Electronics
How To Calculate Resistivity
Resistance describes a particular object; resistivity describes the material it is made of. Turning one into the other is a matter of geometry.
Quick Answer
rho = R x A / L
- rho
- Resistivity, the material property independent of the object's shape
- R
- The measured resistance of the sample
- A
- The cross sectional area of the sample
- L
- The length over which the resistance was measured
Multiply the measured resistance by the cross sectional area and divide by the length. A 100 metre wire with 3.4 ohms of resistance and 0.5 square millimetres of cross section has a resistivity of 17 nano ohm metres.
What Is Resistivity?
Resistivity is the electrical resistance a material offers per unit of geometry. It belongs to the material, not to the object, so a thick copper bar and a thin copper wire share the same resistivity while having very different resistances.
The relationship is that resistance grows with length and shrinks with cross sectional area. Doubling the length doubles the resistance; doubling the area halves it, because there is twice as much material in parallel for the current to use.
That is why resistivity carries units that look odd at first. Resistance is ohms, area is metres squared and length is metres, so resistivity comes out in ohm metres rather than ohms.
The numbers are tiny for conductors. Copper sits at about 16.8 nano ohm metres, meaning 16.8 billionths of an ohm metre, which is why the value is usually quoted with the nano prefix rather than in plain ohm metres.
Quoting in nano ohm metres is not just cosmetic. Working in ohm metres means writing numbers with eight leading zeros, which invites exactly the kind of transcription error this topic is prone to.
Conductivity is simply the reciprocal. A material with a resistivity of 17 nano ohm metres has a conductivity of about 58.8235 million siemens per metre, and either number carries the same information.
Temperature is the big confound. Metals become more resistive as they heat, roughly in proportion to the absolute temperature over a wide range, so a resistivity figure without a temperature attached is incomplete.
That is why the resistivity of copper quoted in a table is normally specified at 20 degrees Celsius. Measuring at a different temperature and comparing directly against the table will show a discrepancy that has nothing to do with the material's purity.
Semiconductors behave the opposite way. Their resistivity falls as temperature rises, because heating frees more charge carriers, and that difference in sign is one of the cleanest ways to tell a semiconductor from a metal.
Resistance per unit length is the practical form for cable work. Dividing the measured resistance by the length gives ohms per metre, which is what an electrician actually needs when checking a run.
Comparing a measured value against the standard for pure copper is the usual reason to do this calculation at all. A sample reading appreciably higher than 16.8 suggests an alloy, impurities, or a cross section smaller than the label claims.
For the running example, 3.4 ohms measured over 100 metres with a cross section of 0.5 square millimetres gives 17 nano ohm metres after converting the area to square metres.
The area conversion is where errors concentrate. Half a square millimetre is 0.0000005 square metres, and missing the conversion produces an answer a million times too large.
The result, 17, is 1.0119 times the standard value for copper, which is a plausible measurement on real cable rather than pure annealed copper.
Resistance per metre on this sample is 0.034 ohms, or 34 ohms per kilometre, and that is the figure that determines how much voltage the run will drop.
Formula
rho = R x A / L
Formula page →Scales the measured resistance by the geometry of the sample.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| rho | Resistivity | ohm metres | The material property. |
| R | Resistance | ohms | The resistance measured across the sample. |
| A | Cross sectional area | square metres | The area the current flows through. |
| L | Length | metres | The length over which resistance was measured. |
R = rho x L / A
Formula page →The same relationship rearranged, which is how cable runs are sized.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| R | Resistance | ohms | The resistance of the length in question. |
| rho | Resistivity | ohm metres | The material property. |
| L | Length | metres | The length of the run. |
sigma = 1 / rho
Formula page →The reciprocal view of the same material property.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| sigma | Conductivity | siemens per metre | How well the material conducts. |
| rho | Resistivity | ohm metres | How strongly it resists. |
How To Calculate Resistivity
- 1
Measure the resistance across a known length
The resistance must be measured over exactly the length you put in the formula. Here 3.4 ohms over 100 metres.
- 2
Convert the cross section to square metres
If the area is in square millimetres, divide by a million. Half a square millimetre is 0.0000005 square metres. This is the step that most often goes wrong.
- 3
Multiply resistance by area and divide by length
3.4 times 0.0000005 is 0.0000017, and dividing by 100 gives 0.000000017 ohm metres.
- 4
Rescale to nano ohm metres
Multiply by a billion to get 17. Working in nano ohm metres keeps the figures readable and comparable with published tables.
- 5
Compare against the standard for the material
Copper is about 16.8, so this sample reads 1.0119 times that. A much larger ratio points to an alloy, impurities or an overstated cross section.
Examples
Example 1: Checking a copper cable against the standard
- resistance
- 3.4
- length
- 100
- crossSectionMm2
- 0.5
| Step | Calculation | Result |
|---|---|---|
| Area in square metres | 0.5 / 1000000 | 0.0000005 |
| Resistance times area | 3.4 x 0.0000005 | 0.0000017 |
| Resistivity in ohm metres | 0.0000017 / 100 | 0.000000017 |
| In nano ohm metres | 0.000000017 x 1000000000 | 17 |
| Compared with copper | 17 / 16.8 | 1.0119 |
Result: The resistivity is 17 nano ohm metres, about 1.0119 times the standard for copper. Resistance per metre is 0.034, or 34 ohms per kilometre.
Example 2: The same wire measured over a shorter run
- resistance
- 0.85
- length
- 25
- crossSectionMm2
- 0.5
| Step | Calculation | Result |
|---|---|---|
| Area in square metres | 0.5 / 1000000 | 0.0000005 |
| Resistance times area | 0.85 x 0.0000005 | 0.000000425 |
| Resistivity in ohm metres | 0.000000425 / 25 | 0.000000017 |
| In nano ohm metres | 0.000000017 x 1000000000 | 17 |
| Conductivity | 1000000000 / 17 / 1000000 | 58.8235 |
Result: A quarter length of the same cable gives the same 17 nano ohm metres, confirming that resistivity is a material property and not a property of the particular piece. Its conductivity is 58.8235 megasiemens per metre.
Calculator
Resistivity
17
- Conductivity
- 58.8235
- Resistance per metre
- 0.034
- Resistance per kilometre
- 34
- Compared with copper
- 1.0119
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Resistivity calculator page.
Common Mistakes
Leaving the cross section in square millimetres
It must be in square metres, which means dividing by a million, and forgetting gives an answer a million times too large.
Measuring resistance over one length and putting a different length in the formula, especially when test leads add to the run.
Measuring resistance over one length and putting a different length in the formula, especially when test leads add to the run.
Comparing a value measured at one temperature against a table quoted at another
Metals get more resistive as they heat, so the comparison will not close.
Confusing resistivity with resistance
Resistance belongs to the object and changes with geometry; resistivity belongs to the material and does not.
Using the diameter where area is required
Area is pi times the radius squared, not the diameter, and the factor is not a small correction.
Assuming the result should match pure copper exactly
Real cable is often an alloy or slightly undersized, so a ratio a little above one is normal.
Treating resistivity as fixed with temperature
For metals it rises with temperature; for semiconductors it falls.
FAQ
What is the difference between resistance and resistivity?
Resistance is a property of a particular object and changes with length and thickness. Resistivity is a property of the material, so the same material gives the same resistivity regardless of the shape it is formed into.
Why is resistivity quoted in nano ohm metres?
Because the value in ohm metres is extremely small. Copper is about 0.0000000168 ohm metres, and quoting 16.8 nano ohm metres is far easier to read and far less prone to transcription errors.
How do I convert square millimetres to square metres?
Divide by one million. Half a square millimetre is 0.0000005 square metres, and missing this conversion is the most common source of a wildly wrong answer.
Does temperature affect the result?
Yes, significantly. Metals become more resistive as they heat, so tabulated values are normally quoted at 20 degrees Celsius and should be compared against measurements taken near that temperature.
What should copper read?
About 16.8 nano ohm metres for pure annealed copper. Real cable reading a little above that is normal and reflects alloying or a cross section slightly smaller than the label states.
Why does my sample read high?
The usual causes are an alloy rather than pure metal, a cross section smaller than specified, a measurement taken at a higher temperature, or resistance in the test leads being counted as part of the sample.
References
- [1]Wikipedia, Electrical resistivity and conductivity — https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
- [2]NIST, Copper electrical resistivity — https://www.nist.gov/
- [3]Wikipedia, American Wire Gauge — https://en.wikipedia.org/wiki/American_wire_gauge