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Electrical resistance

2007 Schools Wikipedia Selection. Related subjects: Electricity and
Electronics

   A 750-kΩ resistor, as identified by its electronic color code. A
   multimeter could be used to verify this value.
   A 750-kΩ resistor, as identified by its electronic colour code. A
   multimeter could be used to verify this value.

   Electrical resistance is a measure of the degree to which an object
   opposes an electric current through it. The SI unit of electrical
   resistance is the ohm. Its reciprocal quantity is electrical
   conductance measured in siemens. Electrical resistance shares some
   conceptual parallels with the mechanical notion of friction.

   The resistance of an object determines the amount of current through
   the object for a given voltage across the object.

          I = \frac{V}{R}

   where

          R is the resistance of the object, usually measured in ohms,
          equivalent to J·s/C^2
          V is the voltage across the object, usually measured in volts
          I is the current 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 through or the
   amount of voltage across the object such that the resistance R is a
   constant.

Resistive loss

   When there is a current I 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 energy conversion is useful in applications such as incandescent
   lighting and electric heating but is considered a loss in other
   applications such as power transmission. Ideally, the conductors used
   to connect electrical devices together should have zero resistance but
   in reality, only superconductors achieve this ideal. Common ways to
   combat resistive loss in conductors include using thicker wire and
   higher voltages.

Resistance of a conductor

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.

Causes of resistance

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, 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 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 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.

Resistance of various materials

   Material       Resistivity, ρ
                  ohm-meter
   Metals         10 ^- 8
   Semiconductors variable
   Electrolytes   variable
   Insulators     10^16

Band theory

   Electron energy levels in an insulator.
   Electron energy levels in an insulator.

   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 U-I graph at a particular point, thus:

          R = \frac {\mathrm{d}U} {\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 U-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 U/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.

Frequency Dependent Resistance

   Wire resistance increases at a rate of 10dB/decade for wire radius much
   greater than skin depth.

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