Resistivity Calculator: The Complete Guide to Electrical Resistivity (ρ)
This resistivity calculator finds the electrical resistivity (ρ, "rho") of any conductor from its measured resistance, length, and cross-sectional area using the fundamental relationship ρ = R × A / L. It also solves the equation for any unknown — resistance from resistivity, length, or area — and displays the conductivity (σ = 1/ρ) alongside a live conductor diagram and a comprehensive table of common materials. Use it for cable and busbar sizing, material identification, semiconductor doping checks, heating-element design, electrochemistry, geophysics surveys, and any electrical-engineering problem where the material's intrinsic conductivity matters.
Quick Answer: The Resistivity Formula
| Solve for | Formula | Unit |
|---|---|---|
| Resistivity | ρ = R × A / L | Ω·m (ohm-metre) |
| Resistance | R = ρ × L / A | Ω (ohm) |
| Length | L = R × A / ρ | m (metre) |
| Cross-section area | A = ρ × L / R | m² (square metre) |
| Conductivity | σ = 1 / ρ | S/m (siemens per metre) |
What Is Electrical Resistivity?
Resistivity (symbol ρ, the Greek letter rho) is an intrinsic property of a material — a fixed fingerprint that tells you how strongly a given material opposes the flow of electric current, independent of how it's shaped. Two wires of the same metal will have different resistances if one is longer or thinner than the other, but they share the same resistivity. Resistivity is measured in ohm-metres (Ω·m), sometimes also expressed in Ω·mm²/m (1 Ω·mm²/m = 10⁻⁶ Ω·m) when working with cable cross-sections in square millimetres.
The Resistivity Formula in Detail
The relationship between resistivity and resistance follows from the geometry of a uniform conductor:
• ρ — Resistivity (Ω·m)
• R — Measured resistance of the conductor (Ω)
• A — Cross-sectional area (m²)
• L — Length of the conductor (m)
Rearranged for resistance, the formula shows clearly why short, thick wires conduct better than long, thin ones:
Resistivity Table: Common Materials at 20 °C
Use this reference to identify the conductor you're measuring or to size cables and busbars. Lower resistivity means a better conductor.
| Material | Resistivity ρ (Ω·m) | Conductivity σ (S/m) | Temp. Coeff. α (per °C) |
|---|---|---|---|
| Silver (Ag) — best conductor | 1.59 × 10⁻⁸ | 6.30 × 10⁷ | 0.0038 |
| Copper (Cu) — annealed | 1.68 × 10⁻⁸ | 5.96 × 10⁷ | 0.00393 |
| Gold (Au) | 2.44 × 10⁻⁸ | 4.10 × 10⁷ | 0.0034 |
| Aluminium (Al) | 2.65 × 10⁻⁸ | 3.77 × 10⁷ | 0.00429 |
| Tungsten (W) | 5.60 × 10⁻⁸ | 1.79 × 10⁷ | 0.0045 |
| Brass (70/30) | 7.00 × 10⁻⁸ | 1.43 × 10⁷ | 0.0015 |
| Iron (pure) | 9.71 × 10⁻⁸ | 1.03 × 10⁷ | 0.0050 |
| Platinum (Pt) | 1.06 × 10⁻⁷ | 9.43 × 10⁶ | 0.00392 |
| Tin (Sn) | 1.09 × 10⁻⁷ | 9.17 × 10⁶ | 0.0045 |
| Steel (mild) | 1.43 × 10⁻⁷ | 7.0 × 10⁶ | 0.003 |
| Lead (Pb) | 2.20 × 10⁻⁷ | 4.55 × 10⁶ | 0.0039 |
| Constantan (Cu-Ni) | 4.90 × 10⁻⁷ | 2.04 × 10⁶ | ~0.00001 |
| Manganin (Cu-Mn-Ni) | 4.82 × 10⁻⁷ | 2.08 × 10⁶ | ~0.00002 |
| Mercury (Hg) | 9.80 × 10⁻⁷ | 1.02 × 10⁶ | 0.00090 |
| Nichrome (heating wire) | 1.10 × 10⁻⁶ | 9.09 × 10⁵ | 0.0004 |
| Carbon (graphite) | 3.5 × 10⁻⁵ | 2.86 × 10⁴ | −0.0005 |
| Germanium (intrinsic) | 4.6 × 10⁻¹ | 2.17 | −0.048 |
| Silicon (intrinsic) | 6.40 × 10² | 1.56 × 10⁻³ | −0.075 |
| Glass | 10¹⁰ – 10¹⁴ | 10⁻¹⁴ – 10⁻¹⁰ | — |
| Teflon (PTFE) | 10²³ – 10²⁵ | 10⁻²⁵ – 10⁻²³ | — |
Conductivity (σ): The Reciprocal of Resistivity
For some applications it's more convenient to talk about how well a material conducts rather than how much it resists. Electrical conductivity (σ, sigma) is simply the reciprocal of resistivity:
Unit: siemens per metre (S/m).
Example: copper σ ≈ 5.96 × 10⁷ S/m corresponds to ρ ≈ 1.68 × 10⁻⁸ Ω·m.
How Does Resistivity Change with Temperature?
Resistivity is temperature-dependent. For metals it rises with temperature; for semiconductors it falls. The linear approximation near room temperature is:
• ρ(T) — Resistivity at temperature T
• ρ₀ — Resistivity at reference temperature T₀ (usually 20 °C)
• α — Temperature coefficient of resistance (per °C)
• T − T₀ — Temperature change (°C)
For copper (α ≈ 0.00393 /°C), heating a wire from 20 °C to 100 °C raises its resistivity by 1 + 0.00393 × 80 ≈ 31 %. This effect matters for motor windings, transmission lines, and power resistors that run hot. Special alloys like Constantan (α ≈ 10⁻⁵ /°C) are formulated to keep resistance nearly constant with temperature — ideal for precision measurement shunts.
Worked Examples
Example 1: Finding the Resistivity of a Wire
Problem: A 2.50 m wire of cross-section 1.20 mm² measures R = 12.5 Ω. What is its resistivity, and what material is it likely made of?
Convert area to m²: A = 1.20 mm² × 10⁻⁶ = 1.20 × 10⁻⁶ m²
ρ = R × A / L = 12.5 × (1.20 × 10⁻⁶) / 2.50
ρ = 6.00 × 10⁻⁶ Ω·m — close to nichrome (1.10 × 10⁻⁶ Ω·m, when adjusted for impure alloy), so this is likely a heating-element wire.
Example 2: Finding the Resistance of a Copper Cable
Problem: A 100 m copper cable of cross-section 2.5 mm² is run for a lighting circuit. Find its resistance at 20 °C.
ρ(Cu) = 1.68 × 10⁻⁸ Ω·m, A = 2.5 × 10⁻⁶ m², L = 100 m.
R = ρ × L / A = (1.68 × 10⁻⁸ × 100) / (2.5 × 10⁻⁶)
R = 0.672 Ω — useful for calculating voltage drop along the cable.
Example 3: Resistance Change with Temperature
Problem: A copper motor winding has 4 Ω at 20 °C. What is its resistance when it heats up to 80 °C?
R(T) = R₀ × [1 + α × ΔT] = 4 × [1 + 0.00393 × 60]
R = 4 × 1.236 = 4.94 Ω — about 23 % higher than the cold resistance.
Round Wire: Converting Diameter to Cross-Sectional Area
For a round conductor of diameter d, the cross-sectional area is:
• d = 1.0 mm → A = 0.785 mm² = 7.85 × 10⁻⁷ m²
• d = 2.0 mm → A = 3.14 mm² = 3.14 × 10⁻⁶ m²
• d = 3.0 mm → A = 7.07 mm² = 7.07 × 10⁻⁶ m²
Common Cable Cross-Sections (AWG and mm²)
| AWG | Diameter (mm) | Area (mm²) | R of Copper per 100 m at 20 °C |
|---|---|---|---|
| 20 | 0.812 | 0.519 | 3.24 Ω |
| 18 | 1.024 | 0.823 | 2.04 Ω |
| 16 | 1.291 | 1.31 | 1.28 Ω |
| 14 | 1.628 | 2.08 | 0.81 Ω |
| 12 | 2.053 | 3.31 | 0.51 Ω |
| 10 | 2.588 | 5.26 | 0.32 Ω |
| 8 | 3.264 | 8.37 | 0.20 Ω |
| 6 | 4.115 | 13.3 | 0.13 Ω |
What Affects a Material's Resistivity?
- Material composition: Pure metals conduct best; alloys and impurities raise resistivity.
- Temperature: Metals' resistivity rises with temperature; semiconductors' falls.
- Impurities and defects: Even tiny amounts of foreign atoms scatter electrons and raise resistivity.
- Crystal structure: Annealed copper has lower resistivity than cold-worked copper.
- Strain and mechanical stress: Used in strain-gauge sensors.
- Magnetic field: Magnetoresistance, important in hard-drive read heads and modern sensors.
Practical Applications of the Resistivity Calculator
- Cable and busbar sizing: Compute the resistance of a planned run and the resulting voltage drop and power loss.
- Material identification: Measure R, L, A and compare ρ with the table to identify an unknown conductor.
- Heating elements: Pick high-resistivity alloys (nichrome, Kanthal) for toasters, kilns, soldering irons.
- Strain gauges: Constantan and similar alloys give predictable resistance change with strain.
- Semiconductor doping checks: Doping changes silicon resistivity by many orders of magnitude.
- Earthing / soil resistivity surveys: Critical for grounding design and lightning protection.
- Corrosion testing: Resistivity of soil and water indicates corrosion risk for buried pipelines.
- Geophysical exploration: Subsurface resistivity surveys locate water, ore, and structures.
Resistivity, Resistance, and Conductivity — Side by Side
| Resistance (R) | Resistivity (ρ) | Conductivity (σ) | |
|---|---|---|---|
| Symbol | R | ρ ("rho") | σ ("sigma") |
| Unit | Ohm (Ω) | Ohm-metre (Ω·m) | Siemens per metre (S/m) |
| Depends on shape? | Yes | No | No |
| Depends on material? | Yes | Yes | Yes |
| Used in | Ohm's Law | Material properties | Material properties |
Frequently Asked Questions
What is the formula for resistivity?
ρ = R × A / L, where R is the resistance, A is the cross-sectional area, and L is the length of the conductor.
What is the resistivity of copper?
The resistivity of pure annealed copper at 20 °C is approximately 1.68 × 10⁻⁸ Ω·m.
What is the resistivity of aluminium?
Aluminium's resistivity at 20 °C is approximately 2.65 × 10⁻⁸ Ω·m — about 60 % higher than copper but much lighter, which is why overhead transmission lines use aluminium.
What unit is resistivity measured in?
The SI unit is the ohm-metre (Ω·m). Engineers sometimes use Ω·mm²/m (= 10⁻⁶ Ω·m) when working with cables, or Ω·cm in semiconductor work.
What is the difference between resistance and resistivity?
Resistance depends on both the material and the conductor's shape and size. Resistivity depends only on the material (and temperature). Two copper wires of different dimensions have different resistances but the same resistivity.
Which material has the lowest resistivity?
Among pure metals at room temperature, silver has the lowest resistivity (~1.59 × 10⁻⁸ Ω·m), followed by copper and gold. Superconductors have effectively zero resistivity below their critical temperature.
Does resistivity change with temperature?
Yes. For metals, resistivity rises with temperature; for semiconductors and most insulators, it falls. The linear approximation is ρ(T) = ρ₀ × [1 + α × (T − T₀)], where α is the material's temperature coefficient.
How do I convert resistivity to conductivity?
Take the reciprocal: σ = 1 / ρ. Conductivity is measured in siemens per metre (S/m).
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