Resistance dependance on temperature for ohmic resistors

[luxux]

Hello,
I was giving a random thought to Ohm's Laws and thermal dilation.

As a general rule we are told that ohmic resistors will increase their resistivity if their temperature increases. Does this take into account the fact that its dimensions will change due to thermal dilation?
This would make sense given the similarity between the mathematical expressions of the two phenomena.

The usual conclusion is that therefore resistance of a wire of that material must increase.

But shouldn't we take into account the fact that by warming up a wire we are also increasing its length and cross-sectional area? The former effect would contribute to an increase in resistance, but the latter would reduce the resistance.

Do we usually say that increasing temperature increases resistivity and thus resistance only because we neglect thermal dilation effects? Or is it because since the ratio length/c.s. area is usually high the reduction of resistance due to an increase in c.s. area is much smaller than its increase due to the increase in length?

 

[.Scott]

In general, expansion of the metal is treated as an entirely mechanical concern. The increase in electrical resistance (dTemp) is generally much greater than the increase in length. If part of the change is due to the mechanical changes, that seldom affects the electrical considerations because you almost always have the full component in the electrical circuit regardless of any change there may be in its dimensions. For example, if power lines expand in the heat - then they simply droop - but current still flows through the length of the entire cable. So the tables that show resistivity changes due to temperature presume that to be the condition.

Invar:
http://www.matweb.com/search/datasheet.aspx?matguid=b6fb00b235f0442da4d31a0cd04671c9&ckck=1
Electrical Resistivity: 0.0000820 ohm-cm
Electrical Temperature Coefficient: 0.0011/K (21-100°C)
Mechanical Temperature Coefficient: 1.30 µm/m-°C @Temperature 93.0 °C

Copper:
https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
Electrical Resistivity: 0.00000168 ohm-cm
Electrical Temperature Coefficient: 0.003862/K
Mechanical Temperature Coefficient: 17 µm/m-°C

 

[nasu]

It may help using similar "units" for the temperature coefficients (the unit for both is inverse degree Celsius). The coefficients for resistivity are of the order of 10^-3 whereas those for thermal expansion are of the order of 10^-5-10^-6 . So the effect of the change in dimensions on electrical resistance is several order of magnitude smaller than that of change in resistivity.