# Resistivity

For a piece of material with cross sectional area A square metres and length l metres, the resistance R of the piece is proportional to the length, and inversely proportional the the area, i.e.

```R ∝ l
R ∝ 1/A
```

We can write R=ρl/A where ρ is the appropriate constant of proportionality for the material. The value ρ has a name —resistivity— and the unit of measurement is ohm metres (Ωm).

The reason for the variation of resistance with the cross sectional area is that the larger the area, the larger the current, and the larger the current, the smaller the resistance. Decreasing the area decreases the current and this increases the resistance.

When the distance from a charge changes so does the potential difference and this explains the variation of resistance with the length. We know that W/q=Ed where W/q is just V (volts) and d is the distance from the charge. Now, increasing d, in our case this is increasing l, will decrease the voltage and this in turn decreases the drift velocity of the electrons. The lower the drift velocity, the lower the current, and the lower the current, the higher the resistance. Obviously, decreasing l will decrease the resistance.

The value of ρ is different for different materials and it also depends on temperature. When the temperature increases so does the thermal motion of the atoms in the material. This movement increases the chance of an electron colliding and collisions decrease the drift velocity. As in the case above, the lower the drift velocity, the lower the current. The table below shows the resistivity for some materials at 20 degrees Celsius:

Material ρ (Ωm)
silver1.64×10-8
copper1.72×10-8
aluminium2.83×10-8
tungsten5.5×10-8

We can also say that the larger the number of mobile charge carriers in a material, the lower the resistivity of that material. We can compare insulators which have few such carriers and high resistance, with conductors which have many such carriers and low resistance.

## References

Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.