4098:, 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.
4592:
4557:, 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. A consequence of this is that an electric current flowing in a loop of
4003:, 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 and their distribution within the crystal.
4534:
3923:
557:
3009:
2011:
3391:
4043:
2733:
2543:
1732:
829:, 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
8902:
3084:
2230:
8970:
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
573:
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
4576:
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
4132:
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
4019:
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
976:
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
9376:
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
7037:
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
4598:
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. Plasma temperatures in lightning might approach 30,000 kelvin (29,727 °C) (53,540 °F), and electron densities may exceed
7057:
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
4027:
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
1627:
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
9385:
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
7423:
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
3772:
8194:
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
4141:
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,
9547:
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
4074:, 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. This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (a
8962:
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
4142:
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.
7739:. 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
3004:{\displaystyle {\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}}}
7613:
2724:
1691:
4572:, 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
5060:
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
8782:
4045:
4049:
4048:
4044:
2006:{\displaystyle {\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}},}
697:
1612:, 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
8971:
and oxygen (and lack of physical strength). Aluminium is far more stable. Toxicity excludes the choice of beryllium. (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.
4050:
4828:
1337:
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.
8195:
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.
4093:
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
3386:{\displaystyle {\begin{aligned}J_{x}&=\sigma _{xx}E_{x}+\sigma _{xy}E_{y}+\sigma _{xz}E_{z}\\J_{y}&=\sigma _{yx}E_{x}+\sigma _{yy}E_{y}+\sigma _{yz}E_{z}\\J_{z}&=\sigma _{zx}E_{x}+\sigma _{zy}E_{y}+\sigma _{zz}E_{z}\end{aligned}}.}
4047:
3600:
813:
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
7848:
4843:
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
9377:
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.
4023:
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.
7073:
worse insulator than oven-dry. 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.
4737:
420:
7433:
2538:{\displaystyle {\begin{aligned}E_{x}&=\rho _{xx}J_{x}+\rho _{xy}J_{y}+\rho _{xz}J_{z},\\E_{y}&=\rho _{yx}J_{x}+\rho _{yy}J_{y}+\rho _{yz}J_{z},\\E_{z}&=\rho _{zx}J_{x}+\rho _{zy}J_{y}+\rho _{zz}J_{z}.\end{aligned}}}
216:
2577:
1633:
1353:
Assume the geometry has a uniform cross-section and the resistivity is constant in the material. Then the electric field and current density are constant and parallel, and by the general definition of resistivity, we obtain
3075:
2221:
8391:
9703:
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
9481:
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
1314:
619:
4973:
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.
7058:
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.
10091:
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
7689:
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:
4394:
4010:
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
8775:
8161:
564:
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
624:
8074:
4133:
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
4744:
3089:
2235:
10105:
8268:
917:(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.
8897:{\displaystyle \mathbf {J} (\mathbf {r} )=\sigma (\mathbf {r} )\mathbf {E} (\mathbf {r} )\,\,\rightleftharpoons \,\,\mathbf {E} (\mathbf {r} )=\rho (\mathbf {r} )\mathbf {J} (\mathbf {r} ),}
1052:
4028:
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.
4164:) traveling, each carrying an electrical charge. The resistivity of ionic solutions (electrolytes) varies tremendously with concentration – while distilled water is almost an insulator,
1536:
960:
615:
1616:
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.
1593:
883:
333:
136:
4090:
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.
3437:
3593:
9381:
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:
4046:
3544:
3415:
2092:
1438:
1390:
4183:
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
4082:
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
11243:
Mutual
Impedivity Spectrometry and the Feasibility of its Incorporation into Tissue-Diagnostic Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements
7175:
8599:
7775:
1478:
289:
4292:
11352:
8720:
4662:
7395:
2116:
2039:
7683:
8946:. 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
7770:
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:
3864:
3834:
3804:
3494:
833:
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.
8508:
7258:
7644:
1087:
7958:
7870:
7368:
7348:
7327:
7305:
7197:
4517:
4314:
4201:
10511:
8566:
8535:
8461:
7285:
7231:
7069:
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
4497:
4470:
4443:
4248:
4015:. 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
3884:
3018:
911:
749:
8481:
2161:
1189:
1138:
9372:, that purportedly showed this astonishing fact: Although rarified air conducts electricity better than common air, a vacuum does not conduct electricity at all.
1608:
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
11242:
4985:
Collective: waves, or organized motion of plasma, are very important because the particles can interact at long ranges through the electric and magnetic forces.
8685:
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
8434:
8414:
7938:
7918:
7894:
4416:
4221:
1213:
1158:
1107:
775:
723:
4545:
The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In normal (that is, non-superconducting) conductors, such as
7414:
499:) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter
9386:
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.
7735:
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
1228:
346:
9422:
4997:
4889:. 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:
10523:
4867:. 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
3767:{\displaystyle \sigma _{xx}={\frac {\rho _{xx}}{\rho _{xx}^{2}+\rho _{xy}^{2}}},\quad \sigma _{xy}={\frac {-\rho _{xy}}{\rho _{xx}^{2}+\rho _{xy}^{2}}}.}
5021:
is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of
149:
1326: — 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
4058:. Like the balls in Newton's cradle, electrons in a metal quickly transfer energy from one terminal to another, despite their own negligible movement.
822:
than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper.
4321:
8732:
8087:
7397:, and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the
4606:
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
4137:(i.e., without introducing further complications, the density of electrons) in the conduction band increases. In extrinsic (doped) semiconductors,
4951:
so that they behave independently in many circumstances, with different bulk velocities and temperatures, allowing phenomena such as new types of
8293:
7728:
in the 1860s) states that the total resistance can be approximated by adding up several different terms, each with the appropriate value of
11280:. Vol. 21, January 1, 1774, through March 22, 1775. Yale University Press. pp. 147–149 – via Founders Online, National Archives.
7713:
The Bloch–Grüneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the first
11382:
10057:
8009:
7608:{\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,}
3443:
of each other. However, in the most general case, the individual matrix elements are not necessarily reciprocals of one another; for example,
4614:. If an electrode is inserted into a plasma, its potential generally lies considerably below the plasma potential, due to what is termed a
10769:
2719:{\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}}}.}
1686:{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\mathbf {E} \,\,\rightleftharpoons \,\,\mathbf {E} ={\boldsymbol {\rho }}\mathbf {J} ,}
9412:
8665:
An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real
4622:, which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma (
885:
The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, if
10132:"The Feynman Lectures in Physics, Vol. III, Chapter 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity"
8726:, where the material is inhomogeneous (different resistivity in different places), and the exact paths of current flow are not obvious.
8209:
4909:
Very low: air is an excellent insulator until it breaks down into plasma at electric field strengths above 30 kilovolts per centimetre.
9605:
4541:
showing the resistance of a mercury wire as a function of temperature. The abrupt drop in resistance is the superconducting transition.
990:
10699:
4871:. The dynamics of plasmas interacting with external and self-generated magnetic fields are studied in the academic discipline of
1495:
929:
587:
17:
4577:
nominal superconducting transition, these vortices can become frozen so that the resistance of the material becomes truly zero.
4078:) is applied across the metal, the resulting electric field causes electrons to drift towards the positive terminal. The actual
1559:
849:
10026:
4970:: collisions usually lead to a Maxwellian velocity distribution of all gas particles, with very few relatively fast particles.
4618:. The good electrical conductivity of plasmas makes their electric fields very small. This results in the important concept of
692:{\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}}\\{}\Leftrightarrow \rho &=R{\frac {A}{\ell }},\end{aligned}}}
11367:
8669:. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more
11304:
10826:
10649:
10433:
10353:
10306:
10203:
10167:
10036:
9921:
9844:
9810:
9773:
536:
35:
10536:
4006:
The material's electrons seek to minimize the total energy in the material by settling into low energy states; however, the
10785:
10500:
10470:
9990:
8674:
1407:
1359:
798:(Ω⋅m) — i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length).
11017:
809:
describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an
450:. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the
11345:
7098:
10751:
10558:
9834:
7646:
is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the
4967:
1447:
7410:
5037:
makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a
4255:
11116:
11043:
10920:
10729:
10679:
10227:
10115:
9901:
9880:
9427:
4172:
is also controlled by the movement of ions, but here we are talking about molten salts rather than solvated ions. In
2547:
Since the choice of the coordinate system is free, the usual convention is to simplify the expression by choosing an
10846:
7697: = 5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
7703: = 3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
4651:
The magnitude of the potentials and electric fields must be determined by means other than simply finding the net
10902:
ASTM D5391 Standard Test Method for
Electrical Conductivity and Resistivity of a Flowing High Purity Water Sample
10859:
R. M. Pashley; M. Rzechowicz; L. R. Pashley; M. J. Francis (2005). "De-Gassed Water is a Better
Cleaning Agent".
9472:
The atomic number is the count of electrons in an atom that is electrically neutral – has no net electric charge.
8641:). The magnitude of impedivity is the square root of sum of squares of magnitudes of resistivity and reactivity.
8618:
8078:
An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the
4569:
3991:
305:
8723:
7350:
is different for different reference temperatures. For this reason it is usual to specify the temperature that
4823:{\displaystyle \mathbf {E} =-{\frac {k_{\text{B}}T_{\text{e}}}{e}}{\frac {\nabla n_{\text{e}}}{n_{\text{e}}}}.}
105:
8943:
3420:
1442:
Since the current density is constant, it is equal to the total current divided by the cross sectional area:
31:
11313:
10484:
3549:
11387:
9552:
silver is a tolerable conductor, but corroded copper is a fairly good insulator, like most corroded metals.
8658:. Admittivity is the sum of a real component called the conductivity and an imaginary component called the
7755:
3503:
3398:
2075:
6777:
10327:
9397:
8079:
7724:
If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated by
4859:
prevents electric fields from directly affecting the plasma over large distances, i.e., greater than the
4151:
10935:"Effect of relative humidity and sea level pressure on electrical conductivity of air over Indian Ocean"
10612:
8571:
11392:
11140:
9407:
8967:, aluminium is frequently used rather than copper (Cu) because it is lighter for the same conductance.
6985:
3977:
3935:
339:
262:
142:
11111:. James R. Chelikowsky, Steven G. Louie. Boston: Kluwer Academic Publishers. 1996. pp. 219–250.
9437:
4007:
3927:
8688:
1623:
means that the material has different properties in different directions. For example, a crystal of
11338:
11060:
8958:
In some applications where the weight of an item is very important, the product of resistivity and
4856:
1723:
are vectors. These tensors can be represented by 3×3 matrices, the vectors with 3×1 matrices, with
778:
11297:
Physics for
Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics
10183:
9495:
portion of the resistivity. For example, at 30 °C (303 K), the resistivity of silver is
7373:
2099:
2022:
11273:
10420:
Douglas
Giancoli (2009) . "25 Electric Currents and Resistance". In Jocelyn Phillips (ed.).
8638:
8614:
8191:
7989:
7973:
7767:
7661:
7201:
5160:
4863:. However, the existence of charged particles causes the plasma to generate, and be affected by,
4741:
Differentiating this relation provides a means to calculate the electric field from the density:
4111:
3985:
3965:
1727:
used on the right side of these equations. In matrix form, the resistivity relation is given by:
10345:
9713:
This value range is typical of high quality drinking water and not an indicator of water quality
3839:
3809:
3779:
3469:
10766:
10658:
9873:
Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, and applications
9369:
8633:). Impedivity is the sum of a real component, the resistivity, and an imaginary component, the
8486:
8203:
7747:
7236:
7049:
tidily summed up the nature of the metals' dealings with electrons in his popular science book
5283:
5034:
4912:
Usually very high: for many purposes, the conductivity of a plasma may be treated as infinite.
4565:
4538:
4134:
4115:
30:
This article is about electrical conductivity in general. For other types of conductivity, see
10298:
10292:
10003:
9800:
9763:
7620:
7415:
Free electron model § Mean free dependence of the resistivity of gold, copper and silver.
1063:
10976:"What we can learn from measurements of air electric conductivity in 222Rn - rich atmosphere"
9561:
Copper is widely used in electrical equipment, building wiring, and telecommunication cables.
8947:
8933:
7961:
7943:
7855:
7353:
7333:
7312:
7290:
7182:
5134:
5007:
4644:
4502:
4299:
4186:
1724:
726:
478:
443:
10991:
10816:
10217:
10078:
9935:"Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media"
4160:, electrical conduction happens not by band electrons or holes, but by full atomic species (
11368:
https://edu-physics.com/2021/01/07/resistivity-of-the-material-of-a-wire-physics-practical/
11198:
11072:
10987:
10946:
10386:
8630:
8544:
8513:
8439:
7873:
7843:{\displaystyle {\kappa \over \sigma }={\pi ^{2} \over 3}\left({\frac {k}{e}}\right)^{2}T,}
7725:
7398:
7263:
7209:
7092:
5045:
4872:
4558:
4475:
4448:
4421:
4226:
3869:
896:
734:
570:
10491:. (PDF format; see page 2, table in the right lower corner). Retrieved on 2011-12-17.
8466:
7087:
The electrical resistivity of most materials changes with temperature. If the temperature
1165:
1114:
8:
10449:
9631:
9601:
9588:
9574:. The unit for expressing the conductivity of nonmagnetic materials by testing using the
9452:
9447:
8964:
8923:
7063:
7045:
The extremely low resistivity (high conductivity) of silver is characteristic of metals.
4845:
4037:
11202:
11076:
10950:
10390:
11219:
11186:
11134:
10639:
10402:
10332:
9951:
9934:
9417:
8419:
8399:
8199:
8001:
7923:
7903:
7897:
7879:
5073:
is normally used to measure conductivity in a solution. A rough summary is as follows:
4656:
4401:
4206:
4087:
4055:
1198:
1143:
1092:
810:
760:
708:
540:
532:
11326:
10691:
10538:. (Calculated as "15% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
10502:. (Calculated as "28% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
10472:. (Calculated as "56% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12.
10377:
Matula, R.A. (1979). "Electrical resistivity of copper, gold, palladium, and silver".
10244:
4732:{\displaystyle n_{\text{e}}\propto \exp \left(e\Phi /k_{\text{B}}T_{\text{e}}\right).}
415:{\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}
11300:
11224:
11122:
11112:
11088:
11039:
10916:
10893:
ASTM D1125 Standard Test
Methods for Electrical Conductivity and Resistivity of Water
10876:
10822:
10747:
10725:
10675:
10645:
10594:
10429:
10359:
10349:
10302:
10223:
10199:
10163:
10111:
10032:
9917:
9897:
9876:
9840:
9806:
9769:
9365:
7997:
7751:
7740:
7686:
6196:
4528:
4095:
4000:
3931:
3440:
2153:
826:
439:
11255:
10858:
10798:"Simplified 2D Bidomain Model of Whole Heart Electrical Activity and ECG Generation"
10547:
10406:
7746:
An investigation of the low-temperature resistivity of metals was the motivation to
4982:
Binary: two-particle collisions are the rule, three-body collisions extremely rare.
11214:
11206:
11187:"Stability and conductivity of self-assembled wires in a transverse electric field"
11080:
10995:
10954:
10868:
10584:
10394:
9967:
9946:
9432:
8670:
7976:
resistivity decreases with increasing temperature. The electrons are bumped to the
7709: = 2 implies that the resistance is due to electron–electron interaction.
6408:
6059:
5038:
4886:
4586:
4573:
447:
11061:"Die Abhängigkeit des elektrischen Widerstandes reiner Metalle von der Temperatur"
10131:
7419:
In general, electrical resistivity of metals increases with temperature. Electron–
4591:
211:{\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}
11021:
10773:
10722:
Protective coatings and thin films: synthesis, characterization, and applications
10488:
10153:
8284:
7977:
6748:
6160:
5835:
5094:
4952:
4948:
4885:
after solid, liquids and gases. It is distinct from these and other lower-energy
4881:
4603:
Plasmas are very good conductors and electric potentials play an important role.
4125:
4083:
4067:
3900:
3894:
3866:, is enough to describe the electrical resistivity. It is then written as simply
1192:
982:
841:
11038:. McGraw-Hill technical education. Maidenhead, UK: McGraw-Hill. pp. 36–40.
3976:; however, in semiconductors the bands are near enough to the Fermi level to be
11024:. US Dep. of Commerce. National Bureau of Standards Handbook. February 21, 1966
11014:
10053:
9361:
8645:
8622:
7714:
7655:
6637:
5144:
4864:
4652:
4129:
4128:(see Band Theory, above) giving rise to free conduction electrons. However, in
4079:
2225:
In either case, the resulting expression for each electric field component is:
1334: — because even a small electric field pulls a lot of current through it.
1223:
Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by:
978:
965:
535:
of materials, giving the opposition of a standard cube of material to current.
524:
504:
451:
3070:{\displaystyle \mathbf {J} _{i}={\boldsymbol {\sigma }}_{ij}\mathbf {E} _{j},}
11376:
11346:
Comparison of the electrical conductivity of various elements in
WolframAlpha
11092:
11084:
10598:
10425:
10341:
9674:
9402:
8975:
Resistivity, density, and resistivity-density products of selected materials
7993:
7981:
7647:
7401:
is inadequate and a more detailed analysis and understanding should be used.
5167:
Resistivity, conductivity, and temperature coefficient for several materials
5114:
5049:
5018:
4852:
4554:
4176:, currents are carried by ionic salts. Small holes in cell membranes, called
4173:
4107:
3981:
3969:
2216:{\displaystyle \mathbf {E} _{i}={\boldsymbol {\rho }}_{ij}\mathbf {J} _{j}~.}
11126:
10573:"Free-surface instabilities during electromagnetic shaping of liquid metals"
10363:
11228:
10880:
10589:
10572:
10512:
IITC – Imec Presents Copper, Cobalt and Ruthenium Interconnect Results
9575:
8919:
8666:
8602:
8538:
8386:{\displaystyle \rho (T)=\rho _{0}+aT^{2}+bT^{5}+c_{m}\ln {\frac {\mu }{T}}}
7985:
7651:
7046:
6956:
5929:
4868:
4860:
4833:
4639:
4615:
4169:
4165:
1609:
579:
458:
84:
8680:
11334:
11256:"Berryllium (Be) - Chemical properties, Health and Environmental effects"
11106:
11000:
10975:
10959:
10934:
10797:
10297:(2nd ed.). Fort Worth, Texas; London: Saunders College Pub. p.
9492:
9442:
8659:
8649:
8613:
When analyzing the response of materials to alternating electric fields (
7718:
6857:
6362:
5124:
5026:
4956:
4648:
are formed, the charge separation can extend some tens of Debye lengths.
4533:
4177:
4157:
4121:
4016:
4012:
3955:
3917:
3463:
477:
solid cube of material has sheet contacts on two opposite faces, and the
9894:
An introduction to Tensor Analysis: For Engineers and Applied Scientists
7980:
by thermal energy, where they flow freely, and in doing so leave behind
5003:
A conductor such as a metal has high conductivity and a low resistivity.
4180:, are selective to specific ions and determine the membrane resistance.
8654:
8185:
6124:
5057:
3922:
3496:
is nonzero. In the Hall effect, due to rotational invariance about the
1620:
1613:
1394:
Since the electric field is constant, it is given by the total voltage
556:
11210:
10872:
9578:
method. Generally used for temper and alloy verification of aluminium.
4553:, this decrease is limited by impurities and other defects. Even near
3595:, so the relation between resistivity and conductivity simplifies to:
1309:{\displaystyle \sigma (x)={\frac {1}{\rho (x)}}={\frac {J(x)}{E(x)}}.}
27:
Measure of a substance's ability to resist or conduct electric current
10398:
10159:
10054:"Questions & Answers – How do you explain electrical resistance?"
9623:
9213:
9185:
9159:
9133:
6552:
6522:
6290:
6261:
5995:
5662:
5358:
4925:
4595:
3951:
10196:
The Fourth State of Matter: An Introduction to the Physics of Plasma
9587:
Despite being less conductive than copper, gold is commonly used in
7428:
of a metal can be approximated through the Bloch–Grüneisen formula:
4991:
1220:
The current density is parallel to the electric field by necessity.
10184:
Flashes in the Sky: Earth's Gamma-Ray Bursts Triggered by Lightning
9678:
8198:
In non-crystalline semiconductors, conduction can occur by charges
6460:
6357:
6320:
6088:
6022:
5763:
5488:
5070:
4932:
4837:
3973:
3934:
for a certain energy in the material listed. The shade follows the
1624:
566:
560:
A piece of resistive material with electrical contacts on both ends
543:
that give the opposition of a specific object to electric current.
8644:
Conversely, in such cases the conductivity must be expressed as a
3926:
Filling of the electronic states in various types of materials at
10711:
Probably, the metal with highest value of electrical resistivity.
10672:
Schaum's outline of theory and problems of basic circuit analysis
10481:
10460:– via Internet Archive - archive.org (archived 2001-03-10).
9107:
9081:
8992:
8959:
7233:
is a fixed reference temperature (usually room temperature), and
6835:
6682:
6232:
5900:
5864:
5691:
5459:
5423:
5042:
4944:
4921:
4086:. The mechanism is similar to transfer of momentum of balls in a
4075:
245:
63:
10932:
10524:"Temperature Coefficient of Resistance | Electronics Notes"
9722:
Conductivity is lowest with monatomic gases present; changes to
523:) are sometimes used. The SI unit of electrical conductivity is
9627:
9619:
9267:
9239:
9055:
7988:, which also flow freely. The electric resistance of a typical
7420:
6935:
5626:
5589:
5286:
5247:
5211:
4940:
4655:. A common example is to assume that the electrons satisfy the
4550:
4546:
4389:{\displaystyle \sigma =q\left(b^{+}+b^{-}\right)\alpha N_{0}~,}
4138:
1708:
752:
9355:
4920:
One: all gas particles behave in a similar way, influenced by
3895:
Relation between current density and electric current velocity
2094:
is the resistivity tensor, in general a three by three matrix;
10155:
SQUIDS, the Josephson Effects and Superconducting Electronics
9652:
Nickel-iron-chromium alloy commonly used in heating elements.
9012:, i.e. cross-sectional area required to give same conductance
8770:{\displaystyle J=\sigma E\,\,\rightleftharpoons \,\,E=\rho J}
8156:{\displaystyle {\frac {1}{T}}=A+B\ln \rho +C(\ln \rho )^{3},}
7330:. Because the linear approximation is only an approximation,
6806:
5560:
5394:
5163:
of various materials at 20 °C (68 °F; 293 K).
5104:
5053:
5022:
5011:
4223:
to the concentration of molecules of the dissolved substance
4063:
3947:
795:
528:
520:
512:
470:
9323:
9295:
7006:
6718:
6613:
6491:
5959:
5727:
5524:
5322:
4071:
2152:
Equivalently, resistivity can be given in the more compact
11350:
10973:
10842:
10840:
10838:
10818:
Crystal Growth and Evaluation of Silicon for VLSI and ULSI
8276:= 2, 3, 4, depending on the dimensionality of the system.
3776:
If the electric field is parallel to the applied current,
1089:
is the resistivity of the conductor material at the point
582:) is the constant of proportionality. This is written as:
11274:"From Benjamin Franklin to Jan Ingenhousz, 18 March 1774"
8069:{\displaystyle \rho =\rho _{0}e^{\frac {E_{A}}{k_{B}T}}.}
6878:
5799:
4936:
4499:: concentration of molecules of the dissolved substance,
4161:
2118:
is the electric current density vector, with components (
791:
466:
10422:
Physics for Scientists and Engineers with Modern Physics
9738:
upon equilibration to the atmosphere due to dissolved CO
8648:(or even as a matrix of complex numbers, in the case of
8287:
are materials where the resistivity follows the formula
4642:
there can be charge imbalance. In the special case that
34:. For specific applications in electrical elements, see
10835:
9932:
8681:
Resistance versus resistivity in complicated geometries
8608:
787:
574:
cross-sectional area, where the electrical resistivity
462:
9690:
Corresponds to an average salinity of 35 g/kg at
9026:
7307:
is an empirical parameter fitted from measurement data
7066:
gives a basic description of electron flow in metals.
4472:: mobility of positively and negatively charged ions,
2952:
2799:
2742:
1951:
1798:
1741:
1346:
Derivation of the constant case from the general case
9643:
18% chromium and 8% nickel austenitic stainless steel
8785:
8735:
8691:
8574:
8547:
8516:
8489:
8469:
8442:
8422:
8402:
8296:
8212:
8202:
from one localised site to another. This is known as
8090:
8012:
7946:
7926:
7906:
7882:
7858:
7778:
7664:
7623:
7436:
7376:
7356:
7336:
7315:
7293:
7266:
7239:
7212:
7185:
7101:
4747:
4665:
4505:
4478:
4451:
4424:
4404:
4324:
4302:
4258:
4229:
4209:
4189:
3872:
3842:
3812:
3782:
3603:
3552:
3506:
3472:
3423:
3401:
3087:
3021:
2736:
2580:
2233:
2164:
2102:
2078:
2025:
1735:
1636:
1562:
1498:
1450:
1410:
1362:
1231:
1201:
1168:
1146:
1117:
1095:
1066:
993:
932:
899:
852:
763:
737:
711:
622:
590:
349:
308:
265:
152:
108:
9491:
The numbers in this column increase or decrease the
9423:
Electrical resistivities of the elements (data page)
9368:
relates an experiment by another British scientist,
7964:. The ratio on the rhs is called the Lorenz number.
4998:
Electrical resistivities of the elements (data page)
4947:
can be distinguished by the sign and value of their
4168:
is a reasonable electrical conductor. Conduction in
3889:
10637:
10079:"Electromigration : What is electromigration?"
4101:
10802:Computational and Mathematical Methods in Medicine
10331:
9805:. New Delhi: Laxmi Publications. pp. 280–84.
8896:
8769:
8714:
8593:
8560:
8529:
8502:
8475:
8455:
8428:
8408:
8385:
8263:{\displaystyle \rho =A\exp \left(T^{-1/n}\right),}
8262:
8155:
8068:
7952:
7932:
7912:
7888:
7864:
7842:
7677:
7638:
7607:
7389:
7362:
7342:
7321:
7299:
7279:
7252:
7225:
7191:
7169:
4822:
4731:
4511:
4491:
4464:
4437:
4410:
4388:
4308:
4286:
4242:
4215:
4203:, which is the ratio of the concentration of ions
4195:
3878:
3858:
3828:
3798:
3766:
3587:
3538:
3488:
3431:
3409:
3385:
3069:
3003:
2718:
2537:
2215:
2110:
2086:
2033:
2005:
1685:
1587:
1530:
1472:
1432:
1384:
1308:
1207:
1183:
1152:
1132:
1101:
1081:
1046:
954:
905:
877:
769:
743:
717:
691:
609:
414:
327:
283:
210:
130:
11314:Measuring Electrical Resistivity and Conductivity
10849:. Transmission-line.net. Retrieved on 2014-02-03.
10290:
10215:
9408:Classification of materials based on permittivity
9360:In a 1774 letter to Dutch-born British scientist
8997:
8991:
8982:
8621:, it is convenient to replace resistivity with a
5200:
5187:
5174:
4992:Resistivity and conductivity of various materials
818:, but a long, thin copper wire has a much larger
11374:
11184:
10577:Proceedings in Applied Mathematics and Mechanics
10424:(4th ed.). Upper Saddle River, New Jersey:
10419:
10340:(3rd ed.). Upper Saddle River, New Jersey:
9968:"The Quantum Hall Effect: TIFR Infosys Lectures"
7750:'s experiments that led in 1911 to discovery of
5065:, relative to the conductivity of pure water at
4145:
10913:Diamond: electronic properties and applications
10720:Y. Pauleau, Péter B. Barna, P. B. Barna (1997)
10570:
10379:Journal of Physical and Chemical Reference Data
10326:
9768:. Cambridge University Press. pp. 254–55.
4561:can persist indefinitely with no power source.
2041:is the electric field vector, with components (
11162:A. Matthiessen, Progg. Anallen, 122, 47 (1864)
11153:A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)
10776:. Kayelaby.npl.co.uk. Retrieved on 2011-12-17.
10151:
10058:Thomas Jefferson National Accelerator Facility
8953:
5029:and composition of the semiconductor material.
3886:, and this reduces to the simpler expression.
1047:{\displaystyle \rho (x)={\frac {E(x)}{J(x)}},}
10933:S. D. Pawar; P. Murugavel; D. M. Lal (2009).
10814:
10746:(3rd ed.). Academic Press. p. 561.
10741:
1318:For example, rubber is a material with large
515:) (especially in electrical engineering) and
11245:. otto-schmitt.org. Retrieved on 2011-12-17.
10571:Karcher, Ch.; Kocourek, V. (December 2007).
10452:. United States National Bureau of Standards
10322:
10320:
10318:
5695:
5666:
5630:
5593:
5564:
5528:
5492:
5463:
5427:
5362:
5290:
5251:
5215:
5014:has low conductivity and a high resistivity.
3930:. Here, height is energy while width is the
3899:Electric current is the ordered movement of
2551:-axis parallel to the current direction, so
971:
836:The above equation can be transposed to get
11294:
10633:
10631:
10107:Introduction to the Theory of Liquid Metals
9839:. Prentice Hall Professional. p. 114.
9413:Conductivity near the percolation threshold
9356:John Walsh and the conductivity of a vacuum
6200:
6164:
6128:
6092:
6063:
6026:
5963:
5934:
5904:
5868:
5839:
5803:
5767:
5731:
1531:{\displaystyle \rho ={\frac {VA}{I\ell }}.}
1398:across the conductor divided by the length
955:{\displaystyle \sigma ={\frac {1}{\rho }}.}
786:The resistivity can be expressed using the
610:{\displaystyle R\propto {\frac {\ell }{A}}}
10808:
10286:
10284:
10282:
10280:
10278:
9965:
9914:The Cambridge Handbook of Physics Formulas
6295:
6265:
6236:
5398:
4537:Original data from the 1911 experiment by
3836:are zero. When they are zero, one number,
1588:{\displaystyle \rho =R{\frac {A}{\ell }}.}
878:{\displaystyle R=\rho {\frac {\ell }{A}}.}
485:, then the resistivity of the material is
328:{\displaystyle \sigma ={\frac {1}{\rho }}}
11241:Otto H. Schmitt, University of Minnesota
11218:
11058:
10999:
10958:
10588:
10315:
10276:
10274:
10272:
10270:
10268:
10266:
10264:
10262:
10260:
10258:
9996:
9950:
9867:
9865:
9863:
8841:
8840:
8836:
8835:
8754:
8753:
8749:
8748:
7595:
7077:
6324:
3911:
3906:
1661:
1660:
1656:
1655:
985:of the current it creates at that point:
131:{\displaystyle \rho =R{\frac {A}{\ell }}}
11271:
10628:
10413:
6790:
6704:
6538:
6446:
6146:
6110:
4832:(∇ is the vector gradient operator; see
4590:
4532:
4041:
3921:
555:
531:(S/m). Resistivity and conductivity are
10974:E. Seran; M. Godefroy; E. Pili (2016).
10744:Engineering materials science, Volume 1
10644:(4th ed.). London: Prentice Hall.
10051:
9832:
9669:
9667:
7370:was measured at with a suffix, such as
7082:
6969:
6695:
6343:
6218:
6182:
5999:
5981:
5886:
5821:
5785:
5749:
5713:
5648:
5611:
5546:
5510:
5445:
5380:
5344:
5326:
5308:
5269:
5233:
4564:In a class of superconductors known as
3432:{\displaystyle {\boldsymbol {\sigma }}}
3425:
3403:
3039:
2182:
2080:
1671:
1646:
14:
11375:
10735:
10376:
10255:
10145:
9933:Josef Pek, Tomas Verner (3 Apr 2007).
9860:
9761:
9572:International Annealed Copper Standard
8986:
7761:
6960:
6781:
6761:
6526:
6429:
6366:
6333:
6304:
6274:
6245:
6209:
6173:
6137:
6101:
6072:
6035:
6008:
5972:
5943:
5913:
5877:
5848:
5812:
5776:
5740:
5704:
5675:
5639:
5602:
5573:
5537:
5501:
5472:
5436:
5407:
5335:
5299:
5260:
5224:
4296:The specific electrical conductivity (
3588:{\displaystyle \rho _{yx}=-\rho _{xy}}
1490:into the first expression, we obtain:
465:unit of electrical resistivity is the
401:
387:
370:
353:
194:
177:
163:
155:
11383:Electrical resistance and conductance
11267:
11265:
10641:Physics: Principles with Applications
10103:
9798:
9615:
9613:
9031:
9024:
7020:
6994:
6944:
6919:
6891:
6866:
6844:
6819:
6752:
6686:
6495:
6475:
5371:
3539:{\displaystyle \rho _{yy}=\rho _{xx}}
3410:{\displaystyle {\boldsymbol {\rho }}}
2087:{\displaystyle {\boldsymbol {\rho }}}
1603:
1433:{\displaystyle E={\frac {V}{\ell }}.}
1385:{\displaystyle \rho ={\frac {E}{J}},}
729:of a uniform specimen of the material
537:Electrical resistance and conductance
36:Electrical resistance and conductance
11033:
10242:
10024:
9794:
9792:
9677:depends strongly on the presence of
9664:
8675:Mathematical descriptions of opacity
8609:Complex resistivity and conductivity
7011:
6989:
6939:
6910:
6861:
6839:
6810:
6732:
6658:
6596:
6579:
6556:
6505:
6466:
6412:
5077:Resistivity of classes of materials
4522:
11185:Stephenson, C.; Hubler, A. (2015).
10861:The Journal of Physical Chemistry B
10674:, p. 19, McGraw-Hill Professional,
9802:Comprehensive Physics for Class XII
9591:because it does not easily corrode.
8673:the material is). For details, see
8279:
8206:and has the characteristic form of
8184:This equation is used to calibrate
7170:{\displaystyle \rho (T)=\rho _{0},}
6882:
6723:
6641:
6565:
6383:
4519:: the coefficient of dissociation.
3964:lies inside at least one band. In
2728:Conductivity is defined similarly:
1628:the three-dimensional tensor form:
1140:is the electric field at the point
24:
11288:
11262:
10081:. Middle East Technical University
10028:Electronics All-in-One For Dummies
9952:10.1111/j.1365-246X.1997.tb05314.x
9610:
8594:{\displaystyle \ln {\frac {1}{T}}}
7666:
7513:
7481:
7260:is the resistivity at temperature
5155:This table shows the resistivity (
5056:, and other chemical species that
4791:
4693:
268:
25:
11404:
11319:
9939:Geophysical Journal International
9789:
9661:Graphite is strongly anisotropic.
9428:Electrical resistivity tomography
8729:In cases like this, the formulas
7967:
3890:Conductivity and current carriers
1473:{\displaystyle J={\frac {I}{A}}.}
1351:We will combine three equations.
924:, is the inverse of resistivity:
284:{\displaystyle \mathrm {s} ^{-1}}
11351:Partial and total conductivity.
11276:. In Willcox, William B. (ed.).
11108:Quantum theory of real materials
10767:Physical properties of sea water
10661:. hyperphysics.phy-astr.gsu.edu)
10198:, Publisher: Adam Hilger, 1989,
9730:upon complete de-gassing, or to
8922:. This equation, along with the
8884:
8876:
8868:
8851:
8843:
8828:
8820:
8812:
8795:
8787:
8192:Extrinsic (doped) semiconductors
5836:Phosphor Bronze (0.2% P / 5% Sn)
4749:
4570:high-temperature superconductors
4287:{\displaystyle N=\alpha N_{0}~.}
4102:In semiconductors and insulators
3395:Looking at the two expressions,
3054:
3024:
2197:
2167:
2104:
2027:
1676:
1663:
1651:
1638:
442:of a material that measures its
11299:(5th ed.). W. H. Freeman.
11278:The Papers of Benjamin Franklin
11248:
11235:
11178:
11165:
11156:
11147:
11099:
11052:
11027:
11008:
10967:
10939:Journal of Geophysical Research
10926:
10911:Lawrence S. Pan, Don R. Kania,
10905:
10896:
10887:
10852:
10790:
10779:
10760:
10714:
10705:
10684:
10664:
10605:
10564:
10552:
10548:Material properties of niobium.
10541:
10530:
10516:
10505:
10494:
10475:
10464:
10442:
10370:
10338:Introduction to Electrodynamics
10236:
10209:
10194:Yaffa Eliezer, Shalom Eliezer,
10188:
10176:
10124:
10097:
10071:
10045:
10018:
9984:
9716:
9707:
9697:
9684:
9655:
9646:
9637:
9594:
9581:
9564:
9555:
9541:
9538:is the temperature coefficient.
9485:
9475:
8619:electrical impedance tomography
3681:
2624:
11036:Electrical engineering science
10219:Fundamentals of Plasma Physics
10110:. Cambridge University Press.
9959:
9926:
9916:, Cambridge University Press,
9906:
9886:
9875:, p. 61, William Andrew, 1993
9826:
9755:
9466:
9048:
9035:
9009:
8944:partial differential equations
8888:
8880:
8872:
8864:
8855:
8847:
8837:
8832:
8824:
8816:
8808:
8799:
8791:
8750:
8724:spreading resistance profiling
8715:{\displaystyle R=\rho \ell /A}
8568:a lattice vibrations term and
8306:
8300:
8141:
8128:
8000:with temperature following an
7633:
7627:
7589:
7567:
7564:
7545:
7461:
7455:
7446:
7440:
7161:
7158:
7139:
7127:
7111:
7105:
1657:
1297:
1291:
1283:
1277:
1262:
1256:
1241:
1235:
1178:
1172:
1127:
1121:
1076:
1070:
1035:
1029:
1021:
1015:
1003:
997:
656:
436:specific electrical resistance
13:
1:
10787:. chemistry.stackexchange.com
9748:
6464:perpendicular to basal plane
4917:Independently acting species
4316:) of a solution is equal to:
4146:In ionic liquids/electrolytes
1540:Finally, we apply Ohm's law,
964:Conductivity has SI units of
551:
546:
11272:Franklin, Benjamin (1978) .
10638:Douglas C. Giancoli (1995).
10245:"Dielectric Strength of Air"
10004:"Current versus Drift Speed"
9836:Signal Integrity: Simplified
9634:fabricated in advanced nodes
9570:Referred to as 100% IACS or
7756:History of superconductivity
7390:{\displaystyle \alpha _{15}}
5025:, and, most important, with
4418:: module of the ion charge,
4031:
3989:
3972:the Fermi level is inside a
2111:{\displaystyle \mathbf {J} }
2034:{\displaystyle \mathbf {E} }
1342:
7:
10559:AISI 1010 Steel, cold drawn
9398:Charge transport mechanisms
9390:
8954:Resistivity-density product
8617:), in applications such as
8510:the residual resistivity,
8179:Steinhart–Hart coefficients
7678:{\displaystyle \Theta _{R}}
7658:of electrons in the metal.
7411:Bloch–Grüneisen temperature
5048:is highly dependent on its
4878:Plasma is often called the
4638:), but on the scale of the
4152:Conductivity (electrolytic)
3932:density of available states
446:or how strongly it resists
10:
11409:
10336:. In Alison Reeves (ed.).
10291:Raymond A. Serway (1998).
10216:Bittencourt, J.A. (2004).
9765:Fundamentals of Geophysics
9626:are considered to replace
9350:
7408:
7091:does not vary too much, a
7051:One, Two, Three...Infinity
4995:
4584:
4526:
4149:
4105:
4035:
3915:
3859:{\displaystyle \rho _{xx}}
3829:{\displaystyle \rho _{xz}}
3799:{\displaystyle \rho _{xy}}
3489:{\displaystyle \rho _{xy}}
3462:. This can be seen in the
1482:Plugging in the values of
481:between these contacts is
473:(Ω⋅m). For example, if a
29:
11353:"Electrical conductivity"
11327:"Electrical Conductivity"
11034:Ward, Malcolm R. (1971).
10915:, p. 140, Springer, 1994
10031:. John Wiley & Sons.
9438:SI electromagnetism units
9322:
9294:
9266:
9238:
9212:
9184:
9158:
9132:
9106:
9080:
9054:
9016:
9005:
9002:
8979:
8503:{\displaystyle \rho _{0}}
8483:are constant parameters,
7404:
7253:{\displaystyle \rho _{0}}
7005:
6984:
6955:
6934:
6907:
6877:
6856:
6834:
6805:
6776:
6747:
6717:
6681:
6636:
6576:
6551:
6521:
6490:
6459:
6407:
6356:
6319:
6289:
6260:
6231:
6195:
6159:
6123:
6087:
6057:
6021:
5994:
5958:
5928:
5899:
5863:
5834:
5798:
5762:
5726:
5690:
5661:
5625:
5588:
5559:
5523:
5487:
5458:
5422:
5393:
5357:
5321:
5282:
5246:
5210:
5205:
5171:
5143:
5133:
5123:
5113:
5103:
5093:
5084:
5081:
4580:
4008:Pauli exclusion principle
972:General scalar quantities
338:
295:
254:
244:
230:
225:
141:
95:
82:
72:
62:
50:
45:
11339:University of Nottingham
11085:10.1002/andp.19334080504
10821:. CRC Press. p. 7.
10333:"7 Electrodynamics"
9799:Kumar, Narinder (2003).
9762:Lowrie, William (2007).
9503:. This is calculated as
9459:
9008:Resistivity relative to
7920:is the electron charge,
7639:{\displaystyle \rho (0)}
4906:Electrical conductivity
4020:electrons at all times.
3999:According to elementary
3986:intrinsic semiconductors
3936:Fermi–Dirac distribution
1082:{\displaystyle \rho (x)}
10992:2017E&SS....4...91S
10980:Earth and Space Science
10847:Transmission Lines data
10222:. Springer. p. 1.
10152:John C. Gallop (1990).
10006:. The physics classroom
9966:David Tong (Jan 2016).
8948:finite element analysis
8615:dielectric spectroscopy
8080:Steinhart–Hart equation
7974:intrinsic semiconductor
7953:{\displaystyle \sigma }
7865:{\displaystyle \kappa }
7363:{\displaystyle \alpha }
7343:{\displaystyle \alpha }
7322:{\displaystyle \kappa }
7300:{\displaystyle \alpha }
7202:temperature coefficient
7192:{\displaystyle \alpha }
5161:temperature coefficient
4840:for more information.)
4566:type II superconductors
4512:{\displaystyle \alpha }
4309:{\displaystyle \sigma }
4196:{\displaystyle \alpha }
4112:Insulator (electricity)
3984:. "intrin." indicates
3946:: no state filled). In
1695:where the conductivity
493:Electrical conductivity
250:siemens per metre (S/m)
18:Electrical conductivity
11139:: CS1 maint: others (
11059:Grüneisen, E. (1933).
10815:Eranna, Golla (2014).
10742:Milton Ohring (1995).
10692:"Resistivity of steel"
10590:10.1002/pamm.200700645
10583:(1): 4140009–4140010.
9892:J.R. Tyldesley (1975)
9833:Bogatin, Eric (2004).
9604:with steel reinforced
9530:is the resistivity at
9388:
9379:
9003:Resistivity × density
8898:
8779:must be replaced with
8771:
8722:above. One example is
8716:
8652:materials) called the
8595:
8562:
8531:
8504:
8477:
8457:
8430:
8410:
8387:
8264:
8204:variable range hopping
8157:
8070:
7978:conduction energy band
7954:
7934:
7914:
7890:
7866:
7844:
7748:Heike Kamerlingh Onnes
7679:
7640:
7609:
7391:
7364:
7344:
7323:
7301:
7281:
7254:
7227:
7193:
7171:
7078:Temperature dependence
7062:More technically, the
7060:
5017:The conductivity of a
4964:Velocity distribution
4824:
4733:
4600:
4568:, including all known
4542:
4539:Heike Kamerlingh Onnes
4513:
4493:
4466:
4439:
4412:
4390:
4310:
4288:
4244:
4217:
4197:
4135:charge carrier density
4116:Charge carrier density
4059:
3996:
3942:: all states filled,
3912:Band theory simplified
3907:Causes of conductivity
3880:
3860:
3830:
3800:
3768:
3589:
3540:
3490:
3433:
3411:
3387:
3071:
3005:
2720:
2539:
2217:
2112:
2088:
2035:
2007:
1687:
1589:
1532:
1474:
1434:
1386:
1310:
1209:
1185:
1154:
1134:
1103:
1083:
1048:
956:
907:
879:
771:
745:
719:
693:
611:
561:
428:Electrical resistivity
416:
329:
285:
212:
132:
10670:John O'Malley (1992)
10294:Principles of Physics
9383:
9374:
8899:
8772:
8717:
8596:
8563:
8561:{\displaystyle T^{5}}
8532:
8530:{\displaystyle T^{2}}
8505:
8478:
8458:
8456:{\displaystyle c_{m}}
8431:
8411:
8388:
8265:
8158:
8071:
7962:electric conductivity
7955:
7935:
7915:
7891:
7867:
7845:
7719:Debye phonon spectrum
7680:
7641:
7610:
7392:
7365:
7345:
7324:
7302:
7282:
7280:{\displaystyle T_{0}}
7255:
7228:
7226:{\displaystyle T_{0}}
7194:
7172:
7055:
5035:semiconductors doping
4825:
4734:
4630:= ⟨Z⟩ >
4594:
4536:
4514:
4494:
4492:{\displaystyle N_{0}}
4467:
4465:{\displaystyle b^{-}}
4440:
4438:{\displaystyle b^{+}}
4413:
4391:
4311:
4289:
4245:
4243:{\displaystyle N_{0}}
4218:
4198:
4053:
3925:
3881:
3879:{\displaystyle \rho }
3861:
3831:
3801:
3769:
3590:
3541:
3491:
3434:
3412:
3388:
3072:
3006:
2721:
2540:
2218:
2113:
2089:
2036:
2008:
1725:matrix multiplication
1711:, and electric field
1688:
1590:
1533:
1475:
1435:
1387:
1311:
1210:
1186:
1155:
1135:
1104:
1084:
1049:
957:
908:
906:{\displaystyle \ell }
880:
772:
746:
744:{\displaystyle \ell }
727:electrical resistance
720:
694:
612:
559:
444:electrical resistance
417:
330:
286:
213:
133:
11295:Paul Tipler (2004).
11173:Physical Electronics
11001:10.1002/2016EA000241
10960:10.1029/2007JD009716
10724:, p. 215, Springer,
10696:The Physics Factbook
10659:Table of Resistivity
10450:"Copper wire tables"
10249:The Physics Factbook
10243:Hong, Alice (2000).
10104:Faber, T.E. (1972).
8965:overhead power lines
8783:
8733:
8689:
8631:electrical impedance
8572:
8545:
8514:
8487:
8476:{\displaystyle \mu }
8467:
8440:
8420:
8400:
8294:
8210:
8088:
8010:
7944:
7940:is temperature, and
7924:
7904:
7880:
7874:thermal conductivity
7856:
7776:
7737:residual resistivity
7726:Augustus Matthiessen
7662:
7621:
7434:
7399:linear approximation
7374:
7354:
7334:
7313:
7291:
7264:
7237:
7210:
7183:
7099:
7093:linear approximation
7083:Linear approximation
6577:Swimming pool water
5159:), conductivity and
5063:specific conductance
4873:magnetohydrodynamics
4745:
4663:
4559:superconducting wire
4503:
4476:
4449:
4422:
4402:
4322:
4300:
4256:
4227:
4207:
4187:
4174:biological membranes
3870:
3840:
3810:
3780:
3601:
3550:
3504:
3470:
3452:may not be equal to
3421:
3399:
3085:
3019:
2734:
2578:
2231:
2162:
2100:
2076:
2023:
1733:
1717:and current density
1634:
1560:
1496:
1448:
1408:
1360:
1229:
1199:
1184:{\displaystyle J(x)}
1166:
1144:
1133:{\displaystyle E(x)}
1115:
1093:
1064:
991:
930:
897:
850:
779:cross-sectional area
761:
735:
709:
620:
588:
541:extensive properties
533:intensive properties
497:specific conductance
347:
306:
263:
150:
106:
11388:Physical quantities
11203:2015NatSR...515044S
11175:, chapter 2, Pitman
11171:J. Seymour (1972)
11077:1933AnP...408..530G
10951:2009JGRD..114.2205P
10690:Glenn Elert (ed.),
10391:1979JPCRD...8.1147M
10206:, 226 pages, page 5
10025:Lowe, Doug (2012).
9673:The resistivity of
9632:integrated circuits
9602:overhead power line
9589:electrical contacts
9534:(in this case) and
9453:Dielectric strength
9448:Spitzer resistivity
8976:
8924:continuity equation
7768:Wiedemann–Franz law
7762:Wiedemann–Franz law
7531:
7095:is typically used:
7064:free electron model
5168:
5078:
4846:electrostatic force
4038:Free electron model
3978:thermally populated
3757:
3736:
3674:
3653:
3079:both resulting in:
438:) is a fundamental
11065:Annalen der Physik
11020:2010-08-21 at the
11015:Copper Wire Tables
10772:2018-01-18 at the
10487:2011-11-23 at the
10482:Physical constants
10162:. pp. 3, 20.
9600:Commonly used for
9418:Contact resistance
9017:Approx. price, at
8974:
8934:Poisson's equation
8894:
8767:
8712:
8591:
8558:
8527:
8500:
8473:
8453:
8426:
8406:
8383:
8260:
8200:quantum tunnelling
8177:are the so-called
8153:
8066:
7950:
7930:
7910:
7898:Boltzmann constant
7886:
7862:
7840:
7754:. For details see
7675:
7636:
7605:
7502:
7387:
7360:
7340:
7319:
7297:
7277:
7250:
7223:
7189:
7167:
6409:Carbon (amorphous)
5166:
5076:
4928:with one another.
4820:
4729:
4657:Boltzmann relation
4601:
4543:
4509:
4489:
4462:
4435:
4408:
4386:
4306:
4284:
4240:
4213:
4193:
4060:
3997:
3980:with electrons or
3876:
3856:
3826:
3796:
3764:
3740:
3719:
3657:
3636:
3585:
3536:
3486:
3429:
3407:
3383:
3378:
3067:
3001:
2995:
2941:
2785:
2716:
2535:
2533:
2213:
2108:
2084:
2031:
2003:
1994:
1940:
1784:
1683:
1604:Tensor resistivity
1585:
1528:
1470:
1430:
1402:of the conductor:
1382:
1306:
1205:
1181:
1150:
1130:
1099:
1079:
1044:
952:
903:
875:
811:intrinsic property
767:
741:
715:
689:
687:
607:
562:
539:are corresponding
432:volume resistivity
412:
325:
281:
208:
128:
86:SI base units
11393:Materials science
11306:978-0-7167-0810-0
11211:10.1038/srep15044
10873:10.1021/jp045975a
10828:978-1-4822-3281-3
10651:978-0-13-102153-2
10435:978-0-13-149508-1
10355:978-0-13-805326-0
10308:978-0-03-020457-9
10204:978-0-85274-164-1
10169:978-0-7503-0051-3
10038:978-0-470-14704-7
9922:978-0-521-57507-2
9871:Hugh O. Pierson,
9846:978-0-13-066946-9
9812:978-81-7008-592-8
9775:978-05-2185-902-8
9366:Benjamin Franklin
9348:
9347:
9014:
8950:may be required.
8589:
8429:{\displaystyle b}
8409:{\displaystyle a}
8381:
8099:
8060:
7933:{\displaystyle T}
7913:{\displaystyle e}
7889:{\displaystyle k}
7822:
7807:
7787:
7752:superconductivity
7741:superconductivity
7687:Debye temperature
7593:
7490:
7035:
7034:
6461:Carbon (graphite)
6358:Carbon (graphite)
5153:
5152:
4989:
4988:
4815:
4812:
4801:
4786:
4779:
4769:
4718:
4708:
4673:
4574:magnetic vortices
4529:Superconductivity
4523:Superconductivity
4411:{\displaystyle q}
4382:
4280:
4216:{\displaystyle N}
4051:
4001:quantum mechanics
3994:
3759:
3676:
2711:
2671:
2663:
2619:
2209:
2154:Einstein notation
1599:
1598:
1580:
1523:
1465:
1425:
1377:
1301:
1266:
1208:{\displaystyle x}
1153:{\displaystyle x}
1102:{\displaystyle x}
1039:
968:per metre (S/m).
947:
870:
827:hydraulic analogy
770:{\displaystyle A}
718:{\displaystyle R}
680:
648:
605:
440:specific property
425:
424:
323:
221:
220:
126:
16:(Redirected from
11400:
11359:
11357:
11342:
11310:
11282:
11281:
11269:
11260:
11259:
11252:
11246:
11239:
11233:
11232:
11222:
11182:
11176:
11169:
11163:
11160:
11154:
11151:
11145:
11144:
11138:
11130:
11103:
11097:
11096:
11056:
11050:
11049:
11031:
11025:
11012:
11006:
11005:
11003:
10971:
10965:
10964:
10962:
10930:
10924:
10909:
10903:
10900:
10894:
10891:
10885:
10884:
10856:
10850:
10844:
10833:
10832:
10812:
10806:
10805:
10794:
10788:
10783:
10777:
10764:
10758:
10757:
10739:
10733:
10718:
10712:
10709:
10703:
10698:, retrieved and
10688:
10682:
10668:
10662:
10655:
10635:
10626:
10625:
10623:
10622:
10617:
10609:
10603:
10602:
10592:
10568:
10562:
10556:
10550:
10545:
10539:
10534:
10528:
10527:
10520:
10514:
10509:
10503:
10498:
10492:
10479:
10473:
10468:
10462:
10461:
10459:
10457:
10446:
10440:
10439:
10417:
10411:
10410:
10399:10.1063/1.555614
10374:
10368:
10367:
10335:
10324:
10313:
10312:
10288:
10253:
10252:
10240:
10234:
10233:
10213:
10207:
10192:
10186:
10180:
10174:
10173:
10149:
10143:
10142:
10140:
10138:
10128:
10122:
10121:
10101:
10095:
10094:
10088:
10086:
10075:
10069:
10068:
10066:
10064:
10049:
10043:
10042:
10022:
10016:
10015:
10013:
10011:
10000:
9994:
9988:
9982:
9981:
9979:
9977:
9972:
9963:
9957:
9956:
9954:
9930:
9924:
9910:
9904:
9890:
9884:
9869:
9858:
9857:
9855:
9853:
9830:
9824:
9823:
9821:
9819:
9796:
9787:
9786:
9784:
9782:
9759:
9742:
9737:
9735:
9729:
9727:
9720:
9714:
9711:
9705:
9701:
9695:
9693:
9688:
9682:
9681:in the material.
9671:
9662:
9659:
9653:
9650:
9644:
9641:
9635:
9617:
9608:
9598:
9592:
9585:
9579:
9568:
9562:
9559:
9553:
9545:
9539:
9537:
9533:
9529:
9520:
9502:
9500:
9489:
9483:
9479:
9473:
9470:
9433:Sheet resistance
9050:
9037:
9028:
9011:
9006:
8999:
8988:
8977:
8973:
8942:, form a set of
8941:
8931:
8917:
8911:
8903:
8901:
8900:
8895:
8887:
8879:
8871:
8854:
8846:
8831:
8823:
8815:
8798:
8790:
8776:
8774:
8773:
8768:
8721:
8719:
8718:
8713:
8708:
8625:quantity called
8600:
8598:
8597:
8592:
8590:
8582:
8567:
8565:
8564:
8559:
8557:
8556:
8536:
8534:
8533:
8528:
8526:
8525:
8509:
8507:
8506:
8501:
8499:
8498:
8482:
8480:
8479:
8474:
8462:
8460:
8459:
8454:
8452:
8451:
8435:
8433:
8432:
8427:
8415:
8413:
8412:
8407:
8392:
8390:
8389:
8384:
8382:
8374:
8366:
8365:
8353:
8352:
8337:
8336:
8321:
8320:
8285:Kondo insulators
8280:Kondo insulators
8275:
8269:
8267:
8266:
8261:
8256:
8252:
8251:
8247:
8176:
8172:
8168:
8162:
8160:
8159:
8154:
8149:
8148:
8100:
8092:
8075:
8073:
8072:
8067:
8062:
8061:
8059:
8055:
8054:
8044:
8043:
8034:
8028:
8027:
7959:
7957:
7956:
7951:
7939:
7937:
7936:
7931:
7919:
7917:
7916:
7911:
7895:
7893:
7892:
7887:
7871:
7869:
7868:
7863:
7849:
7847:
7846:
7841:
7833:
7832:
7827:
7823:
7815:
7808:
7803:
7802:
7793:
7788:
7780:
7731:
7708:
7702:
7696:
7684:
7682:
7681:
7676:
7674:
7673:
7645:
7643:
7642:
7637:
7614:
7612:
7611:
7606:
7594:
7592:
7588:
7587:
7557:
7556:
7543:
7542:
7533:
7530:
7526:
7521:
7520:
7510:
7501:
7500:
7495:
7491:
7489:
7488:
7476:
7427:
7396:
7394:
7393:
7388:
7386:
7385:
7369:
7367:
7366:
7361:
7349:
7347:
7346:
7341:
7329:
7328:
7326:
7325:
7320:
7306:
7304:
7303:
7298:
7287:. The parameter
7286:
7284:
7283:
7278:
7276:
7275:
7259:
7257:
7256:
7251:
7249:
7248:
7232:
7230:
7229:
7224:
7222:
7221:
7198:
7196:
7195:
7190:
7176:
7174:
7173:
7168:
7157:
7156:
7126:
7125:
7090:
7072:
7041:
7027:
7023:
7018:
7014:
6997:
6992:
6976:
6974:
6967:
6965:
6947:
6942:
6926:
6922:
6917:
6913:
6908:Wood (oven dry)
6899:
6895:
6889:
6885:
6869:
6864:
6848:
6842:
6836:Carbon (diamond)
6826:
6822:
6817:
6813:
6797:
6795:
6788:
6786:
6768:
6766:
6759:
6757:
6739:
6735:
6730:
6726:
6711:
6709:
6702:
6700:
6693:
6691:
6673:
6671:
6665:
6663:
6656:
6654:
6648:
6646:
6628:
6626:
6603:
6599:
6594:
6592:
6586:
6584:
6568:
6563:
6561:
6545:
6543:
6533:
6531:
6512:
6508:
6502:
6498:
6482:
6480:
6473:
6471:
6453:
6451:
6444:
6442:
6436:
6434:
6427:
6425:
6419:
6417:
6398:
6396:
6390:
6388:
6381:
6379:
6373:
6371:
6350:
6348:
6340:
6338:
6331:
6329:
6311:
6309:
6302:
6300:
6281:
6279:
6272:
6270:
6252:
6250:
6243:
6241:
6225:
6223:
6216:
6214:
6207:
6205:
6189:
6187:
6180:
6178:
6171:
6169:
6153:
6151:
6144:
6142:
6135:
6133:
6117:
6115:
6108:
6106:
6099:
6097:
6079:
6077:
6070:
6068:
6060:electrical steel
6051:
6049:
6042:
6040:
6033:
6031:
6015:
6013:
6006:
6004:
5988:
5986:
5979:
5977:
5970:
5968:
5950:
5948:
5941:
5939:
5920:
5918:
5911:
5909:
5893:
5891:
5884:
5882:
5875:
5873:
5855:
5853:
5846:
5844:
5828:
5826:
5819:
5817:
5810:
5808:
5792:
5790:
5783:
5781:
5774:
5772:
5756:
5754:
5747:
5745:
5738:
5736:
5720:
5718:
5711:
5709:
5702:
5700:
5682:
5680:
5673:
5671:
5655:
5653:
5646:
5644:
5637:
5635:
5618:
5616:
5609:
5607:
5600:
5598:
5580:
5578:
5571:
5569:
5553:
5551:
5544:
5542:
5535:
5533:
5517:
5515:
5508:
5506:
5499:
5497:
5479:
5477:
5470:
5468:
5452:
5450:
5443:
5441:
5434:
5432:
5414:
5412:
5405:
5403:
5387:
5385:
5378:
5376:
5369:
5367:
5351:
5349:
5342:
5340:
5333:
5331:
5315:
5313:
5306:
5304:
5297:
5295:
5276:
5274:
5267:
5265:
5258:
5256:
5240:
5238:
5231:
5229:
5222:
5220:
5203:coefficient (K)
5197:
5191:
5184:
5178:
5169:
5165:
5158:
5088:
5079:
5075:
5068:
4892:
4891:
4887:states of matter
4829:
4827:
4826:
4821:
4816:
4814:
4813:
4810:
4804:
4803:
4802:
4799:
4789:
4787:
4782:
4781:
4780:
4777:
4771:
4770:
4767:
4760:
4752:
4738:
4736:
4735:
4730:
4725:
4721:
4720:
4719:
4716:
4710:
4709:
4706:
4700:
4675:
4674:
4671:
4637:
4608:plasma potential
4587:Plasma (physics)
4518:
4516:
4515:
4510:
4498:
4496:
4495:
4490:
4488:
4487:
4471:
4469:
4468:
4463:
4461:
4460:
4444:
4442:
4441:
4436:
4434:
4433:
4417:
4415:
4414:
4409:
4395:
4393:
4392:
4387:
4380:
4379:
4378:
4366:
4362:
4361:
4360:
4348:
4347:
4315:
4313:
4312:
4307:
4293:
4291:
4290:
4285:
4278:
4277:
4276:
4249:
4247:
4246:
4241:
4239:
4238:
4222:
4220:
4219:
4214:
4202:
4200:
4199:
4194:
4054:An animation of
4052:
3990:
3901:electric charges
3885:
3883:
3882:
3877:
3865:
3863:
3862:
3857:
3855:
3854:
3835:
3833:
3832:
3827:
3825:
3824:
3805:
3803:
3802:
3797:
3795:
3794:
3773:
3771:
3770:
3765:
3760:
3758:
3756:
3751:
3735:
3730:
3717:
3716:
3715:
3699:
3694:
3693:
3677:
3675:
3673:
3668:
3652:
3647:
3634:
3633:
3621:
3616:
3615:
3594:
3592:
3591:
3586:
3584:
3583:
3565:
3564:
3545:
3543:
3542:
3537:
3535:
3534:
3519:
3518:
3499:
3495:
3493:
3492:
3487:
3485:
3484:
3461:
3451:
3438:
3436:
3435:
3430:
3428:
3416:
3414:
3413:
3408:
3406:
3392:
3390:
3389:
3384:
3379:
3375:
3374:
3365:
3364:
3349:
3348:
3339:
3338:
3323:
3322:
3313:
3312:
3293:
3292:
3279:
3278:
3269:
3268:
3253:
3252:
3243:
3242:
3227:
3226:
3217:
3216:
3197:
3196:
3183:
3182:
3173:
3172:
3157:
3156:
3147:
3146:
3131:
3130:
3121:
3120:
3101:
3100:
3076:
3074:
3073:
3068:
3063:
3062:
3057:
3051:
3050:
3042:
3033:
3032:
3027:
3010:
3008:
3007:
3002:
3000:
2999:
2992:
2991:
2978:
2977:
2964:
2963:
2946:
2945:
2938:
2937:
2923:
2922:
2908:
2907:
2891:
2890:
2876:
2875:
2861:
2860:
2844:
2843:
2829:
2828:
2814:
2813:
2790:
2789:
2782:
2781:
2768:
2767:
2754:
2753:
2725:
2723:
2722:
2717:
2712:
2710:
2709:
2700:
2699:
2690:
2685:
2684:
2672:
2669:
2664:
2662:
2661:
2652:
2651:
2642:
2637:
2636:
2620:
2618:
2617:
2608:
2607:
2598:
2593:
2592:
2571:
2550:
2544:
2542:
2541:
2536:
2534:
2527:
2526:
2517:
2516:
2501:
2500:
2491:
2490:
2475:
2474:
2465:
2464:
2445:
2444:
2428:
2427:
2418:
2417:
2402:
2401:
2392:
2391:
2376:
2375:
2366:
2365:
2346:
2345:
2329:
2328:
2319:
2318:
2303:
2302:
2293:
2292:
2277:
2276:
2267:
2266:
2247:
2246:
2222:
2220:
2219:
2214:
2207:
2206:
2205:
2200:
2194:
2193:
2185:
2176:
2175:
2170:
2146:
2117:
2115:
2114:
2109:
2107:
2093:
2091:
2090:
2085:
2083:
2069:
2040:
2038:
2037:
2032:
2030:
2012:
2010:
2009:
2004:
1999:
1998:
1991:
1990:
1977:
1976:
1963:
1962:
1945:
1944:
1937:
1936:
1922:
1921:
1907:
1906:
1890:
1889:
1875:
1874:
1860:
1859:
1843:
1842:
1828:
1827:
1813:
1812:
1789:
1788:
1781:
1780:
1767:
1766:
1753:
1752:
1722:
1716:
1706:
1701:and resistivity
1700:
1692:
1690:
1689:
1684:
1679:
1674:
1666:
1654:
1649:
1641:
1594:
1592:
1591:
1586:
1581:
1573:
1553:
1537:
1535:
1534:
1529:
1524:
1522:
1514:
1506:
1489:
1485:
1479:
1477:
1476:
1471:
1466:
1458:
1439:
1437:
1436:
1431:
1426:
1418:
1401:
1397:
1391:
1389:
1388:
1383:
1378:
1370:
1343:
1333:
1329:
1325:
1321:
1315:
1313:
1312:
1307:
1302:
1300:
1286:
1272:
1267:
1265:
1248:
1214:
1212:
1211:
1206:
1190:
1188:
1187:
1182:
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1108:
1106:
1105:
1100:
1088:
1086:
1085:
1080:
1053:
1051:
1050:
1045:
1040:
1038:
1024:
1010:
961:
959:
958:
953:
948:
940:
923:
916:
912:
910:
909:
904:
892:
888:
884:
882:
881:
876:
871:
863:
776:
774:
773:
768:
750:
748:
747:
742:
724:
722:
721:
716:
698:
696:
695:
690:
688:
681:
673:
655:
649:
641:
616:
614:
613:
608:
606:
598:
577:
518:
510:
502:
488:
487:1 Ω⋅m
484:
476:
456:
448:electric current
421:
419:
418:
413:
411:
410:
405:
404:
397:
396:
391:
390:
383:
382:
374:
373:
366:
365:
357:
356:
334:
332:
331:
326:
324:
316:
299:other quantities
297:Derivations from
290:
288:
287:
282:
280:
279:
271:
240:
223:
222:
217:
215:
214:
209:
207:
206:
198:
197:
190:
189:
181:
180:
173:
172:
167:
166:
159:
158:
137:
135:
134:
129:
127:
119:
99:other quantities
97:Derivations from
87:
78:s (Gaussian/ESU)
58:
43:
42:
21:
11408:
11407:
11403:
11402:
11401:
11399:
11398:
11397:
11373:
11372:
11363:
11355:
11325:
11322:
11307:
11291:
11289:Further reading
11286:
11285:
11270:
11263:
11254:
11253:
11249:
11240:
11236:
11183:
11179:
11170:
11166:
11161:
11157:
11152:
11148:
11132:
11131:
11119:
11105:
11104:
11100:
11057:
11053:
11046:
11032:
11028:
11022:Wayback Machine
11013:
11009:
10972:
10968:
10931:
10927:
10910:
10906:
10901:
10897:
10892:
10888:
10857:
10853:
10845:
10836:
10829:
10813:
10809:
10796:
10795:
10791:
10784:
10780:
10774:Wayback Machine
10765:
10761:
10754:
10740:
10736:
10719:
10715:
10710:
10706:
10689:
10685:
10669:
10665:
10656:
10652:
10636:
10629:
10620:
10618:
10615:
10611:
10610:
10606:
10569:
10565:
10557:
10553:
10546:
10542:
10535:
10531:
10522:
10521:
10517:
10510:
10506:
10499:
10495:
10489:Wayback Machine
10480:
10476:
10469:
10465:
10455:
10453:
10448:
10447:
10443:
10436:
10428:. p. 658.
10418:
10414:
10375:
10371:
10356:
10328:David Griffiths
10325:
10316:
10309:
10289:
10256:
10241:
10237:
10230:
10214:
10210:
10193:
10189:
10181:
10177:
10170:
10150:
10146:
10136:
10134:
10130:
10129:
10125:
10118:
10102:
10098:
10084:
10082:
10077:
10076:
10072:
10062:
10060:
10050:
10046:
10039:
10023:
10019:
10009:
10007:
10002:
10001:
9997:
9989:
9985:
9975:
9973:
9970:
9964:
9960:
9931:
9927:
9912:G. Woan (2010)
9911:
9907:
9891:
9887:
9870:
9861:
9851:
9849:
9847:
9831:
9827:
9817:
9815:
9813:
9797:
9790:
9780:
9778:
9776:
9760:
9756:
9751:
9746:
9745:
9741:
9733:
9731:
9725:
9723:
9721:
9717:
9712:
9708:
9702:
9698:
9691:
9689:
9685:
9672:
9665:
9660:
9656:
9651:
9647:
9642:
9638:
9618:
9611:
9599:
9595:
9586:
9582:
9569:
9565:
9560:
9556:
9546:
9542:
9535:
9531:
9528:
9522:
9519:
9504:
9498:
9496:
9490:
9486:
9480:
9476:
9471:
9467:
9462:
9457:
9393:
9358:
9353:
9046:
9041:
9033:
9020:
9019:9 December 2018
9018:
8995:
8984:
8956:
8937:
8927:
8913:
8907:
8883:
8875:
8867:
8850:
8842:
8827:
8819:
8811:
8794:
8786:
8784:
8781:
8780:
8734:
8731:
8730:
8704:
8690:
8687:
8686:
8683:
8637:(in analogy to
8629:(in analogy to
8611:
8581:
8573:
8570:
8569:
8552:
8548:
8546:
8543:
8542:
8521:
8517:
8515:
8512:
8511:
8494:
8490:
8488:
8485:
8484:
8468:
8465:
8464:
8447:
8443:
8441:
8438:
8437:
8421:
8418:
8417:
8401:
8398:
8397:
8373:
8361:
8357:
8348:
8344:
8332:
8328:
8316:
8312:
8295:
8292:
8291:
8282:
8273:
8243:
8236:
8232:
8228:
8211:
8208:
8207:
8174:
8170:
8166:
8144:
8140:
8091:
8089:
8086:
8085:
8050:
8046:
8045:
8039:
8035:
8033:
8029:
8023:
8019:
8011:
8008:
8007:
8002:Arrhenius model
7970:
7945:
7942:
7941:
7925:
7922:
7921:
7905:
7902:
7901:
7881:
7878:
7877:
7857:
7854:
7853:
7828:
7814:
7810:
7809:
7798:
7794:
7792:
7779:
7777:
7774:
7773:
7764:
7729:
7706:
7700:
7694:
7669:
7665:
7663:
7660:
7659:
7622:
7619:
7618:
7580:
7576:
7552:
7548:
7544:
7538:
7534:
7532:
7522:
7516:
7512:
7511:
7506:
7496:
7484:
7480:
7475:
7471:
7470:
7435:
7432:
7431:
7425:
7417:
7407:
7381:
7377:
7375:
7372:
7371:
7355:
7352:
7351:
7335:
7332:
7331:
7314:
7311:
7310:
7308:
7292:
7289:
7288:
7271:
7267:
7265:
7262:
7261:
7244:
7240:
7238:
7235:
7234:
7217:
7213:
7211:
7208:
7207:
7184:
7181:
7180:
7152:
7148:
7121:
7117:
7100:
7097:
7096:
7088:
7085:
7080:
7070:
7039:
7025:
7021:
7016:
7012:
6995:
6990:
6972:
6970:
6963:
6961:
6945:
6940:
6924:
6920:
6915:
6911:
6897:
6893:
6887:
6883:
6867:
6862:
6846:
6840:
6824:
6820:
6815:
6811:
6793:
6791:
6784:
6782:
6778:Ultrapure water
6764:
6762:
6755:
6753:
6749:Deionized water
6737:
6733:
6728:
6724:
6707:
6705:
6698:
6696:
6689:
6687:
6669:
6667:
6661:
6659:
6652:
6650:
6644:
6642:
6624:
6622:
6601:
6597:
6590:
6588:
6582:
6580:
6566:
6559:
6557:
6541:
6539:
6529:
6527:
6513:
6510:
6506:
6503:
6500:
6496:
6478:
6476:
6469:
6467:
6463:
6449:
6447:
6440:
6438:
6432:
6430:
6423:
6421:
6415:
6413:
6399:
6394:
6392:
6386:
6384:
6377:
6375:
6369:
6367:
6360:
6346:
6344:
6341:
6336:
6334:
6327:
6325:
6307:
6305:
6298:
6296:
6277:
6275:
6268:
6266:
6248:
6246:
6239:
6237:
6221:
6219:
6212:
6210:
6203:
6201:
6185:
6183:
6176:
6174:
6167:
6165:
6161:Stainless steel
6149:
6147:
6140:
6138:
6131:
6129:
6113:
6111:
6104:
6102:
6095:
6093:
6075:
6073:
6066:
6064:
6058:Grain oriented
6047:
6045:
6038:
6036:
6029:
6027:
6011:
6009:
6002:
6000:
5984:
5982:
5975:
5973:
5966:
5964:
5946:
5944:
5937:
5935:
5916:
5914:
5907:
5905:
5889:
5887:
5880:
5878:
5871:
5869:
5851:
5849:
5842:
5840:
5824:
5822:
5815:
5813:
5806:
5804:
5788:
5786:
5779:
5777:
5770:
5768:
5752:
5750:
5743:
5741:
5734:
5732:
5716:
5714:
5707:
5705:
5698:
5696:
5678:
5676:
5669:
5667:
5651:
5649:
5642:
5640:
5633:
5631:
5619:
5614:
5612:
5605:
5603:
5596:
5594:
5576:
5574:
5567:
5565:
5549:
5547:
5540:
5538:
5531:
5529:
5513:
5511:
5504:
5502:
5495:
5493:
5475:
5473:
5466:
5464:
5448:
5446:
5439:
5437:
5430:
5428:
5410:
5408:
5401:
5399:
5383:
5381:
5374:
5372:
5365:
5363:
5347:
5345:
5338:
5336:
5329:
5327:
5311:
5309:
5302:
5300:
5293:
5291:
5272:
5270:
5263:
5261:
5254:
5252:
5236:
5234:
5227:
5225:
5218:
5216:
5202:
5195:
5193:
5189:
5182:
5180:
5176:
5156:
5145:Superinsulators
5095:Superconductors
5086:
5066:
5000:
4994:
4882:state of matter
4865:magnetic fields
4857:Debye screening
4848:dissipates it.
4809:
4805:
4798:
4794:
4790:
4788:
4776:
4772:
4766:
4762:
4761:
4759:
4748:
4746:
4743:
4742:
4715:
4711:
4705:
4701:
4696:
4689:
4685:
4670:
4666:
4664:
4661:
4660:
4636:
4629:
4623:
4620:quasineutrality
4612:space potential
4589:
4583:
4531:
4525:
4504:
4501:
4500:
4483:
4479:
4477:
4474:
4473:
4456:
4452:
4450:
4447:
4446:
4429:
4425:
4423:
4420:
4419:
4403:
4400:
4399:
4374:
4370:
4356:
4352:
4343:
4339:
4338:
4334:
4323:
4320:
4319:
4301:
4298:
4297:
4272:
4268:
4257:
4254:
4253:
4234:
4230:
4228:
4225:
4224:
4208:
4205:
4204:
4188:
4185:
4184:
4154:
4148:
4126:conduction band
4120:In metals, the
4118:
4106:Main articles:
4104:
4088:Newton's cradle
4084:current density
4056:Newton's cradle
4042:
4040:
4034:
3995:
3963:
3920:
3914:
3909:
3897:
3892:
3871:
3868:
3867:
3847:
3843:
3841:
3838:
3837:
3817:
3813:
3811:
3808:
3807:
3787:
3783:
3781:
3778:
3777:
3752:
3744:
3731:
3723:
3718:
3708:
3704:
3700:
3698:
3686:
3682:
3669:
3661:
3648:
3640:
3635:
3626:
3622:
3620:
3608:
3604:
3602:
3599:
3598:
3576:
3572:
3557:
3553:
3551:
3548:
3547:
3527:
3523:
3511:
3507:
3505:
3502:
3501:
3497:
3477:
3473:
3471:
3468:
3467:
3459:
3453:
3449:
3444:
3424:
3422:
3419:
3418:
3402:
3400:
3397:
3396:
3377:
3376:
3370:
3366:
3357:
3353:
3344:
3340:
3331:
3327:
3318:
3314:
3305:
3301:
3294:
3288:
3284:
3281:
3280:
3274:
3270:
3261:
3257:
3248:
3244:
3235:
3231:
3222:
3218:
3209:
3205:
3198:
3192:
3188:
3185:
3184:
3178:
3174:
3165:
3161:
3152:
3148:
3139:
3135:
3126:
3122:
3113:
3109:
3102:
3096:
3092:
3088:
3086:
3083:
3082:
3058:
3053:
3052:
3043:
3038:
3037:
3028:
3023:
3022:
3020:
3017:
3016:
2994:
2993:
2987:
2983:
2980:
2979:
2973:
2969:
2966:
2965:
2959:
2955:
2948:
2947:
2940:
2939:
2930:
2926:
2924:
2915:
2911:
2909:
2900:
2896:
2893:
2892:
2883:
2879:
2877:
2868:
2864:
2862:
2853:
2849:
2846:
2845:
2836:
2832:
2830:
2821:
2817:
2815:
2806:
2802:
2795:
2794:
2784:
2783:
2777:
2773:
2770:
2769:
2763:
2759:
2756:
2755:
2749:
2745:
2738:
2737:
2735:
2732:
2731:
2705:
2701:
2695:
2691:
2689:
2677:
2673:
2670: and
2668:
2657:
2653:
2647:
2643:
2641:
2629:
2625:
2613:
2609:
2603:
2599:
2597:
2585:
2581:
2579:
2576:
2575:
2572:. This leaves:
2569:
2560:
2552:
2548:
2532:
2531:
2522:
2518:
2509:
2505:
2496:
2492:
2483:
2479:
2470:
2466:
2457:
2453:
2446:
2440:
2436:
2433:
2432:
2423:
2419:
2410:
2406:
2397:
2393:
2384:
2380:
2371:
2367:
2358:
2354:
2347:
2341:
2337:
2334:
2333:
2324:
2320:
2311:
2307:
2298:
2294:
2285:
2281:
2272:
2268:
2259:
2255:
2248:
2242:
2238:
2234:
2232:
2229:
2228:
2201:
2196:
2195:
2186:
2181:
2180:
2171:
2166:
2165:
2163:
2160:
2159:
2150:
2145:
2136:
2127:
2119:
2103:
2101:
2098:
2097:
2079:
2077:
2074:
2073:
2068:
2059:
2050:
2042:
2026:
2024:
2021:
2020:
1993:
1992:
1986:
1982:
1979:
1978:
1972:
1968:
1965:
1964:
1958:
1954:
1947:
1946:
1939:
1938:
1929:
1925:
1923:
1914:
1910:
1908:
1899:
1895:
1892:
1891:
1882:
1878:
1876:
1867:
1863:
1861:
1852:
1848:
1845:
1844:
1835:
1831:
1829:
1820:
1816:
1814:
1805:
1801:
1794:
1793:
1783:
1782:
1776:
1772:
1769:
1768:
1762:
1758:
1755:
1754:
1748:
1744:
1737:
1736:
1734:
1731:
1730:
1718:
1712:
1702:
1696:
1675:
1670:
1662:
1650:
1645:
1637:
1635:
1632:
1631:
1606:
1572:
1561:
1558:
1557:
1541:
1515:
1507:
1505:
1497:
1494:
1493:
1487:
1483:
1457:
1449:
1446:
1445:
1417:
1409:
1406:
1405:
1399:
1395:
1369:
1361:
1358:
1357:
1331:
1327:
1323:
1319:
1287:
1273:
1271:
1252:
1247:
1230:
1227:
1226:
1218:
1200:
1197:
1196:
1193:current density
1167:
1164:
1163:
1145:
1142:
1141:
1116:
1113:
1112:
1094:
1091:
1090:
1065:
1062:
1061:
1025:
1011:
1009:
992:
989:
988:
974:
939:
931:
928:
927:
921:
914:
898:
895:
894:
890:
886:
862:
851:
848:
847:
842:Claude Pouillet
784:
781:of the specimen
762:
759:
758:
755:of the specimen
736:
733:
732:
710:
707:
706:
686:
685:
672:
662:
654:
651:
650:
640:
630:
623:
621:
618:
617:
597:
589:
586:
585:
575:
554:
549:
516:
508:
500:
486:
482:
474:
454:
406:
400:
399:
398:
392:
386:
385:
384:
375:
369:
368:
367:
358:
352:
351:
350:
348:
345:
344:
315:
307:
304:
303:
300:
298:
272:
267:
266:
264:
261:
260:
257:
236:
233:
199:
193:
192:
191:
182:
176:
175:
174:
168:
162:
161:
160:
154:
153:
151:
148:
147:
118:
107:
104:
103:
100:
98:
85:
75:
68:ohm metre (Ω⋅m)
56:
53:
39:
28:
23:
22:
15:
12:
11:
5:
11406:
11396:
11395:
11390:
11385:
11371:
11370:
11361:
11360:
11348:
11343:
11321:
11320:External links
11318:
11317:
11316:
11311:
11305:
11290:
11287:
11284:
11283:
11261:
11247:
11234:
11177:
11164:
11155:
11146:
11117:
11098:
11071:(5): 530–540.
11051:
11044:
11026:
11007:
10966:
10945:(D2): D02205.
10925:
10904:
10895:
10886:
10851:
10834:
10827:
10807:
10789:
10778:
10759:
10753:978-0125249959
10752:
10734:
10713:
10704:
10683:
10663:
10650:
10627:
10604:
10563:
10551:
10540:
10529:
10515:
10504:
10493:
10474:
10463:
10441:
10434:
10412:
10369:
10354:
10314:
10307:
10254:
10235:
10228:
10208:
10187:
10175:
10168:
10144:
10123:
10116:
10096:
10070:
10044:
10037:
10017:
9995:
9983:
9958:
9945:(3): 505–521.
9925:
9905:
9885:
9859:
9845:
9825:
9811:
9788:
9774:
9753:
9752:
9750:
9747:
9744:
9743:
9739:
9715:
9706:
9696:
9683:
9675:semiconductors
9663:
9654:
9645:
9636:
9609:
9593:
9580:
9563:
9554:
9540:
9526:
9517:
9484:
9474:
9464:
9463:
9461:
9458:
9456:
9455:
9450:
9445:
9440:
9435:
9430:
9425:
9420:
9415:
9410:
9405:
9400:
9394:
9392:
9389:
9362:Jan Ingenhousz
9357:
9354:
9352:
9349:
9346:
9345:
9343:
9341:
9338:
9335:
9332:
9329:
9326:
9320:
9319:
9316:
9313:
9310:
9307:
9304:
9301:
9298:
9292:
9291:
9288:
9285:
9282:
9279:
9276:
9273:
9270:
9264:
9263:
9260:
9257:
9254:
9251:
9248:
9245:
9242:
9236:
9235:
9233:
9231:
9228:
9225:
9222:
9219:
9216:
9210:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9182:
9181:
9179:
9177:
9174:
9171:
9168:
9165:
9162:
9156:
9155:
9153:
9151:
9148:
9145:
9142:
9139:
9136:
9130:
9129:
9127:
9125:
9122:
9119:
9116:
9113:
9110:
9104:
9103:
9101:
9099:
9096:
9093:
9090:
9087:
9084:
9078:
9077:
9075:
9073:
9070:
9067:
9064:
9061:
9058:
9052:
9051:
9043:
9038:
9030:
9022:
9021:
9015:
9004:
9001:
8990:
8981:
8955:
8952:
8893:
8890:
8886:
8882:
8878:
8874:
8870:
8866:
8863:
8860:
8857:
8853:
8849:
8845:
8839:
8834:
8830:
8826:
8822:
8818:
8814:
8810:
8807:
8804:
8801:
8797:
8793:
8789:
8766:
8763:
8760:
8757:
8752:
8747:
8744:
8741:
8738:
8711:
8707:
8703:
8700:
8697:
8694:
8682:
8679:
8646:complex number
8610:
8607:
8588:
8585:
8580:
8577:
8555:
8551:
8541:contribution,
8524:
8520:
8497:
8493:
8472:
8450:
8446:
8425:
8405:
8394:
8393:
8380:
8377:
8372:
8369:
8364:
8360:
8356:
8351:
8347:
8343:
8340:
8335:
8331:
8327:
8324:
8319:
8315:
8311:
8308:
8305:
8302:
8299:
8281:
8278:
8259:
8255:
8250:
8246:
8242:
8239:
8235:
8231:
8227:
8224:
8221:
8218:
8215:
8152:
8147:
8143:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8098:
8095:
8065:
8058:
8053:
8049:
8042:
8038:
8032:
8026:
8022:
8018:
8015:
7969:
7968:Semiconductors
7966:
7949:
7929:
7909:
7885:
7861:
7839:
7836:
7831:
7826:
7821:
7818:
7813:
7806:
7801:
7797:
7791:
7786:
7783:
7763:
7760:
7715:Brillouin zone
7711:
7710:
7704:
7698:
7672:
7668:
7656:number density
7635:
7632:
7629:
7626:
7604:
7601:
7598:
7591:
7586:
7583:
7579:
7575:
7572:
7569:
7566:
7563:
7560:
7555:
7551:
7547:
7541:
7537:
7529:
7525:
7519:
7515:
7509:
7505:
7499:
7494:
7487:
7483:
7479:
7474:
7469:
7466:
7463:
7460:
7457:
7454:
7451:
7448:
7445:
7442:
7439:
7406:
7403:
7384:
7380:
7359:
7339:
7318:
7296:
7274:
7270:
7247:
7243:
7220:
7216:
7204:of resistivity
7199:is called the
7188:
7166:
7163:
7160:
7155:
7151:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7124:
7120:
7116:
7113:
7110:
7107:
7104:
7084:
7081:
7079:
7076:
7033:
7032:
7030:
7028:
7019:
7010:
7003:
7002:
7000:
6998:
6993:
6988:
6982:
6981:
6979:
6977:
6968:
6959:
6953:
6952:
6950:
6948:
6943:
6938:
6932:
6931:
6929:
6927:
6918:
6909:
6905:
6904:
6902:
6900:
6890:
6881:
6875:
6874:
6872:
6870:
6865:
6860:
6854:
6853:
6851:
6849:
6843:
6838:
6832:
6831:
6829:
6827:
6818:
6809:
6803:
6802:
6800:
6798:
6789:
6780:
6774:
6773:
6771:
6769:
6760:
6751:
6745:
6744:
6742:
6740:
6731:
6722:
6715:
6714:
6712:
6703:
6694:
6685:
6679:
6678:
6676:
6674:
6657:
6640:
6638:Drinking water
6634:
6633:
6631:
6629:
6620:
6617:
6609:
6608:
6606:
6604:
6595:
6578:
6574:
6573:
6571:
6569:
6564:
6555:
6549:
6548:
6546:
6537:
6534:
6525:
6519:
6518:
6516:
6514:
6504:
6494:
6488:
6487:
6485:
6483:
6474:
6465:
6457:
6456:
6454:
6445:
6428:
6411:
6405:
6404:
6402:
6400:
6382:
6365:
6354:
6353:
6351:
6342:
6332:
6323:
6317:
6316:
6314:
6312:
6303:
6294:
6287:
6286:
6284:
6282:
6273:
6264:
6258:
6257:
6255:
6253:
6244:
6235:
6229:
6228:
6226:
6217:
6208:
6199:
6193:
6192:
6190:
6181:
6172:
6163:
6157:
6156:
6154:
6145:
6136:
6127:
6121:
6120:
6118:
6109:
6100:
6091:
6085:
6084:
6082:
6080:
6071:
6062:
6055:
6054:
6052:
6043:
6034:
6025:
6019:
6018:
6016:
6007:
5998:
5992:
5991:
5989:
5980:
5971:
5962:
5956:
5955:
5953:
5951:
5942:
5933:
5926:
5925:
5923:
5921:
5912:
5903:
5897:
5896:
5894:
5885:
5876:
5867:
5861:
5860:
5858:
5856:
5847:
5838:
5832:
5831:
5829:
5820:
5811:
5802:
5796:
5795:
5793:
5784:
5775:
5766:
5760:
5759:
5757:
5748:
5739:
5730:
5724:
5723:
5721:
5712:
5703:
5694:
5688:
5687:
5685:
5683:
5674:
5665:
5659:
5658:
5656:
5647:
5638:
5629:
5623:
5622:
5620:
5610:
5601:
5592:
5586:
5585:
5583:
5581:
5572:
5563:
5561:Brass (30% Zn)
5557:
5556:
5554:
5545:
5536:
5527:
5521:
5520:
5518:
5509:
5500:
5491:
5485:
5484:
5482:
5480:
5471:
5462:
5456:
5455:
5453:
5444:
5435:
5426:
5420:
5419:
5417:
5415:
5406:
5397:
5391:
5390:
5388:
5379:
5370:
5361:
5355:
5354:
5352:
5343:
5334:
5325:
5319:
5318:
5316:
5307:
5298:
5289:
5280:
5279:
5277:
5268:
5259:
5250:
5244:
5243:
5241:
5232:
5223:
5214:
5208:
5207:
5204:
5199:
5188:Conductivity,
5186:
5173:
5151:
5150:
5147:
5141:
5140:
5137:
5131:
5130:
5127:
5121:
5120:
5117:
5115:Semiconductors
5111:
5110:
5107:
5101:
5100:
5097:
5091:
5090:
5083:
5033:The degree of
5031:
5030:
5015:
5004:
4996:Main article:
4993:
4990:
4987:
4986:
4983:
4980:
4976:
4975:
4971:
4965:
4961:
4960:
4931:Two or three:
4929:
4918:
4914:
4913:
4910:
4907:
4903:
4902:
4899:
4896:
4819:
4808:
4797:
4793:
4785:
4775:
4765:
4758:
4755:
4751:
4728:
4724:
4714:
4704:
4699:
4695:
4692:
4688:
4684:
4681:
4678:
4669:
4653:charge density
4634:
4627:
4585:Main article:
4582:
4579:
4527:Main article:
4524:
4521:
4508:
4486:
4482:
4459:
4455:
4432:
4428:
4407:
4385:
4377:
4373:
4369:
4365:
4359:
4355:
4351:
4346:
4342:
4337:
4333:
4330:
4327:
4305:
4283:
4275:
4271:
4267:
4264:
4261:
4237:
4233:
4212:
4192:
4150:Main article:
4147:
4144:
4130:semiconductors
4103:
4100:
4080:drift velocity
4066:consists of a
4036:Main article:
4033:
4030:
3970:semiconductors
3961:
3913:
3910:
3908:
3905:
3896:
3893:
3891:
3888:
3875:
3853:
3850:
3846:
3823:
3820:
3816:
3793:
3790:
3786:
3763:
3755:
3750:
3747:
3743:
3739:
3734:
3729:
3726:
3722:
3714:
3711:
3707:
3703:
3697:
3692:
3689:
3685:
3680:
3672:
3667:
3664:
3660:
3656:
3651:
3646:
3643:
3639:
3632:
3629:
3625:
3619:
3614:
3611:
3607:
3582:
3579:
3575:
3571:
3568:
3563:
3560:
3556:
3533:
3530:
3526:
3522:
3517:
3514:
3510:
3483:
3480:
3476:
3457:
3447:
3441:matrix inverse
3427:
3405:
3382:
3373:
3369:
3363:
3360:
3356:
3352:
3347:
3343:
3337:
3334:
3330:
3326:
3321:
3317:
3311:
3308:
3304:
3300:
3297:
3295:
3291:
3287:
3283:
3282:
3277:
3273:
3267:
3264:
3260:
3256:
3251:
3247:
3241:
3238:
3234:
3230:
3225:
3221:
3215:
3212:
3208:
3204:
3201:
3199:
3195:
3191:
3187:
3186:
3181:
3177:
3171:
3168:
3164:
3160:
3155:
3151:
3145:
3142:
3138:
3134:
3129:
3125:
3119:
3116:
3112:
3108:
3105:
3103:
3099:
3095:
3091:
3090:
3066:
3061:
3056:
3049:
3046:
3041:
3036:
3031:
3026:
2998:
2990:
2986:
2982:
2981:
2976:
2972:
2968:
2967:
2962:
2958:
2954:
2953:
2951:
2944:
2936:
2933:
2929:
2925:
2921:
2918:
2914:
2910:
2906:
2903:
2899:
2895:
2894:
2889:
2886:
2882:
2878:
2874:
2871:
2867:
2863:
2859:
2856:
2852:
2848:
2847:
2842:
2839:
2835:
2831:
2827:
2824:
2820:
2816:
2812:
2809:
2805:
2801:
2800:
2798:
2793:
2788:
2780:
2776:
2772:
2771:
2766:
2762:
2758:
2757:
2752:
2748:
2744:
2743:
2741:
2715:
2708:
2704:
2698:
2694:
2688:
2683:
2680:
2676:
2667:
2660:
2656:
2650:
2646:
2640:
2635:
2632:
2628:
2623:
2616:
2612:
2606:
2602:
2596:
2591:
2588:
2584:
2565:
2556:
2530:
2525:
2521:
2515:
2512:
2508:
2504:
2499:
2495:
2489:
2486:
2482:
2478:
2473:
2469:
2463:
2460:
2456:
2452:
2449:
2447:
2443:
2439:
2435:
2434:
2431:
2426:
2422:
2416:
2413:
2409:
2405:
2400:
2396:
2390:
2387:
2383:
2379:
2374:
2370:
2364:
2361:
2357:
2353:
2350:
2348:
2344:
2340:
2336:
2335:
2332:
2327:
2323:
2317:
2314:
2310:
2306:
2301:
2297:
2291:
2288:
2284:
2280:
2275:
2271:
2265:
2262:
2258:
2254:
2251:
2249:
2245:
2241:
2237:
2236:
2212:
2204:
2199:
2192:
2189:
2184:
2179:
2174:
2169:
2149:
2148:
2141:
2132:
2123:
2106:
2095:
2082:
2071:
2064:
2055:
2046:
2029:
2017:
2002:
1997:
1989:
1985:
1981:
1980:
1975:
1971:
1967:
1966:
1961:
1957:
1953:
1952:
1950:
1943:
1935:
1932:
1928:
1924:
1920:
1917:
1913:
1909:
1905:
1902:
1898:
1894:
1893:
1888:
1885:
1881:
1877:
1873:
1870:
1866:
1862:
1858:
1855:
1851:
1847:
1846:
1841:
1838:
1834:
1830:
1826:
1823:
1819:
1815:
1811:
1808:
1804:
1800:
1799:
1797:
1792:
1787:
1779:
1775:
1771:
1770:
1765:
1761:
1757:
1756:
1751:
1747:
1743:
1742:
1740:
1682:
1678:
1673:
1669:
1665:
1659:
1653:
1648:
1644:
1640:
1605:
1602:
1601:
1600:
1597:
1596:
1584:
1579:
1576:
1571:
1568:
1565:
1527:
1521:
1518:
1513:
1510:
1504:
1501:
1469:
1464:
1461:
1456:
1453:
1429:
1424:
1421:
1416:
1413:
1381:
1376:
1373:
1368:
1365:
1348:
1347:
1305:
1299:
1296:
1293:
1290:
1285:
1282:
1279:
1276:
1270:
1264:
1261:
1258:
1255:
1251:
1246:
1243:
1240:
1237:
1234:
1217:
1216:
1204:
1180:
1177:
1174:
1171:
1161:
1149:
1129:
1126:
1123:
1120:
1110:
1098:
1078:
1075:
1072:
1069:
1058:
1043:
1037:
1034:
1031:
1028:
1023:
1020:
1017:
1014:
1008:
1005:
1002:
999:
996:
979:electric field
973:
970:
951:
946:
943:
938:
935:
920:Conductivity,
902:
874:
869:
866:
861:
858:
855:
838:Pouillet's law
832:
821:
817:
783:
782:
766:
756:
740:
730:
714:
703:
684:
679:
676:
671:
668:
665:
663:
661:
658:
653:
652:
647:
644:
639:
636:
633:
631:
629:
626:
625:
604:
601:
596:
593:
578: (Greek:
553:
550:
548:
545:
423:
422:
409:
403:
395:
389:
381:
378:
372:
364:
361:
355:
342:
336:
335:
322:
319:
314:
311:
301:
296:
293:
292:
291:(Gaussian/ESU)
278:
275:
270:
258:
255:
252:
251:
248:
242:
241:
234:
232:Common symbols
231:
228:
227:
219:
218:
205:
202:
196:
188:
185:
179:
171:
165:
157:
145:
139:
138:
125:
122:
117:
114:
111:
101:
96:
93:
92:
89:
80:
79:
76:
73:
70:
69:
66:
60:
59:
54:
52:Common symbols
51:
48:
47:
26:
9:
6:
4:
3:
2:
11405:
11394:
11391:
11389:
11386:
11384:
11381:
11380:
11378:
11369:
11366:
11365:
11364:
11354:
11349:
11347:
11344:
11340:
11336:
11332:
11331:Sixty Symbols
11328:
11324:
11323:
11315:
11312:
11308:
11302:
11298:
11293:
11292:
11279:
11275:
11268:
11266:
11257:
11251:
11244:
11238:
11230:
11226:
11221:
11216:
11212:
11208:
11204:
11200:
11196:
11192:
11188:
11181:
11174:
11168:
11159:
11150:
11142:
11136:
11128:
11124:
11120:
11118:0-7923-9666-9
11114:
11110:
11109:
11102:
11094:
11090:
11086:
11082:
11078:
11074:
11070:
11066:
11062:
11055:
11047:
11045:9780070942554
11041:
11037:
11030:
11023:
11019:
11016:
11011:
11002:
10997:
10993:
10989:
10986:(2): 91–106.
10985:
10981:
10977:
10970:
10961:
10956:
10952:
10948:
10944:
10940:
10936:
10929:
10922:
10921:0-7923-9524-7
10918:
10914:
10908:
10899:
10890:
10882:
10878:
10874:
10870:
10867:(3): 1231–8.
10866:
10862:
10855:
10848:
10843:
10841:
10839:
10830:
10824:
10820:
10819:
10811:
10803:
10799:
10793:
10786:
10782:
10775:
10771:
10768:
10763:
10755:
10749:
10745:
10738:
10731:
10730:0-7923-4380-8
10727:
10723:
10717:
10708:
10702:16 June 2011.
10701:
10697:
10693:
10687:
10681:
10680:0-07-047824-4
10677:
10673:
10667:
10660:
10653:
10647:
10643:
10642:
10634:
10632:
10614:
10608:
10600:
10596:
10591:
10586:
10582:
10578:
10574:
10567:
10560:
10555:
10549:
10544:
10537:
10533:
10525:
10519:
10513:
10508:
10501:
10497:
10490:
10486:
10483:
10478:
10471:
10467:
10451:
10445:
10437:
10431:
10427:
10426:Prentice Hall
10423:
10416:
10408:
10404:
10400:
10396:
10392:
10388:
10384:
10380:
10373:
10365:
10361:
10357:
10351:
10347:
10343:
10342:Prentice Hall
10339:
10334:
10329:
10323:
10321:
10319:
10310:
10304:
10300:
10296:
10295:
10287:
10285:
10283:
10281:
10279:
10277:
10275:
10273:
10271:
10269:
10267:
10265:
10263:
10261:
10259:
10250:
10246:
10239:
10231:
10229:9780387209753
10225:
10221:
10220:
10212:
10205:
10201:
10197:
10191:
10185:
10179:
10171:
10165:
10161:
10157:
10156:
10148:
10133:
10127:
10119:
10117:9780521154499
10113:
10109:
10108:
10100:
10093:
10080:
10074:
10059:
10055:
10052:Keith Welch.
10048:
10040:
10034:
10030:
10029:
10021:
10005:
9999:
9992:
9987:
9969:
9962:
9953:
9948:
9944:
9940:
9936:
9929:
9923:
9919:
9915:
9909:
9903:
9902:0-582-44355-5
9899:
9895:
9889:
9882:
9881:0-8155-1339-9
9878:
9874:
9868:
9866:
9864:
9848:
9842:
9838:
9837:
9829:
9814:
9808:
9804:
9803:
9795:
9793:
9777:
9771:
9767:
9766:
9758:
9754:
9719:
9710:
9700:
9687:
9680:
9676:
9670:
9668:
9658:
9649:
9640:
9633:
9629:
9625:
9621:
9616:
9614:
9607:
9603:
9597:
9590:
9584:
9577:
9573:
9567:
9558:
9551:
9544:
9525:
9516:
9512:
9508:
9494:
9488:
9478:
9469:
9465:
9454:
9451:
9449:
9446:
9444:
9441:
9439:
9436:
9434:
9431:
9429:
9426:
9424:
9421:
9419:
9416:
9414:
9411:
9409:
9406:
9404:
9403:Chemiresistor
9401:
9399:
9396:
9395:
9387:
9382:
9378:
9373:
9371:
9367:
9363:
9344:
9342:
9339:
9336:
9333:
9330:
9327:
9325:
9321:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9297:
9293:
9289:
9286:
9283:
9280:
9277:
9274:
9271:
9269:
9265:
9261:
9258:
9255:
9252:
9249:
9246:
9243:
9241:
9237:
9234:
9232:
9229:
9226:
9223:
9220:
9217:
9215:
9211:
9207:
9204:
9201:
9198:
9195:
9192:
9189:
9187:
9183:
9180:
9178:
9175:
9172:
9169:
9166:
9163:
9161:
9157:
9154:
9152:
9149:
9146:
9143:
9140:
9137:
9135:
9131:
9128:
9126:
9123:
9120:
9117:
9114:
9111:
9109:
9105:
9102:
9100:
9097:
9094:
9091:
9088:
9085:
9083:
9079:
9076:
9074:
9071:
9068:
9065:
9062:
9059:
9057:
9053:
9044:
9039:
9023:
9013:
8994:
8978:
8972:
8968:
8966:
8961:
8951:
8949:
8945:
8940:
8935:
8930:
8925:
8921:
8920:vector fields
8916:
8910:
8904:
8891:
8861:
8858:
8805:
8802:
8777:
8764:
8761:
8758:
8755:
8745:
8742:
8739:
8736:
8727:
8725:
8709:
8705:
8701:
8698:
8695:
8692:
8678:
8676:
8672:
8668:
8663:
8661:
8657:
8656:
8651:
8647:
8642:
8640:
8636:
8632:
8628:
8624:
8620:
8616:
8606:
8604:
8586:
8583:
8578:
8575:
8553:
8549:
8540:
8522:
8518:
8495:
8491:
8470:
8448:
8444:
8423:
8403:
8378:
8375:
8370:
8367:
8362:
8358:
8354:
8349:
8345:
8341:
8338:
8333:
8329:
8325:
8322:
8317:
8313:
8309:
8303:
8297:
8290:
8289:
8288:
8286:
8277:
8270:
8257:
8253:
8248:
8244:
8240:
8237:
8233:
8229:
8225:
8222:
8219:
8216:
8213:
8205:
8201:
8196:
8193:
8189:
8187:
8182:
8180:
8163:
8150:
8145:
8137:
8134:
8131:
8125:
8122:
8119:
8116:
8113:
8110:
8107:
8104:
8101:
8096:
8093:
8083:
8081:
8076:
8063:
8056:
8051:
8047:
8040:
8036:
8030:
8024:
8020:
8016:
8013:
8005:
8003:
7999:
7998:exponentially
7995:
7994:semiconductor
7991:
7987:
7983:
7979:
7975:
7965:
7963:
7947:
7927:
7907:
7899:
7883:
7875:
7859:
7850:
7837:
7834:
7829:
7824:
7819:
7816:
7811:
7804:
7799:
7795:
7789:
7784:
7781:
7771:
7769:
7759:
7757:
7753:
7749:
7744:
7742:
7738:
7733:
7727:
7722:
7720:
7716:
7705:
7699:
7693:
7692:
7691:
7688:
7670:
7657:
7653:
7649:
7648:Fermi surface
7630:
7624:
7615:
7602:
7599:
7596:
7584:
7581:
7577:
7573:
7570:
7561:
7558:
7553:
7549:
7539:
7535:
7527:
7523:
7517:
7507:
7503:
7497:
7492:
7485:
7477:
7472:
7467:
7464:
7458:
7452:
7449:
7443:
7437:
7429:
7422:
7416:
7412:
7402:
7400:
7382:
7378:
7357:
7337:
7316:
7309:, equal to 1/
7294:
7272:
7268:
7245:
7241:
7218:
7214:
7205:
7203:
7186:
7177:
7164:
7153:
7149:
7145:
7142:
7136:
7133:
7130:
7122:
7118:
7114:
7108:
7102:
7094:
7075:
7067:
7065:
7059:
7054:
7052:
7048:
7043:
7031:
7029:
7008:
7004:
7001:
6999:
6987:
6983:
6980:
6978:
6958:
6954:
6951:
6949:
6937:
6933:
6930:
6928:
6906:
6903:
6901:
6880:
6876:
6873:
6871:
6859:
6855:
6852:
6850:
6837:
6833:
6830:
6828:
6808:
6804:
6801:
6799:
6779:
6775:
6772:
6770:
6750:
6746:
6743:
6741:
6720:
6716:
6713:
6684:
6680:
6677:
6675:
6639:
6635:
6632:
6630:
6621:
6618:
6616:
6615:
6611:
6610:
6607:
6605:
6575:
6572:
6570:
6554:
6550:
6547:
6535:
6524:
6520:
6517:
6515:
6493:
6489:
6486:
6484:
6462:
6458:
6455:
6410:
6406:
6403:
6401:
6364:
6359:
6355:
6352:
6322:
6318:
6315:
6313:
6292:
6288:
6285:
6283:
6263:
6259:
6256:
6254:
6234:
6230:
6227:
6198:
6194:
6191:
6162:
6158:
6155:
6126:
6122:
6119:
6090:
6086:
6083:
6081:
6061:
6056:
6053:
6044:
6024:
6020:
6017:
5997:
5993:
5990:
5961:
5957:
5954:
5952:
5931:
5927:
5924:
5922:
5902:
5898:
5895:
5866:
5862:
5859:
5857:
5837:
5833:
5830:
5801:
5797:
5794:
5765:
5761:
5758:
5729:
5725:
5722:
5693:
5689:
5686:
5684:
5664:
5660:
5657:
5628:
5624:
5621:
5591:
5587:
5584:
5582:
5562:
5558:
5555:
5526:
5522:
5519:
5490:
5486:
5483:
5481:
5461:
5457:
5454:
5425:
5421:
5418:
5416:
5396:
5395:Brass (5% Zn)
5392:
5389:
5360:
5356:
5353:
5324:
5320:
5317:
5288:
5285:
5281:
5278:
5249:
5245:
5242:
5213:
5209:
5175:Resistivity,
5170:
5164:
5162:
5148:
5146:
5142:
5138:
5136:
5132:
5128:
5126:
5122:
5118:
5116:
5112:
5108:
5106:
5102:
5098:
5096:
5092:
5085:Resistivity,
5080:
5074:
5072:
5064:
5059:
5055:
5052:of dissolved
5051:
5050:concentration
5047:
5044:
5040:
5036:
5028:
5024:
5020:
5019:semiconductor
5016:
5013:
5009:
5005:
5002:
5001:
4999:
4984:
4981:
4979:Interactions
4978:
4977:
4972:
4969:
4966:
4963:
4962:
4958:
4957:instabilities
4954:
4950:
4946:
4942:
4938:
4934:
4930:
4927:
4923:
4919:
4916:
4915:
4911:
4908:
4905:
4904:
4900:
4897:
4894:
4893:
4890:
4888:
4884:
4883:
4876:
4874:
4870:
4869:Debye lengths
4866:
4862:
4858:
4854:
4853:astrophysical
4849:
4847:
4841:
4839:
4835:
4830:
4817:
4806:
4795:
4783:
4773:
4763:
4756:
4753:
4739:
4726:
4722:
4712:
4702:
4697:
4690:
4686:
4682:
4679:
4676:
4667:
4658:
4654:
4649:
4647:
4646:
4645:double layers
4641:
4633:
4626:
4621:
4617:
4613:
4609:
4604:
4597:
4593:
4588:
4578:
4575:
4571:
4567:
4562:
4560:
4556:
4555:absolute zero
4552:
4548:
4540:
4535:
4530:
4520:
4506:
4484:
4480:
4457:
4453:
4430:
4426:
4405:
4396:
4383:
4375:
4371:
4367:
4363:
4357:
4353:
4349:
4344:
4340:
4335:
4331:
4328:
4325:
4317:
4303:
4294:
4281:
4273:
4269:
4265:
4262:
4259:
4251:
4235:
4231:
4210:
4190:
4181:
4179:
4175:
4171:
4170:ionic liquids
4167:
4163:
4159:
4153:
4143:
4140:
4136:
4131:
4127:
4123:
4117:
4113:
4109:
4108:Semiconductor
4099:
4097:
4091:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4057:
4039:
4029:
4025:
4021:
4018:
4014:
4009:
4004:
4002:
3993:
3987:
3983:
3979:
3975:
3971:
3967:
3960:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3924:
3919:
3904:
3902:
3887:
3873:
3851:
3848:
3844:
3821:
3818:
3814:
3791:
3788:
3784:
3774:
3761:
3753:
3748:
3745:
3741:
3737:
3732:
3727:
3724:
3720:
3712:
3709:
3705:
3701:
3695:
3690:
3687:
3683:
3678:
3670:
3665:
3662:
3658:
3654:
3649:
3644:
3641:
3637:
3630:
3627:
3623:
3617:
3612:
3609:
3605:
3596:
3580:
3577:
3573:
3569:
3566:
3561:
3558:
3554:
3531:
3528:
3524:
3520:
3515:
3512:
3508:
3481:
3478:
3474:
3465:
3460:
3450:
3442:
3393:
3380:
3371:
3367:
3361:
3358:
3354:
3350:
3345:
3341:
3335:
3332:
3328:
3324:
3319:
3315:
3309:
3306:
3302:
3298:
3296:
3289:
3285:
3275:
3271:
3265:
3262:
3258:
3254:
3249:
3245:
3239:
3236:
3232:
3228:
3223:
3219:
3213:
3210:
3206:
3202:
3200:
3193:
3189:
3179:
3175:
3169:
3166:
3162:
3158:
3153:
3149:
3143:
3140:
3136:
3132:
3127:
3123:
3117:
3114:
3110:
3106:
3104:
3097:
3093:
3080:
3077:
3064:
3059:
3047:
3044:
3034:
3029:
3014:
3011:
2996:
2988:
2984:
2974:
2970:
2960:
2956:
2949:
2942:
2934:
2931:
2927:
2919:
2916:
2912:
2904:
2901:
2897:
2887:
2884:
2880:
2872:
2869:
2865:
2857:
2854:
2850:
2840:
2837:
2833:
2825:
2822:
2818:
2810:
2807:
2803:
2796:
2791:
2786:
2778:
2774:
2764:
2760:
2750:
2746:
2739:
2729:
2726:
2713:
2706:
2702:
2696:
2692:
2686:
2681:
2678:
2674:
2665:
2658:
2654:
2648:
2644:
2638:
2633:
2630:
2626:
2621:
2614:
2610:
2604:
2600:
2594:
2589:
2586:
2582:
2573:
2568:
2564:
2559:
2555:
2545:
2528:
2523:
2519:
2513:
2510:
2506:
2502:
2497:
2493:
2487:
2484:
2480:
2476:
2471:
2467:
2461:
2458:
2454:
2450:
2448:
2441:
2437:
2429:
2424:
2420:
2414:
2411:
2407:
2403:
2398:
2394:
2388:
2385:
2381:
2377:
2372:
2368:
2362:
2359:
2355:
2351:
2349:
2342:
2338:
2330:
2325:
2321:
2315:
2312:
2308:
2304:
2299:
2295:
2289:
2286:
2282:
2278:
2273:
2269:
2263:
2260:
2256:
2252:
2250:
2243:
2239:
2226:
2223:
2210:
2202:
2190:
2187:
2177:
2172:
2157:
2155:
2144:
2140:
2135:
2131:
2126:
2122:
2096:
2072:
2067:
2063:
2058:
2054:
2049:
2045:
2019:
2018:
2016:
2013:
2000:
1995:
1987:
1983:
1973:
1969:
1959:
1955:
1948:
1941:
1933:
1930:
1926:
1918:
1915:
1911:
1903:
1900:
1896:
1886:
1883:
1879:
1871:
1868:
1864:
1856:
1853:
1849:
1839:
1836:
1832:
1824:
1821:
1817:
1809:
1806:
1802:
1795:
1790:
1785:
1777:
1773:
1763:
1759:
1749:
1745:
1738:
1728:
1726:
1721:
1715:
1710:
1705:
1699:
1693:
1680:
1667:
1642:
1629:
1626:
1622:
1617:
1615:
1611:
1595:
1582:
1577:
1574:
1569:
1566:
1563:
1555:
1552:
1548:
1544:
1538:
1525:
1519:
1516:
1511:
1508:
1502:
1499:
1491:
1480:
1467:
1462:
1459:
1454:
1451:
1443:
1440:
1427:
1422:
1419:
1414:
1411:
1403:
1392:
1379:
1374:
1371:
1366:
1363:
1355:
1350:
1349:
1345:
1344:
1341:
1340:
1339:
1335:
1316:
1303:
1294:
1288:
1280:
1274:
1268:
1259:
1253:
1249:
1244:
1238:
1232:
1224:
1221:
1202:
1195:at the point
1194:
1175:
1169:
1162:
1147:
1124:
1118:
1111:
1096:
1073:
1067:
1060:
1059:
1057:
1054:
1041:
1032:
1026:
1018:
1012:
1006:
1000:
994:
986:
984:
980:
969:
967:
962:
949:
944:
941:
936:
933:
925:
918:
900:
872:
867:
864:
859:
856:
853:
845:
843:
840:(named after
839:
834:
830:
828:
823:
819:
815:
812:
808:
804:
799:
797:
793:
789:
780:
764:
757:
754:
738:
731:
728:
712:
705:
704:
702:
699:
682:
677:
674:
669:
666:
664:
659:
645:
642:
637:
634:
632:
627:
602:
599:
594:
591:
583:
581:
572:
568:
558:
544:
542:
538:
534:
530:
526:
522:
514:
506:
498:
494:
490:
480:
472:
468:
464:
460:
453:
449:
445:
441:
437:
433:
430:(also called
429:
407:
393:
379:
376:
362:
359:
343:
341:
337:
320:
317:
312:
309:
302:
294:
276:
273:
259:
253:
249:
247:
243:
239:
235:
229:
224:
203:
200:
186:
183:
169:
146:
144:
140:
123:
120:
115:
112:
109:
102:
94:
90:
88:
81:
77:
71:
67:
65:
61:
55:
49:
44:
41:
37:
33:
19:
11362:
11330:
11296:
11277:
11250:
11237:
11194:
11190:
11180:
11172:
11167:
11158:
11149:
11107:
11101:
11068:
11064:
11054:
11035:
11029:
11010:
10983:
10979:
10969:
10942:
10938:
10928:
10912:
10907:
10898:
10889:
10864:
10860:
10854:
10817:
10810:
10801:
10792:
10781:
10762:
10743:
10737:
10721:
10716:
10707:
10695:
10686:
10671:
10666:
10640:
10619:. Retrieved
10607:
10580:
10576:
10566:
10554:
10543:
10532:
10518:
10507:
10496:
10477:
10466:
10454:. Retrieved
10444:
10421:
10415:
10382:
10378:
10372:
10337:
10293:
10248:
10238:
10218:
10211:
10195:
10190:
10178:
10154:
10147:
10135:. Retrieved
10126:
10106:
10099:
10090:
10083:. Retrieved
10073:
10061:. Retrieved
10047:
10027:
10020:
10008:. Retrieved
9998:
9993:. ibchem.com
9991:Bonding (sl)
9986:
9974:. Retrieved
9961:
9942:
9938:
9928:
9913:
9908:
9893:
9888:
9872:
9850:. Retrieved
9835:
9828:
9816:. Retrieved
9801:
9779:. Retrieved
9764:
9757:
9718:
9709:
9699:
9686:
9657:
9648:
9639:
9596:
9583:
9576:eddy current
9571:
9566:
9557:
9549:
9543:
9523:
9514:
9510:
9506:
9487:
9477:
9468:
9384:
9380:
9375:
9359:
9007:
8983:Resistivity
8969:
8957:
8938:
8928:
8914:
8908:
8905:
8778:
8728:
8684:
8667:permittivity
8664:
8660:susceptivity
8653:
8643:
8634:
8626:
8612:
8603:Kondo effect
8539:Fermi liquid
8395:
8283:
8271:
8197:
8190:
8183:
8178:
8164:
8084:
8077:
8006:
7992:(non doped)
7986:valence band
7972:In general,
7971:
7851:
7772:
7765:
7745:
7736:
7734:
7723:
7712:
7652:Debye radius
7616:
7430:
7424:resistivity
7418:
7200:
7178:
7086:
7068:
7061:
7056:
7050:
7047:George Gamow
7044:
7036:
6957:Fused quartz
6612:
6361:parallel to
6293:(0 °C)
5930:Carbon steel
5154:
5125:Electrolytes
5062:
5032:
4879:
4877:
4861:Debye length
4850:
4842:
4834:nabla symbol
4831:
4740:
4650:
4643:
4640:Debye length
4631:
4624:
4619:
4616:Debye sheath
4611:
4607:
4605:
4602:
4563:
4544:
4397:
4318:
4295:
4252:
4182:
4178:ion channels
4158:electrolytes
4155:
4124:lies in the
4119:
4092:
4061:
4026:
4022:
4005:
3998:
3958:
3943:
3939:
3898:
3775:
3597:
3455:
3445:
3394:
3081:
3078:
3015:
3012:
2730:
2727:
2574:
2566:
2562:
2557:
2553:
2546:
2227:
2224:
2158:
2151:
2142:
2138:
2133:
2129:
2124:
2120:
2065:
2061:
2056:
2052:
2047:
2043:
2014:
1729:
1719:
1713:
1703:
1697:
1694:
1630:
1618:
1607:
1556:
1550:
1546:
1542:
1539:
1492:
1481:
1444:
1441:
1404:
1393:
1356:
1352:
1336:
1317:
1225:
1222:
1219:
1055:
987:
975:
963:
926:
919:
846:
837:
835:
824:
806:
802:
800:
785:
700:
584:
563:
496:
492:
491:
452:Greek letter
435:
431:
427:
426:
246:SI unit
237:
226:Conductivity
64:SI unit
40:
32:Conductivity
11335:Brady Haran
10613:"JFE steel"
10385:(4): 1147.
10137:26 December
9896:, Longman,
9704:particles).
9493:significand
9443:Skin effect
8655:admittivity
8650:anisotropic
8627:impedivity
8186:thermistors
6858:Hard rubber
6363:basal plane
5201:Temperature
5027:temperature
4122:Fermi level
4017:Fermi level
4013:Fermi level
3956:Fermi level
3928:equilibrium
3918:Band theory
3464:Hall effect
1707:are rank-2
1621:anisotropic
1614:anisotropic
816:resistivity
807:resistivity
256:Other units
74:Other units
46:Resistivity
11377:Categories
10657:(see also
10621:2012-10-20
10456:3 February
10344:. p.
9749:References
9692:20 °C
9679:impurities
9532:20 °C
9370:John Walsh
8635:reactivity
7996:decreases
7409:See also:
6125:Constantan
5206:Reference
5196:20 °C
5183:20 °C
5135:Insulators
5067:25 °C
4968:Maxwellian
4926:collisions
4166:salt water
3966:insulators
3952:semimetals
3916:See also:
1330:and large
1322:and small
820:resistance
803:resistance
571:conductors
552:Ideal case
547:Definition
479:resistance
11197:: 15044.
11135:cite book
11093:1521-3889
10599:1617-7061
10330:(1999) .
10160:CRC Press
10010:20 August
9852:March 24,
9818:March 24,
9781:March 24,
9624:ruthenium
9214:Magnesium
9186:Aluminium
9160:Beryllium
9134:Potassium
9032:Relative
8980:Material
8862:ρ
8838:⇌
8806:σ
8762:ρ
8751:⇌
8743:σ
8702:ℓ
8699:ρ
8639:reactance
8579:
8492:ρ
8471:μ
8376:μ
8371:
8314:ρ
8298:ρ
8238:−
8226:
8214:ρ
8138:ρ
8135:
8120:ρ
8117:
8021:ρ
8014:ρ
7990:intrinsic
7948:σ
7860:κ
7796:π
7785:σ
7782:κ
7667:Θ
7625:ρ
7582:−
7574:−
7559:−
7514:Θ
7504:∫
7482:Θ
7453:ρ
7438:ρ
7379:α
7358:α
7338:α
7317:κ
7295:α
7242:ρ
7187:α
7146:−
7137:α
7119:ρ
7103:ρ
7040:0 °C
7009:(teflon)
6553:Sea water
6523:Germanium
6291:Plutonium
6262:Manganese
5996:Galinstan
5663:Ruthenium
5359:Aluminium
5172:Material
5129:Variable
5119:Variable
5082:Material
5008:insulator
4933:electrons
4855:plasmas,
4792:∇
4757:−
4694:Φ
4683:
4677:∝
4596:Lightning
4507:α
4458:−
4368:α
4358:−
4326:σ
4304:σ
4266:α
4191:α
4096:interfere
4032:In metals
3874:ρ
3845:ρ
3815:ρ
3785:ρ
3742:ρ
3721:ρ
3706:ρ
3702:−
3684:σ
3659:ρ
3638:ρ
3624:ρ
3606:σ
3574:ρ
3570:−
3555:ρ
3525:ρ
3509:ρ
3475:ρ
3426:σ
3404:ρ
3355:σ
3329:σ
3303:σ
3259:σ
3233:σ
3207:σ
3163:σ
3137:σ
3111:σ
3040:σ
2928:σ
2913:σ
2898:σ
2881:σ
2866:σ
2851:σ
2834:σ
2819:σ
2804:σ
2675:ρ
2627:ρ
2583:ρ
2507:ρ
2481:ρ
2455:ρ
2408:ρ
2382:ρ
2356:ρ
2309:ρ
2283:ρ
2257:ρ
2183:ρ
2081:ρ
1927:ρ
1912:ρ
1897:ρ
1880:ρ
1865:ρ
1850:ρ
1833:ρ
1818:ρ
1803:ρ
1672:ρ
1658:⇌
1647:σ
1610:Ohm's law
1578:ℓ
1564:ρ
1520:ℓ
1500:ρ
1423:ℓ
1364:ρ
1254:ρ
1233:σ
1068:ρ
995:ρ
945:ρ
934:σ
901:ℓ
865:ℓ
860:ρ
739:ℓ
678:ℓ
660:ρ
657:⇔
643:ℓ
638:ρ
600:ℓ
595:∝
567:resistors
377:−
360:−
340:Dimension
321:ρ
310:σ
274:−
201:−
184:−
143:Dimension
124:ℓ
110:ρ
11337:for the
11229:26463476
11191:Sci. Rep
11127:33335083
11018:Archived
10881:16851085
10770:Archived
10700:archived
10561:. Matweb
10485:Archived
10407:95005999
10364:40251748
10063:28 April
9550:corroded
9391:See also
9045:Relative
9042:per kg)
8932:and the
8918:are now
7654:and the
7053:(1947):
6321:Nichrome
6089:Manganin
6023:Titanium
5764:Platinum
5489:Tungsten
5284:Annealed
5071:EC meter
5046:solution
5010:such as
4945:neutrons
4895:Property
4838:gradient
3974:band gap
3466:, where
3439:are the
1625:graphite
915:1 m
913: =
891:1 m
889: =
483:1 Ω
475:1 m
91:kg⋅m⋅s⋅A
11341:. 2010.
11220:4604515
11199:Bibcode
11073:Bibcode
10988:Bibcode
10947:Bibcode
10387:Bibcode
10092:metals.
10085:31 July
9482:fields.
9351:History
9318:19,000
9315:39,000
9108:Calcium
9082:Lithium
8993:Density
8960:density
8623:complex
7984:in the
7960:is the
7896:is the
7872:is the
7685:is the
6721:(damp)
6683:Silicon
6233:Bismuth
6197:Mercury
5932:(1010)
5901:Niobium
5865:Gallium
5692:Lithium
5460:Rhodium
5424:Calcium
5043:aqueous
4941:protons
4924:and by
4922:gravity
4901:Plasma
4880:fourth
4076:voltage
4068:lattice
3500:-axis,
1709:tensors
1191:is the
983:density
981:to the
966:siemens
777:is the
751:is the
725:is the
525:siemens
519: (
511: (
507:), but
503: (
461:). The
457: (
238:σ, κ, γ
11303:
11227:
11217:
11125:
11115:
11091:
11042:
10919:
10879:
10825:
10750:
10728:
10678:
10648:
10597:
10432:
10405:
10362:
10352:
10305:
10226:
10202:
10166:
10114:
10035:
9976:14 Sep
9920:
9900:
9879:
9843:
9809:
9772:
9628:copper
9620:Cobalt
9606:(ACSR)
9521:where
9340:5.727
9331:7.874
9312:1.319
9303:19.30
9300:22.14
9284:0.946
9275:10.49
9272:15.87
9268:Silver
9244:16.78
9240:Copper
9230:2.616
9218:43.90
9202:1.579
9190:26.50
9176:2.122
9150:4.291
9124:2.002
9098:5.531
9072:2.843
9056:Sodium
9027:g·mΩ/m
8906:where
8671:opaque
8396:where
8272:where
8165:where
7852:where
7717:and a
7650:, the
7617:where
7421:phonon
7405:Metals
7179:where
6936:Sulfur
5627:Nickel
5590:Cobalt
5287:copper
5248:Copper
5212:Silver
5198:(S/m)
5185:(Ω·m)
5105:Metals
5089:(Ω·m)
5058:ionize
4949:charge
4581:Plasma
4551:silver
4547:copper
4398:where
4381:
4279:
4139:dopant
4114:, and
3948:metals
2208:
2015:where
1619:Here,
1056:where
831:solely
794:
753:length
701:where
11356:(PDF)
10616:(PDF)
10403:S2CID
9971:(PDF)
9460:Notes
9337:505%
9328:96.1
9309:285%
9281:111%
9253:100%
9247:8.96
9221:1.74
9208:0.16
9193:2.70
9167:1.85
9164:35.6
9141:0.89
9138:72.0
9115:1.55
9112:33.6
9089:0.53
9086:92.8
9063:0.97
9060:47.7
7982:holes
6807:Glass
6706:−75.0
6540:−48.0
6448:−0.50
6148:0.008
6112:0.002
5069:. An
5054:salts
5039:water
5023:light
5012:glass
4953:waves
4610:, or
4599:10 m.
4072:atoms
4064:metal
3982:holes
3944:white
3940:black
825:In a
801:Both
796:metre
790:unit
529:metre
521:gamma
513:kappa
505:sigma
471:metre
11301:ISBN
11225:PMID
11141:link
11123:OCLC
11113:ISBN
11089:ISSN
11040:ISBN
10917:ISBN
10877:PMID
10823:ISBN
10748:ISBN
10726:ISBN
10676:ISBN
10646:ISBN
10595:ISSN
10458:2014
10430:ISBN
10360:OCLC
10350:ISBN
10303:ISBN
10224:ISBN
10200:ISBN
10182:See
10164:ISBN
10139:2021
10112:ISBN
10087:2017
10065:2017
10033:ISBN
10012:2014
9978:2018
9918:ISBN
9898:ISBN
9877:ISBN
9854:2019
9841:ISBN
9820:2019
9807:ISBN
9783:2019
9770:ISBN
9622:and
9497:1.65
9334:757
9324:Iron
9306:427
9296:Gold
9287:456
9278:166
9259:6.0
9250:150
9227:51%
9205:2.0
9199:48%
9173:44%
9147:43%
9121:35%
9095:33%
9069:31%
9040:(USD
8998:g/cm
8987:nΩ·m
8936:for
8926:for
8912:and
8601:the
8537:the
8463:and
8173:and
7766:The
7413:and
7007:PTFE
6792:5.49
6783:1.82
6719:Wood
6697:4.35
6619:166
6614:Bone
6602:0.30
6598:0.25
6536:2.17
6492:GaAs
6439:2.00
6431:1.25
6345:0.40
6335:6.70
6306:6.85
6276:6.94
6247:7.75
6220:0.90
6211:1.02
6202:98.0
6184:0.94
6175:1.45
6166:69.0
6139:2.04
6130:49.0
6103:2.07
6094:48.2
6074:2.17
6065:46.0
6046:3.80
6037:2.38
6028:42.0
6010:3.46
6001:28.9
5983:3.90
5974:4.55
5965:22.0
5960:Lead
5945:6.99
5936:14.3
5915:7.00
5906:14.0
5888:4.00
5879:7.10
5870:14.0
5850:8.94
5841:11.2
5823:4.50
5814:9.17
5805:10.9
5787:3.92
5778:9.43
5769:10.6
5751:5.00
5742:10.3
5733:9.70
5728:Iron
5715:6.00
5706:10.8
5697:9.28
5677:14.1
5668:7.10
5650:6.00
5641:14.3
5632:6.99
5613:7.00
5604:16.0
5595:6.24
5575:16.7
5566:5.99
5548:3.70
5539:16.9
5530:5.90
5525:Zinc
5512:4.50
5503:17.9
5494:5.60
5474:23.1
5465:4.33
5447:4.10
5438:29.8
5429:3.36
5409:33.4
5400:3.00
5382:3.90
5373:37.7
5364:2.65
5346:3.40
5337:41.1
5328:2.44
5323:Gold
5310:3.93
5301:58.0
5292:1.72
5271:4.04
5262:59.6
5253:1.68
5235:3.80
5226:63.0
5217:1.59
4955:and
4943:and
4937:ions
4836:and
4445:and
4162:ions
3992:edit
3968:and
3954:the
3950:and
3806:and
3546:and
3417:and
1486:and
805:and
569:and
527:per
495:(or
11215:PMC
11207:doi
11081:doi
11069:408
10996:doi
10955:doi
10943:114
10869:doi
10865:109
10585:doi
10395:doi
10346:286
10299:602
9947:doi
9943:128
9732:7.5
9630:in
9515:T ρ
9290:84
9224:76
9196:72
9170:66
9144:64
9118:52
9092:49
9066:46
9047:to
9034:to
8223:exp
7024:to
7015:to
6986:PET
6971:1.3
6962:7.5
6923:to
6914:to
6896:to
6886:to
6879:Air
6823:to
6814:to
6763:4.2
6754:1.8
6736:to
6727:to
6688:2.3
6666:to
6649:to
6600:to
6589:4.0
6587:to
6581:3.3
6567:4.8
6558:2.1
6528:4.6
6509:to
6499:to
6477:3.3
6468:3.0
6437:to
6422:0.8
6420:to
6414:0.5
6391:to
6376:500
6374:to
6368:250
6326:110
6297:146
6267:144
6238:129
5800:Tin
5194:at
5181:at
5139:10
5109:10
5006:An
4898:Gas
4851:In
4680:exp
4549:or
4156:In
4070:of
3013:or
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844:):
792:ohm
580:rho
467:ohm
459:rho
434:or
83:In
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11333:.
11329:.
11264:^
11223:.
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9791:^
9736:10
9728:10
9724:12
9666:^
9612:^
9509:=
9501:10
9364:,
9262:1
9256:1
9049:Cu
9036:Cu
9029:)
9010:Cu
9000:)
8989:)
8677:.
8662:.
8605:.
8576:ln
8436:,
8416:,
8368:ln
8188:.
8181:.
8169:,
8132:ln
8114:ln
8082::
8004::
7900:,
7876:,
7758:.
7743:.
7732:.
7721:.
7383:15
7206:,
7071:10
7042:.
7026:10
7022:10
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6991:10
6975:10
6966:10
6946:10
6941:10
6925:10
6921:10
6916:10
6912:10
6898:10
6894:10
6888:10
6884:10
6868:10
6863:10
6847:10
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6825:10
6821:10
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6796:10
6787:10
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6729:10
6725:10
6710:10
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6330:10
6310:10
6301:10
6280:10
6271:10
6251:10
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6224:10
6215:10
6206:10
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6179:10
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6098:10
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6032:10
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5099:0
4959:.
4939:,
4935:,
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8852:r
8848:(
8844:E
8833:)
8829:r
8825:(
8821:E
8817:)
8813:r
8809:(
8803:=
8800:)
8796:r
8792:(
8788:J
8765:J
8759:=
8756:E
8746:E
8740:=
8737:J
8710:A
8706:/
8696:=
8693:R
8587:T
8584:1
8554:5
8550:T
8523:2
8519:T
8496:0
8449:m
8445:c
8424:b
8404:a
8379:T
8363:m
8359:c
8355:+
8350:5
8346:T
8342:b
8339:+
8334:2
8330:T
8326:a
8323:+
8318:0
8310:=
8307:)
8304:T
8301:(
8274:n
8258:,
8254:)
8249:n
8245:/
8241:1
8234:T
8230:(
8220:A
8217:=
8175:C
8171:B
8167:A
8151:,
8146:3
8142:)
8129:(
8126:C
8123:+
8111:B
8108:+
8105:A
8102:=
8097:T
8094:1
8064:.
8057:T
8052:B
8048:k
8041:A
8037:E
8031:e
8025:0
8017:=
7928:T
7908:e
7884:k
7838:,
7835:T
7830:2
7825:)
7820:e
7817:k
7812:(
7805:3
7800:2
7790:=
7730:n
7707:n
7701:n
7695:n
7671:R
7634:)
7631:0
7628:(
7603:,
7600:x
7597:d
7590:)
7585:x
7578:e
7571:1
7568:(
7565:)
7562:1
7554:x
7550:e
7546:(
7540:n
7536:x
7528:T
7524:/
7518:R
7508:0
7498:n
7493:)
7486:R
7478:T
7473:(
7468:A
7465:+
7462:)
7459:0
7456:(
7450:=
7447:)
7444:T
7441:(
7426:ρ
7273:0
7269:T
7246:0
7219:0
7215:T
7165:,
7162:]
7159:)
7154:0
7150:T
7143:T
7140:(
7134:+
7131:1
7128:[
7123:0
7115:=
7112:)
7109:T
7106:(
7089:T
6973:×
6964:×
6892:~
6845:~
6794:×
6785:×
6765:×
6756:×
6708:×
6699:×
6690:×
6670:×
6668:5
6662:×
6660:5
6653:×
6651:2
6645:×
6643:2
6625:×
6623:6
6591:×
6583:×
6560:×
6542:×
6530:×
6479:×
6470:×
6450:×
6441:×
6433:×
6424:×
6416:×
6395:×
6393:3
6387:×
6385:2
6378:×
6370:×
6347:×
6337:×
6328:×
6308:×
6299:×
6278:×
6269:×
6249:×
6240:×
6222:×
6213:×
6204:×
6186:×
6177:×
6168:×
6150:×
6141:×
6132:×
6114:×
6105:×
6096:×
6076:×
6067:×
6048:×
6039:×
6030:×
6012:×
6003:×
5985:×
5976:×
5967:×
5947:×
5938:×
5917:×
5908:×
5890:×
5881:×
5872:×
5852:×
5843:×
5825:×
5816:×
5807:×
5789:×
5780:×
5771:×
5753:×
5744:×
5735:×
5717:×
5708:×
5699:×
5679:×
5670:×
5652:×
5643:×
5634:×
5615:×
5606:×
5597:×
5577:×
5568:×
5550:×
5541:×
5532:×
5514:×
5505:×
5496:×
5476:×
5467:×
5449:×
5440:×
5431:×
5411:×
5402:×
5384:×
5375:×
5366:×
5348:×
5339:×
5330:×
5312:×
5303:×
5294:×
5273:×
5264:×
5255:×
5237:×
5228:×
5219:×
5190:σ
5177:ρ
5157:ρ
5087:ρ
5041:/
4818:.
4811:e
4807:n
4800:e
4796:n
4784:e
4778:e
4774:T
4768:B
4764:k
4754:=
4750:E
4727:.
4723:)
4717:e
4713:T
4707:B
4703:k
4698:/
4691:e
4687:(
4672:e
4668:n
4635:i
4632:n
4628:e
4625:n
4485:0
4481:N
4454:b
4431:+
4427:b
4406:q
4384:,
4376:0
4372:N
4364:)
4354:b
4350:+
4345:+
4341:b
4336:(
4332:q
4329:=
4282:.
4274:0
4270:N
4263:=
4260:N
4236:0
4232:N
4211:N
3962:F
3959:E
3938:(
3852:x
3849:x
3822:z
3819:x
3792:y
3789:x
3762:.
3754:2
3749:y
3746:x
3738:+
3733:2
3728:x
3725:x
3713:y
3710:x
3696:=
3691:y
3688:x
3679:,
3671:2
3666:y
3663:x
3655:+
3650:2
3645:x
3642:x
3631:x
3628:x
3618:=
3613:x
3610:x
3581:y
3578:x
3567:=
3562:x
3559:y
3532:x
3529:x
3521:=
3516:y
3513:y
3498:z
3482:y
3479:x
3456:ρ
3446:σ
3381:.
3372:z
3368:E
3362:z
3359:z
3351:+
3346:y
3342:E
3336:y
3333:z
3325:+
3320:x
3316:E
3310:x
3307:z
3299:=
3290:z
3286:J
3276:z
3272:E
3266:z
3263:y
3255:+
3250:y
3246:E
3240:y
3237:y
3229:+
3224:x
3220:E
3214:x
3211:y
3203:=
3194:y
3190:J
3180:z
3176:E
3170:z
3167:x
3159:+
3154:y
3150:E
3144:y
3141:x
3133:+
3128:x
3124:E
3118:x
3115:x
3107:=
3098:x
3094:J
3065:,
3060:j
3055:E
3048:j
3045:i
3035:=
3030:i
3025:J
2997:]
2989:z
2985:E
2975:y
2971:E
2961:x
2957:E
2950:[
2943:]
2935:z
2932:z
2920:y
2917:z
2905:x
2902:z
2888:z
2885:y
2873:y
2870:y
2858:x
2855:y
2841:z
2838:x
2826:y
2823:x
2811:x
2808:x
2797:[
2792:=
2787:]
2779:z
2775:J
2765:y
2761:J
2751:x
2747:J
2740:[
2714:.
2707:x
2703:J
2697:z
2693:E
2687:=
2682:x
2679:z
2666:,
2659:x
2655:J
2649:y
2645:E
2639:=
2634:x
2631:y
2622:,
2615:x
2611:J
2605:x
2601:E
2595:=
2590:x
2587:x
2567:z
2563:J
2558:y
2554:J
2549:x
2529:.
2524:z
2520:J
2514:z
2511:z
2503:+
2498:y
2494:J
2488:y
2485:z
2477:+
2472:x
2468:J
2462:x
2459:z
2451:=
2442:z
2438:E
2430:,
2425:z
2421:J
2415:z
2412:y
2404:+
2399:y
2395:J
2389:y
2386:y
2378:+
2373:x
2369:J
2363:x
2360:y
2352:=
2343:y
2339:E
2331:,
2326:z
2322:J
2316:z
2313:x
2305:+
2300:y
2296:J
2290:y
2287:x
2279:+
2274:x
2270:J
2264:x
2261:x
2253:=
2244:x
2240:E
2211:.
2203:j
2198:J
2191:j
2188:i
2178:=
2173:i
2168:E
2143:z
2139:J
2134:y
2130:J
2125:x
2121:J
2105:J
2066:z
2062:E
2057:y
2053:E
2048:x
2044:E
2028:E
2001:,
1996:]
1988:z
1984:J
1974:y
1970:J
1960:x
1956:J
1949:[
1942:]
1934:z
1931:z
1919:y
1916:z
1904:x
1901:z
1887:z
1884:y
1872:y
1869:y
1857:x
1854:y
1840:z
1837:x
1825:y
1822:x
1810:x
1807:x
1796:[
1791:=
1786:]
1778:z
1774:E
1764:y
1760:E
1750:x
1746:E
1739:[
1720:J
1714:E
1704:ρ
1698:σ
1681:,
1677:J
1668:=
1664:E
1652:E
1643:=
1639:J
1583:.
1575:A
1570:R
1567:=
1551:R
1547:I
1545:/
1543:V
1526:.
1517:I
1512:A
1509:V
1503:=
1488:J
1484:E
1468:.
1463:A
1460:I
1455:=
1452:J
1428:.
1420:V
1415:=
1412:E
1400:ℓ
1396:V
1380:,
1375:J
1372:E
1367:=
1332:σ
1328:ρ
1324:σ
1320:ρ
1304:.
1298:)
1295:x
1292:(
1289:E
1284:)
1281:x
1278:(
1275:J
1269:=
1263:)
1260:x
1257:(
1250:1
1245:=
1242:)
1239:x
1236:(
1215:.
1203:x
1179:)
1176:x
1173:(
1170:J
1160:,
1148:x
1128:)
1125:x
1122:(
1119:E
1109:,
1097:x
1077:)
1074:x
1071:(
1042:,
1036:)
1033:x
1030:(
1027:J
1022:)
1019:x
1016:(
1013:E
1007:=
1004:)
1001:x
998:(
950:.
942:1
937:=
922:σ
887:A
873:.
868:A
857:=
854:R
765:A
713:R
683:,
675:A
670:R
667:=
646:A
635:=
628:R
603:A
592:R
576:ρ
517:γ
509:κ
501:σ
469:-
455:ρ
408:2
402:I
394:3
388:T
380:3
371:L
363:1
354:M
318:1
313:=
277:1
269:s
204:2
195:I
187:3
178:T
170:3
164:L
156:M
121:A
116:R
113:=
57:ρ
38:.
20:)
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