# Resistors, Resistance and Resistivity

## Resistors, Resistance and Resistivity

When there is a potential difference across two ends of a conductor, electrons move and the rate of flow of the electrons is what is called current. However, electrons move more easily through some conductors than others for the same a p.d.. The amount of opposition the electrons faces when they move through a conductor is called its resistance.

Resistance to a current is caused by the repeated collisions between the charge carriers and with the fixed positive ions of the material. A good conductor has a low resistance and a poor conductor has a high resistance.

## Resistance

Resistance is the opposition to the flow of current. According to Ohm’s law, the resistance of a conductor is the ratio of the voltage across the conductor to the current flowing through it. Resistance is measured in ohms (Ω)

### The ohm (Ω)

According to Ohm’s law:

$latex \displaystyle V = IR$

Which means that:

$latex \displaystyle R = \frac{V}{R}$

This means that if there is a current of 1 amp for a potential difference of 1 volt across the ends of a conductor, the resistance of the conductor is given by:

$latex \displaystyle R = \frac{V}{R}$

$latex \displaystyle R = \frac{1V}{1A}$

$latex \displaystyle R = 1 \Omega$

From there we can mathematically define the ohm as:

The ohm is the resistance of a conductor in which the current is 1 ampere when a voltage of 1 volt is applied across it.

## Factors affecting resistance of a material

The resistance of a conductor depends on its size and shape. A longer conductor has a greater resistance than a short one, provided it is of the same thickness and material, whilst a thick wire has less resistance than a thin one.

The resistance of an electrical conductor depends on 4 factors:

• length of the conductor
• cross-sectional area of the conductor
• type of material
• temperature of the material.

### Relationship between resistance and length of a conductor

Resistance, R, of a conductor is directly proportional to the length, l, of the conductor. This is mathematically represented as:

$latex \displaystyle R \propto l$

What this relationship means is:

• the longer the conductor, the higher the resistance.
• if the length of a piece of wire is doubled, then the resistance is doubled.

### Relationship between resistance and cross-sectional of a conductor

Resistance, R, of a conductor is inversely proportional to cross-sectional area, A, of a conductor. This is mathematically represented as:

$latex \displaystyle R \propto \frac{1}{A}$

What this relationship means is:

• the thicker the conductor, the lower the resistance.
• if the cross-sectional area of a piece of wire is doubled then the resistance is halved.

### Combining the two relationships to get the final formula

Since $latex R \propto l$ and $latex R \propto \frac{1}{A}$, we can combine the two relationships to get:

$latex \displaystyle R \propto \frac{l}{A}$

The above relationship means that, at a constant temperature, the resistance of a material is directly proportional to the length of the material and inversely proportional to its cross-sectional area.

After that, we can insert the constant of proportionality which is unique to the type of material used to make the conductor. This constant of proportionality is known as the resistivity of the material and is given the symbol ρ (Greek letter pronounced rho).

We get the formula for resistance as:

$latex \displaystyle R = \frac{\rho l}{A}$

Whereby, ρ is resistivity, measured in ohm metres (Ωm)

## Resistivity

The resistance of a conductor also depends on the material it is made of. For example,

• copper is a better conductor than steel.
• steel is a better conductor than silicon.

The property of a conductor that determines its resistance due to the type of material it is made of is called its resistivity.

The value of the resistivity of a material is the resistance of a unit cube of the material measured between opposite faces of the cube.

The value of resistivity of a material is constant at constant temperature and varies with temperature. For a metal, resistivity increases with temperature because at higher temperatures there are more frequent collisions between the conduction electrons and the vibrating ions of the metal.

Good conductors of electricity have a low value of resistivity while good insulators have a high value of resistivity.

Some typical resistivity values at 20° C are:

Material Resistivity
Copper 1.7 × 10-8 Ωm
Aluminium 2.6 × 10-8 Ωm
Graphite 10 × 10-8 Ωm
Glass 1 × 1010 Ωm
Mica 1 × 1013 Ωm
Silver 1.60 × 10−8 Ωm
Mercury 69.0 × 10−8 Ωm
Nichrome 1.30 × 10−8 Ωm
Germanium 0.65 Ωm
Silicon 2.3 × 103 Ωm
Pyrex glass 1012 Ωm
Manganin 44.0 × 10−8 Ωm
PTFE 1013–1016 Ωm
Eureka 49.0 × 10−8 Ωm
Quartz 5 × 1016 Ωm
• Nichrome is an alloy of nickel, copper and aluminium used in electric heaters because it does not oxidise at 1000 °C.
• Manganin is an alloy of 84% copper, 12% manganese and 4% nickel.
• Eureka (constantan) is an alloy of 60% copper and 40% nickel.
• PTFE is the polymer poly(tetrafluoroethene) or Teflon

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