  4.291  0.43  
Beryllium  35.6  1.85  66  44%  2.122  0.44  
Aluminium  26.50  2.70  72  48%  1.5792  0.48  2.0  0.16 
Magnesium  43.90  1.74  76  51%  2.616  0.51  
Copper  16.78  8.96  150  6.0  
Silver  15.87  10.49  166  111%  0.946  1.11  456  84 
Gold  22.14  19.30  427  285%  1.319  2.85  39,000  19,000 
Iron  96.1  7.874  757  505%  5.727  5.05 

