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Electric current Resistance and resistivity
Bulk properties
Photo of copper electrode holder
Picture 2.2 The high conductivity of copper makes it suitable for this electrode holder which carries enormous currents to a steel making arc furnace.

When we say that copper is a heavier metal than aluminium, we are comparing their densities. In a similar way, when we say that copper is a better conductor than aluminium, we are comparing their resistivities.

Density and resistivity are both bulk properties of a material. Their value doesn't depend on the size or shape of a particular sample – only on the material itself.

Resistivity: depends on the material only
Resistance: depends on size and shape - as well as on the material.

What is resistivity?
In effect, the resistivity represents the resistance across two opposite faces of a cubic metre of material (in the same way that density is the mass of a cubic metre). Resistivity tells how resistive a material is.

Resistivity has the symbol r - (rho to rhyme with snow) and its units are ohm metres.

Quantity Symbol Unit
Resistivity r    (Greek letter rho) W  m   (ohm metre)
Resistance R W


The resistivity of pure copper is 1.7 × 10-8 W  m

Notice that this is a very small number – 0.000 000 017 W  m. This is because the resistance of a cubic metre of copper would be next to nothing. The smaller the resistivity, the better the material is at conducting electricity. Resistivity is the inverse of conductivity (which tells us how good a conductor a material is).

Graph of resistivities
Picture 2.2 Graph of resistivities of some metals.
Some resistivities
Table 3 shows some resistivities. You will see that resistivity varies from about 10-8  W  m to more than 1016  W  m – more than 24 orders of magnitude. No other property of materials varies over such a wide range.

Metal Resistivity /(W  m) Material Resistivity /(W  m)
silver 1.6 × 10-8 carbon 35 to 5000 × 10-8
copper 1.7 × 10-8 graphite 800 × 10-8
aluminium 3.2 × 10-8 germanium 0.65
lead 21.0 × 10-8 silicon 2.3  × 10-3
manganin (alloy) 44.0 × 10-8 pyrex glass 1012
eureka (alloy) 49.0 × 10-8 PTFE 1012 to  × 1016
steel (varies) 10 to 100 × 10-8 quartz 5 × 1016
Table 3. Resistivity values at room temperature. For metals, resistivity increases as temperature increases. For semiconductors and many insulators, the opposite is true.

Graphic of wires
Picture 2.3 Graphics to illustrate how resistance varies with l, A and rho.
Calculating resistance
To calculate the resistance R of a wire, we need to know three things:
  • its length  – the longer the wire, the greater its resistance
  • its cross-sectional area A – the greater the area, the less its resistance
  • the resistivity of the material r  – the greater the resistivity, the greater its resistance.

       resistance = resistivity × length / area

This equation defines resistivity. We can rearrange it to get a formula for resistivity:

       resistivity = resistance × area / length

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Question 3
a) Look at the electrode holder in the photograph at the top of the page. Use the table to compare the resistivities of copper and steel; use this to explain why using copper is an energy-efficient choice.

b) A wire made of a copper alloy is 5 m in length and has a cross-sectional area 1 mm2. Its resistance is 0.15 W. Calculate the resistivity of this alloy.


Tip
1 mm2 = 10-6 m2

c) How does this compare with the resistivity of copper?

d) Calculate the resistance of a copper wire of length 5 m and diameter 2 mm.


Tip
A = p r2