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# Electrical resistance

Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. It is the ratio of the potential difference (i.e. voltage) across an electric component (such as a resistor) to the current passing through that component:

$R = \frac{V}{I}$

where

R is the resistance of the component.

V the voltage across the component, measured in volts

I is the current passing through the component, 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.

Resistance is thus a measure of the component's opposition to the flow of electric charge. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.

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 known as ohmic devices.

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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

$P = {I^{2} \cdot R} \,$

where

P is the power measured in watts

I is the current measured in amps

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 resistance R of a wire can be computed as

$R = {L \cdot \rho \over A} \,$

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.

Additional note: There are two reasons why a small cross-sectional area tends to raise resistance. One is that the electrons, all having the same negative charge, repel each other. Thus there is resistance to many being forced into a small space. The other reason is that the electrons collide with each other, causing "scattering," and therefore they are diverted from their original directions. (More discussion is on page 27 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.)

## 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:

$R = \frac {dV} {dI} \,$

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:

$R = R_0 + aT \,$

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

$R= R_0 + e^{a/T}\,$