# Electrical resistance

Electrical resistance is a measure of the degree to which a body 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.

Resistance is defined as the ratio of the potential difference (i.e. voltage) across the object (such as a resistor) to the current passing through it:

[itex]R = \frac{V}{I}[itex]

where

R is the resistance of the object
V the potential difference across the object, measured in volts
I is the current passing through the object, measured in amperes

V can either be measured directly across the component or calculated from a subtraction of voltages relative to a reference point. The former method is simpler for a single component and is likely to be more accurate. The latter method is useful when analysing a larger circuit or if you want to work one-handed with one lead clipped (which can be a useful safety precaution on systems using dangerous voltages). There may also be problems with the latter method if the system is AC and the two measurements from the reference point are not in phase with each other.

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. This means that voltage is proportional to current and the proportionality constant is the electrical resistance. This case is described by Ohm's law and such materials are described as ohmic.

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## 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

[itex]P = {I^{2} \cdot R} \,[itex]

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 like 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, but may become more common someday.

## Resistance of a wire

The DC resistance R of a wire can be computed as

[itex]R = {L \cdot \rho \over A} \,[itex]

where

L is the length of the wire, measured in metres
A is the cross-sectional area, measured in square metres

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

### AC resistance

If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire available for current conduction is proportionally diminished. (See skin effect).

The Terman formula gives the diameter of wire that will suffer a 10% increase in resistance

[itex]D_w = {200 \over \sqrt{F}} \,[itex]

where

F is the frequency of operation, measured in hertz (Hz)

This formula applies for isolated conductors. In a coil surrounded by other turns the actual resistance will be 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. The outer electrons are free to to dissociate from their parent atoms and travel through the lattice, making the metal a conductor. When an electrical potential (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 real material the atomic lattice is never perfectly regular, so its imperfections scatter the electrons and cause resistance. A rise in temperature causes the atoms to vibrate more strongly, creating even more collisions and increasing the resistance still further.

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.  (http://www.ias.ac.in/resonance/Sept2003/pdf/Sept2003p41-48.pdf)

### In semiconductors and insulators

Semiconductors are part-way between metals and insulators, like glass; a silicon boule has a grayish metallic sheen, like a metal, but is brittle, like glass. It is possible to manipulate the resistive properties of semiconductor materials by doping those materials with atomic elements, such as arsenic or boron, which have electrons or holes which can move across the material lattice.

 Material Resistivity, [itex] \rho [itex]ohm-metre Metals [itex]10^{-8}[itex] Semiconductors variable Insulators [itex]10^{16}[itex]

## Differential resistance

When resistance may depend on voltage and current, differential resistance or incremental resistance is defined as the slope of the V-I graph at a particular point, thus:

[itex]R = \frac {dV} {dI} \,[itex]

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

The electric resistance of a typical metal conductor increases linearly with the temperature:

[itex]R = R_0 + aT \,[itex]

The electric resistance of a typical semiconductor decreases exponentially with the temperature:

[itex]R= R_0 + e^{a/T}\,[itex]

## SI electricity units

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