The
electrical resistance of an
electrical elementElectrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks...
is the opposition to the passage of an
electric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
through that element; the inverse quantity is
electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical notion of
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
. The
SIThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
unit of electrical resistance is the
ohmThe ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...
(
ΩOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to omicron, which means "little O"...
), while electrical conductance is measured in
siemensThe siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...
(S).
An object of uniform cross section has a resistance proportional to its
resistivityElectrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
and length and inversely proportional to its cross-sectional area. All materials show some resistance, except for superconductors, which have a resistance of zero.
The resistance of an object is defined as the ratio of
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
across it to current through it:
For a wide variety of materials and conditions, the electrical resistance
R is
constantIn mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
for a given temperature; it does not depend on the amount of current through or the potential difference (
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
) across the object. Such materials are called Ohmic materials. For objects made of ohmic materials the definition of the resistance, with
R being a constant for that resistor, is known as
Ohm's lawOhm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
.
In the case of a nonlinear conductor (not obeying Ohm's law), this ratio can change as current or voltage changes; the inverse slope of a chord to an
I-V curve is sometimes referred to as a "chordal resistance" or "static resistance".
Conductors and resistors
Objects such as wires that are designed to have low resistance so that they transfer current with the least loss of electrical energy are called
conductorsIn physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...
. Objects that are designed to have a specific resistance so that they can dissipate electrical energy or otherwise modify how a circuit behaves are called
resistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s. Conductors are made of highly conductive materials such as metals, in particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and cost.
DC resistance
The resistance of a given resistor or conductor grows with the length of conductor and
specific resistivityElectrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
of the material, and decreases for larger cross-sectional area. The resistance and conductance of a conductor of uniform cross section, therefore, can be computed as
where
is the length of the conductor, measured in
metreThe metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
s [m],
A is the cross-section area of the conductor measured in
square metreThe square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...
s [m²], σ (sigma) is the electrical conductivity measured in
siemensThe siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...
per meter (S·m
-1), and ρ (
rhoRho is the 17th letter of the Greek alphabet. In the system of Greek numerals, it has a value of 100. It is derived from Semitic resh "head"...
) is the electrical resistivity (also called
specific electrical resistance) of the material, measured in ohm-metres (Ω·m). Resistivity is a measure of the material's ability to oppose electric current. For purely resistive circuits conductance is related to resistance
by:
For practical reasons, any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.
AC resistance
A wire carrying alternating current has a reduced effective cross sectional area because of the
skin effectSkin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...
. Adjacent conductors carrying alternating current have a higher resistance than they would in isolation or when carrying direct current, due to the
proximity effectIn a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed the...
. At
commercial power frequencyThe utility frequency, line frequency or mains frequency is the frequency at which alternating current is transmitted from a power plant to the end-user. In most parts of the world this is 50 Hz, although in the Americas it is typically 60 Hz...
, these effects are significant for large conductors carrying large currents, such as
busbarIn electrical power distribution, a bus bar is a strip of copper or aluminium that conducts electricity within a switchboard, distribution board, substation or other electrical apparatus....
s in an
electrical substationA substation is a part of an electrical generation, transmission, and distribution system. Substations transform voltage from high to low, or the reverse, or perform any of several other important functions...
, or large power cables carrying more than a few hundred amperes.
When an alternating current flows through the circuit, its flow is not opposed only by the circuit resistance, but also by the opposition of electric and magnetic fields to the current change. That effect is measured by electrical reactance. The combined effects of reactance and resistance are expressed by
electrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
.
Measuring resistance
An instrument for measuring resistance is called an
ohmmeterAn ohmmeter is an electrical instrument that measures electrical resistance, the opposition to an electric current. Micro-ohmmeters make low resistance measurements. Megohmmeters measure large values of resistance...
. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use
four-terminal sensingFour-terminal sensing , 4-wire sensing, or 4-point probes method is an electrical impedance measuring technique that uses separate pairs of current-carrying and voltage-sensing electrodes to make more accurate measurements than traditional two-terminal sensing...
.
In metals
A
metalA metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
consists of a lattice of
atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s, each with a shell of electrons. This is also known as a positive ionic lattice. The outer electrons are free to dissociate from their parent atoms and travel through the lattice, creating a 'sea' of electrons, making the metal a conductor. When an electrical potential difference (a
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
) is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the
electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
.
Near room temperatures, the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron waves on non-correlating potentials of ions), and is thus the prime cause of metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is negligible.
The larger the cross-sectional area of the conductor, the more electrons are available to carry the current, so the lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the material, so the higher the resistance. Different materials also affect the resistance.
http://www.ias.ac.in/resonance/Sept2003/pdf/Sept2003p41-48.pdf
In semiconductors and insulators
In metals, the
Fermi levelThe Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...
lies in the conduction band (see Band Theory, below) giving rise to free conduction electrons. However, in semiconductors the position of the Fermi level is within the band gap, approximately half-way between the conduction band minimum and valence band maximum for intrinsic (undoped) semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance is infinite. However, the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors,
dopantA dopant, also called a doping agent, is a trace impurity element that is inserted into a substance in order to alter the electrical properties or the optical properties of the substance. In the case of crystalline substances, the atoms of the dopant very commonly take the place of elements that...
atoms increase the majority charge carrier concentration by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance. Highly doped semiconductors hence behave metallic. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.
In ionic liquids/electrolytes
In
electrolyteIn chemistry, an electrolyte is any substance containing free ions that make the substance electrically conductive. The most typical electrolyte is an ionic solution, but molten electrolytes and solid electrolytes are also possible....
s, electrical conduction happens not by band electrons or holes, but by full atomic species (
ionAn ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
s) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In
biological membranesThe cell membrane or plasma membrane is a biological membrane that separates the interior of all cells from the outside environment. The cell membrane is selectively permeable to ions and organic molecules and controls the movement of substances in and out of cells. It basically protects the cell...
, currents are carried by ionic salts. Small holes in the membranes, called
ion channelIon channels are pore-forming proteins that help establish and control the small voltage gradient across the plasma membrane of cells by allowing the flow of ions down their electrochemical gradient. They are present in the membranes that surround all biological cells...
s, are selective to specific ions and determine the membrane resistance.
Resistivity of various materials
| Material |
Resistivity, 
ohm-metre |
MetalA metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
s |
10−8 |
SemiconductorA semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s |
variable |
| Electrolyte In chemistry, an electrolyte is any substance containing free ions that make the substance electrically conductive. The most typical electrolyte is an ionic solution, but molten electrolytes and solid electrolytes are also possible....
s |
variable |
Insulatorthumb|250px|[[Coaxial Cable]] with dielectric insulator supporting a central coreThis article refers to electrical insulation. For insulation of heat, see Thermal insulation...
s |
1016 |
| Superconductors |
0 (exactly) |
Band theory simplified
Quantum mechanics states that the energy of an electron in an atom cannot be any arbitrary value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible. The energy levels are grouped into two bands: the
valence band and the
conduction band (the latter is generally above the former). Electrons in the conduction band may move freely throughout the substance in the presence of an electrical field.
In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this forbidden gap and into the conduction band. Thus, even large voltages can yield relatively small currents.
Differential resistance
When the current–voltage dependence is not linear,
differential resistance,
incremental resistance or
slope resistance is defined as the slope of the
V-I graph at a particular point, thus:
This quantity is sometimes called simply
resistance, although the two definitions are equivalent only for an ohmic component such as an ideal resistor. For example, a
diodeIn electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...
is a circuit element for which the resistance depends on the applied voltage or current.
If the
V-I graph is not monotonic (i.e. it has a peak or a trough), the differential resistance will be negative for some values of voltage and current. This property is often known as
negative resistanceNegative resistance is a property of some electric circuits where an increase in the current entering a port results in a decreased voltage across the same port. This is in contrast to a simple ohmic resistor, which exhibits an increase in voltage under the same conditions. Negative resistors are...
, although it is more correctly called
negative differential resistance, since the absolute resistance
V/
I is still positive. An example of such an element is the
tunnel diodeA tunnel diode or Esaki diode is a type of semiconductor diode which is capable of very fast operation, well into the microwave frequency region, by using quantum mechanical effects....
.
Differential resistance is only useful to compare a nonlinear device with a linear source/load in some small interval; for example if it is necessary to evaluate a
Zener diodeA Zener diode is a special kind of diode which allows current to flow in the forward direction in the same manner as an ideal diode, but will also permit it to flow in the reverse direction when the voltage is above a certain value known as the breakdown voltage, "Zener knee voltage" or "Zener...
's voltage stability under different current values.
The small-signal modeling technique is commonly used for analysis of non-linear devices, using linearization of equations about the selected DC operating point (bias point).
Temperature dependence
Near room temperature, the electric resistance of a typical metal increases linearly with rising
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, while the electrical resistance of a typical semiconductor decreases with rising temperature. The amount of that change in resistance can be calculated using the temperature coefficient of resistivity of the material using the following formula:
where
T is its temperature,
T0 is a reference temperature (usually room temperature),
R0 is the resistance at
T0, and
α is the percentage change in resistivity per unit temperature. The constant
α depends only on the material being considered. The relationship stated is actually only an approximate one, the true physics being somewhat non-linear, or looking at it another way,
α itself varies with temperature. For this reason it is usual to specify the temperature that
α was measured at with a suffix, such as
α15 and the relationship only holds in a range of temperatures around the reference.
At lower temperatures (less than the Debye temperature), the resistance of a metal decreases as
T5 due to the electrons scattering off of
phononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s. At even lower temperatures, the dominant scattering mechanism for electrons is other electrons, and the resistance decreases as
T2. At some point, the impurities in the metal will dominate the behavior of the electrical resistance which causes it to saturate to a constant value. Matthiessen's Rule
(first formulated by
Augustus MatthiessenAugustus Matthiessen, FRS , the son of a merchant, was a British chemist and physicist who obtained his PhD in Germany at the University of Gießen in 1852 with Johann Heinrich Buff. He then worked with Robert Bunsen at the University of Heidelberg from 1853 to 1856...
in the 1860s; the equation below gives its modern form)
says that all of these different behaviors can be summed up to get the total resistance as a function of temperature,
where
Rimp is the temperature independent electrical resistivity due to impurities, and
a,
b, and
c are coefficients which depend upon the metal's properties. This rule can be seen as the motivation to
Heike Kamerlingh Onnes'sHeike Kamerlingh Onnes was a Dutch physicist and Nobel laureate. He pioneered refrigeration techniques, and he explored how materials behaved when cooled to nearly absolute zero. He was the first to liquify helium...
experiments that led in 1911 to discovery of
superconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
. For details see
History of superconductivityThe history of superconductivity, the property exhibited by certain substances of lacking electrical resistance at temperatures close to absolute zero, began at the end of the 19th century and culminated in Heike Kamerlingh Onnes's 1911 discovery...
.
Intrinsic semiconductorAn intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a pure semiconductor without any significant dopant species present. The number of charge carriers is therefore determined by the properties of the material itself instead of the amount of impurities...
s become better conductors as the temperature increases; the electrons are bumped to the
conduction energy bandIn the solid-state physics field of semiconductors and insulators, the conduction band is the range of electron energies, higher than that of the valence band, sufficient to free an electron from binding with its individual atom and allow it to move freely within the atomic lattice of the material...
by thermal energy, where they flow freely and in doing so leave behind
holesAn electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
in the
valence bandIn solids, the valence band is the highest range of electron energies in which electrons are normally present at absolute zero temperature....
which also flow freely. The electric resistance of a typical
intrinsicAn intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a pure semiconductor without any significant dopant species present. The number of charge carriers is therefore determined by the properties of the material itself instead of the amount of impurities...
(non doped)
semiconductorA semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
decreases
exponentially with the temperature:
Extrinsic (doped) semiconductorsAn extrinsic semiconductor is a semiconductor that has been doped, that is, into which a doping agent has been introduced, giving it different electrical properties than the intrinsic semiconductor....
have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.
The electric resistance of electrolytes and insulators is highly nonlinear, and case by case dependent, therefore no generalized equations are given.
Strain dependence
Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon
strainIn continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...
. By placing a conductor under
tensionIn physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...
(a form of
stressIn continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on
strain gaugeA strain gauge is a device used to measure the strain of an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable...
s for details about devices constructed to take advantage of this effect.