Library / Electrical / ELE-08
Electrical
Resistivity
Resistance of a uniform conductor from its material, length and cross-sectional area.
- Ref
- ELE-08
- Rev
- v1
- Field
- Electrical
- Result
- R [Ω]
ρ in ohm-metres (Ω·m), L in metres, A in square metres. The calculator takes ρ in nΩ·m and A in mm².
A longer wire has more material to push charge through, and a thicker wire gives the charge more room. So resistance rises with length and falls with cross-sectional area. The resistivity ρ says how good the material is.
Calculator
- Resistance per metre6.8 mΩ/m
- Conductivity σ = 1/ρ58.82 MS/m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- R is proportional to L and ρ and inversely proportional to A. Doubling a cable’s cross-section halves its resistance and the voltage dropped along it.
- The resistance per metre (a derived value) sets the voltage drop (V = IR) and the heating (I²R) along the cable.
- For round conductors A = πd²/4, so a small increase in diameter has a large effect. 10% more diameter gives about 21% more area and about 17% less resistance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρ Resistivity | 10 to 1,500 nΩ·m | 0.4 | 60 | 59.6 | +10% | +10 % |
| L Length | 1 to 1,000 m | 0.0068 | 6.8 | 6.793 | +10% | +10 % |
| A Cross-sectional area | 0.5 to 50 mm² | 0.034 | 3.4 | 3.366 | +10% | −9.09 % |
Sweep of Length
| Length (m) | Resistance (Ω) |
|---|---|
| 1 | 0.0068 |
| 2.371 | 0.01613 |
| 5.623 | 0.03824 |
| 13.34 | 0.09068 |
| 31.62 | 0.215 |
| 74.99 | 0.5099 |
| 177.8 | 1.209 |
| 421.7 | 2.868 |
| 1,000 | 6.8 |
Sweep one variable
Variables and how each one contributes
Resistance
What the equation gives you. Everything else feeds it.
Resistivity
Property of the material at a given temperature: about 17 nΩ·m for copper, 27 for aluminium and 1100 for nichrome. Directly proportional: aluminium has about 1.6 times the resistivity of copper, so a cable needs about 1.6 times the area for the same resistance.
Length
Length of the conductor along the current path. Directly proportional: doubling the length doubles the resistance.
Cross-sectional area
Area of the conductor perpendicular to the current. Inversely proportional: doubling the area halves the resistance. For a round wire, area goes as diameter squared, so doubling the diameter divides R by four.
Worked example with the values above
- Convert to SI: ρ = 17 nΩ·m = 1.7×10−8 Ω·m, and A = 2.5 mm² = 2.5×10−6 m².
- R = ρL/A = (1.7×10−8 × 100) / 2.5×10−6 = 0.68 Ω.
Assumptions and limits
- Uniform material and constant cross-section along the whole length.
- Current spreads evenly over the cross-section. This holds for DC and low frequency; at high frequency the skin effect raises the resistance.
- ρ is taken at the working temperature.
Common mistakes
- Mixing area units: 1 mm² = 10−6 m², not 10−3 m².
- Using diameter instead of area in the formula.
- Forgetting the return path. A two-wire circuit has twice the one-way length.
Statistics and uncertainty
The formula is deterministic, but its inputs are uncertain. Because A depends on diameter squared, a 2% error in measured wire diameter gives about a 4% error in R. Resistivity also varies with temperature and purity, so tabulated values are typical, not exact.
Where it is used
- Cable and busbar sizing against voltage drop.
- Heating-element design.
- Comparing copper and aluminium conductors.
Maths you need
- Scientific notation and prefixes: nΩ·m to Ω·m and mm² to m²
- Area of a circle: A = πd²/4 for a round wire