Electrical resistance

Electrical resistance is a measure of the degree to which an object opposes the passage of an electric current. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.

The quantity of resistance in an electric circuit determines the amount of current flowing in the circuit for any given voltage applied to the circuit.
 * $$R = \frac{\Delta V}{I}$$

where
 * R is the resistance of the object, usually measured in ohms, equivalent to J·s/C2
 * ΔV is the potential difference across the object, usually measured in volts
 * I is the current passing through the object, usually measured in amperes

For a wide variety of materials and conditions, the electrical resistance does not depend on the amount of current flowing or the amount of applied voltage. V can either be measured directly across the object or calculated from a subtraction of voltages relative to a reference point. The former method is simpler for a single object and is likely to be more accurate. There may also be problems with the latter method if the voltage supply is AC and the two measurements from the reference point are not in phase with each other.

Resistive loss
When a current, I, flows through an object with resistance, R, electrical energy is converted to heat at a rate (power) equal to
 * $$P = {I^{2} \cdot R} \,$$

where
 * P is the power measured in watts


 * I is the current measured in amperes


 * R is the resistance measured in ohms

This effect is useful in some applications such as incandescent lighting and electric heating, but is undesirable in power transmission. Common ways to combat resistive loss include using thicker wire and higher voltages. Superconducting wire is used in special applications.

DC resistance
As long as the current density is totally uniform in the conductor, the DC resistance R of a conductor of regular cross section can be computed as
 * $$R = {l \cdot \rho \over A} \,$$

where
 * l is the length of the conductor, measured in meters


 * A is the cross-sectional area, measured in square meters


 * ρ (Greek: rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm · meter. Resistivity is a measure of the material's ability to oppose the flow of electric current.

For practical reasons, almost 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
If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire is reduced. This is because of the skin effect.

This formula applies to isolated conductors. In a conductor close to others, the actual resistance is higher because of the proximity effect.

In metals
A metal consists of a lattice of atoms, each with a shell of electrons. This can also be 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 voltage) is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the electric field.

In a metal the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron wave on non-correlating potentials of ions) - 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. 

In semiconductors and insulators
In metals the fermi level lies in the conduction band giving rise to free conduction electrons. However in semiconductors the position of the fermi level is within the band gap, exactly half way between the conduction band minimum and valence band maximum for intrinsic(undoped) semiconductors. This means that at 0 Kelvin, there are no free conduction electrons and the resistance is infinite. However, as the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier 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 dopped 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 electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the salt concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to specific ions and determine the membrane resistance.

Band theory


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, large voltages yield relatively small currents.

Differential resistance
When resistance may depend on voltage and current, differential resistance, incremental resistance or slope resistance is defined as the slope of the V-I graph at a particular point, thus:
 * $$R = \frac {\mathrm{d}V} {\mathrm{d}I} \,$$

This quantity is sometimes called simply resistance, although the two definitions are equivalent only for an ohmic component such as an ideal resistor. 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 resistance, although it is more correctly called negative differential resistance, since the absolute resistance V/I is still positive.

Temperature-dependence
Near room temperature, the electric resistance of a typical metal conductor increases linearly with the temperature:
 * $$R = R_0(1 + aT) \,$$,

where a is the thermal resistance coefficient.

The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:
 * $$R= R_0 e^{a/T}\,$$

Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increased 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.